Keywords: Proportional-integral-derivative controller PSD, Power spectral density QT, Weak equivalence principle WLS, laser ranging

French

C'était une expérience exceptionnelle que de faire une thèse dans le cadre d'une mission spatiale sur le point d'être lancée. Je suis arrivé au moment où le projet MICROSCOPE était proche de son aboutissement, au sein d'une équipe formidable dont les membres avaient travaillé longtemps pour le voir voler. Alors que je menais mes études prospectives sur les futures données, les derniers tests sur l'instrument étaient effectués, on faisait chuter les accéléromètres du haut de la tour de Brême, on tournait les films de vulgarisation scientifique, puis on intégrait la charge utile dans le satellite au centre spatial de Toulouse. Finalement, quelle intense émotion que de voir la fusée VS14 quitter le sol et amener cette belle expérience au dessus de nos têtes, pour son séjour de deux ans autour de la Terre. Je voudrais remercier ici toutes les personnes qui ont fait partie de cette incroyable aventure.

Je voudrais remercier chaleureusement mon directeur de thèse Gilles Métris pour sa bienveillance, sa patience, ses conseils ô combien utiles, sa précision et sa rigueur dans les réponses qu'il a donnée à mes nombreuses questions tout au long de ces trois années. Je voudrais également remercier mon encadrant de laboratoire, Bruno Christophe, qui avait toujours des remarques pertinentes sur les possibles développements de mes recherches, et identifiait toujours les aspects qui nécessitaient une amélioration. Je voudrais exprimer mes remerciements particuliers à Pierre Touboul qui m'a accueilli dans le projet en tant que Chercheur Principal, pour son soutien aux moments où j'en avait besoin, et pour sa passion contagieuse et son enthousiasme lors des réunion du groupe Performance. Je voudrais aussi remercier Manuel Rodrigues, le chef de projet, pour sa bonne humeur et son indulgence envers le doctorant naïf que j'étais.

Par ailleurs j'exprime ici ma profonde reconnaissance aux membres du jury qui ont accepté d'examiner mon travail malgré leur emploi du temps chargé, et en particulier aux rapporteurs Pieter Visser et Martin Hewitson qui se sont déplacés depuis l'étranger. Je voudrais remercier Martin pour nos échanges fructueux sur les données de Lisa Pathfinder, Isabelle Panet pour ses précieux commentaires sur l'étude gradiométrique, et Jean-Luc Starck pour nos discussions sur le problème des données manquantes. Je remercie aussi beaucoup Peter Wolf et François Mignard de leur intérêt pour mon travail.

Je voudrais également exprimer mes chaleureux remerciements à chaque membre de l'équipe MICROSCOPE. Bien sûr, je remercie Bernard Foulon, qui a été comme un mentor pour moi durant ces trois années, pour toutes les questions ayant trait à la physique expérimentale, aux affres de la thèse, et à de nombreux autres sujets. Je remercie particulièrement Joël Bergé pour son aide et ses conseils en rédaction scientifique, en codage, et son introduction aux usages officiels et non officiels du monde scientifique, sans oublier son humour sarcastique. Je voudrais remercier Émilie Hardy, qui a été assez magnanime pour me laisser quelques broutilles à terminer après son doctorat (et merci pour tes fabuleux cannelés). Je voudrais remercier Ratana Chhun, qui a toujours aimablement accepté de répondre à mes questions ennuyeuses, et que je n'ai pas réussi à convaincre que le café bio et équitable est meilleur que la poudre industrielle qu'il a coutume d'acheter. Je remercie aussi Phuong-Anh Huynh, pour toutes les conversations animées sur la politique, la sociologie, les curiosités en biologie, et enfin sa mansuétude envers le manque de culture des jeunes étudiants. Je voudrais aussi remercier Françoise Liorzou qui m'a fait découvrir le monde des salles blanches et la réalité instrumentale. Je remercie aussi Damien Boulanger qui a aimablement pris le temps de me montrer ses travaux de recherche, Pierre-Damien Maigret pour son introduction à la boxe Thaï, et tous les autres membres de l'équipe IEA.

Par ailleurs, je voudrais exprimer des remerciements particuliers à mes compagnons d'épopée doctorale et frères d'armes Karim Douch et Raphaël Chen, avec qui j'ai partagé beaucoup plus qu'un bureau, durant un an pour l'un et trois ans pour l'autre, ce qui a abouti à une durable amitié. Merci pour nos inspirantes discussions philosophiques, économiques, écologiques, et surtout sur la magie des statistiques et de l'analyse de données ! Je remercie les autres membres de l'ONERA comme Patrice Carle et Stéphanie Lala sans qui mes algorithmes n'auraient pu être implémentés dans le centre de mission scientifique. Je voudrais aussi addresser mes remerciement à Sandrine Pires du CEA Saclay, avec qui il était très intéressant de travailler sur l'algorithme inpainting.

Je n'oublie pas mes collègues du CNES qui ont permis à notre expérience d'être envoyée dans l'espace avec succès, et particulièrement Alain Robert et son esprit aiguisé, avec qui nous avons eu des discussions longues, fructueuses et passionnées durant les nombreuses réunions du groupe Performance.

Je voudrais enfin vivement remercier le CNES pour le soutien financier apporté à ces travaux, et en particulier Sylvie Léon. Je remercie aussi la contribution financière du Labex UnivEarthS de l'université Sorbonne Paris Cité qui a soutenu une partie de mon travail et m'a permis de participer à une école d'automne enrichissante.

Conclusion and prospects

Appendices

x Contents A Derivations of useful results for spectral analysis A.1 Bias of the OLS estimator in the presence of a disturbing harmonic signal A.2 Derivation of the envelope function of the maximal projection rate . . . . A.3 Bias of the OLS estimator when fitting two harmonic signals with an error on the frequency . . . . . . . .

It has been an exceptional experience to carry out a Ph.D. thesis related to a space mission which was about to be launched. I arrived at the time where the MICROSCOPE project was close to its culmination, in a formidable team where people had worked a long time to see it fly. While I was doing my prospective work on the future data, the last tests on the instrument were performed, the accelerometers were dropped in the Bremen tower, the science outreach movies were shot, and the payload was integrated in the satellite in the Toulouse Space Centre. Finally, it was an intense emotion to see the VS14 rocket leaving the ground to bring this beautiful experiment above our heads, for his two-year journey around the Earth. Here I would like to thank the people who were part of this incredible adventure.

I would like to warmly thank my thesis supervisor Gilles Métris for his goodwill, his patience, his helpful advices, his precision and his rigour in the answers he gave to my numerous questions throughout these three years. I would also like to thank my local supervisor at ONERA, Bruno Christophe, who always had very relevant remarks about the possible developments of the research, and always spotted the points that needed improvement. I would like to express special thanks to Pierre Touboul who welcomed me in the project as Principal Investigator, for his support at the times where I needed it, and for his contagious passion and enthusiasm during the meetings of the Performance group. I would also like to thank Manuel Rogrigues, the project manager, for his cheerfulness and his indulgence towards the innocent doctoral student that I was.

I express my gratitude and respectful thanks to the jury members who accepted to review my work in spite of their busy schedule, in particular the referees Pieter Visser and Martin Hewitson who came from abroad. I would like to thank Martin for fruitful reflexions on Lisa Pathfinder data, Isabelle Panet for her precious feedback in the study of gradiometric measurements, and Jean-Luc Starck for our discussion on the missing data problem. I also deeply thank Peter Wolf and François Mignard for their interest in my work.

I would like to express my warmest thanks to each member of the MICROSCOPE team. Of course, I would like to thank Bernard Foulon, who has been like a mentor to me during these years, in all questions related to experimental physics, to the hardships of the thesis, and much more. I also particularly thank Joël Bergé, for his precious help and advices in scientific writing, coding, and in all the official and unofficial customs of the scientific world, without forgetting his sarcastic humour. I would like to thank Émilie Hardy, who was magnanimous enough to leave me some tinkers to finish after her Ph.D. thesis (and thank you for your fabulous cannelés). I would like to thank Ratana Chhun, who always kindly accepted to answer my boring questions, and whom I failed to convince that the organic fair-trade coffee was better than the industrial brown powder that he is used to buy. I also thank Phuong-Anh Huynh, for all the lively conversations about science, politics, sociology, and oddities in biology, to finish with her forgiveness towards uncultivated fellows. I would also like to thank Françoise Liorzou who made me discover the world of clean rooms and the instrumental reality. I also thank Damien Boulanger who was kind enough to show me his research works, Pierre-Damien Maigret for his introduction to Thai boxing, and all the members of the IEA unit.

In addition, I would like to specially thank my Ph.D. companions and brothers in arms Karim Douch and Raphaël Chen with whom I shared much more than office space, during one year for the former, and three years for the latter, which ended with a lasting friendship. Thank you for our inspiring discussions about philosophy, art, economy, ecology, and the magics of statistics and data analysis! I thank Patrice Carle and Stephanie Lala without whom my algorithms could not have been implemented in the Scientific Mission Center. I would also like to address my thanks to Sandrine Pires from CEA Saclay, with whom it was very enjoyable to work on the inpainting algorithm.

I do not forget my CNES colleagues who allowed this experiment to successfully go into space, and particularly Alain Robert and his sharpened mind, with whom I had long, fruitful and enjoyable discussions during the Performance meetings.

Finally I would also like to genuinely thank CNES for the financial support that has been granted to me, and in particular Sylvie Léon. I also acknowledge the financial contribution of the UnivEarthS Labex program at Sorbonne Paris Cité which supported some parts of my work and allowed me to participate to an instructive autunm school.

Introduction

Contemporary fundamental physics is able to describe an extremely wide range of phenomena, from the microscopic world of particle physics to the macroscopic universe and its cosmology. However, it is facing key challenges, in particular the one of fully explaining the dynamics of celestial bodies at the scales of galaxies, and the dynamics of spacetime itself at the scales of the universe. To describe these dynamics in a consistent way, the models invoke components which are labelled as "dark". One of them is referred to as "dark matter" and is used to describe the dynamics of stars forming the galaxies, while another ingredient called "dark energy" is introduced to account for the accelerating expansion of the universe that we observe today. The introduction of these components is reflecting the fact that the origin of about 95% of the mass-energy content of the universe is not well understood [START_REF] Ade | Planck 2015 results. XIII. Cosmological parameters[END_REF].

The dynamics of bodies at astrophysical and cosmological distances are driven by the gravitational interaction, also called gravitation. Up to now, the most accurate theory describing the gravitational interaction is Einstein's theory of general relativity [START_REF] Einstein | Die grundlage der allgemeinen relativitätstheorie[END_REF]. Despite the issues that we mentioned, general relativity has proved to be valid with remarkable accuracy through numerous experimental verifications [START_REF] Will | The confrontation between general relativity and experiment[END_REF]. Among them, one of the most recent and important is the direct observation of a gravitational waves generated from a black hole merger [START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF], which are two phenomena predicted by the theory. These tests show that the theory is compliant with observation up to the precision of experimental means.

General relativity is a geometric theory, interpreting gravitation as an effect of the curvature of spacetime due to the mass and energy that it contains. General relativity is based on the hypothesis that in a gravitational field, all bodies fall in the same way, regardless of their composition, their scale (macroscopic or microscopic) or their internal structure. In other words, gravitation acts in the same way on all types of matter or energy. This hypothesis is called the equivalence principle. Its concrete manifestation is the universality of free fall, already formulated by Galileo and Newton, stating that the acceleration undergone by a body due to a gravitational field is independent of the nature of the body. In Newtonian mechanics, this principle translates into the observed fact that the inertial mass of a body, which determines its ability to oppose any movement that is imparted on it, is equal to the gravitational mass, which determines the intensity of the gravitational force that the bodies can undergo. This leads to the assumption that the effects of a uniform gravitational field and the effects of a uniform acceleration are indistinguishable, which is called the weak equivalence principle (WEP). Einstein generalized this concept to the invariance of all laws of physics in free-falling observatories, which is called the Einstein equivalence principle (EEP).

The fact that dark matter has not yet been directly detected, or that the nature of dark energy is not clearly understood, suggests that the standard model of particle physics and general relativity may be incomplete. In addition, the famous "Big Bang" singularity, Introduction also referred to as the origin of the universe, is actually a point of spacetime where the theory breaks down. Besides, the theory describing the microscopic scales, which is the quantum field theory, is based on a very different formalism than general relativity. A way to answer the open questions of fundamental physics could be to harmonize the quantum realm with the theory of gravitation. Efforts are currently carried out by theorists to build such a harmonized theory of both cosmic and quantum scales. Some of them, such as quantum loop gravity, are attempts to quantize general relativity (re-writing it in the quantum formalism), while others start from more fundamental grounds, such as string theories where particles are described as vibrating modes of strings. Some of these theoretical investigations predict the existence of new long-range particle fields, or new interactions, which could be composition-dependent, and hence could call the equivalence principle into question. In order to verify or rule out the new theoretical models, and to gain a more in-depth understanding of what is gravitation, even more precise tests of general relativity must be performed. A way to do it is to test one of its foundations: the weak equivalence principle.

From the 17th century to nowadays, the equivalence principle has been tested by various experiments. Currently, one of the most precise WEP tests is the lunar laser ranging technique, where the fall of the Moon in the gravitational field of the Sun is monitored from the precise measurement of the Earth-Moon distance by means of a laser. But the test is performed in the vicinity of Earth, where the field of the Sun is weak compared to the field of Earth. Other on-ground experiment use Earth as the attractor. This is the case of torsion-balance tests, which consist in measuring the torque of a pendulum made of masses of different compositions. These tests reach a precision of about 10 -13 on the relative differential gravitational signal. However, on-ground experiments are limited by the instability of Earth's environment. Therefore, a space-based experiment would be advantageous in terms of stability.

The objective of the MICROSCOPE space mission is to test the equivalence principle in space for the first time [START_REF] Touboul | The MICROSCOPE experiment, ready for the in-orbit test of the equivalence principle[END_REF]. Planned by the French space agency CNES, the mission was successfully launched on April 2016 from Kourou, French Guyana. The satellite was sent to a quasi-circular orbit around Earth and carries on-board two differential electrostatic accelerometers. Each of them tests the free-fall of two cylindrical and concentric test-masses. In the first accelerometer, the composition of the two test-masses is different: one is made of a platinum-rhodium alloy, whereas the other is made of a titanium alloy. In the second accelerometer the two test-masses are both made of platinum. This instrument thus serves as an experimental reference. To ensure that the test-masses follow the same trajectory and feel the same gravitational field, they are servo-controlled by a set of electrodes so that they stay relatively motionless at the centre of the accelerometer cages. The signal of interest in the experiment is the difference between the electrostatic accelerations applied to the two test-masses needed to maintain them at the centre of the cages. Because of the instrument imperfections, the differential acceleration of the test-masses is coupled to their common acceleration. This effect is nullified by a drag-free system implemented in the satellite. Due to the attitude control of the satellite, the projection of the gravitational acceleration onto the sensitive axis of the instrument varies during the orbit revolution. As a result, a violation of the WEP would be visible through a characteristic periodic difference between the accelerations of the two test-masses measured by the first accelerometer. The objective of the MICROSCOPE experiment is to test the WEP with a target precision of 10 -15 .

In the MICROSCOPE experiment, the measurement consists in time series representing the accelerations of four test-masses on six degrees of freedom as a function of time. A careful and adapted data analysis must be conducted to dig out a possible signal of WEP violation, and even other scientific information with low signal-to-noise ratios. Indeed, the time variations observed in the data are the effect of various physical phenomena. They include systematic perturbations due to the instrument imperfections, which can be finely modelled and calibrated. Other perturbations may arise at different frequencies than the WEP test, which can induce a bias in the measurement. Furthermore, the data is dominated by a stochastic and coloured noise, whose spectrum must be characterized in flight. In addition, unpredictable events in the satellite environment or within the spacecraft itself may induce acceleration peaks or saturations, causing interruptions in the data stream. Such unexpected disturbances and data gaps can affect the accuracy and the precision of the measurement.

The formalism needed to deal with such data belongs to the wide domain of regression analysis in time series, which overlaps many fields of research. To cite a few of them, we can mention biomedical engineering, for the analysis of physiological signals such as heart rate or blood pressure data [START_REF] Chon | Accurate identification of periodic oscillations buried in white or colored noise using fast orthogonal search[END_REF], but also astronomy, for the analysis of the light curves of astronomical objects [START_REF] Israel | A New Technique for the Detection of Periodic Signals in "Colored" Power Spectra[END_REF], econometrics [START_REF] Neusser | Time Series Econometrics[END_REF] for the analysis of macroeconomics figures, climate science for the analysis of seasonal oscillations [START_REF] Allen | Monte Carlo SSA: Detecting irregular oscillations in the Presence of Colored Noise[END_REF], and statistical inference for geophysical data analysis [START_REF] Kitanidis | Introduction to Geostatistics[END_REF]. Thus, the concepts developed in this research work for the analysis of the MICROSCOPE data may find broader applications.

A previous Ph.D. work [START_REF] Hardy | Procédures expérimentales et traitement des données associées pour la mission spatiale MICROSCOPE[END_REF] establishes a calibration process to correct the data for all the modelled and expected perturbations (e.g., gravitational and inertial gradients due to test-mass offcentering, couplings, etc.). This protocol schedules in-flight calibration sessions where the satellite or the test-masses are excited in order to amplify specific disturbances [START_REF] Hardy | Validation of the in-flight calibration procedures for the {MICROSCOPE} space mission[END_REF]. A dedicated data processing is then implemented to estimate the instrument parameters from the calibration sessions. The impact of harmonic disturbances onto the WEP test is also assessed. These periodic signals induce a bias, or "projection", onto the WEP frequency in any finite measurement. A device is implemented to reject the projection when the frequency of the disturbing signal is an integer number of the orbital and spin frequencies of the satellite [START_REF] Hardy | Determination of the equivalence principle violation signal for the microscope space mission: Optimization of the signal processing[END_REF]. The effect of missing data on the projection is also considered. However, this study is limited to harmonic signals with particular frequencies. In addition, the statistical effects due to coloured noise when rejecting harmonic signals or when dealing with missing data is not taken into account. These two kind of errors could respectively limit the accuracy and the precision of WEP test. This is the purpose of this research work. One goal is to develop data analysis methods to reject the bias of arbitrary disturbing harmonic signals. Another goal is to minimize the statistical uncertainty in the estimation of weak signal amplitudes when data are affected by a highly correlated noise and by interruptions. These two objectives are set so as to guarantee the expected precision of the WEP test. An additional objective is to extend the data analysis to another possible scientific product of the mission, which is the estimation of Earth's gravitational gradient.

We review in Chapter 1 the motivations for a WEP test and the state of the art in this experimental field.

In Chapter 2 we describe the satellite and the instrument of the MICROSCOPE mission. From this description we establish the measurement equation and formulate it in a way suitable for the data analysis.

We tackle the core of the study in Chapter 3, where we analyse the bias due to unknown and arbitrary harmonic signals in the framework of least squares with correlated errors. In particular, we assess the impact of time windowing on the uncertainty of the least-squares estimation. Then, we develop a statistical method to detect hidden periodic signals in data affected by a coloured noise of unknown spectrum. The approach is to estimate their frequency and amplitude, so as to decrease their projection onto the WEP violation signal. The relation between the precision on the frequency and the quality of the bias rejection is investigated. In addition, we quantify the achievable precision on the frequency as a function of the signal amplitude and the noise level. Then we test the method by numerical simulations.

Second, we assess in Chapter 4 the impact of missing data on the WEP test. We show that the precision of ordinary least squares is not sufficient to meet the objective of the mission, and that a dedicated method must be developed. To decrease the uncertainty of the test, the correlations of the data must be reduced. To be able to cope with the most general situation, we assume that the noise spectrum is not known. We develop an estimation of the noise correlations operating in the time domain, thanks to an autoregressive model, which avoids the distortions of the spectrum estimate in the frequency domain. We use this estimation to decorrelate the data. Since the number of data points is large, we avoid the storage and the direct inversion of the noise covariance matrix by use of a Kalman filter. However, while sufficient to decorrelate the data, the autoregressive estimation of the noise spectrum may be biased at low frequency. Therefore, we develop a method to better characterize the noise, based on a statistical imputation of missing data. We test the methods on mock differential acceleration data, with different gap patterns, to verify that they allow us to meet the requirement of the mission.

Finally, following the implementation of these analysis tools, we study in Chapter 5 the feasibility to use the data provided by the MICROSCOPE experiment to estimate some components of the Earth's gravitational gradient in the coordinate system of the instrument. We propose to use the two accelerometers on board the satellite to form a gradiometer. Since the experiment is designed to obtain the best sensitivity at low frequencies (around the mHz and below), we motivate the use of MICROSCOPE data to probe the low temporal frequency signature of Earth's gravitational potential, which correspond to the large spatial scales. We derive preliminary figures to assess the performance of the gradient measurement, and we present a protocol to calibrate the gradiometer from the calibration sessions that are already planned during the mission. This study constitutes a first step to broaden the scientific applications of MICROSCOPE.

Statement of the Weak and Einstein's equivalence principle 1.The Equivalence Principle

The motivation of the whole MICROSCOPE space mission is the test of the universality of gravitation, or Principle of Equivalence, which is a phenomenon observed in nature but which has no theoretical basis. This is the principle that leads to the idea that gravitation comes from the structure of space and time themselves, which is the fundamental idea of General Relativity (GR), namely the theory of gravitation as stated by Albert Einstein. The aim of this section is to define the Principle of Equivalence, and to explain why it is the first pavement leading to GR.

1. An experimental test of a gravitational theory

The first manifestation of the Principle of Equivalence is what we call the Weak equivalence principle (WEP), according to which the degree of resistance of all materials to motion is related to the magnitude of the gravitational attraction that they may undergo, in the same way for all of them. In other words the effect of gravitation is independent of the composition of the bodies. In Newtonian mechanics this means that the inertial mass m I and the gravitational mass m g are proportional, with the same proportionality constant for all bodies.

The inertial mass is involved in the resistance of objects to any force applied on them, according to Newton's second law:

F = m I a, (1.1)
where m i is independent of the force F , and a is the acceleration of the body. The gravitational mass is involved in the intensity of the gravitational force experienced by a body in a gravitational field of potential Φ, according to Newton's law of gravitation:

F = -m g ∇Φ. (1.2)
At first glance, the nature of m g is very different from the nature of m I , and there is no reason for them to be equal (or rather, universally proportional). Nevertheless, Galileo showed in 1610 that this is experimentally the case. The legend goes that he used to drop weights off the Tower of Pisa, but he actually merely rolled balls down inclined planes. More precise tests have been performed by now, as we will see in Sec. (1.3). Formally, the deviation from proportionality can be quantified with the Eötvös parameter defined for two bodies labelled 1 and 2 by

η 1,2 = 2 m g,1 m I,1 - m g,2 m I,2 m g,1 m I,1 + m g,2 m I,2
.

(1.

3)

The WEP thus simply translates into the following equality:

η i,j = 0 ∀ bodies i, j (1.4)
In practice the proportionality constant is absorbed in the definition of the potential through the value of the gravitational constant G, such that m I = m g .

The universality of free fall suggests that there are special trajectories, that we will later call geodesics, where particles at rest travel. Here, "at rest" does not mean "motionless", but rather "subject only to gravity". These particles are said to be "free falling".

The WEP implies that one cannot tell whether one is in an uniformly accelerated frame or in a homogeneous gravitational field by simply observing freely-falling particles.

To follow [START_REF] Carroll | Spacetime and Geometry: An Introduction to General Relativity[END_REF], the WEP can be formulated this way: Principle 1. WEP: The motions of freely-falling particles are locally the same in a gravitational field and in a uniformly accelerated frame.

We have added the word "locally" to account for possible inhomogeneities in the gravitational field, leading to tidal forces whose detection could make the difference between gravitation and a uniformly accelerated frame.

Before 1915 and his theory of gravitation, Einstein had set up special relativity (SR), which is a model describing flat spacetime (with no gravity). This model brought major changes in physics, in rupture with Newtonian mechanics. Among them are the relativity of time which is not universal any more, but simply a convenient coordinate to describe events in a four-dimensional set (which is called "spacetime", implying that the notions of space and time are not separated). Another is the nature of mass about which special relativity says that it is a manifestation of energy. In addition SR describes a particular structure of spacetime where the speed of light is constant for all observers, leading to the fact that coordinate changes are given by Lorentz transformations.

Einstein meant to encompass special relativity in his theory of gravitation. To do so he generalized the WEP by saying that an observer cannot tell whether he is in an uniformly accelerated frame or in a homogeneous gravitational field, whatever experiment he performs (not only in experiments involving freely-falling particles). This led to Einstein's equivalence principle (EEP) which is a pillar of general relativity:

Principle 2. EEP : Locally the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments.

In practice, the EEP means that gravity interacts with all forms of energy and momentum, and not only with mass. Indeed, the gravitational mass of a hydrogen atom includes its negative binding energy (and hence is not equal to the sum of the masses of the electron and the proton).

We see that a theory which violates the WEP automatically violates the EEP. However, a theory satisfying the WEP but violating the EEP could be possible, but hard to imagine (Carroll [18] gives the example of particles that would spin while falling, betraying the effect of gravitation).

Consequences

The EEP has immediate consequences on the description of spacetime.

A first consequence is that gravity is not a force any more. A force is something that produces an acceleration. However, the EEP implies that it is impossible for a particle to escape gravitation provided that it has a mass, or an energy, hence it is impossible to isolate the "acceleration" due to gravity only. In General Relativity "no acceleration" is rather defined as the possibility to move freely either in the presence or the absence of a gravitational field.

Suppose that we set up an inertial frame, and let a particle move freely, starting at the origin of the frame. If the gravitational field has inhomogeneities, the freely falling particle will move away from the origin, and when far enough it will look like it is accelerated with respect to the reference frame. Therefore an inertial frame can be constructed only locally. A possible explanation for this impossibility to construct extended inertial frames is the idea that spacetime is curved, which Einstein had. In other words, gravitation can be thought of as a manifestation of the curvature of spacetime.

Without developing the whole theory of GR, some experimentally verifiable consequences of the EEP can be directly derived:

1. An experimental test of a gravitational theory 𝒂 𝒈 Figure 1.1: Bending of light by gravitation, as a consequence of the EEP. At the left-hand side, the reference frame is linked to a spacecraft uniformly accelerating towards a travelling photon. At the right-hand side, it is at rest, embedded in a uniform gravitational field of corresponding magnitude. According to the EEP, the two situations are indistinguishable by an observer inside the spacecraft.

Gravitational deviation of light

Let us imagine a photon travelling in straight line in an empty space-time (with no gravity). If an observer lies in a reference frame moving with constant acceleration a directed perpendicularly toward the trajectory of the photon, from his point of view the photon will follow a curved trajectory, as illustrated in Fig. 1.1. According to the EEP, this situation is not distinguishable from the situation where a photon is travelling in a uniform gravitational field, and where the reference frame of the observer is at rest. Therefore, similarly, the observer will see that the trajectory of the photon is bended. This leads to the prediction that lights rays are deviated by gravitation, i.e., by energy and mass.

Gravitational redshift

Consider two rockets separated by a distance z, moving on the same trajectory and undergoing the same constant acceleration a. Assume that the first rocket emits a photon of wavelength λ 0 toward the second one. The photon reaches the second rocket within a time ∆t = z/c. During the same amount of time the rockets have gained an additional velocity of ∆v = a∆t = az/c. Therefore, when reaching the second rocket the photon will be redshifted due to Doppler effect, by an amount ∆λ/λ 0 = ∆v/c = az/c 2 (to first order). Now consider the same photon, but emitted on Earth from a source at rest on the ground, towards a receiving station located at the top of a tower whose height is equal to z. According to the EEP, this situation is equivalent to the one of the accelerating rockets, and hence the photon will be redshifted by an amount ∆λ/λ 0 = gz/c 2 . More generally, if Φ is the gravitational potential, the redshift is proportional to the potential variation ∆λ/λ 0 = ∆Φ/c 2 . This is what we call the gravitational redshift.

As a result, we saw that the equivalence principle has not only profound implications on the nature of spacetime, but leads to physical effects that can be inferred from it without Gravitational redshift, as a consequence of the EEP. Left-hand side: a photon is emitted from a rocket moving with constant acceleration a towards another rocket undergoing the same acceleration. When reaching the leading rocket the photon will be redshifted because of conventional Doppler effect. Right-hand side: if the photon is emitted from the ground, in an homogeneous gravitational potential, according to the EEP it will be equivalently redshifted, leading to a gravitational version of the Doppler effect.

writing any relativistic equation. One of the aims of GR is to describe the physical situations identified as equivalent by the EEP (i.e., uniformly accelerated reference frame, and reference frame at rest in a homogeneous gravitational field) with the same mathematical formalism, which is Riemannian geometry. GR is the most accurate theory of gravitation up to now, and allows us to describe numerous experimental facts, both quantitatively and qualitatively, which were confirmed in the 20th century and after, with a remarkable accuracy [START_REF] Will | Was Einstein right?[END_REF]. Among them we can cite the precession of the orbital plane of orbiting objects (advance of Mercury perihelion [START_REF] Einstein | Erklärung der perihelbewegung des merkur aus der allgemeinen realtivitätstheorie[END_REF]), gravitational lensing [START_REF] Einstein | Lens-like action of a star by the deviation of light in the gravitational field[END_REF], gravitational redshift [START_REF] Pound | Gravitational red-shift in nuclear resonance[END_REF], cosmological expansion [START_REF] Friedmann | Über die Krümmung des Raumes[END_REF][START_REF] Friedmann | Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes[END_REF][START_REF] Lemaître | Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae[END_REF], and more recently, the existence of black holes and gravitational waves [START_REF] Abbott | Observation of gravitational waves from a binary black hole merger[END_REF].

However, some observational facts, especially at large astronomical scales, remain unexplained by the theory of gravitation as stated. This suggests to perform ever more accurate tests of general relativity, as we will see next.

Motivation of the test 1.2.1 The unexplained observations

Beyond the extraordinary precision of the predictions of general relativity, modern astrophysics suffer from lacuna that have not been bridged at the time of writing. They can be summarized in four main issues.

The first one is that general relativity itself predicts the existence of regions in spacetime where some quantities, such as curvature, become infinite. The first example is the prediction of the existence of black holes, which are regions of the universe where the mass-energy content is so large and so dense that the curvature of spacetime becomes tremendously large. This kind of region is often referred to as a singularity, because it is a point where infinities appear. Another singularity, even more famous, arises when applying the Einstein's field equations to a homogeneous and isotropic universe at large scales. This leads to the Friedmann-Lemaître-Robertson-Walker (FLRW) metric equations [START_REF] Friedmann | Über die Krümmung des Raumes[END_REF][START_REF] Friedmann | Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes[END_REF][START_REF] Lemaître | Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae[END_REF], describing the time evolution of the universe. Under some hypothesis on the energy content of the universe, the FLRW equations predict that the universe is expanding. However, at the time origin, the term corresponding to curvature becomes infinite, and is known as the "Big Bang".

The second problem is the apparent need for an invisible ingredient in the universe, in order to explain the evolution of the universe across cosmological times, but also the dynamics of stars in galaxies and clusters of galaxies. This ingredient is called "dark matter" and is supposed to interact only gravitationally. It would be electromagnetically neutral, and could not be detected by conventional astronomical means of observation. The dynamic of stars or clusters can be described by general relativity only if we assume that they lie in halos of invisible but massive matter [START_REF] Zwicky | Die Rotverschiebung von extragalaktischen Nebeln[END_REF]. Likewise, the spectrum of temperature anisotropies of the cosmological microwave background can only be fully explained if a component like dark matter is present. Today, cosmological data are compatible with a total matter content in the universe being 85% dark (according to the last results of Plank mission [START_REF] Ade | Planck 2015 results. XIII. Cosmological parameters[END_REF]). The remaining 15% is called baryons, and correspond to the conventional, visible matter.

The third issue is the acceleration of the expansion of the universe. The measurement of the apparent receding velocities of distant galaxies suggests that the universe is in accelerated expansion. The only way to explain this behaviour is to add a term in the FLRW, or rather, in the Einstein's field equation from which they are derived. This is often referred to as the cosmological constant, and could correspond to the energy density characteristic to empty space. While the value of this vacuum energy can be estimated from observation, its nature and origin are currently unknown, and is called "dark energy". Cosmological observations are consistent with a universe where dark energy represents 70 % of its energy content. A quick order of magnitude calculation shows that vacuum quantum fluctuations cannot explain the value that is extracted from cosmological observations, which leads us to the last, more general issue.

The last issue is the compatibility of general relativity with the theory describing microscopic scales, which is the relativistic quantum fields theory. In particular, general relativity is a classical theory, in the sense that quantities such as energy can take continuous values, unlike in quantum mechanics. This problem might be at the origin of the other ones, that is why building a harmonized theory of both astrophysical and quantum scales could address some of the observational issues of contemporary physics.

Theoretical paths leading to a violation of the equivalence principle

Two different descriptions of the world are available nowadays: the geometric theory of gravitation, and the quantum field theory, both proving their predictive power at their respective scales. Several attempts have been made to harmonize them. Among these efforts is loop quantum gravity which tries to quantize general relativity. Others are string theory models which try to unify all fundamental interactions (and not only gravitation) and postulate that particles can be modelled by one-dimensional strings whose vibrational state determines the properties of the particles. One of these states gives rise to the graviton, a particle carrying the gravitational interaction. Loop quantum gravity attempts to build a Hamiltonian version of GR (the formalism of quantum mechanics) and hence preserves the equivalence principle. However string theory models start from a different formulation (vibrating strings), and a spacetime of higher dimension (more than 4), and can lead to violations of the equivalence principle at different levels. In this section we focus on string theory, to give insight into the rationale leading to such a violation.

Original formulation of general relativity

As we pointed out in Sec. 1.1.2, the equivalence principle leads to the idea that spacetime may be curved, and that the gravitational force is replaced by the curvature of spacetime. A curved spacetime (or manifold) is described by a symmetric (0,2) tensor called the metric, and labelled g µν , where the indices µ, ν run across the number of dimensions. Hence the metric will somehow replace the classical Newtonian gravitational potential Φ. The metric determines, inter alia, how path lengths, proper time, and "scalar products" are calculated. It also determines the shortest distance between two points. In addition, the metric defines the velocity limit above which no particle can travel, i.e. the speed of light c. This is already the case in special relativity which describes flat spacetime, also called Minkowski space, where the coordinate transformations preserve the speed of light. The consequence is that time is treated as a coordinate similar to the three spatial coordinates. In SR the coordinate transformations under which the speed of light is maintained constant are Lorentz transformations. This framework is generalized in curved space time, where a metric-related quantity, the line element ds 2 , is invariant within any change of coordinates. Formally, this means that for any change of coordinates x µ → x µ , we have ds 2 = g µν dx µ dx ν = g µ ν dx µ dx ν , (1.5) where dx µ are the gradients of the coordinate functions. Note that there is an implicit summation over repeated indices (Einstein's notation). Although this is not rigorously true, the line element can be interpreted as an infinitesimal length in the four-dimensional spacetime.

The curvature of spacetime is quantified by the Riemann tensor R ρ σµν , which is constructed from the second-order derivatives of the metric. By contraction (i.e., partial summation over its indices) we can construct two others quantities, the Ricci tensor R µν and its trace, the Ricci scalar R.

The metric and the Riemann tensor are tools to describe the spacetime geometry. In order to derive the actual geometry of a given region of the universe, what is left to have is a way to describe how the metric responds to energy and momentum. This is governed by Einstein's field equations, which can be viewed as a tensorial, relativistic generalization of Poisson's equation ∇ 2 Φ = 4πGρ. We saw that what plays the role of Φ is the metric, and that the Ricci tensor contains second-order derivative of g µν . The right-hand side is replaced by the energy-momentum tensor T µν which is the generalization of the mass density. When adding the additional constraint of energy conservation ∇ ν T µν = 0 , the simplest choice for the field equations is:

R µν - 1 2 Rg µν = 8πG c 4 T µν , (1.6)
where the constant 8πG/c 4 ensures that this equation reduces to Poisson's in the weak field limit, that is, for small potential (Φ c 2 ), small velocities (v c) and time derivatives much smaller than spatial derivatives. The left-hand side is the geometrical part, involving the contractions of the curvature tensor and the metric, while the right-hand side is proportional to energy and momentum. The idea conveyed by the field equations is that gravitation, i.e., spacetime geometry, is driven by the mass-energy content of the universe. In turn, the motion of particles is driven by the spacetime geometry.

Lower and upper indices in tensors correspond to their co-variant and contra-variant form defining the way they transform under change of coordinates, but this aspect is not important in this discussion.

General relativity, and physical theories in general, can be formulated in a Lagrangian way. This formulation allows the encapsulation of the field equation (1.6) into a single dimensionless scalar S, called the action, which is related to the Lagrangian density L:

S ≡ L √ -gd 4 x, (1.7)
where g is the determinant of the metric, and √ -gd 4 x is the generalization of the volume element in curved spacetime. Small variations of the metric should leave the action unchanged, which is called the principle of least action. We can define an action corresponding to the gravitational part, the Hilbert action S H :

S H = c 4 16πG R √ -gd 4 x, (1.8)
Similarly, the matter part can be modelled by an action S M . Then the total action is:

S = S H + S M .
(1.9)

The action can be varied with respect to the metric and it turns out that by writing δS = 0 we can derive the fields equations (1.6), if we define the energy-momentum tensor as

T µν = - 2 √ -g δS M δg µν , (1.10)
where g µν is the inverse of the metric g µν .

Alternative formulation and violation of the equivalence principle

In this section we give insight into what could be the meaning of a violation of the equivalence principle, and relate this violation to possible extensions of general relativity. The equivalence principle is supported by well observed experimental facts, but is not an unwavering principle nor a statement that is theoretically and rigorously derived. It is likely that the metric, and gravitation as the effects of spacetime geometry, are concepts which are derived from a more complete description of the universe, as quantum field theory might be. A call into question of the equivalence principle would not undermine general relativity, but would rather suggest either the existence of an additional ingredient in the theory, or a manifestation of this more comprehensive framework that remains to be found.

Since an EP violation has not been detected so far, a possible (but not the only one) source of violation is the existence of a field that either weakly couples to (i.e. interacts with) mass or a field that resembles gravity, i.e. which is long-range and couples almost universally to mass, so that we were not able to make the difference by means of experiment up to now. This is often referred to as the discovery of a "fifth force".

Let us now see how the theory could be extended, and how this new field would act. A category of extension is called scalar-tensor field theories, involving the metric tensor g µν and a scalar field λ, representing a new interaction, whose value depends on the localization in space and time. They amount to formally modifying the Hilbert action in Eq. (1.8) by replacing c 4 16πG by some function f (λ) of the field. A new term S λ corresponding to a pure scalar part is also added in the action in Eq. (1.9). This can be written as

S = S f H + S λ + S M , (1.11)
where S f H is the modified Hilbert action as described above.

A convenient way to deal with scalar-tensor theories is to perform transformations so that the new theory can be written in a way similar to GR. The first transformation to apply is called a conformal transformation and corresponds to a multiplication of the metric by a function depending on the coordinates. By setting q = 4π Gc -4 (where G is the appropriate constant for the new equations to reduce to Newtonian ones in the weak field limit) this transformation can be written as gµν = 4qf (λ)g µν .

(1.12)

Another convenient operation to perform is a change of variable of the form ϕ = Z 1/2 (λ)dλ, so that we can write the gravitational and pure scalar pieces of the action as (see e.g. [START_REF] Carroll | Spacetime and Geometry: An Introduction to General Relativity[END_REF], pp. 183-185)

S f H + S λ = -gd 4 x 1 4q R + K(ϕ, ∇ µ ϕ) -V (ϕ) , (1.13)
where K represents the "kinetic and gradient energy" part of the scalar field Lagrangian, whereas V represents the potential energy. This transformed frame in which the equations looks like GR is called the Einstein's frame. The kinetic part can be written as

T (ϕ, ∇ µ ϕ) ≡ - 1 2q gµν ∇µ ϕ ∇ν ϕ = - 1 2q ( ∇ϕ) 2 . (1.14)
General relativity can also be modified by postulating the existence of extra spatial dimensions, says d, in a way such that spacetime appears to be four-dimensional on large scales whereas it has actually 4 + d dimensions. This is the case of string theory models. The line element in this extended universe can be split into a 4-dimensional part and an extra-dimensional part:

ds 2 = g µν (x)dx µ dx ν + b 2 (x)γ ij (x)dy i dy j .
(1.15)

A full treatment of this modification would require to solve the equations of motion in 4 + d dimensions (this is usually done by expanding the dynamical modes on some basis functions). However, since the extra-dimensions are assumed to be "compact" (small) with respect to the other ones, as a first approximation, Eq. (1.15) can be considered as a simplifying ansatz. In this case, one can show that the (4 + d)-dimensional Hilbert action can be reduced to a modified four-dimensional action of the form given by Eq. (1.11). The scalar λ appearing in the action is called the dilaton, often written φ. By a transformation of the form (1.12) writing gµν = CB g (φ)g µν and an appropriate change

1. An experimental test of a gravitational theory of variable φ → ϕ, the modified Hilbert action can be written in the form of Eq. (1.13). This means that the effect of extra-dimensions can be described in the formalism of scalar-tensor field theories. Following Damour and Polyakov [START_REF] Damour | The string dilation and a least coupling principle[END_REF], the function B g (φ) is the dilaton gravity coupling function. The action then involves a gravity term (the modified Hilbert action) and a matter term:

S = -gd 4 x 1 4q R - 1 2q ( ∇ϕ) 2 + S M , (1.16)
where the pure-scalar term is absent in this formulation, and the matter term includes non minimal coupling with φ.

The dynamical implications of this modified theory of gravitation can be studied by writing the field equations deriving from the action Eq. (1.16), which are obtained, again, by varying the action with respect to the metric. After re-arranging Eq. (3.1) of Ref. [START_REF] Damour | The string dilation and a least coupling principle[END_REF] this procedure gives:

Rµν - 1 2 Rg µν = 8π G c 4 Tµν + 2∂ µ ϕ∂ ν ϕ, (1.17a) gµν ∂ µ ∂ ν ϕ = - 4π G c 4 σ, (1.17b)
where T µν is defined as in Eq. (1.10) and σ ≡ 1

√ -g δS M
δϕ is involved in the energy-balance equation ∇ ν T µν = σ∇ µ ϕ. By comparing Eq. (1.17a) to the original field equations (1.6) of GR, we see that a scalar field term is added in the "source side". Eq. (1.17b) is an additional equation describing the coupling between the dilaton field and the matter components. If we consider several gases of condensed particles labelled A, their mass m A [ϕ(x)] depends on the scalar field in the Einstein's frame (and hence depend on space and time), and the dilaton source term can be written as

σ(x) = A α A [ϕ(x)] T A (x), (1.18)
were T A is the contribution of the A-type particles to the total energy-momentum tensor T µν = A T µν A , and α A is a quantity measuring the strength of the interaction between the dilaton and the A-type particles:

α A (ϕ) ≡ ∂ log m A (ϕ) ∂ϕ . (1.19)
It is shown in Ref. [START_REF] Damour | Tensor-multi-scalar theories of gravitation[END_REF] that the solution of the field equations (1.17a,1.17b), taken in the Newtonian approximation (i.e. weak field, small velocities), leads to an interaction potential between two particles labelled A and E which has the Newtonian form, but with a dilaton-dependent modification of the gravitational coupling constant:

Φ = -G 1 + α (0) A α (0) E m A m E x A -x E = 1 + α (0) A α (0) E Φ N (r AE ) , (1.20)
where the (0) indicates that the dilaton coupling strength α(ϕ 0 ) is taken at its present-day value, and m A , m E are the inertial masses of the two bodies (corresponding to ϕ 0 ). We also denoted the Newtonian potential as Φ N . Eq. (1.20) corresponds to an apparent violation of the weak equivalence principle, since the gravitational interaction, because of the coupling to the dilation, depends on the nature of the bodies A and E.

If we consider two test-masses A and B, free-falling in the gravitational field generated by a spherical Earth of mass m E , their respective acceleration will sightly differ from one another (here we ignore the gradient term and assume that the centre of masses coincide in a point of position r):

∆a = a A -a B = - Gm E r 2 1 + α (0) A α (0) E -1 + α (0) B α (0) E = - Gm E r 2 α (0) A -α (0) B α (0) E . (1.21)
We can define the dimensionless quantity:

∆a a ≡ 2 a A -a B a A + a B = 2
α (0) A -α (0) B α (0) E 1 + 1 2 α (0) A + α (0) B α (0) E . (1.22)
As a result, everything is as if the gravitational mass was different from the inertial mass. This is equivalent to an Eötvös parameter given by (see Eq. (1.3)):

η A,B ≈ α (0) A -α (0) B α (0) E . (1.23)
The remaining problem is to know whether the order of magnitude of the dilaton coupling at present day is within reach of experimental means. At first glance, the existence of a massless dilaton violates the equivalence principle in a way which it is strongly coupled to matter, in conflict with current experimental tests of general relativity. The problem is often avoided by postulating that the dilaton has a mass, i.e. that there is a non-zero potential term in the action as in Eq. (1.13) which turns out no have no minimum. Since the field is somewhat related to the scale factor of the extra dimensions ϕ ∝ ln b, it will roll to -∞ and b will become arbitrarily small, so that observable deviations from GR are quenched on large scales.

Nevertheless, a mechanism called "last coupling principle" can make compatible experimental data and the existence of a massless dilaton. The argument involved is cosmological, and is explained by a mechanism which, during the early, inflationary epoch of the universe, drives Φ towards a fixed point at infinity, minimizing the action, hence decoupling the dilaton from matter. However, the value of the field at the present epoch would be such that a residual coupling could still be observable. It is shown by T. Damour et al. [START_REF] Damour | Violations of the equivalence principle in a dilaton runaway scenario[END_REF] that the WEP violation is related to an observable cosmological parameter, which is the matter density fluctuations at the scale corresponding to the present-day horizon (i.e. the maximum distance at which two points of spacetime can be causally related). It is also composition-dependent, as implied by Eq. (1.22). For platinium and titanium, the estimated violation in case of a massless dilaton is within reach of the MICROSCOPE experiment.

Of course, there exists other models introducing new interactions. Although we will not do a comprehensive review here, in the weak field limit, they usually amount to modifying the Newtonian potential in the same way as in Eq. (1.20). We can cite the Yukawa interaction [START_REF] Yukawa | On the interaction of elementary particles[END_REF], carried by new bosons (bosons are integer-spin particles which are mediators of forces, such as the photon is the mediator of the electromagnetic interaction, whereas fermions are half-integer particles which constitute matter). These new bosons would couple with the charge of the test fermionic bodies A or B and the 1. An experimental test of a gravitational theory attractor E. The potential between the test body and the attractor would resemble the gravity potential, but with a multiplicative exponential term:

V AE (r) = ± g 4π qE qA r e -r/λ , ( 1.24)
where qE and qA are the charges (with respect of this new interaction) of the involved fermions, g is a coupling constant, and the parameter λ ≡ h/(m b c) determines the range of the interaction, and depends on the mass m b of the new boson. The Yukawa potential contributes to the total potential in a way formally equivalent to the dilaton modification in Eq. (1.20), where we can substitute

α (0) A α (0) B ↔ g 4πG qE m E qA m A e -r/λ , (1.25)
where a noticeable difference is the exponential scale dependence of the Yukawa modification.

As a conclusion, in order to delimit the way to new physics, and to validate, invalidate, or put more stringent constraints on the theoretical investigations, there is a need for more precise tests of general relativity, and in particular regarding the equivalence principle.

The state of the art

We saw that the efforts to build a universal theory of astronomic and quantum scales lead to a modification of general relativity, sometimes leading to a violation of the equivalence principle, justifying the need for a more precise test than previous experiments. In this section we briefly review the main experimental tests of the WEP that have been performed up to now.

Before being erected as the pillar of Einstein's powerful theory of gravitation, the equivalence principle had been observed as an experimental and well verified fact, and was then formally introduced in the equations of mechanics.

We saw that one of the earliest tests of the equivalence principle was performed by Galileo in the beginning of the 17th century. At the time, the scholar formulated the result of his ball roll experiment by saying that the acceleration of massive bodies due to gravitation does not depend on their mass. At the end of the same century, Newton understood the importance of the phenomenon in mechanics, to the extent that he devoted the beginning of this famous work, Philosophiae naturalis principia mathematica, to it. Newton used pendulum, by comparing the oscillation frequency of pendula with different masses and composition. Later, in the end of the 19th century, Eötvös performed a very precise test using a torsion balance pendulum [START_REF] Eötvös | Beiträge zum gesetze der proportionalität von trägheit und gravität[END_REF]. The principle of this technique, which has been improved throughout time (see Fig. 1.3), provides the best constraint on the universality of free fall up to now. Another modern technique to test the WEP at the scale of the solar system is laser telemetry, and in particular lunar laser ranging (LLR).

A given experimental test has three important characteristics: the gravitational source that is used, the nature of the test-bodies, and the range of the considered interaction (which may be important e.g. to constrain Yukawa-like models).

In the following we describe in more details the principle of torsion balance tests and laser telemetry, which are the most competing techniques nowadays, with precisions of a few 10 -13 .

Torsion balance tests

From 1885, the physicist Loránd Eötvös started to implement a test of the WEP based on a torsion balance. In this experiment, the two test bodies of different composition are attached at the two ends of an ideally massless rod (or frame), which is horizontally suspended by a flexible torsion fibre, as shown in Fig. 1.4. The suspension point on the rod is adjusted so that the rod remains horizontal, i.e., so as to nullify the torque due to the gravitational forces on the test-masses. This adjustment only depends on the gravitational masses of the bodies. If we denote by F 1 and F 2 the external forces applied on each test-mass (excluding the reaction R i of the rod), the torque about the fibre axis can be written as

T z = 1 F 1 + F 2 F 1 × F 2 • r 12 , (1.26)
where r 12 is the vector connecting the two edges of the horizontal rod. The forces F i include the weight parallel to the z-axis, and the axifugal force, i.e., the inertial force due to the non-inertial character of the reference frame of the laboratory at latitude Θ:

F i = F g i + F I i = m g,i g i + m I,i w 2 ⊕ r i , (1.27)
where g i is the gravitational acceleration of mass i induced by the source (e.g., Earth) and w ⊕ is the angular frequency of Earth's rotation, and r i is the vector normal to Earth's rotation axis, pointing towards the location of test-mass i (note that r 1 ≈ r 2 ).

If there is a WEP violation, i.e., a difference between the ratios of the gravitational and inertial forces F g,i / F I,i of the two test-masses, and if this difference is not parallel to the suspension wire, then the resulting forces on the test-masses will not be parallel and a non zero torque T z will be induced on the wire. Hence, the rod will rotate with respect to its original position (until the wire stiffness stops the rotating motion).

An experimental test of a gravitational theory

Torsion pendulum

Inertial force Using this technique, Eötvös and his collaborators obtained a precision of a few 10 -9 on η in 1889 [START_REF] Eötvös | Beiträge zum gesetze der proportionalität von trägheit und gravität[END_REF], followed in 1965 by Dicke's experiment in Princeton with a precision of 10 -11 [START_REF] Roll | The equivalence of inertial and passive gravitational mass[END_REF] and then Braginsky's experiment in Moscow [START_REF] Braginsky | Verification of the equivalence of inertial and gravitational mass[END_REF] where they reached 10 -12 .

Due to the geometrical imperfections of the instrument and the inhomogeneity of the gravitational field, the torsion-balance experiments are limited by gravitational gradients which induce a small difference on the direction of the forces undergone by the two test-bodies. A way to discriminate and better resolve a possible WEP violation is to slowly rotate the torsion pendulum with respect to the attractor, so that the searched signal is sinusoidal (as will be the case in MICROSCOPE). In Princeton's and Moscow's experiments, the sun is chosen as the attractor, so that the rotation is provided by Earth's orbit. However, this configuration has disadvantages. First, the noise sources increase when the frequency decreases, and possible systematic errors may oscillate on a daily basis, making a 24-hours period not optimum. Furthermore, the gravitational source is relatively weak. Finally, the distance between the attractor and the test-mass only constrains long-range Yukawa interactions.

The torsion balance apparatus was improved by the Eöt-Wash group in the end of the 20th century, by using rotating turntables with varying frequency in the field of Earth [START_REF] Adelberger | Testing the equivalence principle in the field of the earth: Particle physics at masses below 1 µev?[END_REF][START_REF] Schlamminger | Test of the equivalence principle using a rotating torsion balance[END_REF], see Fig. 1.5. In addition, the local gravitational gradients were limited by the horizontal and rotational symmetry of the design, and by an appropriate arrangement of masses around the system. Several other devices were implemented to improve the precision, such as the control of the wire alignment with the local vertical (by thermal dilatation control), thermal shields, and a vacuum chamber to eliminate the air drag. The twist of the pendulum is measured by auto-collimation of an optical device based on a laser.

The experiment is carried out with two pairs of different materials: beryllium-titanium and beryllium-aluminium. The precision reached (the best up to now) for these two pairs are σ Be-Ti = 1.8 • 10 -13 and σ Be-Al = 1.5 • 10 -13 . The main limitations of the experiment are thermal fluctuations from the wire damping losses and systematic errors from time-varying local gradients. A solution to better control these two sources of errors is to move away from on-ground thermal and gravity instabilities, that is, going to space.

Lunar laser ranging

The universality of free fall can also be tested by considering astronomical objects. This is the interest of lunar laser ranging (LLR), which considers the Earth-Moon-Sun system. Indeed, Earth and Moon are two celestial bodies of different composition (the mantle of the moon is richer in iron that Earth's) falling in the gravitational field of the Sun. The particularity of this system compared to the experiments discussed in the previous section is that it forms a gravitational experiment involving more massive bodies (Earth and Moon) whose masses are not negligible with respect to the surrounding gravitational field. In this case, a violation of the WEP could be the effect of the internal gravitational energy of the bodies, possibly large enough to be visible. Indeed, the ratio of the gravitational mass to the inertial mass of a body could show a dependence on the body's gravitational self energy U (i.e., the energy holding its components together) as

m g m I = 1 + η U m I c 2 , (1.28)
where the parameter η depends on the theory (η = 0 in general relativity). This dependence is known as the Nordtvedt's effect [START_REF] Nordtvedt | Equivalence principle for massive bodies. i. phenomenology[END_REF]. Such an experiment is actually a test of the EEP as we defined it before (Principle 2), since it involves the binding energy of the test-bodies. However, a distinction is sometimes made between "gravitational laws of physics" and "non gravitational laws of physics". Then the EEP is defined as applying only to the latter, whereas the "Strong equivalence principle" (SEP) is defined as applying to both. Therefore LLR tests are often seen as tests of the SEP. The principle of LLR tests relies on the measurement of the Earth-Moon distance over a long period of time by means of laser telemetry. One measurement consists in the emission and reception (with a telescope) of a laser pulse between a ground-based observatory and a corner-cube retro-reflector located on the surface of the moon. The distance measurement is inferred from the round-trip travel time of the pulse (from 2.34 to 2.71 seconds depending on how far away the moon is at the time of measurement). The moon's reflecting devices were set up by the U.S. manned space missions Apollo 11, 14 and 15 and the Soviet robotic missions Lunokhod 1 and 2 [START_REF] Williams | Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon[END_REF], see Fig. 1.6.

Since the 1960's the lunar retroreflectors have obtained improving precision on the Earth-Moon distance, down to centimetric precision. The main LLR stations, which have provided regular observations for decades, are the McDonald Laser Ranging System (MLRS) in the United States and the station of the Observatoire de la Côte d'Azur in France [START_REF] Samain | Millimetric lunar laser ranging at oca (observatoire de la côte d'azur)[END_REF].

Due to the large width of the beam at reception and the refraction effects, the telescope receives one single photon for 100 laser shots of 10 19 photons each. As a result, the dispersion of the pulse, as well as the slight tilt of the reflector array, introduce an uncertainty on the location of the received photon in the beam. The laser pulse being broader than one millimetre, a millimetre-range precision cannot be achieved with a single photon. Instead, many photons must be collected, so as to get an average roundtrip travel time, thereby statistically improving its precision. However, this requires a larger telescope to collect more photons. This is the goal of the APOLLO project (the Apache Point Observatory Lunar Laserranging Operation) in New Mexico, U.S. This facility, operating since 2005, utilizes a 3.5 meter telescope achieving more than one photon per laser shot, enabling a range precision of 1 mm. The current WEP tests using the LLR technique reach a precision of about 1.8 • 10 -13 [START_REF] Williams | Lunar laser ranging tests of the equivalence principle[END_REF], and it is expected that with improvement such as the APOLLO effort (or others under way at OCA), an order of magnitude should be gained on the WEP test.

However, for this to be possible, it is needed to decrease the modelling errors to the same extent. In particular, the errors on the position of the LLR stations and reflectors (respectively with respect to the centre of the earth and moon) must be carefully modelled. Indeed the geological dynamics induce a slight motion of the stations. In addition, there is a delay in the round-trip time of the pulse due to the atmosphere, depending on the weather and the barometric conditions. This delay must also be accurately modelled. Similarly to the torsion balance test, these errors are inherent to the earth environment, and can be overcome by a space-based experiment.

Summary:

We defined the equivalence principle and gave insight into how it leads to the geometric theory of general relativity. We showed that it is a cornerstone to discriminate between new proposals for unified theories in modern physics, in particular by its relationship with a "fifth interaction". These theoretical developments require a step forward in the precision of the test, in order to better constrain the key-parameters of the theories (in case of a non-violation of the WEP) or to pave the way for new physics (in case of a violation). We saw that the precision of current equivalence principle experiments is a few 10 -13 , however these on-ground experiments are limited by near-Earth environment perturbations or errors in the modelling of Earth dynamics. When thinking about overcoming these limitations the idea to go to space emerges, leading to the design of the MICROSCOPE space mission.

The objective of this chapter is to establish a model for the measurement of the MICROSCOPE instrument which is usable for the data processing. To do so, we first describe the mission rationale leading to a test of the weak equivalence principle with a precision of 10 -15 , which has been motivated in the previous chapter. Then the payload design is justified and described. This description finally serves as a basis to derive the 24 2. The MICROSCOPE experiment and the measurement equation model equations. The model equations are then rearranged in a form convenient for parameter estimation, in particular for the calibration of the instrument.

Description of the mission

In the previous chapter we motivated the needs for an improvement of the test of the equivalence principle. We also described the conditions that must be met to test the WEP. The objective of the MICROSCOPE space mission is a precision of 10 -15 on the Eötvös parameter. This goal leads to the experimental design described in this section, where the motion of two test-masses (TM) with different composition are thrown into orbit around Earth.

Justification for a space-based experiment

There are three major arguments justifying to go to space to achieve the objective of the experiment.

First, the space environment provides soft and stable conditions to perform the experiment, and allows us to be freed from all on-ground perturbation sources. The electromagnetic, thermal and vibrational perturbations, induced by human activity for example, are consistently reduced or can at least be easily controlled in space. In addition, Earth's gravity gradient fluctuations are less than on ground since all the derivatives of the gravitational potential decrease with distance.

Second, an in-orbit experiment provides arbitrarily long free-fall times in steady conditions since the signal can be integrated during several orbits to average the stochastic disturbances. The duration of free-fall can thus be several days.

Finally another advantage of a space-borne set-up is that the frame of observation can be easily rotated with respect to the gravity field which is a good way to discriminate a possible violation signal from other effects by a fine control of its frequency signature.

Mission general rationale

MICROSCOPE is a French acronym for "MICRO Satellite with drag Control for the Observation of the Equivalence Principle". This is a low-earth orbit satellite developed by CNES (the French space agency) which carries on board two differential electrostatic accelerometers developed by ONERA (the French aerospace lab), whose characteristics will be detailed in the next section. These accelerometers are the main and unique payload of the mission and are used as the instrument of the experiment.

As seen in Sec. 1.1.1 of Chap. 1 a proper test of the weak equivalence principle requires that the particles under study (which will be test-masses of different compositions in our case) must follow the same free-fall trajectory. Actually, it is sufficient that they undergo exactly the same non-gravitational accelerations. To meet this condition, the test-masses are protected by a cage (and a housing) and are levitated through an electrostatic set-up which forces them to stay relatively motionless at the centre of the cage. In addition we stated that the experiment must be carried out in a small enough region of spacetime, so that the particles are submitted exactly to the same gravitational field. To fulfil this condition the shapes of test-masses are chosen to be concentric cylinders, so that their centres of inertia are as close as possible.

If we let the test-masses completely free to move inside the volume of the accelerometer cages, we could only measure their position with respect to each other, and take the second derivative to obtain the acceleration. Instead, in order to perform a direct measurement of the acceleration, the experiment is designed to force them to follow the same trajectory with respect to the cage. For this to be possible, their position is detected by capacitive sensing, and a servo-control loop is implemented to maintain them relatively motionless at the centre of the accelerometer cage. This active control is possible thanks to electrodes which surrounds the test-masses and exert electrostatic forces on them. Assuming that the test-masses are perfectly centred, by measuring the necessary restoring force applied to the two test-masses, we check whether gravity acts in the same way on them.

There are two accelerometers functioning on this basis, each one consisting of two test-masses. In the first accelerometer (designated as EP sensor unit) the masses are made of different materials, while in the other one (designated as REF sensor unit for reference) they have the same composition, in order to detect any possible bias in the measurement. A violation of the WEP would result in a slight difference between the accelerations applied to the test-masses of the EP accelerometer, while the accelerations of the REF test-masses would remain the same. The violation signal would be proportional to the gravitational acceleration at the altitude of the satellite g and to the Eötvös parameter δ.

While being in space is advantageous, the satellite must not be too far from the gravitational source for the amplitude of the possible violation signal to be detectable. The chosen altitude is about 700 km above the earth which is a trade-off between the expected signal intensity and the magnitude of the atmospheric drag which decreases when going far from our planet. However, the radiation pressure from the sun and the earth's albedo remain the same. In order to maintain the amplitude of the gravitational signal as constant as possible the satellite was injected into a quasi-circular orbit with an eccentricity below 5 × 10 -3 . With this orbit the gravity intensity is about 8 ms -2 and the required precision expressed in acceleration is δg = 8 × 10 -15 ms -2 . In addition, the orbit is chosen to be sun-synchronous in such a way that the orbital plane precesses in one year, placing the satellite in constant sunlight (except during eclipse phases). Such a configuration guaranties a relative thermal stability and an optimal power supply throughout the mission.

The key idea of the experiment is the ability to finely control the frequency content of a possible violation signal compared to all other disturbances, so as to search for a specific signature. This is done by the mastering of the satellite attitude control. Two pointing modes are provided for this purpose as shown in Fig. 2.1. The first one is the inertial mode where the sensitive axis of the experiment is kept parallel to the orbital plane. The second one is the spin mode where the satellite is slowly rotated around the axis normal to the orbital plane. Because of the projection of the gravitational acceleration onto the sensitive axis of the instrument, the resulting possible violation signal will have a frequency equal to

f EP = f orb + f spin , (2.1)
where f orb is the orbital frequency equal to 1.7 × 10 -4 Hz and f spin is the spin frequency of the satellite which can be adjusted but is usually set to be a certain fraction of the orbital frequency: f spin = 9/2 × f orb = 7.7 × 10 -4 Hz (as we will see in Chap. 3) or f spin = 4/2 × f orb = 3.4 × 10 -4 Hz. Thus the resulting EP frequency is tunable thanks to the attitude control system. For the MICROSCOPE experiment to be valid, important calibration phases are necessary in addition to the sessions during which the WEP is tested. Indeed, the characterization of the instrument is as important as the test itself. During the calibration phases the instrument and the satellite are excited in various ways in order to estimate the characteristics of the experiment. Furthermore, WEP test sessions as well as calibration sessions will be performed several times in different conditions. The mission lifetime will be 1 to 2 years. To analyse the data coming from the satellite and to pilot the operational phases during the mission, a dedicated ground segment has been developed, which is briefly described in the next section.

The ground segment and the scientific mission centre

The mission ground segment is divided in two parts. On the one hand, the first processing chain involves the MIcrosatellite Ground Segment (MIGS) which is common to all satellites of the low-cost Myriade chain. It is composed of a ground station network and of the control and command centre (CCC) receiving the telemetry sent by the satellite. The CCC then transmits the data to the drag compensation system expertise centre (CECT for the French acronym). Both CCC and CECT are located in the CNES centre in Toulouse. On the other hand, the scientific mission centre (CMS for the French acronym) is responsible for the mission and payload monitoring and scientific data processing. It is located in ONERA facilities near Paris. This architecture is summarized by Fig. 2.2.

The scientific mission centre is the heart of the scientific data processing and will also help to make operational decisions during the mission. Part of the algorithmic methods presented in this thesis have been implemented in the CMS. It is interfaced with the CNES centres in the following way.

The CMS, together with the CECT, constructs the mission scenario and the on-board program, and uploads it to the CCC. Once the planned scenario is run in flight, the CCC will download the measured data to the CMS while the CECT will compute the orbit restitution. As illustrated in Fig. 2.2, in addition to their daily operational activities, the CECT and the CMS work in close interaction with two groups: the group of experts (GEX) which has an expertise on the payload environment, and the science performance group (SPG) which makes weakly reports and makes decisions about future operations. Finally, a group of experts and scientists named science working group (SWG) will be in charge of promoting the data analysis, examining the precision of the results and approve the distribution of the data to the scientific community.

The data that will be delivered by the processing chain is organized in different levels. The first level (N0) corresponds to the operational data. The second level of data (N1) is organized by session (calibration or EP test sessions) and corresponds to the measured accelerations per sensor. The third level (N2) gives the common and differential accelerations. The N1 and N2 levels both include 3 sublevels: a) raw data, with some minor corrections from on-ground estimation of instrument parameters; b) calibrated data from in-flight estimation of instrument parameters; c) calibrated data with refined estimation (in particular by taking into account the time variations of the calibration parameters).

The space segment

In this section we describe the satellite platform on which the MICROSCOPE payload is hosted.

The MICROSCOPE space mission uses the Myriade platform initially developed by CNES to send low-cost scientific missions on Earth orbit. It belongs to the microsatellite category with a mass of about 300 kg and a cubic shape of 1m × 1m × 1.5m (see Fig. 2.3). The payload mass is 35 kg with a power consumption of 40 W.

The particularity of the MICROSCOPE satellite is its drag compensation system, which acts continuously, so that the payload and the attitude and orbit control system (AOCS) are closely linked together. The drag-free system is a control loop which measures and attenuates all non-gravitational forces acting on the satellite which are mainly the atmospheric drag force and the solar radiation pressure, whose order of magnitude is 10 -8 ms -2 . For each accelerometer in the experiment, the measurement of interest is the differential acceleration, that is, the difference of the accelerations applied to the two test-masses to maintain them motionless in the cages. Nevertheless, the non-gravitational perturbation forces act in the same way on both test-masses via the satellite, so that their effect would be cancelled by measuring the differential acceleration. Therefore in a perfect world, being drag-free is not a fundamental and necessary condition for such an experiment. However the differences of sensitivity of the two sensors and their imperfections introduce an error term which is proportional to the mean acceleration of the test-masses (we shall call it the common acceleration) and hence to the non-gravitational forces. The drag-free system is therefore implemented to nullify the coupling between the differential and the common accelerations.

The drag-free control loop, along with the attitude control loop, is summarized in Fig. 2.4. The external disturbance forces act on the satellite, whose deviations are measured by the linear common mode of the accelerometer. In addition, the star trackers and the angular rates of the test-masses provide a measurement of the external torque undergone by the satellite. Then these measurements are recorded by the on-board controller which calculates the commands to be sent to the satellite thrusters to correct for the disturbance. The thrusters used in MICROSCOPE are cold gas propellants. They utilize a gas stored at high pressure which is released in a nozzle guiding the output thrust. The cold gas propulsion allows to exert low (micro-newton) thrusts of short duration, and is thus well adapted to drag compensation. In MICROSCOPE the drag-free acceleration residual is less than 10 -12 ms -2 at EP frequency and its stochastic fluctuations are below 10 -10 ms -2 Hz -1/2 .

Description of the payload

In this section we describe the design of the instrument used for the MICROSCOPE experiment. It consists in two differential electrostatic accelerometers which have been developed by ONERA with the technological heritage from geodesy missions such as GRACE (NASA) and GOCE (ESA) [START_REF] Touboul | Accelerometers for CHAMP, GRACE and GOCE space missions: synergy and evolution[END_REF][START_REF] Marque | The ultra sensitive accelerometers of the ESA GOCE mission[END_REF].

Overview

The satellite payload is formed by two pairs of inertial sensors, as illustrated in Fig.

(2.5a).

One pair forms what is called a differential accelerometer. As mentioned in the previous section the first accelerometer is devoted to the WEP test and the other serves as a reference. One accelerometer includes three units:

• the sensor unit (SU), including two cylindrical and co-axial test-masses which are surrounded by four silica cores, the whole being inside the same tight housing (see Fig. 2.5b). Vacuum is maintained in the housing with a pressure below 10 -5 Pa in order to reduce the parasitic forces on the test-mass, such as gas damping and radiometric forces.

• the front end electronic unit (FEEU), including the capacitive sensors (i.e., the position detectors) of both masses, the reference voltage sources and the electronics generating the voltages applied to the electrodes.

• the interface control unit (ICU), formed by the digital electronics implementing the laws of the servo-loops controlling the motion of the test-masses, as well as the data bus which is connected to the satellite.

These units are connected to each other in the following way (see also Fig. 2.7): after receiving the position measurement from the FEEU, the ICU sends the controls back to the FEEU which generates the control voltages applied to the electrodes of the SU. The two inertial accelerometers form the instrument of the MICROSCOPE mission, called Twin Space Accelerometer for Gravitation Experiment (T-SAGE). In the EP accelerometer (devoted to the WEP test), the two test-masses have different compositions: one is made of platinum rhodium alloy (PtRh) and the other of titanium alloy (TA6V) [START_REF] Touboul | The MICROSCOPE experiment, ready for the in-orbit test of the equivalence principle[END_REF]. In the REF accelerometer the test-masses are both made with PtRh.

The two accelerometers can separately be switched on and off, and have three operating modes. The first one is the stand by mode (SBy) in which the sensor is not powered. The second one is the position sensing mode (PSM) in which the servo-control of the test-mass does not operate in closed-loop (there is no levitation), only the position of the test-mass is measured. The third mode is the acceleration sensing mode (ASM): the servo-control loop is closed and the sensor is operating. This latter mode has two configurations: full range mode (FRM) and high resolution mode (HRM). In FRM the measurement range is broader but with a lower resolution, while this is the opposite in HRM. The WEP test session will be carried out in HRM.

The MICROSCOPE experiment and the measurement equation

Before the accelerometers are switched on for the first time in flight, the test-masses must be clamped by the blocking mechanism (indicated in Fig. 2.5b), in order to withstand the vibration during launch.

In order to visualize the orientation of the test-masses with respect to the SU and the configuration of the set of electrodes, we represent the SU's reference frames in Fig. 2.6. The X axis is oriented along the axial direction of the cylindrical mass. The Y axis is oriented along one of the two radial directions of the mass and is normal to the orbital plane (see also Fig. 2.1). The Z axis is oriented so as to form an orthogonal triad with the other axes. The Φ, Θ and Ψ angles are respectively defined about the X, Y and Z axis.

The axial, radial and angular motion of the test-mass is controlled by the electrodes which exert an electrostatic force on it. The axial and the Φ control is done through a change of capacitance by what is called "variation of the covering area", while the radial control is done by "variation of the gap". In the following we explain more precisely the operating principle of the MICROSCOPE accelerometers, and we describe how the electrostatic detection and electrostatic control of the test-mass are performed.

The servo-controlled electrostatic accelerometer

The general concept of an accelerometer is the measurement of the motion of a test-mass inside a cage. The simplest design to which one can think of is a test-mass connected by a spring to the cage wall, whose position is measured in some way and gives access to its acceleration. But the measurement performed by such a simple device will be limited by the spring stiffness. In addition, the measurement will be easy to do along the spring axis, but measuring the acceleration in any other direction begins to be delicate (one could imagine a net of springs but resonances become an issue). A better measurement of the acceleration of the test-mass would be obtained if there were no direct link between the testmass and the cage wall. This is what is (almost) achieved by an electrostatic accelerometer, where the test-mass is levitating inside the cage thanks to surrounding electrodes. In this device, the acceleration is directly measured by a servo-controlled system. It consists in studying the electrostatic restoring force necessary to maintain the test-mass steady at its balance point, i.e. motionless at the centre of the cage. The control force to be applied is deduced from a measurement of the position of the test-mass through capacitive sensing.

The measurement of the acceleration is obtained at each loop of the servo-system which includes a detection step and an actuation step. The control loop is sketched in Fig. 2.7 where five main blocks are shown. The detection of the test-mass shift is performed by a capacitive sensor. When the test-mass moves inside the cage, it induces a capacity difference between the electrodes (block 1 ). This difference is then converted into an output voltage by the capacitive detector electronics (block 3 ). From this output a digital servo-control loop calculates the control signal (block 4 ). After being amplified (block 5 ), this signal is then applied to the electrodes so as to bring the test-mass back to the centre of the cage, by asymmetrically applying an electrostatic potential to it (block 1 ). The same electrodes are used for both detection and action, through the same electronic interface (block 2 ). Note that for this system to be able to work the potential V m of the test-masses must be controlled. In the T-SAGE instrument, this is done via a thin gold wire connected to the masses, with a thickness of 7 µm.

Detection

In this section we detail the capacitive position detector, and focus on the blocks 1 and 3 of Fig. 2.7.

The test-mass displacement in a given axis is measured by a pair of electrodes facing or surrounding the test-mass along the axis direction. The test-mass forms two capacitors with each facing electrodes, whose capacitances will be denoted C 1 and C 2 . The displacement of the test mass corresponds to a variation of the capacitance.

The detection signal is then obtained as an output of block 3 in Fig. 2.7. In this scheme the pairs test-mass/electrode are viewed as equivalent capacitors. The key idea is to apply a sinusoidal reference voltage V d to the test-mass. Its frequency is typically 100 kHz (we will see later why). This alternating voltage generates currents across the capacitors C i . Then the superposition network (block 2 ) extracts the difference of currents running in the capacitors.

In the end, the detector output voltage is proportional to the difference of the capacitance

V det = G det (C 1 -C 2 ) , (2.2)
where G det is the sensitivity gain of the capacitive sensor. This factor is determined by the potential of the test-mass V d and the mean capacitance [START_REF] Josselin | Capacitive detection scheme for space accelerometers applications[END_REF]:

G det = 2V d /C eq
where C eq is the capacitance of the capacitor formed by the proof-mass and the sensing electrodes when the proof-mass is at the centre of the cage. The output voltage is then amplified by an alternative current selective amplifier which has a gain of about 2.3 dB at f d = 100 kHz. The signal is then demodulated by lock-in detection (V d ) to provide a low-frequency signal. This is finally amplified by a DC amplifier to be within a voltage range suitable for the control loop (typically ±5 V).

The capacitance depends on the insulating material and the geometry. If we assume that the test-mass and the electrode i is an ideal planar capacitor in vacuum it can be approximated by where 0 is the dielectric permittivity of vacuum, A i is the electrode area facing the test-mass, and e i is the gap between the test-mass and electrode i.

C i = 0 A i e i , ( 2.3)
There are two ways of performing a capacitive detection: detection by variation of the gap e i , or detection by variation of the covering area A i . These two techniques are detailed below.

Detection by gap variation

The gap variation technique is used for the Y and Z axis. In this case, the electrodes are facing the test-mass, and its displacement induces a variation of the gap e i , as shown in Fig. 2.8.

The gap of each capacitor is given by

e i = e + y if i = 1, e -y if i = 2.
Then the detector output voltage is

V det = G det 0 A 1 e -y - 1 e + y ≈ 2G det 0 A e 2 • y + o y e 3 , ( 2.4)
where we have assumed that the displacement y is much smaller than the equilibrium gap e. Therefore at first order in y e the output voltage delivered by the position sensor is proportional to the displacement y with a gain depending on the sensor sensitivity and the instrument geometry. These parameters must be well known in order to accurately infer the position.

Detection by area variation

We saw in the previous paragraph a method to detect the displacement of the testmass based on a variation of the gap between the electrodes and the test-mass. The geometrical equation (2.3) of the capacitance suggests another approach, which consists in detecting the displacement of the test-mass through the variation of the covering area A of the capacitance.

We consider a pair of cylindrical and concentric electrodes and a test-mass of similar shape sharing the same symmetry axis represented in Fig. 2.9. At rest the test-mass lies in the centre of the cage, at position x = 0, and each of the electrode surfaces covers the test-mass over a distance h.

h -x h + x X Y V 1 V 2 V p F el,1 F el,2 O Figure 2
.9: Representation of the capacitive detection of the axial displacement of the test-mass by variation of the covering area between the electrodes and the test-mass.

In spite of the boundary effects, the covering area of electrode i can be approximated by

A i ≈ 2πrh i , (2.5)
where r is the radius of the cylindrical electrode, h i is the covering distance, which reads, according to Fig. 2.12:

h i = h -x if i = 1, h + x if i = 2.
Therefore the detection voltage expressed in Eq. (2.6) gives

V det = - G det 0 4πr e • x. (2.6)
This last equation does not stem from a Taylor expansion at first order and is "exact" (provided that all the other hypotheses are true: perfectly cylindrical shape, perfect alignment, etc.). Therefore, in the case of the detection by area variation, the detection voltage is directly proportional to the displacement.

Amplification and digital conversion

The servo-control loop operates at 1 kHz and is digital. The analogue signal V det must therefore be converted. This is done by a sigma-delta ADC [START_REF] Kester | ADC Architectures IV: Sigma-Delta ADC Advanced Concepts and Applications[END_REF] which converts the analogue voltage into a bit stream. The general idea of this device is illustrated in Fig. 2.10. A latch synchronizes the whole set-up at an oversampling frequency (20 MHz), and delivers a bit stream (that is, 0 or 1) whose frequency depends on the input analogue signal. These bits are then converted into pulses of electric voltages by the 1-bit DAC: a 1 is converted into V ref and a zero is converted to -V ref (where V ref is the maximum voltage that the input can take). This pulse stream is then subtracted (delta) to the input analogue voltage, so that the subtractor measures the difference between the input analogue signal and the result of the conversion. After this difference is made, an integrator (sigma) sums the variation over time of the difference between the input and the output signal. If the absolute value of this difference is high, the output signal of the integrator will quickly increase (or decrease, according to the sign of the difference). The analogue signal is then represented by the bit stream at the output of the latch. In this type of coding, the value of the digitized signal is a function of the number of bits 1 per time period. If there are only bits equal to 1 in some time interval, the signal will be close to V ref , and if there are only bits equal to 0, the signal will be close to -V ref . If there is a regular alternation of 1 and 0 the signal will be close to 0. At the output a digital filter averages the bit stream and gives a more precise signal between 0 and 1.

Action

Following the scheme in Fig. (2.7) the detector output voltage is then digitized and treated by the servo-control loop. The goal is to nullify V det by applying an actuation voltage to the electrodes in order to compensate for the displacement of the test-mass, and bring it back to the centre of the cage.

Principle

Let us consider the system defined by a single capacitor of capacitance C i . We assume that one plate is fixed (which is, in the case under study, the electrode) and that the other plate can move. To simplify we assume that the medium in-between is vacuum, such that the capacitance only depends on the displacement u. We further assume that the charge ±Q on each plate is constant. The movable plate is subject to a force F el .

On the one hand, according to Newton's second law (1.1) and to the definition of the kinetic energy E k , an elementary displacement du of the movable plate corresponds to a work of the force equal to

dE k = F el,i • du.
(2.7)

On the other hand, the voltage of each plate of the capacitor is connected to one pole of an electrical generator which maintains their potential constant. The potential energy stored in the capacitor (which can be derived as the work needed to establish the 2. The MICROSCOPE experiment and the measurement equation electric field between the plates) is expressed as a function of the constant voltage V between the plates:

E p = 1 2 C i V 2 .
(2.8)

The power supplied by the generator is dE dt = V I = V dQ dt , therefore the infinitesimal energy generated during dt is

dE = V dQ = V 2 dC i , ( 2.9)
where the last equality comes from the definition of the capacitance

C ≡ Q V .
(2.10)

We now assume that there is no loss by Joule heating, therefore the energy conservation implies that the energy supplied by the generator is either stored in the capacitor or transformed into work:

dE k + dE p = dE, (2.11)
with

dE p = 1 2 V 2 dC i . ( 2.12)
Then Eq. (2.11) yields

F el,i • du + 1 2 V 2 dC i = V 2 dC i .
and hence

F el,i = 1 2 V 2 ∇C, (2.13)
where ∇C is the spatial gradient of the capacitance. In order to control the applied force, the potentials of the electrode and the test-mass must be controlled. The potential of the test-mass is maintained constant at a value V p called polarization. The potential V i of the electrodes is tuned by the servo-control loop. The voltage V is equal to the potential difference between the plates

V = V i -V p .
As in the case of the detection, the test-mass control depends on whether the capacitance varies with the gap e between the electrodes or with the covering area A facing the electrodes. We describe these two different cases in the following.

Gap control

The capacitance of the capacitor formed by the test-mass and one electrode can depend on the gap between them. This is the technique used to control the radial motion (Y and Z axis) as shown in Fig. 2.6b.

We now consider the system defined by the test-mass alone. The resulting force acting on it is the sum of the electrostatic forces from the two electrodes:

F el,y = F el,1,y + F el,2,y .
(2.14) Thanks to the geometric expression of the capacitance in Eq. ( 2.3) we can calculate the gradient of the capacitance with respect to y and the expression of the force in Eq. 2.13 to get

F el,y = 1 2 0 A - (V 1 -V p ) 2 (e + y) 2 + (V 2 -V p ) 2 (e -y) 2 , ( 2.15)
which can be developed to first order:

F el,y = 1 2 0 A e 2 V 2 2 -V 2 1 + 2V p (V 1 -V 2 ) + V 2 1 + V 2 2 + 2V 2 p -2V p (V 2 + V 1 ) 2y e + o y e 2 .
(2.16)

If we control the potential of the electrodes to be the same value but with opposite sign such that V 2 = V e and V 1 = -V e as shown in Fig. 2.11, then Eq. (2.15) becomes

F el,y = 2 0 A e 2 -V p V e + (V 2 e + V 2 p ) y e + o y e 2 .
(2.17)

To first order the acceleration applied to the test-mass can therefore we written as

Γ y = F el,y m I ≈ -G act V e + w 2 p   1 + V e V p 2   y, ( 2.18)
where we defined:

• the sensitivity factor G act = 2 0 AVp m I e 2 ;

• the stiffness coefficient

w 2 p = 2 0 AV 2 p m I e 3
If we consider only the first term of the right-hand-side (RHS) of Eq. (2.18) we see that if we know the value of G act well enough, the acceleration of the test-mass can be measured through the voltage V e necessary to apply the restoring force. This measurement is perturbed by the second term proportional to the electrostatic stiffness w 2 p which introduces a bias if the test-mass is not servo-controlled to the equilibrium point

The MICROSCOPE experiment and the measurement equation

(y = 0). Nevertheless, the asymmetry in the design of the electrostatic configuration and the displacement amplitude are sufficiently small to ignore this term with respect to the first one. Note that in the case of a cylindrical test-mass as shown in Fig. 2.6 the expression for the sensitivity factor is slightly different, but the general form of Eq. (2.18) is preserved. Note that a similar device exists to ensure the angular control of the test-mass. However, to be able to induce a torque, 2 pairs of electrodes are necessary. This is e.g. the way the angle θ of the test-mass about the X axis is controlled.

Covering area control

As for the detection process, the control action can be done by using the covering area geometry, as represented in Fig. 2.12.

h -x h + x X Y V 1 V 2 V p F el,1 F el,2 O Figure 2
.12: Representation of the axial control of the test-mass by variation of the covering area.

Eqs. (2.5) and 2.6, and the differentiation of Eq. (2.3) with respect to the displacement x gives

dC i dx = (-1) i 2πr 0 e . ( 2.19)
Then by using Eq. (2.13) we can write the total electrostatic force undergone by the test-mass, projected onto the X axis:

F el,x = F el,1,x + F el,2,x = πr 0 e -(V p -V 1 ) 2 + (V p -V 2 ) 2 = 2πr 0 e V p (V 1 -V 2 ) + V 2 2 -V 2 1 2 . (2.20)
If the control voltages are such that V 1 = -V e and V 2 = +V e , then the applied acceleration is

Γ el,x = - 4πr 0 V p m I e V e .
(2.21)

Therefore for an axial control using the covering area the acceleration applied to the test-mass is directly proportional to the potential of the electrodes. Unlike the control by gap variation there is no perturbation term proportional to the displacement, hence no electrostatic stiffness. If the geometry of the instrument is well known, this provides a more accurate way to measure the acceleration of the test mass.

A similar device exists for the angular control. For example, the rotation angle about the X axis is also controlled by variation of the covering area.

Management of action and detection

In the T-SAGE instrument, the same electrodes are used for both action and detection. This is possible thanks to a difference between the cadence of the detection potential V d and the one of the actuation potential V p of the test-mass. The former varies at a frequency of about 100 kHz whereas the latter is held at a constant value (near DC). This allows the decoupling of the action and detection processes. While V d is used to produce a current across the capacitors, V p is used to tune the potential of the electrodes. The actual value of the mass potential is thus

V m = V p + V d .

Digital control loop

In this section we detail the block 4 of Fig. 2.7 describing the digital control loop of the test-mass. This loop operates at a 1 kHz frequency, and is digital, so that the signal delivered by the position detector is digitized by an analogue to digital converter (ADC) described in Sec. 2.2.4. By combination of all sensors responses, a six-degree of freedom position is measured and transmitted to a proportional-integral-derivative (PID) controller which computes the actuation control signals. These signals are recombined to obtain the final voltages that must be applied to the electrodes to compensate the motion of the test-mass. The voltages are converted into an analogue signal by a digital-to-analogue converter (DAC) and are then amplified by a digital voltage amplifier (DVA) to be finally transmitted to the electrodes. The measurement of the acceleration is obtained by reading the PID output thanks to a read-out circuit. For the X axis, there is also an output after the DVA, with a read-out and an ADC. This signal is low-pass filtered (to reduce aliasing effects), averaged and decimated down to 4 Hz. Note that an arbitrary external command can be added at the input of the PID to control the motion of the test-mass. This will be used to calibrate the instrument in flight, but is set to zero for nominal WEP test sessions.

Derivation of the measurement equation

So far we have described the trajectory of the satellite carrying the instrument, as well as its operating process. Based on this description we now establish the physical equations modelling the in-orbit measurement provided by the sensor units. This means, on the one hand, to derive the dynamics of the test-mass, the electrostatic acceleration that is applied to it, and the dynamics of the satellite. On the other hand, we will refine the actual measured acceleration by including the instrument defects. This comprehensive model will then allow us to identify the sources of perturbations disturbing the measurement of the Eötvös parameter. In the clockwise direction, starting from the top left: the external acceleration Γ ext corresponds to a position of the test-mass (by double integration), which is converted into a detection voltage by the capacitive sensor, then transmitted to the controller which calculates the actuation voltage necessary to apply the corrective acceleration Γ app via the actuation gain G act . The measured acceleration Γ meas is given by the controller output after filtering and proper re-scaling.

Derivation of the equations of dynamics

We saw in Sec. 2.2 that a differential accelerometer is made of a pair of inertial sensors, each one containing a test-mass. Thus we consider the system defined by a test-mass i of a single inertial sensor, whose inertial mass is denoted m i and gravitational mass mi , with a centre of inertia O i . We use the Newtonian formalism to derive the equations of dynamics, since the relativistic effects will be negligible.

The system is subject to forces which can be classified in three types:

• the gravitational force F g,i = mi g

• the electrostatic force f e,i

• the sum of all other disrupting forces f p,i

We begin to apply Newton's second law to the test-mass in the inertial reference frame R in (for example the geocentric reference frame J2000 of origin Earth's centre O in ):

m i γ(O i )| R in = F g,i + f e,i + f p,i .
(2.22)

The main gravitational sources are Earth and the satellite carrying the instrument. The effect of distant astrophysical sources like Sun and Moon can be included in the calculation of the gravitational acceleration g, but it can be shown that their effect is negligible in the measurement (for instance the influence of the Sun represents an error of less than 1% on a non-zero Eötvös parameter). Therefore we omit their contribution in the following. Hence the gravitational acceleration of a point P of the test-mass can be decomposed as the sum of two contributions g = g E + g S . In addition, the test-mass is not a point-mass but occupies a volume in space. Therefore the gravitational attraction acting on it is expressed by the integral over all the infinitesimal mass elements forming the test-mass:

F g,i = P ∈ mass i g • d m(P ) = P ∈ mass i (g E + g S ) • d m(P ) = P ∈ mass i g E • d m(P ) + mi g S,i , (2.23)
where in the last line we separated the effect of the gravitational field generated by the satellite onto the test-mass by setting

g S,i ≡ 1 mi P ∈ mass i g S • d m(P ). (2.24)
By construction this quantity is maintained under an acceptable level. We now consider the system formed by the satellite of mass M (excluding the testmasses) and of centre of inertia O s . This system undergoes the gravitational force F g,sat , all the non-gravitational external forces F, as well as the electrostatic forces from all the N tm test-masses. Thus in the inertial reference frame R in Newton's second law applied to the centre of inertia of the satellite writes

M γ(O s )| R in = F g,sat + F - Ntm j=1 f e,j .
(

As for the test-mass, the gravitational attraction undergone by the satellite can be expressed as an integral over all the infinitesimal masses forming the satellite

F g,sat = P ∈sat g E • d m(P ). (2.26)
The integrals in the first term of Eq. (2.23) and in Eq. (2.26) can be expanded around a fixed point A. This is done by calculating the Taylor expansion of the gravitational acceleration

g E (P ) = g E (A) + [T](A) • AP + o ( AP ) , ( 2.27)
where [T](A) is the gravitational gradient calculated at point A, and is defined as the 3 × 3 Jacobi matrix of the gravitational potential U :

[T] = ∇ 2 U, (2.28)
which contains the gradients of the gravitational acceleration g. If P is a point of a solid body (either the test-mass or the satellite) whose centre of inertia is labelled O, taking the integral of Eq. (2.27) gives

P ∈ solid g E (P ) • d m(P ) ≈ M g E (A) + [T](A) • P ∈ solid AP • d m(P ). (2.29)
Then we can decompose AP = AO + OP and use the definition of the centre of inertia

P ∈ solid OP • d m(P ) = 0, (2.30)
to get

P ∈ sat g E (P ) • d m(P ) ≈ M g E (A) + [T](A) • AO . (2.

31)

The closer A is to O, the more accurate this expansion will be. For example, A can be chosen to be the middle of the geometric centres of the test-masses. Now we use Eq. (2.31) to expand around the same point A the gravitational attraction acting on the test-mass on the one hand, and the attraction acting on the satellite on the other hand:

1 m i F g,i ≈ (δ i + 1) • g E (A) + [T](A) • AO i + g S,i , 1 M F g,sat ≈ (δ S + 1) • g E (A) + [T](A) • AO s , ( 2.32)
where we introduced the parameters δ i and δ S defined as

δ i = mi m i -1, δ S = M M -1, (2.33)
which quantify the violation of the WEP in the conventional unit system. Note that the Eötvös parameter can be expressed as a function of δ since

η 1,2 = 2 (δ 1 -δ 2 ) (2 + δ 1 + δ 2 ) ≈ δ 1 -δ 2 .
(2.34)

The expanded expressions of Eqs. (2.32) will be useful in the following.

Derivation of the kinematic equations

So far we have expressed Newton's laws of dynamics applied to the test-mass and the satellite in the inertial reference frame R in . However, the test-mass is not located at the centre of inertia of the satellite, which introduces inertia terms. Therefore we must compare the acceleration of the test-mass and the acceleration of the centre of mass of the satellite in the inertial reference frame. We define the satellite reference frame, labelled R sat , as rigidly fixed to the satellite platform, which is in rotation with respect to R in with the angular velocity Ω = Ω Rsat/R in . The law of the acceleration transformation then gives

γ(O i )| R in = γ(O s )| R in + Ω × (Ω × O s O i ) + Ω × O s O i + 2Ω × Ȯ s O i + Ös O i , (2.35)
where the time derivatives are taken with respect to the satellite reference frame:

Ȯ s O i ≡ dO s O i dt Rsat and Ö ss O i ≡ d 2 OsO i dt 2
Rsat . By noting Ω x , Ω y and Ω z the components of the angular velocity vector Ω, the last equation can be simplified by introducing the angular velocity matrix [Ω] and the inertia gradient matrix [In], which are respectively defined as

[Ω] ≡    0 -Ω z Ω y Ω z 0 -Ω x -Ω y Ω x 0    , (2.36) and [In] ≡ [Ω 2 ] + [ Ω], (2.37)
with

[Ω 2 ] ≡    -Ω 2 y -Ω 2 z Ω x Ω y Ω x Ω z Ω x Ω y -Ω 2 z -Ω 2 x Ω y Ω z Ω x Ω z Ω y Ω z -Ω 2 x -Ω 2 y    , ( 2.38)
and

[ Ω] ≡    0 -Ωz Ωy Ωz 0 -Ωx -Ωy Ωx 0    .
(2.39)

Eq. (2.35) is then rewritten as

γ(O i )| R in -γ(O s )| R in = [In]O s O i + 2[Ω] Ȯ s O i + Ös O i , ( 2.40)
which gives the kinematic equation of the test-mass in the satellite reference frame.

Derivation of the applied electrostatic acceleration

We now consider the electrostatic force applied to the test-mass, which provides a measurement of its acceleration. In the previous section we derived the dynamic and kinematic equations for the motion of the test-mass. By plugging the expression of the test-mass acceleration and the satellite accelerations respectively given by Eq. (2.22) and Eq. (2.25) into Eq. (2.40), we obtain

[In]O s O i + 2[Ω] Ȯ s O i + Ös O i = 1 m i f e,i + 1 M Ntm j=1 f e,j + 1 m i F g,i - 1 M F g,sat + 1 m i f p,i - 1 M F. (2.41)
By using the Taylor expansions of Eqs. (2.32), we can rewrite the second line of Eq. (2.41) as

1 m i F g,i - 1 M F g,sat ≈ g S,i + (δ i -δ S ) g E (A) +[T](A) • O s O i + (δ i AO i -δ S AO s ) . (2.42)
The last term in parenthesis can be neglected. Indeed the order of magnitude of the gradient is typically ∼ 10 -6 s -2 at a 700 km altitude and we saw in Chap. 1 that previous EP experiments showed that δ < 10 -12 . This leads to an acceleration error less than 10 -18 ms -2 to be compared to the required precision on δ i g E which is 8 × 10 -15 ms -2 . We obtain

[In]O s O i + 2[Ω] Ȯ s O i + Ös O i = 1 m i f e,i + 1 M Ntm j=1 f e,j +g S,i + (δ i -δ S ) g E (A) + [T](A) • O s O i + 1 m i f p,i - 1 M F. (2.43)
To simplify the equations it is useful to define the mechanical bias, gathering the mechanical perturbation on the masses and the self-gravity of the satellite:

b i ≡ - 1 m i f p,i -g S,i . (2.44)
We also define the offcentreing vector as

∆ i ≡ O s O i . ( 2
i M Γ App,i = -[T](A) • ∆ i -(δ i -δ S ) g E (A) +[In]∆ i + 2[Ω] ∆i + ∆i +b i + 1 M F - Ntm j =i m j M Γ App,j . (2.47)
We note that the acceleration applied to one test-mass depends on the reaction of the satellite to the electrostatic forces applied to all other masses through the last term of this equation. The interest of the MICROSCOPE experiment is to monitor the relative free-fall of two test-masses, therefore we are interested in the difference of their acceleration. Thus we define the differential acceleration as half the difference of the accelerations applied to the two test-masses of one sensor unit:

Γ App,d ≡ 1 2 (Γ App,1 -Γ App,2 ) . (2.48)
It is also useful to define the common acceleration defined as the half-mean of the individual accelerations:

Γ App,c ≡ 1 2 (Γ App,1 + Γ App,2 ) . (2.49)
More generally for any quantity x i we will respectively denote the corresponding differential and common modes by

x d = 1 2 (x 1 -x 2 ) , ( 2.50a
)

x c = 1 2 (x 1 + x 2 ) . (2.50b)
We adopt the convention where label 1 (respectively 3) refers to the internal mass and label 2 (respectively 4) refers to the external mass of the ep accelerometer (respectively the ref accelerometer). Using Eq. (2.47) and (2.48) we derive the expression of the differential acceleration:

Γ App,d = -([T] -[In]) • ∆ d -δ d g E +2[Ω] ∆d + ∆d + b d , (2.51)
where we defined the differential EP parameter δ d which is approximately related to the Eötvös parameter in Eq. (1.3) by the relation δ d ≈ 1 2 η 1,2 . To lighten the notation we have also dropped the information about the point A at which the gravitational gradient and acceleration are calculated. Note that in the MICROSCOPE official documentation the differential offcentring vector is defined as ∆ = O 1 O 2 . In this thesis we will therefore keep the subscript "d" not to confuse ∆ d with ∆ = -2∆ d . We note that the applied differential acceleration does not depend on the non-gravitational forces F nor the reaction of the satellite to the electrostatic forces f e,j from the masses of the other accelerometer. Hence if the instrument were perfect, being drag-free would be unnecessary for the experiment, since both test-masses should in principle undergo the same non-gravitational forces.

Using Eq. (2.47) and (2.49) we also derive the expression of the common acceleration:

Γ App,c = 1 1 + α c -([T] -[In]) • (∆ c -α d ∆ d ) -(δ c -α d δ d ) g E + δ S g E +2[Ω] ∆c -α d ∆d + ∆c -α d ∆d +b c -α d b d + 1 M F - Ntm j =1,2 m j M Γ App,j , (2.52)
where we set α c ≡ (m 1 + m 2 ) /M and α d ≡ (m 1 -m 2 ) /M . Note that since M ∼ 10 3 m i these ratios are small with respect to one. Therefore a crude approximation of the common acceleration is

Γ App,c ≈ -([T] -[In]) • ∆ c -(δ c -δ S ) g E +2[Ω] ∆c + ∆c +b c + 1 M F - Ntm j =1,2 m j M Γ App,j . (2.53)
We note that the common acceleration contains the effect of the non-gravitational forces and the reaction of the satellite to the accelerations applied to the other masses.

Instrument imperfections

In Sec. 2.3.3 we derived the measurement equation in the case of a perfect instrument.

However, in practice an inertial sensor does not directly measure the accelerations Γ App,i .

Indeed, this acceleration is deduced from the voltage applied to the electrodes, which is subject to all the slight defects in the measurement process, including systematic errors and noise. The systematic errors (i.e., the deterministic errors) include scaling differences, axis couplings, axis misalignment and non-linearity. The noise (i.e., the stochastic errors) include stochastic variations of the forces applied to the test-masses, electronic noise and thermal random fluctuations.

Systematic errors

We list and formalize below the systematic instrumental defects (following [START_REF] Métris | Formulation de l'équation de mesure microscope pour une utilisation dans les logiciels d'analyse[END_REF]):

1. the scale factors of the measurement (close to 1), modelled by a diagonal matrix [K 1i ]:

[K 1i ] ≡    K 1xi 0 0 0 K 1yi 0 0 0 K 1zi    ; (2.54)
2. the coupling between axes, because of non exact orthogonality between the test-mass faces and the cage, modelled by a symmetric matrix [η i ]:

[η i ] ≡    0 η zi η yi η zi 0 η xi η yi η xi 0    ;
(2.55)

3. the electrical bias (additive constant term), modelled by a vector b 0i ;

4. the small rotations (or misalignment) of the instrument with respect to its assumed orientation, i.e., the rotations between the coordinate system in which the acceleration Γ App,i is modelled, denoted SU0 , and the coordinate system proper to the inertial sensor, denoted SUi. The "real" coordinate system of the inertial sensor is the one where the measurement is actually performed and depends on the mass geometry and the fine configuration of the electrodes. This is modelled by a rotation matrix [Θ i ] which is written as

[Θ i ] ≡    1 θ zi -θ yi -θ zi 1 θ xi θ yi -θ xi 1    ; (2.56)
5. the coupling with the angular acceleration, reflecting the fact that the angular control might slightly affect the linear control. This term is modelled by a coupling matrix [C i ];

6. non-linear effects, for example in Sec. 2.2.5 we saw that particularly in the case of varying gap control the applied acceleration is not exactly linear with the voltage. This is modelled by a quadratic term Q i , described by a 3-components vector K 2i , and proportional to the squared applied acceleration projected in the sensor frame:

Q i = K 2i • [Θ i ]Γ 2 App,i
, where • denotes the element-wise multiplication and Γ 2 App,i is the vector composed by the squared components of the applied acceleration. The quadratic coefficient vector has the form

K 2i ≡ K 2ixx K 2iyy K 2izz T .
(2.57)

Stochastic errors

In Sec. 2.2 we described the whole process providing the measurement of the test-mass acceleration. But each elementary part of this process, for instance the blocks of the digital control loop in Fig. 2.13, introduces some random disturbances, that we call instrumental stochastic noise. The sum of all these contributions is modelled by a random variable n i which is added to the measurement equation. We list below the main noise sources.

A stationary stochastic noise is described by a quantity called power spectral density (PSD) S(f ), whose formal definition will be given in Sec. 3.1.3.1. For a given frequency f , S(f )df represents the amount of power contained in the interval [f ; f + df ]. If A is the unit of the quantity of interest (e.g. the acceleration), then S(f ) is expressed in A 2 Hz -1 . The PSD may depend on the frequency f , then the noise is said to be coloured. If the PSD is independent of frequency, the noise is said to be white.

We now list the main sources of noise affecting the control loop of Fig.

(2.7), and give their PSD as a function of the instrument features.

Thermal acceleration noise

We first consider the noise coming from the physics of the test-mass inside its cage. A major source is the radiometric noise, which comes from the exchange of momentum between the test-mass and the particles remaining in the cage. This perturbation depends on the temperature, and is expressed as a function of the pressure P inside the cage, the mass m i of the test-mass, the collision cross-section A s , and the temperature T . If there is a stochastic temperature variation ∆T of PSD S ∆T , the corresponding noise on the acceleration is proportional to:

S R = 1 2 P A s m i T S ∆T . (2.58)
The typical values for the MICROSCOPE accelerometers are P = 10 -5 Pa , A s = 4 cm 2 , T = 300 K, and m = 100 g. The satellite and the orbit are designed to have a temperature stability below 10 mK • Hz -1/2 , leading to a radiometric noise of about √ S R ∼ 6.7 • 10 -13 ms -2 Hz -1/2 at f EP .

Another random force applied to the test-mass is the radiation pressure, which comes from the thermal radiation of the environment of the test-mass. If there is a small stochastic difference of temperature ∆T between the electrodes for instance, then a pressure difference is induced on each side of the test-mass, leading to a random variation of the acceleration proportional to:

S RP ∝ 16 3
σA s T 3 m i c S ∆T , (2.59)
where σ is the Stefan-Boltzmann constant. The order of magnitude of the radiation pressure noise is √ S RP ∼ 10 -12 ms -2 Hz -1/2 .

Capacitive detector electronic noise

There is a measurable noise coming from the detector electronics (e.g., the thermodynamic noise of the differential transformer, the ADC conversion noise, etc.) which corresponds to a noise in position by multiplication by 1/G det , according to Eq. (2.4). The PSD of this position noise is specified to be equal to

√ S det ∼ 10 -11 mHz -1/2 .
By double derivation the position noise is translated into a noise in acceleration which is proportional to

S pos ∝ (2πf ) 2 S det , (2.60)
where f is the frequency. Therefore this noise will be dominant at high frequency, but its contribution at the WEP frequency is low (a few 10 -16 ms -2 Hz -1/2 ).

Gold wire

The gold wire which maintains the test-mass at a constant potential can be seen as a thin spring with a stiffness k w such as the resulting force is F = k w x. Hence the position noise generates a stochastic force on the test-mass, whose PSD is

S stiff = k w m i S det . (2.61)
Since k w is required to be a few 10 -5 Nm -1 , this stiffness noise shall be quite low, with a level of about √ S stiff ∼ 10 -15 ms -2 Hz -1/2 . However, part of the bending energy of the wire is dissipated into heat or friction. This results in a fluctuating force, hence an acceleration, whose PSD can be written as

S damp = 1 m i 4k B T k w 2πf Q , ( 2.62)
where k B is the Boltzmann constant and Q is the quality factor of the oscillating system defined by the wire. This damping noise, calculated at the frequency of the WEP test of about 1 mHz, is expected to be S damp ∼ 2 • 10 -12 ms -2 Hz -1/2 , and is the main contributor of the uncertainty of the measurement. It is interesting to note that wire damping is also one of the main limitations of the Eöt-Wash experiment (although the wire plays a different part in that case).

Other sources of noise are present which are not detailed above, such as fluctuations in the potential V p , read-out noise, variations of the magnetic field (which is decreased by magnetic shielding), gas damping, etc.

The total noise on the acceleration of test-mass i is given by the sum of all contributions, and if the various sources are assumed to be independent, then their PSDs can be summed. The final noise model is obtained by taking into account the control loop transfer function H(f ), as well as the anti-aliasing filter and the averaging filter. The anti-aliasing filter is a Butterworth filter of order 5 with a cut-off frequency of f BW = 1 Hz. The averaging filter computes the mean on 256 points of the data provided at 1 kHz (1024 Hz to be exact) before the decimation down to 4 Hz.

This results in the PSD model plotted in Fig. 2.14. In this plot we distinguish the filtered and unfiltered PSD, which will be useful in the following. The low frequency f -1 slope of the PSD is due to the wire damping noise of Eq. (2.62), while the f 4 slope comes from the position detector noise of Eq. (2.60).

The PSD model can be approximated by a power law with three main terms, corresponding to the white part of the noise, the low frequency part and the high frequency dependence, all multiplied by a term accounting for the control loop transfer function: Model of the noise power spectral density on the X axis taking into account all the contributions, for the internal (dashed red) and external (dashed blue) masses, and for the differential mode (solid black). The plot on the left is obtained by ignoring the filters (there is no cut-off, only the attenuation due to the control loop transfer function) whereas the plot on the right takes the effect of the Butterworth and averaging filters into account. The fact that the black curve is lower than the red and the blue curves comes from the factor 1/2 in the definition of differential acceleration.

S i u (f ) = α i 0 + α i -1 f -1 + α i 4 f 4 |H(f )| 2 , ( 2
where the subscript u stands for "unilateral". This indication will be explained in Sec. 3.1.3.1 of Chap. 3.

For an inertial session the frequency of the test is f EP = 1.7•10 -4 Hz which corresponds to a noise level in the differential acceleration of about 3.1 • 10 -12 ms -2 according to Fig. 2.14. When the satellite is spinning, the WEP frequency is increased to about f EP = 9.4 • 10 -4 Hz which is a less noisy region of the spectrum, with a PSD of about 1.3 • 10 -12 ms -2 .

The noise level can be reduced down to the desired detection limit of σ EP = 4 • 10 -15 ms -2 by integrating several data points during a long enough period of time. At first approximation, assuming that the WEP violation signal is a pure sine wave, the uncertainty of the test will be given by

σ 2 EP = S d (f EP ) T , ( 2.64)
where S d is the differential acceleration noise PSD, given by S d = 1 4 (S 1 + S 2 ), and T is the integration time, related to the number of points N and the sampling frequency f s as T = N/f s .

With Eq. (2.64) we can calculate the integration time needed to reach the required precision. This gives about 102 orbits for the inertial mode, and 18 orbits for the spin mode. These sessions are actually planned to last respectively 120 and 20 orbits, because of margins.

Derivation of the measured acceleration

Now that we have listed and modelled all sources of systematic and random errors, we can formalize their impact on the measurement equation.

The MICROSCOPE experiment and the measurement equation

The measurement affected by all these defects is noted Γ meas,i , and can be written as

Γ meas,i = b 0i + [A i ]Γ App,i + Q i + [C i ] Ωi + n i , (2.65)
where Γ App,i is the applied acceleration required to keep the test mass at rest in the cage, and is given by Eq. 2.47. Thus Γ meas,i is the actually measured version of Γ App,i .

The matrix [A i ] includes the effect of scale factors, couplings and misalignment. It first transforms the real applied acceleration into the SU i reference frame by applying matrix [Θ i ], and then apply the scale-factor and coupling matrix [M i ] defined as

[M i ] ≡    K 1xi η zi η yi η zi K 1yi η xi η yi η xi K 1zi    , (2.66) so that the matrix [A i ] ≡ [M i ][Θ i
] can be approximated to first order by

[A i ] ≈    K 1xi η zi + θ zi η yi -θ yi η zi -θ zi K 1yi η xi + θ xi η yi + θ yi η xi -θ xi K 1zi    .
(2.67)

It will be useful to express the differential and common modes of the measured acceleration. By applying the formulas (2.50) to the measured accelerations in Eq. (2.65), we obtain after some re-arrangements

Γ meas,c = b 0c + [A c ]Γ App,c + [A d ]Γ App,d + Q c + n c , (2.68a) Γ meas,d = b 0d + [A c ]Γ App,d + [A d ]Γ App,c + Q d + n d . (2.68b)
To simplify the above equations, we neglected the couplings with the angular accelerations for simplicity (in the following analysis if we needed to include them again we would simply have to add a term in the coefficients proportional to the components of Ω which arise from inertial gradient terms in Γ App ).

Reformulation of the equation for parameter estimation

In the previous section we derived the measured differential and common accelerations as a function of the "real" applied acceleration and the parameters of the instrument. But the main goal of the data processing of the MICROSCOPE mission is to estimate the amplitude δ d (or η 12 ) of a possible WEP violation signal. To achieve this objective, the measurement must be described by a model equation where we are able to well describe all of the terms. This is done by decomposing it into a signal that we look for, and a sum of disturbing signals that either can be modelled, or measured, or controlled to remain at an acceptable value. Then these disturbances will be characterized and rejected in order to minimize the error on the WEP violation signal. The purpose of this section is therefore to set up a formalism where the measured accelerations can be decomposed with signals whose variations in time are known, but whose amplitudes are to be determined.

Expression as a function of known signals

We start from Eqs. (2.68), and inspect the measured differential acceleration which is of interest for the WEP test. It depends on the applied differential acceleration Γ App,d for which we have already a model, given by Eq. (2.51). Most of the components of this equation can be easily estimated provided that we have a good knowledge of the orbit restitution and the attitude of the satellite, for example to estimate the value of the gravitational and inertial gradients. We also note that Eq. (2.68b) contains a term proportional to the applied common acceleration Γ App,c whose expression is given by Eq. (2.52). However in this last equation some terms are difficult to model, especially the non gravitational forces (e.g. the atmospheric drag and the solar radiation pressure) whose variations are not known with a sufficient precision. Therefore it is needed to replace this term by something we know. The easiest way is to take advantage of the measurement of the common acceleration, which is given by Eq. (2.68a).

Before doing so, we note that in Eqs. (2.68), Γ App,c is given in the coordinate system SU0, but since we are unable to model it, first we will rather express it in the mean coordinate system of the sensor units, that we denote SU, which is obtained from SU0 by the infinitesimal rotation through the application of the matrix

[Θ c ] = 1 2 ([Θ 1 ] + [Θ 2 ]). Thus we use Γ App,c = [Θ c ]Γ App,c . By defining the matrices [A c ] ≡ [A c ][Θ c ] -1 , (2.69a) [A d ] ≡ [A d ][Θ c ] -1 , (2.69b)
we can rewrite Eq. (2.68) as

Γ meas,c = b 0c + [A c ]Γ App,c + [A d ]Γ App,d + Q c + n c , (2.70a) Γ meas,d = b 0d + [A c ]Γ App,d + [A d ]Γ App,c + Q d + n d . (2.70b)
Now the idea is to invert Eq. (2.70a) to get an expression for the common acceleration in the coordinate system SU:

Γ App,c = [A c ] -1 (Γ meas,c -b 0c -Q c -n c -[A d ]Γ App,d ) .
(2.71)

We then plug this last expression into Eq. (2.70b):

Γ meas,d = b 0d + [ Āc ]Γ App,d + [ Ād ] (Γ meas,c -b 0c -n c ) + Qd + n d , (2.72)
where we defined the matrices

[ Ād ] ≡ [A d ][A c ] -1 , (2.73a) [ Āc ] ≡ [A c ] -[A d ][A c ] -1 [A d ], (2.73b)
and we gathered quadratic terms by defining

Qd ≡ Q d -[ Ād ]Q c . (2.74)
The right-hand side of Eq. (2.72) finally contains only signals that can be either modelled or measured. A quick overview of the terms involved in this equation can be done, from left to right:

• the bias b0d is unknown, but can be estimated (after subtracting the mean of the other terms). It is likely to have very slow variations due to thermal effects.

• the applied differential acceleration Γ App,d can be finely modelled from the orbit and attitude restitution (see Eq. 2.51);

• the common mode term (Γ meas,c -b 0c -n c ) cannot be directly measured but we have access to the common acceleration, which is a noisy and biased version of it. To maintain this term to the lowest possible value, the drag compensation system servo-controls the common acceleration to be equal to a control value C (e.g., C = b 0c + n c ). This is possible since according to Eq. (2.52) the common acceleration contains the non gravitational forces which can be split in two terms: the environmental (or external) forces and the satellite thrust:

F = F ext + F th .
Hence the thrusters can be tuned to control the amplitude of Γ meas,c ;

• the quadratic terms Qd are hard to model but have a small magnitude. They can be corrected for by the calibration process;

Definition of the model parameters

In this section we derive a measurement model which is directly usable in the data analysis. To begin with, we substitute Eq. (2.51) into Eq. (2.72):

Γ meas,d = b 0d + [ Āc ]Γ App,d -([T] -[In]) • ∆ d -δ d g E + 2[Ω] ∆d + ∆d + b d (2.75) +[ Ād ] (Γ meas,c -b 0c -n c ) + Qd + n d . (2.76)
It is then useful to separate the gradient term into a symmetric and an antisymmetric part:

([T] -[In]) = [S] + [U], (2.77)
where we have

[S] ≡    T xx + Ω 2 y + Ω 2 z T xy -Ω x Ω y T xz -Ω x Ω z T xy -Ω x Ω y T yy + Ω 2 z + Ω 2 x T yz -Ω y Ω z T xz -Ω x Ω z T yz -Ω y Ω z T zz + Ω 2 x + Ω 2 y    , (2.78a) and [U] ≡    0 Ωz -Ωy -Ωz 0 Ωx Ωy -Ωx 0    = -[ Ω] (2.78b)
We consider the x-component of the measured differential acceleration only. By noting the coefficients of the matrices [ Āc ](i, j) = a cij and [ Ād ](i, j) = a dij , we obtain, after developing and rearranging the terms of Eq. (2.72):

Γ meas,dx = b 0dx -a c11 δ d g E,x -(a c11 ∆ dx S xx + a c12 ∆ dy S yy + a c13 ∆ dz S zz ) -(a c11 ∆ dy + a c12 ∆ dx ) S xy -(a c11 ∆ dz + a c13 ∆ dx ) S xz -(a c12 ∆ dz + a c13 ∆ dy ) S yz + (a c13 ∆ dy -a c12 ∆ dz ) Ωx + (a c11 ∆ dz -a c13 ∆ dx ) Ωy + (a c12 ∆ dx -a c11 ∆ dy ) Ωz + 2 a c13 ∆dy -a c12 ∆dz Ω x + 2 a c11 ∆dz -a c13 ∆dx Ω y + 2 a c12 ∆dx -a c11 ∆dy Ω z + a c11 ∆dx + a c12 ∆dy + a c13 ∆dz + a d11 (Γ meas,cx -b 0cx -n cx ) + a d12 (Γ meas,cy -b 0cy -n cy ) + a d13 (Γ meas,cz -b 0cz -n cz ) + 1 2 K 21xx K 2 11x (Γ meas,1x -b 01x -n 1x ) 2 - 1 2
K 22xx K 2 12x (Γ meas,2x -b 02x -n 2x ) 2 + n dx (2.79)
Therefore the measurement equation can be expressed as a linear combination of known time variations p i (t) with unknown amplitudes β i that we call "calibration parameters":

Γ meas,dx = -a c11 δ d g E,x + Np i=1 β i p i (t) + n d (t).
(2.80)

The variations p i (t) are also referred to as "partial derivatives" since p i (t) = ∂Γ meas,dx ∂β i . If the disturbances β i p i (t) were not removed nor taken into account in the WEP test, they could induce a bias on the estimation of δ d , depending on the projection of p i (t) onto g E,x .

The orders of magnitude of the individual parameters are gathered in Table . 2.1. Note that in the absence of an excitation control, the test-mass is forced to stay at the centre of the cage, and the temporal variations of the offcentrings ∆di are assumed to be zero. In addition, to give insight into the relative orders of magnitude of each term of Eq. (2.79), Table 2.2 shows the upper bound values of the corresponding disturbing signals and their impact onto the WEP frequency (this is done partly according to [START_REF] Hardy | Procédures expérimentales et traitement des données associées pour la mission spatiale MICROSCOPE[END_REF] and [START_REF] Touboul | Microscope instrument in-flight calibration plan[END_REF]).

The last column of Table 2.2 shows that the presence of the disturbing signal introduces a systematic error on the component at f EP , which may bias the estimation of δ d . The cumulative (worst case) error is equal to 10 -13 ms -2 , which is equivalent to an error of about 2.5 • 10 -14 on δ d . This is too high with respect to the 10 -15 objective of the mission. Therefore all the parameters p i 's must be estimated in order to correct the measurement for the disturbing signals. This is done by removing their contribution, which reduces the systematic error down to an acceptable level.

Parameter Range of possible values

δ d < 5 • 10 -13 a c11 1 ± 0.01 a c12 < 2.6 • 10 -3 rad a c13 < 2.6 • 10 -3 rad ∆ d < 20 µm a d11 < 1 • 10 -2 rad a d12 < 1.6 • 10 -3 rad a d13 < 1.6 • 10 -3 rad K 2ixx < 14000 s 2 m -1 b 0dx 2.5 • 10 -8 ms -2
Impact @f EP [ms -2 ] S xx ∼ 2 • 10 -6 s -2 2f EP 2a c11 ∆ dx < 20.2 4 • 10 -14 S yy ∼ 4 • 10 -9 s -2 f EP , 2f EP 2a c12 ∆ dy < 0.05 4 • 10 -16 S zz ∼ 2 • 10 -6 s -2 2f EP 2a c13 ∆ dz < 0.05 3 • 10 -16 S xy ∼ 3 • 10 -8 s -2 2f EP 2a c11 ∆ dy < 20.2 3 • 10 -16 S xz ∼ 2 • 10 -6 s -2 2f EP 2a c11 ∆ dz < 20.2 4 • 10 -14 S yz ∼ 3 • 10 -8 s -2 2f EP 2a c1i ∆ d < 0.05 1 • 10 -18 Ωx ∼ 7 • 10 -11 s -2 nf EP 2a c1i ∆ d < 0.05 2 • 10 -18 Ωy ∼ 2 • 10 -10 s -2 nf EP 2a c11 ∆ dz < 20.2 1 • 10 -15 Ωz ∼ 3 • 10 -10 s -2 nf EP 2a c11 ∆ dy < 20.2 1 • 10 -15 Γ meas,cx ∼ 1 • 10 -12 ms -2 f EP a d11 < 10 -2 1 • 10 -14 Γ meas,cy ∼ 6 • 10 -11 ms -2 2f EP a d12 < 1.6 • 10 -3 rad 2 • 10 -15 Γ meas,cz ∼ 6 • 10 -11 ms -2 2f EP a d13 < 1.6 • 10 -3 rad 2 • 10 -15 Γ 2 meas,1x ∼ 5 • 10 -17 m 2 s -4 2f EP K 21xx < 14000 s 2 m -1 1 • 10 -16 Γ 2 meas,2x ∼ 5 • 10 -17 m 2 s -4 2f EP K 22xx < 14000 s 2 m -1 1 • 10 -16 Total 1 • 10 -13

Summary:

We saw that the MICROSCOPE experiment is composed by two differential electrostatic accelerometers on board a microsatellite orbiting the earth on a quasi-circular orbit. For each accelerometer, the measurement is made from the accelerations applied to two test-masses to maintain them relatively motionless in the centre of the accelerometer cages. Based on the description of this system we derived the physical equations modelling the measurement. An appropriate formulation of this model allowed us to evidence that it involves perturbing signals whose amplitudes depend on the characteristics and the defaults of the instrument, and introduce a bias in the measurement of a possible The aim of this chapter is to develop a data analysis method to assess and possibly correct the bias caused by unexpected signals perturbing the test of the equivalence principle. First, we review the approach to deal with all known signals described by the measurement model derived in the previous chapter. In particular, we describe the linear regression framework and the calibration of the instrument, after which the measurement is corrected for the estimated disturbances. Second, we investigate the bias due to unknown perturbations, i.e., all the remaining perturbations that are not described by the calibration model, in particular the harmonic signals. We review three ways to 58

Treatment of interfering signals and minimization of the measurement bias

reduce the impact of harmonic signals. The first approach assumes that the frequencies of the disturbing signals can take particular values, which can be rejected by tuning the time series length and the WEP frequency. The second approach concerns the rejection of harmonic signals with arbitrary frequencies by time windowing in the least-squares framework. The third approach is to determine them by regression, and is the crux of this chapter. Following this last strategy, we propose a method to statistically detect the harmonic signals, and to characterize their frequencies through maximization of the periodogram. We analytically assess the ability of the method to reduce the bias of the detected signals. This assessment is then verified by numerical tests.

Estimation of parameters by linear regression

In Chap. 2 we established the equation modelling the measurement of the differential acceleration, and we saw that it can be expressed as the sum of a WEP violation term proportional to the Eötvös parameter plus a linear combination of known disturbing signals with unknown amplitudes. The goal is to be able to estimate the amplitude of the WEP term with a precision of δ d g E,x ∼ 4 • 10 -15 ms -2 . Therefore the total error induced by all the disturbances must be less than this value. In this section we present the general method that is used to estimate both the WEP and the instrument parameters.

Estimation principle

The estimation of the WEP parameter δ and the instrument calibration parameters relies on the general form of Eq. (2.80):

Γ meas,dx (t) = 1 2 δg x (t) + Np i=1 β i p i (t) + n dx (t), (3.1)
where we use the parameter δ instead of δ d (since the 10 -15 objective applies to δ). The adopted approach for the whole estimation process is based on a sequence of linear regressions as follows:

1. obtain estimates βi of the instrumental parameters β i by fitting the modelled or measured signals p i to the differential acceleration Γ meas,dx or other variables measured during dedicated calibration sessions;

2. measure the differential acceleration Γ meas,dx during a WEP test session (inertial or spin) and correct it for the disturbance terms by removing Np i=1 βi p i (t) from the signal: we obtain Γ corr,dx ; 3. perform a linear regression of the WEP violation signal on the corrected acceleration Γ corr,dx , to obtain an estimate of δ.

The measured acceleration is actually available at discrete times t n = nτ s , where τ s = 1/f s is the sampling time, which is equal to 0.25 second. Therefore we obtain a time series of sampled acceleration, that we can handle as a vector y of size N whose entries are y n = Γ meas,dx (t n ), ∀n ∈ 0, N -1 . Then it is convenient to rewrite Eq. (3.1) into matrix form:

y = Aβ + n, (3.2)

Treatment of interfering signals and minimization of the measurement bias 59

where we defined the regression vector

β = δ β 1 • • • β Np T
, the design matrix A whose elements contain the model signals (also called explanatory variables) a n0 = 1/2 • g x (t n ) and a nm = p m (t n ) , m ∈ 1, N p . We also denoted the noise vector by n. The simplest way to do this is to use the ordinary least-squares (OLS) estimator, which writes

β = (A * A) -1 A * y, (3.3)
where * denotes the Hermitian conjugate (i.e., the transpose of the conjugate of the matrix). The OLS estimate is the one that minimizes the L 2 -norm y -Aβ 2 .

Estimation bias

The OLS estimator defined in Eq. (3.3) is unbiased if the following conditions are met, which are often referred to as the Gauss-Markov conditions:

1. the matrix model A describes exactly the deterministic part of the signal;

2. A is full rank: its columns are linearly independent;

3. the expected value of the error term is zero:

E [n] = 0; 4. A is deterministic: its columns A i are uncorrelated with the noise term Cov (A i , n) = 0.
In practice, these conditions are not always fulfilled. For example, all the parameters are not estimated at the same time. Indeed, it is usual that some partial derivatives are too similar or too faint to be included in the design matrix. Using all the signals in the model would violate condition 2, which may introduce a higher error on the estimation than ignoring some of them (at the price of a violation of condition 1).

Let p i 1 , . . . , p iq be the partial derivatives that are included in the model (forming the matrix A), and p j 1 , . . . , p jr the partial derivatives that are not taken into account (forming the matrix A ). Thus the deterministic part of the measurement equation in 3.2 includes a modelled part and an unmodelled part:

y = Aβ + A β + n, (3.4)
where

β = β i 1 • • • β iq T
is the vector of parameters to be estimated and β =

β j 1 • • • β jr
is the vector of ignored parameters. Then, when estimating β, there is an error which reads

E β -β = (A * A) -1 A * Aβ + A β -β = (A * A) -1 A * A β . (3.5)
This error is called "bias" or "tone error", and determines the accuracy of the estimation. It is said to be a systematic error since it does not depend on the statistics of the measurement.

The error made in the estimation of the instrument parameters will affect the error made on the estimation of the WEP parameter. After a separate estimation of the instrument parameters during the in-flight calibration sessions, the differential acceleration is corrected to obtain

Γ meas,corr,dx (t) = 1 2 δg x (t) + Np i=1 β i -βi p i (t) + n dx (t). (3.6)
The WEP parameter is then estimated by fitting the gravitational acceleration g x to the calibrated differential acceleration Γ meas,corr,dx , by least-squares regression:

δ = 2 g T x g x -1 g T x Γ meas,corr,dx . (3.7)
Let us note i = β i -βi the error made on the instrument parameter i resulting from the estimation previously performed during the corresponding calibration session. This error depends on the bias and the variance of this estimation. The upper bound of each i can be assessed by a careful analysis of the various error terms involved in each calibration session. This analysis is done in Ref. [START_REF] Touboul | Microscope instrument in-flight calibration plan[END_REF] where they take

¯ i ≡ max ( i ) = max |b β i | + 3σ i , (3.8)
where b β i is the bias in the estimation of the parameter β i , and σ i is its standard deviation.

Then the maximum residual bias can be estimated by summing the bias caused by all of the residual disturbance signals:

bδ = 2 Np i=1 g T x g x -1 g T x p i ¯ i , ( 3.9)
where we used the general expression (3.5) of the bias for an incomplete model.

Estimation uncertainty

Another source of error comes from the noise statistics. We saw in the previous chapter (Sec. 2.3.4.2) that the noise is characterized by its power spectral density (PSD). The covariance of the estimator β depends on this function. In this section, we give the mathematical definition of the PSD, allowing us to derive an expression for the estimator covariance.

Power spectral density

Each source of noise can be modelled by a random variable n, whose realizations will be measured at some times t 0 , • • • , t N -1 . The distribution of n will generally be assumed stationary, meaning that the joint probability distribution of n(t 0 ), . . . , n(t N -1 ) does not depend on time. For example, both the mean and the variance of the vector

n = n(t 0 ) • • • n(t N -1 )
T do not change with time. Among the stationary sources of noise, some are uncorrelated in time. This means that if we choose two times t 1 and t 2 (t 1 = t 2 ), the covariance between n(t 1 ) and n(t 2 ) is equal to zero. Then it is called white noise because it does not favour any frequency.

However, some sources are correlated in time, which means that there exists a duration τ under which the covariance Cov [n(t), n(t + τ )] is different from zero. For a stationary

Treatment of interfering signals and minimization of the measurement bias 61

noise, the covariance only depends on the duration τ separating two points, but not on time t. Therefore we can write:

R(τ ) ≡ Cov [n(t), n(t + τ )] ∀t = E [(n(t) -µ) (n(t + τ ) -µ)] ∀t. (3.10)
The function defined in Eq. (3.10) is the autocovariance function, and characterizes the noise in the time domain.

To characterize the noise in the frequency domain, we also define the PSD function (following Priestley [START_REF] Priestley | Spectral analysis and time series[END_REF]). The PSD of a signal n(t) is the limit when T tends toward infinity of the expectation of the squared modulus of the Fourier transform of the signal truncated on a duration T :

S(f ) ≡ lim T →∞ E   1 T T 2 -T 2 n(t)e -2jπtf dt 2   . (3.11)
The Wiener-Khintchine theorem [START_REF] Wiener | Generalized harmonic analysis[END_REF][START_REF] Khintchine | Korrelationstheorie der stationären stochastischen prozesse[END_REF] shows that the PSD is equal to the Fourier transform of the autocovariance function:

S(f ) = ∞ -∞ R(τ )e -2jπτ f dτ. (3.12)
In the discrete case, when the signal is sampled at a frequency f s , this equation translates into [START_REF] Priestley | Spectral analysis and time series[END_REF]

S(f ) = 1 f s ∞ n=-∞ R (n/f s ) e -j2πf n/fs . (3.13)
Conversely, we obtain the autocovariance from the PSD with the following formula:

R(τ ) = fs 2 -fs 2 S(f )e j2πf τ df. (3.14)
Note that we defined the PSD S(f ) as a function on the interval [-∞, +∞], which is commonly referred to as the bilateral power spectral density. However, the PSD used in the physical noise model presented in Fig. (2.14) in the previous chapter is the unilateral power spectral density, that we call S u (f ), and is defined for positive frequencies only. It is equal to twice the PSD S u (f ) = 2S(f ) to keep the same power.

Covariance of the OLS estimator

We can derive a relationship between the PSD and the covariance of the ordinary leastsquares estimator in the case of stationary time series. By taking the covariance of Eq. (3.3), we obtain

Cov β = (A * A) -1 A * ΣA (A * A) -1 , (3.15)
where we defined the covariance of the noise Σ ≡ Cov [n]. Following the definition (3.10), the entries of the matrix Σ are equal to the autocovariance function calculated at all lags such as

Σ nm = R ((n -m)/f s ) . (3.16)
This defines a Toeplitz matrix, which is a matrix whose diagonal entries are constant.

In addition Σ is also symmetric. We now derive an expression of Σ as a function of the PSD. The integral in Eq. (3.14) can be estimated at the sample times nτ s by its equivalent Riemann sum:

R(nτ s ) = f s N N -1 k=0 S (f k ) e 2jπ nk N , ( 3.17)
for N sufficiently large, where f k are the classical Fourier frequencies

f k =    k N f s if 0 ≤ k ≤ N -1 2 k-N N f s if N -1 2 + 1 ≤ k ≤ N -1. (3.18)
To prevent any unphysical periodicity in R, N must be chosen such that N ≥ 2N . Eq. 3.17 allows us to write the covariance matrix in a convenient way. To do so, we first define the N × N normalized discrete Fourier transform (DFT) matrix F N whose elements are

F N nl = 1 √ N exp -2jπ nl N . ( 3.19)
F N is a unitary matrix since we have F * N F N = I, where I is the identity matrix. This property is important since it means that if we transform the data in the Fourier domain (by applying matrix F N to both the vector y and the matrix A in the OLS estimator), the estimation result will remain unchanged.

We also define the N × N matrix Ω = I N 0 N -N selecting the first N entries of any vector by which Ω is multiplied. This is useful since we need N values of the PSD to compute the N values of the autocovariance. Then the Riemann approximation allows us to write the following set of equations:

Σ = ΩCΩ * , (3.20a) C = F * N ΛF N , (3.20b)
where

Λ is a N × N diagonal matrix defined by Λ = f s × Diag S(f 0 ) ... S(f N -1 ) .
One can verify that this is equivalent to approximating the covariance elements by

Σ nm ≈ R (n -m)τ s , (3.21)
where R is given by Eq. (3.17). This formulation is equivalent to a circulant embedding of the Toeplitz matrix Σ because it is embedded in a matrix C of higher dimension where each row is deduced from the previous one by circular permutation of the coefficients (shift towards the right). Due to this property, C is said to be circulant, and is exactly diagonalizable in Fourier space, according to Eq. (3.20b).

As a result, the OLS covariance in Eq. (3.15) can be written, approximately, as a function of the PSD:

C β ≡ Cov β ≈ (A * A) -1 A * ΩF * N ΛF N Ω * A (A * A) -1 . (3.22)
Actually, when N tends toward infinity, Σ tends toward a circulant matrix. Hence, for N sufficiently large, the covariance matrix is approximately diagonalizable in Fourier space (see e.g. [START_REF] Whittle | Curve and periodogram smoothing[END_REF]).

The knowledge of the covariance allows us to determine whether the estimation of δ as in Eq. (3.7) is significant, i.e., whether it is large enough compared to the uncertainty of the estimation. This can take the form of a statistical test. Let us consider the two hypothesis

H 0 : δ = 0 (null hypothesis) H 1 : δ = 0 (3.23)
We want to test whether δ is non-zero. To do so we consider the following statistics:

Z ≡ δ √ C δ , ( 3.24)
where √ C δ is the covariance of the least-squares estimator given by Eq. (3.15) with A = 1/2 • g x . To assess the significance of the test we must postulate a probability distribution for the noise vector n dx . In the following we assume that it is Gaussian, which is a reasonable assumption considering the MICROSOCOPE noise sources (although this must be checked from the data). If the estimator were unbiased then the random variable Z would follow a Gaussian distribution of mean zero and variance unity. Then the null hypothesis (the equivalence principle is valid within the precision of the test) is true with a 1 -α confidence if

|Z| ≤ z 1-α/2 , ( 3.25)
where z u ≡ Φ -1 (u) is the inverse normal cumulative distribution function. However, we know that even if δ = 0 the estimator has a residual bias, denoted b δ . Since it is possible to get an estimate bδ of its upper bound, the test must be modified by considering that

Z -bδ C -1/2 δ is larger than z 1-α/2 .
Now that we have set up the estimation method and formalized the associated tone error and stochastic error, we examine the case of unknown periodic signal in the data.

Rejection of particular harmonic perturbations

We saw in Sec. 3.1.1 that the data processing relies on in-flight calibration sessions of the instrument, allowing us to reduce the bias of all known physical disturbances by correction of the measurement. This correction mainly concerns signals whose frequency signature is close to f EP . In this section we assume that there are some additive disturbances s p,i (t) in the signal, for which we do not have any particular model. However, practically, we have no choice but to assume some pattern for these perturbations. Given the signature of the modelled signals that we know, it is reasonable to assume that the signals s p,i (t) will be harmonic functions. In case of non-harmonic functions, this would be equivalent to work with the Fourier decomposition of the signal. In Sec. 3.2.1 we assess the impact of harmonic signals onto the WEP frequency in discrete time series, while in Sec. 3.2.2 we review a method which minimizes the projection of specific frequencies.

Projection of a harmonic signal onto the EP frequency

To simplify the analysis, we assume that the WEP violation signal can be well described by a sine function. This is a very reasonable approximation, since the orbit is quasicircular, and the monopolar term of the geopotential is dominant in the computation of the gravitational acceleration. Hence we can neglect the other gravity terms to get

g x = µ r 2 cos (w + v + ϕ) , (3.26)
where w is the argument of perigee, v is the true anomaly (equal to 2πf orb t in the circular case), and ϕ = 2πf spin t + ϕ 0 accounts for the rotation about the axis normal to the orbital plane (due to the attitude control). By setting

f EP = f orb + f spin , φ EP = w + ϕ 0 + π 2
and g 0 = µ r 2 we can rewrite Eq. (3.26) as

g x (t) = g 0 sin (2πf EP t + φ EP ) . (3.27)
We want to estimate the EP parameter δ by fitting g x /2 to the differential acceleration, which is assumed to have been perfectly corrected from all modelled disturbances by prior calibration. We further suppose that the signal contains a harmonic disturbance of amplitude a p of the following form:

s p (t) = a p p(t) = a p sin (2πf p t + φ p ) . ( 3.28)
Thus the measured acceleration vector can be written as

y = 1 2 δg x + s p + n. (3.29)
We saw that estimating δ by the OLS method will result in a bias, which can be computed using Eq. (3.5) by taking A = 1/2 • g x and A = p. Then the hidden parameter is β = a p . In Ref. [START_REF] Hardy | Procédures expérimentales et traitement des données associées pour la mission spatiale MICROSCOPE[END_REF] it is shown that the result of this computation can be written as

E β -β = a p a EP • τ (f EP , f p , φ EP , φ p ) , ( 3.30)
where τ is the projection rate of the disturbing signal onto the WEP violation signal, and where we labelled a EP ≡ g 0 /2 = 4ms -2 the amplitude of the WEP violation signal. The projection rate is a function depending on the frequencies and the phases of the estimated and disturbance signals. An expression for τ is provided in [START_REF] Hardy | Procédures expérimentales et traitement des données associées pour la mission spatiale MICROSCOPE[END_REF] in the case of continuous signals. A similar calculation (see Appendix A.1 ) can be done in the case of discrete signals (sampled at a given frequency f s ) which gives

τ ≈ 1 N cos φ -+ πτ s f -(N -1) sin (πτ s f -N ) sin (πτ s f -) -cos φ + + πτ s f + (N -1) sin πτ s f + N sin (πτ s f + ) , (3.31)
where N is the size of the time series and where we defined The projection rate varies with the phase of the involved signals, as illustrated in Fig. 3.1. It can be maximized with respect to both φ EP and φ p (corresponding to a dark peak in Fig. 3.1). We obtain

φ ± = φ EP ± φ p ; (3.32a) f ± = f EP ± f p . ( 3
τ max = 1 N sin (πτ s f -N ) sin (πτ s f -) + sin πτ s f + N sin (πτ s f + ) . (3.33)
The maximal projection rate is then a function depending only of the frequencies f EP and f p , showing a series of extrema. The envelope function of τ max can be deduced by an integral approximation of Eq. (3.33) which gives (see Appendix A.2)

τmax ≈ f s N π 1 |f EP -f p | + 1 |f EP + f p | . (3.34)
We plotted in panel (a) of Fig. 3.2 the maximum projection rate in the case of an inertial session with f EP = 9.35 • 10 -4 Hz and a duration of 20 orbits.

This plot exhibits a peak at f EP , which means that a disturbance at a frequency close to frequency of the fitted signal will induce a large bias on the estimation. The figure also shows that τ max has periodic minima (where it is actually zero). The curve also has asymptotes with a slope 0 towards 0 and a slope -1 towards +∞. Let us give an example of how the figure must be read. Assume that there is a disturbance of frequency 10 -4 Hz and amplitude a p . In the figure, this corresponds to the low frequency plateau at a level of about τ max = 6 • 10 -3 . This means that the disturbing signal will induce a bias which is at most a p × 6 • 10 -3 /a EP ≈ a p × 1.5 • 10 -3 on the estimation of δ. For a p = 10 -13 ms -2 this gives an error of 1.5 • 10 -16 .

From this first figure we can derive the graph (b) providing the information the other way around. Here we consider the "dangerous" harmonic signals, say, the pairs (f p ,a p ) that can lead to a maximum bias of 10 -16 on the WEP parameter δ (red curve). Therefore, for a given frequency f p , the amplitudes below the red curve induce a negligible bias.

Minimization of the projection rate for particular frequencies

A priori, the zeros of the projection rate depend on the frequencies and the phases of both EP and disturbance signals. However, from Eq. (3.31) we can see that a sufficient condition for τ max to be zero is that both the WEP frequency and the disturbance frequency are exactly a Fourier frequency, i.e,

∃k 0 s.t. f EP = k 0 T ; ∃k 0 = k 0 s.t. f p = k 0 T , ( 3.35)
where T = N/f s is the duration of the signal. In other words, if both the WEP and disturbance waves have an integer number of periods within the measurement duration, the bias is nullified. While the experimental set-up allows us to finely tune the WEP frequency (by adjusting the spin frequency) and/or the duration of the integration, there is no mean to change f p , which is unknown. However, most of the disturbance signals that we are able to model have harmonic peaks which are multiples of the orbital and the spin frequency. Therefore, several unmodelled disturbances are likely to have this kind of frequency signature. The idea proposed in [START_REF] Hardy | Procédures expérimentales et traitement des données associées pour la mission spatiale MICROSCOPE[END_REF] is to tune the integration time and the spin frequency so that all the frequencies of the form

f p = n 1 f orb + n 2 f spin (3.36)
have a null projection onto the WEP frequency. Hence, they are such that

∀n 1 , n 2 ∈ Z, ∃n 0 s.t. n 1 f orb + n 2 f spin = n 0 T . (3.37)
For the condition (3.37) to be realized, it is sufficient that

∃k 1 , k 2 ∈ Z, s.t. T = k 1 f orb f spin = k 2 T
These conditions were implemented in the mission plan, where inertial sessions are configured such that k 1 = 120 (and f spin = 0), whereas the configuration of spin sessions is chosen to have k 1 = 20 and f spin = 9 2 f orb which yields k 2 = 90 or f spin = 7 2 f orb which yields k 2 = 70. By this device, the bias corresponding to all harmonic signals with frequencies of the form (3.37) is nullified.

Rejection of harmonic perturbations by time windowing

We now assume that the frequency f p of the harmonic disturbing signal can take any value, and is not necessarily an integer multiple of the orbital frequency nor the spin frequency. In this case, the projection rate in Eq. (3.31) is generally non-zero. The projection effect is commonly known as spectral leakage in the domain of spectral analysis. Leakage can be understood by considering the discrete Fourier transform of signals with finite duration. In this case the spectral representation of a signal of frequency f p is not exclusively concentrated at a point of the spectrum, but slightly spreads out, leading to non-zero values around f p . We say that its power "leaks" on the surrounding frequencies. The projection rate, or leakage, is inversely proportional to the integration time T , and vanishes when T tends toward infinity (the Fourier representation is then a perfect Dirac function).

A way to attenuate leakage is to use windowing, i.e., to apply a window function to the measurement in the time domain. The window function generally smoothly decays to zero at the edges of the observed time interval. While reducing the power of the main peak, this attenuates the side lobes with respect to the main peak. Some examples are given in Ref. [START_REF] Nuttall | Some windows with very good sidelobe behavior[END_REF]. In this section we study the efficiency of the windowing method.

Bias rejection with windowing

In the least-squares problem that we study we could similarly apply a window to both the measurement vector y and the design matrix A. If w(t) is some window function and w the corresponding vector of entries w n = w(t n ), windowing is introduced by forming the squared diagonal window matrix W = diag (w). Then we apply the window matrix on both sides of Eq. (3.29):

Wy = WAβ + Ws p + Wn, (3.38)
which leads to the windowed version of the least-squares estimator:

β = (A * W * WA) -1 A * W * Wy. (3.39)
Note that in the absence of any perturbation (s p = 0) this estimator is still unbiased (applying a window matrix on both sides of the measurement equation does not change the expectation of the estimator).

3. Treatment of interfering signals and minimization of the measurement bias

For s p = 0, the bias of the new estimator will be given by a windowed version of Eq. (3.5):

E β -β = (A * W * WA) -1 A * W * Ws p .
(3.40)

As an example, we choose the window function to be the Hann function, defined as

w n = 1 2 1 -cos 2πn N -1 , ( 3.41)
and we show in Fig. 3.3 the projection rate as a function of frequency, computed for the ordinary least-squares estimator and for the Hann-windowed estimator. According to the figure, the projection rate envelope is reduced by more than 5 orders of magnitude outside the neighbourhood of f EP . Therefore the windowed least squares (WLS) seems to be a good solution, all the more that windowing does not change the location of the zeros of the projection rate (hence the tuning proposed in Sec. 3.2.2 for the rejection of the particular disturbing frequencies is still valid). We will see that this statement must be qualified when considering the presence of noise.

Impact of windowing on the estimation uncertainty

We saw that windowing is a good way to attenuate the bias of arbitrary harmonics. In this section we consider the impact of this technique on the uncertainty of the fit. Indeed, windowing generally changes the variance of the estimator, as the variance of the WLS depends on the applied weighting. Let us define the weighted design matrix as

A w ≡ WA, (3.42)
and the double weighted design matrix as

A w 2 ≡ W * WA, (3.43)
Then the covariance of the estimator (3.39) is

Cov β = (A * w A w ) -1 A * w 2 ΣA w 2 (A * w A w ) -1 , (3.44)
With this equation we can assess the impact of the window on the variance of the estimate of δ. Let us assume that the model is simply the WEP violation signal A = g x and that the measurement is affected by a coloured noise of PSD given by the model of Fig. (2.14).

By using Eqs. (3.20), the covariance can be written as

Cov β = (A * w A w ) -1 Ã * w 2 Λ Ãw 2 (A * w A w ) -1 . (3.45)
where for any quantity X we note X its discrete Fourier transform (DFT) such that X ≡ F N X. Note that if N > N the DFT is obtained by zero-padding (the calculation amounts to adding N -N zeros to the columns of X before taking the Fourier transform).

If we fit the WEP signal only, Eq. (3.45) simplifies to

Var( δ) ≈ N -1 k=0 | Ãw 2 ,k | 2 f s S(f k ) N -1 k=0 | Ãw,k | 2 2 .
(3.46)

In the case without any windowing (w n = 1 ∀n), we show (see Appendix A.4) that the variance is approximately

Var( δ) ≈ 2S (f EP ) f s N a 2 EP , (3.47)
which is the expected result saying that the error made on the estimation of the amplitude of a sine wave of frequency f is given by the unilateral PSD 2S calculated at this frequency, divided by the integration time. Now we consider the variance in the case of Hann-windowing. We perform a numerical computation of Eq. (3.46) in the case of a spin session of 20 orbits and an inertial session of 120 orbits. The results are gathered in Table 3 We see that applying a window function in the least-squares estimation slightly increases the uncertainty. The explanation can be given by looking at the variance per frequency, that is, the individual terms appearing in the sum of Eq. (3.46):

σ 2 k = | Ãw 2 ,k | 2 f s S(f k ) N -1 k=0 | Ãw,k | 2 2 .
(3.48)

We calculated the sequence σ 2 k for the rectangular window (no apodization) and the Hann window in Fig. 3.4. As expected, the side lobes envelope on both sides of the peak is higher for the rectangular window than for the Hann window, with respect to the maximum. The power is globally more concentrated around the main peak with an apodization window (attenuating power leakage). k given by Eq. (3.48) is plotted with respect to frequency for the OLS estimator (black) and the windowed least-squares estimator of Eq. (3.39) with the Hann window (blue). The calculation is performed for a 20 orbits inertial session, with φ EP = π/4. The area under the curve gives the total standard deviation on the estimation of the WEP parameter.

However, if we take a look at the region in the neighborhood of the peak, we see that the main lobe of the Hann variance is slightly broader than for the rectangular window in the vicinity of the maximum. The OLS regression error is affected by the variance of the frequency points that are close to the peak, since for both windows the side-lobes are anyway negligible compared to the maximum. As a result, while for the rectangular window there is simply one significant point in the DFT of the sine, the error of the Hann-windowed sine regression will be affected by the variance of a few more points around the peak, which are not negligible, yielding an increase of the total variance.

More generally, the fact that the variance is increased by applying an apodization window in the least-squares estimation is related to concept of optimal weighting. Indeed, the best weighting in terms of variance is not given by any apodization window we may choose. There is a single weighting matrix W which minimizes the estimator mean squared error (MSE):

MSE(W) ≡ E ( β -β) * ( β -β) .
(3.49)

The minimizer of the MSE is exactly given by the inverse of the noise covariance:

W * W = Σ -1 . (3.50)
The least-squares estimator with weighting given by Eq. (3.50) is called generalized least squares (GLS) [START_REF] Aitken | On least squares and linear combination of observations[END_REF]. It turns out that when fitting a single sine wave in a coloured noise measurement, the variance of the ordinary least squares with a complete data set is almost the same as the GLS. This is because the signal power is mostly concentrated at a single point in the Fourier domain. Since the GLS variance is the minimum variance that any unbiased estimator can reach (also called Cramér-Rao lower bound, [START_REF] Cramér | Mathematical methods of statistics. Probability and mathematical statistics[END_REF][START_REF] Calyampudi | Information and the accuracy attainable in the estimation of statistical parameters[END_REF]), any other windowing leads to a higher variance. As a conclusion, apodization windowing in the framework of least-squares estimation is efficient to reduce the bias that would be caused by unknown harmonic signals with arbitrary frequencies. However this is done at the price of an increase of the uncertainty, which grows by a factor of about 1.4 with the Hann window. One could think about compensating this increase by an extended integration time. This would represent an increase of a factor 1.4 2 ≈ 2. Therefore combining two WEP-test sessions would be sufficient. The windowing strategy is thus an appealing one, but its loss of precision may not be affordable. Another lead is to model and characterize the harmonic signal, which is what we explore in the next section.

Rejection of harmonic perturbations by inclusion in the model

Another mean to cancel the bias without increasing the variance is to include the disturbing signal in the design matrix A, i.e., to fit a sine wave of frequency f p together with the WEP violation signal. In this section we assess the performance of this approach.

Inclusion of a harmonic signal in the model

While the amplitude of the unknown signal can be estimated by the OLS estimator, the prerequisite is to have a good knowledge of the frequency and phase of the disturbing signal. In fact, only a good knowledge of the frequency is needed, since the disturbing sinusoidal function can be developed in the following form:

a p sin (2πf p t + φ p ) = a p cos(φ) sin (2πf p t) + a p sin(φ) cos (2πf p t) , (3.51)
so that we can fit a sine and a cosine function in addition to the WEP signal, and the total regression vector will be β = δa EP a p cos(φ) a p sin(φ) T (we included the known EP amplitude in the regression vector to simplify the expressions in the following).

We assume again that the measured signal only contains a signature of a WEP violation, and a perturbation, as formalized in Eq. (3.29). We then assume that we perform a linear regression of the WEP signal plus a sine and a cosine as written in Eq. (3.51), but with a frequency fp slightly different from the true one f p . The design matrix is thus

A(f ) = S(f EP , φ EP ) S( fp , 0) C( fp , 0) , (3.52)
where we defined the vectors S(f, φ) and C(f, φ) by their entries

s n (f, φ) = sin (2πf t n + φ) c n (f, φ) = cos (2πf t n + φ) , (3.53)
where t n = n/f s . This means that we perform the linear regression with the OLS estimator constructed with matrix A( fp ) on data which are actually described by the matrix A(f p ), so that

y = A(f p )β + n. (3.54)
Then we must answer two questions. First, we have to find the precision on f p required to have an acceptable error reduction on the estimation of δ. Second, we shall study how to detect and estimate a harmonic signal in the presence of coloured noise, and what is the achievable precision of the estimate of its frequency.

Required precision on the disturbing frequency

In this section we assess the impact of an error on the frequency f p on the bias of the regression. This bias is equal to

b( fp , f p ) ≡ E β -β = A( fp ) * A( fp ) -1 A( fp ) * A(f p )β -β. (3.55)
We are only interested in the bias onto the WEP parameter, namely the first component b 0 ( fp , f p ) of the above bias vector. This component obviously depends on the amplitude and the phase of the disturbing signal. As for the projection rate, we can derive an analytical formula for b 0 ( fp , f p ), which is done in Appendix A.3. It can be maximized with respect to the phase, and the maximum is denoted b 0,max ( fp , f p ) in the following. For a fixed amplitude a p of the disturbance signal, b 0,max ( fp , f p ) reaches local maxima at fp = f EP and f p = f EP . Indeed, fitting a signal close to the WEP frequency induces an ill-posed problem for which the least squares are not well conditioned, and similarly, a disturbing signal oscillating close to the WEP frequency induces a large bias. Besides, b 0,max ( fp , f p ) has a minimum of zero when fp = f p , since there is no bias when the frequency of the disturbing signal is perfectly known. Far from the frequencies f p and f EP the function b 0,max ( fp , f p ) oscillates around the projection rate τ max (f p ) given by Eq. (3.33), corresponding to the situation where we fit an additional harmonic with the wrong frequency.

In order to investigate the impact of the error fp -f p onto the bias rejection, we plot in Fig. 3.5 the bias rate b 0 ( fp , f p ) of Eq. (3.55) obtained by including the harmonic signal in the model, as a function of the true frequency f p . The computation is performed for a spin session (f EP = 9.35 • 10 -4 Hz) and for various values of the frequency error, from the Fourier resolution f s /N ∼ 8.5 • 10 -6 Hz down to 10 -8 Hz. As a comparison, we show the original bias rate τ of Eq. (3.1) without correction (red curve). This plot shows that for a frequency error equal to the Fourier resolution, fitting the harmonic signal does not bring any improvement on the bias rate (the red and black curves are almost coincident). However, when decreasing the error, the rejection begins to be efficient.

The figure shows that from a frequency error of 10 -6 Hz, the maximum bias envelope decreases linearly when the frequency error decreases.

To better understand the non-linear dependence of the bias on the frequency error, we can fix the frequency of the disturbing signal, say, f p = 10 -2 Hz, and concentrate on the dependence of the bias b 0,max ( fp , f p ) with respect to the frequency error fp -f p . For f p = 10 -2 Hz, we plot in Fig. 3.6 the bias (in black) as a function of the frequency error. When the frequency error is larger than a few 10 -6 Hz, the bias is close to the projection rate (in dashed grey, obtained when fitting the WEP signal only). When the frequency error decreases, the bias decreases down to zero when the frequency error is null (at the centre of the plot).

We also show in Fig. 3.6 the "natural" Fourier frequencies f k (blue dashed vertical line) closest to the true frequency of the disturbing signal f p (red vertical line). In general, there is no reason for the closest natural Fourier frequency to lie close to the minimum. In the case shown in the figure, the Fourier frequency is not close enough to the minimum to correspond to a significant rejection of the bias, since it is located outside the narrow "well" where lies the minimum. This confirms the absence of improvement on the bias observed in Fig. 3.5 when the error is equal to the Fourier resolution.

Therefore, the frequency of the harmonic signal must be known with an error at least one order of magnitude lower than the Fourier resolution to be able to reject the bias. In addition, we note that the extrema of the original projection rate τ and the projection after correction b 0 are not located at the same abscissa, and sometimes a maximum of the former corresponds to a minimum of the latter. As a result, for frequencies where the projection rate is already small, the correction is generally inefficient: the bias after correction can be of the same order, or even larger, than the original one. However, at frequencies where the original bias is maximum, the improvement is always significant.

As a conclusion, the natural resolution provided by the Fourier transform on a time series of length N will generally not be high enough to estimate the frequency of a disturbing signal and significantly suppress the bias. Therefore a better precision will be necessary. But the limit on the precision of the frequency estimate is set by the noise level. We investigate the relationship between the frequency precision and the noise in the next section.

f p f k Frequency f -f p [Hz]
Bias rate ( δd

-δ d )/a p τ max (f p ) b 1,max (f p , f )

Harmonic detection in the presence of noise

Now that we have a way to calculate the bias as a function of the error on the frequency, we tackle the problem of the detection and the estimation of harmonic perturbations. We review the framework and the usual tools to deal with it.

Distribution of pure random noise and principle of harmonic testing

Let us first consider a pure noise signal y = n. We define its periodogram as the normalized squared modulus of its Fourier transform:

I N,y (f ) ≡ 1 f s N N -1 n=0 y n e -2πjf nτs 2 .
(3.56)

In the following we will often denote the periodogram as I N (f ) for simplicity. One can show (see e.g. [START_REF] Priestley | Spectral analysis and time series[END_REF], pp. 415-418) that asymptotically, i.e., when N → ∞, for all f ∈]0, f s /2[, the periodogram follows a chi-squared distribution of degree 2, noted χ 2 2 (and of degree 1 for f = 0 or f = f s /2) whose mean and variance are given by the PSD.

Formally, this means that all the random variables I N (f ) are such that:

I N (f ) -→ N →∞ 1 2 S(f )χ 2 2 if f = 0, f s /2 S(f )χ 2 1 if f = 0, f s /2.
(3.57)

The purpose of harmonic detection is to find a frequency where the periodogram is significantly larger than the noise level. It is then convenient to define the statistics

Z ≡ max 1≤k<n I N (f k ) S(f k ) , ( 3.58)
where we set n ≡ N -1

2

, which is the maximum index for which f k is positive. The statistics involves the ratios

Z k ≡ I N (f k ) S(f k ) , ( 3.59)
which compare the magnitude of the periodogram with the theoretical noise power. The goal is to find the frequency f kmax which maximizes the signal-to-noise ratio (SNR)

k max = argmax 1≤k<n Z k . (3.60)
Once the maximum is found, the interest is to know whether it is significant. To do so, we must consider the probability distribution of Z. For k = 0, n, we know from Eq. (3.57) that asymptotically, Z k is proportional to a chi-squared distribution of degree 2, and its cumulative probability distribution function (CDF) is

F Z k (x) = P (X k ≤ x) = 1 -e -x .
(3.61)

In addition, the variables Z k are asymptotically uncorrelated. Then the probability that the maximum of the variables Z k is below some value z is given by the product of the probabilities that all individual Z k are smaller than z:

P (Z ≤ z) = n k=1 P (Z k ≤ x) .
Therefore the probability for the statistics Z to be larger than some value z is given by

P (Z > z) = 1 -1 -e -z n ≈ ne -z . (3.62)
We now allow the possibility that the data contains a disturbing harmonic signal of amplitude a p in addition to the noise.

Then we want to test the following hypotheses against each other:

• H 0 : the data consist only of noise: a p = 0

• H 1 : there is an additive signal in the data, a p = 0 Note that we could even extend H 1 to the more general statement: "there is an additive signal in the data, which is sparse in Fourier space". The test is performed with a 1 -α confidence by imposing a threshold such that the probability that a peak in the spectrum is only due to noise is lower than α:

P (Z > z) ≤ α ⇔ z ≥ log n α ≡ z α . (3.63)
For example, a 99 % confidence test indicates a significant detection for z α ≈ 17.

Review of the literature

An important part of the literature about harmonic testing concerns tests derived from Fisher's test [START_REF] Fisher | Tests of significance in harmonic analysis[END_REF] which is based on the search of the maximum of periodogram ordinates. Several extensions of this test are summarized by Chiu [START_REF] Chiu | Detecting periodic components in a white gaussian time series[END_REF], but they consider white noise only, which is too restrictive for our study.

To deal with noise of continuous and coloured spectrum, other tests based on the periodogram were proposed, see e.g. [START_REF] Priestley | Spectral analysis and time series[END_REF] pp. 616-626 for a review. A first idea proposed by Bartlett [11] is to use blocks of periodogram ordinates to perform local tests, and is referred to as the grouped periodogram test. However, there is no systematic way to choose the size of the blocks. More generally, numerous methods substitute S(f ) for an estimate Ŝ(f ) in Eq. (3.58), which is some smoothed version of the periodogram. One difficulty is to provide a reliable estimate of the PSD in spite of the presence of harmonic peaks in the spectrum. Indeed, the estimate Ŝ(f ) is sensitive to the presence of a signal at a frequency in the neighbourhood of f , more precisely in the interval [START_REF] Hannan | Testing for a jump in the spectral function[END_REF] is a device to reject this bias, based on a tailored modification of the PSD estimate. Similarly, in Ref. [START_REF] Israel | A New Technique for the Detection of Periodic Signals in "Colored" Power Spectra[END_REF] they estimate the PSD at Fourier frequencies by smoothing the neighbour periodogram ordinates, and excluding the ordinate at which the PSD is estimated.

[f -f s /N, f + f s /N ]. Hannan's test
There exist alternative solutions which do not rely on the periodogram ordinates. This is the case of Bartlett's test [START_REF] Bartlett | Probability, Statistics and Time: A collection of essays, chapter Inference and stochastic processes[END_REF], which requires the visual inspection of a diagram. However, its principle relies on criteria that are not formally tractable. Another solution is the P (λ) test proposed by Priestley [START_REF] Priestley | The analysis of stationary processes with mixed spectra-i[END_REF][START_REF] Priestley | Analysis of stationary processes with mixed spectra-ii[END_REF], which considers the expected decaying behaviour of the autocovariance function estimate when the data contains only noise. Once the harmonics are detected, the PSD can then be estimated, e.g. using the double window technique [START_REF] Priestley | Estimation of the spectral density function in the presence of harmonic components[END_REF] which is an approach similar to Hannan's device. However, a limitation of the previously reviewed methods is that the PSD estimate is computed with a windowed-type smoothing procedure, which is usually biased at low frequencies.

Among other types of tests, we can cite maximum likelihood estimation, which is the method adopted by Kay and Nagesha [START_REF] Kay | Maximum likelihood estimation of signals in autoregressive noise[END_REF], where they postulate an autoregressive model for the noise; or Bayesian inference, which is the approach proposed by Carter and Kohn [START_REF] Carter | Semiparametric Bayesian Inference for Time Series with Mixed Spectra[END_REF], based on a Markov chain Monte Carlo algorithm. However, both the maximum likelihood and the Bayesian approaches are computationally demanding for large time series.

All the methods reviewed here suffer from drawbacks, or incompleteness, with respect to the goals of our analysis. Indeed, we would like to 1/ accurately characterize the noise PSD (especially in the low frequency region) and 2/ reject the projection of possible harmonic signals onto the WEP signal. The first requirement implies that the PSD estimator has a low bias in the low frequency region of the spectrum. The second requirement implies that we must estimate the detected frequency with a sufficient accuracy to significantly decrease the bias onto the WEP test (as seen in Sec. 3.4.2), which is not a concern in the methods that we reviewed. A full analysis of the best achievable frequency precision must therefore be carried out, and a frequency detection and estimation method must be developed in the context of coloured noise.

A harmonic detection and estimation method in coloured noise

In Sec. 3.5.1 we build a suitable PSD estimator based on local linear estimation. Then in Sec. 3.5.2 we present a statistical test based on the periodogram and on the constructed PSD estimator to provide a first guess for the hidden frequency. After deriving the optimal uncertainty of the frequency estimation, in Sec. 3.5.3 we propose a refined estimation of the detected frequency by local maximization of the periodogram. We finally test the algorithm with simulated data sets in Sec. 3.5.4.

Estimation of the noise PSD

A PSD estimator is a prerequisite for any detection of periodic signals. Furthermore, we saw in Sec. 3.1.3.2 that the knowledge of the PSD is essential to determine the significance of the WEP test. Although we have at hand a physical model such as the one in Fig. 2.14, the real noise PSD of the in-flight data is not accurately known a priori. Indeed, this model depends on parameters that are not all measured, and thus is not an exact representation of reality. This is all the more the case at low frequency where the effects of the gold wire, of the potential contacts (patch effects) and of thermal noise, dominate. At higher frequency, the accelerometer noise level is better known since it is dominated by the detector electronic noise (which is measured on ground). Nevertheless, the effect of drag free residuals can induce additional high frequency noise due to the thruster control law. Therefore the noise must be characterized from the data, which is the purpose of this section.

Choice and justification of an adequate PSD estimate

The simplest way to assess the noise level is to examine the periodogram, or a windowed version of it (to attenuate the leakage effect), which can be written as

I w,y (f ) = 1 f s C N (w) N -1 n=0 w n y n e -2πjf nτs 2 , ( 3.64)
where w is some apodization window and C N is a w-dependent normalizing constant:

C N (w) = N -1 n=0 w 2 n . (3.65)
The windowed periodogram is an estimator of the PSD. It is usually biased, since we can show (see Appendix A.5) that its expectation is given by the convolution of the PSD of the signal with the periodogram of the window:

E [I w,y (f )] = fs 2 -fs 2 S(f )I w,1 f -f df , ( 3.66)
where I w,1 is the periodogram of the window, obtained by taking y n = 1 ∀n in Eq. (3.64).

The width of the main peak of the window periodogram and its sidelobes is the origin of the leakage effect, meaning that when calculating the periodogram at a frequency f , its value is affected by the contribution of neighboring frequencies (mainly of those lying in the interval [f -f s /N, f + f s /N ] in the case of the rectangular window). If we use an apodization window (see Sec. 3.3), the window periodogram will have lower sidelobes, and thus the bias due to the convolution in Eq. (3.66) will decrease. We saw by Eq. (3.57) that standard deviation of the periodogram is asymptotically equal to the PSD. As a result, its variance does not decrease with the number of data points. In order to reduce the variance of the estimation, the usual way to estimate a PSD from regularly sampled data is to use periodogram smoothing techniques. This is based on the assumption (asymptotically true) that the periodogram points are locally homoscedastic, i.e., that neighbour points have approximately the same finite variance. The basic concept is to convolve the periodogram values with a smoothing kernel. Theses estimators are referred to as "kernel smoother". Smoothing decreases the variance of the estimate, since in this case the variance is inversely proportional to the size N of the data ( [START_REF] Priestley | Spectral analysis and time series[END_REF], p. 454-455). The estimator is said to be consistent. But the variance reduction is done at the cost of the bias, which depends on the curvature, or variability, of the PSD. The more "curved" the PSD, the higher the bias, since the smoothing tends to reduce the local features of the PSD. Both the variance and the bias depend on a smoothing parameter p controlling the amount of smoothing. When the amounts of smoothing increases, the variance decreases while the bias gets larger.

Another popular estimator is the Welch's method [START_REF] Peter | The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. Audio and Electroacoustics[END_REF] which slices the data in K overlapping intervals of size L, compute the K weighted periodogram obtained from these intervals, and finally average them. The averaging process decreases the variance, at the cost of a reduction of the frequency bandwidth.

However, both kernel smoothers and the Welch's technique suffer from drawbacks. First, they generally show a bias appearing at the boundaries (especially around zero frequency). Second, they depend on a smoothing parameter (p in the case of kernel smoothers, K in the case of Welch's periodogram) which may not be appropriate for all the regions of the spectrum. For example, this is typically the case for the PSD of Fig. 2.14 which varies from several orders of magnitude and has different slopes. The estimation of the PSD in the high frequency region of the spectrum will require more smoothing than the low frequency part. Instead, we need an accurate estimate on the whole bandwidth. We require that the estimator must have the following characteristics:

• smoothing prior: the PSD is assumed to be a smooth function, typically a sum of power laws. Therefore the estimator must provide a smooth result, with an adaptive smoothing parameter depending on the frequency region.

• minimal bias: we look for a technique that minimizes the boundary effects.

• computationally fast: the method must be designed to deal with typical data sets of N = 10 6 points.

An ideal method which meets these requirements is local polynomial fitting, and more particularly local linear smoothing, which consists in locally fitting polynomial functions to the periodogram, or rather, to the log-periodogram. This method was initially designed to estimate probability densities by smoothing the histogram obtained from data sets. It has been further developed and adapted to spectral analysis by Fan and Yao [START_REF] Fan | Nonlinear Time Series[END_REF].

The advantage of the local linear smoother is two-fold. First, this estimator is nearly unbiased for estimating steep trends, while kernel smoothers usually show a large bias when the spectrum is steep near the boundary. Second, for our application it makes sense to apply this kind of smoothing to the log-periodogram. Indeed, we saw in Chap. 2 that in a broad frequency region the PSD is likely to have the form of a power law (see Eq. 2.63), thus the log-spectrum will locally be a linear function. That being said, the local linear estimator can be used for very general shapes.

Construction and justification of an adequate PSD estimate

In the following we summarize the basics of the local linear fit starting from [START_REF] Fan | Nonlinear Time Series[END_REF]. Then we will adapt the technique to the needs of large data sets and adaptive smoothing.

Local polynomial fitting

The analysed data are denoted Y i , and can be rather general data. For our PSD application this will be the log-periodogram. Let us consider a time series of N regularly spaced data points y q sampled at a frequency f s . Let Y i be the quantity

Y i ≡ log(I N (f i )) -C 0 , ( 3.67)
where f i are the Fourier frequencies defined as f i = if s /N , and C 0 is the Euler's constant, which is introduced so that Y i has zero mean (see [START_REF] Fan | Nonlinear Time Series[END_REF], p. 285). Indeed the random variable log(I N (f i )) has approximately the distribution f I (y) = exp (y -m i -e y-m i )

where m ≡ log S(f i ), with mean C 0 and variance π 2 /6. We say that taking the logarithm of the periodogram has a stabilizing effect on the variance.

We look for an estimator of the logarithm of the power spectral density, that we denote g(f ) = log S(f ), where S(f ) is the theoretical PSD. The idea of the local polynomial fit is that locally, the function g can be approximated by a Taylor expansion. Thus, for a frequency f at the neighbourhood of f 0 the linear approximation is

g(f ) = g(f 0 ) + g (f 0 )(f -f 0 ) + O (f -f 0 ) 2 . (3.68)
By noting a ≡ g(f 0 ) and b ≡ g (f 0 ), the estimate of g is given by the local intercept a of the best fitted line a + b(f -f 0 ) going through the points Y i in a neighbourhood h around f 0 . This is actually a least-squares problem for each frequency point f 0 , whose solution can be formulated as (see Appendix A.6 for the derivation)

â = n i=1 K T f i -f 0 h , f 0 Y i , ( 3.69)
where K T (x, f 0 ) is a function proportional to a kernel K(x), determining the "range" of the local fit. We obtain the final PSD estimate by taking the exponential

Ŝ(f 0 ) = exp{â(f 0 )}. (3.70) 80

Treatment of interfering signals and minimization of the measurement bias

Smoothing adaptation and frequency grid

In this section we address the problem of adaptive smoothing and the ability to handle large data sets. Once the periodogram is computed, the computation of the linear kernel estimator at a given frequency has a complexity linear in O(N ). Therefore evaluating the PSD at all Fourier frequencies would require a complexity in O(N 2 ), which is too large for our purpose. However, it is possible to compute the estimate on the whole Fourier grid on a single calculation using FFT techniques (see [START_REF] Hardle | Algorithm as 222: Resistant smoothing using the fast fourier transform[END_REF]). The resulting algorithm scales as O(N log N ), but this is possible only for one given bandwidth h. Since we require that the bandwidth changes with frequency, this method is not suitable.

The idea that we retained is to estimate the PSD only at frequencies f j on a downsampled grid of size J, with J N . Then the desired values of the PSD (for example, at Fourier frequencies) are obtained by linear interpolation. For the interpolation to be precise within all the frequency decades of the measurement bandwidth, the initial frequency grid is taken to be logarithmic, as well as the smoothing parameter h j , in a way similar to the LPSD method proposed by [START_REF] Tröbs | Improved spectrum estimation from digitized time series on a logarithmic frequency axis[END_REF]. Thus the sequence f j is chosen such that

f j = f min f max f min j/(J-1) , ( 3.71)
where f min , f max are the boundaries of the interval of interest (typically f s /N, f s /2).

We thus obtain the values Ŝ(f j ). The estimator at intermediate frequencies is then given by linear interpolation.

Considering that the amount of smoothing is controlled by the parameter h, it allows to modify the trade-off between the bias and the variance of the estimator. The authors of [START_REF] Fan | Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation[END_REF][START_REF] Fan | Automatic local smoothing for spectral density estimation[END_REF] implement a systematic method to estimate the optimal bandwidth as a function of the frequency. The idea is to estimate the mean squared error (MSE) at each frequency f 0 where the PSD is estimated, and to minimize it with respect to the bandwidth parameter h. This process may be computational demanding for large data sets like the ones that must be processed in MICROSCOPE, since it requires us, for each frequency f j , to estimate the PSD along with its bias and variance several times, by varying h on a gridded interval in order to minimize the MSE. Instead, we adopt the same variation law for h as for f j in Eq. (3.71), with the parameters h min and h max to be chosen.

Furthermore, the number of operations to compute the local linear estimator is reduced by choosing K(x) = 0 (and hence K T (x) = 0) for x > 1 according to the definition in Eq. (A.37) of Appendix A.6). In this work we use the Epanechnikov kernel [START_REF] Epanechnikov | Non-Parametric Estimation of a Multivariate Probability Density[END_REF], which is optimal in the sense of the mean integrated squared error (MISE):

K(x) = 3 4 1 -x 2 if |x| ≤ 1, 0 otherwise.

Performance of the local linear estimator

In this section we assess the accuracy and the precision of the constructed PSD estimator.

Here we extend the results of [START_REF] Fan | Nonlinear Time Series[END_REF]. We consider that the considered frequency is one of the subgrid, i.e., f 0 = f j . From Eq. (3.69) we can calculate the expectation of â(f 0 ):

µ(f 0 ) = E [â(f 0 )] = n i=1 K T f i -f 0 h , f 0 log S(f 0 ), (3.72)
and its variance

σ 2 (f 0 ) = Var [â(f 0 )] = π 2 6 n i=1 K 2 T f i -f 0 h , f 0 , ( 3.73)
where we assumed that the variables Y i are uncorrelated (which is true asymptotically).

According to [START_REF] Fan | Nonlinear Time Series[END_REF] the estimator â is asymptotically normal, and a (1 -α)-confidence interval for the frequency for this estimator is approximately given by:

I α = e µ(f 0 )-z 1-α/2 σ(f 0 ) , e µ(f 0 )+z 1-α/2 σ(f 0 ) , ( 3.74)
where z u = Φ -1 (u) and Φ is the Normal cumulative distribution function. In Fig. (3.7) we plot the theoretical performance of the constructed PSD estimator (at the frequencies f j only) for an inertial session (red) and a spin session (blue), along with the 99%-confidence interval computed with Eq. (3.74) represented by the colour-filled areas. Although the smoothing parameter h is not optimized, the bias of the estimator is low, and similar for both sessions. The estimator variance is higher for the spin session since the integration time is shorter. From this analysis we can derive theoretical figures giving the order of magnitude of the error made on the PSD for each session. By comparing the boundaries of the confidence interval with the true value of the square root of the PSD, the error at f EP is about 2 • 10 -12 ms -2 Hz -1/2 for an inertial session and 6 • 10 -13 ms -2 Hz -1/2 for a spin session, which is better because f EP is higher for a spin session and thus lies in a less noisy region of the spectrum. These theoretical results will be confirmed by the numerical simulations performed in Chap. 4.

Detection test based on the periodogram

In this section we propose a new test of harmonic detection based on the periodogram, using the following statistics:

Z ≡ max 1≤k≤n I N (f k ) Ŝ(f k ) , ( 3
Z k ≡ I N (f k ) Ŝ(f k )
is now a random variable itself, they do not follow a chi-squared distribution any more. A full derivation of the distribution of Z k would involve the ratio of two random variables, where the numerator follows a chi-squared distribution whereas the numerator follows (approximately) a log-normal distribution. However they can be considered as approximately chi-squared distributed, as we check by plotting the distribution of Z k for f k = 10 -2 Hz in Fig. 3.8, computed from a set of 400 simulated noise realizations. We also checked other frequencies from 10 -4 Hz to 1 Hz. However, below 10 -4 Hz, the distribution is more concentrated close to zero, and the chi-square approximation is not valid any more. An important quantity to consider in hypothesis testing is the power of the test, defined as the probability to reject H 0 when H 1 is true (true detection). In our case, this is the probability to detect a significant peak when there is effectively one. Formally, the test power writes

B(α, a p , f 0 ) = P (Z > z α |H 1 : a p , f p ) .
(3.76)

The test power depends on the chosen confidence level, and on the amplitude and the frequency of the harmonic signal. Rigorously it may also depend on the phase, but weakly, since we are dealing with periodogram ordinates, involving the modulus of the DFT. To facilitate the calculation, one can assume that the disturbance signal has a random phase, i.e., that φ p is a random variable in the interval [-π, π]. In this case s p can be treated as a stationary signal (which is the assumption adopted by Priestley [START_REF] Priestley | Spectral analysis and time series[END_REF] in the chapter 6 devoted to mixed spectra). This means that we can define its PSD S p (f ). Then, by considering that n and s p are independent, we can write an effective total PSD for the measured signal y:

S y (f ) = S n (f ) + S p (f ). (3.77)
It can be shown that S p (f ) is given by the expression in Eq. (A. [START_REF] Carter | Semiparametric Bayesian Inference for Time Series with Mixed Spectra[END_REF]) after dropping the third term involving the phase.

To derive an expression for the power we have to inspect the CDF of Z k in the presence of a harmonic signal (under H 1 ). Considering the numerator, thanks to Eq. (3.77), we know that under H 1 , the periodogram of y is distributed as

I N,y (f ) ∼ 1 2 S y (f )χ 2 2 .
Regarding the denominator Ŝ(f ), it is a random variable whose mean is approximately the exponential of the log-spectrum estimate expectation given by Eq. (3.72), depending on the effective PSD S y . As previously, we consider that the ratio Z k is approximately proportional to a chi-squared distribution:

Z k ∼ 1 2 S y (f k )e -µ(f k ) χ 2 2 .
The resulting CDF is thus

F Z k |H 1 (x) ≈ 1 -exp - e µ(f k ) S y (f k ) x . ( 3.78)
The theoretical power of the test can then be computed with the formula

B(α, a p , f p ) = 1 - n k=1 F Z k |H 1 (x). (3.79)
In this way, the power of the test takes the bias of the PSD estimator due to the harmonic component into account (at the neighbourhood of f p the estimator will overestimate the PSD, hence decreasing the power of the test).

We plot in Fig. 3.9 the theoretical power of the proposed test with α = 0.01, corresponding to a detection threshold of z α ≈ 17, as a function of various amplitudes and frequencies of the disturbance signal. The computation is done based on the theoretical PSD model of the accelerometer noise.

As expected, the power of the test globally increases with increasing values of a p , meaning that the amount of smoothing of the PSD estimator is sufficient to attenuate the effect of harmonic peaks on the PSD estimation. However, a remarkable feature of the graph is that the power is very poor around zero frequency. There are three reasons for this behaviour. The first one is that signals whose frequencies are not significantly higher than the Fourier resolution have a broader Fourier representation and are not well identified. The second one is that the smoothing parameter h of the local linear estimator is minimal in this region, hence the PSD estimate tends to be more biased by the presence of the peak at low frequencies. The third reason is that the chi-squared approximation breaks down, as mentioned earlier.

Except for the lowest and highest frequencies, the surface levels follow the shape of the PSD, which determines the threshold above which the test is said to be powerful. ). However, if a sine with such power is present in the data, it will have between 10% and 20% chance to be detected, since it lies in the dark blue region. The white solid line defines the region of high power (amplitudes of the harmonic signals which, if present, will be detected with a probability larger than 90% in most of the spectral band).

More precisely, we know that the test rejects the null hypothesis if Z > z α . Yet, the ratio Z of the periodogram to the PSD is maximum at f = f p , and we have

I N,sp (f p ) = N 4f s a 2 p . (3.80)
Therefore the detection threshold for the periodogram roughly translates into an amplitude threshold by the inequality

a α (f p ) > 2 z α f s S(f p ) N . (3.81)
This amplitude detection threshold is represented in Fig. 3.9 by the white dashed line. We see that this limits is not in a region where the test is powerful. However, in most of the frequency band, the test has an acceptable power (almost equal to 100%) for amplitude of the form a p ∼ c • a α (f p ) with c ≈ 2.5. The curve defined by this relation is called "powerful limit" in the following, and corresponds to amplitudes for which the harmonic signal is detected with a 100% probability, provided that its frequency is in the interval 10 -4 < f p < 1 Hz). It is represented by the white solid line in Fig. 3.9.

Refinement of the frequency estimate of a detected harmonic signal

Now that we have a device to detect the approximate localization of harmonic peaks in the spectrum, we must determine what is the best achievable precision one can reach on the estimation of their frequency, and implement a precise enough estimator. Given the study of Sec. 3.4.2, this information will allow us to know whether it is possible to reject the bias of any detected signal.

The likelihood of Gaussian stationary time series

In order to find the optimal precision on the frequency, we must postulate a statistical model for the data, and resort to the concept of likelihood. For a given distribution, the likelihood is the probability to observe the data y given the values of all the parameters θ describing the model:

L : Θ → R θ → L y (θ) ≡ f θ (y) , ( 3.82)
with f θ being the probability density function of the distribution of the random variable y. The likelihood is a function from the parameter space to the set of real numbers.

It is also convenient to define the log-likelihood function, as the natural logarithm of the likelihood l(θ) ≡ log L(θ).

A useful distribution model when analyzing data is the Gaussian one, since numerous physical processes are Gaussian, or approximately Gaussian. Indeed, when many noise contributors are involved, their sum always tend to the Gaussian distribution because of the central limit theorem. The likelihood of a N × 1 Gaussian random field y (for instance, a piece of measured time series) of mean µ and variance Σ writes

L y (θ) = 1 (2π) N |Σ| exp - 1 2 (y -µ) * Σ -1 (y -µ) . ( 3.83)
In Eq. (3.83) the dependence on the parameter θ is implicit through µ(θ) and Σ(θ). In the linear regression framework (with a model of the type of Eq. (3.2)), and in the case of stationary random processes, the set of model parameters will consist of the regression parameter vector β and the P parameters S k = S(f k ) necessary to describe the variance (according to Eq. (3.20)). Hence we will have θ = (β, S) where S is a P × 1 vector whose entries are the sequence S k , and µ = Aβ. Furthermore, in case of a hidden harmonic, the frequency f p of the harmonic component is another parameter of the likelihood: θ = (β, S, f p ).

Cramér-Rao lower bound on the frequency

Now that we postulated a model for the data, the tool to find the best achievable precision on the frequency is given by the Cramér-Rao lower bound (CRLB). The CRLB gives, for a given probability distribution and a given parameter, the minimum achievable variance of any unbiased estimator of this parameter. In our study we consider the parameter f p of the Gaussian random field whose likelihood is given by Eq. (3.83), with mean µ = A(f p )β and covariance Σ. We show in Appendix A.7 that the CRLB is

Var fp ≥ v * Σ -1 -Σ -1 A A * Σ -1 A -1 A * Σ -1 v -1 , ( 3.84)
with v ≡ ∂A ∂fp β, and A is the model defined in Eq. (3.52). In crude approximation, the variance of f p will be proportional to the PSD, and inversely proportional to the squared amplitude of the harmonic signal and to the integration time: Var fp ∼ S(f p )f 2 s N -1 a -2 p . The CRLB is shown in Fig. 3.10, as a function of the frequency and the amplitude of the harmonic signal. The contour lines correspond to specific values of the CRLB, expressed in Hz. We recognize the shape of the noise PSD on which the optimal variance depends, and the decrease of the CRLB with the amplitude of the signal is confirmed.

Consequence on the bias rejection

By comparing Fig. 3.10 to Fig. 3.9, we see that the amplitudes above which the test is powerful, i.e., where the unknown harmonics are almost always detectable, approximately corresponds to a standard deviation of a few 10 -7 Hz on the estimation of the frequency. We saw in Fig. 3.6 that with such a precision, if the frequency corresponds to a region where the bias is maximum (i.e., close to the envelope), then the rejection is significant. When the amplitude of the harmonic signal increases, the projection onto f EP is larger, but the precision of the frequency estimation increases by the same amount, yielding a better bias rejection. Therefore, for frequencies of largest projection the bias can theoretically be rejected by the technique of harmonic detection, provided that we have an estimator whose variance is close to the CRLB, and has the same dependence on the amplitude.

We can combine these results to assess the theoretical performance of the bias rejection. We would like to know what is the improvement on the bias that we can hope for when the harmonic signal has a "detectable" amplitude, where "detectable" means "with a probability of almost 100%". To do so, we first compute the detectable amplitudes with respect to the frequency, given by 2.5 • a α (f p ). Then, we compute the Cramér-Rao lower bound σ(f p ) ≡ Var fp corresponding to these amplitudes. Finally, we compute the bias (maximized with respect to the phases) that we would have by inclusion of the harmonic signal in the model with an error on the frequency equal to 3σ(f p ). We perform the same analysis for amplitudes an order of magnitude larger, i.e., 25 • a α (f p ). We restrict the analysis to the frequency interval [10 -4 , 1] Hz since we saw that for lower or larger frequencies the test is not powerful, hence the estimator of f p will be biased (the higher frequencies are usually filtered out by the anti-aliasing filters of the system, while the lower frequencies can be treated by a longer integration time or by polynomial fitting). These computations are shown in Fig. 3.11. We see that for amplitudes at the high power limit -panel (a) -the bias is about a factor 10 less if we compare the bias after correction (black) to the original bias (red). If we consider the harmonic signals with amplitudes 10 times larger than the high power limit -panel (b) -the bias is obviously increased by the same amount, but the bias envelope after correction almost remains at the same level. We would obtain a similar result if we increased the amplitude by an arbitrary amount. In addition, the level at which the bias is maintained is acceptable with respect to the objective of the mission, since it remains below 10 -16 on the whole spectrum (except around f EP ). Therefore, this analysis suggests that the bias of harmonic signals can be maintained down to an acceptable level by the harmonic detection technique. Since the variance of any estimator will be larger than (or equal to) the CRLB, we must construct an estimator to assess the actual performance of the proposed approach.

Refined frequency estimation

Before verifying the theoretical results derived in the previous sections, we must choose an estimator of the frequency of the harmonic signal, which provides a better precision than the frequency obtained by the detection test.

Let f 0 p be the frequency detected by the periodogram test presented in Sec. 3.5.2. This test has a frequency resolution of f s /N . This means that the true frequency lies in the interval

I 0 = f (0) 0 -f s /(2N ), f (0) 0 + f s /(2N
) . The idea is to set a finer frequency grid covering this interval, and to find the frequency that maximizes the periodogram. Thus we propose to use fp = argmax

f ∈I 0 I N,y (f ). (3.85)
This approach is justified by two arguments. First, the noise PSD is considered constant in the interval I 0 (smooth condition), therefore maximizing the SNR amounts to maximizing the periodogram. Second, the works of Lomb [START_REF] Lomb | Least-squares frequency analysis of unequally spaced data[END_REF] and Scargle [START_REF] Scargle | Studies in astronomical time series analysis. II -Statistical aspects of spectral analysis of unevenly spaced data[END_REF] showed that finding the maximum of the periodogram (or a time-delayed version of it) is exactly equivalent to performing a least-squares fit of a cosine and a sine function to the data, and finding the frequency which minimizes the residuals. While not formulated for coloured noise, the result is extended to irregular sampling, and sophisticated routines such as famous [START_REF] Mignard | Frequency analysis mapping on unusual samplings[END_REF] rely on a least-squares approach.

Note that for the method to be as precise as possible, the frequency grid must be chosen to provide a higher resolution than the CRLB corresponding to the amplitude and frequency of the detected signal. To adjust the frequency grid covering I 0 , the CRLB can be estimated from the data, since fitting a signal of frequency f 0 p gives a first estimate of its amplitude, which in turns enables us to compute a reliable estimate of the CRLB.

In the next section we test the theoretical results on the frequency estimation and bias rejection by applying this estimator to numerically simulated data.

Numerical simulations

In order to verify the performance of the harmonic detection method with respect to the analytical approach, we test the developed algorithm with simulated acceleration data. In addition, we check the validity of the theoretical minimal variance by implementing the refined frequency estimator.

The data

We generate synthesized times series, each one containing a noise component and one single harmonic component. The noise is generated from the PSD model of Eq. (2.63) with a code based on the method described in [START_REF] Timmer | On generating power law noise[END_REF]. We generate 400 realizations of the noise, corresponding to a 20-orbit spin session sampled at 4 Hz. This number of draws is chosen such that, when computing the sample standard deviation, the error with respect to the true value of the standard deviation is less than 10% with a 99% confidence.

We generate sets of 20 harmonic signals with frequencies f p in the interval [10 -5 , 2] Hz. There are four sets of such signals, that we label A, B, C, D.

Set A: in this set of simulations, the amplitudes are taken equal to the detection threshold a α (f p ) defined by Eq. 3.81, and the frequencies are equally spaced on a logarithmic grid from 10 -5 Hz to 2 Hz. In this set, the phase of the signals is taken equal to φ p = 0 for simplicity.

Set B: in this set, the amplitudes are equal to 2.5 times the detection threshold, which theoretically corresponds to a test with a power close to 100%, with arbitrary frequencies and phases such as in set A.

Set C: the amplitudes are equal to 2.5 times the detection threshold as in set B, but the frequencies f p and phases φ p are chosen such as the bias of the harmonic signals is maximum. The frequencies have the form f p = f s (k + 1/2)/N and the phase is φ p = π/2 for f p < f EP and φ p = 0 for f p > f EP . Indeed, these values maximize the projection rate in Eq. (3.31).

Set D: The amplitudes are equal to 25 times the detection threshold and the frequencies and phases are chosen to maximize the bias, as in set C.

Results of the detection tests

We apply the detection test of Sec. 3.5.2 on the 4 × 20 × 400 simulated time series.

To provide a visual example, we plot in Fig. 3.12 an example of periodogram (black), and PSD estimation (blue), with a periodic signal at frequency f p = 2.1 • 10 -2 Hz extracted from sets A and B. For the amplitude equal to the detection threshold (a), there is no detection for this realization of the noise. For an amplitude 2.5 times larger (b), the test is positive. In the first case the peak is not visible, and the algorithm does not detect the signal. In the second case the detection is positive with a confidence close to 100% (the estimated SNR is 43).

For each value of f p , we count the number of positive detections. This is done for sets A and B of harmonic signals, with amplitudes a α (f p ) and 2.5 × a α (f p ). The results of the empirical power are shown in Fig. 3.13.

At the detection limit (blue bars) the empirical power of the test is not high, which confirms the theoretical analysis of Fig. 3.9 where the threshold (dashed line) corresponds to power below 20%. However, when increasing the value of the signal amplitude (red bars), the empirical power is found to be 100%, which corresponds to a shift towards the red region in Fig. 3.9. The empirical power also confirms the theoretical drop in power at low (< 10 -4 Hz) and large (close to Nyquist) frequency.

For sets C and D the empirical results of the test power are similar to set B since the amplitude of the signals are equal or larger than in set B.

As a result, the constructed test has a power of about 100% for signals corresponding to an average signal-to-noise ratio larger than about c 2 Z ∼ 100 in power (20 dB). This is valid for frequencies which are higher than 10 -4 Hz, and not too close to the Nyquist frequency.

Results of the frequency estimation

In this section we test the refined frequency estimator introduced in Eq. (3.85), and compare its results with the formula of the Cramér-Rao lower bound on the frequency that we derived in Sec. 3.5.3. We consider signals of amplitudes equal to 2.5 × a α (f p ) of set B, corresponding to the power represented by the red bars in Fig. 3.13, which is close or equal to 100% for nearly all tested frequencies. For each frequency f p , and for each realization of the noise, if the test is positive, the maximization algorithm is run, starting from the frequency f 0 p = f kmax found with Eq. (3.60) as first guess. The results are gathered in panel (a) of Fig. 3.14 showing the average error made on the frequency (grey). The sample standard deviation of the frequency estimates is plotted in blue, and remains above or is coincident to the theoretical lower bound (dashed red), confirming the validity of the CRLB formula. In addition, it is clear that the frequency estimation algorithm brings an improvement in the estimation of f p with respect to the natural Fourier resolution provided by the periodogram ordinates (black dashed line). For each tested frequency f p , we compute the bias on the WEP signal before any correction (black dots) on panel (b) of the figure. For each estimated frequency fp , we compute the absolute value of the bias b( fp , f p ) that we would have by correcting the data for the harmonic disturbance (by including it in the model). Then we calculate the average absolute bias over all the frequency estimates (blue dots). We also show the maximum bias after correction of the harmonic model (black dashed line), obtained with the highest error on the frequency. The shaded area corresponds to the possible values that the bias can take after the correction process.

As expected, for this set of frequencies and amplitudes, the bias is not always lowered. This is particularly the case for frequencies where the bias is already small with respect to the maximum shown by the red curve. A good example is the lowest point around 4 • 10 -2 Hz, where f p fortuitously fall close to a minimum of a the projection rate. This is because the grid of the tested frequency in set B is arbitrary (it is simply tailored to be regularly sampled on a logarithmic axis). Now we perform the same simulations for the harmonic signals of set C, i.e., with frequencies f p and phases φ p corresponding to a maximum of the original bias. The results of this new simulation are plotted in the panels (a) and (b) of Fig. 3.15. We see that the black dots now all coincide with the red curve, confirming that the harmonic signal maximizes the bias. In this case, the average error is always improved by the correction step.

To verify that the bias level is maintained even for larger amplitudes, the simulation is also done with the signals of set D, whose amplitudes are 10 times larger. The results are plotted in panels (c) and (d) of Fig. 3.15, confirming that the rejection is maintained, and even improved, for larger amplitudes (keeping in mind that the rejection is not a linear function of the frequency precision, as shown by Fig. 3.6). As a result, the bias of detected and undetected signals will always be below the maximum level represented in red in the panel (b) of Fig. 3.15. Hence, the error on the WEP parameter due to harmonic disturbances will always be smaller than 10 -16 . This shows that the harmonic detection method is efficient to maintain the bias below an acceptable level.

The method can be easily extended to multiple detections, where we iterate between frequency detection and removal of the detected signals. Of course, there can be several undetected signals, whose cumulative projection can lead to a larger error. However, to induce a false WEP detection (say, 3 • 10 -15 ) due to projection only, at least 30 undetected periodic signals would be necessary, which is an unlikely situation.

Summary

We saw that the bias caused by all the modelled perturbations can be reduced through instrumental calibration, which mainly concerns all the signals whose frequencies are close to the WEP frequency. We then considered the bias of unexpected harmonic perturbations. We showed that windowing is a good way to reject the bias of disturbing harmonics in the least-squares framework. But the price to pay for this technique is an increase of the statistical uncertainty of a factor 1.5. Furthermore, we investigated the detection and estimation of unknown harmonics in the presence of coloured noise. We showed that to reject their projection onto the WEP frequency, the maximum error on the estimation of the harmonic frequency must generally be much lower than the Fourier resolution 1/T . Therefore we implemented a statistical method to detect the harmonics, and to estimate their frequencies with a precision close to the minimal variance bound. We showed that the precision of this frequency estimator is sufficient to maintain the bias below 10 -16 on the Eötvös parameter. However, so far we dealt with the systematic errors on the WEP test, and we did not consider its statistical uncertainty. In particular, ordinary least squares may be imprecise when there is a lack of several data points in the recorded time series. Therefore a regression method adapted to measurements with missing data must be developed, as well as a way to estimate the precision of the fit.

4

Treatment of missing data and minimization of the measurement uncertainty

Treatment of missing data and minimization of the measurement uncertainty

This chapter aims at developing data processing methods to deal with interruptions which may occur in the measurement. The prerequisites of any such method is that the invalid data must be correctly identified. Once the invalid or missing data are flagged, the objective of the analysis is three-fold. First, it must maintain the precision objective of the mission. Second, it must allow the estimation of the confidence of the test. Third, it must provide consistent reconstructed data sets which are easily usable by the scientific community. To achieve the first goal we must ensure the lowest possible variance in the estimation of the WEP violation parameter and the calibration parameters, in spite of the loss of data. After assessing the impact of missing data on the least-squares precision, we set up a protocol to construct the data mask by an analysis of the mechanisms leading to the missing or outlier data. Then a linear regression algorithm with gapped data is designed to minimize the uncertainty. This is done by constructing an approximate general least-squares estimator, which uses an autoregressive estimate of the noise covariance. To achieve the second and third goals the covariance of the WEP and calibration estimates are evaluated. This is done by a refinement of the estimation of the noise power spectral density (PSD) through iterative reconstructions of the missing data.

Assessment of the impact of the missing data on the precision of the measurement

Various physical or operational phenomena can lead to the absence, or corruption, of some data in the time series that are measured in flight. Both cases (absence or invalidity of the information) will be referred to as "missing data" or "data gaps". After reviewing the most likely causes of these interruptions, we assess their impact on the performance of the estimation method described in Sec. 3.1 of Chap. 3.

Origin of missing data

Missing data can result from several causes. A detailed review can be found in Ref. [START_REF] Hardy | Procédures expérimentales et traitement des données associées pour la mission spatiale MICROSCOPE[END_REF], but we recall the main phenomena below.

Telemetry losses

The telemetry is transmitted during the flight of the satellite over the ground stations. Each flight lasts 8 minutes. In this time span, approximately 6 hours of data are downloaded. Errors can occur in the down-link transmission, which may lead to data losses. The maximum loss fraction is specified to be 1.5%. However the recovery rate has proven to be better than the specification in previous Myriad-based missions such as PICARD [START_REF] Rouzé | The picard scientific mission: status of the program[END_REF]. Indeed, even in case of transmission failure, the data remain stored in the satellite during about 9 hours, which is enough to be downloaded again. If the command control centre (see Sec. 2.1.3) detects some data loss, a new download of the telemetry can be ordered, increasing the theoretical recovery rate to 99.93%. The telemetry losses are therefore likely to be rare. By analysing the telemetry thread of the PICARD mission, we observe that the standard duration of such losses is about 1 minute, and that the loss a data segment longer than 1 minute happens every four days.

Instrument saturation

Another cause of missing data are instrument saturations. The measurement range of the accelerometers being limited, some events can induce acceleration peaks exceeding the saturation threshold, hence leading to an absence of information, or corrupted data.

The corresponding interval must be discarded in the data analysis. There are three identified possible sources of saturation. First, cracks in the cold gas tanks of the thrusters are triggered by decreasing pressure as they empty. An on-ground experiment was performed in CNES centre in Toulouse [START_REF] Ledeuff | Rapport d'essai de caractÉrisation des microperturbations du reservoir[END_REF], where one tank was emptied while several sensors where measuring the local and global distortions of the structure, as well as the wall vibration (by means of a laser). The measurements reveal short events corresponding to acceleration levels that can be larger than the saturation threshold (> 10 -7 ms -2 ). By taking into account the number of tanks on-board the satellite, the number of measured events was extrapolated to be 260 events per orbit, each one lasting 0.5 second.

Second, cracks may also occur in the multilayer insulation (MLI) coating of the satellite, which ensures the thermal insulation of the platform (the gold-coloured coating visible in Fig. 2.3). Some faces of the satellite are successively oriented towards the space vacuum or towards the earth, generating temperature variations in flight. These variations may cause momentum changes whose measurement can saturate the accelerometers. Onground investigations have been carried out to evaluate the number of occurrences of MLI cracking, showing that 6 cracking events lasting half to three quarter of a second occur in average during one thermal cycle (which will correspond, in flight, to one WEP period). By multiplying by the number of involved faces, we obtain 24 cracks per WEP period.

Finally, CNES experts forecast a non-negligible flux of micrometeorites (or small debris) hitting the satellite. A micrometeorite impact induces a small velocity change in the satellite, corresponding to an acceleration peak measurable by the accelerometers. A statistical study [START_REF] Lefebve | Impact des chocs de micro-meteorites en mca[END_REF] shows that there may be about 0.4 saturation events per orbit of less than 1 second.

All saturated data are clearly identified by a flagging system in the telemetry, which is delivered every quarter of second (at the sampling frequency f s ). The accelerometer control loop operates at 1024 Hz, while the measurement is obtained at 4 Hz. Thus one measurement point is computed by averaging about 256 values. The flagging system is such that if at least one value among the 256 points saturates in a block of the control loop, a flag is raised for the corresponding 4 Hz measurement point. The full impact of saturation on the measurement process will be detailed in Sec. 4.2. In this section we simply assume that all corrupted or missing data are correctly flagged.

Uncertainty of the OLS estimator with missing data

In this section we assess the impact of missing data on the performance of the linear regression presented in Sec. 3.1. We assume that the flagging information is given by a mask vector labelled w, which is such that:

∀ n ∈ 0; N -1 , w n ≡
1 if the data at time t n is available 0 if the data at time t n is missing.

We also define the mask matrix as the diagonal matrix constructed from the mask vector W = diag (w).

Note that we assume that the missing data mechanism is completely independent of the data itself. This situation is often refereed to as "missing completely at random" (MCAR) in the literature (see e.g. [START_REF] Roderick | Wiley Series in Probability and Statistics[END_REF], pp. 11-13). In the following we denote N m the number of missing data points, and N o the number of observed data points, such that N = N m + N o .

When performing a least-squares estimation from a time series, the obvious way to deal with missing data is to simply ignore the measurement points that are missing, and to fit the model to the available data only. In the least-squares formalism, this amounts to applying the windowed least squares of Sec. 3.3, with the window matrix W constructed from the mask defined in Eq. (4.1). The estimator is thus given by Eq. (3.39).

We saw that if the noise has mean zero, this estimator is unbiased. Hence the missing data do not induce any bias. However, like any windowing, they affect the variance of the estimation, according to the general formula (3.44) of the variance of the windowed least squares. To give insight into the qualitative impact of missing data on the uncertainty, we start from Eq. (3.44):

Cov β = (A * w A w ) -1 A * WΣW * A (A * w A w ) -1 . (4.1)
where we used the fact that W * W = W in the case of the mask window (since it is binary with entries equal to 0 or 1). This equation highlights an effective noise covariance Σ w ≡ WΣW * , which is a masked version of the original covariance. We saw from Eq. (3.20b) that the matrix F N ΣF * N is approximately diagonal. In general this will not be true for Σw ≡ F N Σ w F * N , although the diagonal terms are dominant with respect to off-diagonal terms. To illustrate this we plot in Fig. The entries of matrix Σw are given by the following expression:

Σw (l, m) = 1 N N -1 p=0 N -1 k=0 w p w k R(p -k)e -2iπ N (pl-km) . (4.2)
Since by definition R(p -k) = E [y p y k ] for a pure zero-mean noise, the diagnoal entries (obtained for l = m) are equal to the expectation of the periodogram of the windowed noise (up to the constant f s ):

Σw (l, l) = f s E [I N,Wn (f l )] = f s fs 2 -fs 2 S(f )I N,w f l -f df . (4.3)
Going back to the expression of the covariance, we can introduce the masked covariance matrix in the Fourier domain (that we labelled Σw ):

Cov β = (A * w A w ) -1 A * F * N Σw F N A (A * w A w ) -1 . (4.4)
We now consider the fit of a single signal, say the WEP violation model, A = 1/2 • g x . Then A is a N × 1 vector and the covariance is given by Eq. (3.46). In addition, we assume that the off-diagonal terms of Σw are negligible compared to the diagonal terms. Then the equation can be rewritten as

Var( δd ) ≈ N -1 k=0 | Ãk | 2 Σw (k, k) N -1 k =0 | Ãw (k )| 2 2 (4.5)
First we consider the denominator. If we assume that the WEP violation signal 1/2•g x is a pure sine wave of frequency f EP and amplitude a EP , then we have

N -1 n=0 | Ãw (n)| 2 = N -1 n=0 w 2 n a 2 EP sin 2 (2πnf EP τ s + φ EP ) = a 2 EP 2 N o - N -1 n=0 w n cos (4πnf EP τ s + 2φ EP ) = a 2 EP 2 N o - N -1 n=0 {cos (4πnf EP τ s + 2φ EP ) + (1 -w n ) cos (4πnf EP τ s + 2φ EP )} ≈ a 2 EP 2 N o . ( 4.6)
The last approximate equality relies on the hypothesis that the sum is negligible, which is valid under two conditions. The first one is that there is an integer number of WEP periods in the signal. Indeed, the first sum can be computed with Eq. (A.3) in Appendix A.1, and is zero in this case. The second condition is that N o N m , in which case the second sum can be neglected, since it is always smaller than N m .

Considering the numerator, as for Eq. (3.47) we assume that f EP = k 0 /T and we obtain

Var( δ) ≈ N 2 a 2 EP Σw (k 0 , k 0 ) No 2 a 2 EP 2 = 2f s N a 2 EP N 2 o fs 2 -fs 2 S(f )I N,w f EP -f df . (4.7)
This result shows that the uncertainty of the OLS estimator is (approximately) related to the convolution of the noise PSD with the periodogram of the mask w. When fitting a sine wave at a given frequency, the variance is proportional to the convolution term calculated at the frequency of the sine. In other words, when data are missing and when the noise is coloured, there is an increase of uncertainty, not only because of the loss of signal-to-noise ratio, but also because of the power leakage due to the noise coloration and the missing data pattern. At f EP , the leakage term is dominant. Now that we have an analytical expression for the OLS variance in the presence of missing data and coloured noise, we evaluate the impact of various missing data patterns.

Impact assessment as a function of the missing data pattern

In this section we study the effect of 3 missing data patterns, corresponding to different possible physical situations.

The first one is a "tank cracking type" window w a which simulates random and unpredictable saturations in flight. It is generated so that all gaps are of equal duration (0.5 second) and their positions are randomly distributed over the sample. The position of the gaps follows a uniform probability distribution with 260 gaps per orbit.

The second pattern is a "telemetry losses type" window w b where the gaps durations are drawn from a probability distribution similar to the telemetry thread of the PICARD mission [START_REF] Rouzé | The picard scientific mission: status of the program[END_REF], with a standard duration of one minute. Their positions are distributed in the same way as for the first window.

The third pattern is a "periodic window" w c and simulates the data unavailability that could occur at a special frequency (due to periodic temperature changes for example). In this case there exists a period T g such that w n+Tgfs = w n . The period of the interruptions chosen in the simulation is the orbital period, which is a likely periodicity for an experiment on-board an orbiting satellite.

The gaps that would be induced by micrometeorite impacts are not considered as a special pattern here, as it would be similar to w a .

Each window represents the same fraction of missing data, of about α m = 2% (or about N m = 10 4 points for a spin session). Thus window w a comprises more gaps than windows w b and w c but gaps are shorter in average. To illustrate this, we plot in Fig. 4.2 an extract of the windows. The gaps (i.e., when the mask is zero) are represented by coloured areas. The effect of masking on the periodogram is shown in Fig. 4.3 where we plot the periodogram mean given by Eq. (4.3). Fig. (4.3a) shows the case where the mask is applied before the filtering process. This is the typical situation where a saturation event occurs inside the control loop, before the signal is filtered. The resulting expectation of the periodogram can be approximated by

E [I N,Wn (f )] ≈ |H filter (f )| 2 fs 2 -fs 2 S(f )I N,w f -f df . (4.8)
In this calculation, the convolution term is the expectation of the periodogram of the masked noise, and the PSD S(f ) is given by the model of Fig. 2.14a (unfiltered PSD).

Then the filter transfer function H filter (f ) is applied. This is not exactly the description of what happens in reality. Rigorously, the right-hand side of Eq. (4.8) would be the PSD of the product of two stochastic processes of PSD S(f ) and I N,w , which would then be filtered. Therefore, we implicitly assumed that the mask is a stationary process (which is not one in general, except for purely random masks). In spite of this approximation, the curves give a good representation of the final noise level in the periodogram. In Fig. (4.3b) the mask is applied after the anti-aliasing filters. In this case the high frequency noise power of the effective PSD (black) is lower, leading to a smaller leakage effect. Here the calculated periodogram expectations are exact, because the filtered noise is a stationary process, and the expectation of the periodogram is exactly given by the convolution of the mask periodogram and the PSD of the filtered noise as

E [I N,Wn (f )] = fs 2 -fs 2 S(f ) H filter (f ) 2 I N,w f -f df . (4.9)
The largest leakage effect is obtained for window w a which corresponds to short and numerous gaps. The other windows include less gaps and yield a smaller leakage, with a similar magnitude for windows b and c. The periodic window induces small artefacts (spurious peaks) in the periodogram. In the case of mask a, the power leakage increases the apparent noise level in the low frequency region by about a factor 60 for the pre-filter PSD, and by a factor 30 for the post-filter PSD. Consequently, the least-squares uncertainty is increased by approximately the same amount. Note that even in the absence of gaps, the periodogram expectation is not exactly equal to the original PSD, because of an aliasing effect due to the convolution with the rectangular window, also called Féjer kernel.

The fact that the amount of leakage depends on the number of gaps is confirmed theoretically. Indeed, we show (see Appendix A.8) that for a mask where the gaps all have the same duration and the gap positions are randomly distributed (identically and independently) the periodogram mean is given by

E [I N,Wn (f )] = µ 2 w • E [I N,n (f )] + 1 f s σ 2 w σ 2 , (4.10)
where the expectation is computed from the joint distribution of the noise and of the window. We defined the mean µ w ≡ E [w n ] and the variance σ 2 w ≡ Var [w n ] of the window for all n, and the variance σ 2 ≡ R(0) of the noise. The first term of the right-hand side term of Eq. (4.10), which is proportional to the expectation of the periodogram of complete data, accounts for the loss of noise power due to the missing data, whereas the right-hand side term is what we may call the "leakage term", and does not depend on frequency, but is proportional to the average noise power, which is also the noise variance.

The increase of the uncertainty due to the leakage effect from missing data is assessed by computing the variance of the masked OLS estimator with Eq. (3.46). The results are gathered in Table

Window

Standard deviation [×10 We also assess the impact of the number of gaps on the OLS uncertainty. To this aim, we generate several random masks having the same distribution as (w a ), but with an increasing fraction of missing data. We then use Eq. This behaviour is not acceptable with respect to the objective of the mission, and a better estimation method must be developed. Ordinary least-squares uncertainty as a function of the number of gaps per orbit, computed empirically (red curve) and analytically (black curve). The computation is done for a random mask, corresponding to tank cracking gaps distributed over a spin session. The empirical curve is obtained by averaging 400 estimates obtained from 400 independent realizations of the noise vector. The theoretical curve is obtained with Eq. (4.4).

Mask construction

Up to now we assumed that the mask vector w indicating if the data are valid or invalid was given and well defined. In practice, the mask vector must be constructed by a careful analysis of the data. The accuracy of the regression method particularly depends on the definition of the mask.

We called the data "missing" when it is either unavailable, or corrupted by some saturation, or by some sharp acceleration peak. Some indicators in the telemetry already flag the data points that are saturated or unavailable. But the effect of the whole processing chain must be taken into account when constructing the mask. We can distinguish two situations. The first one is when there is really no information at some data point, because it is flagged as saturated or unavailable in the telemetry, and is therefore not exploitable. The other situation is when the data point is neither saturated nor lost, but disturbed by some localized event or outlier. In this case, we can choose to discard the point when performing a regression, in order to avoid a bias on the measurement. In this section we show how to construct the mask in both situations, and we test the method on realistic simulated data.

Key steps of the measurement process

We consider the control loop in Fig. 2. [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF]. Let e i be the 1024 Hz digital signal, expressed in volt, at the output of the PID of sensor i.

If the value of the input signal is too large, e i can be saturated. There are three main levels where the signal can be saturated in the loop: at the capacitive detector level, at the PID level, and at the DVA and read-out level. Each saturation level ±E is expressed in volts (where E = 2.5 V for the PID). We label ēi the signal after saturation. After read-out, the signal is filtered by a filter of impulse response h and cut-off frequency f c which averages the data and prevents aliasing of high frequencies. We obtain an output signal denoted s i , that is finally decimated to build a time series s i,n sampled at f s = 4 Hz and sent to the telemetry transmitter. The differential signal is obtained by computing the half-difference s d,n = (s 1,n -s 2,n )/2. This processing chain is represented by Fig. 4 Usually the gain G i is applied after the filtering process but these two box functions are presented in the reverse order for convenience to derive the equations (this choice has no importance since they are linear operations).

We assume that the signals e i contain an unmodelled short acceleration perturbation p u,i (for instance the response of the sensors to a micrometeorite hitting the satellite) which leads to voltages above the saturation threshold. We also assume that the perturbations p u,i are simultaneous.

Then the voltage e i will represent a measurement of p u,i , which, to simplify, is assumed to be proportional to the true acceleration, such that e i = p u,i /G i , where G i is the sensor gain converting voltages into acceleration.

Due to the saturation threshold, ēi (t) = E for times t such that |e i (t)| ≥ E, and ēi (t) = e i (t) otherwise. We can define the mask w i (t) (which is here sampled at 1024 Hz) such that

w i (t) ≡ 0 ∀ t / |e i (t)| ≥ E, 1 otherwise.
We can then decompose the output signal ēi into two parts:

ēi (t) = w i (t) • e i (t) + (1 -w i (t)) • E. (4.11)
The first term is the masked signal, while the second term is a square-wave-like signal, alternatively equal to E or 0. Therefore, the saturation will induce a leakage effect, as studied in Sec. 4.1.3, which will spread the power of the original signal throughout the spectrum. If w i (t) is random, the second term has the effect of an additive white noise term (with a non-zero mean).

The saturated differential signal is then

ād (t) = 1 2 (w 1 (t) • a 1 (t) -w 2 (t) • a 2 (t)) + 1 2 (γ 1 -γ 2 + γ 2 w 2 (t) -γ 1 w 1 (t)) , ( 4.12)
where we defined the signal expressed in acceleration a i ≡ G i e i and the saturation level expressed in acceleration γ i ≡ G i E.

If we take into account the effect of the filter, the final differential acceleration that we actually measure is

s d (t) = 1 2 h * (w 1 • a 1 -w 2 • a 2 ) (t) + 1 2 h * (γ 1 -γ 2 + γ 2 w 2 -γ 1 w 1 ) (t), (4.13)
where h is the impulse response of the filters, and * is the convolution operator.

Strategy of the mask construction in case of saturation

At times where w 1 = 0 or w 2 = 0, the measured data is invalid and has no meaning. It is then convenient to define the mask w ≡ w 1 w 2 which is zero whenever the data is invalid. By using the fact that w 2 i = w i and by multiplying Eq. (4.12) by w(t) we get

w(t)ā d (t) = 1 2 w(t) (a 1 (t) -a 2 (t)) . (4.14)
As a result, by using the mask that is equal to the multiplication of masks indicating the saturation for the internal and external mass accelerations, we come back to the situation of a masked differential acceleration considered in the previous sections. Therefore any method dealing with missing data can take this mask into account. However, up to now we did not consider the effect of the filter. Indeed, after each saturation, there is a distortion of the signal due to the filtering process. The convergence time of the distortion depends on the cut-off frequency of the filter.

To circumvent the effect of the filter, the proposed solution is to consider a mask w i that is an extended version of w i . Each gap is then slightly lengthened so that the filter response perturbation is negligible in the data outside the gaps. Then, the final mask for the differential acceleration is w = w 1 w 2 .

We can determine the minimal duration of the mask extension necessary to reject the bias from the filter response by the following approach. First, we simulate 1024 Hz spin session time series including the acceleration noise, with random saturations. Then the data is filtered and decimated to obtain the 4 Hz time series. The information about the saturation events is available through the masks w i of the individual test-masses (sampled at 4 Hz), from which we construct the mask for the differential acceleration. Our approach is to compute the standard deviation estimated from the residuals of the least-squares regression when fitting the WEP violation signal, for various time extension. Then, we choose the mask extension that corresponds to the minimum of the estimated standard deviation. Indeed, for shorter extensions the distortions of the signal due to the filters are dominant, whereas for longer extensions the error is dominated by the loss of SNR due to the discarded data points.

The results are given in Fig. 4.6, for two values of the filter cut-off frequency. The standard deviation is estimated from the regression method that will be described in Sec. 4.3 (with Eq. (4.55)). The knowledge of the method is not needed to understand the plot. The key information is that the estimated standard deviation is proportional to the squared residuals of the regression, which can be viewed as a goodness-of-fit test. The rejection of high frequencies is greater with f c = 1 Hz than with f c = 2 Hz, but the filter response to the saturations is longer, leading to the masking of more data after each saturation. According to the figure, for the value of the cut-off frequency of the Butterworth filter which is actually implemented (2 Hz), an extension of each gap by at least 1 second is required to cancel the filtering bias.

Strategy of the mask construction in case of outliers

Some high amplitude events may bias a linear regression estimate, but may not be large enough to saturate the instrument. In this case, the corresponding data intervals should be detected and discarded.

For each mass acceleration s i , the outliers detection is done by a "sigma-clipping" procedure where we flag all data which are 3σ away from the mean calculated over the whole time series. This is done in an iterative way where we loop between the steps: 1/ estimation of the sample mean and standard deviation from the data considered as valid (discarding outliers), 2/ detection of outliers based on the previous estimation. We obtain the masks w i , which are combined to come back to the situation of Eq. (4.14). In case where no saturation is flagged, the effect of the filters generally does not matter. However, the events are often followed by a transient period, which can be removed by extension of the mask as in Sec. 4.2.2.

Development of an approximate general least-squares estimator with missing data: the KARMA method

The failure of the OLS estimator to maintain an acceptable precision level when data are missing comes from the fact that it is not optimal in terms of the variance when the noise is coloured, and even less when data are missing. We first give the expression of the theoretical optimal estimator. Then we review the literature exploring the problem of regression analysis, coloured noise and missing data. Finally we construct an estimator which approximates the optimal one when the noise covariance is unknown.

The theoretical best linear unbiased estimator

When data are missing, the model describing the remaining observed data is

y o = A o β + n o , ( 4.15)
where y o is the vector gathering the observed data only, and A o is the corresponding design matrix. Thus, if t(n 0 ), . . . , t(n No-1 ) are times where the data are observed, we have y o (j) = y(n j ) for all j ∈ 0; N o -1 , and A o (j, k) = A(n j , k) for all j and all k ∈ 0; N p -1 . Then, the OLS estimator in Eq. (3.39) can be written in the following equivalent way:

β = (A * o A o ) -1 • A * o y o . (4.16)
Formally, the observed data and their model can be defined by use of a N o ×N indicator matrix W o , which "maps" the vector of observed data from the original complete data vector y. The indicator matrix is such that:

∀ (i, j) ∈ 0; N o -1 × 0; N -1 , (4.17) W o (i, j) = 1 if j = n i , 0 otherwise.
To guarantee optimality, a weighting of the form given in Eq. (3.50) must be applied. This is equivalent to minimizing the weighted residuals

χ 2 = (y o -A o β) * Σ -1 oo (y o -A o β) , ( 4.18)
where Σ oo is the covariance of the observed data vector, defined by

Σ oo ≡ Cov [y o ] = W o ΣW * o . (4.19)
By minimizing the quantity in Eq. (4.18) with respect to β we end up with the following estimator:

βGLS = A * o Σ -1 oo A o -1 A * o Σ -1 oo y o , (4.20)
This is the generalized least-squares estimator [START_REF] Aitken | On least squares and linear combination of observations[END_REF] (GLS), and one can show that this is the best linear unbiased estimator (BLUE), where "best" means "with minimal variance". The covariance of the GLS estimator is given by

Cov βGLS = A * o Σ -1 oo A o -1 . (4.21)
The above equation is the Cramér-Rao lower bound (CRLB) for the parameter β, for a Gaussian random field of mean A o β and covariance Σ oo . Actually, since the regression and noise parameters are well separated (for instance the covariance does not depend on β, and µ does not depend on the parameters describing the covariance), Eq. (4.21) is the minimum reachable variance even if the covariance is unknown. Constructing the BLUE estimator raises two problems. First, the covariance of the observed data Σ oo is not a Toeplitz matrix, contrary to the covariance of the complete data given by Eq. (3.16). Indeed, the entries are given by:

Σ oo (i, j) = R ((n i -m j )/f s ) , ( 4.22)
whose diagonals are not constant. This matrix is not approximately diagonalisable in Fourier space, and storing the N 2 o entries of matrix Σ oo is difficult while computing its inverse is computationally demanding, requiring O(N 3 o ) operations. Second, the covariance (or equivalently the noise PSD) being generally unknown, it must be estimated from the data. Unfortunately the log-periodogram smoothing technique presented in Sec. 3.5.1.2 cannot be used since the observed data is not regularly sampled any more.

Therefore we have to construct an estimation method that: (1) is able to estimate the noise PSD when data are missing; (2) computes efficiently the matrix product Σ -1 oo A o .

Review of the literature

We saw that when fitting a sine function, the ordinary least squares or equivalently Lomb-Scargle periodogram [START_REF] Lomb | Least-squares frequency analysis of unequally spaced data[END_REF][START_REF] Scargle | Studies in astronomical time series analysis. II -Statistical aspects of spectral analysis of unevenly spaced data[END_REF], which is a formulation of the periodogram adapted to irregular sampling, fail in retrieving the optimal precision [START_REF] Zechmeister | The generalised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms[END_REF][START_REF] Pires | Gap interpolation by inpainting methods : Application to Ground and Space-based Asteroseismic data[END_REF], mainly because these approaches rely on a white noise assumption. In order to increase the precision of the fit, we saw that the noise correlation matrix must also be estimated, which, in the stationary case, amounts to estimate the noise PSD. Some methods are already implemented to extract unknown coloured spectral densities, especially in the domain of gravitational waves detection (see for example [START_REF] Röver | Modelling coloured residual noise in gravitational-wave signal processing[END_REF][START_REF] Vitale | Data series subtraction with unknown and unmodeled background noise[END_REF][START_REF] Littenberg | Bayesian approach to the detection problem in gravitational wave astronomy[END_REF][START_REF] Littenberg | Bayesline: Bayesian inference for spectral estimation of gravitational wave detector noise[END_REF]), but they do not tackle the problem of gapped time series. Some works use least-squares iterative adaptive approaches (IAA) to estimate harmonic and noise parameters iteratively [START_REF] Stoica | Missing data recovery via a nonparametric iterative adaptive approach[END_REF], but again require to store and invert the covariance matrices, hence do not overcome the computational obstacle.

The problem that we tackle covers three domains: statistics, time series analysis, and spectral analysis. Extensive literature in the field of statistics with missing data is now available. An important approach is missing data imputation, that is, estimation, or attribution of statistically consistent values to the missing data (see e.g. [START_REF] Roderick | Wiley Series in Probability and Statistics[END_REF], and more specifically [START_REF] Carpenter | [END_REF], pp. [START_REF] Eliasson | Numerical study of mode conversion between lower hybrid and whistler waves on short-scale density striations[END_REF][START_REF] Epanechnikov | Non-Parametric Estimation of a Multivariate Probability Density[END_REF]. This allows us to come back to the situation where complete data methods are usable. The general idea consists in two steps: (1) draw the missing data using their conditional distribution with respect to the observed data, (2) estimate the parameters from the reconstructed data set using standard techniques adapted to complete data sets. Then these two steps are repeated so that the final estimate is given by the average of the estimates obtained from the various independent reconstructions. This is called multiple imputation, and allows the assessment of the variability of the reconstructed data and its impact on the final result.

Another general approach is to postulate a probability distribution, and to maximize the likelihood function of the observed data with respect to the regression parameters (δ in our problem) and the noise correlation matrix Σ. The original maximum likelihood problem of observed, irregularly sampled data, is generally a complex optimization problem with a large number of parameters, requiring tremendous computational efforts. A way to indirectly perform this maximization is to use the Expectation-Minimization (EM) procedure like MAPES algorithms [START_REF] Wang | Nonparametric spectral analysis with missing data via the em algorithm[END_REF]. However, as is, their convergence may be very slow, and require large matrix inversion to impute the missing data, which is computationally expensive especially for large data samples like in the MICROSCOPE case (about 10 6 points). Later we will come back to maximum likelihood estimation (MLE) with the EM algorithm.

A different type of algorithms based on missing data imputation is the "inpainting" technique, which is based on a sparsity-prior to fill the gaps [START_REF] David | Uncertainty principles and ideal atomic decomposition[END_REF][START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (mca)[END_REF]. Their adaptation to general noise spectra has been studied [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF], and is discussed in Appendix D.

In the first place, we would like to construct a precise regression method in the presence of coloured noise and missing data, but we would like to rather focus on an approach which avoids filling the gaps. Indeed, a procedure which does not rely on artificial data is interesting, above all when scientific results are concerned.

Our technique is based on the estimation of the noise spectrum by using a high-order autoregressive (AR) model. The advantage of such a model is that it can be fitted in the time domain, thus avoiding the leakage distortions of the Fourier domain. The result of the AR fit is used to weight the data through an orthogonalization of the covariance matrix. This leads to an approximation of the best estimator in terms of the variance, which is the GLS estimator in a linear regression context. While in the literature there exist distinct methods to orthogonalize the autoregressive data on the one hand, and efficiently estimate autoregressive parameters on the other hand, the main idea in the proposed approach is to take advantage of both. This is done in an iterative procedure where the AR coefficients and the regression parameters are successively estimated conditionally on each other, avoiding the use of non-linear optimization algorithms.

The proposed approach, that we dub "Kalman-AR Model Analysis" or "KARMA" for short, is divided in three steps. The first step consists in estimating the AR parameters describing the noise. This is done by using Burg's algorithm adapted to discontinuous data [START_REF] De | The burg algorithm for segments[END_REF]. The second step is carried out via a Kalman filter algorithm based on the AR model which allows us to compute the weights, as shown by Ref. [START_REF] Richard | Maximum likelihood fitting of arma models to time series with missing observations[END_REF]. In the third step we finally compute an approximation of the GLS estimator of the regression parameters, in a way similar to maximum likelihood computation methods applied to regression models [START_REF] Kohn | Efficient estimation and prediction in time series regression models[END_REF][START_REF] Gómez | Estimation, prediction, and interpolation for nonstationary series with the kalman filter[END_REF]. These steps can be reproduced to converge to the maximum likelihood estimator (MLE) of the parameters.

Autoregressive estimation

The autoregressive model of order p

The first idea is to model the noise n by an autoregressive (AR) process z of order p. In such a model, the signal at time t i linearly depends on the data at previous times until t i -pτ s , up to a random variable:

z i + a 1 z i-1 + • • • + a p z i-p = i , (4. 23
)
where i is a zero mean Gaussian process of variance σ 2 :

i ∼ N 0, σ 2 ∀i ∈ 0, N -1 . (4.24)
If the roots of the polynomial Φ(x) = 1 -a 1 x -• • • -a p x p are outside the unit circle, an autoregressive process is stationary and its PSD can be written as a function of the AR coefficients:

S(f ) = τ s σ 2 |1 + a 1 e -2iπf τs + • • • + a p e -2iπpf τs | 2 , (4.25)
where in the literature the sampling time τ s is often normalized to 1. Therefore, according to this last equation, the AR process can be viewed as the output of an infinite impulse response filter which would take a white noise of variance σ 2 as input. Hence, estimating the parameters a k and σ 2 amounts to estimating the PSD of the output noise.

The choice of this model is motivated by the following arguments. First, the estimation is performed in the time domain, avoiding the leakage problem of the discrete Fourier transform in case on an irregular sampling. Second, the use of a parametric model consistently reduces the number of noise parameters to estimate (p instead of N ), and therefrom the computational cost. Third, choosing an AR model rather than a more general class such as autoregressive-moving average models (ARMA) allows us to easily estimate the parameters from the discontinuous data, while ARMA models usually involve computationally expensive optimization procedures, or the direct estimation of the autocovariance function which is generally not accurate when data are missing. Furthermore any moving-average model can be approximated by a high order AR model as discussed in Ref. [START_REF] Durbin | Efficient estimation of parameters in moving-average models[END_REF].

As an example, we take the PSD model of the form given by Eq. (2.63), and we compute what would be the autoregressive approximation of this PSD, for different values of the order. The AR spectra are computed theoretically using the true autocovariance obtained from the true PSD. Then we use the matrix formulation of the Yule-Walker equations [START_REF] Yule | On a method of investigating periodicities in disturbed series, with special reference to wolfer's sunspot numbers[END_REF][START_REF] Walker | On Periodicity in Series of Related Terms[END_REF], which are solved for (a i , σ 2 ) by the Levinson-Durbin recursion [START_REF] Levinson | The wiener (root mean square) error criterion in filter design and prediction[END_REF][START_REF] Durbin | The fitting of time-series models[END_REF]. The results are shown in Fig. 4.7. We see that the high frequency slope f 4 is well described by the AR model, while the f -1 low frequency variation is not well captured. However, for increasing values of p, we can approximate the original PSD with arbitrary accuracy. Indeed, for p = 10000 we see that the low frequency bias is significantly reduced.

A standard method to estimate the AR coefficients is based on the estimate of the autocorrelation function, using the Yule-Walker equations or their variants [START_REF] Friedlander | The modified yule-walker method of arma spectral estimation[END_REF]. However estimating the autocovariance function is difficult when data are missing and can lead to a large bias. It is more appropriate to use Burg's method [START_REF] Burg | Maximum entropy spectral analysis[END_REF] which directly estimates the parameters from the data, and for which there are fast implementations for regular Then we have to minimize what we call the forward and backward residuals, which are the difference between the data and the forward and backward models:

F p = N -1 n=k ẑF n -z n 2 , ( 4.28
)

B p = N -k-1 n=0 ẑB n -z n 2 . (4.29) (4.30)
Burg's algorithm is a recursive method, which consists in finding the values of the AR coefficients of order k minimizing the residuals F k and B k from the values of the coefficients of order k -1. The process is repeated from k = 1 to k = p.

This algorithm must be adapted to the problem where data are missing on a grid which is initially regular, which is done in Ref. [START_REF] De | The burg algorithm for segments[END_REF]. More details can be found in Appendix B.1, where we also give the main step of the algorithm that we implemented.

Choice of the AR order

Before starting the whole estimation process, the order of the AR model must be chosen. The choice of the order depends on the PSD of the noise affecting the measurement, and on the missing data pattern. A way to properly choose the order p is to minimize the Akaike's information criterion [START_REF] Akaike | A new look at the statistical model identification[END_REF] defined as AIC(p) = 2p -2 log (L max (p)) , (4.31) where L max is the maximized log-likelihood. In the case of an AR model, the log-likelihood can be expressed in terms of the estimate of the AR residual variance σ2 which is directly computable from the residuals of the Burg's algorithm:

AIC(p) = 2p + N o log(σ 2 ). (4.32)
Applying Burg's algorithm to the residual series ẑ = y -A β with increasing order k allows us to find the order that minimizes the AIC.

Data orthogonalization

Once the AR coefficients are estimated, the noise covariance matrix Σ oo can be approximately characterized. However, the problem is to find a device that allows to efficiently compute the weighted data Σ -1 oo y and the weighted model Σ -1 oo A o . The covariance matrix is a symmetric, positive definite matrix. Hence it admits a Cholesky decomposition

Σ oo = L o L * o , (4.33)
where L o is a lower triangular matrix. This formulation is useful since we can rewrite the GLS estimator in Eq. (4.20) by introducing the weighted data vector and the weighted design matrix:

e o = L -1 o y o , E o = L -1 o A o . (4.34)
The above equations are equivalent to performing a change of basis where the observed data would be orthonormal. Indeed, if we consider the zero-mean noise vector n, it can be viewed as belonging to a vector space with the inner product X, Y = E [XY ]. Then it is easy to verify that e o is orthonormal for this scalar product, i.e. that e o has a covariance equal to identity (the data is decorrelated in the new basis).

Actually, when the covariance is described by an AR model, the Kalman filter can be used as a device to perform the orthogonalizing operations (4.34) without computing L o . To see this, we must write the AR process as an equation of state. Then we must understand the relationship between the Kalman filter and the likelihood formalism on the one hand, and how the outputs of the Kalman filter can be used to compute the weighted residuals on the other hand.

The Kalman filter for the AR model

First, let us write the AR process in the Kalman filtering formalism, by rewriting and detailing the works of Jones (1980) [START_REF] Richard | Maximum likelihood fitting of arma models to time series with missing observations[END_REF] and Gomez & Maravall (1994) [START_REF] Gómez | Estimation, prediction, and interpolation for nonstationary series with the kalman filter[END_REF]. The Kalman filter is an estimator of the dynamical states of a system. Assume that we have some measurement z of a process, which depends on some state variable x. The relationship between the observations and the state variable is described by a design matrix H. The typical application of the Kalman filter is the case of radar tracking, where the purpose is to estimate in real-time the position and velocity of a target from noisy radar measurements. The Kalman filter allows the prediction of the state at a given time based on the observations recorded at the previous times, and the modeling of the measurement uncertainty, as well as the model uncertainty. In the case of our autoregressive problem, the measured time series of the residuals z = y -Aβ plays the role of the observations, and is assumed to verify Eq. 4.23.

The state equation (describing how the state vector evolves in time) and the observation equation (describing how the state vector relates to observations) of the Kalman filtering can be written as

x(n) = Fx(n -1) + G (n), (4.35
)

z n = H T x(n) + v. (4.36)
If z is an AR process, then the vectors and matrices involved in these equations all depend on the AR coefficients, and are defined as follows. The state vector is

x(n) = z n z n+1|n . . . z n+p-1|n T , ( 4.37)
where z n+k|n is the conditional variable, interpreted as "z at time t n+k given z at times t s ≤ t n ". H is the matrix relating the state vector to the observations:

H = 1 0 . . . 0 T . ( 4

.38)

F is the matrix relating the current state to the previous state:

F =         0 1 0 . . . 0 0 0 1 . . . 0 . . . . . . . . . . . . . . . 0 0 0 . . . 1 -a p -a p-1 -a p-2 . . . -a 1         . (4.39)
For an AR process, G = 1 g 1 . . . g p-1

T is a p × 1 vector defining the noise on the state vector. Its exact expression as a function of the AR coefficients is given in Appendix B.2, and the full Kalman filter equations can be found in Appendix B.3.

The Kalman filter and the likelihood function

Now we can highlight the relationship between the Kalman filter and the likelihood function.

The key concept is conditional probability. As we saw previously, the Kalman filter aims at computing, for each time t n , an estimation of the state vector (and thus, of the true data) knowing all the data observed until time t n-1 . In other words, we want to calculate:

ẑn|n-1 ≡ E [z n |z 1 , z 2 , . . . , z n-1 ] , σ2 (n|n -1) ≡ Var [z n |z 1 , z 2 , . . . , z n-1 ] ≡ E z n -ẑn|n-1 2 .
To make these quantities show up in the likelihood function, we will decompose it by using the basic relation between joint probability and conditional probability:

P (A|B) = P (A, B) P (B) . (4.40)
By this simple relation, we can factorize the joint probability of all observations as

P (z 0 , . . . , z N -1 ) = P (z 0 ) N -1 n=1 P (z n |z 0 , . . . , z n-1 ) , ( 4.41)
Since we deal with a Gaussian random field, all the conditional probabilities can be written as

P (z n |z 0 , . . . , z n-1 ) = 1 2πσ 2 (n|n -1) exp - (z n -ẑn|n-1 ) 2 2σ 2 (n|n -1) . ( 4

.42)

Given that we have observations only for n ≥ 0 the expectation and the variance of z 0 given the previous times reduce to the expectation and the variance of z 0 . For a zero-mean AR process, these initial conditions write:

ẑ0|-1 = E [z 0 ] = 0, (4.43) σ2 (0| -1) = Var [z 0 ] = σ 2 . (4.44)
Therefore we can rewrite the likelihood L z (θ) ≡ P θ (z) as a function of the outputs ẑn|n-1 and σ2 (n|n -1) of the Kalman filter:

L z (θ) = 1 N -1 n=0 2πσ 2 (n|n -1) exp - N -1 n=0 (z n -ẑn|n-1 ) 2 2σ 2 (n|n -1) . ( 4

The Kalman filter as an orthogonalization process

Now we would like to use the outputs of the Kalman filter to perform a least-squares estimation. We saw in Sec. 4.3.4.2 that the Kalman filter can be used to compute the likelihood of the AR model. Besides, in Eq. (3.83) of Sec. 3.5.3.1 we gave the expression of the likelihood for a Gaussian random field. Maximizing this likelihood with respect to the regression parameter β yields the GLS estimator of Eq. 4.20. If we apply the Kalman filter to the vector of residuals z, we compute the quantities ẑn|n-1 and σ2 (n|n -1). Now let us compare the equation of the Gaussian likelihood (with z = yµ),

L z (θ) = 1 (2π) N |Σ| exp - 1 2 z * Σ -1 z , (4.46)
with the equation of the Kalman likelihood (4.45) restricted to observed data. Then by identification we have the following equalities:

|Σ| = N -1 n=0 σ2 (n|n -1) (4.47) z * Σ -1 z = N -1 n=0 z n -ẑn|n-1 2 σ2 n|n-1 (4.48)
By introducing the Cholesky decomposition L o of the covariance of z o , we can rewrite the second equation as

(L -1 z) * (L -1 z) = N -1 j=0 (z n -ẑn|n-1 ) 2 σ2 n|n-1 . (4.49)
Therefore, if we define the vector of weighted residuals as

e ≡ L -1 z, (4.50)
then we can identify

e n = z n -ẑn|n-1 σ(n|n -1) . ( 4

.51)

This shows that the Kalman filter allows us to compute the vector of residuals weighted by their variance. Therefore, for a set of given AR parameters, we are able to compute the predicted residuals of any time series. In other words, for any vector y, the Kalman filter is a device to compute L -1 y without computing L nor its inverse. When data are missing, all the sums at the right-hand side of Eqs. (4.46) to (4.49) only involve the observed data, and we have a similar identification by defining the vector of observed weighted residuals as

e o ≡ L -1 o z o ,
e o,j = z n j -ẑn j |n j -1 σ(n j |n j -1) . (4.52)

Least-squares estimation with Kalman outputs

Linear regression

Now we go back to the regression problem for the acceleration time series y. We assume that the noise is characterized by an AR process whose parameters are perfectly known.

If we apply the Kalman algorithm to y, we can compute L -1 y. In addition, as mentioned by Kohn & Ansley [START_REF] Kohn | Efficient estimation and prediction in time series regression models[END_REF], by applying the same algorithm to each column of the design matrix A, we are able to compute L -1 A. Moreover, by replacing Σ oo by its Cholesky decomposition in Eq. (4.20), the solution of the generalized least-squares estimator can be written

β = (E * o E o ) -1 • E * o e o . ( 4

.53)

Therefore we showed how to compute an approximation of the GLS estimator by using a Kalman filter based on an AR model of the noise. Note that the variance of the constructed estimator is only close to minimal. Indeed, it relies on an estimation of the noise covariance from the data, and on an approximation of the noise by an AR process. As a result, there will be a residual correlation between the entries of the vector e o , due to the errors of the estimated noise model.

Uncertainty estimation and detection issues

This is of key interest to be able to assess the statistical uncertainty of a given estimation, especially in a context where the experiment cannot be reproduced a large number of times. In this section we present a tool to quantify the uncertainty of the regression result and to give a confidence threshold for the detection of the signal of interest. To achieve this goal, the estimator variance matrix must be estimated.

The covariance matrix can be evaluated under the assumption that the AR model is a good approximation of the real noise correlations. This hypothesis is equivalent to assuming that the constructed estimator β is exactly the one defined by Eq. (4.53). By denoting C β its covariance, we can write

Ĉ β ≈ σ 2 0 (E o * E o ) -1 , ( 4.54)
where σ 0 accounts for the fact that the covariance might be known up to a proportionality constant. For an unbiased estimator (i.e. the design matrix A o describes all the deterministic components of the signal) this can be estimated by

σ2 0 = ê * z êz N o -N p , (4.55)
where êz is the estimation of the vector of weighted residuals defined by êz ≡ e o -E o β, and N p is the number of estimated parameters. Similarly to Eq. (3.24) in Sec. 3.1.3.2, the statistic to be considered is

Z k ≡ βk Ĉk,k , (4.56)
where k is the index corresponding to the parameter of interest in the vector β. For our application β k is the EP parameter δ d . Then under the assumption that there is no violation signal (hypothesis H 0 ), Z approximately follows a Normal law with mean zero and unit variance. Typically hypothesis H 0 is rejected with a 99% confidence if |Z k | > 2.56.

Summary of the method

We summarize below the strategy to perform an almost-optimal regression, in the case of a Gaussian, stationary and coloured random noise with missing data: 0. Initialization: compute a first crude approximation of β using the OLS estimator, to get βOLS , and apply Burg's algorithm to the estimated residuals ẑ = y -A βOLS to determine the optimal order p of the AR process approximating the noise spectrum; Since at the beginning β may be poorly estimated by OLS, one or several iterations of the last three steps might be necessary for convergence of both AR and regression parameters. The method summarized above is dubbed KARMA for "Kalman AutoRegressive Model Analysis". Its computational complexity scales as O N p N p 2 , where N p is the size of β.

Numerical simulations

To assess its performance, we apply the KARMA method to simulated time series of the differential acceleration. Similarly to Chap. 3, the noise is generated from the PSD model of Eq. (2.63). To consider the worst case scenario, namely that the data loss process occurs before any filtering, we consider the unfiltered noise PSD. We saw in Fig. 4.3a that filtering after masking does not change the noise level at low frequency, so we do not consider the filtering process.

The deterministic part of the signal is simulated from the simula simulator developed at Observatoire de la Côte d'Azur (OCA). First, the attitude of the satellite is computed by a pre-routine called attitu, from a given orbit ephemeris (see Table 4.2 for more details). Then the gravitational acceleration and the gradient tensor are computed in the instrument reference frame, from another code called actens, based on a geopotential model given as input. Finally, the simula code simulates the accelerations measured by the test-masses from a given parametrization of the instrument defects and from the actens outputs. The differential acceleration is then computed.

The simulated time series is sampled at f s = 4 Hz and lasts 20 orbits. This corresponds to a spin session, for which the orbital frequency is equal to 1.7 × 10 -4 Hz and the spin frequency is 7.7×10 -4 Hz. The EP frequency is then equal to the sum f EP = 9.4×10 -4 Hz. To simplify the simulation, we only simulate the disturbances due to the gravity and inertia gradient terms arising because of the slight off-centring of the test-masses along X and Z axis (see Eq. (2.79)). All other disturbances are assumed to be perfectly corrected by previous calibration. In the simulation we assume that the test-masses are off-centred by 20 microns along the X and Z axis which are in the orbital plane. The EP parameter is simulated at a level of 3 × 10 the true acceleration signal g x (t), to which we add the two perturbations modelled with our knowledge of gravity and inertia gradients.

The measurement equation of this simplified simulation is represented by

Γ meas,dx = -a c11 δ d g x -a c11 ∆ dx S xx -(a c11 ∆ dz + a c13 ∆ dx ) S xz + n dx . (4.57)
To simplify further, we assume that a c11 ≈ 1 and a c13 ≈ 0 which gives

Γ meas,dx ≈ -δ d g x -∆ dx S xx -∆ dz S xz + n dx . (4.58)
Thus in this case there are three regression parameters.

The amplitude spectra of each component of the measurement equation are shown in Fig. 4.8. The black spectrum represents the simulated WEP violation, with a peak at f EP = 9.4×10 -4 Hz. The harmonic peaks of highest amplitudes visible around 2f EP = 1.87×10 -3 Hz belong to the gradient components (solid red and dashed blue lines), which oscillate almost at the same frequency, but are in phase quadrature (and hence can be distinguished in the least-squares fit). The other faint gradient harmonic visible at 2f orb = 3.40 × 10 -4 Hz is due to the influence of the second zonal harmonic J 2 of the geopotential.

We consider the same three gap patterns as in Sec. 4.1.3: short and numerous random gaps (tank cracking type), longer and fewer gaps (telemetry type) and periodic gaps. To give insight into the time series and the first two gap patterns, we plot an excerpt of the vector y in Fig. 4.9.

We also plot in Fig. 4.10 the periodogram of the complete generated noise vector (black), and the periodogram of the masked data with mask w a (gray). The simulation confirms the leakage effect from the high to low frequencies that was theoretically derived in Fig. 4.3.

In the following, we present the results of the KARMA method. They are compared to the ordinary least-squares estimate with missing data. We also compare the results to the reference given by the OLS estimator in the case without gaps, and to the minimal possible uncertainty.

PSD estimate from the AR fit

Before starting the whole process, the order of the AR model must be chosen at step 1.

The choice of the order depends on the PSD of the noise affecting the measurement, and on the gap pattern. We use the AIC criterion introduced in Sec. 4.3.3.2. As a example, we plot in Fig. 4.11 the estimation of the AIC obtained from the Burg's algorithm applied with the first mask w a . For this mask, the longest uninterrupted data segment lasts about 3 minutes, corresponding to a maximal order of about 700. Since the noise does not correspond to a particular autoregressive process of fixed order, the AIC is a monotonic function decreasing towards an asymptote, and has minimum at infinity. From a certain value, there is no significant improvement with increasing order. Therefore the optimal order can be chosen as the one for which the AIC is sufficiently close to the asymptote, for instance p ≥ 60 according to the figure. Besides, we could not pick a too high order, because increasing p increases the variance of the AR coefficients I N,W z ) with the tank cracking mask w a . The mask induces a power leakage which tends to relocate the noise power on the whole spectrum, transferring the spectral power from regions where it is high into regions where it is lower. We superimposed the theoretical expectations of the periodograms for the complete (black solid line) and masked (black dashed line) data. Even if the complete data expectation appears slightly higher than the noise, in average for the rectangular window the periodogram converges towards this curve. estimates. This is due to the decrease of the number of usable data segments (with length larger than p). Some authors suggest to modify the AIC (see [START_REF] Bos | Autoregressive spectral estimation by application of the burg algorithm to irregularly sampled data[END_REF]) by introducing a penalty accounting for the increasing estimation variance. For instance, we can replace N o by p p i=1

1 N i -1
in Eq. (4.32), where N i is the number of usable segments to estimate the coefficient a i . When applying this criterion to our simulation with window w a , we find an optimal order of p = 60. However, for the other gap patterns, the modified AIC leads to an underestimation of the optimal AR order, in the sense that the chosen value is too low to properly fit the high frequency slope of the spectrum. Therefore we choose to use the value of p above which there is no significant improvement of the standard AIC.

By using Eq. 4.12 with the AR coefficients computed from the Burg's algorithm applied to the masked residuals, we obtain the spectra of Fig. 4.12. The results of the AR estimation are almost the same as for the complete data case, regardless of the mask.

In all cases, while most of the PSD is well captured by the AR model, it is biased at low frequency. This is due to the limited number of AR coefficients which does not describe the long-range correlations, making the PSD estimate appear flat around zero. We will see that this is not of major importance concerning the result of the regression.

The algorithm is implemented in python language on a typical 2 GHz computer.

For p = 60, N ∼ 5 • 10 5 and N p = 3, each iteration takes less than 10 minutes. For the simulations that we study, 2 iterations are necessary to reach convergence.

Kalman filtering

Once the AR order is chosen and a first estimation of the AR coefficients is done, the decorrelation process can be performed. To verify the behaviour of the Kalman filter, we plot the predicted states ẑn|n-1 (prediction of the data at time n, given the data observed before that time) in Fig. 4.13 (blue curve) along with the "confidence interval" (light blue) defined by the predicted state covariance ẑn|n-1 ± 3σ n|n-1 . The prediction generally tends to follow the real data, which is expected. However, the variance of ẑn|n-1 is lower than the variance of the noise itself, which is normal since the AR process underestimates the low frequencies of the spectrum. We also remark that the confidence interval is broadened at times where the data is missing, and shortly after the gap, which is also expected. Furthermore, we check that the Kalman filter acts as a Gram-Schmidt procedure for the data (that is, orthogonalises the data in terms of the covariance). To do that, we must test the process on a complete time series, to be able to visualize the colour of the noise on a spectrum computed with regularly sampled data. Therefore we apply Burg's algorithm to the complete noise time series z, and then run the Kalman filter. We plot the periodogram of the weighted residuals computed with Eq. 4.51 in Fig. 4.14. We see that the noise is "whitened", reflecting the fact that the filtering process approximately removes the correlations in the time domain. A residual coloration remains visible at low frequency, due to the bias of the AR model in this region.

Linear regression

The results of the linear regression are summarized in Table 4.3, with p = 60. In order to test the precision of our method, we have drawn 400 realizations of the noise and run the KARMA algorithm for each of them, as well as the OLS estimator. The number of draws is chosen such that the error on the true value of the standard deviation of the EP parameter does not exceed 10 -16 with a 99% confidence. The third column of the table indicates the true value of the parameters. Note that we give the values of the "physical" parameters, rather than the one involved in the formal measurement equation:

δ = -2δ d , ∆ i = -2∆ di .
The fourth column indicates the Cramér-Rao lower bound (CRLB), which is the minimum achievable uncertainty, also referred to as the "optimal variance". It is computed with the exact expression of Eq. (4.21) by using the true covariance matrix derived from the true noise PSD S(f ). Note that this theoretical calculation requires the inversion of Σ oo . We will deal with this computational issue later in Sec. 4.4.4.

Columns 5 to 7 show the performance of the OLS estimator: the sample average μ400 , the theoretical standard deviation σ OLS given by Eq. (4.4) and calculated with the true PSD, and the sample standard deviation of the 400 estimates. The last three columns show the results obtained with the KARMA method, and are detailed below.

The sample average μ400 (eighth column of table 4.3) of the estimates obtained with the KARMA method converges to the true value of the parameters, showing that the constructed estimator is unbiased. We also calculate the sample standard deviation of the WEP parameter. For short and numerous gaps (tank cracking window) we find σ = 1.1 × 10 -15 with our method instead of 6.5 × 10 -14 with the OLS estimator. Thus our method enables us to divide the stochastic error by a factor 60 with respect to the OLS. Furthermore, we checked that the choice of an AR order larger than the one found with the AIC has no major impact on the results. With p = 200 for instance, we find the same standard deviation within the precision of 10 -16 .

For fewer and longer gaps, we find σ = 9.8 × 10 -16 instead of 5.1 × 10 -15 with the OLS for the telemetry gaps, and σ = 9.7 × 10 -16 instead of 4.1 × 10 -15 for the periodic gaps. We notice that such gap patterns have less impact on the estimation performance, because they lead to a lower frequency leakage.

By comparing these results to the CRLB, we see that for all gap patterns, the uncertainty of the KARMA estimates is close to the optimal variance. The detection test is positive with a confidence greater than 99% in all cases. The improvement is also significant Table 4.3: Mean and standard deviations on the estimation of the parameters of interest using OLS and the KARMA method. In both cases we present (from left to right) the estimation average calculated on a sample of 400 estimates, the analytical standard deviation, and the sample standard deviation. For OLS, the analytical uncertainty σ OLS is given by Eq. (4.4), which is exact. For the KARMA method, σAR is the average of the uncertainties estimated for each draw with Eq. (4.54). These estimates are compared to the true value of the parameters (second column) and the CRLB (third column), which is the best achievable precision. For each draw, we estimate the uncertainty σAR using the approximate formula (4.54). We then calculate the sample average of this estimate over the 400 draws, and record the results in the table. We find 1.2 × 10 -15 for mask w a , 9.3 × 10 -16 for mask w b and 9.5 × 10 -16 for mask w c . This is close to the calculated sample standard deviation, meaning that when having only one realization at hand, one can estimate the error with an acceptable accuracy. The estimated error does not vary much from one estimation to another, and stays within an interval of ±10 -16 around the mean.

However, the estimate σAR of the real regression error may be biased, depending on the frequency of the estimated signal. This can be explained by the bias of the AR PSD estimate that we observe at low frequency in Fig. 4.12. The lower the signal frequency, the larger the bias on the estimated variance. But the good recovery of the high frequency spectrum is enough to cancel the leakage due to the mask and get a precision of 1 × 10 -15 for the EP estimation, in agreement with the mission requirement.

Reconstruction of the missing data and refinement of the PSD estimate

In the previous section we constructed an estimator with a quasi-optimal variance, based on an autoregressive estimate of the noise, without imputing the missing data. We saw that while the results of the linear regression (i.e., the estimation of the deterministic part of the data) are satisfying, the algorithm does not allow an accurate estimation of the noise PSD at low frequency, which is mainly due to the AR model. The purpose of this section is to develop a method to estimate the PSD on the whole spectrum when data are missing, with a less constrained modelling of the noise correlations. The aim is to be able to directly estimate the PSD itself, along with the regression parameters. The driving idea is to estimate the data inside the gaps to extract the parameters of interest from the data, and to have data series easier to handle.

Review of the literature

A general method to solve the Gaussian regression problem is the maximum likelihood estimation (MLE). When data are missing the direct maximization of the likelihood of the observed data is an optimization problem for which neither the solution nor the gradient of the cost function have an explicit form [START_REF] Szatrowski | Explicit solutions, one iteration convergence and averaging in the multivariate normal estimation problem for patterned means and covariances[END_REF]. This can be circumvented by using iterative procedures where each step increases the likelihood and is convenient to calculate.

The MLE problem can be solved by use of the expectation-maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF] which indirectly maximizes the likelihood of the observed data by computing the expectation of the likelihood of the complete data conditionally on the observed data. This involves the reconstruction of the missing data allowing the use of techniques adapted to regularly sampled data. However each iteration of the EM algorithm still requires O(N 3 o ) operations in its original formulation. Wang et al. [START_REF] Wang | Nonparametric spectral analysis with missing data via the em algorithm[END_REF] proposed an EM algorithm where the model is composed by harmonic functions and a given noise, whose covariance matrix is smoothed by partitioning the data vector y in L overlapping segments of size M . This procedure requires O(LM 3 ) operations at each iteration which can still be demanding if a high frequency resolution is required, and relies on the assumption that the sub-vectors are independent, which is not the case in general.

Storage and inversion of the observed data covariance matrix is a common issue in particular in the field of geostatistics where large spatial data sets may be considered. Several approaches are proposed, consisting in approximating the covariance matrix by a matrix close to it but easily invertible. Two strategies are usually adopted. The first one is covariance tapering (see e.g. [START_REF] Furrer | Covariance tapering for interpolation of large spatial datasets[END_REF]), where the autocovariance function is multiplied by a taper function which vanishes to zero after a certain number of points q. This introduces sparsity in the covariance matrix. The number of entries to be stored is reduced to O(q 2 ) instead of O(N 2 o ) and the complexity of solving for linear equations involving Σ is only linear in N o , instead of being cubic. Another method is to perform a low-rank decomposition of the matrix, reducing the problem to r 2 parameters with r N o (see e.g. [START_REF] Cressie | Fixed rank kriging for very large spatial data sets[END_REF]). The inversion can be performed with a complexity proportional to O(r 2 N o ).

Nevertheless these two methods suffer from drawbacks: covariance tapering captures the short range correlations only, whereas low-rank decomposition usually describes larger scales. In all cases, including the combination of both methods [START_REF] Sang | A full scale approximation of covariance functions for large spatial data sets[END_REF], this approach results in approximations, and it showed not to be efficient for the spectral estimation problem relevant for MICROSCOPE.

More recently fast methods for solving linear systems based on circulant embedding of Toeplitz matrices have been proposed [START_REF] Michael | The role of likelihood and entropy in incomplete-data problems: Applications to estimating point-process intensities and toeplitz constrained covariances[END_REF][START_REF] Fritz | Application of FFT-based Algorithms for Large-Scale Universal Kriging Problems[END_REF] enabling the exact resolution of MLE problems on incomplete data [START_REF] Stroud | Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice[END_REF] with a reasonable computational complexity. But the authors assume that the autocovariance function -or equivalently the PSD -is well known or has a known form.

In the following we tackle the problem of maximum likelihood estimation on time series with missing data and unknown, arbitrary power spectral density. The rationale relies on the assumption that the PSD is continuous and smooth, and that the complete data likelihood can be written in the circulant approximation. The procedure is based on a derivative of the EM algorithm called ECM [START_REF] Meng | Maximum likelihood estimation via the ECM algorithm: A general framework[END_REF], which is modified to introduce smoothing of the spectrum. This is done by a local formulation of the likelihood similarly to a formalism developed by Fan and Kreutzberger [START_REF] Fan | Automatic local smoothing for spectral density estimation[END_REF] for spectral density estimation. This ECM-like algorithm is tailored to the estimation of β and S(f ) through the imputation of missing data, while the KARMA method is used in the first place to find good starting values for the ECM algorithm. We then test our approach with numerical simulations.

As mentioned earlier, an alternate gap-filling approach is adopted by Bergé et al. [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF] and Pires et al. [START_REF] Pires | Dealing with missing data II: An updated version of the inpainting method to the MICROSCOPE space mission[END_REF] where they estimate the missing values using a sparsity-prior with an algorithm commonly named inpainting [START_REF] David | Uncertainty principles and ideal atomic decomposition[END_REF][START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (mca)[END_REF], and then apply a standard least-squares regression on the reconstructed data. The PSD can also be estimated from the reconstruction. This technique is detailed in Appendix D.1. While inpainting is a way to fill the gaps prior to any regression method, it does not treat the Gaussian regression problem with missing data as a unified framework.

Data reconstruction using conditional expectations

We saw in Sec. 4.3 that it is possible to estimate the regression parameters without filling the data gaps. However, performing missing data estimation (also called "data imputation") can be useful for two reasons. First, this is a good way to estimate the noise PSD more accurately. Indeed, while the accuracy of the estimated noise PSD Ŝ0 is sufficient to de-correlate the data and perform a precise regression for β, it may show a bias for certain shapes of S, especially at low frequencies. Secondly, equally spaced data sets are more easily considered for science purpose. The inpainting method is a way to perform data imputation and allows for a posteriori PSD estimation. However in this section we deal with a different approach called Gaussian interpolation which, for any set of estimates β and Ŝ, allows us to estimate the missing data via their approximate conditional expectation.

Description of the data reconstruction process

The indicator matrices introduced in Eq. (4.18) provide a convenient way to define the covariances of vectors y o and y m with themselves and with each other:

Σ oo ≡ W o ΣW * o , Σ mm ≡ W m ΣW * m , Σ mo ≡ W m ΣW * o . (4.59)
By further assuming a Gaussian distribution of the noise, the optimal estimator of the missing data vector is its conditional expectation given the observed data vector:

µ m|o = E [y m |y o , β, S] = µ m + Σ mo Σ -1 oo (y o -µ o ) , (4.60)
where the expectations of the missing and observed vectors are given by the regression model µ o = A o β and µ m = A m β. In our application the first term in Eq. (4.60)

represents the deterministic part of the reconstruction, whereas the second term accounts for the stochastic noise statistics.

If m = y m -µ m|o is the residual of the missing data estimation then the mean squared prediction error (MSE) is (see proof in Appendix C.1)

E [ m * m ] = Σ mm -Σ mo Σ -1 oo Σ * mo = Σ m|o , ( 4.61)
where the last equality means that the MSE is equal to the conditional covariance Σ m|o of y m given y o .

In our study however, both Σ and β are assumed to be unknown. They must therefore be determined beforehand from the observed data. For instance, one can use the estimates provided by the KARMA method. We can then replace the expectations in Eq. (4.60) by the estimated ones, that is μo = A o β and μm = A m β.

As for the covariances involved in Eq. (4.60), they are replaced by their estimate Σ, derived from the estimate of the PSD Ŝ = Ŝ0 and by using Eqs. (3.20) and (4.59).

The uncertainty introduced by the estimation of β leads to an additional error term in the reconstruction [START_REF] Harville | Decomposition of prediction error[END_REF]:

E [ m * m ] = Σ m|o + K m Cov β K * m , (4.62)
with

K m ≡ A m -Σ mo Σ -1 oo A o .
The uncertainty of the spectrum estimate Ŝ also affects the reconstruction error. However the full derivation of the corresponding error term is beyond the scope of this study. See for instance [START_REF] Harville | Accounting for the estimation of variances and covariances in prediction under a general linear model: an overview[END_REF] for a discussion on this aspect.

The preconditioned conjugate gradient method

As pointed out in section 4.3.1, the covariance matrix of the observed data Σ oo looses its useful Toeplitz properties in the presence of missing data, and cannot be inverted exactly and efficiently (see for instance [START_REF] Ammar | Classical foundations of algorithms for solving positive definite toeplitz equations[END_REF][START_REF] Martinsson | A fast algorithm for the inversion of general toeplitz matrices[END_REF] for superfast Toeplitz inversion algorithms). For time series with 10 6 data points, it is not feasible to store and invert the N o × N o covariance matrix.

In the analysis of stationary time series that are originally regularly sampled, the covariance matrix of the observed data is only defined by two vectors of size N : the window w and the PSD S. The product of this matrix by any vector can be calculated using FFT algorithms and element-wise vector multiplications, taking advantages of the covariance form described by Eqs. (3.20) and (4.59). As a result, the linear system Σ oo x = y involved in the estimation of the missing data in Eq. (4.60) can be efficiently solved by iterative algorithms decreasing the residuals r l = y -Σ oo x l at each iteration l leading to an approximate (and sometimes exact) solution. For example Fritz et al. [START_REF] Fritz | Application of FFT-based Algorithms for Large-Scale Universal Kriging Problems[END_REF] suggest to use the conjugate gradient algorithm [START_REF] Hestenes | Methods of Conjugate Gradients for Solving Linear Systems[END_REF]. However this process may be slow, and the use of a preconditioner matrix M is required to reduce the condition number of the linear system, which amounts to solving M -1 Σ oo x = M -1 y.

While Fritz et al. [START_REF] Fritz | Application of FFT-based Algorithms for Large-Scale Universal Kriging Problems[END_REF] use the regularized circulant preconditioner introduced by Nowak [START_REF] Nowak | Geostatistical methods for the identification of flow and transport parameters in the subsurface[END_REF], we choose a tapered covariance as the preconditioner matrix, in order for M to have the same non-Toeplitz structure as Σ oo . To construct M , the correlation is ignored after a certain lag τ 0 by mutiplying the original autocovariance by a taper function K which smoothly goes down to zero at lag τ 0 :

R taper (τ ) = R(τ )K(τ, τ 0 ). (4.63)
The resulting covariance matrix is sparse which allows us to store it and solve the corresponding linear system with an acceptable number of operations. In this study we choose the Wendland 2 taper function [START_REF] Wendland | Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree[END_REF] for K. The taper lag q is chosen equal to the order of the AR model p which is used as initial guess of the PSD estimate, since we saw that this range represents the major noise correlations. As an illustration we plot in Fig. 4.15 the two functions involved in the tapered covariance of Eq. (4.63) until lag p. The preconditioning step adds two operations which are linear in N o at each iteration of the conjugate gradient algorithm. The resulting reconstruction process has a complexity in O(N it N log N ), where N it is the number of iterations needed by the conjugate gradient to reach convergence.

Refinement of regression parameters and noise spectrum estimation through a modified ECM algorithm

In the previous section we showed how to infer the missing data from given estimates of the regression vector β and of the PSD Ŝ. But it is possible to re-estimate these quantities from the reconstructed data, and to iterate between imputation and estimation steps. The objective of such an iterative process is two-fold: on the one hand to improve the estimation of the noise PSD, and on the other hand to obtain a consistent reconstruction of the missing data. These iterations are implemented via a modified ECM algorithm which starts from the initials guesses β0 and Ŝ0 provided by the KARMA method and perform several reconstruction/estimation steps to end up with a converged set of estimates ŷm , β and Ŝ.

Approximate likelihood for complete data

We first consider the case where all data are available. We saw that the covariance of a stationary random process can be approximated by a circulant matrix for large N (which corresponds to P = N and Σ = C in Eqs. (3.20)). Under this assumption, the logarithm of the Gaussian likelihood in Eq. (3.83) can be written as

l y (θ) ≈ - 1 2 N -1 k=0 log Λ k + |z| 2 Λ k , ( 4.64)
where we dropped the constant term and we introduced the vector of residuals z ≡ y -Aβ and the spectrum Λ k = f s S(f k ). This approximation is known as Whittle's likelihood [START_REF] Whittle | Hypothesis testing in time series analysis[END_REF]. It is worth noting that asymptotically, Whittle's likelihood does not rely on the Gaussian hypothesis, but is valid for sufficiently large N , and corresponds to the χ 2 distribution asymptotically followed by the periodogram.

A direct maximization Eq. (4.64) needs high computational resources. Instead, it is easier to iteratively increase l(θ) using several sub-steps. In the Gaussian regression problem two sub-steps are necessary and correspond to conditional and successive updates of the vector β and of the spectrum Λ. These two steps are labelled CM1 and CM2. The value of the parameters at iteration i + 1 are computed from the values of iteration i by • CM1 step:

β i+1 = Ã * Λ i -1 Ã -1 Ã * Λ i -1 ỹi , (4.65)
• CM2 step:

Λ i+1 k = ỹi k -Ãβ i+1 k 2 . ( 4.66)
Besides, the particularity of our problem is that we do not have several realizations of y which we can average (this is usually called a uni-variate problem). The PSD estimate Λ as written in Eq. (4.66) will therefore have a large variance. In order to reduce the variance, the CM2 step must be modified by introducing smoothing. We already constructed a smoothing PSD estimate in Sec. 3.5.1.3. We need to include it in the maximum likelihood framework. We saw that the ideal is to assume that locally, the spectrum can be written as a linear function:

log Λ k ≈ a j + b j (f k -f j ), (4.67)
where a j and b j are coefficients to be estimated for each frequency f j . We saw that the extent of the neighbourhood of f j is determined by a kernel K and a specific bandwidth h, which can depend on f j . In the framework of maximum likelihood estimation this can be expressed as a local formulation of the likelihood which we denote l y j . For a frequency f j of interest we rewrite Eq. (4.64) by using the linear model of Eq. (4.67) and by applying a weight for each frequency:

L j y (a, b) = - 1 2 n k=1 a + b(f k -f j ) + I N,z (k)e -a-b(f k -f j ) K h j (f k -f j ), (4.68)
where we have restricted the summation to the positive frequencies, k running from 0 to n = (N -1)/2 .

Treatment of missing data and minimization of the measurement uncertainty

The maximization of the local likelihood can be done with the Newton-Raphson algorithm [START_REF] Whiteside | The mathematical papers of Isaac Newton[END_REF]. We define the vector X ≡ a b T , and we note L j y the gradient vector and L j y the Hessian matrix of L j y (a, b). If we start from some initial estimate X0 , this algorithm consists in iterating the equation

X1 = X0 -L j y X0 -1 L j y X0 . (4.69)
Actually, the solution of this maximization is exactly the same as the local maximum likelihood estimator proposed by Ref. [START_REF] Fan | Automatic local smoothing for spectral density estimation[END_REF], because the gradient vector is the same (even is the likelihood functions slightly differ). They show that is we take the local linear estimate of Sec. 3.5.1.3 as initial value for X, then the function l j y is nearly maximized by applying Eq. (4.69) once.

Up to now we have written the local log-likelihood in the case of complete data. However in the ECM algorithm the update of the spectrum estimate is performed conditionally on the observed data. This is easily done by taking the conditional expectation of Eq. (4.68) with respect to y o , and then maximizing it with respect to (a, b). It follows that the maximization of the likelihood at iteration i involves the conditional expectation of the periodogram I i z = E I z |y o , θ i . We detail its computation below.

Conditional expectation of the periodogram

Given the definition of the periodogram in Eq. (3.56), its conditional expectation involves cross-products of the data vector y, hence requiring the calculation of the conditional covariance of y given y o . The formal derivation of the conditional expectation of the periodogram is performed in Appendix C.2, and can be written as

E [I z (k)|y o , θ] = I ẑ (k) + σ 2 k . ( 4.70)
The first term of this equation is the periodogram of the reconstructed residual vector ẑ whose entries are equal to z n when the data is observed at time t n , and equal to the conditional expectation of z n given y o and θ i when the data is missing at

t n : ẑ = W * o z o + W * m µ m|o
, where µ m|o is given by Eq. (4.60). The second term accounts for the conditional second order moments. The quantities σ 2 k are given by the entries of the vector:

σ 2 ≡ diag F N W * m Σ m|o W m F * N
, where Σ m|o is the conditional covariance in Eq. (4.61). For large numbers of missing data points, the direct calculation of this term can be computationally demanding, since it involves the matrix product Σ -1 oo Σ * mo , whose complexity is of order O (N it N m N o log N o ) using the conjugate gradient algorithm of Sect. 4.4.4.

To avoid these computational issues σ 2 can be approximated with Monte-Carlo simulations of the vector ẑ as described in Appendix C.3. The idea is to generate several realizations of the reconstructed vector ẑ, to calculate their periodograms, and to compute their sample average. The complexity can then be reduced to O (N it N d N o log N o ) where N d is the number of Monte-Carlo draws. This alternative is obviously of interest when N d < N m . We show below (Sect. 4.4.7) that for our application a rough approximation of σ 2 (i.e. low N d ) is sufficient.

Precision assessment

For given observed data y o , once the PSD is estimated by the M-ECM algorithm the uncertainty of the regression parameters can be approximately assessed. This is very useful since in practice only a few realizations of y o are available. The error is evaluated by estimating the covariance of the GLS estimator with Eq. (4.21), where Σ oo is replaced by its M-ECM estimate, giving an approximate assessment of the precision of the regression. This calculation requires to solve N p linear systems involving the matrix Σ oo . Similarly to the computation of the conditional missing data estimate in Sect. 4.4.4, their solutions are obtained by using an iterative method involving matrix-vector products only. However for this calculation the PCG algorithm shows an irregular convergence behavior. We use the biconjugate gradient stabilized method (BiCGSTAB) [START_REF] Lanczos | Solution of systems of linear equations by minimized iterations[END_REF] instead. While adapted to the more general case of non-Hermitian matrices (unlike the PCG) this variant has better stability properties. To accelerate convergence a preconditioned version of the BiCGSTAB method is used [START_REF] Hogben | Handbook of Linear Algebra[END_REF], with the same sparse preconditioner as in Sect. 4.4.4. The iterations are stopped when the residuals r have decreased by some specified amount.

The estimate of the standard deviation of β is obtained by taking the square root of the diagonal of the estimated GLS covariance in Eq. (4.21). In the following we call this the "M-ECM error estimate" and denote it σECM .

Final formulation of a modified ECM algorithm

This section summarizes the main steps of the likelihood maximization procedure. An iteration i of the developed algorithm can be summarized as follows: 0. Initialization: calculation of the first guesses β0 , Ŝ0 with the KARMA method described in Sect. 4.3;

1. E step: calculation of the terms involved in the conditional expectation of the likelihood E1 Calculation of the conditional data vector given the observed data y i = E y|y o , θ i with Eq. (4.60); E2 Calculation of the conditional periodogram given the observed data I i z = E I z |y o , θ i following the method described in Sect. 4.4.5.2 which gives an approximation of Eq. (4.70);

CM step: conditional maximization of the likelihood

CM1 Calculation of the new estimate β i+1 with Eq. (4.65). CM2 Calculation of the new estimate of the spectrum Λ i+1 . To do this, we first compute the reconstructed residuals z i = y i -Aβ i+1 . Then, after replacing the periodogram I z by its conditional estimate I i z in Eq. (4.68), l y j is maximized with respect to (a, b) for each frequency f j to obtain (a j , b j ), where the spectrum estimate is Λ i+1 = a j . The values of the spectrum at all Fourier frequencies f k are deduced by linear interpolation from the values at f j .

It must be noted that this algorithm is not exactly an ECM because we introduced a local maximization at the CM2 step, so that the form of the likelihood slightly changes with respect to the CM1 step. We will therefore designate it as a "modified" ECM algorithm, or M-ECM.

Numerical tests

We apply the M-ECM method to mock in-flight data. We successively analyse the result of the missing data reconstruction, the PSD estimation, the regression parameters estimation, and the error assessment.

Data

We consider exactly the same data sets as the ones used for the KARMA method in Sec. 4.3.7. Since we showed that the periodic and telemetry gaps have a similar impact on the regression results, we restrict the study to the tank crackle mask w a and to the periodic mask w c , which is deterministic and non-stationary. Thus in the following w a is referred to as "random gaps" and w c is referred to as "periodic gaps".

Parametrization of the modified ECM algorithm

In this section we give some details about how the M-ECM algorithm is parametrized.

For each draw of z the observed vector y o is constructed according to Eq. (4.15).

After obtaining the KARMA estimates β0 and Ŝ0 , the M-ECM algorithm is run to obtain the final estimates β, Ŝ, and ŷm . This scheme is repeated for the 400 draws.

The iterations of the M-ECM algorithm is stopped when the difference between the current and the previous estimation of the Eötvös parameter δiδi-1 is less than 10 -17 .

At each iteration i of the ECM, the preconditioned conjugate gradient (PCG) algorithm described in Sect. 4.4.4 is run to calculate x = Σ -1 oo z o . This process itself involves N it iterations, with typically N it ∼ 100. The taper length q = τ 0 f s for the preconditioner is selected to be equal to the order of the AR process used in the KARMA estimation. The latter is found by minimizing the Akaike information criterion determined from the data, namely p = 60 in the case of the simulated noise (see [START_REF] Baghi | Regression analysis with missing data and unknown colored noise: Application to the microscope space mission[END_REF] for more details). The choice q = p allows us to represent the noise correlation to a range sufficient to approximate the GLS estimator in the KARMA method. The iterations of the PCG algorithm are stopped when the norm of the solution residuals r = z o -Σ oo x reaches the threshold , chosen to be the standard deviation of the residuals z o of the observed data model.

At each iteration, the conditional periodogam given by Eq. (4.70) is approximated by

N d = 5
Monte-Carlo draws. N d is taken to be sufficiently low to reduce the computational cost. We will see next that this choice does not have a major impact on the result.

Nevertheless when processing real experimental data for which few realizations are usually available, it is obviously safer to use a higher number of conditional draws.

The algorithm is implemented in python language and the DFTs are computed using the python wrapper around the efficient FFTW library [START_REF] Frigo | The design and implementation of fftw3[END_REF]. Each iteration of the M-ECM procedure takes almost 3 minutes for N ∼ 5 × 10 5 on a typical 2 GHz computer.

For the data under study the convergence is obtained within less than 10 iterations.

Treatment of missing data and minimization of the measurement uncertainty 133

1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 For clarity we plotted a sample which is longer for periodic gaps (lasting more than one orbital period T orb , or about 6 WEP periods 1/f EP ) than for random gaps (which are 260 times shorter and more scattered). Observed data are in black and reconstructed data from one Monte-Carlo conditional draw are in blue. The missing data spans are indicated by gray areas.

Results of a single reconstruction

We present here the result of a reconstruction obtained on a single simulation.

To show the noise dominating the data, the gap distribution and the reconstruction of missing values, we have plotted in Fig. 4.16 an extract of the observed time series (in black) as well as the estimated missing data (in blue) in the case of the periodic and the random gaps. In the first case we see that the periodic gaps are concentrated whereas the random gaps are distributed throughout the time series. The pattern that we see on the top of Fig. 4.16 for the periodic window is repeated every T orb .

In Fig. 4.17 the Lomb-Scargle periodogam of the observed data is plotted without reconstruction (in light grey). This power spectrum estimate is adapted to unevenly spaced data and constructed so as to have similar statistical properties as the classical periodogram of Eq. (3.56) in the case of white noise. Then we plot the periodogram of the original complete data (black) and the periodogram of the reconstructed data (blue). We see that the reconstruction allows a more faithful visualization of the real noise level. Indeed with respect to the true original periodogram for complete data, the Lomb-Scargle periodogram exhibits spurious peaks in the case of periodic gaps while it shows a noise level which is higher by almost two orders of magnitude in the case of random gaps. The leakage level is not the same for random and periodic gaps, although the fraction of missing data is the same. This is due to the number of gaps as detailed in [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF]. In comparison the data reconstruction cancels the leakage effect that is present in the Lomb-Scargle periodogram when data are missing. We study the average behaviour of the reconstructed periodogram in the next section. Fig. 4.18 shows that on average the conditional periodogram converges toward the true periodogram with no missing data. For random gaps however the conditional periodogram looks slightly biased between 3 and 30 mHz. This is mainly due to the low number of Monte Carlo (MC) draws used to approximate the corrective term in Eq. (4.70). To verify this explanation with a reasonable CPU time we have used a toy PSD model of almost similar shape with less data points and we have evidenced a decrease of the bias when increasing the number of MC draws to N d = 100 (see Appendix C.4 for more details). Taking such a number of MC draws with N = 4.7 × 10 5 data points would be computationally expensive but could be achieved by use of parallel computing. However this is not necessary for our purpose, as the signals of interest are located at lower frequencies than the biased interval. Fig. 4.19 shows the average (blue) and the confidence level (light blue) of the PSD estimates. This average estimate is compared to the original PSD from which the noise is generated (black), and to the average estimate from the AR model of the KARMA method used as initial guess (dashed red).

Convergence of the periodogram and the PSD estimate

The final PSD estimate brings an improvement with respect to the autoregressive one by reducing the bias in the low frequency part of the spectrum.

In the case of random gaps, again a residual bias of less than 3 × 10 -13 ms -2 Hz -1/2 remains in the band between 3 and 30 mHz (where the PSD is minimum). Since the PSD is estimated from the periodogram, the bias in the periodogram whose origin is explained above has an impact on the bias of the PSD. Another bias of order 10 -12 ms -2 Hz -1/2 is still visible on both graphs (periodic and random gaps) at lower frequencies. This comes from another source of error: smoothing in the frequency domain. While reducing the variance, smoothing inevitably introduces a small bias visible even for the complete data case (we checked it by inspecting the PSD estimate obtained for complete data). A lead to slightly reduce this bias is to choose the optimal smoothing bandwidth h j as in [START_REF] Fan | Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation[END_REF] instead of the logarithmic function mentioned in Sect. 3.5.1.4. The optimal bandwidth minimizes the Mean Squared Error (MSE) between the estimator and the true spectrum, but the minimization would be done at a higher computational cost since it requires to estimate the MSE on a grid of varying h for each frequency.

Finally, to give an insight of the short-range correlations of the noise and their estimation, we show in Fig. 4.20 the estimate of the normalized autocovariance function R(τ )/R(0) obtained from the M-ECM PSD estimate with Eq. (3.17), in the case of random gaps only since it corresponds to the largest error on the PSD. We can see an irregular pattern taking successively positive and negative values, which prevents us from fitting smooth autocovariance models commonly used in geostatistics as in [START_REF] Stroud | Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice[END_REF]. According to the figure, although it is model-independent, the M-ECM algorithm provides an estimation of the covariance with a low bias. Sample average of the normalized autocovariance estimated from random-gapped data with the M-ECM algorithm (blue) along with its 99% confidence interval (dashed blue) and the true autocovariance (black). The autocovariance alternates between positive and negative values. The bar heights of the histogram corresponds to the values of R(τ )/R(0) at lags τ located at the left-hand-side edge of the bars. To be comprehensive the confidence interval should include both the upper and lower bounds for positive and negative values of the autocovariance. For the sake of clarity, only the upper bound of positive values and the lower bound of negative values are displayed.

Performance of the regression parameters estimation

Here we examine the empirical mean and variance of the regression vectors β obtained with the M-ECM algorithm applied to the 400 simulated samples.

We gather in Table . (4.4) the sample average of the 400 estimates μ400 of the regression parameter vector β, as well as their sample standard deviation σ400 . Three estimation methods are compared: the ordinary least squares from the observed data described by Eq. (3.39), the KARMA method (Sect. 4.3), and the M-ECM algorithm (Sect. 4.4.5). We check that the KARMA and M-ECM estimates both converge towards the true expectation value and therefore are unbiased. In addition the standard deviation of the M-ECM regression is comparable to KARMA's. A slight increase of uncertainty is observed for the M-ECM algorithm but this is not significant with respect to the precision of the sample standard deviation. Therefore the reconstruction is consistent and there is no loss of precision nor additional bias when performing a linear regression on data reconstructed with the M-ECM algorithm. Like the KARMA method the M-ECM algorithm allows us to improve the precision of the regression with respect to the OLS: the uncertainty is decreased by a factor 4 in the case of periodic gaps and by nearly a factor 60 in the case of random gaps. That is not because the M-ECM algorithm is less efficient with periodic gaps, but because the leakage effect is smaller with periodic gaps than with random gaps (see comment in Sect. 4.4.6.3).

We indicate the value of the CRLB in the third column of Table 4.4: both for the KARMA and M-ECM methods the estimation of δ has an uncertainty close to the CRLB within a difference less or equal to 10 -16 . Table 4.4: Mean and standard deviations of the regression parameters with OLS, KARMA, and the M-ECM method. The first four columns indicate respectively the gap pattern, the parameter, its true value and the Cramer-Rao lower bound (minimal achievable standard deviation). Then from left to right there are 3 column groups corresponding to the 3 estimation methods. For all methods the quantities μ400 and σ400 respectively correspond to the sample average and the sample standard deviation of the 400 estimates. For OLS the term σ OLS is the theoretical error calculated with Eq. (4.4), for the KARMA method σAR is the average of the error estimates (see [START_REF] Baghi | Regression analysis with missing data and unknown colored noise: Application to the microscope space mission[END_REF] for details) and for the M-ECM algorithm σECM is the average of the error estimates calculated with Eq. (4.21). The barsat the top-right of the table stand for the fact that it does not really make sense to apply the ECM algorithm to complete data (since there is no E step any more).

OLS

Blind simulation including mask construction

In the simulations presented in the previous section we assumed that the mask was given, and we concentrated on the regression results. In this section we apply both the strategy of the data-driven mask construction implemented in Sec. 4.2, and the regression methods developed in Secs. 4.3 and 4.4 to simulated data. In addition, the time series considered here are generated differently than the previous ones. They stem from a simulation campaign named "technical qualification", (QT for the French acronym) which was done before the satellite launch. The QT simulation aims at validating the data processing pipeline (but not the performance of the calibration itself, which was tested by the "operational qualification" campaign). This is a good exercise not only to test the mask construction process but also the developed tools, with more complex and independently generated simulations.

The data

The QT data are generated with an end-to-end simulator named suson and developed by CNES with ONERA support. In these simulations, some parts of data generation process comes from real hard-ware signals, such as the front-end electronics. The simulation inputs, such as instrument defects, were parametrized by the CNES team. The philosophy of this mock data procedure is that it is devoted to blind estimation, i.e. the data analysis team was not aware of the actual input parameters chosen for the simulation.

The QT data are a way to validate the strategy of the mask construction, as well as the developed estimation algorithms. As an example, we study a time series simulating an inertial session dedicated to the calibration of the offcentrings along the X and Z axis. No previous calibration has been made, which means that all the effects of the instrument imperfections are present in the data.

In the simulation, micrometeorites impacts (about 10 impacts per orbit) are simulated, which induce spurious acceleration peaks (see Fig. 4.21).

Mask construction

We use the protocol described in Secs. 4.2.2 and 4.2.3 to construct the mask. For each mass i, a flagging vector indicating unavailable or saturated data is provided. Then we construct a second mask by detection of the acceleration peaks. These masks are often coincident, since most of the peaks correspond to saturations. Finally we extend each gap by 2 seconds (to take a margin with respect to the recommendation of Sec. 4.2.2). We multiply the masks of the internal and external mass to obtain the final mask. An excerpt of the result of the mask construction is given by Fig. 4.21 where the masked data are shown in red.

We see that the peaks (red) are correctly flagged by the sigma-clipping technique, and that a sufficient margin covering the transient period after each peak is taken into account. In Fig. 4.22 we plot the periodogram of the original data which includes all the acceleration peaks (dark grey) and we compare it to the periodogram of the masked data (light grey). The original noise level is higher than for the masked data, since the response of the system to the impacts introduces an additional power in the spectrum (since short events lead to a spreading frequency signature). We note that the shape of the noise spectrum is different from the model used so far, because noise sources other than purely "instrumental" are present. The main one is the drag-free residual noise due to the coupling of the differential mode to the common mode. Since the calibration of the a di coefficients has not been done, we see a bump of noise around 30 mHz. We run the KARMA algorithm and the M-ECM algorithm to obtain the estimates of the offcentrings. The reconstructed data is plotted in blue, along with the PSD estimate in red. As will be the case in flight, we have no means to tell whether the reconstructed data is close to reality (since no simulation without meteorite impacts has been run). However, we recover the low-frequency noise level of 10 -12 ms -2 Hz -1/2 which is expected.

Regression results

The results of the estimation are summarized in Table 4.5. While all the instrument perturbations are present in the signal, the model used for the fit is the one of Eq. (4.57). The results of the WEP violation parameter are only given indicatively, since it is not very meaningful for uncalibrated data. It is clear that there is an improvement when masking the outliers, even for the OLS estimator. For a c11 ∆ x , the error of the OLS estimator on the raw data is about 2 µm. When using the masked OLS it is already reduced to 0.3 µm because the bias due to the outlier is avoided. This means that the effect of the bias caused by the outliers is larger than the leakage effect caused by discarding the corresponding data spans. Then the KARMA and M-ECM method achieve an error slightly larger than 0.1 µm, which is the requirement for this parameter according to the in-flight calibration plan [START_REF] Touboul | Microscope instrument in-flight calibration plan[END_REF]. This requirement is usually met after a reprocessing step where each parameter is re-estimated after correction of the other instrument disturbances. We note that the uncertainty estimate from M-ECM is lower than from KARMA, which can be explained by the bias of the AR PSD estimate at low frequency. As a conclusion, discarding the intervals where the data is perturbed by saturations or outliers is essential to decrease the bias when estimating calibration parameters. Since we saw that masking the data increases the estimation uncertainty, such a strategy supposes to use the developed tools for missing data in order to maintain the variance down to the minimum level. For the spikes that do not cause saturations, another approach could be to model them (with, e.g., an exponentially decaying sine) and to remove them from the signal. But since the location of the spikes is not known a priori, this would involve the use of matched-filtering algorithms and Markov chain Monte Carlo techniques, which is computationally expensive if the number of spikes is large.

Summary:

We showed that when the measurement is affected by coloured noise and missing data, the performance of the ordinary least squares (OLS) is not sufficient to maintain the 10 -15 precision level on the Eötvös parameter, with a degraded precision by more than one order of magnitude. The identified reason is that the OLS estimator is not optimal with respect to the variance. As a prerequisite, the mechanism that leads to the data interruptions was analysed in order to derive a reliable protocol to flag the data as "invalid" or "missing". Then, we developed a linear regression method adapted to missing data named KARMA, which performs an autoregressive fit of the noise and use it to de-correlate the data with a Kalman filter. We showed that the resulting estimator has a quasi-minimal variance and allows us to maintain a precision close to 10 -15 in spite of the missing data, which is up to 60 times better than the OLS. In order to improve the quality of the characterization of the noise when data are missing, and hence the confidence of the WEP test, a PSD estimation method relying on statistical Kriging and on a modified ECM algorithm was implemented. The square-root PSD is estimated with an error less than 10 -12 ms -2 Hz -1/2 around the WEP frequency. At the same time, the method provides faithful reconstructed data sets. We finally showed that the flagging procedure and the regression methods were robust by testing them with independently simulated data. Up to now, the developed data analysis tools were applied to the WEP test. However, they may be used to a broader range of applications, within all the fields where regression analysis on time series and spectral estimation are involved. Thereafter, we investigate and motivate the possibility to extract other science with MICROSCOPE, in particular the measurement of the gradient of Earth's gravitational potential. This chapter deals with the possibility to extract other scientific data with MI-CROSCOPE, to extend the data processing to broader applications. We focus on the measurement of Earth's gravitational gradient by using the two accelerometers on-board the satellite as a gradiometer. After providing the scientific motivations for such a new observable, we perform a preliminary analysis of the measurement performance based on the noise spectrum. We then explore the possibility to use the existing calibration sessions already planned for the mission to calibrate the gradiometer. This allows us to refine the error assessment by taking into account both instrumental noise and calibration errors. We finally discuss the possible leads for an improvement.

Motivation

The MICROSCOPE space mission embarks two ultrasensitive differential accelerometers in space, in a stable and soft environment, on a platform equipped with a drag-free control system. This configuration can be interesting for various applications, in addition to the WEP test. The processing of the produced data can therefore be performed with different objectives. Following this philosophy, some proposals have been submitted at the MICROSCOPE Colloquiums III and IV, which were held in November 2014 and 2015.

One of the proposed ideas is to use MICROSOCPE data for the measurement of Earth's gravitational gradient. Other ideas were proposed, such as the test for Lorentz invariance via the standard model extension framework [START_REF] Colladay | Lorentz-violating extension of the standard model[END_REF], the retrieval of densities and winds in the thermosphere [START_REF] Doornbos | Thermospheric Density and Wind Determination from Satellite Dynamics[END_REF], the constraint of the neutrino exchange forces and spin-dependent interactions [START_REF] Fischbach | Higher order weak interactions and the equivalence principle[END_REF], the constraint of the chameleon screening mechanism [START_REF] Khoury | Chameleon cosmology[END_REF], the study of the sensitivity of the drag-free mode to very low frequency phenomena in the ionosphere [START_REF] Eliasson | Numerical study of mode conversion between lower hybrid and whistler waves on short-scale density striations[END_REF], or the study of the cold damping effect in MICROSCOPE experiment by a quantum mechanical analysis of noise [START_REF] Grassia | Quantum theory of fluctuations in a cold damped accelerometer[END_REF].

In this chapter, we concentrate on the measurement of the gradient of Earth's gravity field, which is also called "gradiometry".

Gradiometry is interesting when the aim is to identify sources geometries, because it is sensitive to the derivatives of the field, and hence to the spatial heterogeneity of Earth's mass distribution. Several space missions have been already launched to this aim. Among them is the Gravity

Measurement principle

The T-SAGE instrument includes two differential accelerometers, labeled su-ep and su-ref, separated by a 17 cm-length arm. We propose to use the differential acceleration between two masses, each one belonging to a different accelerometer, to form a gradiometer (i.e., an instrument measuring the gravitational gradient). The accelerometers configuration is represented in the left-hand side panel of Fig. 5.1.

We consider the gradiometer formed by the internal masses of the two accelerometers (both made of platinum). While another choice can be made, this one is motivated by the fact that the expected sensitivity of such a gradiometer configuration on the Y axis is better than the other configurations.

We call ∆ g the vector linking the internal mass of the su-ep and the internal mass of the su-ref. We can write a simplified measurement equation, by ignoring for the time being the defects of the instrument, which is formally similar to Eq. (2.51):

2Γ g d = Γ EP,I -Γ REF,I ≈ ([T ] -[In]) ∆ g -2[Ω] ∆g -∆g + 2b g d (5.1)
where [T ] and [In] are respectively the gravitational and inertial gradients in the instrument coordinate system (calculated at some point in the middle of the test-masses forming the gradiometer), and b g d gathers the differential perturbing accelerations on the test-masses. In the instrument coordinate system, the lever arm is oriented with the Y axis, such that

∆ g =    ∆ g x ∆ g y ∆ g z    with : ∆ g y ∆ g x , ∆ g z , (5.2)
where the origin of the coordinate system is taken to be the center of the accelerometer cage of the su-ep and the X and Y axis correspond to the su-ep frame defined in the right-hand side of Fig. 5.1. In this system we have ∆ g y = -d with d = 17 cm while the order of magnitude of ∆ g x and ∆ g z is 10 µm. The MICROSCOPE experiment is not designed for gradiometry. Therefore there are three main limitations to the measurement, by comparison to the GOCE mission for example:

• the most sensitive axis is the X axis in the instrument reference frame. The other axes have a sensitivity about 5 times lower

• the attitude pointing is inertial or spinned: the sensitive axis is either pointed toward a constant direction or slowly rotated about the axis perpendicular to the orbital plane, whereas in GOCE it points towards Earth. This is illustrated in Fig. 5.2.

• the altitude of MICROSCOPE is higher than GOCE: 700 km instead of 270 km, hence the gravitational signal is lower. The consequence of this particular configuration is that possibly 3 components of the gradient in the instrument frame can be measured, which are the ones involving the arm along Y : T xy , T yy and T zy .

Calculation of the expected measured gravity gradient

In this section we detail how to calculate the gravitational gradient in the instrument reference frame, from a model of Earth's gravity field. This summarizes the calculation steps needed to simulate what the instrument will measure.

The gradient is modelled by using a decomposition of Earth's potential in spherical harmonics, that have been determined from previous geodesy measurement (for example, the EGM96 model can be used [START_REF] Lemoine | Gravity[END_REF]). In the coordinate system related to the terrestrial reference frame R T (frame centred on Earth, whose three axes are bound to the globe, see Appendix E.1) this decomposition has the form

U = µ r L 2 l=L 1 l m=0 a e r l P l,m (sin φ) [C l,m cos mλ + S l,m sin mλ] , (5.3)
where µ = GM T , a e is the equatorial reference radius, and C l,m and S l,m are the Stokes coefficients which depend on the mass repartition of the Earth, and are numerical quantities given by the chosen potential model. P l,m are the Legendre function of degree l and order m. The triplet (r,φ,λ) includes the geographic coordinates of the point at which the potential is computed, which are respectively the distance from the Earth's center, the latitude, and the longitude. The full potential is obtained with L 1 = 0 and L 2 = +∞. From this model the value of the gradient at the position and attitude of the satellite is deduced. The general method to do this is performed in two steps:

1. Computation of the gradient in the terrestrial coordinate system by taking the second spatial derivatives of the potential in Eq. ( 5. where R is the rotation matrix from of the instrument frame with respect to the terrestrial frame.

The matrix R includes Earth's rotation, the orbit inclination, the satellite spin, and the proper orientation of the instrument with respect to the satellite. More details can be found in Appendix E.2. The steps enumerated above provide a derivation of the gradient tensor that is observable by the MICROSCOPE instrument.

From the model of Eq. ( 5.3), and ignoring the attitude of the satellite, we can quantify the effect of its altitude on the magnitude of the measured gradient. Each degree contributes to the potential proportionally to (a e /r) l+1 , and proportionally to (a e /r) l+3 to the gradient, where r = a + h and h is the altitude of the satellite. For MICROSCOPE we have a e /r ∼ 0.90 whereas for GOCE we have a e /r ∼ 0.96. To illustrate the contribution of each degree to the gradient amplitude, we plot in Fig. 5.3 the amplitude of the radial gradient T rr = ∂ r ∂ r U as a function of the degree l, using Kaula's rule of thumbs [START_REF] Kaula | Theory of Satellite Geodesy: Applications of Satellites to Geodesy[END_REF], which is an empirical rule followed by the mean amplitude of the Stokes coefficients. From the plot we see that the order of magnitude of the gradients measured by GOCE and MICROSCOPE are comparable for the degree l < 10. As a result, the higher altitude of MICROSCOPE is not a major disadvantage since we are interested in the low frequencies, and hence the low degrees of the potential. In addition, flying at a higher altitude can even be viewed as an advantage when studying the large scales of the gravitational field, since the degree amplitudes decrease faster as a function of l, hence the low degrees better stand out relatively to the high degrees.

Preliminary study of the measurement performance

Now that we have motivated the measurement of Earth's gravity gradient with MI-CROSCOPE and described its principle and the effective observable, we can sketch a preliminary performance assessment. In particular, we derive the expected sensitivity of the gradient measurement, and compare it to the GOCE measurement.

Measurement of the gradient from the differential acceleration

In order to derive the measurement equation for the gradient, we can rearrange Eq. ( 5 Before doing further analysis, we can make two important comments about the gradient estimation via these equations. The first one is about the sensitivity of each estimate. We plot in Fig. 5.4 the expected sensitivity of the measurement for the 3 components. We consider the instrumental noise only, that is, the noise due to the accelerometer via the differential acceleration term in Eqs. (5.5-5.7). We use the Eötvös unit E, where 1E = 10 -9 s -2 which is generally adopted in geophysics. We see that the noise of the accelerometers will be more favorable for the T xy component (black curve), for which the squared root of the PSD is about 5 times lower than the other axes at low frequencies (below 10 mHz). In addition, the noise level on the Y axis (red) and Z axis (blue) is similar.

The second comment that we can make is that the measurement of the Txy and Tzy components involves the angular accelerations, whereas the Tyy components only involves the squared angular velocities. As we will see later, it turns out that the noise is expected to be much larger for the angular accelerations than for the angular velocities.

As a result, the components that will be measured with the best sensitivity a priori are Txy and Tyy , since the former is affected by a lower instrument noise but by the angular acceleration noise, whereas the latter is affected by a higher instrument noise but no angular acceleration noise. The Tzy components combines the two disadvantages, and will therefore we discarded in the following study.

Comparison with GOCE performance

At this point it is interesting to detail the differences between the configuration of MICROSCOPE and GOCE, and to compare their expected sensitivity.

To do this comparison, we express the components of the gradient measured by MICROSCOPE [T ] as a function of the components measured by GOCE [T G ], as if GOCE were flying at the MICROSCOPE's orbit. In other words, we the components of MICROSCOPE gradient are expressed in the along-track oriented, local orbital frame. According to Fig. 5.2, the MICROSCOPE measurement will be a modulation of the GOCE tensor components at the orbital frequency. Indeed, if we apply Eq. 5.4 to go from the GOCE frame to the MICROSCOPE frame, we find that

T xy = cos ω T G xy -sin ω T G yz , T yy = T G yy , T zy = sin ω T G xy + cos ω T G yz , ( 5.8)
where ω = ω+v is the sum of the the true anomaly of the satellite v and the argument of the orbit perihelion ω (see Appendix E.1). In first approximation we can write ω ≈ ω orb t+φ orb . We notice that the T yy component is the same as the one measured by GOCE. With these relations in mind, we can compare the MICROSCOPE expected noise level to the GOCE noise model, which are plotted in Fig. 5.4 (the noise was actually measured to be slightly higher than predicted, see e.g. [START_REF] El | Study of GOCE accelerometers noise during the mission lifetime[END_REF]). These noise models are derived from an internal technical report [START_REF] Bruno | [END_REF] and the measurement equations of the GOCE gradients [START_REF] Stummer | Gradiometer data processing and analysis for the GOCE mission[END_REF]. The GOCE noise model is plotted in dotted line. In GOCE, the T G xy component and the T G yz components (respectively the black and the red dotted lines) are measured by non-sensitive combinations of differential accelerations, yielding a higher noise level than for T G yy . From Eq. 5.8 they can be compared to the T xy component in the MICROSCOPE instrument frame. In the MICROSCOPE measurement, the acceleration noise is expected to be lower than the GOCE level below 100 mHz. Therefore the contribution of the geopotential to the T xy component is likely to be better resolved by MICROSCOPE data.

In addition, we can directly compare the T G yy component of the GOCE gradient (blue dotted line) with the T yy component of the MICROSCOPE gradient (blue solid line, coincident with the red solid line). In GOCE this component is measured by sensitive axes, whereas this is not the case in MICROSCOPE. However, for frequencies below 0.4 mHz, the MICROSCOPE noise level is expected to be more favorable.

Preliminary analysis

We are interested in the low frequency band (below 5 mHz) of the measurement, which is sensitive to the large scales of the gravitational potential. Indeed, the temporal frequency f corresponds to an on-ground distance of dx ≈ R T w orb f -1 , (5.9)

where R T is the earth radius and w orb is the orbital angular frequency. At the MICRO-SCOPE altitude, we have dx ∼ 6.8/f km where f is expressed in Hz. For f = 5 mHz we have dx ∼ 1360 km, so the measurement band of MICROSCOPE will be sensitive to scales above 1000 km. The large scale variations of the geopotential emanate, in particular, from the deep structures of the Earth's mantle (Panet et al. [START_REF] Panet | Mapping the mass distribution of Earth/'s mantle using satellite-derived gravity gradients[END_REF]).

Our strategy is the following. As a first analysis, we investigate what type of signal we can obtain in MICROSCOPE data from different mass distributions of the Earth's deep mantle. The objective is to tell whether the features of the different signals can be distinguished by the instrument, i.e., whether these differences are significant with respect to the noise level.

In the first place, we perform a simulation of the "real" observable T iy on an inertial session of 120 orbits (about 8 days). This duration is chosen because it is the longest session without change of attitude planned in the mission. Of course, several sessions could be combined to increase the SNR (see e.g. Sec. 5.2.5). We check that coverage of Earth is acceptable by plotting the satellite trace during an inertial session in Fig. 5.5. We choose a sampling of 0.1 Hz, which is the upper bound of the GOCE measurement band. We do not expect a significant signal above this frequency.

We perform 3 simulations, each one with a different gravitational model provided by the courtesy of I. Panet from the National Institute of Geographic and Forest Information. These models are the following:

• EGM96: reference model, constructed from satellite tracking data and altimeter data [START_REF] Lemoine | Gravity[END_REF];

• FM_S40RTS_40: model where the internal mass heterogeneity geometry is built from a joint analysis of geoid, gravity and gravity gradients with the seismic velocities [START_REF] Greff-Lefftz | Joint analysis of GOCE gravity gradients data of gravitational potential and of gravity with seismological and geodynamic observations to infer mantle properties[END_REF];

• hVR40_PD: model where the internal mass heterogeneity geometry uses a reconstruction of the movement of tectonic plates from 200 Myr and also includes hot instabilities [START_REF] Rouby | Mantle dynamics, geoid, inertia and TPW since 120 Myr[END_REF].

In this preliminary analysis, we study the large spatial scales corresponding to the deep layers of the Earth's mantle. To this aim, we truncate the three models to order 12, above which the masses of the upper layers are not dominant. The results are presented in the next section.

Simulated time series of the gradient components

In this section we present the results of the simulated gradient observables T xy and T yy , computed with the method described in Sec. (5.1.3). This is the approach adopted by the ACTENS code developed by OCA, which takes as input orbit ephemeris and attitude data.

For the EGM96 model we compute T iy (t) -T REF iy (t) as a function of time, where T REF iy is the gradient corresponding to the reference ellipsoid (spherical rotating Earth). For models FM_S40RTS_40 et hVR40_PD we simply compute T xy (t) since they are already calculated with respect to the reference ellipsoid.

Time series and maps of T xy

A first computation is performed for T xy , whose results are shown in Fig. 5.6. In the top-left panel, we plot an excerpt of the time series of the gradient component in the instrument reference frame for the 3 geopotential models. In the other panels we plot the value of T xy with respect to the position of the satellite. To lighten the figure we plot the points every minute (instead of every 10 seconds). The patterns that we obtain show differences of the order of 0.1 E between the models.

In order to perform a first comparison of the signal that we obtain with the instrumental noise, we plot in Fig. 5.7 the Fourier representation of the time series in amplitude, along with the noise level (corresponding to 120 orbits).

A first conclusion that we can draw from this plot is that the power of the signal coming from the large scales of the potential is mainly concentrated at frequencies below 3 mHz. In this region, the SNR (ratio of the power of the signal and the noise power) is about 100. This means that if we consider the instrument noise alone, the signal is significant. Furthermore, the differences between the 3 models can be distinguished (the frequency peaks have a characteristic distribution from one model to another).

Time series and maps of T yy

The same exercise is performed for the T yy component. The spectrum of the signal in the frequency domain is compared to the noise level in Fig. 5.9. Since the sensitivity is lower for the Y axis, the SNR is about a factor 16 lower than for the T xy component.

Spherical harmonic analysis

Another way to analyse the signal level with respect to the noise is to consider the decomposition of the gravity gradient on spherical harmonics. Indeed, the measured gradient can be written as a linear combination of the Stokes coefficients, similarly to the potential in Eq. (5.3). This allows the construction of a spectrum of the measurement sensitivity based on the modes of the gravitational potential, rather than its variations throughout the satellite orbit. Note that other decompositions are possible, such as wavelet decomposition (see Panet et al. [START_REF] Panet | Modelling the Earth's gravity field using wavelet frames[END_REF]). Therefore, the derivative of the measured gradient with respect to the Stokes coefficients must be calculated. First, we must compute the derivatives of the gradient components in the instrument coordinate system with respect to the gradient components in the terrestrial coordinate system, using Eq. (5.4). These derivatives have the form

∂[T ] ∂T e ij = R ∂[T e ] ∂T e ij R T .
(5.10)

Second, we must calculate the partial derivatives of the gradient in the Earth coordinate system with respect to the Stokes coefficients. The work of Métris et al. [START_REF] Métris | Derivatives of the gravity potential with respect to rectangular coordinates[END_REF] provides analytical expressions to do this. For example, for the component T

where we defined the the coefficients

α l,m = (-1) l (2l + 1)(2 -δ 0m ) (l + m)!(l -m)! , ( 5.12)
and we noted I ≡ √ -1 the complex number. Then, the spherical harmonic noise spectrum can be viewed as the variance of the Stokes coefficients if they were estimated by a least-squares method. This amounts to performing a linear regression on the time series Tiy (t), where the regression parameter estimate writes

β = C S = A T A -1
A T Tiy , (5.13) where C and S are vectors respectively including the C lm and the S lm coefficients.

A is the model matrix, of the form

A = ∂T iy ∂C ∂T iy ∂S . (5.14)
From the PSD of the noise affecting the measurement of Tiy we can compute the covariance of the least-squares estimator using Eq. (3.15). However note that practically a penalized version of the least-squares function is often used to impose a smooth constraint on the solution, which amounts to postulating a prior probability distribution for the Stokes coefficients (see, e.g., Chambodut et al. [START_REF] Chambodut | Wavelet frames: an alternative to spherical harmonic representation of potential fields[END_REF]). To compute the theoretical spherical spectrum variance, we simulate the partial derivatives of Tiy as a function of time and Kaula's empirical rule 10 -5 l -2

Table 5.1: Relevant quantities to estimate the precision of the linear regression on the Stokes coefficients as a function of the degree l, from up to bottom: the mean standard deviation (degree variance), the mean value of coefficients, and Kaula's rule we calculate the estimated variances of C lm et S lm . We plot the results as a function of the degree l as described by Table . 5.1.

As an example, the estimation of the degree RMS error of an inversion performed with the measurement of T yy is shown in Fig. 5.10 for the lowest degrees (l < 12). This spectrum is obtained by combining 4 inertial sessions sampled at 1 Hz, each one lasting 120 orbits. The error is computed with the partial derivatives of T yy with respect to the Stokes coefficients obtained from the EGM96 model. The computation is based on the expected noise PSD of Tyy plotted in Fig. (5.4). Here the inversion of the potential coefficient is only done from one component of the gradient tensor, but in principle one can take advantage of the measurement of all gradient components to perform it.

Noise from angular velocities and acceleration

So far we considered the linear acceleration noise only. However, according to Eq. (5.5), there are perturbations coming from the estimates of the angular velocities and the angular accelerations. In flight, the angular rates are measured by an hybridization of the angular acceleration of the test-masses measured by the accelerometers and star tracker data. These measurement are noisy, because of the intrinsic noise of the attitude control on the one hand, and because of the sensors noise (accelerometer and star trackers) on the other hand.

CNES provided simulations of the measurement of angular rates, both tone errors and stochastic errors. To simplify we focus on the stochastic errors only. In Fig. 5.11 we plot the spectra of the products of angular velocities Ωx Ωy and Ω2

x + Ω2 y and the angular acceleration Ωz involved in the measurement of T xy and T yy . The angular acceleration residuals about the Z axis are plotted in the left-hand side panel of Fig. 5.11. Their level is comprised between 1 and 10 E•Hz -1/2 below 1 mHz, which is 10 to 100 times higher than the expected noise level of the linear acceleration noise on the X axis shown by Fig. 5.4. However, if we consider the noise of the squared acceleration velocities Ω2

x + Ω2 y , it is comprised between 10 -4 E • Hz -1/2 and 10 -3 E • Hz -1/2 , which is far below the linear acceleration noise which remains above 10 -2 E • Hz -1/2 . Therefore the differential acceleration noise along the Y axis is dominated by the accelerometer noise and is likely to provide the most sensitive observable for the measurement of the gradient.

Calibration of the gradiometer

As for the WEP test, the measurement of the gradient may be perturbed by the instrument imperfections. Therefore, the gradiometer must also be calibrated. As mentioned in Chap. 3, in-flight sessions are dedicated to the calibration of the su-ep and the su-ref where some degrees of freedom of the accelerometer and/or the satellite are excited in order to increase the signal-to-noise ratio of the parameters of interest. In this section we give the outline of a calibration protocol which uses the sessions already planned in the mission, and perform a preliminary assessment of its performance.

Since the gradient estimation also involves the differential acceleration, the measurement equation of the gradiometer is formally identical to Eq. (2.79), established for the differential acceleration of the test-masses of the su-ep. The only difference is the value of the parameters and of the measured accelerations involved in the equation. To differentiate them with the quantities related to the su-ep, we label the gradiometer parameters with the subscript g . The measured differential equation along the k axis (k = 1, 2, 3 for X, Y, Z) between the test-masses of the two accelerometers can thus be written

Γ g meas,dk = b g 0dk -a g ck1 δ g d g E,x -a g ck1 ∆ g dx S xx + a g ck2 ∆ g dy S yy + a g ck3 ∆ dz S zz -a g ck1 ∆ g dy + a g ck2 ∆ g dx S xy -a g ck1 ∆ g dz + a g ck3 ∆ g dx S xz -a g ck2 ∆ g dz + a g ck3 ∆ g dy S yz + a g ck3 ∆ g dy -a g ck2 ∆ g dz Ωx + a g ck1 ∆ g dz -a g ck3 ∆ g dx Ωy + a g ck2 ∆ g dx -a g ck1 ∆ g dy Ωz + 2 a g ck3 ∆g dy -a g ck2 ∆g dz Ω x + 2 a g ck1 ∆g dz -a g ck3 ∆g dx Ω y + 2 a g ck2 ∆g dx -a g ck1 ∆g dy Ω z + a g ck1 ∆g dx + a g ck2 ∆g dy + a g ck3 ∆g dz + a g dk1 Γ g meas,cx -b g 0cx -n g cx + a g dk2 Γ g meas,cy -b g 0cy -n g cy + a g dk3 Γ g meas,cz -b g 0cz -n g cz + 1 2 K g 21kk K 2 11k Γ g meas,1k -b g 01k -n g 1k 2 - 1 2
K g 22kk K 2 12k Γ g meas,2k -b g 02k -n g 2k 2
+ n g dk .

(

In Eq. (5.15), for a gradiometry application the term of interest is not the WEP violation any more (which is zero if we use the internal masses of the accelerometers since they are both made of platinum) but the term proportional to S xy or S yy , depending on whether we use Γ g meas,dx or Γ g meas,dy .

Utilization of the existing calibration sessions

In the following we detail each calibration session and describe the signal that are used to estimate the SU's parameters. We show how to adapt the measurement equation of each calibration to perform the estimation of the parameters of the gradiometer. The sessions are labeled by letters corresponding to the CNES nomenclature. In the estimation process, the approach is to fit the partial derivatives of the measured diffential acceleration in Eq. (2.79) or (5.15) with respect to the the excited signals. Even if this aspect is not detailed here, it is often more accurate to simultaneously fit other terms of large amplitude involved in the measurement equation in order to prevent them from biasing the estimation. There are two main differences when considering the differential acceleration constructed from the two accelerometers. First, all the terms proportional to the differential level arm ∆ g dy will be about 4 orders of magnitude larger than for one single accelerometer. Hence, in the estimation it will be necessary to include the terms proportional to the level arm in the model, so as not to bias the estimate. Second, the drag-free point is not locked to nullify the common mode of the gradiometer, but only the common mode of one single accelerometer. While the non-gravitational forces are still well compensated, the drag-free residual is larger than for a single sensor unit. As a consequence, in the calibration protocol it will be necessary to estimate the terms of couplings with the common mode in the first place, so as to be able to correct for them in the subsequent sessions.

Calibration of one sensor unit

To estimate the offcentrings, we have already seen that no artificial excitation is needed, since we can use the signal provided by the gradient oscillating at twice the orbital frequency during an inertial session. The goal of this session is to determine the parameters a c11 ∆ x and a c11 ∆ z .

Since the session is inertial, the components of the matrix [Ω] and its time derivatives are small. In addition, the components of [T ] on Y are 3 orders of magnitude larger than the other components.

Indeed, the gravitational gradient tensor can be approximated by [START_REF] Métris | Expression du vecteur accélération de gravité et du tenseur gradient de gravité en repère instrument[END_REF] [T ] ≈

µ r 3    3 2 cos(2(ω + v + ϕ)) + 1 2 0 3 2 sin(2(ω + v + ϕ)) 0 -1 0 3 2 sin(2(ω + v + ϕ)) 0 -3 2 sin(2(ω + v + ϕ)) + 1 2   +O r -5 , (5.16)
which oscillates at twice the EP frequency. We noted O r -5 the 3 × 3 matrix term depending on the higher degrees of the potential, which are dominated by the zonal harmonic coefficient of degree 2, J 2 = -C 20 .

The partial derivatives of the measured differential acceleration with respect to the offcentrings along X and Z are given by

∂Γ meas,dx ∂a c11 ∆ dx ≈ -S xx ∂Γ meas,dx ∂a c11 ∆ dz ≈ -S xz , ( 5.17)
and are included in the model matrix of the linear regression.

Adaptation to the gradiometer

The gradient signal can also be used to estimate the offcentrings a g c11 ∆ g x and a g c11 ∆ g z by using the X component of the differential acceleration. It may seems surprising to use the gradient signal to calibrate a gradiometer. However, only the components of the degree zero (or at most degree 2), which are well known, are sufficient to estimate the offcentrings. As a result, this calibration technique does not compromises the further measurement of the gradient (since from the point of view of gravimetry, only the deviations from the perfect rotating sphere are interesting). Therefore the transposition of the partial derivatives in Eq. (5.17) to the case of a g c11 ∆ g x and a g c11 ∆ g z is valid. However, a major difference is that the terms proportional to ∆ g dy are now dominant, and cannot be neglected. For example, the terms in S yy , S xy and Ωz must be included in the regression model.

For the differential acceleration along the Y axis, it is impossible to determine the individual offcentrings a g c22 ∆ g x and a g c22 ∆ g z with accuracy, because whenever they appear in the signal amplitudes of Eq. (5.15), they are dominated by coefficients proportional to ∆ g y . However, we can use the estimation of the offcentrings performed from the X axis, by making the approximation a g c22 ∆ g j ≈ a g c11 ∆ g j (hypothesis of equal scale factors). The resulting error is a few percents of the true value, in addition to the estimation error of ∆ g j .

Offcentring along Y (session B)

In the calibration session to determine a c11 ∆ dy , the satellite is oscillated about the Z axis (in the instrument coordinate system) at a frequency f cal/ang with an amplitude α 0 . This excitation magnifies the inertial acceleration proportional to Ω z , with typical values of α 0 = 0.05 rad and f cal/ang = 1.3 mHz. Thus we have

α(t) = α 0 sin(2πf cal/ang t), Ω z (t) = α 0 (2πf cal/ang ) cos(2πf cal/ang t), Ωz (t) = -α 0 (2πf cal/ang ) 2 sin(2πf cal/ang t).
(5.18)

The symmetric part of the gravitational and inertia gradient matrix is then

[S] ≈    T xx + Ω 2 z T xy T xz T xy T yy + Ω 2 z T yz T xz T yz T zz    , ( 5.19)
and the anti-symmetric part [U ] is dominated by the angular acceleration along the Z axis Ωz .

Calibration of one sensor unit

There are two terms proportional to the offcentring along Y , and the partial derivative used for the fit writes ∂Γ mes,dx ∂a c11 ∆ dy ≈ -S xy + Ωz .

(5.20)

Note that we implicitly assumed that the term S xy -Ωz proportional to a c12 ∆ dx will not perturb the estimation, since a c11 ∆ dy a c12 ∆ dx (∆ g dy ∼ 10 cm while ∆ g dx ∼ 10 µm, and a c11 ∼ 1 whereas a c12 ∼ 10 -3 ).

Adaptation to the gradiometer

For the gradiometer differential acceleration along the X axis, the calibration equation is more favorable to estimate the offcentring along the Y axis (that is, the length of the gradiometer arm), since the comparison a g c11 ∆ g dy a g c12 ∆ g dx is even more valid. Therefore, the calibration equation is formally identical to Eq. (5.20).

Regarding the differential acceleration along the Y axis, the leading term is the one proportional to S yy , which is also amplified since it contains the squared angular rate Ω 2 z . This gives access to the parameter a g c22 ∆ g dy . In addition, we can benefit from this session to estimate the parameters in front of S xy and S yz , because these terms are also amplified by about 2 order of magnitudes due to the rotation of the satellite. The measurement equation can be summarized by

Γ g mes,dx ≈ -a g c22 ∆ g dy S yy -a g c21 ∆ g dy + a g c22 ∆ g dx S xy -a g c22 ∆ g dz + a g c23 ∆ g dy S yz + a g c22 ∆ g dx -a g c21 ∆ g dy Ωz .
(5.21)

Parameters of the differential sensitivity matrix (sessions F, H, and G)

We consider the calibration session designed to determine the parameters a d1j , which quantify the coupling of the differential mode with the common mode. To amplify this coupling, the common mode is excited via the control command of the drag-free system, so that the satellite is oscillated linearly along the X, Y and Z axis in order to respectively estimate a d11 , a d12 and a d13 .

The term proportional to a d1j is amplified by oscillation of the satellite along the axis j such that

C i = C 0i + α sin(2πf cal/lin ), (5.22)
where the amplitude of the oscillation is typically α = 5 × 10 -8 ms -2 and its frequency is f cal/lin = 1.3 × 10 -3 Hz.

Calibration of one sensor unit

The partial derivative of Eq. (2.79) with respect to a d1j is the common acceleration along the axis j and writes

∂Γ meas,dx ∂a d1i ≈ (Γ meas,cj -b 0cj -n cj ) . ( 5.23)
Adaptation to the gradiometer Contrary to the calibration of a single accelerometer, when exciting the satellite along the axis j, the common mode of the gradiometer along the other axis is not completely nullified anymore. Nevertheless, the common mode amplitude along the excited axis is about 2 order of magnitude larger than the others, which allows us to perform a fair estimation of the parameters a g dkj , with an equation formally equivalent to Eq. (5.23).

Parameters of the common sensitivity matrix (sessions E2 and D2)

The session dedicated to the estimation of the parameters a c12 and a c13 (called E2 and D2) is based on a modulation of the Coriolis term by both an oscillation of the test-masses and an oscillation of the satellite in opposite phase.

In the Coriolis-type session, a command control is sent to the test-masses via the digital control loop to respectively oscillate them along the Y and Z axis. Simultaneously, a command is sent to the attitude control system of the satellite to oscillate it about the X axis.

The oscillating command controlling the motion of the test-masses has the form

∆ d (t) = ∆ d,bias + ∆ d,TM sin(2πf TM t), ( 5.24)
which is expressed in the instrument coordinate system, and where we have ∆ d,TM = 12 µm and f TM = 8.2 • 10 -3 Hz. Note that to simplify, we neglect the scale factor on the position command. It is necessary to reorganize the measurement equation because the offcentrings ∆ dj that were considered as fixed in the original formulation are now time-varying signals.

All the gradient terms in the right-hand side of Eq. (2.79) or Eq. (5.15) are expressed in the satellite coordinate system. Thus, the parameters that we labeled ∆ dj up to now are the offcentrings expressed in this system. However, when imparting an oscillation to the testmasses through a command injected in the servo-control loop, the motion is triggered in the 5. Application of the data processing to geodesy 161 real instrument coordinate system. To establish the relationship between the offcentrings in the satellite coordinates and in the instrument coordinates (labeled ∆ d,ins,j ), which correspond to the effective motion imparted to the test-masses, we need to apply a rotation.

Thus, due to the misalignment of the instrument, a motion ∆ d,ins of the test-mass will translate into a motion ∆ d in the satellite coordinate system, such as

∆ d,sat = [Θ c ] -1 ∆ d,ins , (5.25)
where Θ c is equal to the mean of the misalignment matrices for each test-mass (see Sec. 2.4.1). Rigorously, Θ c is not a rotation matrix, but has the same properties as a rotation matrix in case of infinitesimal rotations. We can then write the approximate relationships

∆ dx = ∆ dx,sat ≈ ∆ dx,ins -θ cz ∆ dy,ins + θ cy ∆ dz,sat , ∆ dy = ∆ dy,sat ≈ ∆ dy,ins + θ cz ∆ dx,ins -θ cx ∆ dz,sat , ∆ dz = ∆ dz,sat ≈ ∆ dz,ins -θ cy ∆ dx,ins + θ cx ∆ dy,ins . (5.26)
Equation for a c12 : oscillation of the test-masses along the Z axis (session E2)

In case of an oscillation of the test-masses along the Z axis, we can write Eq. (5.26) as

∆ dx (t) ≈ ∆ dx + θ cy ∆ d (t), ∆ dy (t) ≈ ∆ dy -θ cx ∆ d (t), ∆ dz (t) ≈ ∆ dz + ∆ d (t), (5.27)
where, to simplify, we defined ∆ dj as gathering all the static part of the offcentring along the axis j, and ∆ d (t) is the control command of the test-mass defined in Eq. (5.24). By decomposing the static part of the offcentrings and the dynamical part proportional to ∆ d (t), and by taking into account the Coriolis terms involving the angular rate of the satellite about the X axis, we obtain

Γ meas,dx ≈ B dx -(a c11 ∆ dx -a c13 ∆ dz ) S xx -(a c11 ∆ dz + a c13 ∆ dx ) S xz -(a c12 ∆ dz -a c13 ∆ dy ) Ω x -(a c12 + a c13 θ cx ) ∆ d (t) Ωx -2 (a c12 + a c13 θ cx ) ∆d (t)Ω x + (a c11 θ cy + a c13 -a c12 θ cx ) ∆d (t).
(5.28)

By considering the partial derivative of the differential equation with respect to a c12 alone we have

∂Γ meas,dx ∂a c12 ≈ -∆ d (t) Ωx + 2 ∆d (t)Ω x .
(5.29)

Equation for a g ck2 : application to the gradiometer We have an equation formally equivalent to Eq. (5.29) when calibrating the differential acceleration of the gradiometer along the X axis. Thus, the estimation of a g c12 is still possible even if in this case only one single mass is oscillated. To reject the couplings with the common mode (a g d1j , they must be corrected for prior to the estimation. However, we cannot estimate a g c21 this way, since it would require an oscillation of the satellite about the Y axis. Nevertheless, we can use an approximate equality relating the components of the matrix [ Ā] to the misalignment and the couplings. From the definition that we gave in Eq. (2.73b) in Chap. 2 and the expression of the matrix [A i ] in Eq. (2.67), we have

a g c21 ≈ η g cz -θ g cz ≈ -θ g cz ≈ -a g c12 , ( 5.30)
which is true up to an error of about 10 -4 rad.

Equation for a c13 : oscillation of the test-masses along the Y axis (session D2)

The oscillation of the test-masses along the Y axis of the instrument is projected as

∆ dx (t) ≈ ∆ dx -θ cz ∆ d (t), ∆ dy (t) ≈ ∆ dy + ∆ d (t), ∆ dz (t) ≈ ∆ dz + θ cx ∆ d (t). ( 5

.31)

As for the session E2, we take into account the Coriolis terms involving the rotation of the satellite about the X axis:

Γ meas,dx = B dx -(a c11 ∆ dx -a c13 ∆ dz ) S xx -(a c11 ∆ dz + a c13 ∆ dx ) S xz -(a c12 ∆ dz -a c13 ∆ dy ) Ωx + a c13 ∆ d (t) Ωx -2 (a c12 θ cx -a c13 ) ∆d (t)Ω x -(a c11 θ cz -a c12 -a c13 θ cx ) ∆d (t), (5.32)
where the partial derivative with respect to a c13 (alone, when not involved in products of parameters) is

∂Γ meas,dx ∂a c13 ≈ 2 ∆d (t)Ω x + ∆ d (t) Ωx . (5.33)
Fitting such a partial derivative works well since in line 4 of Eq. (5.32) we can neglect the parameter a c12 θ cx (which is of order 2) compared to a c13 (order 1).

Equation for a g ck3 : adaptation to the gradiometer For the gradiometer along the X axis, a similar partial derivative as Eq. (5.33) can be fitted to the data to estimate a g c13 , along with the terms of comparable magnitude. However, when considering the differential acceleration along the Y axis, the terms a g c22 θ g cx is of the same order of magnitude as a g c23 . In addition, with a similar rationale as in Eq. (5.30) we have the approximate expression a g c23 ≈ a g c22 θ g cx , and hence the term a g c22 θ g cx -a g c23 involved in line 4 of the gradiometer version of Eq. (5.32) is small. Therefore we cannot use the Coriolis term to estimate a g c23 . Again, the approach that we adopt is to estimate combinations of parameters instead of directly estimating a g c23 , which is involved in the term proportional to the gradients S xz , S yz and to the angular accelerations Ωx and Ωy . The terms in S xz and Ωy are too faint to be properly estimated, and they are not amplified in any planned session. However, we saw that the parameters proportional to S yz can be estimated in session B. Similarly, we can use the oscillations about the X axis in section E2 or D2 to estimate the factor of Ωx .

Quadratic parameters (sessions F and J)

Two calibration sessions are planed to determine the quadratic parameters K 21xx /K 2 11x and K 22xx /K 2 12x . It is convenient to introduce the differential and common quadratic parameters defined with the usual equations (2.50), and respectively labeled K 2dxx and K 2cxx . We can make them appear by transforming the before-last line of Eq. (2.79) as

Qdx ≈ K 2dxx K 2 1cx (Γ meas,dx -b 0dx -n dx ) 2 + (Γ meas,cx -b 0cx -n cx ) 2 +2 K 2cxx K 2 1cx (Γ meas,dx -b 0dx -n dx ) (Γ meas,cx -b 0cx -n cx ) .
(5.34)

Calibration of one sensor unit

To estimate the differential quadratic parameter K 2dxx /K 1cx , it is planed to use exactly the same equation as for a d11 (session F) to take advantage of the excitation of the common mode (and hence its square) through the oscillation of the satellite along the X axis.

The strategy adopted to estimate the common quadratic parameter K 2cxx /K 2 1cx is to separately estimate K 21xx /K 2 11x and K 22xx /K 2 11x by dedicated sessions labeled J1 and J2. During these sessions, the satellite drag compensation system is locked on one sensor while the value of the quadratic term of the other sensor is assessed. The assessment is done by forcing a high frequency sine motion of the test-mass along the X axis, modulated by a square signal. The signal of interest is ∆dx . The high frequency oscillation is chosen to be above the measurement bandwidth, so that the sinusoidal part of ∆2 dx arising in the quadratic term is filtered out. The output signal is then a square signal whose amplitude is proportional to K 2jxx /K 2 1jx . This method has been implemented recently and is not considered in this study. However, as shown in the following paragraph, this estimation can directly be used for the calibration of the gradiometer.

Simulation results

The results of the calibration of the gradiometer are shown in Table 5.2 for the X axis and in Table 5.3 for the Y axis. We present the estimation of the parameters involved in each term of the measurement equation (5.15), except the time derivatives of the offcentrings. Some of them are directly estimated from dedicated sessions, while others are a combination of estimations from different sessions.

In each table, the two last columns respectively show the estimation bias and the estimation standard deviation. As we saw in the protocol outlined in Sec. 5.3.1, concerning the X axis, all parameters and combinations of parameters can be estimated, while some information are missing for the Y axis.

For the X axis, the parameters are determined with a good accuracy and precision. By defining the error as equal to bias + 3σ, the order of magnitude of the error is 10 nm for the offcentrings along X and Z, and 0.4 µm for the effective level arm of the gradiometer a g c11 ∆ y . The common sensitivity coefficients a c1j are determined with an error less than 0.1 mrad, and the error made on the differential sensitivity coefficients is about 5 µrad. The differential quadratic parameter is estimated with an accuracy of 50 s 2 m -1 . These figures almost correspond to the expected performance of the calibration of the individual accelerometers [START_REF] Touboul | Microscope instrument in-flight calibration plan[END_REF].

For the Y axis, when the estimation is possible, the estimation errors are comparable to the X axis, except for two important parameters. The effective level arm a g c22 ∆ y is estimated with a lower accuracy, equal to a few fractions of millimeters. The estimation error of a d22 is also large, of about 40%, mainly because of the bias due to the strong common quadratic term which is not corrected for in this study. From these tables we can calculate a calibration residual for any session that we want to correct for the instrument imperfections. We consider an inertial session of about 15 orbits, and we simulate the differential acceleration Γ g d,meas measured during this session. Again, to isolate the systematic errors, we simulate the deterministic part of Γ g d,meas only. Then, we correct the data for the estimated instrumental perturbations as follows. Let A be the full model matrix containing all the time-varying partial derivatives with respect to the parameters, where the gradient terms are computed from a different model than the one used for the simulation of Γ g d,meas . We denote βcal the vector of estimated parameters from all the calibration sessions. The calibrated gradient estimate can be written Tky = -1 âckk ∆ dy Γ meas,dk -A βcal .

Parameter

3.4 • -6 a g c12 ∆ g z -a g c13 ∆ g y m D2,B -4.9508 • 10 -4 -5.1079 • 10 -4 1.6 • 10 -5 3.4 • -6 a g c13 ∆ g x -a g c11 ∆ g z m A -5.1518 • 10 -6 -5.6040 • 10 -6 4.5 • 10 -7 9.9 • -9 a g c11 ∆ g y -a g c12 ∆ g x m B -1.7466 • 10 -1 -1.7466 • 10 -1 4.2 • 10 -7 5.3 • -9 a g c12 rad E2 3.0118 • 10 -3 3.0034 • 10 -3 8.3 • 10 -6 2.0 • -5 a g c13 rad D2 -2.9124 • 10 -3 2.9245 • 10 -3 1.3 • 10 -5 2.0 • -5 a g d11 rad F -7.2928 • 10 -3 -7.2924 • 10 -3 4.1 • 10 -7 1.2 • -7 a g d12 rad H 1.0717 • 10 -3 1.0760 • 10 -3 4.3 • 10 -6 1.2 • -7 a g d13 rad G -1.1781 • 10 -3 -1.1764 • 10 -3 1.7 • 10 -6 1.9 • -7 K g dxx /K g2 1cx s 2 m -1 F 9.4718 • 10 2 9.9474 • 10 2 4.8 • 10 1 3.7 • -5
(5.36)

Then we can compute a residual calibration bias with the difference between the estimated gradient and the true (simulated) one Tky -T ky . The statistical uncertainty of the correction (only the uncertainty coming from the calibration process, excluding the stochastic noise of the current measurement) can be quantified by

Cov A βcal ≈ 1 ( âckk ∆ dy ) 2 ACov βcal A * , ( 5.37)
where the approximate equality stands for the fact that we neglected the variance of the prefactor, since the error due to the variance of (a ckk ∆ dy ) -1 is of order 5 • 10 -7 /( âckk ∆ dy ) 2 ∼ 10 -5 × T ky . To simplify, we assume that the covariance of the vector of calibration parameters βcal is diagonal (which is true if all the individual parameters are estimated from independent calibration sessions), whose entries are given by the last column of Tables 5.2 or 5.3. The Fourier representation of the residual error is shown in Fig. 5.12 for the measurement of T xy (upper panel) and T yy (bottom panel). The original bias is represented by the black dashed line, while the bias after calibration is represented by the black solid line. The blue line shows the 3σ statistical error above the bias, computed with Eq. (5.37). By comparing the black and blue solid lines in both figures, we see that the contribution of the statistical error is small with respect to the systematic error.

If we compare the upper and bottom panels of Fig. 5.12, we see that not surprisingly, the improvement between the calibrated and uncalibrated errors better for T xy . This is consistent with the better performance of the parameter estimation along the X axis, as shown by Tables 5.2 and 5.3, which results from the fact that initially the calibration sessions are tailored to calibrate the differential acceleration along the X axis. However, the errors before any calibration are smaller for T yy than for T xy , which results in a comparable post-calibration residuals for both components. In most of the frequency band, the errors from instrument defects can be reduced by at least one order of magnitude.

The periodogram of the residuals shows two main peaks at f EP and 2f EP (equal to the once and twice the orbital frequency in the case of an inertial session). This is because almost all the disturbance terms present the same peaks with various amplitudes.

Discussion

From the analysis carried out in the previous section, it follows that the calibration sessions that are scheduled in the in-flight mission plan can be useful to calibrate the gradiometer formed by the two accelerometers of the T-SAGE instrument. The preliminary results that we obtain suggests that the systematic errors can be maintained below a level of 10 mE, for both the measurement of T xy and T yy . As a comparison, the accelerometer amplitude noise level for an inertial session is less than 2 mE for T yy according to Fig. 5.9.

The approach adopted in the presented simulations gives a first idea of the performance one can reach in the calibration of the gradiometer, which is mainly focused on the estimation bias. However, it has several limitations. The first one is that the calculated values of the bias are valid for a given set of instrumental parameters. A full analysis would require to draw multiple values of the parameters in their respective possible physical intervals, in a Monte Carlo approach. A second limitation is that the noise from the drag-free and attitude control residuals is not taken into account in the assessment, which is involved in both the matrix of partial derivatives and the observations themselves. Another limitation related to the latter is that the correlations between the matrix of partial derivatives and the observations is not taken into account, which may affect both the bias and variance of the estimation. However, we performed a simulation (not shown here) with a synthetic noise simulating the accelerometer stochastic errors and the drag free residuals, which do not show a significant discrepancy with the simple assessment that we made. Nevertheless, further investigations must be carried out to confirm this result.

Summary:

We showed that the data produced by the MICROSCOPE space mission could be used in another field of gravitational science, which is gradiometry. We studied the possibility to measure the Earth's gravitational gradient by using the ep and ref accelerometers, and showed that the low frequency signature of Earth's potential as seen by the instrument has a sufficient signal-to-noise ratio for a significant measurement. Taking advantage of a different measurement bandwidth than GOCE, MICROSCOPE could provide an additional observable in gradiometry to discriminate between different models describing the large scales of the mass distribution in the Earth's deep mantle. We demonstrated that the already existing calibration sessions can be utilized to calibrate the gradiometer. The study also suggests that the estimation of the angular accelerations is the main limitation of the measurement for the T xy component, while the measurement of the T yy component is dominated by the accelerometer noise. For the latter, the sensitivity is expected to be below 1 E • Hz -1/2 at frequencies below 1 mHz, which is a better performance than GOCE in this frequency band. The systematic errors from calibration residuals are estimated to be below 10 mE at the orbital frequency. A more extended analysis remains to be made to assess the performance of the calibration and to optimize it, as well as the precision of the estimation of the angular acceleration. More sophisticated methods could also be tested such that pattern recognition techniques to compare the large-scale gravity models to MICROSCOPE data.

Conclusion and prospects

The MICROSCOPE space mission aims at testing the weak equivalence principle in space with a target precision of 10 -15 . The justification and the estimation of this precision level in flight will be as important as the result itself. The estimation and rejection of instrumental errors, the characterization of the measurement noise, the reproducibility of the measurement, are crucial tasks to validate the objective.

The main data produced by the mission consists of accelerations measured on six degrees of freedom, dominated by a coloured noise integrated over several orbits. The measured data include various modelled or possibly unknown perturbation signals and may be frequently interrupted during the long integration period. Therefore the cautious analysis of the time series that will be obtained is of utmost importance for the validation of the mission. To deal with this type of data, analysis methods were developed, which gather tools of spectral analysis, linear regression, Kriging and statistics. This research work brings out three main results that can be summarized as follows.

We showed that while the time windowing by apodization functions is effective to reject the bias caused by periodic signals with arbitrary frequencies on the WEP test, it may increase its statistical uncertainty. An alternative approach was therefore developed to reject the bias, based on a statistical test to detect the periodic signals, which is adapted to any stationary coloured noise of unknown power spectral density. Thanks to a model of the expected noise level, we show that all signals which have a significant impact on the test (i.e., inducing a bias above 10 -16 ) can be detected with almost 100% probability in most of the measurement band. As a result, their projection onto the possible WEP violation signal can be maintained below the 10 -16 level.

In a second study, we assessed the impact of data gaps which can arise in the measured time series. By relating the least-squares estimation and the spectral convolution effect, we showed that they could induce an increase of the uncertainty up to a factor 60 for random and short gap patterns, in contradiction with the mission objective. We proposed a method that we dubbed "KARMA", which provides a general way to perform precise linear regressions with large and incomplete data sets affected by unknown coloured noise.

However, the autoregressive PSD estimate may be biased at low frequency, inducing an inaccurate estimation of the precision of the WEP test. To improve the characterization of the noise, we implemented a Gaussian regression method which is valid in the context of gridded stationary Gaussian processes with missing data and unknown, arbitrary smooth PSD. It is based on a reconstruction of the data in the missing intervals by conditional expectation of their deterministic and stochastic parts. To provide consistent reconstructed time series, the reconstruction and estimation steps are reproduced several times until convergence using an ECM-like algorithm. In order to adapt the ECM algorithm to spectral density estimation, the PSD update step is performed using spectrum smoothing 172 Conclusion and prospects by a local formulation of the likelihood. This modification is referred to as the M-ECM algorithm. This leads to maximize the full likelihood by using a more general, modelindependent description of the residuals in comparison to the autoregressive approach. Still, the KARMA method is valuable since it does not rely on any imputation of missing data, and is used to provide a first guess for the PSD estimate and the regression parameter for the M-ECM algorithm.

We applied the developed tools to representative simulated data of the MICROSCOPE space mission and we demonstrated their accuracy and precision for two relevant types of gap patterns that can be encountered during the satellite flight, namely periodic gaps and stationary random gaps. Both KARMA and the M-ECM algorithm reach a precision close to the minimal variance bound, which is below 1.1 • 10 -15 with 2% data losses. This represents an improvement of a factor 4 to 60 with respect to the ordinary least squares depending on the gap pattern. We showed that in the low frequency region of the spectrum the result of the PSD estimation is improved compared to the initial autoregressive guess, leading to a better characterization of the noise with an error less than 10 -12 ms -2 Hz -1/2 . In addition, the modified ECM algorithm provides reconstructed data which are faithful to the true complete data. Indeed the reconstructed periodograms are on average consistent with the original ones within a few 10 -13 ms -2 Hz -1/2 .

Concerning the scientific objective of the MICROSCOPE mission, the results demonstrate that based on the current noise model of the accelerometers, we will be able to get a 99% (resp. 68%)-confidence level detection of a 3 × 10 -15 (resp. 1 × 10 -15 ) EP violation signal, even in the presence of missing data, for a 20 orbit-measurement session (completed in 1.4 days).

Beyond the WEP test, this research work was extended to the estimation of Earth's gravity gradient, following the philosophy to broaden the scientific return of the MICRO-SCOPE space mission. In cooperation with the National Geographic Institute (IGN), the interest of such a measurement was motivated, and a prospective study was lead to assess its performance. By using the test-masses of the two different accelerometers on board the satellite, we showed that the sensitivity of MICROSCOPE to the gravity gradient at frequencies below 1 mHz is better than the sensitivity of GOCE, and can therefore complement the data already provided by this gradiometry mission. We found that a sensitivity of 1 E • Hz -1/2 is achieved for the T yy component of the gradient tensor, where the Y axis is perpendicular to the orbital plane. We tested the signal observed by three different models of the mass structure of the deep earth mantle, and showed that they can be distinguished with a SNR of 30 dB with respect the expected noise level of a 120-orbit inertial measurement session. We also outlined a protocol to calibrate the parameters of the gradiometer whenever it is possible. This preliminary study suggests that the residual systematic error shall be below 10 mE.

Several areas of development can be foreseen to extend this research work. One of them is the possible combination of the analysis methods that were developed. Indeed, the problem of harmonic detection with missing data and coloured noise remains to be treated. This issue can easily be tackled based on the building blocks that have already been implemented. By combining the detection method of Chap. 3 and the statistical data imputation method of Chap. 4, this problem can be dealt with in the maximum likelihood framework.

Another aspect which can be deepened is the possible hybridization of the sparse inpainting technique and the M-ECM algorithm developed in this work. Such an approach would combine the flexibility of inpainting and the suitability of the M-ECM to the case of coloured noise.

Furthermore, the generality of the analysis tools that were implemented make them easily applicable to other types of analysis. In particular, a cooperation was established during this thesis with the Albert Einstein Institute in Hanover, Germany, to test the methods on data simulated for the LISA Pathfinder mission. In this technological demonstrator mission for space-based gravitational wave detection, free-flight experiments were scheduled, where the test-masses undergo periodic kicks, during which the data is highly perturbed. The aim was to recover the noise spectrum in spite of these events, which was done successfully.

Finally, the application of the data processing to gradiometry must be investigated more in-depth. The measurement of the gravity gradient with MICROSCOPE must be validated by more extensive simulations taking into account all the error sources. It must also be adjusted in the light of the real upcoming data, and more thematic-oriented algorithms must be tested. Besides, the way paved by the calibration protocol of the gradiometer can be extended to the individual accelerometers. This would meet the requirement of a 3-axis calibration which was expressed at the MICROSCOPE colloquium, due to its potential usefulness for other scientific purposes.

The KARMA and M-ECM methods were implemented in the scientific mission centre, which defines the data processing pipeline of the mission. At the time of writing, the commissioning phase of the mission has been carried out, and the first stable measurements are scheduled for mid-August 2016. As a conclusion, the application of the implemented tools on real in-flight data is the natural outcome of this thesis work. To quote a referee of a submitted paper, in such sensitive space missions, "the data analysis techniques often make the difference in extracting the maximum science return, and MICROSCOPE will likely be no exception."

A. Derivations of useful results for spectral analysis

We use the following result:

N -1 n=0 cos (αn + φ) = cos φ + α 2 (N -1) sin α 2 N sin α 2 (A.3) to get SS(f EP , f p , φ EP , φ p ) = 1 2 cos π f s (f EP -f p ) (N -1) + φ EP -φ p sin π fs (f EP -f p ) N sin π fs (f EP -f p ) -cos π f s (f EP + f p ) (N -1) + φ EP + φ p sin π fs (f EP + f p ) N sin π fs (f EP + f p ) (A.4)
We can also write

A * A = g 2 0 • SS(f EP , f EP , φ EP , φ EP ) = g 2 0 2 N -cos 2π f s f EP (N -1) + 2φ EP sin 2π fs f EP N sin 2π fs f EP . (A.5)
The second term in this expression is zero when there is an integer number of EP period in the signal, giving A * A ≈

A.2 Derivation of the envelope function of the maximal projection rate

In this section we derive an approximate expression of the envelope function of the maximum projection rate of Eq. (3.33), that we reproduce here for convenience: Our approach is to calculate the envelope function of g. Then the envelope of τ max will the sum of the envelopes of the two terms in Eq. (A.6). The function g can be written as a discrete sum, by using Eq. (A.

τ max =

A.5 Expectation of the periodogram as a function of PSD

The aim of this section is to compute the expectation of the windowed periodogram I N,w defined in Eq. (3.64) of a time series y as a function of the power spectral density (PSD). In the following we assume that y is zero-mean.

First of all we need to define the autocovariance estimate of a signal x, which is:

Rx (n) ≡ 1 N N -1-|n| i=0 x i x i+|n| (A.21)
Similarly as the relationship between the PSD and the autocovariance in Eq. (3.13), one can show (see e.g. [START_REF] Priestley | Spectral analysis and time series[END_REF] pp. 398-399) that the periodogram of any signal x can be written on the form: This is what we wanted to show: the expectation of the periodogram of a windowed signal is equal to the convoltion of the PSD of this signal with the periodogram of the window.

I N,x (f ) = 1 f s (N -

A.6 Derivation of the local linear estimator

The aim of this section is to derive the expression of the local linear estimator, that is, the least square estimator that minimizes the following quantity for each frequency f 0 :

n i=1 (Y i -a -b (f i -f 0 )) 2 K f i -f 0 h , (A.30)
where • Y i is the observed data (log-periodogram minus C 0 in our case)

• f i are the Fourier frequencies of the sample

• n is the number of strictly positive Fourier frequencies n ≡ N -1

2

• f 0 is the frequency at which the PSD estimator is calculated

• K is the kernel function (to be chosen)

• h is the smoothing parameter, or bandwidth. It determines the amount of neighboring data that will used to compute the estimate.

The vector to be estimated involves the coefficient of a simple affine function:

β = g(f 0 ) g (f 0 ) = a b (A.31)
The solution of the least squares problem for each frequency point f 0 is given by β = X T W X -1

X T W y, (A.32)

Where:

• X is the design matrix defined by X lj = (f l -f 0 ) j-1

• W is the diagonal weight matrix defined by W ii = K f i -f 0 h

• y is the data vector defined by y = Y 1 . . . Y n T

We can write the matrix S n = X T W X with entries:

S n (f 0 ) = s n,0 s n,1 s n,1 s n,2 (A. [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF] with:

s n,j = n i=1 (f i -f 0 ) j K f i -f 0 h (A.34)
The inverse of the matrix S n can then be easily computed:

S -1 n = 1 s n,0 s n,2 -s 2 n,1
s n,2 -s n,1 -s n,1 s n,0 (A.35)

Hence, according to Eq. (A.32) the expression of the estimator is given by:

â = β0 = 1 s n,0 s n,2 -s 2 n,1 n i=1 s n,2 Y i -s n,1 Y i (f i -f 0 ) K f i -f 0 h = n i=1 K T f i -f 0 h , f 0 Y i . (A.36)
where we defined K T as the effective kernel involved in the first line of Eq. (A.36):

K T ≡ 1 s n,0 s n,2 -s 2 n,1 n i=1 s n,2 -s n,1 (f i -f 0 ) K f i -f 0 h (A.37)

A.7 Cramér-Rao lower bound for the estimation of the frequency of unknown harmonics

We consider the Gaussian model of Eq. (3.83) where the covariance describes a stationary noise with PSD S(f ) and where the mean µ describes a harmonic signal of unknown amplitude A p , phase φ p and frequency f p . The aim of this section is to calculate the minimum achievable variance for any unbiased estimator of f p (Cramér-Rao lower bound). For a general probability distribution p θ (y), depending on a set of parameters θ, for any unbiased estimator of a function g(θ) of the parameters, the CRLB is such that: Var [g(θ)] ≥ ∇g T I(θ) -1 ∇g (A. [START_REF] Durbin | The fitting of time-series models[END_REF] where I(θ) ≡ E s θ (y)s θ (y) T is the Fisher information matrix, and s θ (y) is the gradient of the log-likelihood function:

s θ (y) ≡ ∇l y (θ) = ∇ ln p θ (y) (A.39)

For the Gaussian probability distribution, one can show that the elements of the Fisher information matrix can be written as [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF] I Thus the matrix I µ can be written as a four-block matrix:

I µ = A * Σ -1 A A * Σ -1 v v * Σ -1 A v * Σ -1 v , (A.44)
where we defined v ≡ ∂A ∂f p β.

(A.45)

The inverse of the Fisher matrix can be written as

I(θ) -1 = I -1 µ 0 0 I -1 Σ , (A.46)
where an expression of I -1 µ can be found by using the formula for inverse block matrices applied to Eq. (A.44). We are interested in the frequency of the harmonic signal only. So we choose g(θ) = f p to select the entries of I(θ) -1 corresponding to the parameter f p . Then by using Eq. (A.38) and the block decomposition of I(θ) -1 , we obtain an expression of the CRLB:

Var fp ≥ v * Σ -1 v -v * Σ -1 AC β A * Σ -1 v -1 , (A.47)
where C β = A * Σ -1 A -1 is the CRLB of β. Eq. (A.47) provides an expression of the optimal variance of an unbiased estimate of f p .

When n = n , E w 2 n = σ 2 w + µ 2 w . When n = n , the value of the mask being independent of time, E [w n w n ] = µ 2 w . Plugging these results into Eq. (A. [START_REF] Fisher | Tests of significance in harmonic analysis[END_REF] gives

E [I w (f )] = 1 f s σ 2 w + µ 2 w • 1 N f s N -1 n=0 N -1 n =0 e -2πf (n-n )/fs = 1 f s σ 2 w + µ 2 w • I 1 (f ) (A.53)
were we noted 1 the vector whose entries are all equal to 1. Thus I 1 (f ) is the periodogram of the rectangular window (which is actually equal to the Fejér kernel). Then we plug this result into Eq. (A.51) to obtain

ĪWy (f ) = fs 2 -fs 2 S(f ) µ 2 w • I 1 (f ) f -f + 1 f s σ 2 w df . (A.54)
We note that the left-hand side term is proportional to the convolution of the PSD with the periodogram of the rectangular window. Therefore, this term is proportional to the expectation of the complete data periodogram. As for the right-hand side term, it is proportional to the average power, which is equal to the variance σ 2 of the noise. where we defined the quantities f p n ≡ ẑF n -z n and b p n ≡ ẑB n -z n . Burg's algorithm is a recursive method, since we can find the estimates of the coefficients of an AR(k) model from the estimates of the fit of an AR(k -1) model. The recursion formula has the form

a k+1 = a k 0 + µJ a k * 0 (B.1)
where a is the vector of entries a i ∀i ∈ 0, k (setting a 0 = 1) where µ is called the reflexion coefficient, and is found by minimizing the residuals corresponding to order k + 1. J is the exchange matrix, reversing the order of the components of any vector:

J ik = 1 if i = N -k -1, 0 otherwise.
One can show [START_REF] Collomb | Burg's method, algorithm and recursion[END_REF] that the reflexion coefficient minimizing the sum of the residuals F k + B k is

µ k = - 2b k * f k f k * f k + b k * b k , (B.2)
where f k and b k are vectors of size N -k defined by

f k ≡ f k k+1 f k k+2 . . . f k N -1 * b k ≡ b k 0 b k 1 . . . b k N -k-1 * (B.
3)

The estimation of the variance scale is given by the mean residuals:

σ2 = 1 N -k N -1 n=k f p k 2 (B.4)
Now we adapt the Burg's recursion to the case where data are missing. Consider a time series with missing data, with S -1 data gaps. Then there are S segments of observed data in-between the gaps. The idea is, for a given order k in the recursion, to fit the AR coefficients using all the segments of observed data. Therefore we simultaneously minimize the residuals s F s k + s B s k for each segment s of size N s larger than the order k. As a result, at each iteration of the recursion, the reflexion coefficient is now

µ k = - S s=1 2b k,s * f k,s S s=1 f k,s * f , s k + b k,s * b k,s , (B.5)
We see that the highest order of the AR model that can be fitted to the data is limited by the size of the longest segment of consecutive observed data.

B.2 Details of the Kalman filter for othogonalization

The goal of this section is to establish the state equations that describe an autoregressive process of order p. We want to show that:

x(n) = Fx(n -1) + G (n) z(n) = H T x(n)
where F and H are respectively defined by Eqs. (4.38,4.39) and the state vector is defined as

x(n) = z n z n+1|n . . . z n+p-1|n T ,
The state vector contains the observation z n and the predictions until time t n+(p-1) Thus, the covariance matrix Q = GG T of the model depends on this definition, and must be calculated.

We recall that the AR model writes

z n = p k=1 a k z n-k + n ,
where n is a white zero-mean Gaussian process with variance σ 2 . The prediction at the following time n + 1 given the data until time n is defined by the expectation E [z n+1 |z s , s ≤ n]. There fore it writes:

z n|n-1 = p k=1 a k z n-k
The "j-step forward" prediction given the data until time t n is written as a function of the predictions and the data already available as follows: We set g j such that t g j ≡ z n+j-1|n -z n+j-1|n-1 .

The series g j actually follows the recursion g 1 = 1, (B.8)

g j = j-1 k=1
a k g j-k for 2 ≤ j ≤ p. (B.9)

We have just shown that for j < p we have z n+j-1|n = z n+j-1|n-1 + g j t , (B.10) where g j is a series defined by the recursion (B.9). For j = p we also have z n+p-1|n = z n+p-1|n-1 + g p t , = p k=1 a k z n+p-k-1|n-1 + g p t .

By defining G = (g 1 , . . . , g p ) we obtain the state equation (B.6), which is what we wanted to get.

B.3 Kalman filter equations with missing data

In this section we review the Kalman equations adapted to the autoregressive problem, with an emphasis on missing observations. No new derivations are presented here, this is a summary of Refs. [START_REF] Gómez | Estimation, prediction, and interpolation for nonstationary series with the kalman filter[END_REF][START_REF] Richard | Maximum likelihood fitting of arma models to time series with missing observations[END_REF]. We review the initialization process, the update equation, the prediction equation, and the observation equation.

Initialization

We denote x(n|q) the expectation of the state vector at time t n given all the data observed until time t q :

x(n|q) ≡ E [x(n)|z 1 , . . . , z q ] . The initialization of the state vector is the null vector in the case of a zero-mean AR process:

x(1|0) = 0.

We denote Σ(n|q) the covariance of the state vector at time t n given all the data observed until time t q : Σ(n|q) ≡ Var [x(n)|z 1 , . . . , z q ] . The calculation of the initial covariance involves the AR parameters and is not detailed here, see the work of Jones [START_REF] Richard | Maximum likelihood fitting of arma models to time series with missing observations[END_REF] for a more complete derivation.

The iterative equation of the Kalman filtering are described below, for n < N :

Prediction equation

x(n|n -1) = F x(n -1|n -1), Σ(n|n -1) = F Σ(n -1|n -1)F T + Q, where Q = GG T .

Update equation

K(n) = Σ(n|n -1)H H T Σ(n|n -1)H -1
.

If z n is observed, we update the state vector and its variance with Σ(n|n) = Σ(n|n -1) -K(n)H T Σ(n|n -1),

x(n|n) = x(n|n -1) + K(n) z n -H T x(n|n -1) .

If the observation z n is missing, then the information from times until t n is the same as the information until time t n-1 , and we have Σ(n|n) = Σ(n|n -1),

x(n|n) = x(n|n -1).

Predicted state mean and variance

The outputs of the Kalman filter that are useful to the computation of the likelihood or the pseudo generalized least squares are the predicted state at time t n and its covariance ẑn|n-1 = H T x(n|n -1), σ2

n|n-1 = H T Σ(n|n -1)H.

Note that these outputs are ignored if the data is missing at time t n .

C

Conditional distribution in spectral analysis

C.1 Calculation of the prediction error

In this section we derive the proof of Eq. (4.61) giving the prediction error when estimating the missing data vector y m by its conditional expectation on the observed data vector y o . We assume that the expectation of the complete data µ and its covariance Σ are perfectly known. We want to calculate the mean squared error E [ m * m ] where the vector m = y m -µ m|o is the residual of the missing data estimation. We can develop: which is what we wanted to prove. It turns out that the prediction error is equal to the conditional covariance of y m given y o .

C.2 Calculation of the conditional expectation of the periodogram

Here we derive the expression of the conditional expectation of the periodogram of the noise, which is computed by using the model residuals

z = y -Aβ. (C.7)
The periodogram of the noise calculated at the Fourier frequencies f k then writes

I z (k) = 1 N N -1 n=0
z n e -2Iπkn/N 2 .

(C.8)

We first note that I z (k) is the diagonal element of the matrix z z † where z = F N z is the normalized discrete Fourier transform of the residual vector z. This vector can be decomposed into an observed part and a missing part:

z = W † o z o + W † m z m , (C.9)
where W o and W m are indicator matrices defined in Eq. (4.18). Thus we have:

zz † = W † o z o z † o W o + W † m z m z † m W m + W † o z o z † m W m + W † m z m z † o W o .
By taking the conditional expectation of this equation we get: Injecting this expression into Eq. (C.10) we obtain:

E zz † |y o = W † o z o z † o W o + W † m E z m z † m |y o W m +W † o z o µ † zm|o W m + W † m µ zm|o z † o W o , (C.
E zz † |y o = W † o z o z † o W o + W † m µ zm|o µ † zm|o W m +W † m Σ m|o W m + W † o z o µ † zm|o W m +W † m µ zm|o z † o W o .
To simplify this equation, we define the reconstructed residuals as:

ẑ ≡ µ z|o = W † o z o + W † m µ zm|o , (C.12)
whose elements are equal to z i when y i is observed and to its conditional expectation µ z|o,i when y i is missing. Using this definition we get:

E zz † |y o = ẑ ẑ † + W † m Σ m|o W m . (C.13)
Thus the conditional expectation of z z † is given by the matrix:

E z z † |y o = z z † + F N W † m Σ m|o W m F † N . (C.14)
The conditional expectation of the noise periodogram is finally given by the diagonal elements of the matrix C. where we defined σ 2 k as being the diagonal element of the matrix

F N W † m Σ m|o W m F † N .

C.3 Conditional generation of the periodogram

The purpose of this appendix is to efficiently compute realizations of a random vector whose mean is the conditional expectation of the periodogram given by Eq. (C.15). We consider the vector z * of size N drawn from the distribution N (0, Σ). As mentioned in [START_REF] Dietrich | A fast and exact method for multidimensional Gaussian stochastic simulations: Extension to realizations conditioned on direct and indirect measurements[END_REF], the vector of size N m constructed as

z * m|o = µ z|o + W m z * -Σ mo Σ -1 oo W o z * (C.16)
has mean µ zm|o and covariance Σ m|o . We then construct the vector:

z * |o = W † o z o + W † m z * m|o (C.17)
We verify that this vector has the desired covariance as given by Eq. (C. To compute a reliable approximation of the expectations involved in the above equation we need to draw several realizations of the vector y. For each realization the conditional expectation E [I y |y o ] requires to generate N d conditional draws of the missing data. We aim to check that increasing N d improves the ability of the left-hand side of Eq.(C.18) to approximate the right-hand-side.

For the computational cost of this simulation to be acceptable we reduce the size of the problem to N = 1000 data points, and we modify the PSD model in Eq. (2.63) to shift its minimum to higher frequency in order for it to be visible in the shortened observation bandwidth. In the following we label the PSD of this "toy model" S (f ) and all related quantities are identified with a prime.

As explained in Sect. 4.4.5.2 multiple imputation of missing data allows us to approximate the second term σ 2 in Eq. (4.70). To amplify its relative amplitude with respect to the first term I ẑ we sharpen the "hollow" of the PSD shape by increasing the high frequency slope:

S (f ) = α 0 + α -1 f -1 + α 5 f 5 (C.19)
We choose a random gap window w representing 10% data losses -still with the idea of increasing the difference between the two terms of Eq. (4.70). To simplify and focus on the PSD estimation we remove the deterministic part of the model (i.e. A = 0, so that y = z).

We generate 400 realizations of a noise vector z whose PSD is given by Eq. (C. [START_REF] Carter | Semiparametric Bayesian Inference for Time Series with Mixed Spectra[END_REF]). Then we run the M-ECM algorithm for all realizations of the observed data z o to obtain the conditional periodograms and the PSD estimates. We do it twice: for N d = 5 and for N d = 100.

We finally calculate the sample average of the conditional periodograms and of the PSD estimates obtained with the two numbers of MC draws and show the result in

Principle of sparse reconstruction

We already mentioned that inpainting, which is a very different method than the M-ECM algorithm, was also applied to MICROSCOPE mock data. In this section we give some details about this technique.

In this section, we denote x(t) the inpainting solution (the complete signal we are looking for), y(t) the masked measured time series and w(t) the mask function. y(t) is only available at times where w(t) is non zero, such as y(t) = w(t)x(t). The sparse inpainting method is based on the assumption that there exists a representation Φ * of x where most coefficients α = Φ * x are close to zero.

The inpainting algorithm solves the following problem: The solution to problem (D.1) is found by a recursion which begins to transform the masked data into the dictionary, and then finds the parameters α i which are above a certain threshold λ. The other parameters are set to zero. Then this minimal sparse vector α is used to reconstruct the data in the time domain. The reconstructed data is then transformed again in the dictionary, in order to select more coefficients, above a lowered threshold. The recursion goes on this way with an increasing number of selected coefficients and a decreasing selection threshold, until the final iteration. If the dictionary Φ * is a Vandermonde matrix (if there exists a constant c such as Φ * Φ = cI), the algorithm can be computationally efficient.

From the formulation of problem (D.1) we see that the algorithm is based on the assumption that the noise is white (it is only defined by its zero-lag variance), and that the deterministic part of the signal is sparse in the dictionary Φ * . In Ref. [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF] we apply the inpainting algorithm to colored noise by taking the discrete cosine transform (DCT) matrix for Φ * . In spite of the violation of the white noise hypothesis, the algorithm then well recovers the original noise level. This can be interpreted as follows. The algorithm treats each noise spikes as a harmonic signal, allocating a non-zero coefficient in the spectrum α. This results in a representation of the signal which is not sparse, but allows a fair reconstruction of the original signal. In Ref. [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF] we apply a least-squares estimator to the inpainted data to measure a simulated WEP violation. We showed that the KARMA and the inpainting methods have similar performances in terms of variance. However, the estimates obtained from inpainted data show a slight bias, depending on the amplitude of the simulated signal (see also Ref. [START_REF] Pires | Dealing with missing data II: An updated version of the inpainting method to the MICROSCOPE space mission[END_REF]). In any case, the inpainting technique provides another reconstruction process based on a modeling which is different from, and more general than, the presented M-ECM algorithm.

D. Sparse inpainting and coloured noise

As an example, we applied the inpainting algorithm to the noise data used in the former studies. The inpainted data is shown in Fig. D.1. The noise level of the reconstructed time series is significantly reduced in the low frequency region with respect to the masked data. While the original spectrum is not fully recovered, the computation is very efficient and scales with O(N it N log(N ), where N it is the number of iterations, which is typically N it ∼ 100.

An improved version of the inpainting algorithm, dubbed "Inpainting for COlored-Noise dominated signals" (ICON), was implemented to better deal with the colored shape of the spectrum [START_REF] Pires | Dealing with missing data II: An updated version of the inpainting method to the MICROSCOPE space mission[END_REF]. The major improvement consists in adding a prior on the noise power spectrum directly derived from the data. This version enables us to lower the level of the residual noise by a factor 20 in PSD (which corresponds to an improvement of 4.5 with respect to the level of This effort was still made with the objective to inpaint non-sparse data (a colored noise) with an algorithm based on a white noise assumption. The bias observed in Refs. [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF][START_REF] Pires | Dealing with missing data II: An updated version of the inpainting method to the MICROSCOPE space mission[END_REF] may come from the invalidity of this hypothesis. Another strategy would be to inpaint the "sparsified" data, by pre-whitening the data, to come back to problem (D.1). In the following section we explore the possibility to adapt the inpainting algorithm to colored noise, and we link this analysis with the M-ECM framework.

D.1.2 Inpainting and EM algorithms: an adaptation to colored noise

Fadili, Starck and Murtagh [START_REF] Fadili | Inpainting and Zooming Using Sparse Representations[END_REF] showed that the sparse inpainting algorithm of problem (D.1) can be formulated as an EM algorithm in the white noise case. Indeed, this problem can be reformulated as a penalized maximum likelihood problem, where the function to minimize is the normalized residuals plus a penalty function: α = argmin where we used the subscript o introduced in Sec. 4.3.1 which designates the observed data (multiplication by the indicator matrix W o ). We also used the norm z 2 Σ ≡ z * Σ -1 z which normalizes by the inverse of the noise covariance matrix. Note that in the formulation of Eq. (D.3) we implicitly defined the covariance of the observed residuals as Cov (z o ) = σ 2 Σ oo .

We note that the ECM algorithm developed in Sec. 4.4 solves a similar problem as in Eq. (D.3), but with no penalty (λ = 0) and a much restrictive dictionary of signals (Φ = A).

Therefore, we can think of an adaptation to the case of correlated noise, by combining the ECM algorithm developed in this thesis and the sparse inpainting algorithm.

D.2 Sparse inpainting and the EM framework in the case of couloured noise

In this section we propose an adaptation of the sparse inpainting to the colored case by an ECM algorithm solving the penalized maximum likelihood problem stated in Eq. (D.3) We modify the formulation of the algorithm proposed in Ref. [START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF][START_REF] Fadili | Inpainting and Zooming Using Sparse Representations[END_REF] to the case where the observations are correlated. The important point is to note that solving problem (D.3) amounts to solving the original sparse inpainting problem (D.1) applied to e ≡ L -1 x is one, if L is the Cholesky decomposition of the covariance of x. The penalized likelihood problem that we want to solve can then be written as α = argmin where we defined Φ = L -1 Φ. This sparse inpainting problem can then be solved by a recursive soft-thresholding process (see, e.g., [START_REF] Starck | Sparse Image and Signal Processing[END_REF][START_REF] Elad | Sparse and Redundant Representations[END_REF]).

D. Sparse inpainting and coloured noise

In the white noise inpainting, the conditional expectation of the data at iteration i reads

y i = wy + (1 -w)x i (D.5)
where w is the mask window and x i is simply given by the inverse transformation x i = Φα i . However, in the colore noise case, the noise correlations must be taken into account and we have

x i = E y m |y o , θ i = Φ m α i + Σ mo Σ -1 oo y o -Φ o α i (D.6)
Initially, the noise covariance is unknown, and hence L is unknown. It must be estimated by the algorithm itself. We see that based on the previous work, an hybridization of the PSD estimation process and the inpainting algorithm can be constructed.

The modified inpainting recursion would be CM2 Obtain the new estimate of the spectrum Ŝi+1 .

3. Repeat this scheme until convergence, i.e., until y i+1 -y i Σ is lower than some specified amount.

E

Useful information for gradiometry

E.1 Useful coordinate systems

To describe the transformation from the terrestrial coordinate system to the instrument coordinate system, it is convenient to define four frames, illustrated in

E.2 Rotation matrix

In this section we detail the computation of the matrix R to transform the gradient from the terrestrial frame into the instrument frame, according to Eq. (5.4). R includes several rotations, which, as a first approximation, can be reduced to four main rotations: where we set ω e = Ω -θ.

We find that the T xy component of the gravitational gradient measured in the (perfect) instrument reference frame is and we have similar expressions for the T yy and T zy components. We see that the measured gradients in the instrument frame are a linear combination of the gradients components in the terrestrial frame.

T xy = -

E.3 Orders of magnitude of the terms in the measurement equation

In this section we give a rough order of magnitude of the various terms involved in the measurement equation, which allows us to determine the dominant term which must be fitted in the calibration sessions. On étudie en premier lieu la détection et l'estimation des perturbations harmoniques. On montre qu'en les incluant dans le modèle de la mesure, leur projection sur le signal de violation du PE peut êtrerejetée. On analyse ensuite l'impact des pertes de données sur la performance du test du PE. On montre qu'avec l'hypothèse pire cas sur la fréquence des interruptions de données l'incertitude des moindres carrés ordinaires peut croître de plus d'un ordre de grandeur. Pour compenser cet effet, une méthode de régressionlinéairebasée sur une estimation autorégressive du bruit est développée, qui permet de décorréler efficacement les observations disponibles, et de garantir la précision du test au niveau attendu. On met également en place une méthode pour caractériser la densité spectrale de puissance du bruit à partir des données disponibles, grâce à une modification de l'algorithme espérance-maximisation. En dernier lieu, on étend les applications de l'analyse de données en démontrant la faisabilité de la mesure du gradient de gravité terrestre avec MICROSCOPE.

Mots Clés

Test expérimental de la gravitation, instrumentation spatiale, analyse de données, séries temporelles, données manquantes.

  Figure 1.2:Gravitational redshift, as a consequence of the EEP. Left-hand side: a photon is emitted from a rocket moving with constant acceleration a towards another rocket undergoing the same acceleration. When reaching the leading rocket the photon will be redshifted because of conventional Doppler effect. Right-hand side: if the photon is emitted from the ground, in an homogeneous gravitational potential, according to the EEP it will be equivalently redshifted, leading to a gravitational version of the Doppler effect.

Figure 1 . 3 :

 13 Figure 1.3: Precision of the WEP test as a function of the epoch, excerpted from [147].

Figure 1 . 4 :

 14 Figure 1.4: Principle of the torsion balance test as performed by Eötvös and his collaborators, excerpted from [137]. On the left:-→ F 1 and -→ F 2 represent the external forces applied on each test-mass, and include the weight and the axifugal force (typically the inertial force due to Earth's rotation). A violation of the WEP would result in a torque induced on the suspension wire, leading to a deviation angle of the rod with respect to its original position. On the right: representation of the inertial force due to the earth rotation at latitude Θ.

Figure 1 . 5 :

 15 Figure 1.5: Scheme of the Eöt-Wash WEP experiment. Credits: [137].

Figure 1 . 6 :

 16 Figure 1.6: Apollo 15 LLR retroreflector array (left) and the LLR retroreflector sites on the moon (right). Credits: [148].

Figure 2 . 1 :

 21 Figure 2.1: Illustration of the principle of the MICROSCOPE space mission. Two cylindrical and concentric test-masses of different compositions (represented in grey) are freely falling around Earth. The gravitational acceleration g is shown by the red arrows. If the WEP is violated the test-masses will feel this acceleration differently, with a periodic difference due to the projection of g on the sensitive x axis. This period is different for an inertial session (left-hand side) and a spin session (right-hand side). The orientation of the instrument axis is shown by the blue arrows.

Figure 2 . 2 :

 22 Figure 2.2: Architecture of the MICROSCOPE ground segment. The vertical layers correspond to centres and working teams. The horizontal layers indicate the type of analyses which are carried out by the teams and their time basis.

Figure 2 . 3 :

 23 Figure 2.3: Artist view of the MICROSCOPE satellite. CNES courtesy

Figure 2 . 4 :

 24 Figure 2.4: Scheme of the attitude and orbit control loop

  One sensor's core

Figure 2 . 5 :

 25 Figure 2.5: Left-hand side: picture of the two sensor units; right-hand side: scheme of the differential accelerometer core including a tight housing with a ring of hermetic connectors, four cylindrical silica supports for electrodes, two concentric test-masses and a blocking system.

  Y and Z axis

Figure 2 . 6 :

 26 Figure 2.6: Configuration of the electrodes of one test-mass of the T-SAGE instrument. The axial and Φ control uses variations of the surface of the test-mass facing the electrodes (left) whereas the radial control is based on a variation of the gap between the test-mass and the electrode (right). From [71].

3 Figure 2 . 7 :

 327 Figure 2.7: Scheme of the test-mass control loop of an electrostatic accelerometer.

Figure 2 . 8 :

 28 Figure 2.8: Representation of the capacitive sensor, where the test-mass is shifted by a displacement y with respect to its balance point. Extract of Ref. [75].

Figure 2 . 10 :

 210 Figure 2.10: Scheme of a sigma-delta analog-to-digital converter

Figure 2 . 11 :

 211 Figure 2.11: Representation of the test-mass control by variation of the gap. The electrostatic forces pull the mass towards the electrodes on both side of the cage but with a different magnitude. The resulting force tends to drive the mass at its equilibrium point.

Figure 2 . 13 :

 213 Figure 2.13:Representation of the digital control loop. In the clockwise direction, starting from the top left: the external acceleration Γ ext corresponds to a position of the test-mass (by double integration), which is converted into a detection voltage by the capacitive sensor, then transmitted to the controller which calculates the actuation voltage necessary to apply the corrective acceleration Γ app via the actuation gain G act . The measured acceleration Γ meas is given by the controller output after filtering and proper re-scaling.

.63) 2 .Figure 2 .

 22 Figure 2.14:Model of the noise power spectral density on the X axis taking into account all the contributions, for the internal (dashed red) and external (dashed blue) masses, and for the differential mode (solid black). The plot on the left is obtained by ignoring the filters (there is no cut-off, only the attenuation due to the control loop transfer function) whereas the plot on the right takes the effect of the Butterworth and averaging filters into account. The fact that the black curve is lower than the red and the blue curves comes from the factor 1/2 in the definition of differential acceleration.

3 3 . 1

 31 Treatment of interfering signals and minimization of the measurement bias Contents Estimation of parameters by linear regression . . . . . . . . . 58 3.1.1 Estimation principle . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.2 Estimation bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1.3 Estimation uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2 Rejection of particular harmonic perturbations . . . . . . . . 63 3.2.1 Projection of a harmonic signal onto the EP frequency . . . . . . 64 3.2.2 Minimization of the projection rate for particular frequencies . . 66 3.3 Rejection of harmonic perturbations by time windowing . . . 67 3.3.1 Bias rejection with windowing . . . . . . . . . . . . . . . . . . . . 67 3.3.2 Impact of windowing on the estimation uncertainty . . . . . . . . 68 3.4 Rejection of harmonic perturbations by inclusion in the model 71 3.4.1 Inclusion of a harmonic signal in the model . . . . . . . . . . . . 71 3.4.2 Required precision on the disturbing frequency . . . . . . . . . . 72 3.4.3 Harmonic detection in the presence of noise . . . . . . . . . . . . 74 3.5 A harmonic detection and estimation method in coloured noise 77 3.5.1 Estimation of the noise PSD . . . . . . . . . . . . . . . . . . . . 77 3.5.2 Detection test based on the periodogram . . . . . . . . . . . . . 82 3.5.3 Refinement of the frequency estimate of a detected harmonic signal 85 3.5.4 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . 88

Figure 3 . 1 :

 31 Figure 3.1: Absolute value of the projection rate of a disturbance signal of frequency f p = 10 -2 Hz onto the WEP signal at f EP = 9.35 • 10 -4 Hz in a 20-orbit spin session as a function the phases φ EP and φ p expressed in fractions of π. The projection rate shows minima and maxima depending of the phases.

2 ]Figure 3 . 2 :

 232 Figure 3.2: (a) Maximum of the projection rate for a spin session (black) as a function of the frequency of the disturbing signal, and its envelope function (red). The plot is log-scaled, hence the zeros of τ max are not visible. (b) amplitudes of disturbance signals inducing a bias of 10 -16 on the estimation of δ. The gray area corresponds to signal amplitudes inducing a bias lower than 10 -16 .

10 - 5 6 Figure 3 . 3 :

 5633 Figure 3.3: Comparison of the projection rates obtained with the OLS estimator (black) and the windowed least-squares estimator of Eq. (3.39) with the Hann window (blue). The calculation is performed for a 120 orbits inertial session, with φ EP = φ p = π/4.

12 FrequencyFigure 3 . 4 :

 1234 Figure 3.4: Impact of apodization on the least-squares variance per frequency. The standard deviation σ 2k given by Eq. (3.48) is plotted with respect to frequency for the OLS estimator (black) and the windowed least-squares estimator of Eq. (3.39) with the Hann window (blue). The calculation is performed for a 20 orbits inertial session, with φ EP = π/4. The area under the curve gives the total standard deviation on the estimation of the WEP parameter.

Figure 3 . 5 :

 35 Figure 3.5: Bias rate b 0 ( fp , f p ) as a function of the true frequency of the harmonic signal f p , for different values of the error ∆f = fp -f p : ∆f = f s /N , 10 -6 , 10 -7 , 10 -8 Hz (increasingly light grey scale). The original bias (before any correction) is shown by the red line. The calculations are done with φ p = 0, f EP = 9.35 • 10 -4 Hz, N = 470602 (20-orbit-long spin session). Note that here the bias rate is not maximized with respect to the phase, contrary to Fig. 3.2.

Figure 3 . 6 :

 36 Figure 3.6: Bias rate on f EP as a function of the fitting frequency for f p = 10 -2 Hz, for a spin session (f EP = 9.35 • 10 -4 Hz). The grey dashed line represents the projection rate of the OLS estimator, when fitting the WEP signal only. The black solid curve is the bias of OLS estimator when fitting the WEP signal and a harmonic signal of frequency fp . The red line indicates the location of the disturbing frequency, the blue dashed line indicate the natural Fourier frequencies closest to f p .

Figure 3 . 7 :

 37 Figure 3.7: Expectation (dashed lines) and 99%-confidence interval (shaded areas) of the local linear estimator applied to the log-windowed-periodgram (with Hann windowing), for a spin (blue) and inertial (red) session. Since spin sessions are shorter, the variance of the PSD estimate is increased at low frequency.

. 75 )

 75 where Ŝ(f k ) is the local linear PSD estimator defined by Eq. (3.70) in Sec. 3.5.1.3. Since the denominator of the individual ratios

Figure 3 . 8 :

 38 Figure 3.8: Histogram of the distribution of the statistics Z k for f k = 10 -2Hz under H 0 (gray) computed with 400 samples compared with the probability density function of half the chi-squared distribution (black solid curve). The empirical distribution is well described by the chi-squared model.

Figure 3 . 9 :

 39 Figure3.9: Power of the 99%-confidence test for harmonic peaks as a function of the amplitude and frequency of the harmonic signal. The white dashed line represents the amplitude detection threshold of Eq. (3.81). It is the line where the ratio of the peak power over the PSD is larger than the 99%-detection threshold of Eq. (3.63). However, if a sine with such power is present in the data, it will have between 10% and 20% chance to be detected, since it lies in the dark blue region. The white solid line defines the region of high power (amplitudes of the harmonic signals which, if present, will be detected with a probability larger than 90% in most of the spectral band).

Figure 3 . 10 :

 310 Figure 3.10: Cramér-Rao lower bound for the frequency estimation of unknown harmonics signals as a function of their amplitude a p and frequency f p . Some contour lines are shown for particular values of the CRLB (in Hz).

Figure 3 . 11 :

 311 Figure 3.11: Theoretical rejection of the bias by harmonic detection, for amplitudes corresponding to the powerful limit (a) and 10 times the powerful limit (b). The red curve is the original bias on the WEP parameter estimate as a function of the frequency f p of the harmonic signal. The black curve shows the bias after detection and estimation of the harmonic signal, assuming an error on f p equal to 3 times the CRLB.

Figure 3 . 12 :

 312 Figure 3.12: Example of the detection of a harmonic signal at f p = 2.1 • 10 -2Hz for an amplitude equal to the detection threshold (a) and to 2.5 times the threshold (b). In the first case the peak is not visible, and the algorithm does not detect the signal. In the second case the detection is positive with a confidence close to 100% (the estimated SNR is 43).

90 3 .FrequencyFigure 3 . 13 :

 3313 Figure 3.13: Empirical power of the test (probability to have a positive detection when H 1 is true). Blue bars: signals of set A with amplitudes equal to the detection threshold. Red bars: signals of set B with amplitudes equal to 2.5 times this threshold.

  correction Original bias at fp Maximum bias after correction (a) Frequency estimation (b) Bias rejection

Figure 3 . 14 :

 314 Figure 3.14: Detection and estimation of the frequency of hidden harmonics from set B, for amplitudes 2.5 × a α (f p ) and arbitrary frequencies f p . Panel (a): estimates of the frequencies. The gray dots are the average frequency error computed over all of the detected frequencies among the 400 tests. The blue dots represent the sample standard deviation. The dashed black line indicates the Fourier frequency resolution, and the red dashed line is the theoretical minimum standard deviation on the frequency. Panel (b): consecutive bias rejection vs. frequency. The black dots show the original bias, the blue dots show the bias after correction, and the red curve is the maximum bias. The shaded area represents the range of possible values of the bias after correction.

25 Figure 3 . 15 :

 25315 Figure 3.15: Detection and estimation of the frequency of hidden harmonics. The frequencies and phases are chosen so that the original bias is maximum (simulation sets C and D). Amplitudes are of the form c • a α (f p ) with c = 2.5 (top panel) and c = 25 (bottom panel). The meaning of the curves is the same as for Fig. 3.14.

Contents 4 . 1

 41 Assessment of the impact of the missing data on the precision of the measurement . . . . . . . . . . . . . . . . . . . . . . 96 4.1.1 Origin of missing data . . . . . . . . . . . . . . . . . . . . . . . . 96 4.1.2 Uncertainty of the OLS estimator with missing data . . . . . . . 97 4.1.3 Impact assessment as a function of the missing data pattern . . . 100 4.2 Mask construction . . . . . . . . . . . . . . . . . . . . . . . .

96

 96

  (4.1) the covariance matrix of the 50 consecutive points of a time series whose PSD is given by Fig.2.14a.

Figure 4 . 1 :

 41 Figure 4.1: Covariance matrix Σw in the Fourier domain of a time series of size N = 50 sampled at f s = 4 Hz with a PSD of the form given by Fig. 2.14a, obtained for a complete data set (i.e. w n = 1 ∀n) (left) and for random missing data with a 8.5% loss rate (right). The values are given in arbitrary units.

Figure 4 . 2 :

 42 Figure 4.2: Representation of different gap patterns corresponding to mask a (tank cracking type) in blue, b (telemetry loss type) in red, and c (periodic type) in green. The gap distribution of mask a is more spread than the two other masks, but the individual duration of gap is shorter. Note that the blue-shaded areas often represent several type-a gaps which are close to each other.

102 4 .

 4 Treatment of missing data and minimization of the measurement uncertainty

Figure 4 . 3 :

 43 Figure 4.3:The leakage effect in the presence of data gaps. The periodogram expectation ĪN,Wn of a noise n of PSD S(f ) (black curve) is plotted in gray, while the coloured curves represent the periodogram expectation of the masked noise vector with windows w a (blue), w b , (red) and w c (green). The left-hand side figure corresponds to a mask applied before the filtering process, and the right-hand side corresponds to a mask applied after the anti-aliasing filtering.

  .

  4.1, showing the values of the OLS standard deviation σ δ ≡ Var( δ) computed with the PSDs of Fig. 2.14, with and without filtering. Mask w a , corresponding to numerous and short gaps, leads to the largest increase of uncertainty, with an increase consistent with what is observed on the periodogram expectations of Fig. 4.3.

  (3.46) to compute the OLS standard deviation corresponding to each missing fraction. To verify the theoretical covariance formula, we also generate 400 realizations of the noise vector and perform 400 OLS estimations of the WEP violation signal, and compute their sample standard deviation. The results are shown in Fig.4.4, where the empirical (red) and the theoretical (black) uncertainties both rapidly increase with the number of gaps, and exceed by one order of magnitude the complete data level from 8 gaps per orbit.

Figure 4 . 4 :

 44 Figure 4.4:Ordinary least-squares uncertainty as a function of the number of gaps per orbit, computed empirically (red curve) and analytically (black curve). The computation is done for a random mask, corresponding to tank cracking gaps distributed over a spin session. The empirical curve is obtained by averaging 400 estimates obtained from 400 independent realizations of the noise vector. The theoretical curve is obtained with Eq. (4.4).

Figure 4 . 5 :

 45 Figure 4.5: Simplified scheme of the read-out processing chain.Usually the gain G i is applied after the filtering process but these two box functions are presented in the reverse order for convenience to derive the equations (this choice has no importance since they are linear operations).

Figure 4 . 6 :

 46 Figure 4.6: Estimated standard deviation of the KARMA method (see Sec. 4.3) as a function of the mask extension in case of random saturations, for filter cut-off frequencies f c = 1 Hz (red) and f c = 2 Hz (black). All the gaps last half a second and are randomly distributed over the time series.

Figure 4 . 7 :

 47 Figure 4.7: Approximation of the PSD with an autoregressive model, for increasing values of the order.

Figure 4 . 8 :Figure 4 . 9 :

 4849 Figure 4.8: Amplitude spectra of the gradient terms (red and dashed blue lines) with 20 µm off-centrings and the simulated WEP violation with δ = 3 • 10 -15 .

120 4 .

 4 Treatment of missing data and minimization of the measurement uncertainty

Figure 4 . 10 :

 410 Figure 4.10: Periodogram of the complete and masked noise vector (respectively I N,z and I N,W z) with the tank cracking mask w a . The mask induces a power leakage which tends to relocate the noise power on the whole spectrum, transferring the spectral power from regions where it is high into regions where it is lower. We superimposed the theoretical expectations of the periodograms for the complete (black solid line) and masked (black dashed line) data. Even if the complete data expectation appears slightly higher than the noise, in average for the rectangular window the periodogram converges towards this curve.

Figure 4 . 11 :

 411 Figure 4.11: Akaike information criterion (solid line) as a function of the order of the AR model, obtained from a 20-orbit session, with random gaps obtained with mask w a (tank cracking type). The dashed line corresponds to the modified AIC taking into account the increasing variance of the AR coefficients with the order.

Figure 4 . 12 :

 412 Figure 4.12: Autoregressive estimate of the PSD obtained with Burg's algorithm applied to the model residuals with orders p = 60 (red) and p = 200 (blue), compared to the true PSD (black) and the periodogram of the masked residuals (gray). The estimate is computed from the complete data (a), the tank cracking mask (b), the telemetry mask (c) and the periodic mask (d).

Figure 4 . 13 :

 413 Figure 4.13: Predicted states of the Kalman filter (dark blue line), along with the 3σ-confidence interval (light blue aera), computed from the estimated covariance of the predicted states. The observed data are plotted in black, and the missing intervals are represented by gray-shaded areas.

4 .

 4 Treatment of missing data and minimization of the measurement uncertainty 123 (a) Original noise (b) Weighted residuals

Figure 4 . 14 :

 414 Figure 4.14: (a) Periodogram of the complete noise vector z as a function of frequency, expressed in ms -2 Hz -1/2 . (b) Periodogram of the weighted residuals e z expressed in Hz -1/2 (e z has no dimension). The filtering process decorrelates the data, flattening the spectrum.

128 4 .

 4 Treatment of missing data and minimization of the measurement uncertainty

Figure 4 . 15 :

 415 Figure 4.15: Autocorrelation function ρ(τ ) ≡ R(τ )/R(0) (a) and Wendland 2 function (b) as a function of the time lag. The tapered autocovariance function is the product of the functions (a) and (b).

130

 130

Figure 4 . 16 :

 416 Figure 4.16: Extract of the time series obtained for a 20 orbit spin session sampled at 4 Hz, for the periodic gaps (a) and for the random gaps (b).For clarity we plotted a sample which is longer for periodic gaps (lasting more than one orbital period T orb , or about 6 WEP periods 1/f EP ) than for random gaps (which are 260 times shorter and more scattered). Observed data are in black and reconstructed data from one Monte-Carlo conditional draw are in blue. The missing data spans are indicated by gray areas.

Figure 4 . 17 :

 417 Figure 4.17: Lomb-Scargle periodograms of the observed data (grey), of the reconstructed data (blue) and of the original data (black) for a 20 orbits spin session and for the periodic (left) and random (right) gap patterns. In order not to overload the figure the values have been plotted on a sub-grid of about 700 Fourier frequencies. The harmonic peaks of highest amplitudes visible around 2f EP = 1.87 × 10 -3 Hz and common to the blue and black periodograms are not an artefact but are due to the gravitational gradient perturbation T xx and T xz (which are included in the model A). This is also the case for the peak at f orb = 1.70 × 10 -4 Hz.

Fig. 4 .

 4 Fig. 4.18 shows the average of the periodograms over the 400 simulations: the Lomb-Scargle periodogram of the observed data (without any reconstruction), the complete data periodogram I y and the conditional periodogram E [I y |y o ]. Note that they include the deterministic part of the signal since we have E [I y |y o ] = E [I z |y o ] + I A β .Fig.4.18 shows that on average the conditional periodogram converges toward the true periodogram with no missing data. For random gaps however the conditional periodogram looks slightly biased between 3 and 30 mHz. This is mainly due to the low number of Monte Carlo (MC) draws used to approximate the corrective term in Eq. (4.70). To verify this explanation with a reasonable CPU time we have used a toy PSD model of almost similar shape with less data points and we have evidenced a decrease of the bias when increasing the number of MC draws to N d = 100 (see Appendix C.4 for more details). Taking such a number of MC draws with N = 4.7 × 10 5 data points would be computationally expensive but could be achieved by use of parallel computing. However this is not necessary for our purpose, as the signals of interest are located at lower frequencies than the biased interval. Fig.4.19 shows the average (blue) and the confidence level (light blue) of the PSD estimates. This average estimate is compared to the original PSD from which the noise is generated (black), and to the average estimate from the AR model of the KARMA method used as initial guess (dashed red).The final PSD estimate brings an improvement with respect to the autoregressive one by reducing the bias in the low frequency part of the spectrum.In the case of random gaps, again a residual bias of less than 3 × 10 -13 ms -2 Hz -1/2 remains in the band between 3 and 30 mHz (where the PSD is minimum). Since the PSD is estimated from the periodogram, the bias in the periodogram whose origin is explained above has an impact on the bias of the PSD.

Figure 4 . 18 :

 418 Figure 4.18: Average of 400 Lomb-scargle periodogram of the observed data (gray), of the periodogram of the reconstructed data (blue) and of the original data (black) for a 20 orbits spin session and for the periodic (left) and random (right) gap patterns. The same remarks made in Fig. 4.17 about the peaks of the gravitational gradient perturbation apply. Averaging reveals the peak at f EP = 9.35 × 10 -4 Hz that is due to the simulated WEP violation. It also makes another faint gradient harmonic visible at 2f orb = 3.40 × 10 -4 Hz which is due to the influence of the second zonal harmonic J 2 .

Figure 4 . 19 :

 419 Figure 4.19: Sample average of the PSD estimates (blue) along with its 99% confidence interval (light blue area), autoregressive estimate (dashed red) and true PSD (black) for a 20 orbits spin session and for the periodic (left) and random (right) gap patterns.

Figure 4 .

 4 Figure 4.20: Sample average of the normalized autocovariance estimated from random-gapped data with the M-ECM algorithm (blue) along with its 99% confidence interval (dashed blue) and the true autocovariance (black). The autocovariance alternates between positive and negative values. The bar heights of the histogram corresponds to the values of R(τ )/R(0) at lags τ located at the left-hand-side edge of the bars. To be comprehensive the confidence interval should include both the upper and lower bounds for positive and negative values of the autocovariance. For the sake of clarity, only the upper bound of positive values and the lower bound of negative values are displayed.

  20.0 0.290 0.296 20.0 0.006 0.004 20.0 0.003 0.007 ∆ z [µm] 20 0.004 20.0 0.290 0.314 20.0 0.006 0.005 20.0 0.004 0.007

Figure 4 . 21 :

 421 Figure 4.21: Simulated time series of the differential acceleration with peaks due to impacts of micrometeorites. The valid data are plotted in black and the discarded data corresponding to the impacts are plotted in red. The data correspond to an uncalibrated inertial session generated in the framework of the technical qualification A1.

Figure 4 . 22 :

 422 Figure 4.22: Periodogram of the differential acceleration from the simulated inertial QT session, for the raw data (dark gray), the masked data (light gray) and the reconstructed data (blue). The red curve is the M-ECM PSD estimate computed from the reconstructed data. This computation is made for N = 524288 points sampled at 4 Hz.

144 5 .

 5 Application of the data processing to geodesy5.1 Measurement of the Earth gravitational gradient: mo-

Figure 5 . 1 :

 51 Figure 5.1: Left-hand side: the two sensor units su-ep and su-ref of the T-SAGE instrument, separated by an arm of 17 cm, with which we can measure a gradient. Right-hand side: axis position and orientation of the two sensor units. The Y axes are perpendicular to the orbit.

Figure 5 . 2 :

 52 Figure 5.2: Left-hand side: attitude pointing of the GOCE instrument. Right-hand side: attitude pointing of the MICROSCOPE experiment during an inertial session.

2 .

 2 3): [T e ] = ∇ 2 U ; Transformation of [T e ] into the instrument coordinate system (O, X, Y, Z) by applying the following matrix transformation: [T] = R[T e ]R T , (5.4)

Figure 5 . 3 :

 53 Figure 5.3: Mean amplitude of the radial gradient as a function of the potential degree at MICROSCOPE and GOCE altitudes.

10 - 5 Figure 5 . 4 :

 554 Figure 5.4: PSD of the noise on the gradient estimates (in Eötvös per√ Hz, 1E = 10 -9 s -2 ) derived from the expected instrument noise model on X (black), Y (blue) and Z (red) axis, for MICROSCOPE (solid line) and GOCE (dotted line). All the spectra are given in the instrument frame, which is different for the two satellites.

Figure 5 . 5 :

 55 Figure 5.5: Satellite trace sampled every minute during 120 orbits as a function of the latitude (y axis) and of the longitude (x axis) in degrees.

Figure 5 . 6 :

 56 Figure 5.6: Top-left: one-day exerpt of the time series of the theoretical observable T xy (in Eötvös) obtained for the 3 models EGM96 (blue), FM_S40RTS_40 (green), and hVR40_PD (black) described in Sec. 5.2.3 for an inertial session. Top-right and bottom: observable T xy -T REF xy

Figure 5 . 7 :

 57 Figure 5.7: Periodograms of the T xy component computed on an inertial session of 120 orbits for models EGM96 (blue), FM_S40RTS_40 (green), and hVR40_PD (black). The instrument noise level integrated over the duration of the session is shown by the red line.

Figure 5 . 8 :Figure 5 . 9 :

 5859 Figure 5.8: Top-left: one-day excerpt of the time series of the theoretical observable T yy (in Eötvös) obtained for the 3 models EGM96 (blue), FM_S40RTS_40 (green), and hVR40_PD (black) described in Sec. 5.2.3 for an inertial session. Top-right and bottom: observable T yy -T REF yy

P

  l,m (sin φ) e Imλ , (

Figure 5 . 10 :

 510 Figure 5.10: Spherical harmonic spectrum computed from a simulation of the component T yy measured in the MICROSCOPE instrument frame, and the expected accelerometer noise PSD.We plot the root mean squared error (red) of the Stokes coefficients, their average amplitude (blue dots) along with Kaula's rule (as defined by Table.5.1) as a function of the degree l in the spherical harmonics decomposition.

  Figure 5.10: Spherical harmonic spectrum computed from a simulation of the component T yy measured in the MICROSCOPE instrument frame, and the expected accelerometer noise PSD.We plot the root mean squared error (red) of the Stokes coefficients, their average amplitude (blue dots) along with Kaula's rule (as defined by Table.5.1) as a function of the degree l in the spherical harmonics decomposition.

Figure 5 . 11 :

 511 Figure 5.11: Left-hand side: periodograms of the angular acceleration noise on the 3 axis. Righthand-side: periodograms of the products of angular velocities involved in the inertia gradient. The spectra are computed from a simulation of the attitude control of an inertial session lasting 20 orbits.

158 5 . 5 . 3 . 1 . 1

 55311 Application of the data processing to geodesy Offcentrings along X and Z axis (session A)

Figure 5 . 12 :

 512 Figure 5.12: Amplitude periodograms in Eötvös of the systematic calibration residuals along the X axis (upper panel) and along the Y axis (bottom panel). We represent the error before calibration (dashed black line), the residual error after calibration (solid black line) and the statistical upper bound computed with the diagonal values of the covariance in Eq. (5.37) (blue solid line).

g 2 0 2 N

 22 . We finally obtain the desired resultE β -β = a p N g 0 cos π f s (f EP -f p ) (N -1) + φ EP -φ p sin π fs (f EP -f p ) N sin π fs (f EP -f p ) -cos π f s (f EP + f p ) (N -1) + φ EP + φ p sin π fs (f EP + f p ) N sin π fs (f EP + f p ), which is reformulated in Eq. (3.31).

3 ): N - 1 n=0 2 EP 4 (N 4 a 2 EP ( 2 ×N 4 a 2 EP 2 = 2 EP,

 3124422222 cos (2xn + φ) = cos (φ + x(N -1)) sin (xN ) sin (x) . (A.8)which is the Fejér kernel.To simplify we assume that there is an integer number of EP periods in the signal (f EP = k 0 /T ), thus the WEP wave is orthogonal to all Fourier frequencies. In this case | Ãk | is zero everywhere, except for k = k 0 and k = N -k 0 . Then we have| Ãk | 2 = N a δ k 0 + δ N -k 0 ) and Eq. (3.46) simplifies to Var( δ) ≈ f s S(f EP )) 2 × 2S (f EP ) f s N awhich is the expected result.

1 )ws fs 2 -fs 2 Ss fs 2 -fs 2 S- 1 ) 2 -fs 2 S

 12222122 n=-(N -1)Rx (n)e -2jπf n/fs (A.[START_REF] Chiu | Detecting periodic components in a white gaussian time series[END_REF] In particular we can apply this relation for the vector x = Wy. By taking the expectation of the windowed periodogram, we can write:E [I N,w,y (f )i w i+|n| R(nτ s ) (A.25)where R(nτ s ) is the theoretical autocovariance function of y. If we plug Eq. (A.25) into Eq. (A.22) we get:E [I N,w,y (f )] = N Cf s (N -1) n=-(N -1)   1 N N -1-|n| i=0 w i w i+|n| R(nτ s )   e -2jπf n/fs . (A.26)Now we can express R as a function of the PSD using Eq.(3.14):E [I N,w,y (f )] = N Cf 2jπ(f -f )n/fs df (A.27)where we recognize the expression of the autocovariance estimate Rw (n) of the window function:E [I N,w,y (f )] = N Cf Rw (n)e -2jπ(f -f )n/fs df , (A.28) By using the relation in Eq. (A.22) applied to the window function, we obtain E [I N,w,y (f )] = N C fs (f )I N,w f -f df . (A.29)

  µ = A(f p )β, where A(f p ) is the harmonic model of unknown phase given by Eq. (3.52), including (possibly) a WEP violation signal, and a sine and a cosine describing the perturbation signal. Thus the vector of parameters θ to consider includes possibly the amplitude of the WEP violation, the amplitudes of the sine and cosine parts of the disturbing harmonic signal, its frequency f p , and potentially the PSD parametersθ = β 0 β 1 f p S T. Since the covariance matrix does not depend on the 3 first parameters, the Fisher information matrix has the formI(θ) = I µ 0 0 I Σ (A.41)where I µ involves only the parameters describing the harmonic signal. Starting from Eq. (A.40) and expressing in vector form we findI µ = B * Σ -1 B (A.42)where we definedB ≡ A ∂A ∂fp β . (A.43)

  Finally we findE [I Wy (f )] = µ 2 w • E [I y (f )background for of the KARMA method

z n+j-1|t- 1 = j- 1 k=1a

 11 k z n+j-k-1|n-1 + p k=j a k z n+j-k-1 . (B.6)The state equation involves x(n -1) therefore we must calculatez n+j-1|n = j k=1 a k z n+j-k-1|n + p k=j+1 a k z n+j-k-1 . (B.7)Then we calculate the difference between Eq. (B.6) and Eq. (B.7) to getz n+j-1|n -z n+j-1|n-1 = j-1 k=1 a k z n+j-k-1|n -z n+j-k-1|n-1 -a j z n-1 + a j z n-1|n = j k=1 a k z n+j-k-1|n -z n+j-k-1|n-1 .

E

  [ m * m ] = E y m y * m + µ m|o µ * m|o -µ m|o y * m -y m µ * m|o (C.1)By definition of the covariance, the first term of the above equation isE [y m y * m ] = Σ mm + µ m µ * m . (C.2)In addition, we know from Eq. (4.60) that the conditional expectation of the missing vector writesµ m|o = µ m + Σ mo Σ -1 oo (y o -µ o ) . (C.3)With this relation, we can calculate the second term of Eq. (C.1):E µ m|o µ * m|o = µ m µ * m + Σ mo Σ -1 oo E [(y o -µ o ) (y o -µ o ) * ] Σ -1 oo Σ * mo +µ m E [(y o -µ o ) * ] + E [(y o -µ o )] µ * m = µ m µ * m + Σ mo Σ -1 oo Σ * mo . (C.4)We can also derive the third and fourth terms of Eq. (C.1):E µ m|o y * m = E [µ m y * m ] + Σ mo Σ -1 oo E [(y o -µ o ) y * m ] = µ m µ * m + Σ mo Σ -1 oo Σ * mo , (C.5) 194 C. Conditional distribution in spectral analysis and the fourth term of Eq. (C.1) is deduced by taking the transpose of the third term (they are actually symmetric, hence equal). Then we can plug the results of Eqs. (C.2), (C.4) and (C.5) into Eq. (C.1) to find E [ m * m ] = Σ mm + µ m µ * m + µ m µ * m + Σ mo Σ -1 oo Σ * mo -2µ m µ * m -2Σ mo Σ -1 oo Σ * mo = Σ mm -Σ mo Σ -1 oo Σ * mo , (C.6)

10 )

 10 where we have set µ zm|o = E [z m |y o ]. By definition of the conditional covariance of the missing residuals, we have: E z m z † m |y o = µ zm|o µ † zm|o + Σ m|o . (C.11)

  [START_REF] Bos | Autoregressive spectral estimation by application of the burg algorithm to irregularly sampled data[END_REF]:E [I z (k)|y o ] = E z z † |y o k,k

C. 4

 4 [START_REF] Bergé | Dealing with missing data: An inpainting application to the MICROSCOPE space mission[END_REF]):E (z * |o)(z * |o) † = W † o z o z † o W o + W † m E z * m|o z * m|o † W m +W † o z o µ zm|o W m + W † m µ zm|o z † o W oBy using the definition of the reconstructed residual vector ẑ in Eq. (C.12) and the definition of the covariance E z * m|o z * m|o † = µ zm|o + Σ m|o we end up with the right-hand side of Eq. (C.13). Thus the periodogram of z * |o will have its expectation given by Eq. (C.15), which is what we want. Impact of the number of Monte-Carlo draws on the estimation of the conditional periodogram In Appendix C.3 we showed how to approximate the conditional expectation of the periodogram by multiple imputations (i.e. Monte-Carlo draws) of the missing data. Here we check that the number of Monte-Carlo draws drives the accuracy of the conditional periodogram with respect to the original one. In other words, the objective is to check that the mean of the conditional periodogram converges towards the mean of the original periodogram. Formally we verify the equality: E [E [I y |y o ]] = E [I y ] . (C.18)

  Fig. C.1. By comparing them to the true PSD we check that the mean of the estimate obtained with the largest N d is less biased than the other, which is what we meant to prove. D Sparse inpainting and coloured noise D.1 The inpainting method D.1.

min α 1

 1 subject to W (x -Φα) 2 2 ≤ σ 2 , (D.1) where ||.|| 1 is the convex l 1 norm (i.e. ||z|| 1 = k |z k |), ||.|| 2 is the classical l 2 norm (i.e. ||z|| 2 = k |z k | 2) and σ is the standard deviation of the noise in the observed time series.

Figure D. 1 :

 1 Figure D.1: Periodogram of the reconstructed data with inpainting algorithm (blue), compared to the periodogram of masked data (gray), and of the original data (black) for a 20 orbits spin session and for the random tank cracking gap pattern.

  Fig. D.1, where the plot is expressed in √ PSD). However, the slight amplitude-dependent bias of the post-reconstruction regression is still present.

α 1 2 W

 2 (x -Φα) 2 2 +λΨ (α) , (D.2)where the function Ψ(.) is the penalty function, which is taken to be the l 1 norm Ψ(.) = . 1 . From this formalism we can take into account the noise correlation by solving the new problem α = argmin α 1 2 x o -Φ o α 2 Σo +λΨ (α) , (D.3)

Φ o α 2 Σo +λ α 1 ⇔

 21 α = argmin α,Σ 1 2 e o -Φo α 2 2 +λ α 1 (D.4)

0. 2 .

 2 Initialization: calculation of the first guesses of the PSD Ŝ0 and calculate L o . Set α 0 = 0 1. E step: calculation of the terms involved in the conditional expectation of the penalized likelihood E1 Calculation of the conditional data vector given the observed data y i = E y|y o , θ i = wy + (1 -w)x i ; E2 Calculation of conditional periodogram given the observed data I i z = E I z |y o , θ i ; CM step: conditional maximization of the penalized likelihood CM1 Obtain the new estimate of the dictionary coefficients α i+1 : Compute e i = L -1 y i andn for each columns l of Φ do • Compute the transform coefficient Φ * l e i -Φα i + α i l • Obtain α i+1 by soft thresholding of the transform coefficient.

  Fig. E.1). These frames are described by Table E.1 and Fig. E.1. Reference frame Description Inertial (O, x I , y I , z) Non-rotating, centered on the center of Earth, the z-axis pointing toward the poles. Terrestrial (O, x T , y T , z) Deduced from the inertial reference frame (roughly) by the rotation of the Earth θ(t) about z. Nodal (O, x N , y N , z N ) Deduced from the inertial frame by a rotation of angle Ω (argument of the ascending node) about z I and I (orbit inclination) about x N . Orbital Deduced from the nodal frame by rotation of angle ω = ω + v about z N where v is the true anomaly of the satellite, and ω is the argument of the orbit perihelion.

Figure E. 1 :

 1 Figure E.1:Orbital reference frames. The orbital trajectory is represented by the dashed black line, and the satellite is located at the intersect between the y N -axis of the nodal frame and the trajectory. The position of the orbital plane is moving by an angle Ω depending on time while the Earth is rotating by an angle θ with respect to the inertial frame.

-

  3. a rotation corresponding to the attitude of the satellite (for example, in spin mode, a rotation of angle ϕ = 2πf spin t + ϕ 0 about the z N -axis)4. a rotation to end up in the instrumental reference frame (O, X, Y, Z), deduced by applying a matrix P which simply consists in reversing the x-axis and exchanging the y and z-axis.In an inertial session, the instrument frame is almost equal to the nodal frame (up to the matrix P which takes into account the proper orientation of the axes). Then we have O, X, Y, Z) = (O, -x N , z N , y N ).The rotation matrix is thus cos ω e -sin ω e 0 sin I sin ω e -sin I cos ω e cos I cos I sin ω e cos I cos ω e sin I

1 2 (

 2 sin 2ω e sin I) T e xx + 1 2 ((cos 2ω e + 1) sin I) T e yy + (cos 2ω e sin I) T e xy -(cos ω e cos I) T e xz -(sin ω e cos I) T e yz ,

Table 2 . 1 :

 21 Range of values of the main model parameters

	Partial derivative	Amplitude	Main freq.	Param. [µm]

Table 2 . 2

 22

: Review of the order of magnitude and impact of the disturbing signals in the measurement equation

(2.79)

. The columns include, from left to right: disturbing signal (partial derivative), amplitude of the main harmonics, frequency of the main harmonics, maximum value of the corresponding dominant parameter, and impact of the disturbing term onto the WEP frequency (i.e amplitude of the component of the partial derivative at f EP multiplied by the value of the dominant parameter).

  .1.

	Window	Inertial session Spin session
	Rectangular	0.98 • 10 -15	0.93 • 10 -15
	Hann window	1.35 • 10 -15	1.29 • 10 -15
	Blackman window	1.41 • 10 -15	1.48 • 10 -15

Table 3 . 1 :

 31 Impact of apodization on the least-squares variance: the values correspond to the standard deviation of the least-squares estimation, with and without windowing. This computation is done based on Eq. (3.44) using the PSD model of Fig.2.14b and taking N to be the next power of two greater than 2N for the Riemann approximation of the covariance.

4.3 Development of an approximate general least-squares esti- mator with missing data: the KARMA method . . . . . . . . 107 4

  .3.1 The theoretical best linear unbiased estimator . . . . . . . . . . . 107 4.3.2 Review of the literature . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.3 Autoregressive estimation . . . . . . . . . . . . . . . . . . . . . . 110 4.3.4 Data orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3.5 Least-squares estimation with Kalman outputs . . . . . . . . . . 116 4.3.6 Summary of the method . . . . . . . . . . . . . . . . . . . . . . . 117 4.3.7 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . 117 4

. . 104 4.2.1 Key steps of the measurement process . . . . . . . . . . . . . . . 104 4.2.2 Strategy of the mask construction in case of saturation . . . . . . 105 4.2.3 Strategy of the mask construction in case of outliers . . . . . . . 106

.4 Reconstruction of the missing data and refinement of the PSD estimate

  Review of the literature . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.2 Data reconstruction using conditional expectations . . . . . . . . 126 4.4.3 Description of the data reconstruction process . . . . . . . . . . . 126 4.4.4 The preconditioned conjugate gradient method . . . . . . . . . . 127 4.4.5 Refinement of regression parameters and noise spectrum estimation through a modified ECM algorithm . . . . . . . . . . . . . . 128 4.4.6 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.4.7 Convergence of the periodogram and the PSD estimate . . . . . 134 4.4.8 Performance of the regression parameters estimation . . . . . . . 136 4.4.9 Blind simulation including mask construction . . . . . . . . . . . 139

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.4.1

Table 4 . 1 :

 41 Standard deviation of the OLS estimator as a function of the observation mask. The second and third columns respectively corresponds to noise levels of Fig. 4.3a and Fig. 4.3b.

  .5.

	e i	Saturation	ēi	G i	āi	Filter	s i	Decimation	s i,n

  1. Apply Burg's algorithm to compute the AR parameters a 1 , . . . , a p and σ 2 (as described in Sec. 4.3.3; 2. Apply the Kalman filter to the observed data of y and to the deterministic design matrix A (as described in Sec. 4.3.4); 3. Use the Kalman residuals e o and E o to calculate the estimate β (as described in Sec. 4.3.5).

Table 4 . 2 :

 42 -15 . Thus the regression model A contains Main parameters of the orbital simulation. The dates are provided in CNES Julian day, where 21107 days correspond to 16 October 2007. The intervals given for the orbital parameters correspond to their standard deviation during the flight.

		Eccentricity	(5.314 ± 0.779) • 10 -3
		Orbital period	5882.521 ± 0.059 s
	Orbital parameters	Semi-major axis	7041.594 ± 0.070 km
		Inclination	98.05 •
		Local hour of ascending node 6 h
	Earth's parameters	Potential model Maximal degree	GRIM4-S4 50
	Date (CNES Julian day)	Begin End	21107 days 00000 s 21108 days 50000 s

Table 4 . 5 :

 45 Estimation of the offcentrings from an inertial session generated in the framework of the technical qualification campaign.

	Parameter True value OLS Masked OLS KARMA σAR M-ECM σECM
	a c11 ∆ x [µm]	-39.60	-37.68	-39.87	-39.73	0.019	-39.73	0.004
	a c11 ∆ z [µm]	19.80	20.11	19.85	19.82	0.019	19.81	0.004
	δ [10 -15 ]	5	-175.2	81.0	7.4	4.0	4.6	0.8

  Measuring gravity gradients with high accuracy over the whole spectrum would allow a better identification of mass signals, for example through pattern recognition techniques.The present study aims to explore whether MICROSCOPE data can complement GOCE to measure Earth's gravity gradients at large scales, i.e., above 800 km.

	Field and Steady-State Ocean Circulation Explorer
	(GOCE) mission, launched in 2009 at an altitude of 270 km and deorbited in 2013. GOCE
	already provided valuable gradiometry data in the band from 5 to 100 mHz. The mission
	carried onboard 6 accelerometers built by ONERA, and measured the full gradient tensor.
	Another gravimetry mission, the Gravity Recovery And Climate Experiment (GRACE),
	launched in 2003, was designed to map the variations of Earth's gravity field by the
	measurement of the gravitational acceleration with respect to time. The mission is formed
	by two satellites following each other on the same orbit, at a distance of about 220 km
	[1], measuring the variations of the inter-satellite distance. Therefore GRACE is not
	a direct measurement of the gravitational gradient.
	The small scale spatial variations of the gravity gradients are usually estimated with
	GOCE, while the large-scale gravity gradients are reconstructed from GRACE and orbit
	data.

Table 5 . 2

 52

	Parameter	Unit	Session True value	Estimate	Bias	σ
	a g c21 ∆ g x a g c22 ∆ g y a g c23 ∆ g z a g c21 ∆ g y + a g c22 ∆ g x a g c21 ∆ g z + a g c23 ∆ g x a g c22 ∆ g z + a g c23 ∆ g y a g c22 ∆ g z -a g c23 ∆ g y a g c23 ∆ g x -a g c21 ∆ g z a g c21 ∆ g y -a g c22 ∆ g x a g c21 a g c23 a g d21 a g d22 a g d23 K g dyy /K g2 1cy	m m m m m m m m m rad rad rad rad rad s 2 m -1 H A, E2 B . . . B . . . B D2 E2,A E2,B E2 . . . F H G	-1.4562 • 10 -8 -1.7466 • 10 -1 1.5059 • 10 -8 5.0023 • 10 -4 4.9694 • 10 -10 -5.0685 • 10 -4 5.1713 • 10 -4 2.9621 • 10 -8 4.8996 • 10 -4 -2.9124 • 10 -3 3.0118 • 10 -3 -1.1781 • 10 -3 -7.2928 • 10 -3 1.0716 • 10 -3 -9.4718 • 10 2	-1.5458 • 10 -8 -1.7443 • 10 -1 . . . 4.9953 • 10 -4 4.9694 • 10 -10 . . . 9.0 • 10 -10 1.6 • -9 2.3 • 10 -4 4.9 • -7 1.5 • 10 -8 . . . 7.0 • 10 -7 6.6 • -7 . . . -5.0627 • 10 -4 5.8 • 10 -7 7.2 • -7 5.1622 • 10 -4 9.1 • 10 -7 2.3 • -8 . . . 3.0 • 10 -8 . . . 5.2390 • 10 -4 3.4 • 10 -5 6.4 • -6 -3.0034 • 10 -3 9.1 • 10 -5 3.7 • -5 . . . . . . . . . -1.1718 • 10 -3 6.2 • 10 -6 7.5 • -6 -5.1249 • 10 -3 2.2 • 10 -3 6.8 • -6 -1.0723 • 10 -3 6.1 • 10 -7 4.0 • -7 -9.5987 • 10 2 1.3 • 10 1 1.1 • 1

: Results of the simulations of the gradiometer calibration along the X axis. From left to right columns: estimated parameter, physical unit, calibration session used for the estimation (among those already planed in the mission), true value of the parameter, estimated value, estimation bias, and theoretical statistical uncertainty.

Table 5 . 3 :

 53 Results of the simulations of the gradiometer calibration along the Y axis.

  1 N sin (πτ s (f EP -f p )N ) sin (πτ s (f EP -f p )) + sin (πτ s (f EP + f p )N ) sin (πτ s (f EP + f p )) .

				(A.6)
	This expression involves the following function
	g(x) ≡	sin(N x) x	.	(A.7)

Table E . 1 :

 E1 Definition of useful reference frames

  TermMain peak SU-EP Γ meas,dx [ms -2 ] Γ g meas,dx [ms -2 ] Γ g meas,dy [ms -2 ] a c11 ∆ x S xx 2f EP 10 -11 10 -11 10 -14 a c12 ∆ y S yy 2f EP 10 -14 10 -10 10 -7 a c13 ∆ z S zz 2f EP 10 -14 10 -14 10 -14 (a c11 ∆ y + a c12 ∆ x ) S xy 2kf orb + pf spin 10 -14 10 -10 10 -13 (a c11 ∆ z + a c13 ∆ x ) S xz 2f EP 10 -11 10 -11 10 -14 (a c12 ∆ z + a c13 ∆ y ) S yz 2kf orb + pf spin 10 -17 10 -13 10 -10(a c12 ∆ z -a c13 ∆ y ) Ωx f EP 10 -17 10 -13 10 -11 (a c13 ∆ x -a c11 ∆ z ) Ωy f EP 10 -15 10 -15 10 -18 (a c11 ∆ y -a c12 ∆ x ) Ωz f EP 10 -15 10 -11 10 -15 a d11 (Γ mescx -b 0cx ) f EP ,2f EP 10 -12 10 -12 10 -12 a d12 Γ mescy -b 0cy f EP ,2f EP 10 -24 10 -12 10 -12 a d13 (Γ mescz -b 0cz ) f EP ,2f EP 10 -24 10 -12 10 -12

Table E . 2 :

 E2 Order of magnitude of the terms of the measurement equation for a session F (estimation of the differential sensitivity parameter a d11 ) Le Principe d'Equivalence (PE) est un pilier fondamental de la relativité générale, et il est aujourd'hui remis en question par certaines tentatives d'élaborer des théories plus exhaustives en physique fondamentale. Dans ce contexte, la mission spatiale MICROSCOPE vise à tester ce principe à travers l'universalité de la chute libre, avec un objectif de précision de 10 -15 , soit un gain de deux ordres de grandeur par rapport aux expériences actuelles. Le satellite embarque deux accéléromètresélectrostatiques, chacun intégrant deux masses-test. L'objectif est de comparer la chute libre de masses de compositions différentes dans le champ gravitationnel de la Terre, en mesurant leur accélérationdifférentielle. Compte tenu de la grande sensibilité de l'instrument, diverses perturbations sont à prendre compte, comme Ce contexte expérimentalnécessite le développement d'outils adaptés pour l'analyse de données, qui s'inscrivent dans le cadre général de la régressionlinéaire multiple de séries temporelles.

	le	bruit	stochastique,	de	possibles
	harmoniques	parasites	ou	des	pics
	d'accélérations dus à l'environnement du
	satellite.

The MICROSCOPE experiment and the measurement equation

• the stochastic noise term n d is by definition not deterministic. However the corresponding uncertainty can be decreased by a sufficient long integration time, measuring the differential acceleration during several orbits.Some of the terms that we reviewed are multiplied by unknown matrices. The purpose of the whole in-flight calibration process will be to estimate the coefficients of these matrices, or their products with other involved parameters, and to correct the acceleration for the corresponding disturbances. This allows us to isolate the possible WEP violation signal δ d g E . In the next section we bring up this signal along with the disturbance terms so as to identify the relevant parameters to characterize.

Treatment of missing data and minimization of the measurement uncertainty

Acknowledgements

English

55

WEP violation. Their amplitudes form a set of parameters which must be estimated to reduce the bias. In addition, the impact of possible hidden parameters -corresponding to possible signals which have not been characterized in the measurement equationmust be assessed. This is the purpose of the next chapter.

Calibration of the gradiometer

The estimation of the differential quadratic parameter can be done with the same session as for the individual sensor unit. For K g 2dxx / (K g 1cx ) 2 we use the session F whereas for

we use session G where the common mode along the Y axis is excited.

Regarding the common quadratic parameter of the gradiometer, we can derive it from the values estimated during the individual calibration of the su-ep and suref by the expressions

where K EP 21xx and K REF 21xx respectively correspond to the quadratic parameters of the internal test-masses of the su-ep and the su-ref.

An error of about 2% will come from the discrepancy between K g 21x and K 21cx . As for the quadratic terms along the Y axis K g 2jyy , it is impossible to determine it since no excitation of the test-masses along the Y axis is scheduled in the current in-flight calibration plan.

Numerical simulations

We derived the calibration equations of the gradiometer. The purpose of this section is to test them on simulated data generated by the simula code which has already been described in Sec. 4.3.7. The goal is to have a first indication of the feasibility of the proposed calibration process.

Approach

We assume a given set of instrumental parameters. We first simulate the accelerations measured by both accelerometers during each calibration session, lasting 10 orbits. Then we construct the differential accelerations and the partials derivatives that are needed for the regression. Finally we perform an ordinary least-squares estimation to estimate the calibration parameters of the gradiometer. In order to separate the systematic and stochastic errors, we do not introduce any synthetic noise in the simulation, such that only the deterministic variations remain. The statistical uncertainty due to the noise is derived afterwards from the theoretical covariance of the least-squares estimator given by Eq. (3.15) computed with the expected noise PSD shown in Fig. 5.4.

After running all the calibration sessions, the calibration performance is tested on an inertial session which is corrected for the contribution of all the determined parameters.

The difference between the calibrated gradient measurement and the true gradient is then computed, and compared to the uncalibrated measurement.

A

Derivations of useful results for spectral analysis

In this appendix we give some results of spectral or harmonic analysis that are particularly useful for the rationale of Chapt. 3.

A.1 Bias of the OLS estimator in the presence of a disturbing harmonic signal

In this section we calculate the bias of the OLS estimator when fitting a harmonic signal of frequency f EP in data including another harmonic signal of frequency f p . The general expression of this bias is given by Eq. (3.5) that we reproduce here for convenience:

In this case the size of the model matrix A is N × 1 (as well as A ) and we have

We first calculate

where we defined

By choosing φ = -x(N -1), we find that

This sum can be seen as the Riemann approximation of the integral

which can be calculated:

Thus we obtain an oscillating function whose envelope is 1/x. As a result, the envelope of the maximum projection rate is approximately equal to

A.3 Bias of the OLS estimator when fitting two harmonic signals with an error on the frequency

In this section we derive an analytical expression for the maximum bias given in Eq. (3.55) on the WEP parameter when fitting the WEP violation signal (approximated by a simple sine wave) and another harmonic component, in the case where the frequency f p of the latter is not well known, and approached by a frequency f . We aim at computing

where A :0 denotes the first column of A. We will also compute he maximum of b 0 (f ) with respect to the phase φ p of the harmonic signal. The subscript 0 means that we consider only the bias on the amplitude of the first sine wave (the WEP violation signal in our study).

For convenience, we reproduce the definition (3.52) of the model matrix

where the vectors S(f, φ) and C(f 1 , φ 1 ) are defined by Eq. (3.52). We can then write

(A.14) where we defined the following quantities:

whose analytical expression can be easily calculated following the example of Eq. (A.4).

In the following we will drop the phase arguments for the sake of clarity. The expression of A(f

where we also defined

The expression in Eq. (A.18) can be maximized with respect to φ p to find

A.4 Variance of the OLS estimator when fitting a sine wave

In this section we compute an approximate expression for the variance of the OLS estimate when fitting a sine wave of frequency f EP and amplitude a EP to a stationary time series with colored residuals of PSD S(f ). We start from Eq. (3.46) giving the variance of the windowed least squares:

We consider the case without any windowing (w n = 1 ∀n), and we assume that the WEP violation signal is a pure sine wave of the form (3.27), whose discrete Fourier transform is then given by

where we defined the frequency ν k = kf s /N and the function

A.8 PSD deformation in the case of random missing data patterns

We derive here the PSD of the masked data in the case where the gaps positions in the time series are drawn from a uniform distribution. Let N be the length of the time series, N g the number of gaps, and n b,i the indices indicating the location of the beginning of each gap (such that w n b,i = 0). Each gap ends at the location n b,i + dn i (we adopt the convention w n b,i +dn i = 1). By uniformly distributed, we mean that n b is a random variable following a discrete uniform distribution on the interval 0, N -1 .

We also allow the gap duration dn to be randomly distributed. The window vector is then generated by drawing N g realizations of n b and dn.

The probability p to observe a data at a time n is calculated as follows:

where in the third line we use the fact that all the variables n b,i are identically and independently distributed. The cumulative probability function of n b is given by

In the case where the duration of the gaps is fixed (i.e. dn i = dn 0 ∀i), Eq. (A.48) gives:

Therefore we have a discrete probability law such that P(w n = 1) = p and P(w n = 0) = 1p. We deduce that w n is a Bernoulli's law of parameter p. Its expectation is µ w = p and its variance is σ 2 w = p(1 -p). We notice that p is independent of time, and the autocovariance function of w is simply R w (n) = σ 2 w δ(n) where δ(n) is the delta Dirac function. Then we use Eq. (3.66) to calculate the expectation of the masked periodogram, conditional to the mask vector:

where we dropped the N in the notation of the periodogram to lighten the equations.

If we now consider w as a random variable, we can compute the expectation of the above expression with respect to the distribution of the mask itself. Indeed, in average, the periodogram of the mask is

B

Theoretical background for of the KARMA method In this appendix we give some useful theoretical results in the framework of Chapt. 4.

B.1 A derivation of Burg's algorithm and its application to missing data

In this section we give details on the Burg's algorithm, and in particular on its adaptation to the problem of missing data. The aim of the Burg's algorithm is to minimize the forward and backward residuals defined by Eq. 4.30, that we reproduce here for convenience:

Publications

Publications in peer-reviewed journals We first study the statistical detection and estimation of unknown harmonic disturbances. We show that by including them in the measurement model, their projection onto the WEP violation signal can be rejected. Secondly we analyze the impact of the data unavailability on the performance of the EP test. We show that with the worst-case hypothesis on the data gaps occurrences, the uncertainty of the ordinary least squares can be increased more than one order of magnitude. To counterbalance this effect, a linear regression method based on an autoregressive estimation of the noise is developed, which allows a proper decorrelation of the available observations. The variance of the constructed estimator is close to the optimal value, allowing us to perform the EP test at the expected level. In addition, we implement a method to more accurately characterize the noise power spectral density when data are missing. The approach is based on modified expectationmaximization algorithm. Finally, we extend the applications of the data analysis by demonstrating the feasibility of the measurement of Earth's gravitational gradient with MICROSCOPE data.