Y Nishikawa
email: nishikawa@ipgp.fr

P Lognonné

T Kawamura

A Spiga

E Stutzmann

M Schimmel

T Bertrand

F Forget

K Kurita

Y Suffix

Bertrand Particle

T Suffix

F Suffix

K Suffix

William B Banerdt

Christopher T Russell

• P Lognonné

• T Kawamura

• A Spiga

• E Stutzmann

• M Schimmel

• K Kurita

B Y Nishikawa

Mars' Background Free Oscillations

Keywords: Mars -Planetary free oscillation -GCM -Seismometer -Normal mode -InSight Mars, Planetary free oscillation, GCM, Seismometer, Normal mode, InSight

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Introduction

When a large earthquake occurs, seismic waves propagate around the entire planet for tens of hours following the event. These waves can generate standing oscillations if they fulfill the boundary conditions for specific frequencies; the standing modes are referred to as Earth's free seismic oscillations and their associated frequencies are called Earth's seismic eigenfrequencies.

The first search for Earth's free oscillations started in the late 1950 [START_REF] Benioff | Searching for the Earth's free oscillations[END_REF]) these oscillations were first observed following the great Chilean earthquake of May 1960 [START_REF] Benioff | Excitation of the free oscillations of the Earth by earthquakes[END_REF]. Further developments in long-period seismometers and global networks then allowed the normal mode frequencies to be determined, and this information was used to invert the structure models of Earth's interior, including the famous Preliminary Earth Reference Model [START_REF] Dziewonski | Preliminary reference Earth model[END_REF]. Further inversions were done by using the splitting properties of the normal modes, which provided three-dimensional models of the Earth's structure. [START_REF] Woodhouse | Theory and Observations -Earth's Free Oscillations[END_REF]; [START_REF] Laske | Treatise on Geophysics, Theory and Observations: Normal Mode and Surface Wave Observations[END_REF]; [START_REF] Montagner | Treatise on Geophysics[END_REF] and [START_REF] Romanowicz | Treatise on Geophysics, Deep Earth Structure: Q of the Earth from Crust to Core[END_REF], provide an extensive description of observations of Earth's normal modes and their use in determining Earth's interior structure, while [START_REF] Dahlen | Theoretical Global Seismology[END_REF] and [START_REF] Millot-Langet | Realistic long period synthetic seismograms using normal modes of a 3d anelastic elliptic rotating Earth[END_REF] provide a theoretical description of the same.

Thus, determining Mars' normal modes has been proposed as one of the overarching goals of a seismological mission to Mars from the early times of space exploration [START_REF] Kovach | The interiors of the terrestrial planets[END_REF][START_REF] Bolt | Free bodily vibrations of the terrestrial planets[END_REF] to Viking [START_REF] Okal | Theoretical models for Mars and their seismic properties[END_REF] as well as the more recent projects such as Mesur [START_REF] Solomon | ) A. Spiga, Elements of comparison between Martian and terrestrial mesoscale meteorological phenomena: katabatic winds and boundary layer convection[END_REF], Intermarsnet [START_REF] Lognonné | Ultra broad band seismology on InterMarsNet[END_REF] and NetLander [START_REF] Lognonné | The NetLander very broad band seismometer[END_REF]. Several papers detail detection techniques have been detailed in several papers (Lognonné and Mosser 1993;[START_REF] Zharkov | On the dissipative factor of the Martian interiors[END_REF][START_REF] Gudkova | Mars: interior structure and excitation of free oscillations[END_REF][START_REF] Lognonné | Planetary seismology[END_REF]Lognonné andJohnson 2007, 2015). The detection of the normal modes of Mars is one of the goals of the upcoming seismic monitoring mission "InSight" which should land and deployment in 2019 and of the SEIS experiment [START_REF] Tong | Extraterrestrial Seismology[END_REF]Lognonné et al., this issue).

Planetary free oscillations constitute a powerful tool to investigate the internal structure of planets, as illustrated by [START_REF] Panning | Planned products of the Mars structure service for the InSight mission to Mars[END_REF] with the synthetic inversion of the structure of the fundamental spheroidal normal modes. However making such long-term seismic observations on Mars obliges us to overcome several difficult problems.

One issue is related to Mars itself, which lacks plate tectonics and therefore is devoid of the large-magnitude quakes (M > 7) the type of which are generally used on the Earth to analyze the normal modes. All seismicity models of Mars [START_REF] Phillips | Martian seismicity[END_REF][START_REF] Golombek | A prediction of Mars seismicity from surface faulting[END_REF][START_REF] Knapmeyer | Working models for spatial distribution and level of Mars' seismicity[END_REF] suggest that the largest quake annually is in the moment-magnitude range of 5.2-6 (i.e., 10 17 to 10 18 Nm). Larger quakes, although very rare, cannot be excluded.

The second difficulty is the seismic noise expected on the Martian surface, which is known to be subject to large variations in temperature and wind that generate long-period noise much greater than anything measured in Earth's seismic vaults, where Earth's normal modes are typically observed. The InSight SEIS noise model, described in detail by [START_REF] Mimoun | The seismic noise model of the InSight mission to Mars[END_REF], suggests that, over long periods, most of the SEIS instrument noise will be related to the thermal noise, which is expected to grow below 10 mHz as f -2.5 , with amplitude at 10 mHz in the range of 1 to 2.5 × 10 -9 m s -2 Hz -1/2 depending on the local time.

As shown by [START_REF] Lognonne | GEP-ExoMars: a geophysics and environment observatory on Mars[END_REF], seismic signals would have larger amplitudes globally were a quake with a moment larger than 2 × 10 17 Nm to occur on Mars. Even if such a moment corresponds to the upper limit of the annual predicted seismicity, the Poisson statistics of such a rare event suggests that several years of operation are required to attain a significant occurrence probability. Therefore, we consider herein the nonseismogenic sources of normal mode excitation.

One possible source of normal mode excitation comes from atmospheric phenomena (see [START_REF] Spiga | Atmospheric science with InSight[END_REF], for a detailed review). Mars has a very active atmosphere, with weather processes generating high winds, dust devils, and storms [START_REF] Read | The Martian Climate Revisited: Atmosphere and Environment of a Desert Planet[END_REF][START_REF] Spiga | A new model to simulate the Martian mesoscale and microscale atmospheric circulation: validation and first results[END_REF]. Cloud tracking shows that Martian wind speeds easily reach 80 m/s above 30 km [START_REF] Kaydash | Measurements of winds on Mars with Hubble Space Telescope images in 2003 opposition[END_REF], whereas surface wind can be as high as 20 m/s, according to measurements by the Viking landers (e.g., [START_REF] Murphy | Observations of Martian surface winds at the Viking Lander 1 site[END_REF]). Measurements by the Martian landers also show that the Martian atmosphere has a regular diurnal cycle of wind and pressure [START_REF] Schofield | The Mars Pathfinder atmospheric structure investigation/meteorology (ASI/MET) experiment[END_REF][START_REF] Martínez | The modern near-surface Martian climate: a review of insitu meteorological data from viking to curiosity[END_REF].

The nonacoustic pressure variations associated with the global weather and the Reynolds stress associated with the wind, be it turbulent gusts in the planetary boundary layer (PBL) or high-altitude planetary-scale wind, all constitute forces to excite planetary free oscillations (Kobayashi and Nishida 1998b,a), with theoretical details in [START_REF] Tanimoto | Cause of continuous oscillations of the Earth[END_REF]. Part of the excitation forces has frequencies within the bandwidth of the normal modes and that corresponds to the eigenfrequencies of Martian normal modes between 0.4 and 20 to 30 mHz. These particular excitation forces can globally and continuously excite the corresponding normal modes. This continuous excitation force is a possible source of Mars' background free oscillations (MBFs).

This idea for the observation of MBF was actually been suggested shortly after the first observation of Earth's background free oscillations (EBFs) [START_REF] Suda | Earth's background free oscillations[END_REF]. Oceanic excitation was initially proposed by [START_REF] Watada | Oceanic excitation of the continuous oscillations of the Earth[END_REF]. The first interpretations proposed that the entire excitation is produced by atmospheric turbulence in the Earth's PBL, which suggests that a simple scaling may exist between Earth and Mars (Kobayashi and Nishida 1998b,a). The seasonal variations in the continuous excitation later observed by [START_REF] Nishida | Resonant oscillations between the solid earth and the atmosphere[END_REF] also support an atmospheric origin, while [START_REF] Tanimoto | Cause of continuous oscillations of the Earth[END_REF] proposed a simplified theory. However, the most recent studies show that the major source of continuous excitation on Earth is over the oceans [START_REF] Tanimoto | The oceanic excitation hypothesis for the continuous oscillations of the Earth[END_REF]Rhie andRomanowicz 2004, 2006) and that, in this process, infragravity waves over the continental shelves are much more efficient seismic sources [START_REF] Webb | The Earth's hum is driven by ocean waves over the continental shelves[END_REF]) than atmospheric turbulence. However, excitation by atmospheric sources remains significantly below 5 mHz (Nishida 2013b). Note that coupled modes exist between the atmosphere and the solid earth [START_REF] Lognonné | The seismic optimism experiment[END_REF][START_REF] Lognonné | Modeling of atmospheric-coupled Rayleigh waves on planets with atmosphere: from Earth observation to Mars and Venus perspectives[END_REF], and that pressure sources at the bottom of the ocean [START_REF] Nishida | Source spectra of seismic hum[END_REF]) cannot explain this larger excitation; instead, integrated atmospheric excitations at the base of Earth's atmosphere must be involved.

Observed EBFs are on the order of 0.5 nanogal (1 nanogal = 10 -11 m s -2 ) (Nishida 2013a) per individual mode. Stacking techniques can enhance the peak amplitude of normal modes to overcome instrument-and station-induced noise, which allows the mode eigenfrequencies to be determined. These eigenfrequencies may then be used to invert the internal structure as was done for quakes, as illustrated by [START_REF] Nishida | Global surface wave tomography using seismic hum[END_REF] determine the frequencies of fundamental spheroidal normal modes, which would constrain models for Mars' interior. Kobayashi and Nishida (1998b) estimated the magnitude of MBFs by using a theoretical scaling based on the solar energy received by the planet (related to the planetary albedo and distance to Sun) and on the thickness of the PBL. They assumed that turbulence in the PBL provides the main excitation force for MBFs, which led them to estimate a free oscillation amplitude of several nanogals, a value quite close to that of Earth. [START_REF] Tanimoto | Continuous free oscillations: atmosphere-solid earth coupling[END_REF] estimated relative modal estimated the relative modal amplitude between Earth and Mars by using several parameters for the atmosphere and terrestrial part. Estimates of the MBF amplitude are about 30-50% of the EBFs. [START_REF] Lognonné | Planetary seismology[END_REF] and Lognonné and Johnson (2007) focused on very-long-period MBFs and, by using more realistic Martian-climate models and normal mode summations, produced MBF seismograms for periods ranging 300 to 400 s, which correspond to angular orders 12 to 15. This approach is based on the assumption that, for these very long periods and wavelengths, the major source of excitation is not Reynolds stresses or nonacoustic pressure related to the PBL, but the nonacoustic pressure related to global atmospheric circulation. They estimated the free-oscillation amplitude to be several hundredths of nanogals, which corresponds to amplitudes typically ten times smaller than those of EBFs of the same angular order.

The generation of seismic waves by atmospheric activity is not only associated with nonacoustic pressure acting on the surface, but also with the acoustic waves generated in the atmosphere and converted into seismic waves at the surface. [START_REF] Lognonné | The seismic optimism experiment[END_REF], Watada and[START_REF] Watada | Acoustic resonant oscillations between the atmosphere and the solid earth during the 1991 Mt. Pinatubo eruption[END_REF][START_REF] Lognonné | Modeling of atmospheric-coupled Rayleigh waves on planets with atmosphere: from Earth observation to Mars and Venus perspectives[END_REF] simplified this concept by demonstrating that the overall excitation in the atmosphere may be estimated based on the normal modes of the coupled solid-atmosphere system and on the atmospheric force volumetric density, which acts throughout the atmosphere and is subject to wind and nonacoustic pressure fluctuations. This approach requires integrating over the full atmospheric seismogenic volume to calculate the excitation force of the atmosphere. As shown more precisely by [START_REF] Nishikawa | Excitation of jovian seismic waves by the Shoemaker-Levy 9 cometary impact[END_REF] the latter is associated with seismic forces related to nonacoustic pressure and nonlinear Reynolds stresses in the atmosphere and can be expressed as

Π j i = -p glut δ j i -ρv i v j , ( 1
)
where p glut nonacoustic pressure, which is defined as p glut = p truep Hooke , where p Hooke = -κ∇.v. v is the atmospheric wind, ρ is atmospheric density, and p Hooke is the pressure modeled by the Hooke pressure, which involves the linear model used in linear acoustics or seismology. Therefore, following the proposition made by [START_REF] Backus | Moment tensors and other phenomenological descriptions of seismic sources-I. Continuous displacements[END_REF] for localized sources in Earth seismology and by [START_REF] Goldreich | Solar seismology. II. The stochastic excitation of the solar p-modes by turbulent convection[END_REF] for globally distributed sources in Solar seismology, we generalize here the concept of stress glut to the pressure induced by nonlinear Reynolds stresses that occur in the Martian PBL. This generalization considers that all true volumetric forces, apart from those generated by Hooke pressure, are source terms captured by the seismic moment of Eq. (1). Thus, we consider that the pressure glut and wind are respectively written as

p glut = p glut,global + p glut,PBL , v = v global + v PBL . ( 2
)
Practically speaking, the wind and pressure glut associated with the PBL turbulence will be at much higher frequencies and on a smaller scale than those associated with the global circulation, but both will contribute to the overall excitation processes through Eq. ( 1). Therefore, the two estimates seem to provide the two end members of the general case, with Lognonné and Johnson (2007) focusing on the global part of the excitation and Kobayashi and Nishida (1998b) focusing on the PBL part.

Major differences in PBL thickness exist between the Earth PBL, which ranges from 0.2 to 2 km and the Martian PBL, which is much thicker (about 5 to 6 km, typically) [START_REF] Petrosyan | The Martian atmospheric boundary layer[END_REF]Spiga 2011). Because the speed of sound is less on Mars than on Earth (220 m/s compared with 340 m/s), the volumetric excitation is likely much more important on Mars than on Earth where, moreover, most of the energy of the atmospheric winds is transferred to oceanic waves, which then play a major role in the excitation of EBFs. In other words, the Earth view, where most EBF excitation may be modeled by invoking bottom-oceanic and near-surface forces, is not likely to be valid on Mars.

A first key difference is the important daily repetition of weather patterns on Mars, which is mostly related to the importance of solar flux and to the lack of a major source of other atmospheric energy, such as Earth's humidity and water. Although this initially appears in the temperature variations, which are fairly well represented by a Fourier series (see, e.g., [START_REF] Van Hoolst | Tidally induced surface displacements, external potential variations, and gravity variations on Mars[END_REF], it is also the case for the pressure field, as illustrated in Fig. 1, which shows the pressure variations at zero longitude as a function of local time and of latitude over two sols, as generated by a typical GCM. Such sol repetition also appears in the signal, when the latter is filtered in the frequency bandwidth of the normal modes. This is shown in Fig. 1, which corresponds to the signals of Fig. 1, but is filtered between 1 and 2 mHz (i.e., periods between 1000 and 500 s) and by Fig. 3, where the bandwidth is higher and corresponds to 2 to 3 mHz (i.e., periods between 330 and 500 s).

In this paper, we expand these past studies by using higher-resolution GCMs developed recently by the Laboratoire de Météorologie Dynamique (LMD) (see e.g., Forget and Lebonnois 2013) to better estimate normal mode amplitudes and to analyze whether the repeating atmospheric sources significantly impact the excitation of normal modes, including terms of stacking strategies. These GCMs can resolve all the large-scale Martian atmospheric processes and variabilities such as thermal tides, baroclinic waves, and planetary-scale waves [START_REF] Haberle | General circulation model simulations of the Mars Pathfinder atmospheric structure investigation/meteorology data[END_REF][START_REF] Forget | Improved general circulation models of the Martian atmosphere from the surface to above 80 km[END_REF]. Turbulent motions in the PBL are, by design, left unresolved in GCMs; their study requires large-eddy simulations [START_REF] Spiga | Martian mesoscale and microscale wind variability of relevance for dust lifting[END_REF]. This will not be included for the excitation force investigated herein [note that atmospheric excitation by small-scale turbulence in the PBL is addressed by [START_REF] Kenda | Modeling of ground deformation and shallow surface waves generated by Martian dust devils and perspectives for near-surface structure inversion[END_REF] and [START_REF] Murdoch | Estimations of the seismic pressure noise on Mars determined from Large Eddy Simulations and demonstration of pressure decorrelation techniques for the InSight mission[END_REF].

Normal mode amplitudes are computed from GCM results by summing the normal modes, which estimates the signals that may be recorded by the InSight SEIS VBB seismometer (Lognonné et al. 2015;Lognonné et al., this issue). Compared with previous studies, the present study provides not only better lower estimates of MBF normal modes and We applied a band-pass filter between 2.0 and 3.0 mHz to the raw pressure data (shown in Fig. 1). The large pressure fluctuations in this band occur twice a day (at sunrise and sunset). The frequency band corresponds to middle degree MBFs (angular order 9 to 15). Normal modes at this frequency are sensitive to the upper mantle (see Fig. 5) extension to higher frequencies than Lognonné andJohnson (2007, 2015), but also pathways for the future processing of InSight measurements. Our synthetic signals will be compared with the expected noise and used to test stacking strategies, with a special emphasis on possible coherent stacking techniques made possible by the daily repetition of Martian weather. Finally, we conclude by estimating the probability of detecting MBFs with the SEIS seismometer.

AUTHOR'S PROOF

Journal To predict the atmospheric excitation force, we used Mars Global Climate Model (GCM) which is designed to simulate large-scale atmospheric dynamics over an entire planetary sphere. The characteristics of the model are described in detail in [START_REF] Forget | Improved general circulation models of the Martian atmosphere from the surface to above 80 km[END_REF]. We use the latest, most realistic, version of the model, which features interactive dust transport [START_REF] Madeleine | Revisiting the radiative impact of dust on Mars using the LMD global climate model[END_REF], radiatively active water and ice clouds [START_REF] Navarro | Global climate modeling of the Martian water cycle with improved microphysics and radiatively active water ice clouds[END_REF], and a thermal plume model for the boundary layer [START_REF] Colaïtis | A thermal plume model for the Martian convective boundary layer[END_REF].

Computing normal mode excitation requires a GCM simulation that is able to characterize the Martian atmospheric temporal variability of pressure and winds in the Martian atmospheric at 0.01 Hz (period 100 s), with a horizontal resolution suitable for normal modes, (i.e., 10 • latitude and longitude). These requirements, dictated by seismic computations, are somewhat paradoxical from a meteorological point of view. A timescale of 100 s is associated with atmospheric circulations evolving over typical spatial scales of tens to hundreds of meters, which are challenging to resolve with GCMs, even with the best available supercomputing cluster. Most of the 100 s variability of the Martian atmosphere is instead captured through limited-area, turbulence-resolving modeling [large-eddy simulations; see, e.g., [START_REF] Spiga | Martian mesoscale and microscale wind variability of relevance for dust lifting[END_REF]; see also [START_REF] Kenda | Modeling of ground deformation and shallow surface waves generated by Martian dust devils and perspectives for near-surface structure inversion[END_REF], [START_REF] Murdoch | Estimations of the seismic pressure noise on Mars determined from Large Eddy Simulations and demonstration of pressure decorrelation techniques for the InSight mission[END_REF] for a discussion of local seismic signature]. The remainder of this variability (i.e., the contribution that is not caused by microscale turbulence) is associated with mid-latitude baroclinic fronts and regional slope winds (which impact the global dynamics) and can be satisfactorily simulated through GCMs with a horizontal resolution of 1 • . Conversely, nearly three decades of Mars GCM studies have consistently demonstrated that the global dynamics (at spatial scales of about 10 • latitude and longitude) are well simulated by using a typical time step of 925 s in the GCM.

We also ran Martian GCM simulations specifically tailored for the seismic computations. The LMD GCM simulations used a 60 s time step, 360 longitude grid points, and 180 latitude grid points, corresponding to a mesh spacing of 1 • × 1 • and a horizontal resolution of 60 km [in a setting akin to the GCM simulations described in [START_REF] Pottier | Unraveling the Martian water cycle with high-resolution global climate simulations[END_REF]]. In the vertical direction, 25 levels are typically used with most of the levels located in the first 15 km to ensure a suitable resolution in the lower troposphere and in the boundary layer. Above 10 km, the vertical resolution is about one scale height and the altitude of the top level is about 60 km, which corresponds to about 6 scale heights. This vertical grid thus offers both the refined near-surface resolution and the accounts for the vast majority of the atmospheric mass, which allows us to deal with all possible seismic coupling in subsequent computations. It is important to note that turbulent motions developing at high frequency are not resolved by the GCM: the PBL mixing they cause is parameterized in the LMD GCM by dedicated schemes [START_REF] Colaïtis | A thermal plume model for the Martian convective boundary layer[END_REF]. The GCM results used to compute normal modes thus only feature atmospheric variability at scales ranging from regional to global, and frequencies typically of the order 10 -3 Hz (and below).

Typical GCM simulations of the Martian climate are needed so that the resulting analysis of normal modes applies to the conditions of the InSight mission. However, this does not require, however, as many simulations as would be expected from equivalent terrestrial studies. Both the low thermal inertia of the Martian surface and the fast radiative timescale of the thin Martian atmosphere imply a very low Martian-climate inter-annual variability (except during the dust storm season, but InSight will land in 2018 at the end of this season) [START_REF] Read | The Martian Climate Revisited: Atmosphere and Environment of a Desert Planet[END_REF]. Furthermore, given the key role played by the atmospheric-dust loading in driving the Martian climate as well as the small inter-annual variability of this parameter in the first half of the Martian year [START_REF] Montabone | Eight-year climatology of dust optical depth on Mars[END_REF] ). The results differ by an order of magnitude, implying that GCMs cannot fully explain the observed energy, which is likely due to inadequate modeling in GCMs of local-scale turbulences. Thus, MBFs calculated by using GCMs should be regarded as a lower limit of possible excitations simulation for the year in which the InSight mission occurs will provide sufficiently accurate predictions for all Martian years (outside the dust storm season). The dust scenario used thus corresponds to the Martian Year 27 which is a typical clear atmosphere scenario (dust visible opacity around 0.2) with no effects from a global dust storm. GCM simulations start at Ls = 0 • (northern spring), which corresponds to the first months of the InSight mission on Mars, and are performed for 20 Martian sols with a fixed dust profile. Given that gravity waves are partly resolved in our 1 • × 1 • GCM simulation, we do not use the subgrid-scale scheme to model the effect of those waves on the large-scale flow. The outputs of the model, which are used as inputs for the modeling of the seismic moment density, as defined by relation Eq. ( 1), and are surface pressure and temperature, as well as atmospheric density, temperature, and winds. The observed atmospheric status differs from that predicted by the GCM, despite the location being the same. This is illustrated in Fig. 4 for the Pathfinder location. Parts of these discrepancies are related to both the pitot pressure (associated with local wind) and to the local-scale eddies,which are not resolved by the GCM but are discussed above.

As we mentioned above, this is a challenging problem because our approach relies on GCMs. It is dependent on the environment in which the observation is made, the season, and other factors, notably, that GCMs lose energy at Martian normal mode frequencies (Fig. 4). Large-amplitude areas concentrate the models energy and sensitivity. Low-degree waves (angular order 2 to 9) are sensitive at the core and deep mantle. High-degree waves (angular order 10 to 39) are sensitive at the middle and shallow part of the mantle. Each mode is sensitive at a different depth [START_REF] Sohl | The interior structure of Mars: implications from SNC meteorites[END_REF][START_REF] Spohn | Geophysical constraints on the evolution of Mars[END_REF][START_REF] Rivoldini | Geodesy constraints on the interior structure and composition of Mars[END_REF]). However, only normal modes with very long periods are sensitive to the core size and, furthermore, their amplitude near the surface depends only weakly on core size. Consequently, the choice of internal model is not critical for our estimate of MBF amplitude, and all models will provide similar results. In the present work, we use the Sohl model to generate a reference internal structure. We calculate fundamental spheroidal normal modes longer than 120 s, and the limitation of the period is dictated by the GCM time step. This corresponds to calculating spheroidal normal modes of angular orders between = 2 and = 39 and of radial order n = 0 which constrains the core and the deep, middle mantle (Fig. 5).

Estimating the Amplitude of Martian Normal Modes

Normal Modes Excitation

The theory of normal mode excitation and of normal mode summation is described in several papers (see, e.g., [START_REF] Lognonné | Normal modes and seismograms in an anelastic rotating Earth[END_REF][START_REF] Dahlen | Theoretical Global Seismology[END_REF][START_REF] Lognonné | Normal modes of the Earth and planets[END_REF]) so we only briefly summarize the approach here. Martian global oscillations are governed by the gravito-elastic equation (when attenuation is neglected) or by the gravito-anelastic equation, when anelastic processes are considered. Atmospheric normal modes may also be computed by using the theory developed by Lognonné et al. (1998a). [START_REF] Lognonné | Modeling of atmospheric-coupled Rayleigh waves on planets with atmosphere: from Earth observation to Mars and Venus perspectives[END_REF] studied attenuation processes in the Mars atmosphere due to both the viscosity and CO 2 molecular relaxation. They concluded that, for long periods (e.g., T ≥ 50 s), no significant atmospheric attenuation should be expected in the lower atmosphere where the greatest excitation force is concentrated, suggesting that an adiabatic approximation of the atmosphere suffices for our modeling. The starting equation is

-ω 2 k ρ 0 u k + A(u k ) = f, (3)
where ω k is the angular eigenfrequency, u k is the associated displacement eigenfunction, A is the gravito-elastic operator (or gravito-anelasto operator), ρ 0 is the unperturbed density and f is the external force. The gravito-elastic operator A is defined for the internal structure model and is given by: for the solid part, where Φ is the mass-redistribution potential, g is the gravity acceleration due to gravity, T 0 is the pre-stress and δT elastic is the elastic stresses. The operator in the atmospheric part can be written by using T 0 = -p 0 I and

A(u) = ∇(δT elastic -u • ∇T 0 ) -g∇ • (ρ 0 u) + ρ 0 ∇Φ, (4)
T elastic = -p Hooke I = κ∇.v I, ( 5
)
where I is the identity tensor. The result is

A(u) = ∇(κ∇ • u + u • ∇p 0 ) -g∇ • (ρ 0 u) + ρ 0 ∇Φ, ( 6
)
where κ is the bulk modulus and p 0 is the equilibrium pressure. Following [START_REF] Nishikawa | Excitation of jovian seismic waves by the Shoemaker-Levy 9 cometary impact[END_REF], we can write the momentum-density field ρv in the form

ρ(r, t)v(r, t) = ρ 0 (r) k ċk (t)u k (r), (7)
which provides the displacement field u, which can be expressed to first order (and therefore sufficiently far enough from the sources when on location) as

u = k c k (t)u k (r). ( 8
)
We get the differential equation with respect to time and the source function as

∂ 2 c k (t) ∂t 2 + ω 2 k c k (t) = Ψ k (t), (9)
where Ψ k is the excitation force for each mode. To compute the normal mode excitation forces, we consider the nonlinear equation of momentum conservation,

∂ ∂t (ρv) = -∇p + ρg -∇ • (ρvv), = -∇p Hooke -∇ • (ρu)g -∇δp glut -∇ • (ρvv), = -A(u) -∇δp glut -∇ • (ρvv), ( 10
)
δp glut = p -p 0 + κ∇ • u + u • ∇p 0 , δp Hooke = -κ∇ • u -u • ∇p 0 ,
where p 0 is the equilibrium pressure and p glut = p 0 + δp glut is the GCM pressure as the GCM filter acoustic waves. We assume here that the gravity term is linear and therefore excluded from this term excitation processes associated with nonlinearity [START_REF] Julián | Canonical piecewise-linear approximation of smooth functions[END_REF]). If existing in the Martian atmosphere, (e.g., for atmospheric gravity waves), this might be therefore an additional excitation term for frequencies below the 2 mHz acoustic cutoff [START_REF] Lognonné | Modeling of atmospheric-coupled Rayleigh waves on planets with atmosphere: from Earth observation to Mars and Venus perspectives[END_REF]). However, transport terms are fully included in the Reynolds stress term ρvv. Note also that the GCM pressure variations are much greater than those generated by the flow, so that δp glut ≈ pp 0 . We project Eq. ( 10) onto a given normal mode, which gives

V dV u k • ∂ ∂t (ρv) = ck , = - V dV u k • A(u) - V dV u k • ∇δp glut + ∇ • (ρvv) ,
V dV u k • ∂ ∂t (ρv) = ck , = -ω 2 k c k (t) - V dV u k • ∇δp glut + ∇ • (ρvv) . ( 11
)
Note that, in this projection, we shall decompose the volumetric integration for all partial volumes (in practical terms, the atmosphere and interior) i separated by discontinuities j , such that

V dV u k • ∇δp glut + ∇ • (ρvv) = i S i dS [δp glut u k • n i + ρu k • v v • n i ] - V i dV ∇u k • m = - V dV ∇u k • m + j S j dS [u k • n i δp glut ] + - ( 12
)
where we assume no vertical winds at the interface. (Note that this approach is here limited to a spherical surface and that additional excitation terms could occur due to topography, in way similar to what occurs for Earth's hum [see, e.g., [START_REF] Nishida | Ambient seismic wave field[END_REF]]. The vector n i is the normal to the surface, leaving the volume i, denoted + andthe volume against +, for which the normal vector is oriented in the opposite directions. Following Lognonné and Mosser (1993), we define here the flux-glut moment density tensor as

m ij = (p + κ∇ • u)δ ij + ρv i v j . ( 13
)
Equation ( 12) simplifies further because we have not only continuity of pressure at the atmosphere-solid interface, but also continuity of the Hooke pressure and the vertical amplitude of modes across this interface. We therefore identify the second term as the excitation coefficient, which can be rewritten as

Ψ k (t) = - V dV u k • ∇ δp glut + •(ρvv) = V dV k : m, ( 14
)
where k is the strain tensor of the normal mode k. Figure 6 shows the pressure and Reynolds stress spectra for an arbitrary point on the surface of Mars, as calculated by the GCM. Note that the amplitudes decrease as 1/f . Most of the variability in the normal mode bandwidth is associated with the Reynolds stress. All terms in Eq. ( 13) can be derived from the values calculated by the GCM, thereby giving us a full description of the moment tensor based on the GCM results.

Normal Modes Amplitudes in Atmosphere

Estimating the normal modes excitation coefficients Ψ k (t) requires knowledge of the normal mode amplitude in the atmosphere. Given the exponential decay of the density and the a priori thickness of the Martian boundary layer, most of the pressure glut release will occur in the first 10 km so we focus on computing the normal modes at these relatively low altitudes. and in the bandwidth of normal modes, viscous and relaxation processes can be neglected, as done by Lognonné and Johnson (2007). However both Lognonné andJohnson (2007, 2015) and [START_REF] Lognonné | Modeling of atmospheric-coupled Rayleigh waves on planets with atmosphere: from Earth observation to Mars and Venus perspectives[END_REF] assume the atmosphere to be spherically, symmetric and rigorous. The three-dimensional computation of the normal modes requires a prohibitive amount of computing time. Although the interior structure of Mars can be considered spherically symmetric, the acoustic properties of the Martian atmosphere depend strongly on local time, with large variations between the night and day for the height scale, temperature, and density, and therefore for the sound speed and acoustic impedance as well as the coupling between ground and atmosphere. This is illustrated in Fig. 7, which shows the variation of atmospheric scale height. Relative variations of ±20% with LS (or during one day at the same longitude) are observed, which suggest that lateral variation of the atmospheric coupling should be accounted for in MBF calculations. Furthermore, Mars has larger lateral atmospheric variations than does Earth. The main driver of atmospheric variation is temperature: on Mars, temperature variations are much greater than on Earth because of the absence of oceans and atmospheric water vapor, which serve as effective thermal reservoirs and stabilize the temperature. Thus, the MBF with three-dimensional atmospheric structure must be estimated to better understand the signal levels of MBFs and perfect the simulations. However, to account for the lateral variation while maintaining a realistic calculation time requires some modeling and simplification. Thus we simplified the atmospheric structure by neglecting viscosity, radiative boundaries, relaxation, and resonance.

In this paper, we propose to compute the amplitude of the atmospheric normal modes by propagating the normal mode amplitude at the surface into the atmosphere. In other words, instead of directly calculating normal modes in the atmosphere, we extrapolate the modes calculated for the solid part of the planet into the atmosphere. For this purpose, we make the following assumptions:

-the normal mode phase velocity is much greater than acoustic-wave velocity, leading to a vertically longitudinal upward-moving acoustic wave; -the normal modes interact asymptotically with the atmosphere and without significant resonance between solid and atmosphere, which limits this approach to normal modes above the atmospheric cutoff frequency.

AUTHOR'S PROOF

Journal Given these two assumptions, the wave-propagation normal mode is

u k (z) = u k (0) • e -ikzz × ρ(0)c(0) ρ(z)c(z) , ( 15
)
where k z is a vertical wave number and z is the altitude above the ground. The first factor on the right side of the equation is the wave-propagation factor and the second factor is the wave transmission ratio. With the wave-propagation method, we can describe the local time and, more generally, how the normal mode amplitude in the atmosphere depends on geography.

We compare the resulting amplitudes with those computed by using the more precise model of [START_REF] Lognonné | Modeling of atmospheric-coupled Rayleigh waves on planets with atmosphere: from Earth observation to Mars and Venus perspectives[END_REF]. For angular orders ≥ 10, the difference is less than the effect of the lateral variation (see Fig. 8). However, for smaller angular orders and for modes with frequencies close to or below the atmospheric cutoff, this simple propagation fails and large discrepancies appear because of acoustic-wave reflection and resonance in the atmosphere. Figure 8 shows the relative atmospheric energy for the fundamental spheroidal modes. The energy peaks around 2.2 mHz, which corresponds to the angular order = 9. This large coupling of the normal mode with angular order = 9 is normal if also seen on the mode amplitude (Fig. 9).

Such modes, thus require exact computation. Note, however, that these modes are also difficult to observe because of the expected high very low-frequency modes associated with temperature fluctuation below 5 mHz.

AUTHOR'S PROOF

Results and Discussion

Normal Modes Amplitude

The MBF is obtained by summing the fundamental Rayleigh normal modes, with the amplitude of each computed by using Eqs. ( 8) and ( 11). For this study, we summed the fundamental normal mode for degrees = 2 to = 39. The amplitude of the MBF signal is a few nanogals (10 -11 m/s 2 ), as shown in Fig. 11. This amplitude is greater than that reported by Lognonné andJohnson (2007, 2015), which is likely because of the larger angular orders and therefore larger frequencies that they used. Since the signal is the sum of all normal modes, specific peak amplitudes are more clearly shown in Fig. 12. The lowest-frequency peak (0.4 mHz) is a normal mode of angular order 2, whereas the highest-frequency peak (8.3 mHz) is a normal mode of angular order 39. Note, however, that the amplitudes are well modeled only for frequencies greater than 2 mHz. We also observe a gradual increase of the amplitude with frequency up to 5 mHz, after which the amplitude of the peaks remains constant. Each peak is a normal mode; there are 38 peaks in this figure, ranging from angular order 2 to 39. The amplitude grows with frequency up to 5 mHz and is then remains constant for frequencies ≥5 mHz

AUTHOR'S PROOF

Normal Modes: Detection and Seismometer Performance

MBF amplitudes are quite small and so can be detected only by using instruments with high sensitivity and low noise. Let us consider whether these modes can be detected in the POS output of the InSight SEIS VBB sensors. This output has a flat acceleration gain in the frequency bandwidth of normal modes (see Lognonné et al., this issue).

The least significant bit (LSB) of the InSight SEIS seismometer for differential output is

LSB = 2 × Voltage Gain . ( 16
)
The seismometer voltage is ±25 V, the acquisition dynamics is 24 bit, and the gain is ≈10 4 V/DU for low gain mode and about 4.5 times greater for high gain mode. Therefore, the LSBs are

LSB LG = 2 × 25 10 4 × 2 24 = 30 nanogal, (17)
LSB HG = 2 × 25 4.5 × 10 4 × 2 24 = 6.6 nanogal.

The amplitude of MBFs is only nanogals and the LSBs are therefore greater. The real ground acceleration signal contains MBF, as well as thermal drift, thermal and instrument noise, and possibly seismic signal. For a more realistic evaluation, we approximate the daily temperature variation associated with the temperature sensitivity of the sensors with an ideal sinusoidal curve and superpose it on the MBFs. Because this drift is much greater than 1 LSB, it enhances the MBFs up to a detectable level. The sinusoidal thermal-noise model is given by

A temp (t) = γ δT 1 -cos 2πt D sol , ( 18
)
where A temp is the equivalent acceleration of temperature variations, t is local time, D sol is the duration of a Martian day, γ is the sensitivity per degree of the VBB in the acceleration unit, and δT is the temperature variation over one Martian day. The temperature sensitivity γ is set to 10 -5 m/s 2 /K and daily temperature variation δT is 4 K, which corresponds to the variations expected during winter [START_REF] Mimoun | The seismic noise model of the InSight mission to Mars[END_REF]Lognonné et al., this issue). The amplitude of this thermal daily variation is therefore 4.0 × 10 6 nanogal, which is six orders of magnitude greater than MBF signals.

We simulate the signals by adding the MBF signal to the thermal drift, and then simulate the digitization by converting this acceleration signal into bits. Subtracting from the signal a sol-periodic sine wave with amplitudes computed by using the least-squares method simulates data processing. For both low gain and high gain, clear MBFs signal are retrieved from the residual, which demonstrates the capability of the acquisition system to use stochastic amplification of the noise to detect signals smaller than the LSB.

The signal remains much smaller than the expected instrument noise, which is the superposition of the instrument self-noise and of the residual of both the subtracted temperature drift and the potential pressure decorrelation [START_REF] Murdoch | Estimations of the seismic pressure noise on Mars determined from Large Eddy Simulations and demonstration of pressure decorrelation techniques for the InSight mission[END_REF]. This is illustrated in Fig. 13, which compares the MBF spectrum to the instrument self-noise in both low gain and high gain and to the expected thermal-drift signal. In the 5 to 10 mHz band, which contains a large part of the target bandwidth for the Martian hun search (5 to 20 mHz) and the bandwidth in which our modeling hypothesis and the GCMs have acceptable errors, the MBF signal is expected to be 5 to 10 below the instrument self-noise, depending on the gain. Stacking this signal over the mission duration will then allow successful detection.

The amplitude of ideal thermal noise model is 4.0 × 10 6 nanogal. The acceleration is 0.4 × 10 6 DU in low-gain mode or 1.4 × 10 6 DU in high gain mode. The thermal noise acceleration is far greater than the MBF signal. Thus, no signal is detected by the InSight SEIS seismometer in low-gain mode. In high-gain mode, we can detect 1 DU signal several times per half Martian day. However, ideal thermal noise can kick up to 1 DU after the decimal point. After subtracting the digitized thermal noise, more MBF signals are detected using the InSight SEIS seismometer in both low-and high-gain mode. Although the MBF signal is hidden by the large thermal noise, after subtracting the digitized thermal noise, we capture a greater part of the MBF signal.

To improve the signal to noise ratio and detect the MBF normal modes, we further process data as follows. First, we create a trace that contains MBF signal and noise. As explained previously, noise can be decomposed as self-noise, thermal noise and LSB noise. We com- pute self-noise as the inverse Fourier transform of a spectrum whose amplitude follows the station sensitivity as a function of frequency and whose phase is random. This self-noise is added to the MBF seismogram to create a trace S1. Thermal noise is then computed as a sinus function over one sol with an amplitude of 10 -4 m/s 2 to simulate the seismic acceleration generated by thermal drift. This thermal noise is added to trace S1 to create trace S2. LSB noise takes into account the least significant bit (5.81/4 10 -11 m/s 2 ). Trace S2 is modified so that the only signal above the LSB is kept and this trace is called S3. Finally, we assume that the thermal drift is known and it is subtracted to trace S3 to generate to the final trace S4. The seismogram S4 is then cut into segments of two sols with one sol overlap, starting at sol 4. Figure 14 (top) displays spectra amplitude of each two sol-long segments for the MBF signal (blue lines) and self-noise (red lines) and confirms that the amplitude of the MBF normal modes is below the self-noise. Figure 14 (bottom) shows the spectra amplitude of each segments of trace S4 (black lines). Normal modes are not visible in the amplitude spectra of the different data segments.

AUTHOR'S PROOF

Data are then processed as follows. Phase auto-correlation is computed for each segment and then stacked using the phase weighted stack method following the method developed in [START_REF] Schimmel | Using instantaneous phase coherence for signal extraction from ambient noise data at a local to a global scale[END_REF][START_REF] Ventosa | Extracting surface waves, hum and normal modes: time-scale phase-weighted stack and beyond[END_REF], [START_REF] Schimmel | Low-frequency ambient noise autocorrelations: waveforms and normal modes[END_REF]. For comparison, we apply the same processing to MBF traces, S1 traces (MBF+self noise) and S4 traces (MBF+self noise+LSB). Figure 15 (top) show the amplitude spectra of each stack. Normal mode peaks are clearly visible for the MBF stack amplitude spectrum between 1.5 and 8 mHz. Some modes can also be detected for the stack of traces S4 at frequencies higher than 4 mHz.

In order to enhance the signal to noise ratio, we further select the Rayleigh wave train windows on the stack of auto-correlograms by setting to 0 the rest of the signal as in [START_REF] Deen | First observation of the Earth's permanent free oscillations on ocean bottom seismometers[END_REF]. We keep signal around 0-time lag and around each surface wave train R1 to R3. We keep 6 minutes around 0 lag time and select R1 to R3 wave trains considering that their group velocities are between 3.8 and 5.7 km/s. Figure 15 (bottom) displays the new amplitude spectra, and we observe that the normal mode peaks are now clearly visible for frequencies above 2.5 mHz for the MBF stack and also for data with realistic noise (self-noise or MBF+self noise). Adding longer time series will improve the signal to noise ratio and therefore we expect that normal modes excited by the Mars atmosphere should be detectable.

AUTHOR'S PROOF

Estimate of Seismic Magnitude of MBF Signal

We now not only estimate the MBF amplitude in terms of seismic magnitude, but also check that our estimate of the MBFs is comparable to that obtained by applying a simple energetic budget, following the approach of Kobayashi and Nishida (1998b).

For a first estimate, we assume that the release of acoustic and seismic energy into atmospheric activities is driven by solar flux and, furthermore, that all the energy of planetary background free oscillations comes from solar flux. This maximum energy may be expressed as

W M = F E × D E D M 2 × (1 -A M ) × πR 2 M , (19)
where W M is the rate at which the Sun transfers energy to the Mars surface, F is the solar flux above Earth's atmosphere (F E = 1360 W/m 2 ), D is the distance from the Sun (D M = 1.52D E ), A is the bound albedo (A M = 0.25), subscripts E and M refer to Earth and Mars, respectively, and R M is the radius of Mars. This solar energy is converted to seismic energy, with a coherency duration of T √ Q, where Q is the quality coefficient of the mode and T is its period. Thus the MBF's energy (E MBF ) is

E MBF = W M × s × Q × T × C, ( 20
)
where s = 0.01 is an acoustic efficiency (which can be achieved for high-altitude winds of 40 to 50 m/s, assuming an efficiency depending on M 3 , where M is Mach number, [see, e.g., [START_REF] Goldreich | The interaction of acoustic radiation with turbulence[END_REF]]). The constant C is the energy coupling ratio between the atmosphere and the solid part of Mars (a typical value is C = 5 10 MBFs (typical value T = 200 s). We then use the energy-magnitude relation of [START_REF] Gutenberg | Earthquake magnitude, intensity, energy, and acceleration: (second paper)[END_REF]:

log E = 4.8 + 1.5M, ( 21
)
where M is the magnitude of the seismic event. A rough estimation of the MBF magnitude gives M = 4.9. We now compare the amplitude obtained by our GCM modeling with that excited by seismic activity. This is achieved by comparing the spectrum of the MBFs with those of quakes that occur at an epicentral distance of 90 • . We find amplitudes of MBFs close to those generated by a M5.0 free oscillation (Fig. 16), while the estimates based on the GCM calculation are consistent with those based on solar flux. The results are also consistent with those estimated from Earth's hum, which has been estimated to be equivalent to a daily earthquake of magnitude 5.75-6 [START_REF] Rhie | Excitation of Earth's continuous free oscillations by atmosphere-ocean-seafloor coupling[END_REF]. Given the earthbound albedo of 0.306, the solar flux for Earth is eightfold that of Mars, which corresponds to magnitude 0.6 greater and therefore an extrapolation of 5.3. Nevertheless, all these estimates of magnitudes, although comparable, are less than the magnitude of 5.9 (i.e., 10 18 Nm) which is considered by several studies as a prerequisite for detecting normal modes. Therefore, we will focus our next analysis on the possibility of coherent stacking for MBFs, which, for a Mars year of 687 days, might lead to an increase in amplitudes by ≈26 , where is the fraction of daily coherent hum. For a sol-to-sol coherency exceeding 0.4, this might lead to an order of magnitude increase in signal, allowing signals to possibly peak out of the noise after stacking data over a year.

Weather Correlation and Stacking Method

The Martian surface has low thermal inertia because it is devoid of oceans. Thus, the diurnal surface-temperature cycle is very large. Combined with the very thin atmosphere, this makes thermal tides a prominent element of diurnal variability on Mars. This is further reinforced by the equatorial location of InSight. Mars has a clear dichotomy whereby low lands dominate the southern hemisphere, while the northern hemisphere is dominated by highlands. The boundary of the dichotomy is close to the equator and this dichotomy contributes to the diurnal cycle through slope-induced circulations, although the day-to-day variability associated with baroclinic waves is small [START_REF] Spiga | Atmospheric science with InSight[END_REF]. Thus, provided we consider atmospheric variability within a given season, a significant daily repetition of atmospheric temperature, wind, and pressure should be the norm. This will correlate strongly with the diurnal range in our seismic computations derived from GCM simulations (Fig. 17), which means that stacking is likely to be an efficient way to increase the MBF amplitude above that of the self-noise or other noise sources. We tested this hypothesis with a one-week stack from the GCM simulation. The 7 sol-long data were divided into 7 time series of 24.6 hours. The results shown in Fig. 18 indicate a threefold increase in the amplitudes of stacks over 7 sols.

Although extrapolation over a long period will require further GCM modeling, and because the sol-to-sol correlation might weaken over weeks, these preliminary results suggest that stacking the seismic signal over the sol time might enable significant to very significant amplification of the Mars hum. Furthermore, detecting the normal mode frequencies will allow the inversion of the Mars upper mantle, as already illustrated by [START_REF] Panning | Planned products of the Mars structure service for the InSight mission to Mars[END_REF]. The position of the normal mode peaks depends on the internal structure, as shown on Fig. 19. These simulations were done with two different internal structure models for Mars and the results show that the resolution after the stacking suffices to adequately shift associated with the structures.

Conclusion

This study evaluates the magnitude of the signal of Mars' background free oscillations by using high-precision GCMs. Given the limitation of GCMs, the values obtained should be regarded as a lower limit of the possible amplitude of background free oscillations expected on Mars.

To obtain a realistic evaluation, we make several assumptions that are based on observations and theory to simplify the model and take into account the three-dimensional atmospheric structure of Mars.

Given these models and assumptions, the results indicate that MBF amplitudes are likely to be on the order of nanogals, which is consistent with previous studies, such as Kobayashi and Nishida (1998b), Lognonné and Johnson (2007). We also confirmed that the values obtained are consistent with solar flux. These results imply that the values obtained represent the typical strength of MBF signals, although the amplitudes may differ because of the assumptions used, such as those involving the internal structure and atmospheric model.

Given the level of MBF signal and the level of noise from the InSight SEIS instruments, detecting MBF signals with the InSight SEIS instruments will remain difficult. However, because the detection strategy exploits the daily repetition of the Martian atmosphere, stacking is found to be a powerful tool to overcome this problem.

The results should be viewed as a minimum estimate for MBF detection by the InSight SEIS seismometer. Knowledge of the Martian MBFs should allow us to estimate the deep internal structure of Mars and can be one of successes of the InSight mission.

  . The same phenomenon should occur on Mars, meaning that the observation of MBFs should allow us to AUTHOR'S PROOF Journal ID: 11214, Article ID: 579, Date: 2019-01-22, Proof No: 1, UNCORRECTED PROOF _####_ Page 4 of 25 Y. Nishikawa et al.

  Journal ID: 11214, Article ID: 579, Date: 2019-01-22, Proof No: 1, UNCORRECTED PROOF _####_ Page 6 of 25 Y. Nishikawa et al.

Fig. 2 Fig. 3

 23 Fig. 2 Pressure fluctuations in Martian atmospheric in frequency band where normal modes concentrate in normal mode band. We applied a 1.0-2.0 mHz band-pass filter to the raw pressure data (shown in Fig. 1). The daily repetition of Martian weather processes appears clearly shown in the pattern. The band of the pattern at 1.0-2.0 mHz corresponds to low MBF frequencies (angular order 2 to 8) MBFs frequencies. The pressure fluctuations in the atmosphere excite daily coherent MBFs

  ID: 11214, Article ID: 579, Date: 2019-01-22, Proof No: 1

  running one GCM AUTHOR'S PROOF Journal ID: 11214, Article ID: 579, Date: 2019-01-22, Proof No: 1, UNCORRECTED PROOF _####_ Page 8 of 25 Y. Nishikawa et al.

Fig. 4

 4 Fig.4Spectrum of GCM pressure and observed pressure. The observational data were acquired by Mars Pathfinder (LS = 170 • ). The GCM spectrum is computed for the same location as the observational data of Mars Pathfinder (LS = 0 • ). The results differ by an order of magnitude, implying that GCMs cannot fully explain the observed energy, which is likely due to inadequate modeling in GCMs of local-scale turbulences. Thus, MBFs calculated by using GCMs should be regarded as a lower limit of possible excitations

3. 1 Fig. 5

 15 Fig.5Vertical amplitude of spheroidal fundamental normal modes in solid part. Low-degree normal modes are low frequency. Large-amplitude areas concentrate the models energy and sensitivity. Low-degree waves (angular order 2 to 9) are sensitive at the core and deep mantle. High-degree waves (angular order 10 to 39) are sensitive at the middle and shallow part of the mantle. Each mode is sensitive at a different depth

Fig. 7

 7 Fig. 7 Daily variation of atmospheric scale height. The red (blue) areas are thick (thin). Because of the large temperature variations, the scale height thickness changes from 8 to 12 km in one day. During the day, the atmosphere expands to 12 km height, whereas it shrinks to 8 km height at night. This variation in scale height directly affects the normal mode structure, and the isosphere model cannot describe this lateral effect

Fig. 8 Fig. 9 Fig. 10

 8910 Fig.8Kernel sensitivity of normal modes in air. Blue curves include the radiative boundary, viscosity, and relaxation. Green curves include only wave propagation and transmission. Solid lines and dotted lines show the real part and imaginary parts, respectively. For order n => 10, the gaps between two adjacent normal modes are much less than the scale height

Fig. 12

 12 Fig.12Calculated spectrum of acceleration power density of MBFs. The MBFs are estimated by using GCMs and the internal structure given by the Sohl model. The normal mode frequencies come from power spectral density calculations. Each peak is a normal mode; there are 38 peaks in this figure, ranging from angular order 2 to 39. The amplitude grows with frequency up to 5 mHz and is then remains constant for frequencies ≥5 mHz

  Journal ID: 11214, Article ID: 579, Date: 2019-01-22, Proof No: 1, UNCORRECTED PROOF _####_ Page 18 of 25 Y. Nishikawa et al.

Fig. 14

 14 Fig. 14 Spectrum amplitude of each two sol long segment for MBF traces (blue) and self-noise traces (red) in the top figure and for S4 traces (black) in the bottom figure

  Fig. 15 Normalized spectrum amplitude of the stack of autocorrelograms without surface wave train selection (top) and after surface wave train selection (bottom). The blue, black and red curves correspond to MBF, MBF+self noise and MBF+self noise + LBS noise (S4) data respectively

Fig. 19

 19 Fig. 19Internal-model-dependent MBFs. We use two internal models of Mars, which results in differing peak positions. The difference in peak positions corresponds to the difference in the internal structure of Mars, and the seismometer on InSight can detect this difference in frequencies. If we can detect these peaks, we can determine the internal structure of Mars

AUTHOR'S PROOF

	Mars' Background Free Oscillations	Page 5 of 25 _####_
	Fig. 1 Pressure fluctuations in
	Martian atmospheric simulated
	by LMD GCM for Mars meridian of longitude 0 • . All pressure
	records are demeaned and only
	pressure variations are shown.
	Vertical axis is latitude;
	horizontal axis is local time [hr].
	The largest pressure fluctuations
	occur at sunrise and sunset. The
	pressure fluctuations repeat to
	high-precision day after day. This
	strong daily repetition is not
	common on Earth
	Journal ID: 11214, Article ID: 579, Date: 2019-01-22, Proof No: 1, UNCORRECTED PROOF

AUTHOR'S PROOF Journal

  ID: 11214, Article ID: 579, Date: 2019-01-22, Proof No: 1, UNCORRECTED PROOF

	Mars' Background Free Oscillations	Page 21 of 25 _####_
	Fig. 17 Pressure correlations in
	normal mode frequency band.
	Large correlations are diurnal and
	semi-diurnal. On a given day, the
	large correlations are the
	sunrise-sunrise and sunset-sunset
	correlations. Semi-diurnal
	correlations are sunrise-sunset
	and sunset-sunrise (semi-diurnal
	thermal tide). The periodic
	excitation force generates
	high-efficiency stacking. This
	precise phenomenon occurs on
	Mars but not on Earth
	Fig. 18 Stacking efficiency for
	one week of data. Blue curve is
	MBF power density. Red line is
	MBF power density after one
	week of stacking. Daily stacking
	leads to large, clear peaks. The
	longer MBFs are stacked, the
	larger and sharper the normal
	mode peaks become. One week
	of stacking already leads to a
	significant improvement in the
	signal-to-noise ratio, which
	demonstrates that Martian
	stacking is far more effective
	than terrestrial stacking. As a
	result of such efficient stacking
	for Mars, some MBF peaks are
	likely to be detected by the
	InSight SEIS seismometer

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Acknowledgements

The authors are grateful for the support of CNES for the development of the SEIS experiment and its scientific support, and to the ANR for supporting the project through ANR SEISMARS. Y.K. acknowledges the support of the CNES and JSPC for his Ph.D. support. P.L. and A.S. acknowledge the support of IUF. This is IPGP contribution number xx and InSight contribution number yy.

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