Valérie Sauvant-Moynot

Jean Kittel

Xavier Lefèbvre

USE OF EIS FOR THE MONITORING AND MODELLING OF MULTI-LAYER ANTICORROSION COATINGS

Keywords: organic coatings, multi-layer, water uptake, polymer degradation

Three-layer polyolefin coatings are being used classically for external pipeline protection in the oil and gas industries over years although cathodic protection is also applied. They are supposed to provide efficient and sustainable corrosion protection to the pipe steel over 20 years service or more, including exposition to humidity and temperature (cycling conditions).

However, the permeability of polyolefin top-coat to water is not negligible and thus does not prevent water ingress into sub-layers after a few months service. Then soaked water in the vicinity to the steel pipe conveying hot effluents possibly induces hydrolysis of the epoxy primer layer and interfacial bonds to metal. This could raise questions over the long term corrosion control. Therefore, the inspection of the water ingress and the investigation of the long-term barrier properties of primer and adhesives layers when exposed to water are mandatory. This paper examines the use of Electrochemical Impedance Spectroscopy (EIS) to monitor durability and anticorrosion properties of three-layer coatings while exposed to water at 60 °C (maximum service temperature). The objective was to investigate the water ingress within a bilayer coating with respect to sublayer thickness. EIS was used to characterise the water uptake but also the long term barrier properties of coatings on industrial tubes. In complement, gravimetry measurements were also performed during ageing on FBE primer coated panels cut from industrially coated tubes to determine the water uptake kinetics using classical destructive techniques.

Degradation mechanisms of 3LPO and potentiality of EIS technique are discussed in the paper.

Introduction

Three-layer polyolefin coatings (3LPO) are being used classically for external pipeline protection in the oil and gas industries over years although cathodic protection is also applied. Such coatings consist of typically an epoxy primer layer (Fusion Bonded Epoxy) providing both adhesion and corrosion protection to steel, overlaid by an intermediate adhesive layer ensuring the proper anchoring of the polyolefin top-coat to the epoxy primer. The use of an external polyolefin top-coat (polyethylen or polypropylen) offers outstanding mechanical resistance to damage occurring during both pipe manufacture and installation. As a consequence, such multi-layer coatings are supposed to provide efficient and sustainable corrosion protection to the pipe steel over 20 years service or more, including exposition to humidity and temperature (cycling conditions).

Recent case studies reported large scale disbondments of 3-LPO coatings, which could favour corrosion beneath the pipe coating and lead to possible leakage or other major damage [START_REF] Roche | Recent experience with pipeline coatings failures, 16 th Pipeline Protection Conference[END_REF][START_REF] Argent | Fitness for purpose issues relating to FBE and 3LPE coatings[END_REF][START_REF] Tandon | Perforamnce of three layer cpolyethylene coating on a cross country pipelinea case study[END_REF][START_REF] Portesan | Cases of massive disbondment with three layer PE pipeline coatings, 3 èmes journées CEFRACOR : Protection cathodique et revêtements associés -Ouvrages enterrés ou exposés au milieu marin[END_REF]. Indeed, polyolefin materials are well known for their very tiny water uptake while exposed to vapor or liquid water but their permeability to water is not negligible [START_REF] Flaconnèche | High pressure permeation of gases in semicrystalline polymers: measurement method and experimental data[END_REF]. Thus polyolefin top-coats do not prevent water ingress into sublayers after a few months service and soaked water in the vicinity to the steel pipe conveying hot effluents possibly induces hydrolysis of the epoxy primer layer and interfacial bonds to metal which, in turn, could raise questions over the long term corrosion control [START_REF] Carpentiers | Corrosion under disbonded cotings of cathodically protected pipelines[END_REF][START_REF] Tsuri | Effect of primer composition on cathodic disbonding resistance and adhesion durability of three layer polyethylene coated steel pipe[END_REF]. This is particularly critical for 3LPO coatings since they are prone to 'shielding effect': when water flows along the metal surface under a disbonded 3LPO, cathodic protection current is likely to be unable to reach the pipe because of the high dielectric strength of polyolefin.

Therefore, the inspection of the water ingress and the investigation of the long-term barrier properties of primer and adhesives layers when exposed to water are mandatory. IFP started to work in this thematic three years ago and proposed a methodology based on the ageing of bi-layer FBE/adhesive coatings as stand alone systems to provide an accelerated measurement of the FBE adhesion to steel during wet ageing [START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings[END_REF][START_REF] Sauvant-Moynot | Nouvelle stratégie de sélection des primairies d'adhésion pour revêtements tricouches, 4 èmes journées CEFRACOR : Protection cathodique et revêtements associés[END_REF]: peel tests are carried out on pipe rings industrially coated with FBE and adhesive to examine the adhesion of sub-layers to steel under wet exposure and directly investigate the effect of water permeation through the coating. This cannot be achieved while submitting 3LPO systems to cathodic disbonding tests since the influence on adhesion performance of FBE wet ageing due to water diffusion through the coating cannot be distinguished from ageing related to cathodic disbonding mechanisms and water entrance in intended defect [START_REF] Meyer | Impact of aging process of three layers extruded polyethylene coatings on their adhesion on pipelines steels[END_REF][START_REF] Coeuille | Revêtements polymères de canalisation de fluide : caractérisation et évolution de l'adhésion en milieu agressif[END_REF].

In addition to a better understanding of failure mechanisms and to the development of new hydrophobic epoxy primers for 3LPO coatings [START_REF] Gaillard | The interest of new hydrophobic epoxy primers for three-layer coatings, 16 th Pipeline Protection Conference[END_REF], there is a need of non destructive methodologies to allow the health monitoring of the coating. The objective of this study was to investigate the potentials of EIS for the monitoring of 3LPO anticorrosion coating. For that purpose, pipe rings coated with the FBE primer layer only were studied as well as bilayer coated pipe rings. EIS characterisation of FBE and FBE/adhesive coatings exposed to water at 60°C are presented in the result part. The influence of the layer thickness is discussed.

Experimental

Coating composition and application

Materials were selected among those currently used for industrial applications. The steel preparation and coating application were performed in a plant according to GDF

Ageing conditions

Coatings as received were aged in deionised water at 60°C on rings. The electrolyte was preferably chosen with high ionic purity to simulate the filtered water by the PE top-coat. Indeed, the polyolefin is supposed to work as a perm-selective membrane, i.e. only the molecules such as water, oxygen, carbon dioxide diffuse but ionic species remain in the external medium. However, continuous water uptake monitoring by EIS on coated pipe rings was performed in 1 % NaCl aqueous solution to ensure sensible impedance measurements.

Small size panels cut into the pipe rings were also used for a classical gravimetric study of water uptake. The rate of the water sorption was measured as the rate of mass change with respect to the initial mass of the dry FBE material (the steel was weighted at the end of sorption tests after burning of the organic material).

EIS tests

Impedance Spectroscopy measurements are carried out by means of a Solartron

Frequency Response Analyzer FRA1260 equipped with a specific dielectric interface 1296 to measure large impedance values. The impedance data are obtained over a frequency range of 100 kHz to 100 mHz using a 100 mV amplitude of sinusoidal voltage. Simple two-electrode configurations electrode/material/electrode are used to characterize the impedance properties of coated rings exposed to a 1% NaCl aqueous solution during ageing (Figure 1). Measurements were performed in isothermal conditions at 60°C with exposed area of 3.1 cm 2 or 10.9 cm 2 .

Figure 1 -Illustration of EIS characterisation of mono / bilayer coated pipe rings

Adhesion tests

Bi-layer coating peel strengths were measured on pipe rings at ambient atmosphere on a tensile machine at constant peeling velocity of 8.33x10 -5 m.s -1 (5 mm.min -1 ). The polymer band cut to metal prior to the peel test was 20mm width. The peel energy in J.m -2 is calculated by dividing the peel strength (N) by the band width (m).

FBE primer coating adhesion was evaluated by pull-off tests specified by ISO 4624 (measurements were performed using 3 dollies).

Results

Short term monitoring

EIS is a powerful tool to perform in situ and non destructive monitoring of the water uptake on organic coatings [START_REF] Brasher | [END_REF][START_REF] Geenen | Characterization of organic coatings with impedance measurements, a study of coating structure, adhesion and underfilm corrosion[END_REF][START_REF] Sauvant-Moynot | Hot wet aging of glass syntactic foam coatings monitored by Impedance Spectroscopy[END_REF], provided relaxations due to the glass transition of the plasticized coating matrix do not occur in the temperature range of the measurement [START_REF] Duval | Water uptake in epoxy-coatings by EIS[END_REF][START_REF] Duval | EIS : a powerful tool for in-situ monitoring of coating performance[END_REF]. This first part deals with the use of EIS to monitor water ingress in FBE primer. The water uptake of B-FBE coating at 60°C as revealed by the capacitance evolution versus the square root of time is displayed in Figure 2a. From the capacitance values at 10000 Hz, the water content evolution was calculated using the Brasher and Kingsbury law [START_REF] Brasher | [END_REF].

 = log (C t /C 0 )/[log( w )]
(equation 1)

where  stands for the mass gain given by EIS, C t stands for the capacitance at t, C 0 stands for the capacitance at time 0 (dry film), is the density of polymer film and  W is the permittivity of water ( W = 63 at 60°C).

In a second step, diffusion coefficient D was evaluated through equation 2 according to a Fickian behavior for the water absorption (18):

                                      0 2 2 2 2 2 ) 1 ( 2 exp ) 1 2 ( 1 8 1 n t l Dt n n M M   (equation 2)
where M t is the mass of water absorbed at time t, M  is the mass of water at infinite time t and l is the thickness of film.

Given the high water content estimated at saturation after 9 hours exposition (Figure 2b), we had some assumptions that the water content could be overestimated due to the plasticization of the glass transition. Thus, the water uptake at 60°C of a B-FBE coated panel was also studied by gravimetry to asses EIS results (Figure 3). In fact, the aforementioned overestimation is likely to occur since the water content after one week exposition is estimated around 1.5% by gravimetry -much lower value than EIS results. Since the long term gravimetric results on FBE coating exhibit a steadily increasing water content, sign of hydrolytic or osmotic degradation, trials were done to fit gravimetric data with the diffusion law of Langmuir (18), equation 3, which should give more appropriate results than the classical diffusion law of Fick :

h t n D e n n t e t e M M t 2 2 ) 1 2 ( 2 0 ) 1 2 ( 2 1 2 8 1                           (equation 3)
where M t is the mass of water absorbed at time t, M  is the mass of water at infinite time t, l is the thickness of film,  is the probability of water to be linked to the polymer,  is the probability of linked water to be released.

Model parameters obtained from in situ impedance and ex situ gravimetry results for B-FBE coating immersed in deionised water at 60°C are presented in Table 2.

Table 2 -Model parameters obtained by fitting experimental ex situ gravimetry and in

situ impedance data of water uptake in B-FBE primer immersed in water at 60°C.

Gravimetry EIS

Fick water content at equilibrium (wt.%) 1.5 6.0

Fick diffusion coefficient (m 2 /s ) 6.0  10 -13 5.4  10 -13

Langmuir (/s ) 10 -6 -Langmuir (/s ) 0.333  10 -6 -Diffusion coefficient and water content at equilibrium are obviously not in close agreement at 60°C for this FBE. Thus, the capacitance change measured at 60°C is not only directly related to the increasing material permittivity with water entrance. It is overlapped by the capacitance increase related to Tg lowering by FBE plasticization. In fact, this is not completely surprising since epoxy resins classically exhibit a lowering of 20°C per percent of water uptake [START_REF] Sauvant-Moynot | Epoxy coatings for anticorrosion challenges: a link between chemistry and performance ? in Eurocorr[END_REF]. Given the initial Tg value and the water content reached within 9 hours (= 1.5 %), then the wet Tg value could be expected around 75°C, i.e. not sufficiently differing from the aging temperature to allow a quantitative use of Brasher law. As a consequence, the langmuir model will be useful to offer a more appropriate simulation of the water ingress into bi-layer coatings than the one proposed in previous works [START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings[END_REF][START_REF] Sauvant-Moynot | Nouvelle stratégie de sélection des primairies d'adhésion pour revêtements tricouches, 4 èmes journées CEFRACOR : Protection cathodique et revêtements associés[END_REF] on the basis of EIS data only.

In conclusion, the use of EIS to quantitatively evaluate the water uptake kinetics in FBE materials having initial Tg values around 100°C should be limited to lower temperature range than 60°C. Nevertheless, as FBE formulations improve towards higher Tg values [START_REF] Gaillard | The interest of new hydrophobic epoxy primers for three-layer coatings, 16 th Pipeline Protection Conference[END_REF], future works to characterize their behavior in water should again investigate the potentiality of EIS.

Long term monitoring

In this part, the potentiality of EIS measurement to detect irreversible damage occurring in the coating material and/or at the metal surface was investigated.

Unidirectional water ingress modelling was performed in B and C bi-layer coatings exposed to water at 60°C. As discussed previously, the Langmuir law to simulate the diffusion of water in the FBE primer layer whereas the Fick law was used to simulate the water diffusion through the adhesive layer, with the hypothesis of 0.4% of water content at saturation and a diffusion coefficient value of 4.10 -12 m 2 /s (8). Simulations

given on Figure 4 show that a three months ageing period should be fully sufficient for systems under consideration to reach a high level of water content in the FBE near the metal surface and investigate the consequences on barrier properties.

B and C bi-layer coated rings were aged in deionised water at 60°C for a 3 months period. B-FBE and C-FBE coatings were also aged in similar conditions. Various coating properties (Table 3) were evaluated prior and after ageing:

-coating aspect (visual inspection), -mechanical properties (adhesion measurement),

-water content (thermo-gravimetric analysis). EIS characterization was performed at 60°C at the beginning and at the end of the ageing time for all coatings under consideration. Results on FBE coatings are presented as Bode spectra on Figure 5. Curves before ageing exhibit an almost capacitive behaviour which is characteristic of an intact coating. Then the apparition of a resistance in both B and C aged FBE coatings means:

Simulation of water diffusion profile in B-bilayer coating at

-the water plasticized the B-coating (see water content) and the Tg becomes closer to the temperature of the measurement, then the film resistance decreases; actually, tiny porosities in the blistered primer B would also lead to similar trends but, unfortunately, no EIS was performed with an electrolyte of higher conductivity to check the second hypothesis [START_REF] Duval | EIS : a powerful tool for in-situ monitoring of coating performance[END_REF].

-a degradation of the C-FBE coating (cracks lower substantially the pore resistance) which dominates the effect of coating plasticization.

. Besides, the high frequency capacitance is strongly diminished. This later could be attributed to thickness increase due to possible swelling and blistering in the primer, as observed in the stand alone B-coating. Actually, no variation of thickness was taken into account in the EIS data processing. In conclusion, EIS results show that damages occurring in B-bilayer coating seem to be less harmful on the impedance response than those occurring in C-bilayer coating. The formation of water blistering in sub-layers in the later case only could explain those observations. Unfortunately no microscopic observation was done to confirm this interpretation. The differences observed between both bilayers are not fully understood yet.

Conclusions

This paper examined the use of Electrochemical Impedance Spectroscopy (EIS) to monitor durability and anticorrosion properties of three-layer coatings while exposed to water at 60 °C (maximum service temperature).

The use of EIS to quantitatively evaluate the water uptake kinetics in FBE primer coatings was investigated. The knowledge of water ingress kinetics is an issue to propose a reliable modeling of the water diffusion process into 3LPO coatings. It was shown that the use of EIS to monitor the water uptake in FBE primer having initial Tg around 100°C should be limited to lower temperature range than 60°C. However, future works to characterize the water uptake of novel high Tg FBE primers should, again, investigate the potentiality of EIS at 60°C or more. Expected benefits from EIS are in situ monitoring, continuous recording, and time saving in comparison to the classical gravimetry.

In addition, EIS appeared to be an appropriate method to detect sub-layer blistering in FBE/adhesive bilayer coatings, even though no macroscopic defect occurred through it (i.e. there is no path flow for the electrolyte to the metal surface). But EIS is obviously not sensitive to the formation of corrosion products at the pipe surface.

Further work is in progress to model the EIS spectra by equivalent electrical circuits in order to improve the understanding of 3LPO degradation mechanisms.

Figure 2 -

 2 Figure 2 -Short term water uptake monitoring by EIS in the B-FBE coating at 60°C a) capacitance (10000Hz) evolution with square root of time; b) water content given by EIS according to Brasher and Kingsbury law (13).

Figure 3 :

 3 Figure 3: Water uptake behavior versus square root of time followed by gravimetry for the B-FBE coating immersed in deionised water at 60°C a) experimental water content (in weight %) and fick modelling; b) reduced water content M t / M ∞ and Langmuir modelling

Figure 4 -

 4 Figure 4 -Simulation of water concentration profiles in B-bilayer and C-bilayer coatings exposed to deionised water at 60°C -Hypothesis of unidirectional diffusion (fick law in adhesive layer / langmuir law in FBE layer)

Figure 5 -

 5 Figure 5 -Evolution with ageing time of the Bode plots for (a) B-FBE coating and (b)C-FBE coating exposed to deionised water at 60°C

Figure 6 -

 6 Figure 6 -Evolution with ageing time of the Bode plots for (a) B-bilayer coating and (b) C-bilayer coating

Table 3 -

 3 Coating properties prior and after 3 months ageing in water at 60°C

	System	Property	Initial state	After 3 months ageing in
				water at 60°C
	B-FBE	Visual inspection	Smooth coating	Numerous blisters
	primer
		Adhesion	18 ± 2 MPa	3 ± 0 MPa
		(pull-off test)	glue rupture	100% adhesive rupture
			White metal	Black metal
		Water content (%)	0	2.0
	C-FBE	Visual inspection	Smooth coating	Cracks to the metal
	primer
		Adhesion	9 ± 1 MPa	5 ± 1 MPa
		(pull-off test)	100% cohesive rupture	100% adhesive rupture
				Black metal
		Water content (%)	0	12.3
	B-bilayer Visual inspection	Rough coating	Rough coating
		Adhesion	29 000J/m 2 *	250J/m 2
		(peel test)	White metal	Grey metal + metallic
				particles on peeled strip
		Water content (%)	0	1.6
	C-bilayer Visual inspection	Rough coating	Rough coating
		Adhesion	29 000J/m 2 *	500J/m 2
		(peel test)	White metal	Grey metal
		Water content (%)	0	2.6
	* Measured in fact on a 3LPO system (9)

Acknowledgments

The authors wish to gratefully acknowledge S. Schweitzer and J. Grenier for their implication to the experimental part of this work.


domain: spi phys math info

text:
Mathieu Brunot

A Gaussian Uniform Mixture Model for Robust Kalman Filtering

Keywords: Kalman filters, Mixture models, Noise robustness, Bayesian estimation

This paper presents a Kalman-type recursive estimator for discrete time systems with a measurement noise modeled by a Gaussian-uniform mixture. The objective is to deal with data containing outliers that degrade the performances of the regular Kalman filter. The introduced non-Gaussian noise model takes into account the reliability of the measurement to be robust with respect to erroneous data. The Kalman-type estimator is based on the Masreliez's formulation that copes with non-Gaussian noise models. Results in different simulated conditions are displayed to evaluate the performances of the introduced algorithm and to compare it to state-of-art alternatives.

I. INTRODUCTION

S INCE the seminal work of R. Kalman and R. Bucy [1]- [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF], the estimation theory has received a large attention from researchers. Consequently, a great number of algorithms have been developed for prediction, filtering and smoothing, for either linear or nonlinear systems. Those algorithms can also be categorized in terms of noise assumption. If the regular Kalman Filter (KF) assumes a Gaussian noise model, many physical systems deviate from this assumption and follow distributions with heavier tails leading to high intensity noise realizations known as outliers.

Many robust approaches have been suggested to deal with non-Gaussian measurements. An intuitive approach is to use a regular KF by adding a detection of the outliers to reduce their weights in the corresponding correction terms [3]- [START_REF] Ting | A kalman filter for robust outlier detection[END_REF]. Several methods directly deal with non-Gaussian distributions. One of them consists in using the Student's t-distribution and performing the statistical inference with an Expectation-Maximization iteration [START_REF] Lange | Robust statistical modeling using the t distribution[END_REF]. That has been extended to sequential filtering and smoothing processes [START_REF] Agamennoni | An outlier-robust filter[END_REF]. Other works consider the variational Bayes approximation to estimate non-Gaussian posterior distribution [START_REF] Piché | Recursive outlier-robust filtering and smoothing for nonlinear systems using the multivariate student-t distribution[END_REF]- [START_REF] Huang | A novel robust student'st-based kalman filter[END_REF]. Without being exhaustive, one can also cite the Particle Filter (PF) and Gaussian Mixture Filter (GMF). Unlike the previously cited methods, they do not assume that the state distribution is Gaussian after each time step. The PF utilizes a set of weighted points, i.e. particles, to approximate the distributions [START_REF] Arulampalam | A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking[END_REF]. The GMF approximates of the noise distribution by a finite sum of Gaussian terms [START_REF] Sorenson | Recursive bayesian estimation using gaussian sums[END_REF]. The drawback of those two methods is that number of particles or Gaussian components have to be large enough to insure a good estimation accuracy, which may lead to unacceptable computing time.

M. Brunot is with ONERA, Toulouse, France e-mail: mathieu.brunot@onera.fr.

In the case of model uncertainties, coming from model errors, unknown input process or noise properties, the risksensitive filter also represents an interesting alternative. Contrary to the regular KF, it minimizes an exponential cost function, tuned by the so called risk-sensitivity parameter; see e.g. [START_REF] Banavar | Properties of risk-sensitive filters/estimators[END_REF]. This parameter can be constant or time-varying. Recent works have extended that filter to a family of robust KF parametrized by the τ -divergence family [START_REF] Zorzi | Robust kalman filtering under model perturbations[END_REF]- [START_REF]Convergence analysis of a family of robust kalman filters based on the contraction principle[END_REF]. In [START_REF] Glover | State-space formulae for all stabilizing controllers that satisfy an h-norm bound and relations to relations to risk sensitivity[END_REF], the authors have highlighted the link between the risksensitive filter and the H ∞ filtering algorithm that may also be appealing. It indeed makes no statistical assumptions on the noise, except that it has finite energy [START_REF] Rawicz | Explicit formulas for two state kalman, h/sub 2/and h/sub/spl infin//target tracking filters[END_REF]- [START_REF] Li | Nonlinear gaussian mixture phd filter with an h criterion[END_REF].

Grandin and Marques [START_REF] Grandin | Robust fusion with reliabilities weights[END_REF] introduced a link between the theory of Probability and the theory of Evidence. The introduction of the reliability of the source in the measurement model led to a robust fusion process. In the specific case where the reliability is not a random variable but a known constant probability, the model developed in [START_REF] Grandin | Robust fusion with reliabilities weights[END_REF] becomes a Gaussian-Uniform (GU) mixture model. In [START_REF] Coretto | Maximum likelihood estimation of heterogeneous mixtures of gaussian and uniform distributions[END_REF], the authors have shown that it is a worthwhile alternative to Gaussian mixture in case of outlier data. Such a modelling has been considered with PF for multiple target tracking in [START_REF] Särkkä | Rao-blackwellized particle filter for multiple target tracking[END_REF] where the uniform distribution represents "clutter" or false alarm measurements. This PF method is referred to as Rao-Blackwellized Monte Carlo Data Association (MCDA) and was designed for multiple target tracking. Few years before, the Probabilistic Data Association Filter (PDAF) have been conceived with the same purpose and noise assumptions [START_REF] Bar-Shalom | Tracking in a cluttered environment with probabilistic data association[END_REF].

In this paper, we consider the GU measurement noise to provide a robust Kalman-type estimator. Masreliez [START_REF] Masreliez | Approximate non-gaussian filtering with linear state and observation relations[END_REF] developed a generic filter formulation to deal with non-Gaussian observation noise. Although the formulation has the reputation for being complex [START_REF] Ting | A kalman filter for robust outlier detection[END_REF]- [START_REF] Kovačević | On robust kalman filtering[END_REF], our goal is to present the calculation of this filter and to illustrate its interest with two examples.

The paper is organized as follows. The formal definition of the GU noise model and Masreliez filter are first given in Section II. Section III is devoted to the development of the filter equations by considering the GU mixture for the measurement noise. A simple test case is firstly considered in Section IV to illustrate the filter characteristics while Section V tackles a more involved multidimensional case. Finally, concluding remarks are provided in Section VI.

Notations General notations a, b, c = Scalars a, b, c = Vectors A, B, C = Matrices Conventions A -1 = Inverse of matrix A A T = Transpose of matrix A |A| = Determinant of matrix A x = Estimated vector x i,k = i th component of vector x k [A] i = i th row of matrix A [A] ij = Element at i th row and j th column of A ∂g(x) ∂x
= Jacobian matrix of function g

II. PRELIMINARIES A. Mixture noise model

To obtain a robust Kalman filter, a non-Gaussian measurement noise is studied. The principle is to take into account outliers by assuming a noise distribution with heavy tails. In one dimension, the following distribution is considered:

G(θ, µ, σ, a) = F (θ)N (µ, σ) + (1 -F (θ)) U(µ, a) (1)
where F is the probability of a correct measurement parametrized by θ, N is a Gaussian distribution (mean µ and standard deviation σ) and U is a centered uniform law (mean µ and half span a). As explained in [START_REF] Grandin | Robust fusion with reliabilities weights[END_REF], the idea of such a model is that the Gaussian term expresses the precision of the instrument, whereas F (θ) assesses if the instrument senses what it is supposed to measure, i.e. the reliability. In case of invalid measurement, a uniform distribution is considered to model the error although a more complex modelling could be applied for some sensors. In this article, the author limits himself to the case where the instrument reliability is known in advance to reduce the complexity of the model. In the sequel, the reliability is a constant known parameter θ. The resulting Probability Density Function (p.d.f.) is a one dimensional GU mixture:

f 1D G (n) = θf 1D N (n) + (1 -θ)f 1D U (n), (2)
where

f 1D N (n) = 1 √ 2πσ exp - 1 2 n -µ σ 2 f 1D U (n) =      0 n ≤ µ -a 1 2a µ -a ≤ n ≤ µ + a. 0 µ + a ≤ n
In m dimensions, a multivariate distribution could be defined as follows

f m G (n) = m i=1 f 1D Gi (n i ) = m i=1 θ i f 1D Ni (n i ) + (1 -θ i )f 1D Ui (n i ) .
Nevertheless, such a distribution assumes that the m components are independent which may be a too strong assumption in practice. Thus, we define the m dimensional probability distribution G:

G(θ, µ, R, a) = F (θ)N (µ, R) + (1 -F (θ))U(µ, a) (3)
Therefore, the p.d.f. is given by

f G (n) = θf N (n) + (1 -θ)f U (n), (4)
with

f N (n) = 1 (2π) m/2 |R| exp - 1 2 (n -µ) T R -1 (n -µ) f U (n) = 1 A n ∈ P (µ, a) 0 otherwise
For the uniform part, P (µ, a) is the m dimensional parallelotope centered in µ with faces defined by a and volume A. By assuming that P is a rectangular parallelotope:

A = µ1+a1 µ1-a1 . . . µm+am µm-am dx 1 . . . dx m = 2 m m i=1 a i . (5)
With (4), the covariance matrix R enables the model to take into account correlated measurements. The price to pay is that the probability of correct measurement F (θ) is the same for each component. However, each component can have a different tail for outliers incorporation by selecting appropriate a i coefficients.

B. Masreliez filter

Like in [START_REF] Masreliez | Approximate non-gaussian filtering with linear state and observation relations[END_REF], in this article, the dynamic system considered are modelled by a linear discrete-time state-space defined as follows

x k+1 = φ k x k + q k (6) z k = H k x k + n k (7)
The plant noise q k is assumed white and Gaussian with covariance matrix Q k . In the following, p Z (z k |Z k-1 ) denotes the observation prediction density, i.e. the density of the observation z k conditioned on prior observations Z k-1 = z 1 , z 2 , ...z k-1 . The predicted state is assumed to be Gaussian with p.d.f. p X (x k |Z k-1 ). According to [START_REF] Masreliez | Approximate non-gaussian filtering with linear state and observation relations[END_REF], this assumption should be satisfied in many practical situations if the state noise is Gaussian or if the state prediction variance is small with respect to the observation variance. Its mean and covariance are given by

xk E (x k |Z k-1 ) = φ k-1 x k-1 (8)
M k E[( xk -x k )( xk -x k ) T |Z k-1 ] (9) = φ k-1 P k-1 φ T k-1 + Q k-1 where x k-1 E (x k-1 |Z k-1
) is the estimated state vector at time k -1 and P k-1 is its conditional covariance:

P k-1 = E[( x k-1 -x k-1 )( x k-1 -x k-1 ) T |Z k-1 ]. (10)
The correction step satisfies:

x k = xk + M k H T k g k (z k ) (11)
P k = M k -M k H T k G k (z k )H k M k (12)
M k+1 = φ k P k φ T k + Q k ( 13
)
where g k (z k ) is a vector valued function given by

g i,k (z k ) = - ∂p Z (z k |Z k-1 ) ∂z i,k p Z (z k |Z k-1 ) -1 . ( 14
)
That function is called the score function in [START_REF] Masreliez | Approximate non-gaussian filtering with linear state and observation relations[END_REF]. The matrix

G k (z k ) is such as [G k (z k )] ij = ∂g i,k (z k ) ∂z j,k (15)
In our case, we assume that n k ∼ G(θ k , 0, R k , a k ) without loss of generality. The mean value µ could indeed be nonzero and estimated in an augmented state vector as a measurement bias. The next section is devoted to the development of g k and G k in closed form.

III. ROBUST KALMAN FILTERING A. Observation prediction density and its derivative

The objective is to obtain the expression of the observation prediction density p Z (z k |Z k-1 ). From ( 7), as the sum of two random variables: H k x k and n k , the p.d.f. of z k is the convolution of their two p.d.f.s:

p Z (z k |Z k-1 ) = R m f X k |Z k-1 (z k -t|Z k-1 )f G (t|Z k-1 )dt where the p.d.f. of the first term f X k |Z k-1 (t|Z k-1 ) is Gaus- sian with mean H k xk and covariance L k = H k M k H T k .
The second p.d.f. comes from our assumptions in the previous section. Therefore, it comes:

p Z (z k |Z k-1 ) = R m (2π) m/2 |L k | 1/2 -1 exp{T k 1 (z k , t)}• θ k (2π) m/2 |R k | 1/2 -1 exp{T k 2 (t)} + (1 -θ k )f U (t) dt = θ k (2π) m/2 |S k | 1/2 -1 exp{T k 3 (z k )} + 1 -θ k A k a k -a k (2π) m/2 |L k | 1/2 -1 exp{T k 1 (z k , t)}dt = θ k f 1 k + 1 -θ k A k f 2 k ( 16
)
where the terms in exponential are

T k 1 (z k , t) = - 1 2 [ k -t] T L -1 k [ k -t] T k 2 (t) = - 1 2 t T R -1 k t T k 3 (z k ) = - 1 2 T k S -1 k k with A k = 2 m m l=1 a l,k , S k = H k M k H T k + R k and k = z k -H k xk . The first term f 1 k is a Gaussian p.d.f.

and given by

f 1 k = (2π) m/2 |S k | 1/2 -1 exp{T k 3 (z k )} (17)
The second term f 2 k is a little more complex:

f 2 k = (2π) m/2 |L k | 1/2 -1 a k -a k exp{T k 1 (z k , t)}dt (18) = (2π) m/2 |L k | 1/2 -1 • J k The term J k is developed in Appendix A. It can be expressed as J k = m l=1 I l,k and its derivative is ∂J k ∂z j,k = J k • γ j,k
, with the intermediate variables I l,k and γ j,k defined by ( 27) and (31) respectively.

The calculation of the derivative is divided into two terms such as

∂p Z (z k |Z k-1 ) ∂z i,k = θ k ∂f 1 k ∂z i,k + (1 -θ k ) A k ∂f 2 k ∂z i,k (19)
= θ k f 1,i k + (1 -θ k ) A k f 2,i k
The first element of the derivative is:

f 1,i k = (2π) m/2 |S k | 1/2 -1 exp{T k 3 (z k )} ∂T k 3 (z k ) ∂z i,k (20)
= -f 1 k S -1 k [z k -H k xk ] i .
Using the results of Appendix A, the second element of the derivative is:

f 2,i k = (2π) m/2 |L k | 1/2 -1 ∂J k ∂z i,k (21)
= (2π) m/2 |L k | 1/2 -1 J k • γ i,k = f 2 k • γ i,k .
In this part, the expression of p Z (z k |Z k-1 ) and its derivative have been developed in closed-form so that the score function can be expressed.

B. Score function and its derivative

From the previous derivations, the score function comes out:

g i,k (z k ) = - ∂p Z (z k |Z k-1 ) ∂z i,k p Z (z k |Z k-1 ) -1 = - θ k f 1,i k + 1-θ k A k f 2,i k θ k f 1 k + 1-θ k A k f 2 k = θ k f 1 k S -1 k [z k -H k xk ] i -1-θ k A k f 2 k γ i,k θ k f 1 k + 1-θ k A k f 2 k = θ k f 1 k S -1 k k i -1-θ k A k f 2 k γ i,k p Z (z k |Z k-1 ) . ( 22
)
Then, the matrix G k is expressed:

[G k (z k )] ij = ∂g i,k (z k ) ∂z j,k = ∂ ∂z j,k θ k f 1,i k + 1-θ k A k f 2,i k p Z (z k |Z k-1 ) = ∂ θ k f 1 k S -1 k k i -1-θ k A k f 2 k γ i,k ∂z j,k 1 p Z (z k |Z k-1 ) - θ k f 1,i k + 1-θ k A k f 2,i k p 2 Z (z k |Z k-1 ) ∂p Z (z k |Z k-1 ) ∂z j,k = 1 p Z (z k |Z k-1 ) θ k ∂f 1 k S -1 k k i ∂z j,k - 1 -θ k A k ∂f 2 k γ i,k ∂z j,k + g i,k (z k )g j,k (z k ) (23)
where, the remaining terms to be estimated are

∂f 1 k S -1 k k i ∂z j,k = f 1,j k S -1 k k i + f 1 k [S -1 k ] ij = f 1 k [S -1 k ] ij -S -1 k k i S -1 k k j ∂f 2 k γ i,k ∂z j,k = f 2,j k γ i,k + f 2 k ∂γ i,k ∂z j,k = f 2 k γ j,k γ i,k + ∂γ i,k ∂z j,k

C. Corner cases

If the previous development is correct in theory, numerical issues may appear in practice in three specific cases, referred to as "Corner Cases" (CCs). Those CCs are:

1) the reliability coefficient is set to one; 2) the Gaussian component, f 1 k , is too weak; 3) both components, f 1 k and f 2 k , are too weak. The first CC would appear in the case where the designer sets θ k = 1. In this case, the filter is equivalent to the regular KF and it is more efficient to apply: 22) and ( 23). The second CC appears in the case where the observation is an outlier for the Gaussian term but explained by the uniform distribution. With such a configuration, the p.d.f. of the Gaussian component may be too low with respect to the floatingpoint relative accuracy leading to numerical instability. In fact, if the Gaussian component f 1 k is neglected, ( 22) and ( 23) can be reduced to g k (z k ) ← γ k and G k (z k ) ← ∂γ k ∂z k . The challenge is now to define when f 1 k can be neglected. As long as the measurement error n k is Gaussian, the term T k S -1 k k follows a chi-square distribution and the limit is evaluated for the χ 2 value with m degrees-of-freedom, such as

g k (z k ) ← S -1 k k and G k (z k ) ← S -1 k , setting θ k = 1 in (
f 1 k > (2π) m/2 |S k | 1/2 -1 exp{- 1 2 x lim }, (24)
where x lim is the inverse of the chi-square cumulative distribution function, selected with a p-value of p = 0.001. One should note that x lim is similar to the gate threshold of the PDAF method; see e.g. [START_REF] Bar-Shalom | The probabilistic data association filter[END_REF].

The remaining CC is when the outlier is not explained by the noise model. The uniform component f 2 k is then too small and a source of instability. Two solutions can be thought to deal with such a situation. First, the regular KF step can be used. Second, the measurement can be discarded and no correction is performed. These two different solutions are referred to as the regular (reg.) and prediction (pred.) variants respectively. The main question left is the detection of an observation that does not follow the model. In practice, the value of p Z (z k |Z k-1 ) is compared with the χ 2 value, such as

f 2 k > (2π) m/2 |L k | 1/2 -1 A k • exp{- 1 2 x lim }. (25)
Although

[ k -t] T L -1 k [ k -t]
is not supposed to follow exactly a chi-square distribution, this criterion gives good results in practice as it is illustrated in the following sections. The pseudo-code of the score function and its derivative calculation is given in Algorithm 1. Compute p Z with [START_REF] Li | Nonlinear gaussian mixture phd filter with an h criterion[END_REF] 11:

Algorithm 1 Kalman-Masreliez Filter with GU Noise Require: L k , S k , k , θ k , a k , A k , x lim 1: if θ k == 1 then 2: g k ← S -1 k k 3: G k ← S -1 k 4: else 5: C k ← chol(L -1 k , upper
f 2,lim k ← (2π) m/2 |L k | 1/2 -1 A k • exp{-1 2 x lim } 12: if p Z < (1 -θ k )f 2,lim k /A k then 13:
Prediction variant:

14:

g k ← zeros(m , 1) 15:
G k ← zeros(m, m)

16:

Regular variant:

17: Compute ∂γ k ∂z k with (34)

g k ← S -1 k k 18: G k ← S -1 k 19:
22:

f 1,lim k ← (2π) m/2 |S k | 1/2 -1 exp{-1 2 x lim } 23: if f 1 k ≥ f 1,lim k then 24:
Compute g k with (

Compute G k with [START_REF] Bar-Shalom | The probabilistic data association filter[END_REF] 26:

else 27:

g k ← γ k 28: G k ← ∂γ k ∂z k return g k , G k

IV. PRELIMINARY TESTS

A. Single state model

For the first evaluation of the introduced filter, a simple one dimensional test case is considered. The constant distance of an object is measured. In practice, one can think about a laser rangefinder used in a dusty environment, i.e. there is probability 1 -θ that a particle positions itself between the object and the sensor. The system properties are as follows

• x * = 100 m, true distance of the target; • x 0 = 50 m, initial distance of the estimators;

• P 0 = 20 2 m 2 , initial variance of the state; • Q = 1 m 2 , variance of the state noise; • x, estimated state; • y = x + n, measured distance with measurement noise n: zero mean and variance σ 2 n . The state model is therefore a random walk. The data are generated with the p.d.f. (2) where the noise parameters are constant: θ = 0.9, µ = 0, σ = 5 and a = 100.

Table I provides three criteria: the Root Mean Square (RMS) error, the Execution Time (ET) and the Average Normalized Equation Estimation Error Squared (ANEES) over 10000 Monte Carlo Simulations (MCS) of a regular KF, the two variants of the introduced Robust KF (RKF), the MCDA and For the first example, the measurement noise is generated as Gaussian with the same standard deviation as the noise models of the KF and RKF filters: σ n = σ = 5. The second example is called "Mixture" because the noise model of the RKF is exactly the one used to generate the data. In both cases, the situation is described as "Nominal" because there is no error introduced in the RKF noise model.

For the Gaussian case, it appears that both RKF variants are equivalent but slightly less precise than the regular KF. That can be explained by a slower convergence due to the noise model. The GU mixture indeed tends to explain more easily the measurement noise and thus corrects less quickly the state. This phenomenon can be observed in Fig. 1. This figure depicts the four estimations for the last Monte Carlo iteration. The MCDA and PDAF algorithms provide equivalent mean estimates in this introductory case. Regarding the ANEES, the PDAF has the closest to 1, which means that its covariance estimate best represents the underlying estimate error [START_REF] Li | Practical measures and test for credibility of an estimator[END_REF].

When the measurement noise is generated with the GU mixture, the KF has difficulty as expected. It seems that the regular variant is more robust than the prediction one. However, this is not so clear by looking at Fig. 2 (last MCS iteration) and hence requires more investigation. In this case, the MCDA algorithm gives a mean RMS error as low as the regular variant. Nonetheless, Table I shows that it requires more execution time. It may be observed that the ratio is more or less the number of particles considered. The difficulties of the PDAF can be explained by the case where the first observation is an outlier, which prevents the estimator to converge and causes large errors due to the coarse initialization.

B. Influence of the outliers model and the reliability coefficient

If the previous subsection illustrated the interest of the RKF in presence of outliers, the current one is devoted to the limit case where the outliers' model is erroneous. The measurement noise is still generated with the same GU mixture but both RKF variants and MCDA have a different range of outliers: a. Table II summarizes the results for 10000 MCS with outliers range errors ∆a = ±50 %. As it could be expected, if the outliers' range is larger than the 'reality', the RKF variants do not suffer. In the opposite, when the range is narrower, the effect is disastrous for the regular variant whereas the prediction one is insensitive. This is also depicted in Fig. 3 that gives the estimation results of the last MCS iteration for ∆a = -50 %. Here also, the MCDA algorithm proves to be robust by providing low RMS estimates, although it has a larger computational cost. Obviously, the results of the PDAF are insensitive since the method does not rely on the range information. With respect to the ANEES, the most credible estimator is the MCDA followed by the prediction variant of the RKF.

The second limitation of the GU mixture lies in the reliability coefficient. To test its influence on the estimation results, RKF variants are considered with reliability errors: ∆θ = ±10 %. The results are provided in Table III. It appears that both variants similarly behave. Here also, the estimations are less accurate in the case where the reliability coefficient is not large enough; i.e. ∆θ = -10 %. Nevertheless, this is not disastrous as the case ∆a = -50 %, which appears to be the limit case. In particular, the RMS errors stay far below the one of the KF. Although the clutter density depends on θ, the PDAF is not really prone to its miscalculation. Therefore, the time histories of the last MCS iteration are not presented because do not provide more information. Finally, by putting aside the regular variant, the same observation can be made for the ANEES criterion as the case ∆a = ±50 %.

A. Simplified car model

For further testing of the introduced a 2-dimensional car model is considered; see Examples 3.6 and 4.3 in Compared with previous example, this one has a real dynamics (i.e. there is a movement) and illustrates a multiestimation. In this case, the outliers could model multipath errors of the GPS receiver for instance. For the sake of clarity, the linear state space model is recalled here:

x k Ax k-1 + q k-1 y k = Hx k + n k q k-1 ∼ N (0, Q), n k ∼ G(θ, 0, R, a),
where x = [x 1 , x 2 , x 3 , 4 ] T with the position of the car (x 1 , x 2 ) and the corresponding velocities (x 3 , x 4 ). The matrices

A    1 0 ∆t 0 0 1 0 ∆t 0 0 0 0 0 0 1    H = 1 0 0 0 0 1 0 0 Q = σ 2 q       ∆t 3 3 0 ∆t 2 2 0 0 ∆t 3 3 0 ∆t 2 2 ∆t 2 2 0 ∆t 0 0 ∆t 2 2 0 ∆t       , R = σ 2 n 1 0 0 1 ,
where ∆t = 0.1 s in the sampling period, σ q = 1 m.s -1 / √ Hz and σ n = 0.5 m. The other parameters of the mixture model are θ = 0.9 and a = [10, 10] T . The initial state is vector x 0 = [0, 0, 1, -1] T and its covariance matrix P 0 is the (4 × 4) identity matrix.

B. Results

Table IV provides the results over 10000 MCS, for the nominal case. It should be noticed that the RMS x and RMS y errors are computed for the whole state vector and only the positions respectively. As expected, with a Gaussian noise, the RKF does not improve the estimation. That is confirmed by Fig. 4. This figure illustrates the vehicle position, the measurements and the estimates for the last MCS iteration. With the GU mixture noise, the prediction variant outperforms the KF whereas the regular one has major difficulties. When the outlier is not explained by the noise model, the application of the regular KF correction step appears to be counterproductive. Hence, the corresponding trajectory is not depicted in Fig. 5, where it is visible that the RKF provides a more reliable estimated trajectory compared to the regular filter. The MCDA algorithm gives results (RMS x , RMS y and ANEES) similar to the prediction variant but has the advantage of providing narrower confidence ellipsoids, i.e. smaller covariances, although the computational cost is still larger. The PDAF algorithm has also RMS x and RMS y equivalent to the prediction variant with a lowest computational cost, however, it is more optimistic according to the ANEES criterion. One should note here that there is no initialization issue.

Tables V and VI give the criteria values in the case of errors in the error model. Those results definitively disqualify the regular variant which gives worse results than the regular KF. On the contrary, the prediction variant appears to be robust to the errors and always provides mean RMS errors lower than the regular KF. Furthermore, its estimation is similar to the reference MCDA algorithm, though its confidence ellipses are larger. Here also, the PDAF provides estimates with RMS errors comparable to those of prediction variant and MCDA, with a lowest computational cost. Nonetheless, with a slightly larger amount of time, the prediction variant appears to be credible with respect to the ANESS criterion.

From those different observations, the following comments can be drawn

• the regular variant should be dropped;

• compared to the regular KF, the RKF is more robust to outliers but has a slower convergence and provides larger states covariances; • compared to the MCDA, the RKF has a lower computational cost, provides as accurate estimates but is less optimal (larger covariances); • compared to the PDAF, the RKF provides as accurate estimates with more credibility (ANEES closer to 1), is less sensitive to rough initialisation, but has a slightly larger computational cost; • the worst case to test is the narrower outlier range (∆a < 0); • the RKF is not too sensitive to the reliability coefficient.

VI. CONCLUSION

This paper suggests the use of a Gaussian-uniform mixture for the observation noise model in a Kalman-type recursive estimator. Such a noise model encompasses the outliers by expressing the instrument reliability. Based on the Masreliez's work, a closed-form formulation of the filter is drawn. Although this formulation is established, that may lead to numerical instability in several cases: (i) the filter is used as regular KF, (ii) the Gaussian component of the error model is too weak, (iii) both components of the error model are too weak. The resulting algorithm is able to detect those cases and adapts itself to provide an accurate estimation. Though case (iii) led to two variants, the simulation results indicate that the best solution is to perform only the prediction when the measurement is not explained at all by the GU mixture.

Regarding the prospects, the next step is to undertake a real-time implementation in order to evaluate the solution in realistic experimental situations.

APPENDIX A DEFINITION AND DERIVATIVES OF J k

From (18), the term J k is defined as follows:

J k = a k -a k exp{- 1 2 T k L -1 k k - 1 2 t T L -1 k t -T k L -1 k t}dt = C k a k -C k a exp{- 1 2 e T k e k -e T k u - 1 2 u T u}du (26) = m l=1 b l,k -b l,k exp{- 1 2 e 2 l,k - 1 2 u 2 l -e l,k u l }du l = m l=1 I l,k
We define the integration variable u = C k t where C k is the upper triangular matrix coming from the Cholesky where erf is the error function.

In order to obtain the derivatives of J k , one should noticed that it is not independent of z k . Thus, the derivative of I l,k is first developed:

= J k γ j,k , by assuming I l,k = 0 and defining the (m × 1) vector γ k such as

γ j,k = m l=1 1 I l,k ∂I l,k ∂z j,k . (31)
With respect to (27), I l,k = 0 implies that b l,k = 0 since erf is a strictly monotonic function. Therefore, one should be cautious to ensure a l,k = 0, ∀l, k. To derive the second order derivatives, it appears in (30) that only ∂γ j,k i,k is missing. It first comes

∂ 2 I l,k ∂z i,k ∂z j,k = ∂ ∂z i,k ([C k ] lj α l,k ) = [C k ] lj ∂α l,k ∂z i,k = -[C k ] lj ∂e l,k ∂z i,k β l,k = -[C k ] lj [C k ] li β l,k (32)

) 6 :b k and e k 7

 67 Compute

else 20 :

 20 Compute γ k with (31) 21:

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Nominal case -Gaussian noise -1D model

Figure 4 .Figure 5 .

 45 Figure 4. Nominal case -Gaussian noise -2D model

factorization of L - 1 k 2 -

 12 = C T k C k . Therefore, the exponential term in the integral can be decomposed axis per axis, by writing b k = C k a k and e k = C k k . By assumption the covariance of the predicted state M k is symmetric positive definite. In addition, H T is assumed to be full column rank; i.e. rank(H T ) = m. Consequently, L k and its inverse are symmetric positive definite, which is the requirement for the Cholesky factorization. Each component of the (m × 1) vector I k can be evaluated separately such asI l,k = + e l,k) 2 }du l (27) = e l,k +b l,k e l,k -b l,k l,k + e l,k ) 1 erf (-b l,k + e l,k ) 1 2

  ,k -e l,k ) 2 } = ∂e l,k ∂z j,k α l,k = [C k ] lj α l,k ,(28)by defining the (m × 1) vector α k such asα l,k = exp{-1 2 (b l,k + e l,k ) 2 } -exp{-1 2 (b l,k -e l,k ) 2 }. (29)The first derivative is thus defined by:

by defining the (m × 1 ) 2 }∂ 2 ∂ 2

 1222 vector β k such asβ l,k = (b l,k + e l,k )exp{-1 2 (b l,k + e l,k ) 2 } (33) + (b l,k -e l,k )exp{-1 2 (b l,k -e l,k )The derivatives of γ k components are thus defined by: I l,k ∂z i,k ∂z j,k I l,k ∂z i,k ∂z j,k

  In our case, we consider only one target and use 10 particles for the MCDA. That number was chosen to have similar performances with the mixture noise case; see TableI. The MCDA software can be downloaded via http://becs. aalto.fi/en/research/bayes/rbmcda/. For the PDAF algorithm, the gate probability is based on x lim defined in section III-C and the clutter density at time k is λ k = (1 -θ)/V k , with the V k the volume of the gate.

				Table I
		NOMINAL CASE -10000 MCS -1D MODEL
		Crit.	KF	RKF (reg.)	RKF (pred.)	MCDA	PDAF
	Gauss.	RMS ET [ms] ANEES	1.74 0.14 0.59	1.77 4.31 0.56	1.77 4.22 0.56	1.77 48.61 0.58	1.75 1.26 0.74
	Mixt.	RMS ET [ms] ANEES	6.24 0.14 8.21	3.84 4.37 3.16	4.10 4.31 5.00	3.36 48.55 2.81	6.02 4.19 56.06
	PDAF algorithms. Both MCDA and PDAF were designed to
	take into account clutter measurements and to track multiple
	targets.

Table II ERRORS

 II IN THE RELIABILITY COEFFICIENT -10000 MCS -1D MODEL

	∆θ	Crit.	KF	RKF (reg.)	RKF (pred.)	MCDA	PDAF
	+10%	RMS ET [ms] ANEES	6.24 0.15 8.21	3.76 4.47 3.45	4.06 4.38 5.73	4.21 50.02 5.28	6.01 4.45 56.14
	-10%	RMS ET [ms] ANEES	6.24 0.14 8.21	3.95 4.46 3.03	4.16 4.36 4.58	3.14 49.15 1.98	6.03 4.33 55.94
				Table III
	ERRORS IN THE OUTLIERS RANGE -10000 MCS -1D MODEL
	∆a	Crit.	KF	RKF (reg.)	RKF (pred.)	MCDA	PDAF
	+50%	RMS ET [ms] ANEES	6.24 0.15 8.21	3.76 4.57 3.44	4.16 4.49 4.48	3.42 51.74 3.05	6.02 4.55 56.06
	-50%	RMS ET [ms] ANEES	6.24 0.14 8.21	8.98 4.37 10.86	3.51 4.28 3.09	3.15 48.92 2.05	6.02 4.34 56.06
			V. COMPLEMENTARY TESTS


Jan Borm

Bernard Cottret

Jean-François Zorn

Paris : Nolin

Andrew Buckler

Can we give the Gospel to those who have not heard it? Reformed theological discourses about the salvation of the pagans in the Early Modern Era

The Protestant Reformation happened almost at the same time as the great voyages around Africa and towards Asia, as well as the discovery of America. Thus, when a new Christian denomination arose, there emerged unprecedented opportunities for evangelizing new peoples. However, all historians agree that, in the 16 th and 17 th centuries, the Catholic Church almost exclusively took up the challenge. Such weakness in Protestant missionary expansion has been accounted for in various ways. It has been suggested that the new Church had other priorities (reforming Christianity, strengthening an alternative doctrine to Roman theology, etc.); that 16 th century Protestant nations were none too familiar with distant overseas lands; that the removal of religious orders led to a lack of missionary personnel, etc. 1 Our contribution to this debate will examine the theological texts dedicated to the evangelization of pagans in order to determine whether New Churches leaders actually strived to boost that effort. We shall limit ourselves to the Calvinist literature tradition, not to embrace too vast a subject.

What evangelizers?

In the Roman Catholic Church, missions used to be, overwhelmingly, performed by members of religious orders. Though a few somewhat adventurous secular priests,

seeking fortune and limiting their ministries to their fellow-countrymen before expanding it to other populations, such as Las Casas in its infancy, 2 and the first modern era missionaries belonged mainly to long-standing religious orders including the mendicant ones (Augustinians, Carmelites, Mercedarians, Franciscans and Dominicans), and later to new orders such as the Jesuits and the Recollects, the Vincentians (Lazaristes), the Sulpicians, the Paris Foreign Missions priests, etc.. However, Protestantism removed the old orders and did not create new ones. Who, then, could evangelize pagans?

Ecclesiology has a very important place in Calvin's theology. The matter of the organization of the Early Church and Church ministries at his time is dealt with extensively in Book IV (the 1559-1560 edition) of the Institution de la Religion Chrétienne. This point was little developed in the early editions of the book and not until 1543, not only from Scripture but not from the Fathers of the Church either. 3 The clearest classification of ministries can be found in Chapter III of Book IV. Calvin believed the early Church had five ministries: apostles, to preach the Gospel to all; prophets, who have special revelations; evangelists, whose "ministries are next in importance to the Apostles', though they are inferior in dignity"; doctors, who are in charge of explaining Scripture; pastors, who are responsible for "discipline", [...] to administer Sacraments, [...] for exhorting and remonstrating, [...] and for explaining Scripture". The first three ministries are not to be perpetual but relevant "to the time it was necessary to establish churches where there were none," a time that, Calvin believed, was gone, though God could still, exceptionally, generate apostles or evangelists. Only pastors are left for the ordinary Church; these are close to apostles, except for one thing: "What Apostles do around the whole world, each Pastor is required to do in his own Church, which he is a representative of. "Every pastor is indeed linked to a particular Church. 4 As can be seen, no ministry was designed to evangelize pagans in Calvin's times. 2 Manuel Giménez Fernández, "Fray Bartolomé de Las Casas: A Biographical Sketch", in: Bartolomé de las Casas in History: Toward This organization is reflected in the various Reformed Churches. For example, the Discipline des Églises réformées de France from 1559 reserves the term "minister" for pastors, and it attributed to the offices of deacon and elder the same tasks as in Geneva; 5 ministers, elected by a consistory, are not allowed to preach elsewhere without prior consent of the minister or the consistory they visitwhich means they can rarely do so in non-Christian lands. 6 Later versions of the Discipline, until 1659, greatly increased the requirements regarding ministers, but they were there again not expected to evangelize outside their own Churches; quite the contrary, actually: "Ministers will not be elected without assigning them a particular herd, and they shall be suited to the herds they have been assigned". 7 Calvin, for his part, was much more discreet than Luther on the notion of universal priesthood, and did not seem to consider that mere faithful can evangelize: according to the Genevan reformer, God can be known only through pastors: "He [God] elects among men those he will make his ambassadors, whose task it is to enlighten the world about His will." 8 Of course, Calvin also stressed in many sermons (especially in sermons on Deuteronomy, preached from 1555 to 1556), the need to "draw all men on earth to God", 9 so as to grow and multiply the kingdom of Christ; note however, that he preached to Geneva citizens, who would never have the opportunity to evangelize pagans: hence, his exhortations were only rhetorical. Nowhere does he tell them to leave their city to make the Word of God available to distant peoples. When he speaks of the whole earth, it is to explain that it is God himself who spreads his doctrine, without us knowing what instruments he might have used: "Suffice it that God spreads the doctrine of salvation throughout the world, that there are small herds assembled here and there [. doctrine of the gospel] must resound throughout the earth, and let our Lord plant Churches everywhere, and be worshiped among barbarous people." 10 Primitive evangelization and the damnation of pagans A second obstacle to the evangelization of pagans is the belief that this work has already been done in apostolic times. That idea was already found in Luther's work, who had taken up the patristic and medieval tradition. It was taken up again, with some nuances, in the Calvinist tradition. Calvin himself was aware, however, that some people had never been evangelized. He knew, because he had followed Admiral Villegagnon's expedition in Brazil quite closely, that the Topinambous Indians had no knowledge of God 11 and, in the Institution de la Religion Chrétienne, he wondered "how is it that cognizance of the pure heavenly truth has never been made known to so many people, and that others have hardly had any opportunity to get a taste of only some rudiments of it?" 12 The answer, he said, commenting on the first letter to Timothy, is that God never said that all men and women were to know the truth, but that it is the sole privilege of all states and all conditions; and when Matthew mentions that the Good News is to be spread throughout the world, he only meant it would simply be found in "the most distant neighborhoods of the world". 13 Augustin Marlorat (1506-1562), a pastor in Rouen, also explained that "all men" actually means "all states & conditions". 14 Calvin himself did not seem to set great store by the actual realization of primitive evangelization because, while he would take a more relative view of it in some passages, he bore it out in others: in his commentaries on the Gospels, he explained that the doctrine had spread throughout the world after the apostles' departure from Galilee, which fulfilled the prophecy in Isaiah 49. 6. After Calvin, Theodore de Bèze expressed the same conviction that the apostles' preaching had reached the other side of the world or the New world in their times, because, as St. Paul wrote, at least a taste of the Gospel had penetrated there. [START_REF] De | Jesu Christi Domini Nostri Novum Testamentum…[END_REF] In the Netherlands, Grotius was also convinced that the Gospel had reached across the ocean in Pope Clement's time. [START_REF] Annotata | Evangelia… Sive Criticorum sacrorum[END_REF] However, Adrian Saravia, professor at Leyden until 1588 (then in exile in England), found that the Gospel had not yet reached all peoples. [START_REF] Saravia | De ministrorum evangelii diversis gradibus liber…[END_REF] As we shall see, many Dutch theologians, rather than question the reality of primitive evangelization, insisted on the need to bring the Good News to pagans. In contrast, Lutheran theologians almost all agreed that this work had already been accomplished in apostolic times: Lossius, Nicolai, Gesner, Gerhard, Johannes Müller, and many others, had not doubt about it either. [START_REF] Krumenacker | La conversion interdite?[END_REF] Equally interesting is the conclusion that most of these theologians derive from the existence of a primitive evangelization. Again, Calvin's ideas were broadly shared. To his mind, pagans had no extenuating circumstances: God "has imprinted some marks of his glory in all his works, and even in such a clear and significant way, that no allowance for ignorance can be allowed to the harshest and least sophisticated peoples in the world." [START_REF][END_REF] The Geneva reformer added in his commentary on Psalm 19 that "the glory of God is not written in cryptic and small letters, but richly engraved in nice big letters for everyone to read, and easily, too." [START_REF]Commentaires de Jehan Calvin sur le livre des Pseaumes[END_REF] For him, pagans have had some knowledge of the truth: "We see that among pagans there had always remained a residue of truth; we will see, from some of the statements pagans have formulated, that it is impossible for natural man to think up by themselves any of those they have brought there. Why then? It is God who has brought them there, for their conviction to be all the greater, so that on the last day it will all be taken into account?" 22 We can better understand why the matter of primitive evangelization was of little importance in Calvin's eyes: even though the Word of God had not been announced to them, they should have known the truth by contemplating creation; and if they remained ignorant, it was because God had doomed them. Predestination does actually extend to particular persons: it is each and every man that God saves or otherwise dooms to damnation. [START_REF][END_REF] It is easy to deduce that trying to convert pagans who had turned away from God could only be a transgression of God's eternal purposes.

Obviously, his successors shared similar views. Marlorat thought knowledge of God through creation "does not come before removing any excuse from men; and is quite different from what is alluded to in John 17, 3; which leads to salvation." 24 Bèze seemed of the same opinion, and hardly considered sending missionaries to the peoples deprived of Revelation because, under predestination, they were just excluded from salvation. [START_REF] Delval | La doctrine du Salut dans l'oeuvre homilétique de Théodore de Bèze[END_REF] Much later, an echo of this teaching can be found in Charles Drelincourt's works, though he was a supporter of sending missionaries to pagans and was sure the whole world had not been evangelized at the time of the apostles. [START_REF] Drelincourt | Les Visites charitables, ou les consolations chrétiennes, pour toutes sortes de personnes affligées[END_REF] The question looks like an open and closed case, then: the belief, a widely shared one, that primitive evangelization has indeed taken place, the assurance that outside knowledge of Jesus-Christ there can be no salvation, and the strength of the doctrine of predestination acted as a powerful restriction to the missionary enterprise of spreading the Word of God. Moreover, it is even more so in the Lutheran world. Georg Möbius went, against all evidence, to the extent of writing, in 1685, that the apostles could only have managed to go to America on foot, in order to accomplish the divine will of offering salvation to all! [START_REF] Möbius | Tractatus Philologico-Theologicus de Oraculorum origine, propagatione, & duratione[END_REF] Lucas Lossius better represented majority opinion regarding the damnation of all pagans and thinking it could only be a consequence of their sins. 28 Their common teaching was to be found in Loci Theologici by Johann Gerhard, who taught that the preaching the Gospel throughout the earth belonged to the apostles and that this duty had not been passed on to the Church; therefore the Gentiles could only be damned. 29 This was the basis of Lutheran teaching for decades. One final example could be a submission written in 1651 by the Faculty of Theology at Wittenberg claiming that God had legitimately removed salvation from the peoples who were still in darkness, on account of their carelessness and ingratitude. 30 Openings for mission Yet there are openings for mission, especially in the Calvinist world, and indeed, though Protestant missions were very rare in the 16 th century, they became slightly more numerous during the next century. Calvin, the first among others, did not close the door on any attempt at evangelization in his time. But can we even say, as do the advocates of Calvin having a missionary vision, that all his actions, the reform of Geneva, the will to bring the powerful to the true faith, sending books and pastors to France or elsewhere, was done out of missionnary intent? 31 This would be consistent with the ancient usage of the word, meaning that God manifests himself, make himself present in this world, as a mission, even though Calvin never used the word "mission" to describe his reformation work. Hold on, are we not going too far when equating preaching within the Christian world and proclaiming the Gospel to those who do not know Christ? Calvin thought the main objective was to reform the Church, to bring it back to true piety, to surrender all its glory to God, but only among peoples that knew him already, even though their faith was deeply clouded. He never ever contemplated starting a new Church, unlike, for example, the Anabaptists. On the other hand, he was convinced that it is always God that leads 28 Annotationum scholasticarum Lucae Lossii Luneburgensis…, Francfort: Chr. Egen, 1562. 29 Johann Gerhard, Loci theologici, 1610-1625 (numerous editions), volume 6, located in 23, ch. V, § 210-225. 30 men. "Missionary activism", like the one that could be found in the Catholic Church, was therefore out of the question. When he commented on the episode when Paul had come to Troas to preach the Gospel, but failed to meet Titus and went on his way to Macedonia instead, where he felt called by God, Calvin declared that "an opportunity had then presented itself to advance the Gospel. For, just as we can enter when the door is open, the Lord's servants also make advancements when the means prove to be available to them. The door is closed when there is no hope of ever making the most of it." 32 When, in 1556, the expedition in Brazil, led by Admiral de Villegagnon, requested pastors, Geneva sent two, but probably more as European explorers.

Nevertheless, one of the members of the expedition, Jean de Lery, began to try and convert the Topinambous Indians who, he thought, had been evangelized by St. Matthew himself. Twenty years later, he had become a pastor and he interpreted his failed attempt as a sign that God had not included the Topinambous among his elect. 33 The opportunity had presented itself to preach the gospel, he had been bound to seize it, but he needed not have persisted when conversion did not happen, because it was evidence it was not God's will. Yet, it was the main and one of the only attempts at evangelization by France or Geneva during the 16 th and 17 th centuries. However, some records have been found of Huguenots lamenting the lack of Protestants on the mission field. The great size of Catholic literature on missions has undoubtedly played a great role. The reports and letters Jesuits would send to report on their missions to their congregations were often published, as early on as the famous one sent by Francis Xavier in 1545; as of 1583, extracts from annual letters were 32 Commentaires sur toutes les epistres de l'apostre sainct Paul, Genève: Estienne Anastase, 1560 (about 2 Cor 2, 12). 33 About that famous expedition, see Franck Lestringant, "Calvinistes et cannibales. Les Écrits protestants sur le Brésil Français (1555-1560)", Bulletin de la Société de l'Histoire du Protestantisme Français, 1980, vol. 126, p. 9-26 and 167-192; id., "Tristes Tropistes: du Brésil à la France, une controverse coloniale à l'aube des guerres de Religion", Revue de l'Histoire des Religions, 1985, vol. CCII, no. 3, p. 267-294; id., Le Huguenot et le sauvage, Paris: Klincksieck, 1990; id., Une sainte horreur ou le voyage en Eucharistie, XVI e -XVIII e siècle, Paris: PUF, 1996; id., "Le huguenot et le sauvage: nouvelles hypothèses, nouveaux documents", dans: La France-Amérique (16th -17th centuries), ed. by Franck Lestringant, Paris: Honoré Champion, p. 131-142 (and, in the same collection, the article by B. Conconi); id., Jean de Léry ou l'invention du sauvage: essai sur l'Histoire d'un voyage faict en la terre du Brésil, Paris: Honoré Champion, 1999; id., D'encre de Brésil: de Léry écrivain, Orléans, Paradigme, 1999; Yves Krumenacker, "Le protestantisme et les découvertes au XVI e siècle"; id., "La conversion interdite? …"; Thierry Wanegffelen, "Rio ou la vraie Réforme", in: Aux temps modernes: naissance du Brésil, Paris: PUPS, 1998, communicated to pupils in Jesuit grammar schools (and we know today that, despite censorship by consistories and synods, many Protestants would send their children there) and to the general public. In the 17 th century, the Relations, whose most famous ones were the Relations de la Nouvelle France (41 volumes, 1632-1673), were a great success.

In addition, missionaries published stories of evangelization, which enjoyed great readership, such as L'Histoire générale des choses de la Nouvelle Espagne by Bernardino de Sahagun; L'Histoire naturelle et morale des Indes by José de Acosta, L'Histoire de l'expédition chrétienne au Royaume de la Chine by Trigault Nicolas, or L'Histoire des Indes orientales et occidentales by Giovanni Pietro Maffei. [START_REF] Comby | Deux mille ans d'évangélisation[END_REF] To which must be added a whole series of Protestant "black literature"; these writings were meant to give Catholic missions a bad namebut at the same time, they made them known as wellsuch as the translation by Urbain Chauveton of the Historia del Mondo Nuovo by Benzoni, De Rebus Gestis by Osorio and translated by Simon Goulart, [START_REF] Bataillon | L'Amiral et les 'nouveaux horizons' français[END_REF] or then again Las Casas' translation of Brevissima Relación de la destruccion de las Indias. [START_REF] Saint-Lu | Les premières traductions françaises de la "Brevissima Relación dela destrucción de las Indias" de Las Casas[END_REF] Though quite a few Protestants were sensitive to the exposure of the cruelties practiced by missionaries and the hypocrisy of Catholic statesarguably more sensitive to economic wealth than to the salvation of pagansor denounced the refusal to accept God's judgment irremediably condemning those who had not known Christ, others took an interest in these remote regions and lamented that Protestants were absent from this evangelistic effort.

We have some examples in France, like Beaulieu Heves, who in 1646 wanted to bring the gospel to the peoples who had never heard of it, [START_REF]Lettre inédite de Beaulieu-Heves à Ph. Vincent ministre de La Rochelle. 1646[END_REF] or like a lawyer at the Paris Parliament, Henry Mouche, Sieur de la Colombière, who sent huge amounts of money in the 1660s and 1670s to the British companies with a view to spreading the faith. [START_REF]Livre des Actes des Eglises Wallonnes aux Pays-Bas[END_REF] Similarly, the faithful person staged in Drelincourt's Visites charitables complains that there are no Protestants preaching the Gospel to pagans. 39 In the Lutheran world, Philipp Nicolaï deemed the Jesuits were Christians doing Christian work by announcing the fall, damnation and salvation through Christ, and that Roman Catholic Churches, be they Ethiopian or Muscovites, were instruments of salvation because they kept planting the Gospel worldwide; Johannes Müller wrote that, when the papists convert pagans to the apostles' faith, it concerns the Church of God, even though the Roman Church is not the true Church. 40 Another route was explored by Adrian Saravia. Born in 1531 in Artois, he was first a Franciscan monk before converting to the Reformation in 1557. After a short stay in Geneva (1558), he went to England where many Dutch Protestants were fleeing Spanish repression. He married when he once returned on the continent and was then appointed curate of the Threadneedle Street French Reformed Church, London. But he returned to Antwerp in 1562, as a minister of the Walloon Church, and worked to spread the Reformation to the Netherlands. But he had to flee in 1563 and took refuge in Guernsey. He then reached Southampton in 1569, where he was headmaster of a school.

The development of the Netherlands revolt allowed him to return to Ghent in 1580 and he became professor of theology (1584) and Dean (1585) of the University of Leiden. But he kept links with Britain and sought British intervention against Spain. Hence, he was suspected of conspiracy and was able to leave Leiden before being stripped of his post and sentenced to death. He fled to England where he became Dean of Tatenhill. This is where he wrote his De ministrorum evangelii diversis gradibus liber in 1590, which was refuted by Bèze and Gerhard among others. He questioned the traditional Calvinist ecclesiology, arguing that the evangelizing mission was not peculiar to the apostles, but was transmitted to their successors, the bishops; since the Church was holding the key to power, it had to send missionaries to unbelieving people. He maintained his position in his response to Bèze, the Defensio tractationis of diversis ministrorum Evangelii gradibus in 1594. He subsequently had a successful career in England, until his death as 39 Laurent Drelincourt, Les Visites charitables. 40 Texts quoted in Raupp, op. cit.

Dean of Westminster in 1613. 41 In his works, he defended Episcopalian ecclesiology, which can obviously be related to its links with England and its integration in the latter part of his life in the Anglican Church. But his word was apparently hardly taken up by Calvinist theologians, perhaps precisely because of its Episcopalian leaning. Bishop & R. Newberie, 1590, and Defensio tractationis de diversis ministrorum Evangelii gradibus, London: reg. Typog., 1594, and by Théodore de Bèze, Ad Tractationem de ministrorum evangelii gradibus, ab Hadriano Saravia Belga editam, Genève: J. Le Preux, 1592. 42 Biographies can be found of these characters on the site of the Boston University School of Theology, History of Missiology: http://www.bu.edu/missiology/about/ of deacons and teachers to Indonesia, and it is deemed that the number of Indonesians baptized between 1708 and 1771 reached 43,748. 43 How to explain this evolution? An essential element is undoubtedly that it kept in touch with reality. The writings that were most hostile to the evangelization of pagans often emanated from theologians who had not left Europe and often not even their countries, either. It dealt with theoretical reflections based on abstract dogmas, which could only spark disputes during the meeting, or through actual readings, with real pagans. One of the first to illustrate it was Jean de Lery: in contact with the Topinambous, he announced the Gospel to them; it was not until much later, after studying theology, that he explained why they had been excluded from salvation.

Similarly, it was probably the encounter with the Malayan archipelago populations which brought theologians as strictly tied to Calvinism as Waleuswho played an important role at the Synod of Dortor Voetius, one of the main representatives of the purest Reformed orthodoxy, to defend the idea of mission-work. Theologically, they could rely on the principle that every person, according to Calvin, must proclaim their faith around them; pastors surrounded by pagans enjoyed total legitimacy to announce the Gospel to them. On the other hand, the fact that these territories were annexed by the Dutch authorities gave the latter the obligation to evangelize their new subjects: it was no longer, strictly speaking, a matter of sending missionaries overseas, but of communicating the Gospel to the inhabitants of a Reformed persuasion territory.

Nevertheless, the Dutch decided to leave their province to serve in very distant areas, where Christianity was almost unheard of. This was then very close to the Catholic missionary paradigm.

It should also be noted that similar efforts were undertaken, in America, by Puritans like Whitaker Mayhew or Eliot, because of their eschatological ideas: they considered themselves as chosen by God to spread his reign by converting the Indians before the advent of end times. Similarly, the protagonists of the Great Awakening in the 18 th century were often millenarians, like Presbyterian Jonathan Edwards. It was the same with Pietists, who had millenarian ideas too, and who wanted to spread the Gospel all 43 Theologische Realenzyklopädie, Berlin: De Gruyter, t. 16 (1987), p. 135-136. over the world. This is probably another reason for the way Reformed missionary thought evolved, which it would be necessary go back on.

Yves Krumenacker

University of Lyon

Institut Universitaire de France

  Another opening to missions took place in Holland, thanks to the development of the colonial empire. Indeed, as of 1595, ships won their fight against the Portuguese monopoly over trade with the Malayan archipelago. In 1602, various commercial companies were merged into the Dutch Vereenigde Oost-Indische Compagnie (VOC), founded in 1602, which had the monopoly of trade with the East, but also the opportunity to take possession of territories such as Java, the Moluccas or Ceylon, which put the Dutch merchants in contact with non-Christian populations. The VOC statutes provided for the promotion and defense of the Dutch Reformed Church and, therefore, pastors accompanied the expeditions and stayed in the colonies. A great number of them were much more supportive of settlers than natives, or baptized the latter only after very superficial evangelization. The experience, however, stirred true missionary zeal among some. Thus, Antoine Waleus founded, in Leiden in 1622, a Seminarium Indicum for VOC pastors, which, however, remained open for only eleven years. He was assisted in his efforts by Justus Heurnius, the author in 1618 of De Legatione evangelica ad Indos capessenda Admonitio and he himself settled in the Dutch East Indies from 1624 to 1639. The great Utrecht theologian, Gisbertus Voetius, defended the idea of missions to unbelievers, but under the authorities of the synod and not of States' or companies'. His pupil, Johannes Hoornbeek, also a professor at Utrecht, then Leiden, wrote an important treatise, De conversione Indorum ac gentilium (posthumous, 1669), where he stressed the responsibility of the Dutch government for advancing the kingdom of Christ. 42 In total, for two centuries, the Dutch Reformed Church sent eight hundred pastors and thousands 41 Willem Nijenhuis, Adrianus Saravia (c. 1532-1613), Leyde : Brill, 1980. The main works on the controversy have been written by Saravia, De ministrorum evangelii diversis gradibus liber…, London: G.

  an Understanding of the Man and his Work, ed. by Juan Friede, Benjamin Keen, DeKalb: Northern Illinois University Press, 1971, p. 67-126; The Life and Writings of Bartolomé de Las Casas, ed. by Henry Raup Wagner, Helen Rand Parish, Albuquerque: University of New Mexico Press, 1967, p. 5-6.

3

Alexandre Ganoczy, Calvin théologien de l'Église et du ministère, Paris: Cerf, 1964. About Calvin in general, see Yves Krumenacker, Calvin au-delà des légendes, Paris: Bayard, 2009.

4

Jean Calvin, Institution de la Religion Chrétienne, ed. by Jean-Daniel Benoit, Paris: Vrin, 1957-1963, IV, III, § 4-9; for the most part, the writing of these paragraphs goes back to 1545.

  Institution de la Religion Chrétienne, IV, III, § 1. 9 Sermon on Deuteronomy, 33, 18-19, quoted by Andrew Buckler, op. cit., p. 91. In this book, Buckler mentions numerous quotations from Calvin, mainly drawn from his Commentaries of Scripture and from his sermons, supporting the advancement of Christ's Kingdom.

	..] It [the
	5 L'Organisation et l'action des églises réformées de France (1557-1563). Synodes provinciaux et autres
	documents, ed. by Philip Benedict, Nicolas Fornerod, Genève: Droz, 2012, p. lii-liii.
	6 Bernard Roussel, "La Discipline des Eglises réformées de France en 1559: un royaume sans clergé?", in:
	De l'Humanisme aux Lumières, Bayle et le protestantisme, ed. by Michelle Magdelaine, Maria-Cristina
	Pitassi, Ruth Whelan, Antony McKenna, Paris: Oxford, Universitas, Voltaire Foundation, 1996, p. 169-
	191.
	7 Isaac D'Huisseau, La Discipline des Eglises réformées de France, Genève et Saumur, ed. by Isaac
	Desbordes, 1667, p. 61.

8

  15 10 Sermon on Isaiah 26, 15, quoted by Andrew Buckler, op. cit., p. 97-98. 11 Olivier Reverdin, Quatorze calvinistes ches les Topinambous. Histoire d'une mission genevoise au Brésil (1556-1558), Genève: Droz, 1957. 12 Institution de la Religion Chrétienne, III, 24, 15.

13

Jean Calvin, Commentaires

[…]

sur toutes les epistres de l'apostre sainct Paul, Genève, Etienne Anastase, 1560 (about 1 Tim 2. 4); id., Concordance qu'on appelle harmonie, Genève: C. Badius, 1555 (about Mt 24. 14). 14 Augustin Marlorat, Le Nouveau Testament, Genève: Jean Bonne-Foy, 1563 (about 1 Tim 2. 4).

15

Commentaires de Jehan Calvin sur le Nouveau Testament, Genève: Conrad Badius, 1561 (about Mc 16. 15 and Mt 28. 19).

  A dissertation presented to Count Erhardt Truchsess zu Wetzhausen in 1651 by the Wittenberg Theological Faculty, in: Mission in Quellentexten, p. 70-71.

31

Philip Hugues, "John Calvin, Director of Missions", in: The Heritage of John Calvin, ed. by John H. Bratt, Grand Rapids: William B. Eerdmans, 1973, p. 40-54; Andrew Buckler, Jean Calvin et la mission de l'Église.