Étienne Bertrand

Rihab Sassi

Hugues Delattre

Marceau Baudin

Loïc Gisselbrecht

Thomas Chartier

Margot Mathey

Marie Foch

Ludmilla Provost

Nathalie Giloy

Lucie Pheulpin

Julien Clément... Laurie Saint Criq

Keywords: site effects, velocity heterogeneities, numerical simulations, seismic ground motion, wave propagation Wave propagation, Site effects, Statistical methods, Numerical modeling, Earthquake ground motions

This thesis results highlight the importance of velocity heterogeneities in understanding the seismic ground motion variability in the sedimentary basin.

Résumé

Lors d'un séisme, le mouvement sismique enregistré en un site donné peut être fortement dominé par la réponse de la géologie de subsurface. En effet, certaines configurations géologiques (par exemple, couche molle sur un encaissant dur, topographie, bassin sédimentaire) sont propices aux piégeages des ondes sismiques conduisant à une amplification du mouvement sismique à certaines fréquences et à un allongement de la durée du signal, on parle dans ce cas d'effets de site. Par ailleurs, les hétérogénéités à petites échelle (quelques mètres à dizaines de mètres) des couches géologiques peuvent redistribuer l'énergie des ondes piégées, ainsi augmentant la variabilité du mouvement sismique enregistré en surface. Les effets de ces hétérogénéités sont très souvent négligés lors du calcul de la réponse d'un site à partir de simulations numériques car la limite de résolution des méthodes classiques d'imagerie géophysique ne permet pas de bien les résoudre. Pour palier aux limites de résolution des connaissances du milieu, il est commun d'employer des méthodes stochastiques dans lesquelles la variation spatiale des hétérogénéités est supposée aléatoire. Ainsi elles peuvent être caractérisées par une fonction d'autocorrélation (ACF) spatiale. L'objectif principal de cette thèse est d'étudier l'effet des hétérogénéités de vitesse dans un bassin sédimentaire sur le mouvement sismique enregistré en surface. Pour cela, des simulations numériques 2D de la propagation des ondes sismiques sont réalisées.

Deux bassins sont considérés : le bassin de Nice, caractérisé par des études précédentes et instrumenté de plusieurs stations du réseau accélérométrique permanent (RESIF-RAP), et un bassin canonique. Les hétérogénéités des bassins sont modélisées comme un champ aléatoire caractérisé par une ACF Von Karman.

Dans un premier temps, des simulations numériques sont réalisées dans le bassin de Nice en considérant une fonction source simple et en faisant varier les paramètres de l'ACF (longueur de corrélation et coefficient de variation) à partir de valeurs issues de la littérature. Les résultats de cette étude de sensibilité montrent que le coefficient de variation contrôle la variabilité du mouvement en surface (valeur maximale de la vitesse, fonction de transfert, spectre de réponse) au premier ordre, devant la longueur de corrélation.

Dans un deuxième temps, nous utilisons les données de forages disponibles dans le bassin sédimentaire de Nice pour contraindre les paramètres de l'ACF, en particulier le coefficient de variation et le coefficient de variation dans la direction verticale. Différentes longueurs de corrélation sont testées dans la direction horizontale. Des nouvelles simulations numériques sont ainsi effectuées en utilisant cette fois-ci comme fonction source un signal sismique enregistré au rocher de Nice. La variabilité du mouvement dans le cas présent est étudiée pour un champ d'onde incident plus complexe. Une comparaison entre la fonction de transfert (FT) calculée à partir des résultats de nos simulations numériques et la FT empirique calculée à une station NLIB située sur le profil du bassin est aussi effectuée. Ces résultats montrent que la réponse du bassin de Nice est contrôlé au premier ordre par le modèle de vitesse de référence, et les hétérogénéités gouverne la variabilité de cette réponse. i Enfin, dans un troisième temps, des simulations numériques sont réalisées dans un modèle canonique plus large et plus profond, afin d'évaluer l'influence de la longueur de corrélation et du coefficient de Hurst pour des ondes qui se propagent plus longtemps dans le bassin. Elles montrent que plus la longueur de corrélation est grande, plus la variabilité de la réponse du bassin l'est.

Ces résultats mettent en lumière l'importance des hétérogénéités du milieu pour comprendre la variabilité du mouvement sismique dans les bassins sédimentaires.

Mots clés: effets de site, hétérogénéités de vitesse, simulation numérique, mouvements sismiques, propagation des ondes.

Abstract

When an earthquake occurs, the ground motion recorded at a given site can be strongly influenced by the subsurface geology's response. Certain geological configurations (e.g., soft sediments on hard bedrock, surface topography, sedimentary basins, etc.) are more likely to trap seismic waves, leading to the amplification of certain frequencies and an increase in the duration of the seismic signal, what we refer to as site effects. Moreover, the small-scale heterogeneities (few meters to tens of meters) of the geological layers can redistribute the trapped seismic energy, thereby increasing the variability of the recorded surface ground motion. The effects of such small-scale heterogeneities are generally ignored when computing the site's response using numerical simulations in part because they can not be resolved using classical geophysical imaging methods. To palliate this limit in the resolution of the medium, stochastic methods are commonly employed. In stochastic methods, the spatial variation of the small-scale heterogeneities is considered to be random. Hence, they can be characterized by an autocorrelation function (ACF). This thesis's main objective is to study the influence of small-scale velocity heterogeneities in sedimentary basins on the surface ground motion. For this, we perform 2D numerical simulations of seismic wave propagation. Two basins are considered: (1) the sedimentary basin of Nice, which has been characterized by previous studies and instrumented by several seismic stations of the french accelerometric network (RESIF-RAP), and (2) a canonical basin model. The small-scale velocity heterogeneities in the basin are modeled as spatially correlated random fields characterized by a Von Karman ACF. This thesis is 3-fold: First, we perform numerical simulations in a 2D cross-section of the Nice sedimentary basin using a simple source-time function. We vary the parameters (correlation length and coefficient of variation) of the ACF based on values reported in the literature. This parameter study shows that the variability of the surface ground (peak ground velocity, Arias intensity, response spectrum) is controlled to the first order by the coefficient of variation of the ACF.

Secondly, we use available borehole data gathered in the Nice sedimentary to constrain the ACF parameters, particularly coefficient of variation and the vertical correlation length. Different values of the horizontal correlation length are used. We then perform a new set of numerical simulations, this time using an earthquake signal recorded on an outcropping bedrock at Nice as the source-time function. We study the variability of the ground motion parameters for this complex source time function. We also compare the transfer function (TF) of the computed ground motion using our numerical simulations to the empirical transfer function computed at the NLIB station located on the basin profile. These results show that the site response of the Nice basin is controlled to the first order by the reference velocity model. The heterogeneities then govern the variability of the response. Finally, we perform numerical simulations in the canonical basin to study the influence of the correlation length and Hurst exponent of the ACF for longer propagation times. The results show that the variability of the basin response increases with an increase in the correlation length. 5.2 Existence conditions of the two-dimensional resonance in the SH case. For a sine-shaped valley having a shape ratio h/l and a velocity contrast C v (after [START_REF] Bard | The two-dimensional resonance of sediment-filled valleys[END_REF]. 123 5.3 One realization of each random model defined in Table 5.1, all the random models have a c

List of Tables

2.1 Characteristics of the ground motion and the ground parameters they reflect (modified after [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]). Note that the table summarizes only the ground motion characteristics we use in this thesis. An exhaustive list of ground motion characteristics and their definition can be found in [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]

General Introduction

It is well known that the characteristics of the local subsurface geology strongly control the ground response at a particular site during earthquake shaking. Their effects are generally perceived as an increase in the amplitude, duration, and spatial variability of the observed motion at the Earth's surface.

These phenomena are commonly known as site effects, and they considerably increase the damage potential of an earthquake. The main physical factor that leads to the amplification of ground motion is the impedance contrast 1 between media, for example between sediments and the underlying bedrock.

Certain structures such as sedimentary basins, alluvial valleys, and hills are favorable to exhibit these effects.

One notable example of site effects was during the Michoacan subduction earthquake of 1985, where large ground motion amplitudes were observed at frequencies corresponding to the resonance frequency of the Mexico city sedimentary basin. The city is located 400 km away from the source epicenter.

On the other hand, the near-source ground motion amplitudes were comparable to the ones recorded in the basin. In particular, the sedimentary basin resonates at 0.5 Hz leading to the damage and collapse of Introduction important for adequate ground motion predictions and designing structures that can resist the expected seismic demand on buildings. (e.g. [START_REF] Bradley | Ground motion and site effect observations in the wellington region from the 2016 mw7.8 kaik ōura, new zealand earthquake[END_REF].

Over the last decades, enormous scientific progress has been made to understand the effects of local and regional site conditions on seismic ground motion. Several studies have investigated the different phenomena that play a major role during site effects, like impedance contrast between the bedrock and overlying layers (e.g. [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF][START_REF] Safak | Local site effects and dynamic soil behavior[END_REF], the constitutive material model and material properties (e.g [START_REF] Bard | Wave propagation in complex geological structures and their effects on strong ground motion[END_REF][START_REF] Bonilla | Hysteretic and dilatant behavior of cohesionless soils and their effects on nonlinear site response: Field data observations and modeling[END_REF][START_REF] Bonilla | Nonlinear site response evidence of k-net and kik-net records from the 2011 off the pacific coast of tohoku earthquake[END_REF], the complexity of the incident motion (e.g. [START_REF] Gélis | 2-d p-sv numerical study of soil-source interaction in a non-linear basin[END_REF][START_REF] Gelis | Influence of a sedimentary basin infilling description on the 2-D P-SV wave propagation using linear and non-linear constitutive models[END_REF], and the site geometry (e.g. [START_REF] Bard | The two-dimensional resonance of sediment-filled valleys[END_REF][START_REF] Graves | Modeling three-dimensional site response effects in the marina district basin, san francisco[END_REF][START_REF] Moczo | Amplification and differential motion due to an antiplane 2d resonance in the sediment valleys embedded in a layer over the half-space[END_REF][START_REF] Olsen | Three-dimensional simulation of earthquakes on the los angeles fault system[END_REF][START_REF] Semblat | Seismic Wave Amplification: Basin Geometry vs Soil Layering[END_REF][START_REF] Imtiaz | Effects of site geometry on short-distance spatial coherency in Argostoli, Greece[END_REF].

The assessment of site effects can be done either (1) Experimentally by using empirical approaches (e.g. [START_REF] Apostolidis | Definition of subsoil structure and preliminary ground response in Aegion city (Greece) using microtremor and earthquakes[END_REF][START_REF] Ktenidou | Directional dependence of site effects observed near a basin edge at Aegion, Greece[END_REF][START_REF] Chávez-Garcìa | Local amplification and subsoil structure at a difficult site: Understanding site effects from different measurements[END_REF], to analyze the recorded earthquakes at a given site, and (2) Theoretically by using numerical methods to simulate seismic wave propagation in a given region (e.g. [START_REF] Bouchon | A simple, complete numerical solution to the problem of diffraction of SH waves by an irregular surface[END_REF][START_REF] Bouchon | A boundary integral equation -discrete wave number representation method to study wave propagation in multi-layered media having irregular interfaces[END_REF][START_REF] Komatitsch | The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF][START_REF] Chaljub | Solving elastodynamics in a solid heterogeneous 3D-sphere: a spectral element approximation on geometrically non-conforming grids[END_REF][START_REF] Moczo | The Finite-Difference Modeling of Earthquake Motions Waves and Ruptures[END_REF].

In regions of low-to-moderate seismicity, the related activity rate makes it difficult to use earthquake data to compute site effects. Thus, methods using seismic ambient noise have been developed to obtain at least the fundamental resonance frequency at a given site (e.g. [START_REF] Nakamura | A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface[END_REF][START_REF] Bonnefoy-Claudet | H/V ratio: a tool for site effects evaluation. Results from 1-D noise simulations[END_REF]. Note, however, that site effects are described by the amplification as a function of frequency.

The fundamental frequency alone does not fully characterize site effects.

Numerical simulations of seismic wave propagation, on the other hand, are of great interest for site effect analysis because they offer the possibility to understand ground motion in complex geological structures and to study past large earthquakes or hypothetical earthquake scenarios and their impact in terms of seismic hazard, building codes and disaster prevention (e.g. Bard et al., 2005a). For site effect assessment, the complex geometries of structures like sedimentary basins, and the heterogeneous nature of their material properties require efficient numerical methods that can handle highly heterogeneous 2D/3D models. Indeed, the phenomena associated with site effects, such as wave focusing, basin edges effects, and resonance effects, require precise modelling techniques that can naturally consider the interface conditions in heterogeneous materials and the free surface associated with topography. The constant increase in computational power and progress in parallel architectures over the last decades has helped to the development of new sophisticated numerical techniques and codes for simulating seismic wave propagation in complex geological structures (e.g. [START_REF] Chaljub | Solving elastodynamics in a solid heterogeneous 3D-sphere: a spectral element approximation on geometrically non-conforming grids[END_REF][START_REF] Komatitsch | Simulations of Ground Motion in the Los Angeles Basin Based upon the Spectral-Element Method[END_REF][START_REF] Maeda | OpenSWPC: an open-source integrated parallel simulation code for modeling seismic wave propagation in 3D heterogeneous viscoelastic media[END_REF][START_REF] Duru | A new discontinuous Galerkin spectral element method for elastic waves with physically motivated numerical fluxes[END_REF].

Performing accurate numerical simulations of seismic wave propagation requires detailed knowledge of the physical parameters (e.g., velocity, density, attenuation) of the medium. The characterization of the material properties is usually done using either (1) direct invasive methods such as downhole logging, and/or (2) indirect or non-invasive geophysical methods (e.g., high-resolution seismic reflection/refraction surveys, spectral analysis of surface waves (e.g. [START_REF] Kalinski | In situ estimate of shear wave velocity using borehole spectral analysis of surface waves tool[END_REF], multichannel analysis of surface waves (e.g. [START_REF] Park | Multichannel analysis of surface waves[END_REF], and horizontal to vertical spectral ratio (e.g. [START_REF] Nakamura | A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface[END_REF][START_REF] Sánchez-Sesma | A theory for microtremor H/V spectral ratio: application for a layered medium[END_REF][START_REF] Bonnefoy-Claudet | H/V ratio: a tool for site effects evaluation. Results from 1-D noise simulations[END_REF]). The first approach provides accurate and direct measurement of the material properties but it is economically costly and difficult to elaborate over large regions. The second method has limited resolution and can not map small-scale variations of the medium properties. For this reason, the models of the propagation media used in numerical simulations, generally consider homogeneous layers or smoothly varying spatial properties in which small fluctuations are neglected. This is generally acceptable when the scale length of the small-Introduction scale variations are smaller than the wavelength being modeled. For wave propagation problems in the frequency range of engineering interest (> 1 Hz), the effects of the small-scale variations of material properties (henceforth regarded as velocity heterogeneities) can no longer be neglected, as they may affect the seismic wave propagation.

Velocity heterogeneities are well known to scatter the seismic wavefield. Scattering includes wave diffraction and conversion and depends on the heterogeneity properties such as the size and perturbation intensity with respect to the background media. Note that the scattering of seismic waves is not only limited to a volume of heterogeneities but can also be due to rough or irregular layering and surface topography (e.g. diffraction phenomena). It was [START_REF] Aki | Analysis of seismic coda of local earthquakes as scattered waves[END_REF] who first interpreted the late arrivals on seismograms (known as coda waves) as being due to the backscattering of the seismic wavefield by uniformly distributed scatterers (velocity heterogeneities) in the Earth's crust. The attenuation or decay of the coda waves with lapse time was found to be related to the distribution of scatterers (e.g. [START_REF] Sens-Schönfelder | Laterally heterogeneous scattering explains Lg blockage in the Pyrenees[END_REF][START_REF] Carcolé | Spatial distribution of scattering loss and intrinsic absorption of short-period S waves in the lithospehre of Japan on the basis of the multiple lapse time window analysis of Hi-net data[END_REF]. Different theories have been developed to explain in more detail the nature of such waves. They include single and multiple scattering theories (e.g. [START_REF] Aki | Origin of coda waves: source, attenuation, and scattering effects[END_REF][START_REF] Sato | Single isotropic scatteringmodel includingwave conversions simple theoretical model of the short period body wave propagation[END_REF], and energy transport theory (e.g. [START_REF] Wu | Multiple-scattering and energy-transfer of seismic-waves separation of scattering effect from intrinsic attenuation[END_REF][START_REF] Hoshiba | Simulation of coda wave envelope in depth dependent scattering and absorption structure[END_REF]. The scattering of seismic waves by velocity heterogeneities results in changes in the amplitude and phase of both the direct waves and later phases. Besides the excitation of long-duration coda waves, other complex features of scattered waves can be noticed on high-frequency seismograms. These features include:

Sens

1. The broadening of the P-and S-wave pulses, known as envelope broadening (e.g. [START_REF] Saito | Unified explanation of envelope broadening and maximumamplitude decay of high-frequency seismograms based on the envelope simulation using theMarkov approximation: forearc side of the volcanic front in northeastern Honshu[END_REF][START_REF] Sato | Broadening of seismogram envelopes in the randomly inhomogeneous lithosphere based on the parabolic approximation: southeastern Honshu, Japan[END_REF][START_REF] Takahashi | Strong inhomogeneity beneath Quaternary volcanoes revealed from the peak delay analysis of S-wave seismograms of microearthquakes in northeastern Japan[END_REF].

2. The distortion of the apparent S-wave radiation pattern2 from the original four-lobe source radiation pattern (e.g. [START_REF] Takemura | Distortion of the apparent S-wave radiation pattern in the high-frequency wavefield: Tottori-ken Seibu, Japan, earthquake of 2000[END_REF][START_REF] Sawazaki | Envelope synthesis of shortperiod seismograms in 3-D random media for a point shear dislocation source based on forward scattering approximation: application to small strike-slip earthquakes in southwestern Japan[END_REF]. The combined analysis of observations and simulations using heterogeneous velocity models have pointed out that the frequency-and distance-dependent characteristics of the apparent S-wave radiation pattern can be explained as a result of seismic wave scattering due to small-scale velocity heterogeneity in the crust (e.g. [START_REF] Takemura | Distortion of the apparent S-wave radiation pattern in the high-frequency wavefield: Tottori-ken Seibu, Japan, earthquake of 2000[END_REF][START_REF] Imperatori | Broad-band near-field ground motion simulations in 3-dimensional scattering media[END_REF][START_REF] Kumagai | Characterization of scattered seismic wavefields simulated in heterogeneous media with topography[END_REF];

3. The appearance of P-wave energy in transverse motion (e.g. [START_REF] Takemura | Scattering of high-frequency P wavefield derived from by the dense Hi-net array observations in Japan and computer simulations of seismic wave propagations[END_REF][START_REF] Nishimura | Heterogeneity of the Japan islands as inferred from transverse component analysis of teleseismic P-waves observed at a seismic station network, Hi-net[END_REF][START_REF] Kubanza | Evaluation of strength of heterogneity in the lithosphere from peak amplitude analysis of teleseismic shot-period P waves[END_REF], and 4. Scattering attenuation of elastic waves (e.g. [START_REF] Hong | Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based[END_REF][START_REF] Frenje | Scattering attenuation: 2d and 3d finite difference simulations vs. theory[END_REF][START_REF] Frankel | Finite difference simulation of seismic scattering: implications for the propagation of short-period seismic waves in the crust and models in crustal heterogeneity[END_REF]. The intensity of the scattering attenuation depends on the spatial and size distribution of heretogeneities, the ratio of P-and S-wave velocities, and the frequency content of the incident waves.

It is therefore important to understand the strength and scale of such velocity heterogeneities for wave propagation and their effect on seismic hazard assessment.

Since it is impossible to obtain detailed mapping of the heterogeneities, they are usually modeled using statistical methods. The spatial distribution can be viewed as stochastic or random processes, such that they can be characterized only by a few statistical parameters like the mean, variance, or correlation length of the fluctuations. The concept of random media was first introduced in seismology by [START_REF] Chernov | Wave propagation in a Random medium[END_REF]. Several studies have considered random velocity fluctuations in numerical simulations to study the amplitude and phase changes due to the velocity heterogeneities (e.g. [START_REF] Frankel | Finite difference simulation of seismic scattering: implications for the propagation of short-period seismic waves in the crust and models in crustal heterogeneity[END_REF][START_REF] Roth | Single scattering-theory versus numerical modeling in 2D random-media[END_REF][START_REF] Hong | Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based[END_REF]Imperatori andMai, 2013, 2015;[START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF][START_REF] Takemura | Scattering of high-frequency P wavefield derived from by the dense Hi-net array observations in Japan and computer simulations of seismic wave propagations[END_REF]. Most of these studies are generally targeted towards (1) better understanding of the scattering process (e.g. [START_REF] Frankel | Finite difference simulation of seismic scattering: implications for the propagation of short-period seismic waves in the crust and models in crustal heterogeneity[END_REF][START_REF] Hong | Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based[END_REF]Introduction Kumagai et al., 2011;[START_REF] Imperatori | Broad-band near-field ground motion simulations in 3-dimensional scattering media[END_REF][START_REF] Takemura | Scattering of high-frequency P wavefield derived from by the dense Hi-net array observations in Japan and computer simulations of seismic wave propagations[END_REF], (2) deciphering the characteristics and distributions of heterogeneities in the Earth subsurface by comparing synthetic data with available theories or observations (e.g. [START_REF] Imperatori | Broad-band near-field ground motion simulations in 3-dimensional scattering media[END_REF][START_REF] Nakata | Stochastic characterization of mesoscale seismic velocity heterogeneity in long beach, california[END_REF][START_REF] Takemura | Scattering of high-frequency seismic waves caused by irregular surface topography and small-scale velocity inhomogeneity[END_REF][START_REF] Takemura | High-frequency seismic wave propagation within the heterogeneous crust: effects of seismic scattering and intrinsic attenuation on ground motion modelling[END_REF], and (3) understanding and quantifying the scattering attenuation (e.g. [START_REF] Frankel | Finite difference simulation of seismic scattering: implications for the propagation of short-period seismic waves in the crust and models in crustal heterogeneity[END_REF][START_REF] Gaebler | Seismic scattering and absorption parameters in the W-Bohemia/Vogtland region from elastic and acoustic radiative transfer theory[END_REF][START_REF] Frenje | Scattering attenuation: 2d and 3d finite difference simulations vs. theory[END_REF][START_REF] Hong | Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based[END_REF][START_REF] Takemura | High-frequency seismic wave propagation within the heterogeneous crust: effects of seismic scattering and intrinsic attenuation on ground motion modelling[END_REF].

In the framework of ground motion prediction, it is important to understand how heterogeneities affect the ground motion intensity measures and ground motion variability. The spatial variability of ground motion is of particular interest in Earthquake engineering since it can significantly affect the dynamic response of structures like dams, nuclear plants, or multiple-supports structures such as bridges.

Structural engineers generally assume spatially uniform ground motion excitations when studying such structures. This assumption may not hold when the structure extends over large distances (e.g. [START_REF] Zerva | Spatial variation of seismic ground motions: An overview[END_REF], hence requiring the consideration of the ground motion variability in their analysis (e.g. [START_REF] Deodatis | EFFECT OF SPATIAL VARIABILITY OF GROUND MOTION ON BRIDGE FRAGILITY CURVES[END_REF][START_REF] Zerva | Spatial variation of seismic ground motions: An overview[END_REF][START_REF] Adanur | Wave-Passage Effect on the Seismic Response of Suspension Bridges Considering Local Soil Conditions[END_REF][START_REF] Svay | Modélisation de la Variabilité Spatiale du Champ Sismique pour les Etudes d'Interaction Sol-Structure[END_REF].

Few studies have been devoted to understand the effect of velocity heterogeneities on ground motion indicators for seismic hazard purposes. [START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF] performed three-dimensional finite-difference simulations to evaluate the effect of random, correlated velocity perturbations in a three-dimensional regional seismic velocity model of the San Francisco Bay Area on predicted ground motion. They used a Von Karman autocorrelation function to characterize the random velocity heterogeneities in their model. Although the frequency range of their simulations was limited to ≤ 1 Hz, their results showed significant deviations in the predicted ground velocities. [START_REF] Imperatori | Broad-band near-field ground motion simulations in 3-dimensional scattering media[END_REF] studied the effect of small-scale heterogeneities on several ground motion parameters -Peak Ground Velocity (PGV), Peak Ground Acceleration (PGA), and spectral acceleration (SA) -at short and intermediate distances from the source using 3D broadband (0 -10 Hz) finite-difference simulations. They found that the ground motion parameters displayed attenuation with increasing epicentral distance from the source produced by the redistribution of energy due to the scattering phenomena.

Recently, [START_REF] Iwaki | Effects of random 3D upper crustal heterogeneity on long-period ( 1s) ground-motion simulations[END_REF] investigated the effect of short-wavelength crustal heterogeneities on the predicted ground motion by conducting 3D finite-difference simulations using a detailed realistic velocity model of the Kanto region in Japan. The short wavelength heterogeneities were also modeled as spatially correlated random fields characterized by exponential-type autocorrelation function with a standard deviation of 5 %. They used different combinations of the random media's correlation lengths and point source models to investigate the variability of the predicted ground motion and the sensitivity of the ground motion parameters -PGV, velocity response spectra (Sv), and Fourier amplitude spectra (FAS). The short wavelength heterogeneities resulted in large variabilities in the ground motion parameters. The relative changes in their amplitude compared to those in the velocity model without heterogeneities were either increased or decreased depending on the spatial distribution of the velocity heterogeneities.

Although these studies provide information to grasp the ground-motion variability due to random heterogeneity, they are either limited to low frequencies (≤ 1 Hz) or, in the case of short period simulations (0 -10 Hz), they do not consider local structures, like sedimentary basins, in which the characteristics of the short-wavelength heterogeneities may differ in nature and scale from those of crustal heterogeneities. Some studies have considered small scale heterogeneity at the sedimentary scale (we consider the sedimentary scale to be of the order of meters to few kilometers).

Introduction [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF] analyzed the soil amplification functions at Kiban-Kyoshin, Japan, and showed that the consideration of small-scale heterogeneities in their numerical model helped reproducing the observed amplification. [START_REF] Pagliaroli | Seismic microzonation of the centeral archaelogical area of Rome: results and uncertainties[END_REF] studied the variations in the Housner intensity for the seismic microzonation of the Central Archaeological Area of Rome. Their study compared the Housner intensity of the simulated seismic wavefield using a deterministic V s profile to those of twenty stochastic V s profiles. They observed that the Housner intensity from the stochastic models, could on average, locally exceed those from the deterministic model by about 50 % in the period range of 0.1 -0.5s. [START_REF] Nour | Finite element model for the probabilistic seismic response of heterogeneous soil profile[END_REF] analyzed the effect of the variability of the shear modulus, fraction of critical damping, and Poisson ratio, modeled as spatial random fields, on the seismic response of heterogeneous soil profiles. They analyzed the influence of the coefficient of variation, inter-property correlation coefficients, and the horizontal and vertical correlation lengths. Their results showed that, as the variability of the soil properties increases, the PGA is attenuated, and maximum spectral amplification is shifted to lower frequencies. They also observed a shift of the fundamental frequency to lower frequencies and an attenuation of the higher frequencies when the variability increases.

More recently, El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] assessed the effect of the 2D spatial variability of the shear wave velocity on the surface ground motion by performing numerical experiments of seismic wave propagation in a sedimentary layer over a half-space. The shear wave velocity in their study was modeled as a spatially correlated random field. They found that the main parameter that controls the variability of the ground motion intensity measures (resonance frequency, spectral amplification, Arias intensity, and duration) is the coefficient of variation.

A common conclusion to all the listed studies is the increase in the spatial variability of the ground motion indicators due to the presence of heterogeneities. However, the way in which the ground motion parameters may vary compared to a reference model is still not well elucidated. These studies show the influence of heterogeneities on the ground motion, confirming the necessity to consider them in seismic hazard assessment studies.

Objectives and overview

In this thesis, we pursue the research in the light of the previously cited studies to investigate the effect of velocity heterogeneities on surface ground motion. In contrast to the other studies, we focus on sedimentary basins, in which wave trapping and reflection/diffraction by the basin edges occurs.

We investigate hereafter the effect of velocity heterogeneities modeled as random fluctuations on the basin response. To that purpose, we study seismic wave propagation in two different basins : (i)

the Nice (France) basin, whose geometry and mechanical properties were characterized in previous projects (e.g. Bard et al., 2005b;[START_REF] Arnal | Projet GEMITIS-Nice : Evaluation des dommages directs d'un séisme sur la ville Nice, scénario de risque, extension des résultats au départements des Alpes Maritimes[END_REF], which is around 50 m deep and 1 km large and has been instrumented by the French Permanent Accelerometric Network (RESIF-RAP) and (ii) a larger and deeper generic basin, 600 m deep and 2 km large. Because of the computational cost of numerical simulations, our investigations are limited to 2D geometries. In order to better understand how the incident wavefield interacts with media heterogeneities, we study their influence on different ground motion parameters throughout this study (i.e. PGV, transfer functions, Arias intensity, duration, and response spectra). We also investigate the influence of the properties of the heterogeneities such as the correlation length and coefficient of variation. Finally, we use two different incoming wavefields, one synthetic impulsive source and one earthquake recording. In this work, the incoming wavefield is modeled as a plane wave with vertical incidence, and the statistical description of the random properties Introduction are modeled using a Von Karman distribution.

This dissertation is organized as follows:

Theoretical background (chapter 2)

In chapter 2, we overview the different concepts used throughout the manuscript. The chapter is divided into three sections.

In the first section, we define the main ground motion parameters used in Seismology and Earthquake engineering. We then provide a brief review of site effects, and we describe the different methods used to characterize them.

The second section tackles the elastic wave equation and the most encountered numerical methods used to solve it. Particular focus shall be paid on the Spectral Element Method (SEM), which is used in this study to simulate seismic wave propagation.

In the last section, we briefly explain the modeling of random fields to approximate the presence of heterogeneities in the propagation media.

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin (chapter 3)

Chapter 3 presents in the form of an article submitted to Geophysical journal international (GJI) a parametric investigation on the effect of 2D random velocity heterogeneities on seismic ground motion in Nice (France), sedimentary basin. The particular choice of the Nice basin was made because it has been the object of previous studies (e.g. Bard et al., 2005b;[START_REF] Arnal | Projet GEMITIS-Nice : Evaluation des dommages directs d'un séisme sur la ville Nice, scénario de risque, extension des résultats au départements des Alpes Maritimes[END_REF][START_REF] Peyrusse | A nodal discontinuous galerkin method for site effect assessment in viscoelastic media -verification and validation in the nice basin[END_REF], where the data are available (CEREMA, personal communication). We study (1) the sensitivity of ground motion indicators (i.e., PGV, response spectra, Arias intensity, and duration) and transfer functions to the statistical parameters of the random velocity models, and (2) understanding the role of heterogeneities compared to the basin geometry and impedance contrast between layers. In this work, we use a synthetic impulsive source. We find that the coefficient of variation is the main controlling parameter compared to the correlation length. Its value dominates the variability on the computed ground motion and the corresponding intensity measures. Finally, some considerations regarding how such random heterogeneities can be taken into account for seismic hazard assessment are also discussed.

Contribution of heterogeneities characterized from borehole data in site response : case study of the

Nice basin (chapter 4)

In chapter 4, we use borehole data collected by CEREMA to obtain the statistical parameters describing the Von Karman distribution for the velocity heterogeneities in Nice (France). Then we generate different 2D velocity models and use the February 25, 2001, Nice earthquake recorded at RAP-NBOR (rock station) as an input for the analyses. We analyze the effects of the velocity heterogeneities on the intensity and duration of the ground motion, the pseudo-spectral acceleration response, Arias intensity, and transfer function (TF) of the basin. Furthermore, using different ground motion indicators, we compare simulated ground motions with the one recorded at the RAP-NLIB station located in the basin during the 2001 Nice earthquake. We perform 1D simulations using soil profiles at NLIB and compare amplifications coming from 1D and 2D models. At last, we investigate whether the variability in the TF caused by the velocity heterogeneities can explain the observed variability in the TF at the station NLIB, located on the basin. We conclude that velocity heterogeneities could be part of the aleatory variability Introduction of the basin response, but it cannot fully explain the earthquake-to-earthquake amplification variability at the NLIB station.

Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response (chapter 5)

The results from chapters 2 and 3 suggest some limitations in the Nice basin model to investigate the effects of the correlation length in stochastic media. Due to the small size of the Nice basin, the influence of the correlation length cannot be fully seen. Therefore, we built a larger model having 600 m depth and 2 km long to better explore the effect of the correlation length size. We use the synthetic impulsive source as incoming wavefield. We find that the Arias intensity and the response spectra are sensitive to larger values of the correlation length, in the order of 50 to 100 m. This remains an exploratory analysis that needs to be pursued in the future.

General conclusions and perspectives

In this chapter we provide a synthesis of the different results obtained in this PhD research. We also enumerate some perspectives to develop in future studies.

Articles / Presentations

The work presented in this dissertation has led to:

• One submitted scientific article under review:

-N.F. Tchawe, C. Gélis, L.F. Bonilla, and F. Lopez-Caballero. Effects of 2D random velocity perturbations on short-period (≤ 1s) ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin. Submitted to: Geophys. J. Int (Accepted with reviews).

• One article under preparation:

-N.F. Tchawe, C. Gélis, L.F. Bonilla, E. Bertrand, O. Rohmer Characterizing velocity fluctuations in the Nice (France) sedimentary basin from boreholes measurements: impact on the site response assessment.

under preparation for submission to: Geophys. J. Int. • Two posters presentations:

-N.F. Tchawe, C. Gélis, L.F. Bonilla, and F. Lopez-Caballero. Numerical analysis of the effect of the spatial variability of soil properties within the Nice sedimentary basin on the free field seismic ground motion.9 e Biennale du RAP, 2018, Lourdes, FRANCE.

-N.F. Tchawe, C. Gélis, L.F. Bonilla, and F. Lopez-Caballero. Effects of 2D random velocity perturbations on short-period (≤ 1s) ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin. Seismological society of America, 2019, Seattle, USA.

-N.F. This chapter provides a brief overview of the theoretical concepts behind the approach used in this thesis. An in-depth treatment of the different concepts covered here can be found in many scientific/review articles and textbooks. Appropriate reference shall be provided for the interested readers.

The chapter is organized into three sections: (1) In section 2.1, we define site-effects and the ground motion parameters commonly used to describe surface ground motion. We also explain the different methods used to estimate site effects; (2) In section 2.2, we derive the equations that govern the propagation of elastic waves in the heterogeneous Earth. We then briefly present the different numerical methods used to solve them, with emphasis on the spectral element method, which is used in this thesis;

(3) Finally, in section 2.3, we cover the modeling of soil heterogeneities (in our case, velocity heterogeneities) following a random field approximation. We describe the different techniques used to determine the soil heterogeneities' random properties from available data.

Theoretical background

Seismic ground motion and site effects

The ground motion or seismogram (G[t], see Figure 2.1 for an illustration) recorded at the earth's surface during an earthquake is influenced by different physical phenomena. It can be mathematically represented as the convolution of; source (S[t]), path (P [t]), site (Si[t]), and instrument effects I[t] [START_REF] Bonilla | Site Amplification in the San Fernando Valley, California: Variability of Site-Effect Estimation Using the S-Wave, Coda, and H/V Methods[END_REF]:

G ij [t] = S i [t] * P ij [t] * Si j [t] * I j [t] (2.1)
where * is the convolution operation, t is time. The indexes i and j represent the ith earthquake (or event) and jth station, such that P ij [t] is the path term between the ith event and jth station. -Path effects comprise any physical phenomena that affect seismic waves' propagation during their path from the seismic source through the earth's crust to the earth's surface. They generally include anelastic attenuation over the source-to-site distance due to the geometric spreading and absorption by the materials that constitute the earth's crust.

-Site effects include any local site conditions such as geology, surface and subsurface geometry, material heterogeneity, among other factors, that modify the seismic waves propagation.

Understanding the extent to which each component on the right-hand-side of equation 2.1 affects ground motion is of great interest in earthquake engineering and seismology.

The research performed in this thesis falls within the scope of site effects. Site effects are of significant importance in seismic hazard assessment. They can considerably intensify an earthquake's destructive potential due to phenomena such as the amplification, the increase in the duration, and spatial variability of the ground motion.

Ground motion parameters

When an earthquake occurs, an accelerometer or seismometer1 located on the Earth's surface records a 3-component motion of the ground shaking. The motion can either be the acceleration, velocity, or displacement time history of the ground shaking. In practice, only one of these quantities is measured directly, and the others are computed from it by either integration and/or differentiation. The ground motion produced by earthquakes is quite complicated and carries enormous information about its source and propagation history. The detailed description of all this information is very difficult to synthesize. Therefore, it is customary to describe only the ground motion characteristics of engineering significance and identify a few ground motion parameters or intensity measures that reflect those characteristics. The three main characteristics of ground motion that are of primary significance for engineering purposes are (1) its amplitude, (2) its frequency content, and (3) its duration [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]. Different ground motion parameters, which provide information about one or more of these characteristics, have been proposed in the earthquake engineering literature. In what follows, we define the ground motion parameters which we use in this dissertation. Table 2.1 summarizes them and the ground motion characteristics they reflect. The readers interested in a detailed explanation of all commonly used ground motion parameters in earthquake engineering literature are referred to chapter 3 of the textbook by [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF].

Peak acceleration and velocity

The amplitude of ground motion is most commonly reflected through the peak acceleration or peak velocity. In earthquake engineering literature, they are termed peak ground acceleration (PGA) and peak ground velocity (PGV), which are the maximum ground acceleration/velocity during an earthquake at a given location. Since earthquake shaking generally occurs in all three spatial directions, the PGA/PGV is often split into horizontal and vertical components. The horizontal components of the ground motion are generally used because of (1) their natural relationship with inertial forces [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF], and (2) the fact that they carry S-waves, which are more energetic and destructive compared to P-waves. The largest dynamic forces induced in certain types of structures are closely related to the PGA. PGV is more likely to characterize ground motion accurately at intermediate frequencies. Hence, for structures and facilities sensitive to loading in the intermediate frequency range, the PGV might provide a much more accurate indication of the potential for damage than the PGA. Both the PGA and PGV can be correlated to the earthquake intensity (e.g. Trifunac and Brady, 1975b;[START_REF] Murphy | The correlation of peak ground acceleration amplitude with seismic intensity and other physical parameters[END_REF][START_REF] Krinitszky | Parameters for specifying intensity-related earthquake ground motions[END_REF]. Although they are very useful parameters, they provide no information on the motion's frequency content or duration. Figure 2.3a shows an illustration of the PGV (red circle) for the February 25, 2001, Nice earthquake.

Seismic response spectrum

The response spectrum is an important parameter used by earthquake engineers for earthquakeresistance design. It describes the maximum displacement, velocity, or acceleration of a single-degreeof-freedom (SDOF) system to a particular input motion as a function of the natural frequency (or natural period) and damping ratio of the SDOF system (Figure 2.2, [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]). They are used for analyzing the performance of structures and equipment to earthquakes since many structures behave principally as simple oscillators. The maximum values of these parameters (displacement, velocity, and Theoretical background Figure 2.2 -Illustration of the concept of the response spectrum. The spectral accelerations are the maximum acceleration amplitudes of SDOF systems in response to the same input motion. The response system is obtained by plotting the spectral accelerations against the periods of vibrations of the SDOF systems (obtained from [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF] acceleration) are referred to as the spectral displacement, spectral velocity, and spectral acceleration, respectively. The spectral values can be affected by the amplitude, frequency content, and to a lesser extent, the duration of the ground motion. Although the response spectrum does not directly describe the actual ground motion, it provides valuable information on its potential effects on structures. In this thesis, we study only the acceleration response spectrum for a damping ratio of 5 %.

Arias intensity

The Arias intensity, proposed by the Chilean engineer and seismologist Arthuro Arias in 1970, is a measure of strength of ground motion. It characterizes the shaking by measuring the acceleration of transient seismic waves. It is a parameter that includes the effects of the amplitude and duration of the ground motion and is defined as [START_REF] Arias | Measure of earthquake intensity[END_REF]:

AI = π 2g T 0 [a(t)] 2 dt (2.2)
where a is the ground motion acceleration, g is the acceleration due to gravity, and T is the total duration of the motion. The Arias intensity has units of velocity and is usually expressed in meters per second.

Some authors (e.g. [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] use the Arias based intensity parameter which is defined by:

A b I = T 0 [v(t)] 2 dt (2.3)
This value is illustrated by the green star in Figure 2.3. In this thesis, we use the Arias based intensity parameter as it is computed directly on the velocity wavefield, which is the output of numerical experiments performed in the subsequent chapters.

Duration

The duration of strong ground motion is proportional to the time required to release the accumulated strain energy by the rupture along the fault [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]. Its value increases as the area of fault rupture increases. As a result, the duration of strong ground motion increases with increasing earthquake magnitude. It can strongly influence earthquake damage since it is proportional to the number of cycles2 during an earthquake. Indeed, many physical processes such as the strength of certain types of structures and the buildup of porewater pressures in loose, saturated sand, are sensitive to the number [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]). Note that the table summarizes only the ground motion characteristics we use in this thesis. An exhaustive list of ground motion characteristics and their definition can be found in [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]. of load or stress reversals that occur during an earthquake [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]. A motion of short duration but high amplitude may not produce enough load reversals for damaging response to buildup in a structure. In contrast, a motion with moderate amplitude but a long duration can produce enough cycles to cause substantial damage [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]. For earthquake engineering purposes, only the strong motion part of the accelerogram is required to estimate the duration. An earthquake accelerogram generally contains all accelerations from the time the earthquake begins until the time the motion has returned to the level of background noise. Several methods have been proposed to evaluate the duration of strong ground motion in an accelerogram or seismogram. [START_REF] Bolt | Duration of strong motion[END_REF] proposed the bracketed duration, which is defined as the time between the first and last exceedances of a threshold acceleration (usually 0.05g). Other authors (Trifunac and Brady, 1975a) define the duration as the time interval between the points at which 5 % and 95 % of the total energy has been recorded. [START_REF] Boore | Stochastic simulation of high frequency ground motions based on seismological models of the radiated spectra[END_REF] defined the duration of strong ground motion to be equal to the corner period (inverse of the corner frequency) of the earthquake spectrum. El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] defines the effective duration (T d ) of an earthquake as the difference between the times where 95 % and 5 % of the total I ab are reached. This is illustrated by the blue arrow in Figure 2.3 for the Nice earthquake.

Theoretical background

Site effects

Site effects can be formally defined as the influence of the local geology and soil conditions on the intensity of ground shaking and earthquake damage [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]. Indeed, the local site conditions can profoundly influence all the essential characteristics (amplitude, frequency content, and duration)

of strong ground motion. The main features that influence the ground motion are the impedance contrast between the soil layers, the subsurface's geometry and material properties, the site topography, and the input motion characteristics. Some significant observational examples of site effects were during the September 19, 1985, Michoacan (Mexico) earthquake [START_REF] Anderson | Strong ground motion from the the Michoacan, Mexico, earthquake[END_REF], the April 6, 2009 L'Aquila (Italy) earthquake (Benessia and De Marchi, 2017, e.g.), the November 14th, 2016 Kaik ōura (New Zealand) earthquake [START_REF] Bradley | Ground motion and site effect observations in the wellington region from the 2016 mw7.8 kaik ōura, new zealand earthquake[END_REF], the April 16, 2016, Kumamoto (Japan) earthquake [START_REF] Hiroaki | Observation of earthquake ground motion due to aftershocks of the 2016 Kumamoto earthquake in damaged areas[END_REF], which led to severe damage of buildings and infrastructures. Figure 2.4 shows the ground motion records of an aftershock of the Michoacan earthquake. The ground motion recorded at station SCT located on the sedimentary basin in Mexico city displays higher amplitudes than the ground motion recorded at the rock sites even in the vicinity of the earthquake's epicenter (station Campos).

The consideration of local site effects is therefore important for earthquake-resistant design and seismic hazard assessment.

A theoretical explanation of why the local site conditions are expected to influence surface ground motion can be done using the principle of conservation of elastic wave energy [START_REF] Kramer | Geotechnical Earthquake Engineering[END_REF]. If the effects of scattering and material damping are neglected, the conservation of elastic energy requires that the flow of energy (energy flux, ρv s u2 , where ρ is the density, v s is the shear wave velocity and u is the particle displacement) from depth to the ground surface be constant. Therefore, since the density and shear wave velocity of materials generally decreases as we get closer to the earth surface, the particle velocity, u, must increase.

In reality, many complex wave propagation phenomena are involved when seismic waves travel through soft sites. Site effects are also related to surface topography and stratigraphic effects (the geometry and sedimentary filling of basins). Topographic effects have received less attention in the literature compared to stratigraphic effect. One of the reasons for this is the difficulty in separating the contribution of topography from other causes, such as stratigraphy and near-surface weathering [START_REF] Lebrun | Experimental study of the ground-motion on a large scale topographic hill a kitherion (greece)[END_REF]. We only provide a brief review of the main phenomena which influence wave propagation in these structures. A review of the origins and early history developments of site effects can be found in Sánchez-Sesma and Crouse (2014).

Surface topography effects

Several authors have attributed the distinctive earthquake-induced damage patterns in topographic regions to topographic ground motion amplification. For example, during the 1985 Chile earthquake [START_REF] Celebi | Topographic and geological amplifications determined from strong-motion and aftershock records of the 3 March 1985 Chile earthquake[END_REF], the Aegion (Greece) 1995 earthquake [START_REF] Bouckovalas | Geotechnical aspects of the 1995 Aegion,Greece,earthquake[END_REF], the L'Aquila (Italy) earthquake [START_REF] Celebi | Recorded motions of the 6 April 2009 Mw 6.3 L'Aquila, Italy, earthquake and implications for building structural damage: Overview[END_REF].

Experimental, theoretical, and numerical analysis have been performed to investigate the parameters contributing to topographic effects (e.g. [START_REF] Bouchon | A simple, complete numerical solution to the problem of diffraction of SH waves by an irregular surface[END_REF][START_REF] Bouchon | Seismic response of a hill : The example of Tarzana, California[END_REF][START_REF] Geli | The effect of topography on earthquake ground motion : a review and new results[END_REF][START_REF] Komatitsch | The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF][START_REF] Kawase | Topography effect at the critical SV-wave incidence : Possible explanation of damage pattern by the Whittier Narrows, California, earthquake of 1 October 1987[END_REF][START_REF] Zhou | A new approach to simulate scattering of SH waves by an irregular topography[END_REF][START_REF] Maufroy | A robust method for assessing 3-d topographic site effects: A case study at the lsbb underground laboratory, france[END_REF][START_REF] Tripe | Slope topography effects on ground motion in the presence of deep soil layers[END_REF]. It has been observed that topographic amplifications computed using numerical simulations often underestimates the actually observed amplifications (e.g. [START_REF] Geli | The effect of topography on earthquake ground motion : a review and new results[END_REF][START_REF] Barani | Topographic effects in probabilistic seismic hazard analysis: The case of narni[END_REF].

The discrepancy is mainly attributed to the oversimplification of numerical models, which are unable to reproduce the complex modifications undergone by the wavefield in the vicinity of steep topography.

The incident wavefield into topographic structures experiences scattering, diffraction, focusing, and defocusing, which either increases and reduces the seismic wave energy, leading to changes in the amplitude, frequency, and duration of motion. The scattering and diffraction of the wavefield leads to late secondary arrivals (coda waves) on seismograms. These diffracted waves have been successfully displayed in numerical experiments by [START_REF] Bouchon | A boundary integral equation -discrete wave number representation method to study wave propagation in multi-layered media having irregular interfaces[END_REF]; [START_REF] Komatitsch | The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF], using a 3D

Gaussian topography model. The authors showed that the diffraction could generate different types of waves (body and surface waves) depending on the incident wavefield's polarization. The diffracted energy displays high directivity during their propagation.

After interaction with the surface topography, the resulting wavefield can be very complex. The type of diffracted waves generated is generally difficult to characterize when the incident wavefield has wavelengths longer than the slope height. For example, [START_REF] Boore | Wave scattering from a step change intopography[END_REF] observed, using numerical simulations, that topographic amplifications are frequency-dependent and significant for wavelengths several times the slope height. [START_REF] Geli | The effect of topography on earthquake ground motion : a review and new results[END_REF] reviews a number of theoretical studies which predict amplification of the wavefield at the crest of topographies (for homogeneous isolated ridges) and the generation of outward propagating waves. This amplification stems from the focusing of seismic energy due to the inwards reflection of the incident waves. Such an increase in the ground motion amplification near the crest of a ridge was measured in five earthquakes in Matsuzaki, Japan [START_REF] Jibson | Summary of research on the effects of topographic amplification of earthquake shaking on slope stability[END_REF]. Bard (1999b) also reviews numerical and experimental studies on topographic effects. The author highlights an important ground motion amplification over convex surfaces, de-amplification over concave surfaces, and a differential motion along the slope in between.

A more recent review on topographic effects, based on both experimental and numerical studies, is provided by [START_REF] Massa | Overview of topographic effects based on experimental observations: meaning, causes and possible interpretations[END_REF]. The authors summarizes the following the important findings on the effect of topographic irregularities on ground motion:

-The ground motion amplification is higher at the top of a topography compared to the base.

-Topographic effects are frequency-dependent, and the maximum amplification is generally ob-Theoretical background served when the incident wavelength is comparable to the width of the morphological relief (Boore et al., 1981, e.g.) -The amplification at the top of a relief generally increases with an increase in the shape ratio SR=H/L (where H is the height of the crest and L is the semi-width of the relief considering its transversal section.) (e.g. [START_REF] Sánchez-Sesma | Elementary solutions for response of a wedge-shaped medium to incident SH and SV waves[END_REF] -The amplifcation of incident P waves by the topography is lower than those of S waves; in particular SH waves amplifications tend to be smaller than those relative to the in-plane solution (SV waves) (e.g. [START_REF] Paolucci | Amplification of earthquake ground motion by steep topographic irregularities[END_REF] -Topographic amplification depends on the incident angle of the seismic wavefield with maximum amplification occuring at vertical incidence (e.g. [START_REF] Bouchon | Seismic response of a hill : The example of Tarzana, California[END_REF].

The complex phenomena induced by the topography on the seismic wavefield make it difficult to define quantitative rules to include topographic effects in seismic hazard assessment. Indeed, the presence of a relief can strongly influence the seismic response of sites, even located at a large distance from the topography. Therefore, site effect analysis requires site-specific analysis, which considers the topography's 3D geometry to better capture their effects.

Although this thesis' focus is not on topographic effects, their effects are implicitly taken into account in the numerical simulations we perform. Indeed, the 2D velocity model of the Nice sedimentary basin, which we will use, displays a hill on the Eastern side.

Effects of the geometry and subsurface sedimentary fillings

Earthquake damage is generally larger over soil sediments than on firm bedrock outcrops. This can easily be observed by looking at the distribution of damage in large cities like Mexico City, San

Francisco, built on soft alluvial deposits following major earthquakes (e.g. [START_REF] Stone | Engineering aspects of the september 19, 1985 mexico earthquake[END_REF][START_REF] Seed | Preliminary report on the principal geotechnical aspects of the october 17, 1989 lomo prieta earthquake[END_REF]. The increase in the intensity of ground motion by soft soils has made the consideration of site effect important, even in moderate seismicity regions. The fundamental phenomenon responsible for amplifying ground motion over soft sediments is the trapping and focusing of seismic waves in the sediments because they are softer than the underlying bedrock. The types of waves trapped depend on the sediments' geometry and incidence angle of the incoming wavefield. For example, horizontally stratified soil layers (a simple 1D configuration) will trap only body waves for vertically incident plane wave on the soil layers. Whereas, when the sediments have a 2D or 3D shape (sedimentary basins), i.e., they present lateral variations at depth, edge-generated wave and surface waves can be generated at the non-planar interfaces (Bard and Bouchon, 1980a,b).

In the early developments of site effects, the analysis of such complex phenomena was mainly done theoretically, by seeking the ground motion solution in simple 1D and 2D geometries using analytical or semi-analytical methods (e.g [START_REF] Aki | Surface motion of a layered medium having an irregular interface due to incident plane SH waves[END_REF][START_REF] Trifunac | Surface motion of a semi-cylindrical Alluvial Valley for incident plane SH waves[END_REF][START_REF] Aki | Local site effects on strong ground motion[END_REF][START_REF] Bouchon | A boundary integral equation -discrete wave number representation method to study wave propagation in multi-layered media having irregular interfaces[END_REF]. Experimental investigations then became possible with the availability of instrumented sites. For example, [START_REF] King | Observed variations of earthquake motion over a sediment-filled valley[END_REF] analyzed the variations in earthquake motion over sediments-filled basins. They observed rapid variations in the transfer functions fundamental modes over the basin, which they interpreted as being due to waves focusing at the center of the basin. Nowadays, the surge in the amount of available data, through seismic data centers around the world, has widely facilitated the development of experimental methods to study sedimentary effects on earthquake ground motion. The analysis of earthquake recordings of arrays located over soft sediments can help decipher the complex phenomena affecting the wave propagation in such structures (e.g. [START_REF] Imtiaz | Effects of site geometry on short-distance spatial coherency in Argostoli, Greece[END_REF][START_REF] Ktenidou | Directional dependence of site effects observed near a basin edge at Aegion, Greece[END_REF]. Numerical simulations of seismic wave propagation have also helped to complement empirical observations to better understand the contribution of sediment fillings or stratigraphic to site effects. The increase in Theoretical background computational power over the last decades has enabled scientist to improve their knowledge on site effects through the ability to perform realistic 2D and 3D numerical simulations (e.g. [START_REF] Frankel | Three-dimensional simulations of ground motions in the San Bernardino Valley, California, for hypothetical earthquakes on the San Andreas fault[END_REF][START_REF] Olsen | Strong shaking in Los Angeles expected from southern San Andreas earthquake[END_REF][START_REF] Delavaud | Simulation numérique de la propagation d'ondes en milieu géologique complexe : application à l'évaluation de la réponse sismique du bassin de Caracas (Venezuela)[END_REF][START_REF] Chaljub | Quantitative comparison of four numerical predictions of 3D ground motion in the Grenoble valley[END_REF].

There is an enormous literature on the role of stratigraphic effect on the site amplification. Totally synthesizing them will be impossible in this subsection. In what follows, we only provide a brief review of their effects with a particular focus on basin-like geometries.

As stated before, sedimentary basins are prone to strong site effects and the nature of their sedimentary filling can lead to complex wave propagation within the basin. [START_REF] Semblat | Seismic Wave Amplification: Basin Geometry vs Soil Layering[END_REF] observed that the combined effect of the basin geometry and soil layering plays a major role on the site amplification functions. The trapping of the seismic waves by low-velocity sedimentary layers can also lead to a considerable increase in the ground motion duration and amplification. For example, [START_REF] Bindi | Site Amplifications Observed in the Gubbio Basin, Central Italy: Hints for Lateral Propagation Effects[END_REF] observed the presence of locally generated basin-edge and surface waves in earthquake recordings at the Gubbio basin, which caused significant increases in the duration and amplitude of the motion.

Seismic waves incident on soft sediments can also lead to surface waves and edge-generated waves that travel across the basin. For example, [START_REF] Imtiaz | Seismic Wave-field Analysis from Dense Seismic Arrays and Implications for Site Effects in Cephalonia, Greece[END_REF] observed the predominance of scattered surface waves generated from the basin edges in the Argostoli (Greece) basin, whose effects play a major role on the observed site amplification functions. [START_REF] Kawase | Constructive Interference of the Direct S-Wave with the Basin-Induced Diffracted/Rayleigh Waves[END_REF] interprets the cause of the Damage Belt in Kobe during the Kobe earthquake as being due to basin-edge induced Rayleigh waves and source directivity effects. Basin-edge induced surface waves and basin-geometry-focusing effects caused by sedimentary fillings can also lead to discrepancies between 2D numerical simulations and observations, as shown by [START_REF] Hartzell | Variability of Site Response in Seattle, Washington[END_REF] for the Seattle basin.

Sedimentary basin geometries can also lead to directivity effects on the ground amplification. Such a directional dependence has been observed by [START_REF] Ktenidou | Directional dependence of site effects observed near a basin edge at Aegion, Greece[END_REF] at the basin-edge of the Aegion, Greece, basin on both observations and 2D numerical simulations using incident SH and SV waves.

The site amplification also appears to be altered by the nonlinear properties of the sediments. Sediments may behave nonlinearly when a certain level of strain is exceeded during a large earthquake.

The nonlinear soil behavior has long been neglected by seismologists, who generally model the earth materials using simple linear viscoelastic models. This is principally due to the lack of satisfactory constitutive model on nonlinearity, and insufficient evidence of nonlinear effect, other than liquefaction, in observed ground motion [START_REF] Riepl | Detailed evaluation of site response estimation methods across and along the sedimentary valley of Volvi (EURO-SEISTEST)[END_REF]. On the other hand, since the pioneering works of [START_REF] Seed | The influence of soil conditions on ground motions during earthquake[END_REF], geotechnical engineers have been convinced for a long time that soft soils exhibit strong inelastic behavior during earthquakes. Such nonlinear soil behavior manifests itself through decreases in shear modulus and increases in material damping as the shear strain increases. Soil nonlinearity can have two origins. On the one hand, they are due to the non-reversible stress/strain relationship caused by the material's partial or total rupture. On the other hand, pore pressure effects can be strong during earthquake shaking in saturated soils. Nonlinear soil behavior has two main consequences on ground motion [START_REF] Bard | Les effets de site d'origine structurale : principaux résultats expérimentaux et théoriques[END_REF]: (1) they generally induce a very significant decrease in the fundamental frequency, which causes changes in the spectral amplification (e.g [START_REF] Bonilla | Nonlinear site response evidence of k-net and kik-net records from the 2011 off the pacific coast of tohoku earthquake[END_REF], and (2) the increased damping leads to substantial reductions in peak acceleration, especially at high frequencies.

Numerous laboratory test performed on soil samples have revealed that the strain threshold above which changes in mechanical properties are observed is very low, on the order of 10 -4 or less.

In this thesis we do not consider the effects of soil nonlinearity in our material models. Indeed, our case study (the Nice sedimentary basin) is located in a region of moderate seismicity and non-linear effects are less expected to develop in the material during earthquakes.

Theoretical background

Besides the listed effects, the heterogeneous nature of the sediments can also affect site effects. [START_REF] Stripajova | Extensive Numerical Study on identification of key structural parameters responsible for site effects[END_REF] observed using 2D numerical simulations that the small-scale velocity heterogeneities can increase the aggravation factor of the cumulative absolute velocity. El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] also showed that velocity heterogeneities in a sedimentary layer could considerably increase the spatial variability of ground motion indicators. There is still a knowledge gap in the physical understanding of their effects on ground motion. The main focus of this thesis is to decipher their effect on ground motion. Details on small-scale heterogeneities shall be provided in the upcoming sections and chapters.

Methods of site effects estimation

Standard spectral ratio

The standard spectral ratio (SSR), which is generally considered a reference site technique, was introduced by [START_REF] Borchedt | Effects of local geology on ground motion near San Francisco Bay[END_REF]. It is calculated by taking the ratio between the spectrum of a ground recording that includes site effects with the spectrum of a recording at a reference site. By rewriting equation 2.1 in the frequency domain for the horizontal component of the ground motion recorded at two stations (one station is located on soft site G site (f ) and the other located on a rock site G rock (f )),

with the same instrument response, we obtain:

G site (f ) = S(f ) P site (f ) Si site (f ) (2.4) G rock (f ) = S(f ) P rock (f ) Si rock (f ) (2.5)
where P site (f ) and P rock (f ) are the path effects encountered by the wavefield during their path from the source to the soft site and rock site, respectively. Si site (f ) and Si rock (f ) are the site effects at the soft site and rock site. The rock site should, in an ideal situation, contain no site effects such that we can remove the term Si rock (f ) in equation 2.5:

G rock (f ) = S(f ) P rock (f ) (2.6)
The standard spectral ratio uses the assumption that the path term is common to the site and rock i.e.

distance between the soft site and the rock site is small compared to the source to site distance, such that P site (f ) = P rock (f ). With this assumption can isolate the site effect Si ef f ect term by taking the ratio between equation 2.4 and 2.6:

Si site (f ) = G site (f ) G rock (f ) (2.7)
Finding an appropriate reference site is difficult in practice (e.g. [START_REF] Hollender | Can We Trust High-Frequency Content in Strong-Motion Database Signals? Impact of Housing, Coupling, and installation Depth of Seismic Sensors[END_REF][START_REF] Perron | Robustness of kappa (κ) measurement in low-to-moderate seismicity areas: Insight from a site-specific study in Provence, France[END_REF], hence several reference sites are often used to compute the SSR (Boore, 2004). The spectral ratio can be computed on different phases of the earthquake signal. In practice, it is generally computed using the S-wave part of the signal.

Horizontal to vertical spectral ratio

The horizontal to vertical spectral ratio (HVSR) is a single-station method which uses the vertical component as a reference. It was initially introduced by [START_REF] Nogoshi | On the amplitude characteristics of microtremor (part 2)[END_REF] but widely popularized by [START_REF] Nakamura | A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface[END_REF]. It can be computed using both earthquake motion (e.g. [START_REF] Lermo | Site effect evaluation using spectral ratios with only one station[END_REF] and ambient noise measurement [START_REF] Nakamura | A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface[END_REF][START_REF] Bonnefoy-Claudet | H/V ratio: a tool for site effects evaluation. Results from 1-D noise simulations[END_REF] Theoretical background e.g.). Some studies have shown that HVSR computed on earthquake motion does not enable the adequate characterization of site effects (e.g. [START_REF] Bonilla | Site Amplification in the San Fernando Valley, California: Variability of Site-Effect Estimation Using the S-Wave, Coda, and H/V Methods[END_REF][START_REF] Bindi | Site Amplifications Observed in the Gubbio Basin, Central Italy: Hints for Lateral Propagation Effects[END_REF]. Nevertheless, the technique is widely used to characterize the 1D site's resonance frequency, which is correlated to the frequency of the curve's first peak.

The generalized inversion technique

The generalized inversion technique (GIT) was first adopted by [START_REF] Andrews | Objective Determination of Source Parameters and Similarity of Earthquakes of Different Size[END_REF]. It enables the separation of the source, path, and site effect of the observed ground motions through the solution of a large inverse problem. The GIT formulation can be derived in a nutshell as follows: First, by rewriting equation 2.1 in the frequency domain and ignoring the instrument response we obtain:

G ij [f ] = S i [f ] • P ij [f ] • Si j [f ] (2.8)
Taking the logarithm of equation 2.8 results in a simple linear equation:

ln G ij [f ] = ln S i [f ] + ln P ij [f ] + ln Si j [f ]
(2.9)

The path effects includes the geometric spreading and other inelastic losses and writes as [START_REF] Yefei | Site effects by generalized inversion technique using strong motion recordings of the 2008 wenchuan earthquake[END_REF]:

P ij [f ] = R -1 ij • exp(-πf (R ij /Q(f )) • V s ) (2.10)
where R ij is the hypocentral distance from the ith source to the jth station, Q(f ) is the frequencydependent quality factor, and V s is the shear wave velocity of the medium.

Including equation 2.10 in equation 2.9 gives:

ln G ij [f ] + ln R ij = ln S i [f ] + ln Si j [f ] -(πf R ij /V s ) • Q(f ) (2.11)
Equation 2.11 can be written in matrix form:

Ax = b (2.12)
where x contains the unknown parameters on the right-hand-side of equation 2.11 to be determined, b contains the terms on the left-hand-side of equation 2.11, and A is a sparse matrix containing three nonzero elements on each row (two 1 and -πf R ij /V s ). The linear system (equation 2.12) can be inverted for x using an appropriate inversion method e.g. the singular value decomposition.

The GIT has widely been used to evaluate the site effects in many studies (e.g. [START_REF] Drouet | Simultaneous inversion of source spectra, attenuation parameters, and site responses: Application to the data of the french accelerometric network[END_REF][START_REF] Tsuda | Inversion Analysis of Site Responses in the Kanto Basin Using Data from a Dense Strong Motion Seismograph Array[END_REF][START_REF] Yefei | Site effects by generalized inversion technique using strong motion recordings of the 2008 wenchuan earthquake[END_REF].

Numerical modeling of seismic wave propagation

Numerical simulations of seismic waves propagation are essential for addressing many outstanding questions in geophysics and seismology. They are a key tool for predicting the ground motion of potential earthquakes (e.g. [START_REF] Komatitsch | Simulations of Ground Motion in the Los Angeles Basin Based upon the Spectral-Element Method[END_REF][START_REF] Honoré | Ground motion simulations of a major historical earthquake (1660) in the french pyrenees using recent moderate size earthquakes[END_REF][START_REF] Maufroy | 3d numerical simulation and ground motion prediction: Verification, validation and beyond -lessons from the e2vp project[END_REF]Jayalakshmi et al., Theoretical background 2020), hence for characterizing site effects in seismic hazard assessment. In seismic inversion and fullwaveform inversion (a recent technique used to image the structure of the subsurface), they are used to estimates the medium's elastic properties (e.g. [START_REF] Virieux | An overview of full-waveform inversion in exploration geophysics[END_REF][START_REF] Fichtner | Full Seismic Waveform Modelling and Inversion[END_REF][START_REF] Fichtner | The deep structure of the north anatolian fault zone[END_REF]. In oil and gas prospecting, they are used to model the seismic response of hydrocarbon reservoirs (e.g. [START_REF] Chopra | Seismic attributes for prospect identification and reservoir characterization[END_REF]. In global geophysics, they are used to image the Earth's interior dynamics by tomography (e.g. [START_REF] Hosseini | Global mantle structure from multifrequency tomography using P, PP and P-diffracted waves[END_REF], and to model wave effects of complex structures (e.g. [START_REF] Komatitsch | Spectral-element simulations of global seismic wave propagation-I. Validation[END_REF]Imperatori andMai, 2013, 2015). More recently, seismic simulations played a major role in the understanding of the novel recordings on Mars, during the InSight mission [START_REF] Van Driel | Preparing for InSight: Evaluation of the blind test for martian seismicity[END_REF]. Just to list a few examples of applications among many.

Simulating seismic wave propagation consists in solving the space-time dependent wave equation in a numerical representation of the medium of interest (e.g. a sedimentary basin, a fault zone etc.). We understand by numerical representation, a finite domain discretization (computer mesh) of the medium in which we seek the solution of the wave equation at every point in the mesh.

In this section, we first introduce the elastic wave equations. Then we briefly present the different time domain numerical methods which can be used to solve them with particular focus on the spectral element method, which we use in this thesis.

The Elastic wave equation

Wave propagation in elastic media is governed by the equations of linear elastodynamics. These equations result from the principles of continuum mechanics, which assure the conservation of momentum, and linearly link displacements, stress, and strain in the materials. In what follows, we only state the general principles that lead to the elastic wave equation. The detail derivation of the elastic wave equation from the principles of elastodynamics can be found in general textbooks on seismology (e.g. [START_REF] Aki | Quantitative seismology, theory and methods[END_REF][START_REF] Stein | An Introduction to Seismology, Earthquakes, and Earth Structure[END_REF][START_REF] Fowler | The Solid Earth: An Introduction to Global Geophysics[END_REF].

Conservation of mass and momentum

We use the Lagrange approach to describe the motion of a particle in a closed volume V (see Figure 2.5) with surface S in the Cartesian coordinate system (x, y, z). We denote by u(x, t) the vector displacement of a particle from a reference position x 0 at time t 0 to the position x at time t. The particle velocity and acceleration are ∂ t u and ∂ 2 t u, respectively.

The law of conservation of mass M (t) on the closed volume V imposes that:

dM (t) dt = 0.
(2.13)

We can now apply both the principles of conservation of mass and conservation of momentum (Newton's second law of motion) to the particles constituting V :

V ρ∂ 2 t udV = V FdV + S TdS, (2.14)
where ρ is the mass density, F is the body force acting per unit volume on a particle, T is the force per unit area (traction) acting across an external surface S of V . Every component i of the traction T can be reformulated as:

T i = σ ij n j , (2.15)
where Einstein summation convention3 is implied. σ ij is the stress tensor and n j is the jth component of the normal vector n to the surface S.

Using Gauss's theorem (or divergence theorem), we can transform the surface integral in equation 2.14 to a volume integral as:

S σ ij n j dS = V ∂ j σ ij dV, (2.16)
where ∂ j is the partial derivate in the direction j.

For finite motions, we write the equation of conservation of the momentum in its local form as:

ρ∂ 2 t u i = F i + ∂ j σ ij (2.17)

Displacement-strain and stress-strain relation

The particle displacements can be linked to the strain tensor ij as follows:

ij = 1 2 ( ∂u i ∂x j + ∂u j ∂x i + ∂u k ∂u k ∂x i ∂x j ) (2.18)
Equation 2.18 defines a nonlinear relation between the strain tensor and displacement. However, seismic wave propagation generally generates small distortions of particles about their equilibrium positions. Hence, it is appropriate to assume small strains when deriving the elastic wave equation. Using

the assumption of small strains and displacements, we can neglect the second-order terms in equation 2.18. As a result, the relation between the displacement and strain fields becomes:

ij = 1 2 ( ∂u i ∂x j + ∂u j ∂x i ) (2.19)
The strain field can in turn be linked to the stress field σ(x, t) based on the constitutive rheological behavior of the material (e.g., elastic, elasto-plastic, and visco-elasto-plastic materials etc.). Following the generalized Hooke's law, the strain field is proportional to the stress field with the general fourthorder tensor of elastic constants C ijkl (x) as proportionality factor:

σ ij = C ijkl kl , i, j, k, l = 1, 2, 3, (2.20)

Theoretical background

The elastic constants and space-dependent density ρ(x) constitute the geophysical properties of an elastic Earth model. Due to the symmetry conditions of the elastic tensor C ijkl = C jikl = C klij and further thermodynamical arguments, it is possible to reduce this tensor to a 6 × 6 matrix. This leads to an elastic tensor with 21 independent elastic coefficients for the most general case of an anisotropic material:

           σ 1 σ 2 σ 3 σ 4 σ 5 σ 6            =            C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66                       1 2 3 4 5 6            (2.21)
where the numerical indices indicate the couples of spatial directions:

(1, 2, 3, 4, 5, 6

) t = (xx, yy, zz, yz, xz, xy) t (2.22)
Extra symmetries can be identified in different materials thereby reducing the number of independent coefficients of C ijkl .

For elastic isotropic materials, which are most generally observed, C ijkl is characterized by two Lamé coefficients; λ and µ (the shear modulus):

           σ xx σ yy σ zz σ yz σ xz σ xy            =            λ + 2µ λ λ 0 0 0 λ λ + 2µ λ 0 0 0 λ λ λ + 2µ 0 0 0 0 0 0 µ 0 0 0 0 0 0 µ 0 0 0 0 0 0 µ                       xx yy zz 2 yz 2 xz 2 xy           
(2.23)

The 3D elastic wave equation

The elastic wave equation for an isotropic medium is obtained by combining equations 2.17, 2.19, and 2.23 to form the following system:

ρ∂ 2 t u x = ∂ x σ xx + ∂ y σ xy + ∂ z σ xz + F x ρ∂ 2 t u y = ∂ x σ xy + ∂ y σ yy + ∂ z σ yz + F y ρ∂ 2 t u z = ∂ x σ xz + ∂ y σ yz + ∂ z σ zz + F z σ xx = (λ + 2µ)∂ x u x + λ∂ y u y + λ∂ z u z σ yy = λ∂ x u x + (λ + 2µ)∂ y u y + λ∂ z u z σ zz = λ∂ x u x + λ∂ y u y + (λ + 2µ)∂ z u z σ xy = µ(∂ y u x + ∂ x u y ) σ yz = µ(∂ z u y + ∂ y u z ) σ xz = µ(∂ z u x + ∂ x u z ), (2.24)
Theoretical background

We note that the displacement u i and stress σ ij fields have space-time dependencies, whereas the physical parameters (λ, µ, ρ) are only space dependent and invariant in time. The exterior force F i (x, t) that initiates the wave propagation are the seismic sources in our case, which can be characterized by the seismic moment tensor M ij (x, t).

The numerical code used for the seismic waves simulations in this thesis is based on the velocity-stress formulation. It is obtained by replacing the displacement field in equation 2.24 by its time derivative, the velocity field v(x, t) = ∂ t u(x, t), to obtain a first-order system of partial-differential equations of hyperbolic form [START_REF] Virieux | P-sv wave propagation in heterogeneous media, velocity-stress finite difference method[END_REF]:

∂ t v x = 1 ρ (∂ x σ xx + ∂ y σ xy ∂ z σ xz ) + f x ∂ t v y = 1 ρ (∂ x σ xy + ∂ y σ yy + ∂ z σ yz ) + f y ∂ t v z = 1 ρ (∂ x σ xz + ∂ y σ yz + ∂ z σ zz ) + f z ∂ t σ xx = (λ + 2µ)∂ x v x + λ∂ y v y + λ∂ z v z ∂ t σ yy = (λ)∂ x v x + (λ + 2µ)∂ y v y + λ∂ z v z ∂ t σ zz = λ∂ x v x + λ∂ y v y + (λ + 2µ)∂ z v z ∂ t σ xy = µ(∂ y v x + ∂ x v y ) ∂ t σ xz = µ(∂ z v y + ∂ y u z ) ∂ t σ xz = µ(∂ z v x + ∂ x v z ), (2.25)
Note that after converting the system into a velocity-stress formulation, the external forces f i become time derivatives of the volumetric forces F i in equation 2.24, divided by the mass density.

The fundamental solution to the equation system 2.25 shows that there are two types of body waves that propagate in infinite homogeneous elastic media. These waves are:

-Compressional P-waves (Figure 2.6a) with velocity V p = λ+2µ ρ . They are the fastest kind of seismic waves and, consequently, the first to arrive at a seismic station. P-wave can propagate through solids and fluids. They cause particles to move in the same direction as the propagating wave.

-Transversely polarized shear (S-) (Figure 2.6c) waves with velocity µ ρ . They are called transverse waves because they cause particles to move in directions perpendicular to the propagating wave's direction. S-waves are slower than P-waves and can only travel through solids.

The 2D approximation of the elastic wave propagation

The velocity-stress formulation of the elastic wave equation (equation 2.25) is expressed for 3D space. In this thesis, we perform 2D simulations of the wave equation. Hence we need a 2D approximation of the 3D elastic wave equation.

The 2D approximation is based on the hypothesis that the wave propagation along one infinite space direction (generally the y-direction in Earth science) is invariant. Indeed, a 2D approximation of wave propagation does not exactly describe the reality since it does not consider (1) the effects of the 3D medium on the wave propagation and (2) the real geometric attenuation of the wavefield. Nevertheless, a 2D approximation is generally made in seismology for simplicity and for saving computational cost without necessarily resulting in non-realistic wave propagation because [START_REF] Brossier | Imagerie sismique à deux dimensions des milieux visco-élastiques par inversion des formes d'ondes : développements méthodologiques et applications[END_REF]:

-Some media may exhibit a certain invariance in their physical properties along one direction of space, thereby partially validating one of the hypotheses of the 2D approximation.

-Solving the system of equations (equation 2.25) using numerical methods requires less computational demand when the number of space dimensions of the problem is reduced.

-The bias induced on the geometric attenuation of wavefield, due to the 2D approximation of the seismic source (a long linear infinite source), can be partially corrected in the simulated wavefield [START_REF] Bleinstein | Two-and-one-half dimensional in-plane wave-propagation[END_REF][START_REF] Williamson | A critical review of 2.5d acoustic wave modeling procedures[END_REF].

The 2D approximation in Earth science is usually obtained by suppressing the spatial derivatives along the y-direction in equation 2.25. This results in two independent systems for two types of wave propagation (P-SV and SH).

For the P-SV, also called in-inplane, wave propagation (compressional wave and shear wave polarized in the x-z plane), the system boils down to:

∂ t v x = 1 ρ (∂ x σ xx + ∂_yσ xy ) + f x ∂ t v z = 1 ρ (∂ x σ xz + ∂ z σ zz ) + f z ∂ t σ xx = (λ + 2µ)∂ x v x + λ∂ z v z ∂ t σ zz = λ∂ x v x + (λ + 2µ)∂ z v z ∂ t σ xz = µ(∂ z v x + ∂ x v z ), (2.26)
Whereas for the SH, also called out-of-plane, wave propagation (shear wave polarized in the ydirection), invariant in the y-direction and independent of the P-SV waves, the system writes as follows:

∂ t v y = 1 ρ (∂ x σ xy + ∂ z σ yz ) + f y ∂ t σ xy = µ(∂ x v y ) ∂ t σ yz = µ(∂ z v y ), (2.27)
This polarization of the seismic waves can lead to important site amplifications in sedimentary Theoretical background basins. [START_REF] Ktenidou | Directional dependence of site effects observed near a basin edge at Aegion, Greece[END_REF] analyzed site effects at the Aegion (Greece) basin using 520 weak motion earthquake records from a vertical array. They observed significant differences in the site amplification in the horizontal direction due to 2D effects. After rotating the components of the seismograms with respect to the basin edge orientation, the amplification n the direction parallel to the basin edge (SH motion) was about twice higher than in the perpendicular direction (SV motion).

It is also important to highlight the existence of surface waves, which may arise due to interference of the body waves and particular geometrical conditions -e.g., the presence of the free surface in a homogeneous elastic half-space. They are called surface waves because their energy is concentrated near the earth's surface. Two type of surface waves, Love waves [START_REF] Love | Theory of the propagation of seismic waves[END_REF] and Rayleigh waves [START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF] named after their discoverers, propagate near the earth's surface.

Rayleigh waves (see Figure 2.6d) are a result of the constructive interference between P and SV waves. Their fundamental mode propagates in an elliptical manner, with a retrograde movement of particles near the surface and a prograde movement at depth. Their depth sensitivity depends on the wavelength of propagation (hence the frequency), and they are therefore naturally dispersive waves when the medium becomes heterogeneous since the velocity of each frequency depends on the depth of the medium.

Love waves (see Figure 2.6b) are a result of the interference between shear waves (S-waves) guided by an elastic layer of velocity v 1 , which is welded to an elastic half-space of velocity v 2 > v 1 on one side while bordering a vacuum on the other side. Love waves require a velocity structure that varies with depth and so cannot exist in a half-space, in contrast to Rayleigh waves.

Numerical methods

The unknown field to be determined in the wave equation is either the displacement field u(x, t) or its time derivative the velocity field v(x, t) = ∂ t u(x, t). The 2D system of the elastic wave equation (just like the 3D system) can be solved analytically or pseudo-analytically under a certain number of simple medium configurations, e.g., in a homogeneous medium, in a half-space with free surface conditions [START_REF] Garvin | Exact transient solution of the buried line source problem[END_REF], in a layered medium [START_REF] Bouchon | A boundary integral equation -discrete wave number representation method to study wave propagation in multi-layered media having irregular interfaces[END_REF]. In heterogeneous media, with structural and rheological complexities, only approximate solutions can be obtained using efficient numerical methods.

Different numerical techniques can be used to model the propagation of seismic waves. Some examples of widely used numerical-modeling methods in seismology are the time-domain finite-difference (FD), finite-element (FEM), Fourier pseudospectral, spectral-element (SEM), and discontinuous Galerkin (DGM) methods. Each method has its own advantages and disadvantages depending on the application context. In fact, a key reason that often governs the choice of a particular method over another is the ability to solve the spatial partial derivatives in the wave equation in complex geometries without loss of precision.

As already stated, we use a numerical code based on the spectral element method to solve the elastic wave equation in this thesis. Hence, we will only present in detail the SEM. The other numerical methods shall only be briefly described.

The content in this section is mainly based on the textbooks by [START_REF] Igel | Computational Seismology A Practical Introduction[END_REF]; Renata and Valerie (2007).

The readers are referred to them for a detailed treatment of the numerical methods discussed here.

Theoretical background

Finite difference method

The FDM is with no doubt the simplest (both theoretical-wise and implementation-wise) amongst the grid numerical methods employed to solve the elastic wave equation. It relies on Taylor series to estimate the finite difference approximations of the partial derivatives between adjacent grid points.

The Taylor series expansion for a scalar quantity q about a point h on the x-axis writes:

q(x + h) = ∞ n=0 h n n! u (n) (x) (2.28)
where h is a parameter that is supposed to be small.

Using equation 2.28 to expanding the field u(x) on a 1D regularly spaced grid, with grid step dx, gives:

u(x + dx) = u(x) + u (x)dx + 1 2 u (x)dx 2 + O(dx 3 ) (2.29)
where O(dx 3 ) is a a quantity which leading term behaves like dx 3 when dx tends toward zero.

We can thus approximate the first derivative u (x) by :

u (x) ≈ u(x + dx) -u(x) dx , ( 2.30)
with an error, (x), of:

(x) = - dx 2 u (x) + O(dx 2 ) (2.31)
whose leading term varies linearly with dx.

Equation 2.30 is called a foward-difference approximation because it involves the values of u at the points x and x + dx, and is said to be of first order (in dx).

A higher-order approximation of the first derivative of u can be obtained by its Taylor series about the points x + dx and x -dx:

u(x + dx) = u(x) + dx u (x) + dx 2 2 u (x) + dx 3 6 (x) + O(dx 4 ) (2.32) u(x -dx) = u(x) -dx u (x) + dx 2 2 u (x) - dx 3 6 (x) + O(dx 4 ) (2.33)
Subtracting equations 2.32 and 2.33 give a central difference approximation of u (x) of order 2:

u (x) = u(x + dx) -u(x -dx) 2dx + O(dx 2 ) (2.34)
Similar reasoning can be made to approximate the higher derivates u. The reader interested in a detailed treatment of FDM is referred to the book chapter by [START_REF] Moczo | The finite-difference time-domain method for modeling of seismic wave propagation[END_REF].

The FDM is generally used to approximate the temporal derivatives in the elastic wave equation, no matter the numerical method used for the spatial derivatives. The FD approximation of the partial derivatives can result in either an explicit or an implicit scheme. In an explicit scheme, the wavefield at a time step t, at any spatial grid point, is computed using the wavefield at the previous time steps.

Whereas in an implicit scheme, the whole linear system must be solved at each time step. Explicit Theoretical background schemes are generally preferred to implicit schemes due to their low computational cost.

The FDM has widely been used in earth science for wave propagation simulations (e.g. [START_REF] Virieux | P-sv wave propagation in heterogeneous media, velocity-stress finite difference method[END_REF][START_REF] Moczo | The Finite-Difference Method for Seismologists An Introduction[END_REF][START_REF] Bohlen | Accuracy of heterogeneous staggered-grid finite-difference modeling of rayleigh waves[END_REF][START_REF] Moczo | The Finite-Difference Modeling of Earthquake Motions Waves and Ruptures[END_REF]. Despite its attractiveness and simplicity, it has some drawbacks which may render its use difficult under certain conditions. The use of regular grids in the FDM makes it arduous to apply for the spatial discretization of complex geometries. In fact, the numerical grid step in the FDM is defined based on the minimum wavelength (which depends on the maximum frequency and minimum propagation velocity) being modeled to avoid numerical dispersion. Hence, for a uniform grid, the spatial discretization is constrained by the medium's minimum velocity. This implies that for media with very low velocity (even if the low velocity is restricted to a small portion), the spatial discretization will lead to very fine meshes, thereby considerably increasing the computational cost. To remedy this problem, FD schemes have been developed for irregular grids (e.g. [START_REF] Moczo | Hybrid modeling of p-sv seismic motion at inhomogeneous viscoelastic topographic structures[END_REF].

Another limitation of the FDM is the representation of the free surface condition in the presence of complex topographies. Indeed, the Cartesian representation of the medium in the FDM approximates topographies using a series of stair-like elements, which may induce spurious diffractions in the wavefield if the grid is too large. This constraint, therefore, leads to the oversampling of the medium in order to properly represent the wavefield. As an example, it has been shown that (e.g. [START_REF] Saenger | Finite-difference modelling of viscoelastic and anisotropic wave propagation using the rotated staggered grid[END_REF][START_REF] Bohlen | Accuracy of heterogeneous staggered-grid finite-difference modeling of rayleigh waves[END_REF] a spatial sampling of at least 60 points per wavelength, in conjunction with a second-order numerical scheme, is required to precisely model Rayleigh waves in a Gaussian topography model. A sampling of 15 points per wavelength is required to model an S-wave in the same configuration, leading to an oversampling by a factor of 16 in 2D and 64 in 3D.

Many authors (e.g. Imperatori andMai, 2013, 2015;[START_REF] Iwaki | Effects of random 3D upper crustal heterogeneity on long-period ( 1s) ground-motion simulations[END_REF] use the FDM to study the effect of random velocity heterogeneities on ground motion simulations.

The Pseudospectral Method

Pseudospectral methods are commonly known as transform methods because their implementation is based on the Fourier transform. They solve partial differential equations in the Fourier domain, where the partial derivative operation is equivalent to a multiplication. When used to solve the wave equation, their attractive property is that the space derivatives can be calculated to at least machine precision. Compared to finite-difference schemes, the pseudospectral method based on Fourier transforms requires substantially less memory, particularly in 3D. This is because a small number of grid points is required due to the high accuracy of the derivative calculations. The drawback of using Fourier transforms is the implicit assumption of periodicity along the spatial dimensions. This implies that boundary conditions, such as the free surface condition, are difficult to implement efficiently. A fix to this problem was achieved using Chebyshev polynomials as the basis function for the interpolation [START_REF] Kosloff | Solution of the equations of dynamics elasticity by a chebyshev spectral method[END_REF]. This formulation allows for the efficient implementation of the free surface or absorbing boundaries through characteristic variables [START_REF] Carcione | A chebyshev collaction method for the wave equaiton in generalized coordinates[END_REF]. This comes with the drawback that the collocations points at which the functions are exactly interpolated are irregular and densifies towards the boundaries.

It is important to note that the implementation of the pseudospectral methods requires global communication; in other words, the future of a certain point in the grid depends on all the other points' current state. This makes it suboptimal for massively parallel computer architectures that favor minimal communications. However, its high accuracy of the spatial differentiation compared to other methods Theoretical background makes it very memory efficient. Readers who are interested in the detailed treatment and theory of pseudospectral methods are referred to the following articles/textbooks [START_REF] Fornberg | A Practival Guide to Pseudospectral Methods[END_REF]; [START_REF] Kosloff | Solution of the equations of dynamics elasticity by a chebyshev spectral method[END_REF]; [START_REF] Igel | Computational Seismology A Practical Introduction[END_REF] An example of applying the pseudospectral method in seismology can be found in the study by [START_REF] Furumura | Seismic wavefield calculation for laterally heterogeneous earth models-ii. the influence of upper mantle heterogeneity[END_REF]. They used the pseudospectral method to simulate seismic wavefields in a 2D cylindrical coordinate system to examine the effect of lateral heterogeneities on seismic wave propagation in the Earth's mantle.

Finite Element Method

The FEM solves the wave equation in its variational form (just as the SEM, which we will see later).

The solution is approximated on basis functions, which are generally polynomials, defined on the localized nodes of the elements which mesh the medium. The communication between the elements is assured by the common nodes which they share at their interfaces. As a result, the solution computed is continuous in space, and hence, they can be termed continuous FEM in contrast to the discontinuous methods such as the discontinuous Galerkin method. An interesting property of the FEM, which makes them useful for seismological applications, is that the free-surface boundary condition is naturally fulfilled. This makes the method useful to study surface wave propagation [START_REF] Lysmer | Evolving geometrical and material properties of fault zones in a damage rheology model[END_REF][START_REF] Schlue | Finite element matrices for seismic surface waves in three-dimensional structures[END_REF]. FEM is adequate for the study of wave propagation in complex geometries as they can adequately mesh complex topographies and geological structures. Nevertheless, they are less used for elastodynamics problems because of the large amount of numerical dispersion related to the loworder polynomial bases used [START_REF] Marfurt | Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations[END_REF], and the large computational efforts required to solve the resulting large linear systems.

An example of the application of FEM in seismology can be found in [START_REF] Bao | Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers[END_REF]. The authors used the FEM to simulate the seismic response of the San Fernando Valley in Southern California to an aftershock of the 1994 Northridge Earthquake. An advanced implementation of the FEM for large-scale earthquake ground motion simulation in realistic basins can be found in [START_REF] Bielak | Parallel octree-based finite element method for large-scale earthquake ground motion simulation[END_REF].

Discontinuous Galerkin Method

The DGM is a variant of the classical FEM in which there is no continuity between the mesh elements. It was introduced in the 70s to solve neutron transport equations [START_REF] Reed | Triangular mesh methods for the neuron transport equation[END_REF]. The first application to the elastic wave equation was published by [START_REF] Käser | An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes, I: the two-dimensional isotropic case with external source terms[END_REF] for the 2D case and later extended to 3D by [START_REF] Käser | An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes III: viscoelastic attenuation[END_REF]. The DGM can be developed on both triangular and tetrahedral meshes. Since the wavefield is computed in a discontinuous manner, the mesh elements communicate through numerical fluxes without necessarily sharing common nodes. This characteristic gives the DGM high flexibility in the choice of the interpolation method between neighboring elements and the possibility to use non-conforming meshes (e.g. [START_REF] Fah | Numerical evaluation of a non-conforming discontinuous galerkin method on triangular meshes for solving the time-domain maxwell equations on non-conforming meshes[END_REF][START_REF] Fah | Convergence and stability of a high-order leap-frop based discontinuous galerkin method for the maxwell equations on non-conforming meshes[END_REF]. The geometrical flexibility of the discontinuous Galerkin method is an attractive feature for kinematic and dynamic rupture simulation problems. For elastic wave-propagation problems without strong geometrical complexity or material heterogeneities, the DGM is most likely not the method of choice, as FDM or SEM provide more efficient solutions. However, for dynamic rupture problems, the DGM is currently the most accurate solver, in particular, for complicated fault models.

An example of the application of the DGM in seismology can be found in [START_REF] Peyrusse | A nodal discontinuous galerkin method for site effect assessment in viscoelastic media -verification and validation in the nice basin[END_REF] A detailed treatment of the DGM for solving the elastic wave equation can be found in Igel (2017).

The spectral element method

The numerical code used in this thesis to perform elastic wave propagation simulations is based on the spectral element method (SEM). The SEM is generally considered as a higher-order classical finiteelement method due to their similarity with the concept of basis functions. The SEM was developed to take advantage of the exponential convergence properties4 of the spectral basis functions. The term spectral element was coined in relation to pseudospectral methods, which use the concepts of exact interpolation on collocation points with spectral convergence properties. The method was first introduced in computational fluid dynamics [START_REF] Patera | A spectral element method for fluid dynamics: laminar flow in a channel expansion[END_REF]. The spectral element formulations for elastic wave problems were first published by (e.g. [START_REF] Priolo | Numerical simulation of interface waves by high-order spectral modeling techniques[END_REF][START_REF] Seriani | Spectral element method for acoustic wave simulation in heterogeneous media[END_REF][START_REF] Facioli | Spectral-domain decomposition methods for the solution of acoustic and elastic wave equations[END_REF], in which they used Chebyshev polynomials to approximate the unknown fields. The breakthrough in seismology came with the work by [START_REF] Komatitsch | The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF], which introduced the combination of Lagrange polynomials as interpolants and an integration scheme based on Gauss quadrature defined on the Gauss-Lobatto-Legendre (GLL) points for the elastic wave equation. This led to a diagonal mass matrix that can be trivially inverted. Since then, the SEM is currently one of the most widely used numerical approaches for seismic wave-propagation problems. Various implementations of SEM codes have been developed, and are mostly open-source. [START_REF] Fichtner | Efficient numerical surface wave propagation through the optimization of discrete crustal models ---a technique based on non-linear dispersion curve matching (DCM)[END_REF] implemented a spectralelement method for wave propagation in spherical coordinates (3D spherical sections) that led to the first-ever application of the adjoint inversion method to regional earthquake data.

In what follows, I will provide a walk through the theory of the method to solve the elastic wave equation (equation 2.25). The details of the method can be found in numerous scientific articles (e.g. [START_REF] Komatitsch | The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF][START_REF] Komatitsch | Introduction to the spectral element method for three-dimensional seismic wave propagation[END_REF][START_REF] Chaljub | Solving elastodynamics in a solid heterogeneous 3D-sphere: a spectral element approximation on geometrically non-conforming grids[END_REF] or in textbooks on computational seismology (e.g. [START_REF] Igel | Computational Seismology A Practical Introduction[END_REF][START_REF] Chaljub | Spectral-element analysis in seismology[END_REF], in PhD thesis dissertations that focused on seismic wave propagation modeling (e.g. [START_REF] Komatitsch | Méthodes spectrales et éléments spectraux pour l'équation de l'élastodynamique 2D et 3D en milieu hétérogène[END_REF][START_REF] Chaljub | Modèlisation numérique de la propagation d'ondes sismiques à l'échelle du globe[END_REF][START_REF] Ampuero | Etude physique et numérique de la nucléation des séismes[END_REF][START_REF] Delavaud | Simulation numérique de la propagation d'ondes en milieu géologique complexe : application à l'évaluation de la réponse sismique du bassin de Caracas (Venezuela)[END_REF].

Theoretical formulation of the SEM

The setup of the problem is as shown in Figure 2.7 where we seek to determine the displacement field u (or its time derivative the velocity v) produced by an earthquake in a finite earth model with volume Ω ⊂ d (where is the set of real numbers and d is the dimension of the problem). The boundaries of the volume include a stress-free surface δΩ and artificial absorbing boundaries Γ. In an ideal situation, the free surface will reflect the seismic waves, whereas the artificial absorbing boundaries Γ completely absorb them. The earth model may contain any number of internal discontinuities as shown by the sedimentary layers at the site location in Figure 2.7. A point in the model is denoted by a position vector x = (x, y, z) in 3D. The solution u : Ω x I → d is then going to obtained at discrete positions x j for a given times t ∈ I (where I = [0, T ] ∈ + is the time domain). For simplicity, we shall derive SEM formulation for a simple 1D problem as it allows to fully detail the method. Nevertheless, an extension to higher dimensions is straightforward since it is based on tensorization (separation of variables).

Before we dive into the theory of SEM, it is important to introduce the concept of strong and weak formulation of the wave equation. The strong formulation works directly with the equations of motions

ρ∂ 2 t u = ∂ z (µ∂ z u) + f (2.35)
in which µ is the space-dependent shear modulus, and z is the depth variable. In 1D, the computational domain Ω boils down to: Ω = [0, -L] and we assume here that it is just a 1D soil column, where z = 0 is the free surface and z = -L is soil depth. An important boundary property that has to be obeyed is the stress-free condition that occurs at the Earth's surface. This condition is expressed such that the traction vector t(z, t) = µ(z)∂ z (z, t) must vanish at the surface (z = 0) and for all times.

σ ij n j = 0 (2.36)
where n j is the normal vector at the free surface. At the bottom boundary, we assume that an absorbing boundary condition holds:

t(z = -L, t) = t abs (z -L, t) (2.37)
We will present the different boundary conditions we use in this thesis for a 2D application in the subsequent sections.

The weak (or variational) form is obtained by dotting equation 2.35 with an arbitrary displacement field, w, then integrating over the computational domain Ω:

5. The test function in our case is an arbitrary displacement field, w, which must be continuous throughout the volume.

Theoretical background

Ω w ρ ∂ 2 t u dz = Ω w f dz - Ω w ∂ z (µ∂ z u) dz (2.38)
Integrating by parts the second term on the right-hand side of the above equation gives

Ω w ρ ∂ 2 t u dz + Ω µ ∂ z w ∂ z u dz - ∂Ω µ ∂ z u w dz = Ω w f dz (2.39)
In equation 2.39, the term on the boundary ∂Ω involves the traction vector t(z, t)

∂Ω µ ∂ z u w dz = t(0, t)w(0) -t(-L, t)w(-H) (2.40)
To impose the free-surface boundary condition, we simply cancel the surface traction term in equation 2.40. The implicit fulfillment of the free-surface boundary condition is an extremely attractive feature for 3D problems involving the Earth's surface. We also replace the traction at the bottom boundary using the absorbing condition in equation 2.37.

The weak form or variational form of the elastic wave equation then writes as follows

Ω w ρ ∂ 2 t u dz + Ω µ ∂ z w ∂ z u dz + t abs (-, t)w(-L) = Ω w f dz (2.41)
Note that the variational formulation requires the introduction of the space of all admissible displacements (or velocities), S t :

S t := {u(z, t) : Ω x I | u ∈ H 1 (Ω) d ∀t ∈ I} (2.42)
where H 1 (Ω) is the Sobolev vector space which defines the set of functions in Ω that are together with their first derivative, square integrable. The associated space of admissible displacements δS at a time t is defined as:

δS := {w(z) : Ω → d | w ∈ H 1 (Ω) d } (2.43)
As such, in the variational formulation (equation 2.41), the velocity-displacement couple (u, v) is searched in S t x S t , ∀w ∈ δS, ∀t ∈ I. Now that we have the weak formulation of the elastic wave equation, we are left with the problem of finding solutions of the displacement field u for arbitrary space-dependent test functions w.

At this point, we are still in the continuous world in which solutions are sought by analytical means.

Since we seek to simulate wave propagation in Earth models with heterogeneous distributions of elastic parameters, we need to find appropriate discrete representations of the seismic wavefield u with which we can find solutions by numerical means. This can be achieved by the Galerkin method where the exact solution u(z, t) is approximated by a finite superposition of n basis functions φ i (z) with i = 1, ..., N p weighted by time-dependent coefficients u i (t). The approximate displacement field is denoted by ũ(z, t):

u(z, t) ≈ ũ(z, t) = Np i=1 u i (t)φ i (z) (2.44)
The accuracy of this approximation will depend on the specific choice of basis function and the number of functions superimposed (i.e N p ). We now apply the Galerkin principle i.e. we use the same functions Theoretical background (φ i ) that are used to approximate our unknown fields as our test functions w in equation 2.41:

Ω φ i ρ ∂ 2 t u dz + Ω µ ∂ z φ i ∂ z u dz + t abs (-, t)w(-L) = Ω φ i f dz (2.45)
with the requirement that the medium is at rest at t = 0. Combing equations 2.44 and 2.45 leads to an equation for the unknown coefficients u i (t), where we dropped the absorption term for simplicity in the notation:

Np i=1 ∂ 2 t u i (t) Ω ρ(z)φ j (z)φ i (z)dz + Np i=1 u i (t) Ω µ(z) ∂ z φ j (z) ∂ z φ i (z)dz = Ω φ i f (z, t) dz (2.46)
for all basis functions φ i with j = 1, ..., n. Equation 2.46 is the well-known equation for finite-element problems, and can be written in matrix notation as:

Mδ 2 t u(t) + Ku(t) = f (t), (2.47)
in which implicit matrix-vector operations are assumed.

The mass matrix M is defined as:

M ji = Ω ρ(z) φ j (z) φ i (z) dz, (2.48) the stiffness matrix K is K ji = Ω µ(z) ∂ z φ j (z) ∂ z φ i (z) dz, (2.49)
and the vector containing the volumetric forces f

f j (t) = Ω φ i (z)f (z, t)dz.
(2.50)

All the mathematical developments done till now are akin to the classic finite-element approach.

The problems that remain to be solved are finding appropriate basis functions and integration schemes to efficiently calculate the mass matrix, stiffness matrix, and forces. This is where the particularity of the SEM will come into action.

This matrix equation (equation 2.48) has to be solved for the space-independent but time-dependent coefficients u. The vector of coefficients will take the meaning of the actual displacement values at a global set of points imposed by the specific basis functions to be introduced shortly. The mass matrix in the SEM has the remarkable property of being diagonal by construction (due to the choice of the basis functions and integration rule) as we will see in what follows; thus, its inversion is trivial. The stiffness matrix has a banded structure, in this case, with the bandwidth depending on the number of basis functions that are required inside each element.

The elastic wave equation solution given in equation 2.46 is of global nature where u is the displace-Theoretical background ment field in the complete physical domain. We first need to split the computational domain Ω into elements (which need not be of the same size), that is, generate a mesh of the domain. The meshing process is often referred to, in the finite element literature, as the triangulation phase. We write this formally as: (2.51) where the Ω e are the elements and n e is the number of elements. In the classical SEM, the elements are restricted to quadrangles in 2D and hexahedra in 3D.

Ω = ne e=1 Ω e ,
An example of a 2D spectral element of the Nice sedimentary basin is shown in Figure 2.8. We generated the mesh using the Cubit Trelis mesh generation software [START_REF] Trelis | american fork, ut: csimsoft[END_REF]. This mesh shall be used the subsequent chapters for our 2D application of SEM.

Figure 2.8 -2D spectral element mesh of the Nice sedimentary basin (green area) generated using the Cubit Trelis software [START_REF] Trelis | american fork, ut: csimsoft[END_REF] overly a bedrock (brown zone). The mesh is made of quadrangles that are deformed to follow the geometry of the layers, the surface topography and the basin-bedrock interface.

Dividing the domain Ω into subdomains Ω e allows for the introduction of discontinuities in material parameters, leading to discontinuity of the displacement gradient ∇u. Combining equation 2.51 into equation 2.46 leads to:

Np i=1 ∂ 2 t u i (t) ne e=1 Ωe ρ(z)φ j (z)φ i (z)dz + Np i=1 u i (t) ne e=1 Ωe µ(z) ∂ z φ j (z) ∂ z φ i (z)dz (2.52) = ne e=1 Ωe φ i (z) f (z, t) dz
representing a linear system of N p equations for each j. We note that the coefficients u i depend on a sum over all elements. The matrix notation of algebraic system on the subdomains Ω e then becomes:

M e δ 2 t u(t) + K e u e (t) = f e (t), e = 1, ..., n e (2.55)
To allow for distorted elements in the mesh and to facilitate mathematical operations under the integrals, each element in the mesh is define from a reference undeformed element (a square in 2D and a cube in 3D). We formally write the mapping as:

Ω e = F e ( ) (2.56)
where is the reference element, and F e is the mapping function which can be defined analytically in some cases, but most of the time it is defined as a polynomial.

The principle of the mapping is to choose a number of control nodes, also called anchor nodes, on the reference element and define how they are transported onto the elements. Figure 2.9 shows an example of a 2D mapping on the reference square based upon 9 control or anchor points.

Theoretical background

For the specific choice of the basis function, we use the reference interval [-1,1]. To integrate a function over the reference interval [-1,1], we have to apply the mapping from the global system z ∈ Ω to the local coordinates which we denote ξ ∈ F e . This transformation can be written as:

F e : [-1, 1] → Ω e , z = F e (ξ), ξ = ξ(z) = F e-1 (z), e = 1, ..., n e (2.57)
The physical coordinate z can thus be related to the local coordinate ξ ∈ [-1, 1] as:

z(ξ) = F e (ξ) = h e (ξ + 1) 2 + z e (2.58)
where z e is the coordinate at the top of the element and h e is the element size. The length h e may vary for each element, allowing the adaptation of computational meshes. The inverse mapping is given as

ξ(x) = 2 (z -z e ) h e -1 (2.59)
Every integral over Ω e will therefore require a change of coordinate from z → ξ:

Ωe f (z)dz = 1 -1 f (ξ) dz dξ dξ (2.60)
where the integrand has to be multiplied by the Jacobian J defined as

J = dz dξ = h e 2 (2.61)
Its inverse J -1 = dx dξ = he 2 is also required when the derivatives of the basis functions need to be integrated.

With the mapping being defined, we can assemble our system of equations inside each element as:

Np i=1 ∂ 2 t u e i (t) 1 -1 ρ[z(ξ)]φ e j [z(ξ)]φ e i [z(ξ)] dz dξ dξ + Np i=1 u e i (t) 1 -1 µ[z(ξ)] ∂ ξ φ e j [z(ξ)] ∂ ξ φ e i [z(ξ)] dz dξ 2 dz dξ dξ (2.62) = 1 -1 φ e i [z(ξ)] f [z(ξ), t] dz dξ dξ
This is a system of N p equations for each index j corresponding to one particular basis function for which the wave equation needs to hold.

We now need to find a choice of the basis functions φ i and a numerical integration scheme to calculate the integrals on the reference element.

q(ξ)dξ (2.63)
where = [-1, 1] is the reference interval and q is a scalar quantity. A quadrature rule provides the following approximation for an integral:

q(ξ)dξ ≈ N i=0 ω i q(ξ i ) (2.64)
where the N + 1 points ξ i are called the integration points and the N + 1 scalars ω i are called the integration weights. This kind of formula is in general not exact, but when q is a polynomial it is easy to find exact formulas. For example if q is a polynomial of order N, then any set of N + 1 distinct points provides an exact integration formula. In the Legendre SEM 6 , we use the Gauss-Lobatto-Legendre (GLL) integration rule. When based upon N + 1 integration points, it provides a formula which is exact for polynomials of order up to 2N -1. The GLL points are defined as follows:

i = 0 : ξ 0 = -1 i = 1...N -1 : L N (ξ i ) = 0, where L N is a Legendre polynomial of order N i = N : ξ N = 1 (2.65)
Figure 2.10 shows the GLL points for different polynomial orders. Note that the extremities of the reference segment (i.e. -1 and 1) are always part of the GLL points, and that there is a clustering of points near the edges as the polynomial order increases. It can be shown that the spacing between the points behaves like 1/N in the middle of the reference element, and like 1/N 2 near the edges. Now that we have chosen a quadrature rule, we still need to define the polynomial approximation (basis functions) of the elastic parameters and of the unknown displacement field in equation 2.44. In the Legendre SEM, we use the GLL integration points to define the polynomial basis on the reference element and we choose the Lagrange polynomial as interpolating functions. If we Let (ξ i ) i=0,N be the N + 1 GLL integration points, their Lagrange interpolants are defined as the N + 1 polynomials l i of 6. We precise Legendre SEM to contrast with the Chebychev SEM which is based on the Gauss-Lobatto-Chebychev formula . The corresponding N + 1 = 5 GLL points can be seen along the horizontal axis. All Lagrange polynomials are, by definition, equal to 1 or 0 at each of these points. Note that the first and last points are exactly -1 and 1 (taken from [START_REF] Chaljub | Spectral-element analysis in seismology[END_REF] degree N such that:

l (N ) i (ξ j ) = δ ij (2.66)
where δ ij is the Kronecker index, which is 1 if i = j and 0 otherwise.

The Lagrange interpolants l j are defined as follows:

φ i = l (N ) i (ξ) := N +1 j =i ξ -ξ j ξ i -ξ j , i, j = 1, 2, ..., N + 1 (2.67)
where ξ i are fixed points in the interval [-1,1]. They are represented for N = 4 on Figure 2.11.

Putting the expression of the Lagrange polynomials (equation 2.67) in the general finite-element system (equation 2.62) gives:

N +1 i=1 ∂ 2 t u e i (t) 1 -1 ρ(ξ) l j (ξ) l i (ξ) dz dξ + N +1 i=1 u e i (t) 1 -1 µ(ξ) ∂ ξ l j (ξ) ∂ ξ l i (ξ) dz dξ 2 dz dξ dξ (2.68) = 1 -1 l j (ξ) f (ξ, t) dz dξ dξ
in which the following mappings for density ρ, elastic constant µ, and forces f were used to simply the notation:

ρ(ξ) := ρ[z(ξ)], µ(ξ) := µ[z(ξ)], f (ξ) := f [z(ξ)].
(2.69)

We now use the GLL integration scheme (equation 2.64) to evaluate the integrals at the same points Theoretical background at which we interpolate our unknown function u.

N +1 i,k=1 ∂ 2 t u e i (t) w k ρ(ξ) l j (ξ) l i (ξ) dz dξ | ξ=ξ k + N +1 i,k=1 u e i (t) w k µ(ξ) ∂ ξ l j (ξ) ∂ ξ l i (ξ) dz dξ 2 dz dξ dξ| ξ=ξ k (2.70) ≈ N +1 k=1 w k l j (ξ) f (ξ, t) dz dξ dξ| ξ=ξ k
This is a key step in the SEM as it enables us to make use of the orthogonality of the Lagrange polynomial associated with GLL integration points, hence resulting in a diagonal mass matrix. Equation 2.70 provides the solutions of the unknowns inside one element Ω e . All we need to do is to assemble the solutions of the elements to make up the global solution over the entire domain Ω.

The decomposition of Ω into Ω e required continuity of solution fields at the element boundaries (or collocation points). Hence to build a continuous global solution, we need to first count the degrees of freedom in our grid of points. There are n e elements with N + 1 collocation points in each element. The total number of grid points is thus n e x (N + 1), but because of the choice of the GLL points, we see that the points which belong to two elements are counted twice. The total number of degrees of freedom N is thus only n e x N . The basis of admissible displacements is of size N because there can be only one value at the points belonging to several elements for the sake of continuity. The final global solution at a given time t is thus the sum over the total number of degrees of freedom N :

N k=1 M jk ∂ 2 t u(z k , t) + N k=1 K jk u(z k , t) = N k=1 f jk (t) (2.71)
Rewriting equation 2.71 in matrix notation to form the global algebraic system, we obtain:

M g ∂ 2 t u g (t) + K g u g (t) = f g (t)
(2.72)

Time Marching and stability

The algebraic system (equation 2.72) now needs to be solved for each time step ∆t. As stated before, the time-dependent coefficients u g can be calculated for the next time steps using an implicit or explicit finite different scheme. A simple centered finite-difference scheme with the following mapping [START_REF] Igel | Computational Seismology A Practical Introduction[END_REF]:

u g new → u g (t + ∆t) u g → u g (t)
(2.73)

u g old → u g (t -∆t)
leads to the solution for the coefficient vector u g (t + ∆t) for the next time step:

Theoretical background

u g (t + ∆t) = ∆t 2 M -1 g (f (t) -K g u g (t) + 2u g (t) -u g (t -∆t) (2.74)
The system has to be solved in the time interval I = [0, T ], for every time step dt such that t n = n∆t, ∀n ∈ N.

To ensure the stability of time-marching solver, the maximal time step is imposed by the Courant-Friedrichs-Lewy (CFL) [START_REF] Courant | Über die partiellen differenzengleichungen der mathematischen physik[END_REF] criterion which states that the speed at which information travels cannot exceed one grid cell per time step. This implies:

∆t ≤ C ∆x min v max (2.75)
where ∆x is the minimum distance between adjacent grid points, v max is the maximum velocity of the modeled media, and C is a constant (typically ranging between 0.3 and 0.5) that depends on the geometry of the mesh and on the spatial dimension of the problem.

To avoid numerical dispersion in our application of SEM, we consider C to be 0.3. The element size d is chosen such that d ≤ λ min N/ppw where λ min is the shortest wavelength propagation in the medium, N is the polynomial degree, and ppw is the number of grid points per wavelength, which is chosen to be 5 [START_REF] Priolo | Spectral element method with substructuring: an accurate and efficient highorder finite element approach for wave modeling[END_REF].

Numerical code and boundary conditions

To study 2D wave propagation in linear media, we use the version of the 2D SEM code SEM2DPACK by [START_REF]Multi-dimensional modeling of seismic wave propagation in linear and nonlinear media[END_REF]. The code is based on the 2D velocity-stress formulation [START_REF] Festa | The newmark scheme as velocity-stress timestaggering: an efficient pml implementation for spectral element simulations of elastodynamics[END_REF] of the elastic wave equation. SEM2DPACK was initially developed by [START_REF] Ampuero | Etude physique et numérique de la nucléation des séismes[END_REF], and it models P-SV and SH wave propagation. [START_REF]Multi-dimensional modeling of seismic wave propagation in linear and nonlinear media[END_REF] further implemented additional boundary conditions and soil constitutive models to the code.

One of the soil constitutive model implemented by [START_REF]Multi-dimensional modeling of seismic wave propagation in linear and nonlinear media[END_REF] we use in this thesis is the viscoelasticity model, which enables considering intrinsic attenuation. It is implemented based on the rheological model of the Generalized Maxwell body [START_REF] Emmerich | Incorporation of attenuation into time-domain computations of seismic wave fields[END_REF]. Intrinsic attenuation is then quantified by introducing the quality factor Q parameter, which is a function of the frequency-depend viscoelastic modulus of the material. The Q parameter implemented in the code by using a set of discrete relaxation functions following the procedure of [START_REF] Liu | Efficient modeling of q for 3d numerical simulation of wave propagation[END_REF]. The behavior of these relaxation functions is controlled by the selection of the weight coefficients and relaxation times. In [START_REF] Liu | Efficient modeling of q for 3d numerical simulation of wave propagation[END_REF], the authors define and optimize two sets of weight coefficients and one set of relaxation times to model a constant Q for a sufficiently large range (5 < Q < 5000) in the frequency band 0.01 -50 Hz.

Different boundary conditions are also implemented in SEM2DPACK and need to be specified when creating the numerical domain. For applications that include 2D wave propagation in sedimentary basins, as is our case, we need to define boundary conditions at the free surface, lateral boundaries, and the bottom of the model. The free surface condition that imposes vanishing traction (Neumann condition) is implicitly fulfilled when deriving the SEM.

The second boundary conditions we use in this thesis are the periodic boundary condition. We apply them to the lateral boundaries in our numerical simulations. They enable the creation of an infinite horizontal domain by forcing the left and right boundaries to have identical velocity wavefields.

Theoretical background

The last boundary condition applied in this thesis is the Classical Perfectly Matched Layer (C-PML) [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF], which we apply at the bottom of the computational domain. C-PML is an absorbing layer in which all incoming energy is artificially attenuated to form an infinite domain. They are used with the Dirichlet boundary, where a null displacement is imposed at the absorbing layer's bottom edge. C-PML is implemented as an additional domain in the 2D code we use. Details about their implementation can be found in Oral (2016).

Modeling and characterizing soil heterogeneities

This thesis's main objective is to understand the influence of the spatial variability of linear soil properties on seismic ground motion. By spatial variability of soil properties, we understand the variation in space and time of the soil's mechanical parameter. The term spatial variability of soil properties is generally confused with the spatial variability of ground motion, which is a consequence of soil properties' spatial variability.

It would be synthetically impossible to provide an exhaustive review on the treatment of soil heterogeneities within the scope of this thesis. As a result, in what follows, we only provide a succinct review on the treatment of soil heterogeneities mainly from a seismological and geotechnical perspective.

The rationale is to provide the necessary theory required to understand the techniques (and reason for their choice), which we use to model and characterize the soil heterogeneities in the upcoming chapters. This section is organized as follows:

-First, the evidence of soil heterogeneity and its effects on seismic wave propagation is presented.

-We then discuss the different methods used to model soil heterogeneity, focusing on the modeling of soil heterogeneity as spatially correlated random fields.

-Finally, we describe how the nature and characteristics of the soil heterogeneity can be quantified from available data.

Evidence of soil heterogeneity

Soils, like other naturally occurring Earth media, are variable or heterogeneous because of the way they are formed and the continuous processes of the environment that alter them. Their variability is well recognized in various disciplines, e.g., hydrogeology, geology, geotechnical engineering, geophysics, seismology, petroleum engineering, geostatistics, etc. What generally differs in the treatment of soil heterogeneity between the different disciplines is the soil's mechanical or physical parameter, which is considered to be heterogeneous. For example, in hydrogeology, the heterogeneous parameter is generally the hydraulic conductivity of the soil. In geotechnical engineering, the parameters of interest can be the undrained shear strength, velocity (e.g. [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]. In geophysics and seismology, where the focus is mainly on wave propagation problems, the variable parameters are generally considered to be the wave velocity, density, and attenuation (e.g. [START_REF] Frankel | Finite difference simulation of seismic scattering: implications for the propagation of short-period seismic waves in the crust and models in crustal heterogeneity[END_REF][START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF][START_REF] Plonka | The imprint of crustal density heterogeneities on regional seismic wave propagation[END_REF][START_REF] Nakata | Stochastic characterization of mesoscale seismic velocity heterogeneity in long beach, california[END_REF]. In the remainder of this dissertation, we shall focus principally on velocity heterogeneity and its effect on seismic wave propagation. As a result, the term soil heterogeneity shall be used interchangeably with velocity heterogeneity in what follows.

Direct evidence of velocity heterogeneity can be found in log data from wells or boreholes. As an example, Figure 2.12 [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF] shows wave velocity and density log data from well YT2 drilled through lava, tuff, and volcanic breccia in Kyushu, Japan. The velocity structure in the log was determined from the travel times of ultrasonic waves having frequencies of a few tens of kHz. The rock Some first observations of such variations in the ground motion recorded at sites with small separation distances was reported by [START_REF] Steidl | Variation of Site Response at the UCSB Dense Array of Portable Accelerometers[END_REF]. In their study, they deployed ten accelerometers in two dense arrays following the Landers-Big Bear earthquake sequence. One array was locate on a shallow alluvial soil site. They observed large variations in the ground motion recorded at the stations of the array with separations as small as 80 m. Over the last decade, such evidence have become easily observable due to an increase in the amount of seismic networks, hence an increase in the number of ground motion records.

Effects of soil heterogeneity

The effects of soil heterogeneity on any physical phenomena depend on the domain under consideration. For example, in hydrogeology, the small-scale variations in hydraulic conductivity control the rate of solute spreading or dispersion in the aquifer. In geotechnical engineering, soil heterogeneity may impact engineering-scale systems and therefore needs to be correctly characterized and incorporated in reliability-based geotechnical design approaches.

In seismology, velocity heterogeneity is well known to scatter the propagating waves in the Earth's crust. It is the pioneering work of [START_REF] Aki | Analysis of seismic coda of local earthquakes as scattered waves[END_REF] who recognized the coda 7 waves on seismograms as backscattered seismic energy from uniformly distributed scatterers 8 in the Earth's crust. Since then, 7. Coda waves constitute the late arrivals on a seismogram which arrive after the body and surface waves. They are traditionally defined as the tail of a seismogram.

8. Scatterers here denote the velocity heterogeneity. several theories have been developed to explain the nature of such waves, including single scattering and multiple scattering theories (e.g. [START_REF] Aki | Origin of coda waves: source, attenuation, and scattering effects[END_REF][START_REF] Sato | Single isotropic scatteringmodel includingwave conversions simple theoretical model of the short period body wave propagation[END_REF] and energy transport theory [START_REF] Wu | Multiple-scattering and energy-transfer of seismic-waves separation of scattering effect from intrinsic attenuation[END_REF]. Other complex features of the scattered seismic waves have been noticed on seismograms.

These features include: the broadening of P-and S-wave pulses (e.g. [START_REF] Sato | Broadening of seismogram envelopes in the randomly inhomogeneous lithosphere based on the parabolic approximation: southeastern Honshu, Japan[END_REF][START_REF] Saito | Unified explanation of envelope broadening and maximumamplitude decay of high-frequency seismograms based on the envelope simulation using theMarkov approximation: forearc side of the volcanic front in northeastern Honshu[END_REF][START_REF] Takahashi | Strong inhomogeneity beneath Quaternary volcanoes revealed from the peak delay analysis of S-wave seismograms of microearthquakes in northeastern Japan[END_REF], the distortion of the apparent S-wave radiation pattern from the original four-lobe source radiation pattern (e.g. [START_REF] Takemura | Distortion of the apparent S-wave radiation pattern in the high-frequency wavefield: Tottori-ken Seibu, Japan, earthquake of 2000[END_REF][START_REF] Sawazaki | Envelope synthesis of shortperiod seismograms in 3-D random media for a point shear dislocation source based on forward scattering approximation: application to small strike-slip earthquakes in southwestern Japan[END_REF] and the appearance of P-wave energy in transverse motion (e.g. [START_REF] Nishimura | Heterogeneity of the Japan islands as inferred from transverse component analysis of teleseismic P-waves observed at a seismic station network, Hi-net[END_REF][START_REF] Takemura | Scattering of high-frequency P wavefield derived from by the dense Hi-net array observations in Japan and computer simulations of seismic wave propagations[END_REF].

In seismic hazard assessment, they are well known to result in the spatial variability of surface ground motion (e.g. [START_REF] Imperatori | The role of topography and lateral velocity heterogeneities on near-source scattering and ground-motion variability[END_REF][START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF]. There is currently still a knowledge gap between our physical understanding of small-scale soil heterogeneity and how their variation impacts the ground motion indicators at the sedimentary scale for seismic hazard assessment.

Modeling soil heterogeneity

Soil heterogeneities can be modeled using both traditional statistical methods and spatial correlation analysis.

Traditional statistical methods combine probability and statistics to describe and infer uncertainty in soil data. In contrast, spatial correlation analysis is used to describe the nature and characteristics of the variation of soil heterogeneities in space.

Traditional statistical methods

Traditional statistical methods use the concept of random variables to model soil heterogeneities.

The behavior of the random variable can then be modeled using probability and statistical theory.

Statistical theory encompasses both descriptive and inferential analysis:

Descriptive analysis uses sample statistics to visually inspect and calculate the sample moments of a data set. The use of sample statistics results from the fact that quantitative geotechnical variability investigation relies on sets of measured data which are generally limited in size [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]. For practical considerations, the first four statistical moments of a sample data set, i.e., the mean, variance, Theoretical background skewness, and kurtosis are generally calculated to describe the data [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]. In thesis we also use percentiles to describe the dispersion of a set of stochastic simulations.

The sample mean m φ of a sample set of a random variable φ is given by:

m φ = 1 n n i=1 φ i (2.76)
Its sample variance is given by :

V φ = 1 n -1 n i=1 (φ i -m φ ) 2
(2.77)

The sample standard diviation of the dataset can be otained by taking the square root of the sample variance.

The unbiased estimates of the skewness C sk and kurtosis C ku of the sample set are given by:

C sk = n (n -1)(n -2) n i=1 φ i -m φ V 2 φ 3
(2.78)

C ku =    n(n + 1) (n -1)(n -2)(n -3) n i=1 φ i -m φ V 2 φ 4    - 3(n -1) 2 (n -2)(n -3) (2.79)
The percentile of a dataset simply tells how a value compares to other values. The general rule to calculate the percentile is that if a value X is at the kth percentile, then X is greater than K% of the values in the dataset. The interval between two specified percentile points is the inter-percentile range, which is a stable measure of the spread of the dataset. In the study, the spread of the stochastic simulations shall be estimated by the inner 68th percentile i.e., the difference between the 84th and 16th percentile.

Inferential analysis draws inference on the random variable (target set) based on data from another dataset (source set). It consists of selecting a distribution type for the random variable, estimating the distribution parameters, and calculating the fit between the resulting distribution and the source data.

The probability density function (PDF) is used to specify the probability of a random variable to every interval of the possible values taken by the random variable of interest. It can either be discrete or continuous, depending on whether the random variable takes values from a finite or continuous set.

The continuous PDF of a random variable φ denoted by f φ has the following properties:

f φ (x) ≥ 0 (2.80) +∞ -∞ f φ (x)dx = 1 (2.81) P [a ≤ x ≤ b] = b a f φ (x)dx (2.82)
where x are the values of the random variable.

Several PDFs have been proposed in geotechnical literature, e.g., uniform distribution, triangular distribution, normal distribution, log-normal distribution, beta distribution, etc. There is no generic distribution which is best suited for soil properties, but certain distributions like the normal or lognormal distribution are more frequently employed in geotechnical literature [START_REF] Beacher | Reliability and statistics in geotechnical engineering[END_REF].

Theoretical background

A normally distributed random variable has a probability density function given by:

f φ (x) = 1 σ √ 2π exp -1 2 x -µ σ 2 (2.83)
where the parameters of the distribution, µ and σ, are the mean and standard deviation, respectively. The normal distribution is symetric and it varies in the range -∞ < x < ∞. It is a widely used distribution in geotechnical literature because of its many important properties. Nevertheless, it allow for negative values and since geotechnical parameters can take negative values, they are usually truncated below zero.

The log-normal distribution is generally preferred to the normal distribution because it is consistent with the fact most physical properties are non-negative [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]. A random variable is said to be log-normally distributed if its natural logarithm is normally distributed. Its PDF is given by:

f φ (x) = 1 φ √ 2πξ exp -1 2 ln x -λ ξ 2 , 0 < x < ∞ (2.84)
where the parameters ξ and λ, are the mean and standard deviation of the underlying normal distribution, respectively, and are calculated from the mean and standard deviation of the variable x as:

ξ = ln 1 + σ µ (2.85) λ = ln µ - 1 2 ξ 2 (2.86)
The reader who is interested in a geotechnical perspective on probability theory is referred to the textbooks by Rétháti (1988); [START_REF] Beacher | Reliability and statistics in geotechnical engineering[END_REF].

Spatial correlation analysis

Traditional statistical methods alone are unable to describe the spatial variation of soil properties.

Indeed two sets of measurements may have similar second-moment statistics (i.e., mean and standard deviation) and statistical distributions but could display substantial differences in their spatial distribution. The numerical methods (presented in the previous section) that solve the wave equation to analyze quantitative effects of heterogeneity require a description of the spatial variation of the soil properties.

Characterizing the complete 3D spatial variation of soil properties would require an enormous amount of measurement, which is, however, impossible in practice. Therefore, it is convenient to hypothesize the variation of soil properties as been random [START_REF] Baecher | Simplified geotechnical data analysis[END_REF]. Stochastic methods can, therefore, be employed to describe the random nature of the soil properties. Indeed, the stochastic description of physical systems is useful when the observables are too small and numerous to be studied individually [START_REF] Goff | Stochastic modeling of seafloor morphology: inversion of sea beam data for second-order statistics[END_REF], as it is the case for soil heterogeneity.

The measured soil property (e.g., velocity V ) can therefore be decomposed as a sum of a deterministic 9. The roughness of the plot represents the weak and/or strong nature spatial correlation. The rougher the plot, the weak the spatial correlation and vice-versa (or mean) part V 0 and stochastic part δV which depends on the location x as:

V (x) = V 0 + δV (x) = V 0 [1 + ξ(x)]
(2.87)

where the non-dimensional quantity ξ(x) ≡ δV (x)/V 0 is the fractional fluctuation of wave velocity, which is a function of space [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]. Note that in geotechnical engineering, ξ(x) is usually denoted as the residuals about the trend V 0 (e.g. [START_REF] Beacher | Reliability and statistics in geotechnical engineering[END_REF][START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF].

A heterogeneous medium is classically characterized by an autocorrelation function (ACF) of its fluctuations as:

R(x) ≡ ξ(x)ξ(x + y) (2.88)
where the angular brackets designate the expected value. In such a case, an ensemble, rather than individual properties, is used to characterize the medium. In effect, since the medium is considered to be random, it is preferable to consider a collection of realizations in which the micro-states10 of the random properties vary per realization. In the framework of wave propagation problems, it is the average of different quantities constructed from synthesized waves in many realizations of the random media that is used for comparison with observed quantities. The deterministic part or mean V 0 is chosen such that V 0 = V (x) and ξ(x) = 0 [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF].

Theoretical background

The ACF (equation 2.88) gives a statistical measure of the spatial scale and the magnitude of the medium heterogeneity. The translation symmetry of stationary processes dictates that the ACF depends only on the spatial lag r ≡ |x|. Therefore, stationarity in the soil heterogeneities is an important prerequisite for their modeling because the statistical procedures employed are based on the assumption that the data samples consist of stationary observations. Stationarity can be defined in either a strict or weak sense.

Strictly stationary processes are characterized by the fact that the joint probability density function is independent of the spatial location [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]; [START_REF] Jaksa | Inaccuracies Associated with Estimating Random Measurement Errors[END_REF]. In other words, all the finite-dimensional probability distributions are invariant to translation. Weak stationarity requires that the mean and the variance be independent of the spatial location and the correlation between the values at any two points depends only on their separation distance. Strict stationarity in soil properties is rarely encountered in practice; hence weak stationarity is often an acceptable requirement.

The Fourier transform of the ACF R(x) is the power spectral density function (PSDF). It enables the characterization medium heterogeneity in the spectral domain. Several types of PSDFs of random media have been used to describe soil heterogeities in wave propagation problems. [START_REF] Klimes | Correlation functions of random media[END_REF] proposes some examples of PSDF which are convenient for geophysical applications. The two types of PSDFs which are mostly used in seismology (to represent the velocity heterogeneity in the Earth's crust) are [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]: the Gaussian and Von Karman PSDF.

The Gaussian ACF (R G (x)) is defined by [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]):

R G (x) = R G (r) = 2 exp -r 2 /a 2 (2.89)
Its corresponding 2D PSDF is:

P G (k) = P (m) = 2 √ π 3 a 3 exp -k 2 a 2 /4 (2.90)
where k is the wave number vector and k = |k|, is the mean square (MS) fractional fluctuation ( ≡ R(0)), and a is the correlation length. The Gaussian PSDF is used to describe smooth random media that are poor in short wavelength components.

The 2D Von Karman ACF (R V K (x)) is defined by [START_REF] Goff | Stochastic modeling of seafloor morphology: inversion of sea beam data for second-order statistics[END_REF][START_REF] Carpentier | Underestimation of scale lengths in stochastic fields and their seismic response: a quantification exercise[END_REF]:

R V K (x) = R V K (r) = r ν K ν (r) 2 ν-1 Γ(ν) (2.91)
where ν is the Hurst number, K ν (r) is the second modified Bessel function of fractional order ν, and Γ

is the Gamma function.

The corresponding power spectrum of the Von Karman ACF (equation 2.91) is expressed by:

P V K (k) = P V K (k) = 4πνa x a z 2 ν-1 Γ(ν)(1 + k 2 ) ν+1
(2.92)

where a x and a z are the horizontal and vertical correlation lengths, respectively.

The Von Karman PSDF can be used to characterize a variety of random media. Its spectrum is rich in short wavelength components compared to Gaussian type random media. Hence it is an appropriate ACF to characterize rough media. The degree of roughness is controlled by the Hurst exponent ν (ν > 0), such that P V K (k) decreases as k -2ν-3 for ka >> 1. A medium with ν varying between 0 and 1 is considered to be rich in small wavelengths. A particular case of the Von Karman ACF with ν = 0.5 re- [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF] sults in an exponential ACF. Figure 2.15 [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]) shows 2D plots of a Gaussian ACF and 3 Von Karman ACFs with varying Hurst exponent (ν = 1, 0.5 and 0.1). All four media have the same correlation length and we can observe the increase in the small wavelength content as the ν decreases.

In this thesis, we model the velocity heterogeneity in sediments using a Von Karman type ACF.

Indeed, it has been shown that a Von Karman type ACF is more appropriate for describing random velocity inhomogeneity in real the earth medium compared to Gaussian type ACFs because of their power law characteristics [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]. [START_REF] Shiomi | Broad-band power-law spectra of well-log data in Japan[END_REF] calculated the normalized ACFs R(x)/R(0) of velocity and density in the well-log data from the YT2 well in Kyushu, Japan for a depth range of 600-1700 m. They observed that the shape of the correlation distances near the zero lag distance closely follow exponential or von Karman type ACFs. [START_REF] Sivaji | A physical-model study of the statistics of seismic waveform fluctuations in random heterogeneous media[END_REF] calculated the ACF and PSDF of the fractional fluctuation of the 1-D P-wave velocity distribution in granite samples of 3 different sites (Figure 2.16). They obtained the 1-D velocity distributions by assigning a P-wave velocity of each mineral that composes Granite to its appropriate location on a surface photo image of the granite sample. They observed that each granite sample had different correlation distances but were all well approximated by an exponential ACF. The PSDF is proportional to a power of wavenumber, which is approximated by a straight line in the middle log-log plots of Figure 2.16. [START_REF] Nakata | Stochastic characterization of mesoscale seismic velocity heterogeneity in long beach, california[END_REF] also found that the Von Karman PSDF best fit the 3-D P-wave velocity distribution at Long Beach, California compared to other ACF models.

It is important to note that the ACFs generally encountered in geotechnical literature may differ from those used to model random media in Seismology. [START_REF] Vanmarcke | Random fields: analysis and synthesis[END_REF] provides a geotechnical perspective for the modelling of soils using random field theory. [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF] also presents how the soil variability can be analyzed for geotechnical purposes, and in particular reviews different ACF used in geotechnical practice. To analyse the spatial variability of the soil's geotechnical parameters using geophysical measurements, [START_REF] Salloum | Evaluation de la variabilité spatiale des paramètres géotechniques du sol à partir de mesures géophysiques: application à la plaine alluviale de Nahr-Beyrouth (Liban)[END_REF] uses squared exponential and decreasing exponential ACFs. [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF] notes, however, that no reliable physical correspondence between the soil type and the type of correlation structure seems to be identified and, therefore, the fit between ACF values computed from geotechnical data and from ACF models should be considered to select the best ACF type for a given dataset. Other approaches are proposed that do not rely on the choice of an ACF. For example, [START_REF] Toro | Probabilistic models of site velocity profiles for generic and site-specific ground-motion amplification studies[END_REF] proposes a method to generate stochatic velocity profiles based on the standard deviation of the natural log of the shear-wave velocity and some interlayer correlation parameters. This approach is used for example by [START_REF] Rathje | Site specific validation of random vibration theory-based site response analysis[END_REF] to assess the influence of site property variabilitiy on seismic site response analysis and by [START_REF] Rodriguez-Marek | Application of single-station sigma and site-response characterization in a probabilistic seismic-hazard analysis for a new nuclear site[END_REF] to compute site response in a probabilistic seismic-hazard analysis for a new nuclear site.

Characterizing the soil heterogeneity from data

To characterize the soil heterogeneity from data, one needs to quantify the finite-scale autocorrelation function of the data set. Calculating the real autocorrelation function of the soil heterogeneity is impossible because data sets are, in practice, always limited in size. As a result, it is inevitable to refer the sample autocorrelation function which is expected to be representative of the stochastic process or Theoretical background real autocorrelation function. The discrete sample autocorrelation of a data set is given by [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]:

R(τ j ) = n-j i=1 (s i -m s )(s i+j -m s ) n i=1 (s i -m s ) 2 , j > i (2.93)
where s is the soil property being measured, m s is the mean of the data set, |τ j | = j∆x ,j∈N is the lag distance, n is the number of data points, and ∆x is the sampling interval. An absolute value has been used to denote the separation distance because the ACF is symmetric about zero separation distance.

The above equation requires that the lag distances should have a constant distance, that is, the data points locations should be separated from each other in a line to estimate the autocorrelation function values at each multiple of a separation distance. However, the data points may sometimes have irregular spatial patterns in practice. This increases the difficulty in calculating the sample ACF using equation 2.93 which requires a uniform separation distance between the data points. A common solution to this problem is to bin the data points into bins of similar or equal separation distances before calculating the sample ACF. Nevertheless, even a binning of the data may sometimes be impossible due to a limited number of data points.

Random media generation code and coupling with SEM2DPACK

In this study, we generate the 2D Von Karman random media using the power spectrum methodi.e., in the wavenumber domain (equation 2.92) -following [START_REF] Frenje | Scattering attenuation: 2d and 3d finite difference simulations vs. theory[END_REF]. We then use the inverse 2D Fourier transform to convert the 2D power spectrum to the space domain. The space domain random media are then coupled with the 2D spectral element code (SEM2DPACK) by performing a nearest-neighbor interpolation over the spectral element mesh. In what follows, we enumerate the different steps in the random media generation process.

Generation process

Suppose we want to generate a 2D Von Karman media over spatial grid of dimensions (N x ,N z ) and grid steps dx and dz, respectively. The construction process of the random media can be summarized into 6 steps:

1.

Step 1: We generate the discrete 2D power spectrum, P(k x ,k z ), of the Von Karman ACF in the wavenumber domain. Following [START_REF] Frenje | Scattering attenuation: 2d and 3d finite difference simulations vs. theory[END_REF], for the random media characteristics to be properly sampled by the grid, two conditions must be respected: (1) the minimum wave number must be smaller than the corner wave number, i.e., N x,z > 2πa x,z , and (2) the grid spacing must be at least twice small than the correlation length, i.e., dx < a x /2 and dz < a z /2. This is going to be illustrated in the examples below.

2.

Step 2: We then generate 2D uniform, random phase, θ(k x ,k z ), between 0 and 2π.

3.

Step 3: The random spectrum function is then obtained by multiplying the random phase with the square root of the power spectrum. (kx,kz) (2.94)

P (k x , k z ) = P (k x , k z )e iθ
The random spectrum has to be either tapered or smoothed to remove the unavoidable truncation and round-off errors due to the discretization process.

Theoretical background 4.

Step 4: We now calculate the inverse 2D Fourier transform of P (k x , k z ) to get the random media in the space domain R(x, z).

5.

Step 5: The next step is to standardize the random media by subtracting the mean, µ =< R(x, z) >, and dividing by the the standard deviation ε 2 =< R(x, z) -µ > 2

6. The last step in the generation process is to scale the random media to the desired standard deviation of the velocity fluctuations.

We developed a python code, named RANDOM2D, which generates 2D Von Karman random media as described above. Figure 2.17

Integrating the random media into SEM2DPACK

In this study, the random media generated by RANDOM2D are used to perturb the material velocity of the deterministic model before computing seismic wave propagation using SEM2DPACK. The random media are by construction generated on a regular grid due to the inverse Fourier transform. As seen in the previous section, the spectral element method (SEM) relies on the GLL integration pointswhich are irregularly spaced for polynomial orders > 3 (see Figure 2.10) -to calculate the solution of the wavefield. The simulations performed in this study all use polynomial order > 3 in order to have accurate solutions of the wavefield; hence, the resulting SEM grid after meshing is irregular. This means Theoretical background that to attribute random properties to the GLL points of the SEM grid, we need to interpolate the regular random grid over the SEM grid. This required some implementation adjustments on SEM2DPACK, which are described in appendix A. The verification of SEM2DPACK, after the implementation adjustments, is also described in Appendix A.

The interpolation method used is the Nearest-neighbor interpolation algorithm. In order for the random fields to be properly sampled after interpolation, we made sure that the SEM grid had a finer mesh than random media.

Summary

To summarize this chapter, we have briefly presented the theory of the different aspects or ingredients which are going to be used in this thesis research. We started by presenting seismic site effects and why their proper assessment is important for seismic risk mitigation. Indeed, the ultimate goal of the research performed in this thesis is to better understand site effects to improve seismic hazard assessment. In section 2.2, we explained why numerical simulations of seismic wave propagation is a precious tool for analyzing site effects. We also presented examples of numerical methods, in our case SEM, that can be used to model seismic wave propagation. Finally, in section 2.3, we explained the importance of small-scale velocity heterogeneities on seismic wave propagation and how their characteristics can be inferred from data and modeled. Quantifying the characteristic small-scale velocity heterogeneities of the medium provides more realistic input velocity models for numerical simulations, enabling a better assessment of their effects on seismic wave propagation and consequently on site effects.

In this work, we will use numerical simulations to study wave propagation, and hence site effects,

in

Abstract

Some geological configurations, like sedimentary basins, are prone to strong site effects. Basins are often composed of different layers whose properties are generally considered as spatially homogeneous or smoothly varying. In this study, we address the influence of small-scale velocity fluctuations on the Nice (France) basin's seismic response. For this purpose, we model 2D SH wave propagation using the Spectral Element method in the basin, which is 1.1 km large and ≈ 60 m deep and composed of different sedimentary layers. Anelastic attenuation is introduced and considered as frequency-independent. The velocity fluctuations are modeled statistically as a random process characterized by a Von Karman autocorrelation function and are superimposed to the deterministic model. We assess the influence of the amplitude and correlation length of the velocity fluctuations on the surface ground motion. We vary the autocorrelation function's characteristics and compute seismic wavefields in 10 random realizations of the heterogeneous model. The analyses of our results focus on the Envelope and Phase differences between waveforms computed in the heterogeneous and deterministic models; on the variation of ground motion intensity measures, such as the peak ground velocity (PGV) and the pseudo-spectral acceleration response (PSA); and on the 2D basin response (transfer function). We find that the amplitude of fluctuations has a greater effect on the ground motion variability than the correlation length. Depending on the random medium realization, the ground motion in one heterogeneous model can be locally amplified or de-amplified with respect to the deterministic model. We show that considering the mean amplification of different random media realizations may underestimate the maximum amplification values, compared to the deterministic model, due to the smoothing effect of the average. We discuss how such small-scale velocity fluctuations can be considered for site-specific seismic hazard assessment.

We propose using percentiles of the different random realizations to describe the basin's response when a small number of realizations are made. Such a study highlights the importance of knowing the site properties at different scales, particularly at small scales, for proper seismic hazard assessment.

Introduction

It is well known that local site characteristics may produce large ground motion amplifications during earthquakes (e.g. [START_REF] Trifunac | Surface motion of a semi-cylindrical Alluvial Valley for incident plane SH waves[END_REF][START_REF] Trifunac | Analysis of the Pacoima Dam Accelerogram -San Fernando, California, Earthquake of 1971[END_REF][START_REF] Turker | Observations of Hard-Rock Site Effects[END_REF][START_REF] Bard | The two-dimensional resonance of sediment-filled valleys[END_REF][START_REF] Ktenidou | Directional dependence of site effects observed near a basin edge at Aegion, Greece[END_REF]. The effect of the local geology on the incoming wavefield is commonly denoted as site effects and can lead to important damages of infrastructures and loss of lives during earthquakes (e.g the Great Hanshin earthquake as in [START_REF] Pitarka | Three-Dimensional Simulation of the Near-Fault Ground Motion for the 1995 Hyogo-ken Nanbu (Kobe), Japan, Earthquake[END_REF], and the Michoacan earthquake in Anderson et al. ( 1986)). Their proper assessment is thus important for accurate seismic hazard evaluations.

Investigating site effects can be done by analyzing seismic records or simulating seismic wave propagation in realistic media. On one hand, seismic records provide a basis for theoretical and experimental research and are of great value to understand site effects. Experimental approaches, like the standard spectral ratio technique, which relies on site-specific transfer functions [START_REF] Borchedt | Effects of local geology on ground motion near San Francisco Bay[END_REF], have been developed to evaluate the local site response from earthquake records. However, the scantiness of earthquakes in regions of low to moderate seismicity makes their evaluation difficult. Notwithstanding, seismologists also use ambient noise vibrations to estimate site effects (e.g. [START_REF] Nakamura | A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface[END_REF][START_REF] Kanai | Engineering Seismology[END_REF]Bard, 1999a;[START_REF] Perron | Can broad-band earthquake site responses be predicted by the ambient noise spectral ratio? Insight from observations at two sedimentary basins[END_REF][START_REF] Tchawe | On the use of the coda of seismic noise autocorrelations to compute H/V spectral ratios[END_REF]. On the other hand, the use of numerical simulations of seismic wave propagation allows for direct evaluation of the site response, provided that the input motions, velocity, density, and attenuation of the media are well characterized (e.g. [START_REF] Day | 3d ground motion simulations in bassins: Final report for lifelines project 1a03[END_REF][START_REF] Chaljub | Quantitative comparison of four numerical predictions of 3D ground motion in the Grenoble valley[END_REF][START_REF] Moczo | The Finite-Difference Modeling of Earthquake Motions Waves and Ruptures[END_REF]. The distribution and characteristics of mechanical properties can be inferred using deterministic geophysical imaging methods (e.g., Multichannel Analysis of Surface Waves, seismic reflection, horizontal to vertical spectral ratio, among other techniques), but their limited resolution makes it difficult to map the short-wavelength (or small-scale) variations of the soil properties in the Earth's subsurface. As a result, velocity models often used in ground motion simulations generally assume smooth lateral variations of elastic properties, which may be adequate for long-period ground motion estimations. However, for short-period ground motion simulations, the wavefield can be very sensitive to the small-scale variations in the soil properties (e.g. [START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF][START_REF] Emoto | Statistical parameters of random heterogeneity estimated by analysing coda waves based on finite difference method[END_REF][START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF]. Furthermore, the need for computing high-frequency seismograms in engineering applications requires the description of short-wavelength variations of the media. Since it is impossible to characterize them totally, they are usually approximated using stochastic approaches (e.g. [START_REF] Frankel | A review of numerical experiments on seismic wave scattering[END_REF][START_REF] Hong | Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based[END_REF][START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF].

Understanding the effect of variations in velocity models (henceforth regarded to as random velocity heterogeneities) on the simulated ground motion is a topic under active investigation in seismology.

Most studies that consider random velocity heterogneities often focus on comprehending seismic scattering in the Earth's crust (e.g [START_REF] Frankel | A review of numerical experiments on seismic wave scattering[END_REF][START_REF] Holliger | A stochastic view of lower crustal fabric based on evidence from the ivrea zone[END_REF][START_REF] Roth | Single scattering-theory versus numerical modeling in 2D random-media[END_REF][START_REF] Frenje | Scattering attenuation: 2d and 3d finite difference simulations vs. theory[END_REF][START_REF] Hong | Scattering Attenuation of 2D Elastic Waves: Theory and Numerical Modeling Using a Wavelet-Based[END_REF][START_REF] Imperatori | Broad-band near-field ground motion simulations in 3-dimensional scattering media[END_REF]. Some studies have investigated the random velocity heterogeneities effect on surface ground motion. For example, [START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF] observed that considering random velocity heterogeneities in regional ground motion simulations at San Francisco Bay Area lead to significant deviations in the predicted ground velocities for standard deviations in the velocity of 5 to 10 %. [START_REF] Imperatori | The role of topography and lateral velocity heterogeneities on near-source scattering and ground-motion variability[END_REF] and [START_REF] Iwaki | Effects of random 3D upper crustal heterogeneity on long-period ( 1s) ground-motion simulations[END_REF] also observed that random velocity heterogneities lead to large variabilities in the simulated ground motion parameters.

These studies deal with crustal velocity heterogeneities whose characteristic scales are larger than those

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin observed in sediments. Modeling velocity heterogeneities at the sedimentary scale can be challenging, and few studies have been targeted towards understanding their effect on surface ground motion. In this study, we aim to understand the effect of random velocity heterogeneities on the computation of surface ground motion from a site-effect analysis point of view. We use a 2D velocity model of the Nice (France) sedimentary basin, which is perturbed, to perform 2D viscoelastic numerical simulations of seismic wave propagation. We introduce velocity heterogeneities in the sedimentary basin as 2D saptially correlated random processes characterized by a Von Karman ACF. We investigate the role of the correlation length and the strength of the velocity fluctuations on the simulated ground motion.

The study area is introduced in section 3.3. The stochastic generation of the random velocity models and numerical experiments are presented in section 3.4. Analyses of the ground motion's sensitivity to the variability of the random media are described in section 3.5. We discuss the influence of the velocity fluctuations on the ground motion for site effect assessment in section 3.6. We finish by providing conclusions and perspectives to the current study in section 3.7.

The Nice (France) sedimentary basin and 2D velocity model

The city of Nice is a densely populated area and one of the most seismically active regions in France (e.g [START_REF] Courboulex | Seismic hazard on the french riviera: observations, interpretations and simulations[END_REF]. The city is built on three major geological units; 1) Quaternary alluvial deposits originating mainly from the Var and Paillon rivers, 2) conglomerates of the Pliocene age which form the western hills of the region, and 3) Jurassic and Cretaceous bedrock. Geomorphologically, the area comprises reliefs, valleys, and sedimentary fillings, which are prone to site effects. The region has been the subject of different studies, like the GEMITIS [START_REF] Arnal | Projet GEMITIS-Nice : Evaluation des dommages directs d'un séisme sur la ville Nice, scénario de risque, extension des résultats au départements des Alpes Maritimes[END_REF] and GEMGEP (Bard et al., 2005b) projects, which provided a detailed geological and geotechnical description of the subsurface for site effect analysis and seismic hazard assessment purposes. These led to developing a 3D geotechnical model of the region using borehole data and ambient noise measurements [START_REF] Bertrand | 3d geotechnical soil model of nice, france, inferred from seismic noise measurements for seismic hazard assessment[END_REF]. The model has nine geological layers overlying a bedrock. The lithology and mechanical characteristics of the different layers are summarized in Table 3.1. Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

Methods

Stochastic modeling of random heterogeneities

The distribution of velocity heterogeneities of geological structures can be approximated as a stationary random process. The mathematical description of random media for the modeling of random heterogeneities has widely been described in several publications (e.g [START_REF] Goff | Stochastic modeling of seafloor morphology: inversion of sea beam data for second-order statistics[END_REF][START_REF] Klimes | Correlation functions of random media[END_REF][START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]. In this section, we briefly summarize the essential theory and steps for the generation of random fields.

The heterogeneous velocity (v) of the soil can be expressed as a function of space as:

v(x) = v 0 (1 + (x)) (3.1)
where v 0 is the mean compressional V P or shear wave velocity V S of the medium and (x) = δv(x)/v 0 is the fractional fluctuation of the wave velocity. (x) is a random function of space whose mean value is considered to be zero ( (x) = 0). We express the spatial coordinates in 2D Euclidean space x = (x, z),

where x and z denote the horizontal and vertical space directions, respectively. The auto-correlation function (ACF), C(x), of the fractional velocity fluctuation can be used to characterize the stochastic nature of the random media. When the randomness is stationary and isotropic, the ACF is a function of the spatial lag (r ≡ |x|) distance only. Several types of ACFs of random media have been developed for geophysical applications and wave propagation problems (e.g [START_REF] Klimes | Correlation functions of random media[END_REF][START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF].

Few examples in the literature provide direct empirical observations of the spatial correlation structure of seismic properties. As a result, the correlation function is often constrained by observations of seismic phenomena. For example, [START_REF] Nakata | Stochastic characterization of mesoscale seismic velocity heterogeneity in long beach, california[END_REF] used dense-array recordings of the ambient seismic wavefield to constrain the random-field model representations of a 3D P-wave velocity model in Long Beach, California. In their study, the Von Karman ACF best fits the imaged P-wave velocity model. Several past studies in seismology (e.g. [START_REF] Frankel | A review of numerical experiments on seismic wave scattering[END_REF][START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF][START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF][START_REF] Imperatori | The role of topography and lateral velocity heterogeneities on near-source scattering and ground-motion variability[END_REF] also consider the Von Karman ACF when modeling the Earth's heterogeneities. Furthermore, [START_REF] Thompson | Spatial correlation of shear-wave velocity in the San Francisco Bay Area sediments[END_REF] measured the spatial horizontal correlation function of the near-surface sediments in the San Francisco Bay area. They observed that the horizontal correlation structure of the average S-wave velocity of the upper 10 m of the soil exhibited an exponential correlation function with a range of about 3 km. The exponential correlation function is a Von Karman ACF with a Hurst number of 0.5. In this study, we assume that the spatial distribution of velocity heterogeneities also follows a Von Karman correlation function.

The 2D analytical expression of the Von Karman ACF is expressed in equation 3.2 [START_REF] Goff | Stochastic modeling of seafloor morphology: inversion of sea beam data for second-order statistics[END_REF][START_REF] Carpentier | Underestimation of scale lengths in stochastic fields and their seismic response: a quantification exercise[END_REF] as:

C(r) = r ν K ν (r) 2 ν-1 Γ(ν) (3.2)
where r is the weighted radial autocorrelation lag ( x 2 /a 2 x + z 2 /a 2 z ), ν is the Hurst number, K ν (r) is the second modified Bessel function of fractional order ν, and Γ is the Gamma function. The corresponding power spectrum of the Von Karman ACF (equation 3.2) is expressed by: Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

P (k) = 4πνa x a z 2 ν-1 Γ(ν)(1 + k 2 ) ν+1 (3.3)
where a x and a z are the horizontal and vertical correlation lengths, respectively; k is the weighted radial wavenumber (k = k 2 x a 2 x + k 2 z a 2 z , with k x , k z being the horizontal and vertical wavenumbers, respectively).

The random media are generated by applying a 2D uniform random phase to the 2D power spectrum (equation 3.3) and then transforming back to the space domain using the inverse 2D Fourier transform. The resulting random field is normalized to have zero mean and unit variance. The random field is then scaled to the desired root mean square (RMS) fluctuation using a given coefficient of variation (c v ).

In this work, we generated random fields of the shear wave velocities. All the other material parameters were considered to be deterministic. We truncated the random velocity distributions between -2σ and 2σ (with σ being the standard deviation) to limit the long tail normal distribution. Truncating the distribution of velocities also avoids having negative velocities. It eases the meshing process by avoiding very fine grids due to low-velocity values, thereby ensuring the stability and reducing the computational demands of the numerical scheme. Other studies may use different distributions for the velocity field (e.g., [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF] used the beta distribution to characterize the fluctuation of velocity).

Moreover, El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] observed that simulated ground motion is less sensitive to the velocity probability distributions compared to other parameters of the random medium (correlation length and coefficient of variation). Assessing which distribution best describes the media is not the purpose of this study.

2D random velocity models

We consider random velocity perturbations only within the sedimentary basin to investigate their effect on the computed ground motion. The bedrock is assumed to have a uniform V S of 1000 ms -1 .

Since we conduct a parametric analysis, the statistical parameters -correlation length (a), Hurst exponent (ν), and the coefficient of variation (c v ) -that control the correlation function are chosen based on values reported in the literature. Some studies at the crustal-scale (e.g. [START_REF] Frankel | A review of numerical experiments on seismic wave scattering[END_REF]) report correlation lengths of the order of 10 km and coefficient of variation in the range of 5% -10% in order to replicate teleseismic travel-time anomalies and seismic coda at frequencies up to 30 Hz. Correlation lengths of these orders are unrealistic for small-dimension sedimentary basins. Considering correlation lengths, which are orders of magnitude larger than the medium's size, will have imperceptible effects on the wave propagation in shallow depths, especially for short wavelengths. At the sedimentary scale, [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF] observed that the correlation model must exhibit large c v s, around 25% or larger, and correlation lengths of ≈ 100 m to reproduce the empirical transfer functions at the OKYH07 site of the Kiban-Kyoshin network in Japan. Table 4 of [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF] summarizes the S-wave velocity c v , reported by [START_REF] Holzer | Shear wave velocity of surficial geologic sediments in northern California: Statistical distributions and depth dependence[END_REF] and [START_REF] Wills | Developing a map of geologically defined site-condition categories for California[END_REF], of sedimentary units. The c v values vary in between 14 % and 46 %. Furthermore, table 2 of Sato (2019) summarizes the different statistical parameters of the Von Karman spectrum proposed and measured in the literature. This summary provides useful insights into the order of the statistical parameters exhibited by different geologic materials.

In this study, we simulate seismic wavefields in a shallow basin (< 70 m). For the waves to be sensitive to random velocity heterogeneities in this range, the order of the vertical correlation length must be less the literature for sediments (e.g. [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF][START_REF] Sato | Power spectra of random heterogeneities in the solid earth[END_REF].

Using a combination of these parameters, we build five 2D random media whose characteristics are summarized in Table 3.2. All five random models are assumed to be isotropic such that vertical and horizontal correlation lengths coincide (a x = a z ). Although sediments generally exhibit larger scale lengths in the horizontal direction than vertical direction due to their formation process, we decided to use the same vertical and horizontal correlation lengths in our models for parametric analysis and modeling simplifications.

The value of ν commonly varies between 0 and 0.5 and controls the contribution of short-wavelengths in the medium. Low values increase the variability across high spatial frequencies. We assumed a constant value of ν = 0.3 for all random models. The results of the parametric analysis are going to be based on models R1 -R4. Models R1 and R2 have coefficients of variation of 5 % with correlation lengths of 10 and 50 meters, respectively, while models R3 and R4 have a coefficient of variation of 30 % with correlation lengths of 10 and 50 meters, respectively. We also assume that all the layers have the same random media characteristics to avoid abrupt changes in velocity at the layer's interface. For each random model, we generate 10 realizations with different random seeds for further statistical analysis.

The results for Model R5 with an intermediate c v of 10 % and correlation length of 10 m will only be presented in the discussion.

Numerical simulations of seismic wave propagation

We compute 2D SH seismic wave propagation using the spectral element solver SEM2DPACK [START_REF] Ampuero | Etude physique et numérique de la nucléation des séismes[END_REF]. The computational domain is 2100 m wide and 70 m deep, as shown in Figure 3.2. An unstructured mesh containing 3195 elements with a maximum grid resolution of 10 Hz is generated, taking into account all model interfaces using an external mesh generator software Trelis 16.4 [START_REF] Trelis | american fork, ut: csimsoft[END_REF]. The seismic wavefield is integrated on the mesh domain using 7 Gauss-Lobatto-Legendre (GLL) integration points. The minimum GLL separation distance in the basin's spectral element grid is ≈ 0.1 m. We mimic an infinite lateral domain using periodic boundary conditions. Periodic boundaries were adequately used by [START_REF] Peyrusse | A nodal discontinuous galerkin method for site effect assessment in viscoelastic media -verification and validation in the nice basin[END_REF] to simulate the basin's response. We apply absorbing boundary conditions at the bottom of the model and impose vanishing traction at the surface. Intrinsic attenuation is taken into account by frequency-independent Q values between 5 and 5000 for a frequency range between 0.01 and 50 Hz [START_REF] Liu | Efficient modeling of q for 3d numerical simulation of wave propagation[END_REF], which is based on the rheological model of the Generalized Maxwell body [START_REF] Emmerich | Incorporation of attenuation into time-domain computations of seismic wave fields[END_REF]. This realistic attenuation law was implemented in the spectral element solver by [START_REF] Oral | 2-D P-SV and SH spectral element modelling of seismic wave propagation in non-linear media with pore-pressure effects[END_REF]. For the simulations, we define an Swave quality factor (Qs) of 50. The bedrock is assumed to be purely elastic and homogeneous.

We simulate SH wave propagation (S-waves polarized out-of-plane) generated by a vertical incident plane wave. As a source time function, we use a truncated Gaussian wavelet with a central frequency of 6 Hz (Figure 3.3). The seismic wavefield is computed up to 10 s using a leap-frog time scheme with a calculation time step of 20 µs. The velocity wavefield at the free surface is recorded every 10 ms by 420 surface receivers with an inter-receiver spacing of 5 m.

We compute seismic waves in both the deterministic and random media. To include the random velocity heterogeneities into the velocity model (Figure 3.2), we interpolate the regular grid of the random media over the spectral element mesh of the sedimentary basin using a nearest-neighbor interpolation algorithm. The random media R1 -R4 were all generated on a rectangular grid of 1200 m by 250 m with a constant grid step of 0.2 m in both space directions.

To ease the writing and reading of this paper, simulations in the deterministic media are denoted as model M. In contrast, simulations performed in the random media are labeled following the velocity model and random medium used. Hence, a simulation performed using velocity model M and random medium R1 will be called MR1 in the text. display unphysical variabilities, we decided to maintain them as they will serve as extreme cases.

Results

To reveal the differences in wave propagation and ground motion induced by the random media, we compare the results of simulations from the random media to those from the deterministic model M.

First, we analyze the differences in waveforms by quantifying the relative discrepancies in phase and amplitude between the deterministic and random media seismic signals. We then quantify their effect 3.2 for the characteristics of each model.

on the ground motion in terms of peak ground velocity (PGV), transfer functions (TF), and the pseudospectral acceleration (PSA), which are measurements classically used in Earthquake Engineering. Note that the seismograms recorded at the outcropping bedrock contain only free surface reflections because the medium surrounding the basin has a uniform velocity. The incident wavefield arrives slightly later at the basin's surface than on the outcropping bedrock because the basin has lower velocities than the bedrock. We can observe strong reflected waves from the edges and propagating across the basin for the receivers located within the basin (between 500 m and 1500 m). In particular, the wavefield between 1100 m and 1400 m has higher amplitudes, from 0.4 to 2 seconds, due to the local trapping of seismic waves in the low-velocity layers c1 and c3. These results show that local heterogeneities may strongly affect the seismic wave propagation inside the basin.

Wave propagation in random heterogeneous media

Time-frequency analysis: envelope and phase changes

We quantify the phase and amplitude difference in the surface velocity wavefields between the random media and the deterministic medium. The single-value envelope (EM) and phase (PM) misfit between signals, computed following [START_REF] Kristekova | Misfit criteria for quantitative comparison of seismograms[END_REF], is used as the difference measure. First, the seismograms were bandpass filtered between 0.1 and 10 Hz using a fourth-order non-causal Butterworth filter. The quantities EM and PM vary between 0 and 1, with values close to 0 implying low differences between the signals, and those close to 1 indicate high differences. We do not categorize the misfit values into goodness-of-fit (GOF) categories as in [START_REF] Kristekova | Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals[END_REF], but rather interpret them as relative differences between the wavefield computed in the random media and those computed in the deterministic reference model. As a result, the term misfit in the text denotes the difference with respect to the deterministic model. (2) For a fixed c v , both EM and PM values are positively correlated with the random models' correlation length. Implying that increasing the correlation length will, on average, increase the phase and amplitude differences induced by the velocity heterogeneities.

The average misfits between the 10 realizations of the random models increase between 1100 m and 1400 m, where the basin is the deepest and where velocities near the surface are the smallest, in layers c1 and c3. are approximately one for most receivers and locally vary between 0.8 and 1.3 for the receivers located between 1200 and 1350 m. Whereas for MR3 and MR4, they vary between 0.5 and 2.4. This implies that the intensity of the heterogeneities, or the coefficient of variation for models MR1 and MR2, is not high enough to produce remarkable changes in the PGV values. Yet, the coefficient of variation for models MR3 and MR4 do significantly affect the PGV by a factor of ≈ 2.

Spatial distribution of the peak ground velocity

To quantify in a different way how heterogeneities affect the wave propagation, we plot the PGV arrival time (T pgv ) at each receiver for both the reference and one realization of the random media in Figure 3.9a. In the deterministic medium (black curve), the highest T pgv are observed at the basin edges. This is due to basin edge generated waves, which have higher amplitudes and arrive later at the surface.

The T pgv is constant at the central receivers because the PGVs are all carried by the direct wave. In the random models MR1 and MR2, the pattern remains the same, but the local T pgv at certain receivers between 1250 and 1400 are mildy affected by the velocity heterogeneities. For MR3 and MR4, notable variations in the T pgv are observed. This is because the velocity fluctuations in these models lead to phase conversions, which can have higher amplitudes than the direct arrivals. The resulting T pgv at these receivers thus locally differ from the deterministic case because they are carried by different wave trains. This can be seen in (red curve). The PGVs (denoted by the black and red dots) in both cases are carried by different wave phases, the first arrival phase for the deterministic trace, and by a later arrival for model MR4.

To have a better picture of how the PGV values vary between the different realizations of the random media, we computed a proxy of the coefficient of variation by taking the ratio between the inner-68th percentiles (i.e., the 84th -16th percentiles) and twice the median of the 10 random models (Figure 3.10). The curves show that the variability increases with the amount of perturbation of the velocity heterogeneities. On average, the PGV's variability varies around 5 % for MR1 and MR2 and 30 % for MR3 and MR4.

Spectral ratios

The standard spectral ratio or transfer function (TF) is commonly used to assess site effects as it quantifies the spectral amplification of the site [START_REF] Borchedt | Effects of local geology on ground motion near San Francisco Bay[END_REF]. We calculate the 2D TF of the sedimentary basin by computing the ratio between the Fourier spectrum at each receiver and the average Fourier spectrum of the receivers located on the bedrock. The signals were detrended and bandpass filtered between 0.1 and 10 Hz before computing their Fourier transform. The Fourier spectra were then smoothed using a Konno-Ohmachi smoothing function [START_REF] Konno | Ground motion characteristics estimated from spectral ratio between horizontal and vertical components of microtremors[END_REF], with a bandwidth of 40. To make the 2D TFs of the different models comparable, we used the average Fourier spectra of the receivers located on the deterministic model's bedrock as the common denominator for the spectral ratios.

Pseudo-spectral acceleration

The pseudo-spectral acceleration is the most frequently used intensity measure by engineers to quantify a structure's response to ground motion. We compute the response spectra at two particular periods, 0.5 and 0.17 s. The former is close to the resonance frequency (2 Hz) of the whole basin, while the latter is the predominant frequency (6 Hz) of the input motion.

We plot in Figure 3.12 (top panels), for both periods, the proxy of the coefficient of variation between the 10 realizations each random model (top plots) and the ratio between the median spectral acceleration and the deterministic model (bottom plots). Models MR1 and MR2 show no major variability in the computed PSA, at both frequency, due to velocity heterogeneities. In contrast, the effect of the velocity heterogeneities is remarkable in MR3 and MR4. The coefficients of variation between the random realizations are larger in models MR3 and MR4, at both spectral periods, compared to MR1 and MR2.

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

Besides, for the same coefficient of variation, the effect of correlation length is little on the PSA's spatial distribution. In general, the coefficient of variation for models MR3 and MR4 at 0.17 s is larger than the case of 0.5 s. This is expected because the transfer functions also show strong amplifications at those periods in the basin (see Figures 3.11d and e).

The ratio between the median response spectra of 10 random realizations and the reference case

Discussion

This parametric analysis shows that the ground motion indicators and site response are highly sensitive to the random media's coefficient of variation, and to a lesser extent, the correlation length -for this particular geometry configuration. This corroborates the results of El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] for wave propagation in a model composed of one heterogeneous layer overlaying a bedrock. An increase of the random medium's coefficient of variation leads to an increase in the ground motion's spatial variability and an increase in the variability of the ground motion indicators between the different realizations.

The fact that variations in the correlation length have mild effects on the ground motion indicators does not necessarily mean that the wave propagation is not sensitive to its value. For the sedimentary basin in Nice, the correlation length is limited by the extent of the basin and layers thickness, especially the basin depth. Thus, to better assess the correlation length, deeper models should be used, which we plan to do in the future.

These results show the relevance in considering lateral heterogeneities in site effect assessment studies.

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

Implication of averaging realizations of random media for site effect analysis

When modeling small-scale velocity heterogeneities using stochastic methods, the effects on wave propagation through a random medium are generally described in terms of statistical averages of many observations (e.g. [START_REF] Korn | Seismic waves in random media[END_REF][START_REF] Margerin | Seismic Waves, Scattering[END_REF]. In the context of site effect assessment, the question is whether averaging several realizations of random media provides physically meaningful results, given that there is only one random realization of the subsurface. Moreover, performing a multitude of simulations for different realizations of random media requires high computational demands. In a practical framework, this raises a practical question on the number of simulations necessary to perform reliable analysis.

In our analysis, we considered 10 random media realizations. The variability of the ground motion indicators between the 10 realizations, for high coefficient of variations, creates an ambiguity on how to generalize the results: should the conclusions be based on one realization of randomness or the average of all the 10 realizations?

We plot in Figures 3.13 and 3.14 the geometric mean of the 2D TFs and the associated coefficient of variation between the 10 random realizations. We compute the geometric mean instead of the arithmetic mean because it is conventionally used in practice to compute the average of spectral ratios, which can strongly vary from one earthquake to another. In this case, though, the geometric and arithmetic mean had no major differences. As expected, since the perturbation is low, there is a low variance between the TFs of the 10 realizations of MR1 and MR2 and their averages are comparable to the deterministic case (Figure 3.11a). On the contrary, the large variance between the TFs of models MR3 and MR4 (Figures

3.14b and c

) leads to smoother average responses compared to the deterministic case and individual realizations. The decrease in the mean amplification is because the spatial distribution of the velocity heterogeneities varies from one realization to another. Therefore, averaging the TFs of the 10 realizations produces a smoothing effect on the amplification, leading to a mean TF that largely differs from that of a single realization.

In the context of site effects analysis, the interest is targeted towards understanding the effect of random velocity heterogeneities for seismic hazard and risk mitigation purposes. On one hand, considering an average of several realizations may lead to underestimated results, as shown by the current study. On the other hand, an analysis based on the results of only one realization of the random medium will also lead to a misrepresentation of the uncertainties. We propose to present not only the mean or median transfer function, but also other percentiles. For example, Figures 3.15a and b show the 16th and 84th percentile, respectively, for model MR3. We can clearly see that the variability in the amplification is important. To better visualize this effect, Figure 3.15c shows the site amplification at a surface receiver located at 1215 m on the horizontal profile. We can clearly observe the high variability in the amplification. Moreover, the inner-68th percentiles shows a resonance peak, between 3 and 4 Hz, that is not clearly visible on the median and mean curves.

Link between the coefficient of variation of the random media and velocity contrast

The outstanding feature of the results presented in this study is the high sensitivity of the ground motion indicators to the random velocity heterogeneities' c v , as shown by the results for models MR3 and MR4, with a c v of 30 %. A c v of 30 % appears to be very high in these cases as it results in velocity fluctuations that highly perturbed the deterministic velocity model, in large part because of the low-sedimentary basin velocity contrast between the layers. In essence, the deterministic model should remain the dominant feature1 even after being perturbed by small-scale variations in velocity. Thus one may argue that a c v of 30 % is unapplicable for the current velocity model because of the low-velocity contrast between the layers. In reality, c v s of this order are not unrealistic but rather commonly observed in sediments (e.g. [START_REF] Wills | Developing a map of geologically defined site-condition categories for California[END_REF][START_REF] Sato | Power spectra of random heterogeneities in the solid earth[END_REF].

To validate our observations on the scaling nature of the c v on the ground motion indicators, we performed an extra set of simulations in random model MR5 (see Table 3.2), with an intermediate c v of 10 % and correlation length of 10 m. The plot of one realization of MR5 (Figure 3.4e) shows that the background deterministic structure is still maintained after perturbation by the random velocities.

For brevity, we summarize the results in appendix 3.9.1. Comparison are made with the deterministic case, model MR1, and MR3, which have a correlation of 10 m, to stress on the scaling effect of the c v .

The velocity wavefields of MR5 (Figure 3.16) remain similar to the deterministic case as it was the case for MR1, but the later arrivals are slightly more scattered than MR1 -particularly between 1200 m and 1500 m. The average differences in the phase and amplitude of MR5 with the deterministic model (Figure 3.17) are higher than those of MR1, but lower than MR3. Similarly, the amplification patterns of TF for MR5 are locally more affected than those of MR1, and less affected than MR3. These observations show that the c v act as a scaling factor of the scattering intensity, also shown by previous studies (e.g. [START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF][START_REF] Imperatori | Broad-band near-field ground motion simulations in 3-dimensional scattering media[END_REF][START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF].

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

Conclusions

We performed a series of numerical simulations to investigate how velocity heterogeneities affect seismic ground motion in a complex sedimentary basin. The velocity heterogeneities are modeled as a stationary random process following a Von Karman ACF. We vary the coefficient of variation and the correlation length of the ACF to analyze parametrically how they affect the ground motion and ground motion intensity measures.

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

The presence of velocity heterogeneities in the basin mainly scatter the late arrivals and to a fewer extent the direct waves; hence affecting the resulting ground motion intensity measures. The simulations show that the envelope and phase misfits (EM and PM), between the wavefields of the random and deterministic media, become large as the coefficient of variation of the random velocities increases. The increase in the misfits is spatially correlated with the presence of low-velocity zones in the random models. The misfits also seem to augment with an increase of the random media's correlation length, although less pronounced. This is probably because the size of the basin is relatively small; thus, there is not enough propagation time for their effect to be seen on the travel times.

Regarding the PGV, the presence of random velocity leads to an extended spatial variability, which increases with the coefficient of variation of the velocity fluctuations. The results also show that the PGVs are very sensitive to the random media's realization, as evidenced by the high variability in the PGV computed between the 10 realizations. Similar trends are observed for the pseudo-spectral acceleration. Moreover, the variability in the PSA due to velocity heterogeneities was found to be higher for the shorter periods (0.17 s), which corresponds to the dominant frequency content of the incident wavefield.

The effect of the random velocity heterogeneities on the 2D spectral ratios is less pronounced for the random media with 5% coefficient of variation. Whereas for the 30% case, the basin seismic response becomes quite different from the deterministic model. We also observe a reduction in the maximum amplitude of the mean spectral ratios that is more pronounced with a coefficient of variation of 30%.

This reduction is also reported in the studies by [START_REF] Assimaki | Effects of spatial variability of soil properties on surface ground motion[END_REF] and [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF]. It is due to a smoothing effect in the averaging of the spectral ratios of different realizations of random media. For high coefficient of variation of the velocity fluctuations, the variance between the different realizations of random media shows the difficulty in considering the small-scale velocity fluctuations for site-specific analysis, where there is only one realization of the subsurface. We recommend using the percentiles of the distribution of the amplification functions instead of the mean and standard deviation to describe the site response in random media. Indeed, the mean may underestimate the ground motion amplification at certain frequency bands compared to the deterministic case.

To sum up, these results show that the effect of random velocity fluctuations on ground motion indicators are important, as they increase their variability. The main controlling factor, in this particular case, is the coefficient of variation of the velocity fluctuation. The differences in the values of ground motion intensity measures for the different configurations of the random media highlight the importance of a fine characterize of soil properties in a site.

The current study is by no means exhaustive, and the present conclusions may not be generalized to all cases. Our results show high complexities in the computed ground motion despite the simplifications in the modeling assumptions -we considered isotropic random media and assumed that all the sedimentary layers have the same velocity distributions, which may not be true in reality. The simulations were performed using a wavelet with a dominant frequency as a source time function. As next steps, it would be necessary to perform the analysis using a realistic source time functions, having a broadband frequency content, to conjointly investigate the effects of the source time function and medium properties. It would also be neccessary to model the velocity heterogeneities using anisotropic random media.

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

Acknowledgments

The authors would like to thank Etienne Bertrand for providing us with 2D velocity model of the sedimentary basin. We also thank Emmanuel Chaljub and Jean-Paul Ampuero for constructive discussions that helped to improve this article. We also thank Walter Imperatori for constructive discussions about the generation of random media. The simulations were performed on the IRSN ACALU computer cluster.

3.9 Appendix 3.9.1 Results for model MR5 The average amplitude and phase difference (EM and PM) between MR5 and the reference medium is plotted in Figure 3.17 top and bottom, respectively (cyan curve). The EM and PM for MR1 and MR3

with the same correlation length but different c v are also plotted (red and magenta curve, respectively).

The curves clearly show the dependence of the EM and PM to the c v of the random models. An increase in the c v leads to,on average, an increase in EM and PM. A closer look at the curves between 1000 m and 1300 m shows that both MR3 and MR5 are close. This is probably because the velocity contrast between layers c1 and c3 directly beneath 1000 and 1300 m is 11 %, and is destroyed after perturbation by MR3

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin Complementary work to the article

Effect of the basin and layer's geometry on the TF

This section explores the role of the basin's geometry and layering on the computed ground motion.

Such a sensitivity study has been conducted for example by [START_REF] Semblat | Seismic Wave Amplification: Basin Geometry vs Soil Layering[END_REF] who showed that the basin's geometry mainly controls the spectral amplification functions and that the superficial soil layers mainly influence the high-frequency content of the ground motion.

To investigate the basin layering effect on the computed ground motion in the Nice basin, we performed simulations in two modified versions of the initial deterministic velocity model; (1) model M1 (Fig. 3.20a) in which we consider a homogeneous filling with a velocity of 266 ms -1 (which corresponds to the harmonic mean of the layers in model M), and (2) model M2 (Fig. 3.20b) in which we assume tabular layers. The tabulation was obtained by laterally extending a 1D cross-section, at x = 1200 m, of model M.

This analysis helps us interpret the amplification observed in transfer function computed in the deterministic basin model M by discriminating the effects of the complex layering and basin geometry. It is important to note that the layers in model M have close velocities, hence the conclusion drawn here can not be generalized to other basins.

The computed seismograms in both models are plotted in Figure 3.21. Visually, both models show very similar wavefields, which are also similar to those of model M. This suggests that the incident wavefield is not very sensitive to the layering in the basin. In Figure 3.22, we compute the pairwise amplitude and phase differences (EM and PM) between the wavefields of three velocity models in two frequency bands (0.5 -3 Hz and 3 -10 Hz). The black curve corresponds to the misfits between the pair M1-M2, whereas the red and blue curves correspond to the misfits between the pairs M-M1 and M-M2, respectively. We observe that the misfit between M1 and M2 (black curves) is approximately constant regardless of the frequency band. However, in the low frequencies, the pairs M-M1 and M-M2

(red and blue curve respectively) have misfits that increase to the right, where the basin is deeper. At higher frequencies, the differences between M-M1 and M-M2 are mainly located where model M has a complex layering structure (curved layers acting as secondary basins).

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin suggests that the high-frequency amplification in the Nice sedimentary basin is due to the combined role of the layer geometry and thickness.

We also introduce lateral heterogeneities in models M1 and M2 to investigate whether their role is more important than the medium's detailed layering. The heterogeneities follow the characteristics of the random model R3, with a correlation length of 10 m and a c v of 30 %. We plot in Figures 3.24a We observe for models MR1 and MR2, with a c v of 5 %, that A b I is only slighly changed by the velocity heterogeneities compared to the deterministic case. The mean and median curves all follow the same trend as deterministic case, and the spread between the realizations is low. The corresponding ratio of the random model curves with deterministic case fluctuates between 0.8 and 1.4.

For models MR3 and MR4, with a c v of 30 %, the velocity heterogeneities lead to high spatial variabil- shows that the mean is not adequate in this case to describe the A b I variations. The corresponding ratios show that the velocity heterogeneities, in this case, lead to an increase in the A b I compared to the deterministic case by factors that can locally reach 5.

In Figure 3.27c, we plot the Arias based duration in the four random models MR1 to MR4 (top to bottom plots, respectively). Similarly to A b I plots, the black curve is the duration in the deterministic model. The blue curve is the mean of the 10 realizations of the random models. The red shaded zone is the inner-68th percentile interval, and the red curve is the median.

For models MR1 and MR2, the mean and median duration along the basin are similar, and they follow the same trend as the deterministic case. Their values are overall similar to the deterministic curve, but certain points along the surface show local increases compared to the deterministic case (for example, between 1240 m and 1260 m). The spread between different realizations is quite low.

In models MR3 and MR4, the mean and median curves still have similar spatial variation, but their trend totally differs from the deterministic case. This is because the velocity heterogeneities, in this case, destroy the deterministic structure of the basin layers. The velocity heterogeneities cause the duration along the basin profile to be homogenized, i.e., the curves become flat or uniform around ≈ 2.5 seconds.

The associated spread between the realizations also increases compared to MR1 and MR2.

We can conclude from these figures that, when the c v of the velocity heterogeneities increases, the

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin A b I is also increased compared to the deterministic case, by factors as much as 5. The variability between the realizations of the random models also increases. In this case, the median is more appropriate to describe the central tendency of the variation between the different realizations as the mean can be biased by extreme values of the A b I in the individual realizations. Regarding the duration, for low values of c v , the durations are similar to the deterministic case. In contrast, when the c v is very high such that the basin's deterministic structure is destroyed, the durations along the basin profile are homogenized; that is, they have a uniform value along with the basin profile.

The P-SV case

The article was devoted to SH wave propagation. We performed similar numerical experiments for the P-SV case to assess the effect of the seismic wave polarization. The input motion to the P-SV simulations was a vertically incident SV plane wave. The P-waves velocities (V p) of each layer were inferred from S-waves velocities (V s) by assuming a constant Poisson ratio of 0.4, the P-wave quality factor (Q p ) in the materials was chosen to be 100. In what follows, we present the velocity wavefields, the PGV, and TF computed in the model M, and discuss their differences with respect to the SH case.

Velocity wavefield

We plot in Figure 3.

Peak ground velocity

Similarly to the SH case, we compute and plot in Figure 3.31 the PGV for one realization of the 4 random media and the ratio with respect to the deterministic model. The spatial variability of the PGV increases with the c v of the random medium, as shown by the curves of models MR3 and MR4 (magenta and green curves on Figure 3.31: top, respectively). Models MR1 and MR2 (red and blue curves, respectively) with a c v of 5 % have approximately the same spatial trend and values as the deterministic case.

Their ratio with the deterministic curve is thus approximately unity along the profile. In contrast, the ratio between models MR3 and MR4 and the deterministic curve varies in the range 0.5 to 2.4, implying either an increase or decrease in the PGV compared to the deterministic medium by factors as much as 2. We also plot in Figure 3.32 the coefficient of variation of the PGV between the 10 realizations of the random media. As for the SH case, the variability between the different realizations increases with the c v of the velocity heterogeneities. Hence, models MR3 and MR4 have higher variability than MR1 and MR2.

Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin

Spectral ratio

We compute the 2D TF of the basin for P-SV wave propagation following the same procedure as in the SH case, using the horizontal component of motion. The TF of the deterministic model is plotted in Figure 3.33a. The fundamental resonance frequency of the basin for the P-SV propagation still lies at around 2 Hz, as in the SH case. However, the amplification patterns differ from the SH case due to differences in the propagation. The TFs for one realization of the random models are plotted in Figures

General conclusions to the chapter

In this chapter, we presented a detailed investigation of the effect of the characteristics of random velocity heterogeneities on the seismic wave propagation and ground motion intensity measures in the Nice sedimentary basin. We observed that the dominant characteristic of the velocity heterogeneities that controls the ground motion is the coefficient of variation, which scales the observed variability and scattering. The effect of the correlation length is more visible when the random heterogeneities have a low coefficient of variation. As the coefficient of variation of the random heterogeneities increases, they overshadow the effect of the correlation length on the wave propagation. This is probably due to the small size of the basin, which leads to small total propagation times of the direct waves through the heterogeneities; hence their expression is less observed on the time signals. This effect of the basin size The chapter is an article in preparation to be submitted to Geophysical Journal (GJI).

Abstract

The unmodeled effects of random velocity heterogeneities can lead to either an overestimation or underestimation of the estimated site effects -compared to the reference model -using numerical simulations of seismic wave propagation. This study investigates the impact of velocity heterogeneities on the Nice (France) basin response. The velocity heterogeneities in the basin are modeled as spatially correlated random fields. The characteristics of the random fields (vertical correlation length and coefficient of variation) are determined from available borehole logging data in the basin. Four sets of random models are generated, with varying horizontal correlation length. The random models are then superimposed over a 2D reference velocity model of the basin, constructed from different geophysical measurements and borehole data. We simulate 2D P-SV seismic wave propagation in 20 realizations of the random models, using a record of the February 25, 2001, Nice earthquake (Ml 4.6) at an outcropping bedrock station (NBOR, belonging to the RAP Frence Accelerometric Network) as incident wavefield.

The wavefield has a relatively broadband frequency content between 0.1 and 10 Hz. We then analyze the changes induced by the small-scale velocity heterogeneities (compared to the simulations in the reference velocity model) on the intensity measures of the simulated surface ground motion and the basin response -that is, the spectral ratio, pseudo-spectral acceleration response (PSA), and the Arias based intensity. These simulations show that the velocity heterogeneities lead to an average increase in the surface ground motion intensity in the basin, especially at the basin edges. They also lead to an average increase in the PSA at the basin's resonance frequency and an increase in its spatial variability at higher frequencies (above the resonance frequency). Regarding the spectral ratios, they amplify or de-amplify the high-frequencies amplifications of the basin. A closer look at the spectral ratio computed at the NLIB RAP station -located on the 2D basin profile -showed that the velocity heterogeneities lead to a shift of the fundamental resonance frequency and higher modes to a lower frequency. A similar effect is observed for the 1D stochastic media, extracted from 2D profiles, on spectral ratios computed at NLIB. A comparison between the 2D and 1D spectral ratio at a receiver NLIB showed that the 2D spectral ratio has a higher amplification at the resonance frequency because of locally diffracted waves and lateral propagation in the basin, which is absent in the 1D case. We also investigate whether the earthquake-to-earthquake variability of the observed spectral ratio can be explained by the presence of velocity heterogeneities in the basin. Our results show, in accordance with other studies, that the variability in the spectral ratios induced by the velocity heterogeneities could be part of the aleatory variability of the basin response, but it cannot fully explain the earthquake-to-earthquake amplification variability at the NLIB station.

This study highlights the importance of using in-situ measurements and earthquake recordings to better understand the effects of small-scale velocities heterogeneities for site effect assessment.

Introduction

Seismic waves are modified by the mechanical properties and the geometry of the local sub-surface geological layers as they propagate through them. In the presence of geological configurations like sedimentary basins, the seismic waves can be trapped, and the characteristics of the soft soil deposits can strongly affect the ground motion. A phenomenon known as site effects, which is an important Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin factor that can increase the amplitude, duration, and spatial variability of the incident wavefield.

Several approaches, which include theoretical, empirical, and numerical methods, can be used to estimate site effects. Among them, numerical simulations of seismic wave propagation are of great appeal, as they provide a direct means to quantify the amplification of the seismic waves. Over the last decades, different numerical methods have been developed and applied to compute ground motion in sedimentary basins, e.g., the finite difference method (e.g. [START_REF] Gelis | Influence of a sedimentary basin infilling description on the 2-D P-SV wave propagation using linear and non-linear constitutive models[END_REF], spectral element method (SEM) (e.g. [START_REF] Chaljub | Quantitative comparison of four numerical predictions of 3D ground motion in the Grenoble valley[END_REF], the discontinuous Galerkin method (e.g. [START_REF] Peyrusse | A nodal discontinuous galerkin method for site effect assessment in viscoelastic media -verification and validation in the nice basin[END_REF], the boundary element method (e.g. [START_REF] Semblat | Seismic Wave Amplification: Basin Geometry vs Soil Layering[END_REF], and the finite element method (e.g. [START_REF] Bielak | One-vs two-or three-dimensional effects in sedimentary valleys[END_REF].

Performing realistic ground motion simulations using any of the above-listed methods requires prior knowledge of the medium to use in the computational model: the velocity distribution, the nonplanar interface geometries between the layers, density, and the anelastic attenuation inside the layers. In this study, we do not consider any non-linear soil behavior. The soil properties of a medium can be estimated either directly from logging techniques (e.g. [START_REF] Holliger | Upper-crustal seismic velocity heterogeneity as derived from a variety of p-wave sonic logs[END_REF][START_REF] Shiomi | Broad-band power-law spectra of well-log data in Japan[END_REF] or by using geophysical imaging methods (e.g., spectral analysis of surface waves (e.g. [START_REF] Kalinski | In situ estimate of shear wave velocity using borehole spectral analysis of surface waves tool[END_REF], multichannel analysis of surface waves (e.g. [START_REF] Park | Multichannel analysis of surface waves[END_REF], among other techniques). On the one hand, logging data provide direct measurements of the soil properties but are economically costly and difficult to elaborate over large distances. On other hand, when building models at a basin scale, geophysical imaging techniques are used to determine rough estimates of the properties in each geological layer, typically modeled as piecewise constant or piecewise linear functions of depth. Such a modeling approximation is generally satistifactory for long-period numerical simulations, where the length scale of the variability of the seismic properties about the piecewise function is smaller than the wavelengths being modeled. Indeed, since the the seismic waves propagating within the subsurface interior are always frequency band-limited, any subsurface mechanical properties variations small than a fraction of the shortest propagated wavelength can be viewed as smooth [START_REF] Capdeville | Residual homogenization for seismic forward and inverse problems in layered media[END_REF].

In the frequency range of engineering interest (≥ 1 Hz), the effects of small-scale fluctuations in soil properties on the seismic wave propagation can become important and, therefore, require consideration. Recent studies have shown that they can considerably increase the intensity and variability of ground motion. For example, [START_REF] Pagliaroli | Seismic microzonation of the centeral archaelogical area of Rome: results and uncertainties[END_REF] observed in the period range of 0.1 -0.5 s that the Housner intensity of simulated seismic ground motion in the Central Archaeological Area of Rome could locally increase by 50 % with the consideration of small-scale velocity heterogeneities. [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] and [START_REF] Tchawe | Effects of 2D random velocity perturbations on short-period (≤ 1s) ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin[END_REF] also observed that small-scale velocity heterogeneities at the sedimentary scale could modify the spectral amplification patterns/values and increase the variability of the ground motion intensity measures. Other studies, at the crustal-scale, have also highlighted an increase in the variability of the ground motion due to velocity heterogeneities (e.g. [START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF]Imperatori andMai, 2013, 2015;[START_REF] Iwaki | Effects of random 3D upper crustal heterogeneity on long-period ( 1s) ground-motion simulations[END_REF]. [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF] observed that modeling the full 3D seismic wavefield with spatially variable soil properties could help to explain the observed empirical spectral ratios for sites belonging to the Kiban-Kyoshin network in Japan.

Since it is impossible in practice to detailly measure the small-scale variations of soil properties (henceforth regarded as velocity heterogeneities), they are generally modeled using statistical methods (e.g. [START_REF] Frankel | A review of numerical experiments on seismic wave scattering[END_REF][START_REF] Frenje | Scattering attenuation: 2d and 3d finite difference simulations vs. theory[END_REF][START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF]Imperatori andMai, 2013, 2015;[START_REF] Iwaki | Effects of random 3D upper crustal heterogeneity on long-period ( 1s) ground-motion simulations[END_REF][START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF]. In statistical methods, the small-scale fluctuations in the velocity are generally considered to be random and spatially correlated (e.g. [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF][START_REF] Goff | Stochastic modeling of seafloor morphology: inversion of sea beam data for second-order statistics[END_REF].

As such, they can be characterized using an autocorrelation function of space, which is calibrated using Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin only a few numbers of statistical parameters (the correlation length and coefficient of variation in the most general case) of the fluctuations.

In this study, we analyze the contribution of small-scale velocity heterogeneities to the ground motion intensity measurements and site response in the Nice (France) sedimentary basin. The basin's velocity heterogeneities are modeled as spatially correlated random fields characterized by a Von Karman autocorrelation function. The vertical correlation length and the velocity coefficient of variation in the sedimentary layers are deduced from the available logging data in the basin, and are used to constrain the autocorrelation function. We then simulate 2D P-SV seismic wave propagation of the February 25, 2001 Nice earthquake in both the basin's deterministic and random velocity models. We compare the ground motion inensity measurements with the ones recorded at the NLIB RAP station located on the basin.

This paper is organized as follows; In section 4.3, we present the study site, 2D velocity model, and the available borehole data at the sedimentary basin. In section 4.4, we describe the procedure we employ to characterize and model the small-scale velocity heterogeneities in the basin. The numerical experiments performed are also presented. In section 4.5, we present the results of the numerical simulations, and we discuss them and provide conclusions in section 4.6.

Study site

Seismotectonic context and Nice earthquake (2001)

The city of Nice is located in the South-East of France, and the region resides at the junction between the French-Italian Alps and the offshore Ligurian basin (Figure 4.1b), which is one of the most seismically active areas in western Europe [START_REF] Salichon | A mw 6.3 earthquake scenario in the city of nice (southeast france): Ground motion simulations[END_REF]. The seismicity in the city itself is considered to be low to moderate. The region presents a daily microseismicity, and an earthquake of magnitude (M) 4.5 -5 occurs nearly every 5 years [START_REF] Courboulex | Seismic hazard on the french riviera: observations, interpretations and simulations[END_REF]. The city is subjected to high seismic risk due to its dense population.

A moderate-sized earthquake, that was felt by the population, occurred offshore at about 25 km southeast of Nice (see Figure 4.1b) on February 25th, 2001 (Ml 4.6). The earthquake was generated by a fault roughly parallel to the French Riviera coastline with a reverse source mechanism [START_REF] Courboulex | Seismic hazard on the french riviera: observations, interpretations and simulations[END_REF]. The event's characteristics as reported by [START_REF] Courboulex | Seismic hazard on the french riviera: observations, interpretations and simulations[END_REF] are summarized in Table 4.1.

This event was recorded by a local network of the French permanent accelerometric network (RAP).

This network was particularly designed to study site effects in the region, with some stations located on alluvial deposits while others on hard rocks. Station NBOR was installed from 1998 to 2008 to use it as a reference station for site response assessment in the Nice area. The station was uninstalled because some studies questioned its reliability as a reference station (e.g. [START_REF] Bertrand | Seismological measurements for site effect investigation in Nice, France[END_REF][START_REF] Hollender | Can We Trust High-Frequency Content in Strong-Motion Database Signals? Impact of Housing, Coupling, and installation Depth of Seismic Sensors[END_REF]. Indeed, a reference site should, in theory, exhibit no site effect. However, finding an appropriate reference site is difficult in practice Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin (e.g. St Fleur et al., 2016;Laurendeau et al., 2017b;[START_REF] Hollender | Can We Trust High-Frequency Content in Strong-Motion Database Signals? Impact of Housing, Coupling, and installation Depth of Seismic Sensors[END_REF]. Two other reference stations were installed in the area after NBOR -NCER from 2012 to 2018 and GLAN, currently in activity since 2018. At the moment, only the V s30 value of the site beneath NBOR (927 ± 202 ms -1 ) has been reported in the literature [START_REF] Hollender | Characterization of site conditions (soil class, vs30, velocity profiles) for 33 stations from the french permanent accelerometric network (rap) using surface-wave methods[END_REF]. This value is close to the shear wave velocity of the bedrock (1000 ms -1 ) in the velocity model (which we present in the next section) we use in this study. For this reason, we will use NBOR as the reference station in this study. We compute its horizontal-to-vertical spectral ratio (HVSR) to verify that it exhibits no major site effects in the frequency range (0.1 to 10 Hz) of our study. The HVSR is computed on a 10 seconds S-wave window of the East-West components of 8 earthquakes (see Figure 4.1a, coloured dots), whose characteristics are summarized in Table 4.2, with a signal-to-noise ratio greater than 3. The median HVSR at of the 8 earthquakes is represented by the blue curve in Figure 4.3. We can observe that the curve is approximately flat between 1 and 10 Hz with an amplitude varying between 1 and 2. Between 0.5 and 1 Hz, HVSR values are higher than 2, which is consistent with a local site effect already mentionned. This means that between 1 and 10 Hz, the station displays not major site effect, hence we can adequately use NBOR as a reference station in this study in the frequency range of 1 to 10 Hz. We also plot the HVSR at NLIB in

2D velocity model

A 3D model of the quaternary fillings in the Nice area was proposed by [START_REF] Bertrand | 3d geotechnical soil model of nice, france, inferred from seismic noise measurements for seismic hazard assessment[END_REF] following several research projects (GEMGEP and GEMITIS projects, Bard et al., 2005b;[START_REF] Arnal | Projet GEMITIS-Nice : Evaluation des dommages directs d'un séisme sur la ville Nice, scénario de risque, extension des résultats au départements des Alpes Maritimes[END_REF]. The model was constructed by synthesizing all the available geological, geotechnical, and geophysical data over the city and ambient vibration recordings analysis.

Geotechnical log

A Geotechnical database of the Nice area was constructed by the CETE Méditerranée [START_REF] Bertrand | Etude préalables au PPR sismique de Nice : Modèle géotechnique 3D[END_REF], nowadays called CEREMA, during the GEMGEP project. The database comprises approximately 500 borehole logs. This database was used together with surface waves dispersion analysis (SWDA) to build the 3D deterministic velocity model of the basin, in which the velocities are constant within each layer. Standard penetration tests (SPT) were performed in 73 of the 500 boreholes (red dots in Figure 4.1b). The N-value at each borehole of the database was reported at 1.5 m depth lag. This value can be linked to Vs using empirical correlation equations. Several empirical correlation equations exist in the geotechnical literature. We summarize in Table 4.7 of Appendix 4.7 the different equations used to build the 3D velocity model [START_REF] Bertrand | 3d geotechnical soil model of nice, france, inferred from seismic noise measurements for seismic hazard assessment[END_REF]. The velocity estimates in these boreholes will be used to calculate the vertical correlation length and coefficient of variation in the basin layers in section 4.4.1.1.

2D Random velocity models

Spatial variability modeling

Determining the small-scale spatial variations of a soil property in any direction, in principle, requires a sufficiently high number of measurements. However, this is impossible in practice, and thus the spatial variation of soil properties is usually approximated using statistical models.

The heterogeneous velocity (v) of the medium can therefore be decomposed into a deterministic (or mean) function and fractional fluctuations (assumed to be a random process) about the deterministic trend as follows [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]:

v(x) = v 0 (1 + (x)) (4.1)
where x = (x, z) is the 2-D position vector representing the space direction (horizontal and vertical directions respectively), v 0 is the mean compressional V P or shear wave velocity V S of the medium and (x) = δv(x)/v 0 is the fractional fluctuation of the wave velocity which is a zero mean random process.

If the random or stochastic process is stationary (or at least weakly stationary), it can be characterized by an autocorrelation function (ACF). Stationarity is an important prerequisite for stochastic modeling of material properties because the statistical procedures employed are based on the assumption that data samples consist of stationary observations. The ACF gives a statistical measure of the spatial scale and the strength of the medium heterogeneity. Furthermore, when the randomness is stationary, the ACF becomes a function of separation distance (r ≡ |x|) only [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF].

It is not possible to calculate the real autocorrelation function of a stochastic process because data sets are, in practice, always limited in size [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]. Hence, it is necessary to refer to the sample ACF calculated from a dataset considered representative of the stochastic process. The discrete sample ACF of a data set is given by [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]:

R(|τ j |) = n-j i=1 (s i -m s )(s i+j -m s ) n i=1 (s i -m s ) 2 , j > i (4.2)
where s is the material property being measured (V s in our case), m s is the dataset's mean, |τ j | = j∆x ,j∈N is the separation distance, n is the number of data points, and ∆x is the sampling interval. An Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin absolute value has been used to denote the lag distance because the ACF is symmetric about zero lag distance.

To identify how a given material property varies spatially, one needs to fit an autocorrelation model to the sample ACF calculated by equation 4.2. Different autocorrelation models have been proposed

in the geotechnical (e. [START_REF] Fenton | Random field modelling of cpt data[END_REF][START_REF] Phoon | Identification of statistically homogeneous soil layers using modified barlett statistics[END_REF][START_REF] Uzielli | Random field characterisation of stress-normalised cone penetration testing parameters[END_REF] and seismological literature (e.g [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF][START_REF] Klimes | Correlation functions of random media[END_REF].

In this study, we assume the Von Karman ACF to model the correlation structure of the velocity heterogeneities in the sedimentary basin. The Von Karman ACF has been widely used in Seismology to model the Earth heterogeneities mostly for wave propagation problems at the crustal scale in few studies at the basins scale (e.g. [START_REF] Frankel | A review of numerical experiments on seismic wave scattering[END_REF][START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF][START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF][START_REF] Imperatori | The role of topography and lateral velocity heterogeneities on near-source scattering and ground-motion variability[END_REF][START_REF] Nakata | Stochastic characterization of mesoscale seismic velocity heterogeneity in long beach, california[END_REF]. Even though it is classically used in seismology for wave scattering studies, the ACF does not belong to currently used ACFs in geotechnical engineering (e.g. [START_REF] Vanmarcke | Random fields: analysis and synthesis[END_REF]. [START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF] notes that, no reliable physical correspondence between the soil type and the type of correlation structure seems to be indentified, and there that a purely numerical approach to determine the ACF is justified. [START_REF] Salloum | Evaluation de la variabilité spatiale des paramètres géotechniques du sol à partir de mesures géophysiques: application à la plaine alluviale de Nahr-Beyrouth (Liban)[END_REF] simulates the wave propagation in 1D stochastic models using alternatively the squared exponential and the decreasing exponential ACFs and a Vs coefficient of variation of 20 %. The geometric means of the amplification are very similar in both cases. The ACF choice has an influence, albeit limited on the logarithmic standard deviation of the amplification. In this study, we do not test different ACFs than the Von Karman ACF. Such a sensitivity study should be conducted in the future.

The 2D analytical expression of the Von Karman ACF is expressed in equation 4.3 [START_REF] Goff | Stochastic modeling of seafloor morphology: inversion of sea beam data for second-order statistics[END_REF][START_REF] Carpentier | Underestimation of scale lengths in stochastic fields and their seismic response: a quantification exercise[END_REF] as:

R(r) = r ν K ν (r) 2 ν-1 Γ(ν) (4.3)
where r is the weighted radial autocorrelation lag ( x 2 /a 2 x + z 2 /a 2 z ), ν is the Hurst number, K ν (r) is the second modified Bessel function of fractional order ν, and Γ is the Gamma function. The corresponding power spectrum of the Von Karman ACF (equation 4.3) is expressed by:

P (k) = 4πνa x a z 2 ν-1 Γ(ν)(1 + k 2 ) ν+1 (4.4)
where a x and a z are the horizontal and vertical correlation lengths, respectively; k is the weighted radial wavenumber (k = k 2 x a 2 x + k 2 z a 2 z , with k x , k z being the horizontal and vertical wavenumbers, respectively).

The statistical parameters (correlation length (a) and coefficient of variation (c v )) of the correlation function will be determined from the logging data in the subsequent section.

Spatial scale and magnitude of the velocity heterogeneities

A large part of the GEMGEP project was devoted to deciphering the geological layers crossed by each borehole in Layer 5 is crossed by most boreholes and contains a relatively larger number of data points (59.5 % of the total number of measurements) than the other layers. As a result, the statistics on the data points in layer 5 will be more reliable.

First, we convert the N-values (or SPT resistance) to V s using the correlation equation by [START_REF] Ohta | Empirical shear wave velocity equations in terms of characteristics soil indexes[END_REF] We then fit in Figure 4.6, a Gaussian probability density function (PDF) of each layer's velocity distribution. For this dataset, a Gaussian distribution best fits the data than a log-normal distribution. The mean (µ), standard deviation (std), and coefficient of variation (cov) of each layer are reported on the plot. The dotted vertical line denotes the layer V s in the reference model (Table 4.3). For some layers, the deterministic velocity does not coincide with the mean of the distribution because the deterministic velocity model was constrained using both the ensemble of correlation laws listed in Table 4.7 and surface waves dispersion analysis (SWDA). The velocity distributions of layers 5 and 7, which hold the largest number of data points, adequately follow a Gaussian PDF, as shown by their histograms. The remaining layers have very few data points for the estimated Gaussian distributions to be reliable. We quantify the magnitude of the velocity heterogeneities in each layer by the coefficient of variation (cov)

of the velocity distribution. The cov of layers 1, 2, and 3 are 45 %, 28 %, and 32 %, respectively, whereas layers 4 to 8 have cov values of ≈ 22 %. Despite the lower number of data points in layers 2, 3, 4, 6, and 8, we decided to retain their cov values. Because layer 1 has the lowest number of measurements, we dropped the computed cov value of layer 1, which was extremely high, and replaced it with that of layer 2. Indeed, retaining the cov value of layer 1 would have led to very low velocities in the random model, requiring a very fine mesh for stable numerical simulations, whereas layer 1 is only present in some specific parts of the basin. The final cov values retained for each layer in the 2D velocity model are summarized in Table 4.4. It is important to note that the computed cov values do not characterize only the inherent variability of the soil velocities but also include sources of uncertainties such as (1) the measurement uncertainties, (2) the statistical estimation error, and (3) the error due to the data transformation from SPT resistance to shear velocity. We cannot separate the measurement and statistical estimation errors, but we explore the uncertainties due to data transformation in Appendix 4.7. We observe a variability of 26 % between the cov values computed by the different equations, implying an uncertainty of ± 6 % on the cov values considered in this study, depending on the equation chosen in Table 4.7. In this study, we consider this uncertainty in the cov values to be secondary.

The second parameter to be characterized from the data is the correlation length of the velocity heterogeneity, which designates the spatial extent over which the velocity fluctuations can be considered locally homogeneous. In 2-D, the correlation length is defined by the horizontal and vertical correlation distances (a x and a z ), respectively. The a x and a z values can be estimated by fitting an ACF model to sample ACF (equation 4.2) in both space directions. We shall calculate only a z from the data as the Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin

We applied absorbing boundary conditions at the bottom of the model and imposed vanishing traction at the surface. Intrinsic attenuation was taken into account by the frequency-independent Q values [START_REF] Liu | Efficient modeling of q for 3d numerical simulation of wave propagation[END_REF] modeled following the rheological behavior of the generalized Maxwell body [START_REF] Emmerich | Incorporation of attenuation into time-domain computations of seismic wave fields[END_REF]. We defined the S and P waves quality factor (Qs and Qp) to be 10% the S and P waves velocities of the layers (see Table 4.3).

Following [START_REF] Peyrusse | A nodal discontinuous galerkin method for site effect assessment in viscoelastic media -verification and validation in the nice basin[END_REF], we compute seismic wave propagation in 2D models. This is consistent with the local geology of the Nice basin. Moreover, [START_REF] Ktenidou | Directional dependence of site effects observed near a basin edge at Aegion, Greece[END_REF] showed that the local geology governs the directional effects of the wavefield in a small basin in Greece. Indeed, they find that the maximum discrepancy between amplification of the horizontal components occurs when they are rotated with respect to the orientation of the basin edge. For our simulations, we use as input motion the East-West horizontal component of the wavefield recorded at NBOR (Figure 4.8a) for the February 25, 2001, Nice earthquake. The spectrum of the wavefield has a broadband frequency content (0.1 to 10 Hz), with a dominant content between 0.1 and 2 Hz and is shown in Figure 4.8b. The maximum acceleration of the ground motion at NLIB is 0.056 ms -2 ; hence, the basin's soil behavior is expected to be linear.

The wavefield is injected into the numerical model as a vertically incident plane wave with an inplane (SV) polarization. An array of 623 receivers is placed at the free surface of the model with an inter-receiver spacing of 5 m. We computed the wavefield for 30 secs, with a calculation time step of 20 µs. The surface receivers record the solution at the surface after every 10 ms of wave propagation.

We computed seismic wavefields in both the deterministic and random velocity models of the sedimentary basin. The random media are generated by applying a 2D uniformly distributed random phase to the 2D power spectrum (equation 4.4), and then transforming back to the space domain using the inverse 2D Fourier transform. The resulting random fields were standardized and scaled to the desired coefficient of variation of each layer. Each random medium was generated on a rectangular grid of 1200 m by 250 m with a constant grid step of 0.2 m in both directions. The rectangular grid was then interpolated over the spectral element mesh of the sedimentary basin using a nearest-neighbor interpolation algorithm.

We generated random fields of V s and V p only. Both fields were generated such that a constant Poisson ratio of 0.4 was maintained in the materials. All other material parameters (density and anelastic attenuation) were considered constant. For each random medium, we generated 20 realizations with different random seeds. Figure 4.9 shows one realization of each random medium after interpolation over the sedimentary basin. The spatial correlation of the velocities increases in the horizontal direction as the aspect ratio increases, as expected. jor peak occurs at ≈ 6.3 Hz. The second major peak is, on average, more amplified than the first peak.

Results

Wave propagation in the basin

The TF for the 2001 Nice earthquake (blue curve) has two peaks around the fundamental resonace fre-Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin quency and a third peak at 6.3 Hz. This figure shows that there is a high earthquake-to-earthquake variability between the EFTs. This large variability in the earthquake-to-earthquake ETF has made many authors preconize the computation of the ETF using multiple earthquakes (e.g. [START_REF] Field | Earthquake site response estimation: A weak-motion case study[END_REF]Boore, 2004;[START_REF] Perron | Robustness of kappa (κ) measurement in low-to-moderate seismicity areas: Insight from a site-specific study in Provence, France[END_REF].

For example, [START_REF] Perron | Apport des enregistrements de séismes et de bruit de fond pour l'évaluation site-spécifique de l'aléa sismique en zone de sismicité faible à modérée[END_REF] found that at least 15 earthquakes are required to assess properly a site's ETF in a practical framework. Even though the database used in this study comprises only 8 earthquakes recorded simultaneously at NLIB and NBOR, we discuss plausible sources of the variability observed in the ETF at NLIB.

Uncertaintity in the empirical TF

It is well known that the empirical estimates of a site's TF, using different earthquakes, exhibits large uncertainties (e.g. [START_REF] Field | Earthquake site response estimation: A weak-motion case study[END_REF]Boore, 2004;[START_REF] Perron | Robustness of kappa (κ) measurement in low-to-moderate seismicity areas: Insight from a site-specific study in Provence, France[END_REF]. [START_REF] Maufroy | Source-related variability of site response in the mygdonian basin (greece) from accelerometric recordings and 3d numerical simulations[END_REF] observed, using numerical simulations, that site response estimates are strongly dependent on the source location.

The fact that the azimuth and the epicenters of earthquakes vary can cause the seismic wavefield to illuminate the site at different incidence angles, leading to a different site response depending on the source location. [START_REF] Perron | Robustness of kappa (κ) measurement in low-to-moderate seismicity areas: Insight from a site-specific study in Provence, France[END_REF] also observed a high earthquake-to-earthquake variability on the computed TFs at the Argostoli (Greece) basin due to these effects. This means that two earthquakes of similar magnitudes could produce different site responses depending on the incident wavefield's direction. In his study, he analyzes the different sources of uncertainties in computed earthquake-toearthquake TF and concludes that the uncertainties are not only random but epistemic. He found that these epistemic uncertainties can be related to source parameters such as the back-azimuth and the epicentral distance at the first order, and the earthquake's local magnitude at the second order. He assessed an aleatory uncerntainty of about 20 %.

Though our numerical simulations don't consider these variability parameters, we are interested in investigating to which extent soil heterogeneities can explain part of the observed variability in the ETF at NLIB.

Pseudo-spectral acceleration

We plot the basin's 5 % damped surface acceleration response spectra (PSA) at 2, 4, 6, and 8 Hz in

Discussions and conclusions

We analyzed the effect of Vs heterogeneities in the Nice sedimentary basin on the computed ground motion of the Nice 2001 earthquake. For this, we modeled the Vs heterogeneities in the basin as spatially correlated random fields characterized by a Von Karman ACF. The c v and a z of the ACF were obtained from borehole data available in the basin. We then performed 2D P-SV simulations using the East-West component of the Nice 2001 earthquake ground motion recorded at the outcropping bedrock station NBOR as the incident wavefield. We compared how the Arias based intensity of the ground motion and the basin's response (TF and PSA) varied with and without velocity heterogeneities in the basin.

We also compared the computed quantities with the observed values at the NLIB station located on the velocity profile. The following discussions and conclusions can be made from the current study.

Adding velocity heterogeneities in the basin results in an overall increase in the Arias based intensity of the computed ground motion at the edges, whereas a decrease is observed between 1650 and 1850 m.

We interpret the increase at the basin edges as a superposition of the locally diffracted waves generated by heterogeneities on the main wavefield. The decrease between 1650 and 1850 m is interpreted as being due to scattering of the wavefield by both the velocity heterogeneities and the complex layering that redistributes the energy.

The velocity heterogeneities in the basin affect the TF's high-frequency amplification compared to the reference model TF. The high-frequency amplification in the TF of a single realization of the random models is either locally increased or decreased by the velocity heterogeneities. As previously observed in the parametric study of chapter 3, averaging the TF of the 20 realizations of each random model leads to an average TF with smoother amplification patterns compared to the deterministic model's TF. The coefficient of variation associated with the average TF of the random model was higher for the models with higher a x .

Comparing the 2D TF to the 1D TF at NLIB show that, the fundamental resonance frequency and the higher modes in the 2D TF occur at higher frequencies than in the 1D TF. The 2D TF also results in larger amplifications of the fundamental resonance frequency than the 1D TF. This is due to the presence of locally diffracted and lateral propagation in the 2D case. Similarly to the studies by [START_REF] Nour | Finite element model for the probabilistic seismic response of heterogeneous soil profile[END_REF]; the travel times of the wavefield should be the most relevant parameter to be discretized, since seismic wave propatation is primarily controlled by travel times. In this regard, it could be interesting to link the variability of the observed travel times between two stations (i.e., seismic interferometry) to the perturbations in the 1D and 2D media.

We also investigate whether the velocity heterogeneities in the basin could help explain part of the observed variability in the earthquake-to-earthquake empirical TF at NLIB. [START_REF] Perron | Apport des enregistrements de séismes et de bruit de fond pour l'évaluation site-spécifique de l'aléa sismique en zone de sismicité faible à modérée[END_REF] shows using earthquake recordings in the Argostoli basin in Greece that, among tested parameters, the source distance and back-azimuth have the strongest influence on the amplification. [START_REF] Maufroy | 3d numerical simulation and ground motion prediction: Verification, validation and beyond -lessons from the e2vp project[END_REF] shows the influence of the source location on the numerical response of the Volvi basin in Greece. In this study, we use only 8 earthquakes whose locations are spatially distributed (see Figure 4.1 and Table 4.2 for their localization). Therefore, as found in [START_REF] Perron | Apport des enregistrements de séismes et de bruit de fond pour l'évaluation site-spécifique de l'aléa sismique en zone de sismicité faible à modérée[END_REF] and [START_REF] Maufroy | 3d numerical simulation and ground motion prediction: Verification, validation and beyond -lessons from the e2vp project[END_REF], this explains probably most of the ETF variability. We consider that the variability introduced through stochastic models could contribute to the aleatory variability. We find here that the inter-model variability range from about 20 to 50 % (top pannels in Figure 4.19) when introducing heterogeneities, whereas the total variability is much higher. Therefore, we find that accounting for velocity heterogeneities could explain part of the variability of the basin response, but it cannot fully explain the earthquake-to-earthquake amplification at the NLIB station.

We analyze the surface PSA along the basin at different frequencies, the resonance frequency (≈ 2 Hz) and higher frequencies (4, 6, and 8 Hz). The PSA's spatial variability along the basin increased with increasing frequency in both the deterministic and random models. At the resonance frequency, velocity heterogeneities increases the PSA values compared to the deterministic case. The increase is more pronounced at the basin edges than at the center, as it was the case for the Arias based intensity.

No clear increase or decrease was observed for the higher frequencies. The real observed value of the PSA at NLIB are found to match the computed values at 2Hz, whereas it was under-estimated at 4 Hz and over-estimated at 6 and 8 Hz Due to our inability to appropriately model a x , we considered 4 different values (10, 20, 50, and 100 m) such that the velocity heterogeneities had increasing aspect ratios (5,10,[START_REF] Kawase | Topography effect at the critical SV-wave incidence : Possible explanation of damage pattern by the Whittier Narrows, California, earthquake of 1 October 1987[END_REF]50) to follow the anisotropic nature of sedimentary layers. The overall conclusions on the effect of the velocity heterogeneities on the computed ground motion were similar in all four cases. However, we observed that the variability between the different realizations of the random media increased as a x increased. Implying that there is a higher discrepancy between the velocity distribution of the individual realizations of the random models with higher a x .

We also note that a z (2 m) estimated in this study was lower-bounded by the sampling interval (1.5 m) of the SPT measurements in the boreholes. Even though values of a z less than 1.5 m are plausible, the theoretical Von Karman ACF curve at 1 m suggests that values less than 1.5 m could not fit the sample ACF data points in this study.

To conclude, the study clearly shows the importance of considering random velocity heterogeneities when computing site effects using numerical simulations.

The work presented in this chapter could be considered as a first milestone towards a more extensive study of the site response in the framework of 2D probabilistic modeling of site effects. A next step Figure 5.2 -Existence conditions of the two-dimensional resonance in the SH case. For a sine-shaped valley having a shape ratio h/l and a velocity contrast C v (after [START_REF] Bard | The two-dimensional resonance of sediment-filled valleys[END_REF].

classify them into 4 sets (see Table 5.2) based on their correlation lengths and Hurst exponent:

-In set 1, we put the four random models with equal a z and a x .

-Set 2 contains four random models in which we vary a x in the range 10 to 200 m, and keep a z constant at 10 m.

-In set 3, we also consider four random models in which a z varies between 5 and 40 m, and a x is kept constant at 100 m. The Hurst exponent, ν, of all the models in sets 1, 2, and 3 is 0.3.

-Finally, we consider a last set (Set 4) of three random models with different ν values (0.1, 0.3, and 0.5). The model with ν = 0.5 corresponds to an exponential type ACF, which is a special case of the Von Karman ACF. The a x and a z of the models in set 4 are 50 and 10 m, respectively.

We used a constant c v of 5 % in all 12 random models. Since we are interested in studying the effect of the correlation length over longer propagation times, the c v of the random media has to be low such that scattering of the wavefield is mainly due to the correlation length. Based on the results of chapter 2, the c v of 5 % appeared to be low as the velocity wavefields were only mildly affected. Therefore, a c v of 5 % can also be considered as low in this case since the basin-bedrock velocity contrast is approximately equal to the average basin-bedrock velocity contrast of the Nice sedimentary basin.

In set 1, a x and a z are equal like in chapter 3. Considering equal a x and a z might be unrealistic regarding the fact that the sedimentation process of geologic materials generally dictates that a x be larger a z , we decided to study them here as an extreme case. In contrast, the correlation lengths of all the random model in sets 2, 3, and 4 all have aspect ratios (ar ratio between a x and a z ) greater than 1, thus reflecting the anisotropic nature of geologic materials.

All the random media were generated on a 3000 × 1000 m square grid with a grid step of 1 m, using the random media generation code RANDOM2D [START_REF] Tchawe | Effects of 2D random velocity perturbations on short-period (≤ 1s) ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin[END_REF]. For each random model, 10 different realizations were created with different random seeds.

Numerical experiments

We simulate 2D SH wave propagation in the computational domain shown in Figure 5.1. An unstructured mesh of the computation model, containing 37968 elements, was generated using the Trelis 16.4 [START_REF] Trelis | american fork, ut: csimsoft[END_REF] meshing software. The maximum frequency resolution of the mesh was set to 10 Hz, and the seismic wavefield was integrated over each element of the mesh using 5 Gauss-Lobatto- 5.1, all the random models have a c v of 5 %.

frequency-independent Q parameter (see Table 5.1).

We use the same Gaussian wavelet (Figure 5.4) as in chapter 3 as a source-time function. The wavelet is injected into the computational domain through the bottom boundary as a vertically incident plane wave. This source-time function has a broadband, energetic, frequency content between 2 and 8 Hz, which corresponds to a wavelength (λ) interval of 133 m to 33 m for a Vs of 266 m/s in the basin. Its dominant frequency is ≈ 6 Hz, giving a dominant propagation wavelength of ≈ 45 m. Thus, the basin's maximum depth (600 m) is equivalent to ≈ 13 dominant wavelengths.

The seismic wavefield is computed in the deterministic and random models for 20s using a leap-frog time scheme with a calculation time step of 50 µs. The velocity wavefield at the free surface is recorded Since the scattering by velocity heterogeneities is a frequency-dependent phenomenon, it can be classified by the product of the correlation length a of the fluctuations and the wavenumber, k = 1 λ , of the probing wave (e.g. [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF][START_REF] Korn | Seismic waves in random media[END_REF]. Four scattering regimes can be identified: (1) very weak or negligible scattering occurs when the wavelength is much greater than the correlation length (ka 1). In this case, an equivalent homogeneous medium can be used to replace the random medium;

(2) Rayleigh or weak scattering occurs if ka < 1. In weak scattering, most of the energy of the seismic wave is transmitted through the scatters, i.e., the seismic waves are scattered in the forward direction;

(3) the Mie or strong scattering regime is entered when ka ≈ 1. Scattering is said to be strong when either part of the seismic energy is diffracted backward by the scatterer, or the seismic waves are multiply scattered; and (4) when the scatterer size is much greater than the wavelength (ka 1), the medium becomes piecewise homogeneous and can be described deterministically. There is no deterministic limit to the scattering regimes based on the ka product.

We report the values of the ka z product of the 12 random models in Table 5.1. We use a z here to compute the ka product because the plane wavefield in our simulations is vertically incident. Although, there is no deterministic limit to the scattering regimes, we consider the models for which 0.1 ≤ ka z ≤ 0.5 to be in the Rayleigh scattering regime (weak scattering), and the models for which ka z > 0.5 to be in Mie scattering regime (strong scattering). As such models M3, M4, and M10 are expected to exhibit strong scattering, while the remaining models are in the weak regime.

Results

In this section, we analyze the results of the different sets of simulations. The seismic wavefield simulations performed in the random models of sets 1, 2, and 3 are analyzed conjointly to visualize the effects of the correlation lengths of the random models, whereas the simulations in the random models of set 4 will highlight the effect of ν. The results of the simulation in the deterministic model will serve as a reference for all comparison. Hence, the term effect in what follows will be used to denote any changes -with respect to the deterministic model -induced by velocity heterogeneities on the wavefield or ground motion indicators.

The characteristics of the wavefield and the ground motion intensity measures analyzed are:

1. The envelope and phase differences of the random wavefield with respect to the deterministic wavefield. This will serve as proxy to quantify the degree of scattering of the wavefield by each random model.

Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response 2. The transfer function (TF) of the basin.

3. The pseudo-spectral acceralation (PSA) response.

The Arias based intensity (A b I).

Apart from the TF, all three intensity measures listed above depend on the frequency content of the probing wavefield.

Velocity wavefield

First, we interprete the SH wave propagation in the deterministic basin.

Envelope and phase differences

We quantify the envelope and phase differences between the random and deterministic wavefields using the EM and PM as in chapter 3. This gives us a proxy of the degree of scattering by the random velocity heterogeneities for different correlation length configurations. A larger scattering of the wavefield by the velocity heterogeneities will result in higher phase and amplitude differences between the random models' wavefields and the deterministic model's wavefields. The signals are first filtered between 0.05 Hz and 10 Hz before computing the EM and PM between each random model and the reference. For each random model, the EMs and PMs are averaged over 10 realizations. In Figures 5.7a

-d, we plot the average EM (top panels) and PM (bottom panels) for the random models in sets 1 -4, respectively. The plot is limitted between 500 and 2500 m to show values in the basin only.

In set 1 (Figure 5.7a), where the random models have identical a x and a z , it can be observed that the average EM and PM values increase as the correlation length increases. The amplitude variations are, on average, more affected than the phase changes, as shown by the higher values of the EM curves (e.g. M1, red curve, has an average EM of ≈ 0.1 along the profile, whereas the average PM value is ≈ 0.05). The differences tend to be higher towards the center of the basin. This is because the basin Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response is deepest at the center, and hence the wavefield interacts more with the heterogeneities due to their longer propagation path before reaching the central surface receivers. Note that for models M3 and M4, with a x=z of 50 and 100 m, the EM curves have close values and at some points, the EM curve for M3 is higher than M4. This is probably because the correlation length in M3 is of similar size as the probing (ka z of 1.1), hence an optimal scattering occurs in model M3, whereas in M4 the correlation length is twice larger than probing wavelength and can be considered locally deterministic.

In set 2 (Figure 5.7b), we keep a z constant at 10 m and vary a x by increasing the aspect ratio. The average EM and PM are similar for all the random models, implying that the wavefield is less sensitive

Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response to variations of a x . This is probably because the medium is excited by a vertically incident plane wave that dominates the wavefields (see Figures 5.5 and 5.6) because of intrinsic attenuation. Thus, even if the resonance is 2D in this basin following [START_REF] Bard | The two-dimensional resonance of sediment-filled valleys[END_REF], a x has a lower impact on the wavefield. A closer look at the average PM curves reveals a mild sensitivity of the phase to the variation in a x . The average PM slightly increases as a x increases.

In set 3 (Figure 5.7c), we vary a z and keep a x constant at 100 m. As for the random models in set 1, both the average EM and PM increase as a z increases, hence as ka z draws closer to 1.

For set 4 (Figure 5.7d), all random models have the same a z and a x . We vary their value of ν between 0.1, 0.3 and 0.5. We observe a slight increase in the average EM and PM values as ν increases. This is because, as ν increases the medium becomes smoother and richer in longer wavelengths. The effect of increasing ν is qualitatively similar to an increase in the correlation length of the medium.

The average EM and PM becomes higher for all the random models as we draw closer to the basin's center. This is simply because the distance traveled by the wavefield through the velocity heterogeneities increases towards the center of the basin.

We note that the EM and PM curves are, in theory, expected to be symmetric after averaging over several realizations of randomness. The fact the curves are not necessarily symmetric here implies that 10 realizations of the random medium might not be enough for the average. Nevertheless, a general trend can be observed from the results.

Transfer function

As in the previous chapters, we compute the TF in the basin by taking the ratio between the Fourier Fourier amplitudes are smoothed using a Konno-Omachi smoothing function [START_REF] Konno | Ground motion characteristics estimated from spectral ratio between horizontal and vertical components of microtremors[END_REF], with a bandwidth of 40, before computing the spectral ratio.

Figure 5.8 shows the 2D TF for the deterministic model. The resonance amplification at the basin's center is at ≈ 0.16 Hz. The high-frequency (> 1Hz) amplifications are the harmonics of the fundamental frequency. Beyond 5 Hz, the amplification is mostly significant at the basin center and edges. This is because the reflected waves (as shown in Figure 5.5) that causes the basin to resonate at high frequency travel from the edges to center and vice versa since the basin is symmetric.

We plot in Figures 5.9a and b the 2D TFs for one realization of M1 and M3, with a x=z of 5 and 50 m, respectively. The low frequency amplifications (< 1Hz) in the TFs of both M1 and M3 are unaffected by the velocity heterogeneities. This is because, below 1Hz, the correlation lengths of the velocity heterogeneities in both cases are much lower than the propagating wavelengths. Indeed, below 1 Hz, the propagating wavelengths are ≥ 266 m, which is at least twice higher than the maximum a z (100 m) in all the models. As a result, all the random models in this study have a negligible effect on the wave propagation in this frequency range, hence on the fundamental resonance frequency.

The average 2D TFs between the realizations of models M1 and M3, and their corresponding coefficient of variation are plotted in Figures 5.9c In set 1 (Figure 5.11a), the TF for model M1 and M2 with correlation lengths of 5 and 10m, respectively, are not affected by the velocity heterogeneities. For M3 and M4 with correlation lengths of 50 and 100 m, respectively, the amplification above 2.5 Hz is on average decreased by the velocity heterogeneities. The decrease in the amplification and the variability between the 10 realizations is higher for M3 than M4. We also observed that the reference TF is no more contained in the variability of M3 above 2.5 Hz. Indeed, M3 has a ka z product of 1.1 and exhibits high scattering.

In set 2 (Figure 5.11b), the TFs show no clear changes in a x of the random models.

For set 3 ( 5.11c), the effect of the correlation length becomes noticable above 2.5 Hz for models 9 and 10 (a z = 20 and 40m, respectively).

In set 4 ( 5.11d), it is model M12, with ν = 0.5, that displays effects on the TF amplification above 3

Hz. Increasing a random model's ν increases its long wavelength content, hence increasing the random model's ka z product towards the strong scattering regime.

It is important to note that for all the random models that displayed effects of the TF, the effect was translated by a decrease in the TF amplification at high frequency. This observation has already been seen in other studies (e.g. [START_REF] Nour | Finite element model for the probabilistic seismic response of heterogeneous soil profile[END_REF][START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF].

Pseudo-spectral acceleration

We compute the PSA at the central receiver for 10 different frequencies (1 -10 Hz). The PSAs are obtained using a 5 % damping ratio of the sdof system, and they are plotted in Figure 5.12 for all four sets of random models. The black curve in each plot corresponds to the PSA in the deterministic models, which serves as a reference for the comparison. The red curve is the median PSA over the 10 realizations of each random model, and the red shaded area corresponds to the inner-68th percentile interval. to the dominant wavelength of the probing wavefield, hence the degree of scattering in the different realizations of M3 is higher than in M4. In general, the PSA values (red curves) at the higher frequencies are lower than in the deterministic model. This means that the scattering by the velocity heterogeneities tends to decrease the PSA response.

In set 2 (Figure 5.12b), where we vary a x of the random models, we observe no special dependency of the computed PSA curves to horizontal correlation length. The PSA curves for all the models have low variabilities since a z (10 m) is much smaller than the dominant wavelength (45 m) of the probing wavefield.

In set 3 (Figure 5.12c), it is the value of a z that varies between the random models. We observe an increase in the variability of the PSA curves with an increase in a z , as in set 1. The average PSA curves (red curves) are lower than the deterministic curve (black curve) for the higher a z values. plot, respectively), is higher than M5 with ν = 0.3. The mean PSA curve (red curve) in all three models appear to be lower than deterministic curve (black curve).

Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response

Arias based intensity

We quantify the intensity of the wavefield using the Arias based intensity (A b I ) parameter, which is the integral of the squared velocity over time. We synthesize in Figure 5.13 the results of the A b I at three surface receivers R1, R2, and R3 located at 700, 1000 and 1500 m on the basin, respectively.

R1 is closer to the left edge of the basin, while the R3 is at the basin center. R2 is at an intermediate

Discusions and conclusions

We presented the results of a sensitivity study to investigate the effect of the horizontal and vertical correlation lengths, and the Hurst exponent of the Von Karman ACF on the seismic wave propagation and ground motion intensity measures. We maintain c v of the random models to a low value to mainly highlight the effect of the correlation lengths and Hurst exponent of the Von Karman distribution on the wave propagation. Different random models with different configurations of a x , a z , and ν were considered and grouped into four sets. Sets 1, 2, and 3 were used to analyze the effect of a x and a z , whereas set 4 was used to investigate the effect ν.

The wave propagation in the different random models shows that the scattering of the wavefield by the velocity heterogeneities is a frequency-dependent phenomenon, and increases when the correlation length of the velocity heterogeneities is comparable to the dominant wavelength of the probing wavefield. With this parameter study, we aimed to answer two questions:

1. Which of the three parameters (a x , a z , and ν) of the Von Karman ACF is the wavefield more sensitive to when c v is low ?

2. Which ground motion intensity measures are most affected ?

Based on our results, we found that, for a vertically plane wave with SH polarization, the wavefield is less sensitive to the horizontal correlation length of the velocity heterogeneities. This was shown by the results of set 2 were we varied a x between 10 and 200 m for a constant a z of 10 m. The amplitude difference curves were mainly unaffected by a change in a x , while the phase differences were only mildly increased when a z was increased.

The wavefield is more sensitive to an increase of a z , and the maximum sensitivity was observed when a z was comparable to the dominant wavelength of the incident wavefield.

Regarding the Hurst exponent, its effect on the random medium can be interpreted as a filtering effect.

Increasing the Hurst exponent of the medium filters out the short-wavelength variations in the medium.

As a result, the resulting random medium has a higher effective correlation length. For the random models in set 4, where a z was kept constant at 10 m, increasing the ν value led to an increase in the effective a z value of the medium. Hence, drawing the a z value closer to the dominant wavelength of the probing wavefield. The wavefield, in this case, is more sensitive to an increase in ν.

We analyze the effects of the velocity heterogeneities on three ground motion intensity measuresthe TF, PSA, and A b I. The random models considered in this study did not affect the TF amplification below 2.5 Hz because the correlation lengths of the different random models were not large enough to scatter the wavefield in this frequency range. Previous studies (e.g. [START_REF] Nour | Finite element model for the probabilistic seismic response of heterogeneous soil profile[END_REF][START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] have shown that the resonance frequency of a sedimentary layer decreases when the c v of the velocity heterogeneities increases, and that the effect of the correlation length is negligible. The resonance frequency of the models considered in the cited studies falls in the frequency band of the incident wavefield (for example, in El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] the resonance frequency of their model was 3.54 Hz and the incident wave had a broadband frequency content of 1 -25 Hz). We plan to do further investigation in this direction by considering a basin whose resonance frequency falls in the range of the incident wavefield's dominant wavelength. Nevertheless, the amplification of the average TF amplification beyond 2.5 Hz appeared to decrease for the random models that we considered to exhibit strong scattering (models M3, M4, and M9, with a z of 50, 100, and 40, respectively ). Varying the Hurst exponent only had minor effects on the TF.

The PSA appears to be more sensitive to the velocity heterogeneities than the TF, as shown by the higher variability between the realizations of the random models. The sensitivity tends to increase with an in-Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response crease in a z . The maximum sensitivity was observed for model M3 with a x=z = 50 m, in which all the PSA curves of the different realizations were lower than the deterministic model at high frequencies (> 3Hz). For the random models with a z ≥ 20m, the effect of the velocity heterogeneities tend to, on average, attenuate the PSA curves at high frequencies.

For A b I, higher values were observed at the basin center than at the edges. The two main notable effects observed are (1) the decrease in the A b I value at the center of the basin for model M3 (with a x=z = 50)

compared to the deterministic model, and (2) the increase in the variability (i.e., the error bars) between the different realizations for the random models for higher a z values.

Among all the random models considered in this study, the wavefield and ground indicators were more sensitive to models M3, M4, and M9, which were classified in the strong scattering regime with ka z products of 1.11, 2.22, and 0.88, respectively. We note that models M3 and M4 have equal a x and a z , which is unlikely to be observed in nature.

We note that the classification of the random models based on their ka z product is used here because we have a larger basin model and longer propagation times. We follow a seismological approach of treating the scattering of the wavefield by the velocity heterogeneities. For a smaller basin like the Nice studied in the previous chapters, the effect of the correlation length was not clearly seen.

With theses results, we can conclude that: for a vertically incident plane wave on a random medium with low c v , the scattering of the wavefield is mainly controlled by the ka z product and to lesser extent the ν of the medium. The scattering by the velocity heterogneities tends to attenuate the TF, PSA in the sensitive frequency range. The sensitivity of the PSA to the velocity heterogeneities is more pronounced than the TF. This shows the importance of a good characterization of a region to use the closer parameteres that better represent the stochastic properties of the media.

Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response more, this work is intent to help improving the knowledge of site effect assessment; thus, we summarize the lessons learned from a seismic hazard evaluation perspective.

Effect of the coefficient of variation

In the sensitivity study performed in the Nice basin (chapter 3), we test three different values of c v (5, 10, and 30 %) to model the variability of the velocity heterogeneities, while keeping all the other parameters constant. We observe that the coefficient of variation is the dominant parameter that controls the ground motion variability (Arias intensity, duration, transfer function, and PGV). It acts as a sort of scaling factor of the variations on the computed ground motion intensity measures. In general, larger coefficients of variation produce higher Arias intensity values. On the other hand, the ground motion duration becomes nearly constant all along the basin for large values of coefficient of variation. If c v is high, the deterministic velocity contrast between the sedimentary basin layers can strongly be reduced, and the ground motion variability also becomes high. In this case, the velocity of the heterogeneities dominates over the deterministic structure of the basin. This observation, that c v is the dominant factor that controls the ground motion variability, has also been shown in previous studies (e.g. [START_REF] Hartzell | Effects of 3D random correlated velocity pertubations on predicted ground motions[END_REF][START_REF] Imperatori | Broad-band near-field ground motion simulations in 3-dimensional scattering media[END_REF][START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF]. In this thesis, we have not performed a large number of simulations due to the computational cost. Yet, the influence of the coefficient of variation is clearly seen in the basin transfer function, where the resonance frequencies of the reference model can be modified due to the local media properties. Moreover, when averaging all transfer functions, the transfer function broadens, which makes the average smoother than the individual contributions, even with respect to the reference case. Notwithstanding, we have constructed 2 basin models, and run 340 stochastic models in this dissertation.

Effect of the horizontal and vertical correlation length

In chapter 3, we assume that the velocity heterogeneities have equal a x and a z . The effect of the correlation length is studied keeping the c v equal to 5, 10, and 30 %. The correlation length is varied between 10 and 50 m. We chose these values because at the beginning of this work we did not have the information of the collected borehole data by CEREMA; thus, we decided to use relatively small correlation lengths in the order of 10 m up to a value close to the maximum depth of the basin. We observe minor effect of the correlation length on the variability of the ground motion intensity measures, even for values close to the basin depth. Indeed, to see a pronounced effect of the correlation length, the total propagation distance (that is, the basin size) should be several times larger than the correlation length of the velocity heterogeneities (e.g. [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF][START_REF] Derode | Random multiple scattering of ultrasound. I. Coherent and ballistic waves[END_REF]. To study the effect of the basin dimensions, we use in chapter 5 a larger and deeper basin for a coefficient of variation of 5%.

We see that the Arias intensity, response spectra, and the transfer function present a larger variability when the correlation length increases. This is because the scattering of the wavefield depends on the dimensionless product of the wavenumber of the probing wavefield and the correlation length (k × a), and when k × a ≈ 1 stronger scattering is observed. Increasing the correlation length of the velocity heterogeneities leads to a reduction of the observed ground motion intensity measures, and affects the transfer functions at higher frequencies than the fundamental resonance frequency.

In general, sedimentary layers present larger correlation lengths in the horizontal direction compared to the vertical one due to their geological formation process. This means a x > a z . The results of the sensitivity study show that the ground motion is mainly sensitive to changes of a z rather than a x . Similar results are found in chapter 4. We note that this conclusion corresponds to a vertically incident plane wavefield and should be verified for different angles of incidence. Nevertheless, this observation Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response suggests that more effort should be made to better characterize the vertical velocity profile of a site in a practical framework.

Effect of the frequency content of the incident wavefield

In this study, two source time functions were used as incident wavefields having different dominant frequency content.

In chapters 3 and 5, a truncated Gaussian wavelet having a dominant frequency of 6 Hz was used as a source time function. The results show that for a low c v value (5 %), it is mainly the high frequencies (above the resonance frequencies of the basins) of the wavefield that are affected by the velocity heterogeneities. In the case of the Nice sedimentary basin (chapters 3), the analyzed indicators (PGV, PSA, and TF) had, on average, similar values to the deterministic model. In contrast, for the larger basin (chapter 5) where the wavefield has longer propagation times, scattering by the velocity heterogeneities leads to a reduction of the ground motion intensity measures (TF, A b I, PSA).

In chapter 4, the source time function is an earthquake signal from the 2001 Nice event, which presents a relatively constant frequency content between 0.1 and 2 Hz. For this source, the dominant frequency range coincides with the fundamental resonance frequency of the basin (∼ 2Hz). In this case, the velocity heterogeneities may cause an increase in the PSA values at 2 Hz compared to the reference. At higher frequencies, the PSA is mostly affected by an increase of its spatial variability. In addition, there is an increase in the Arias intensity due to the presence of random heterogeneities. These results show that both the heterogeneities and the frequency content of the incident wavefield control the variability of the time domain intensity measures.

Importance of the basin size and sedimentary filling

When studying the Nice basin, the fact that the original velocity model is already complex and due to its small dimensions, the effect of random heterogeneities is less evident for small coefficients of variation. To further explore the interaction of the incoming wavefield and the heterogeneities, we considered a larger canonical basin model, with a homogeneous filling and having a simpler geometry.

This configuration was chosen so that the random heterogeneities dominate the scattering of the wavefield in the basin. This helps us to distinguish between the effects of the heterogeneities and the basin geometry on the wavefield. Moreover, the scattering theory suggests that the energy of the transmitted wavefield through a scattering medium decays exponentially with the thickness of the medium [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF], hence it is expected that the effect of the heterogeneities is more pronounced in this deeper basin. For a c v of 5 % we observe a higher scattering of the wavefield for increasing correlation lengths.

These results show that the effects of the heterogeneities are more pronounced when the basin is large and has a simple geometry. The structural complexity of a sedimentary basin can overshadow the effect of the small-scale velocity heterogeneities, as seen in the Nice sedimentary basin.

Influence of the polarization of the incoming wavefield

To study the influence of the polarization of the incoming wavefield we modeled the SH and P-SV wave propagation at Nice (chapters 3 and 4). In particular, chapter 3 has a common velocity model for both propagation cases. In general, we observe a similar tendency on the basin response variability. The absolutes values are different, but they follow a similar trend. This point has to be further investigated for inclined waves and 3D velocity models.

Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response

In the framework of seismic hazard analysis, we propose to use several source time functions if the ground motion intensity measures in the time domain are important. This will take into account the aleatory variability of the input signal and its spectral/time characteristics. We note that the correlation lengths and the coefficient of variation seem to control the variability of the resulting ground motion.

Thus, a better characterization of the media to see high frequency effects needs to be done.

Perspectives

This study aims to provide a detailed analysis of the influence of random velocity heterogeneities on the seismic ground motion. There is still much work to be done to better understand these complex phenomena. This section provides future directions to the current study, both on physical and numerical aspects.

In the simulations performed in this thesis, we considered only vertically incident wavefield as input motion. In reality, the incoming seismic wavefield reaching a sedimentary basin may have different angles of incidence. The next step to this study would be to analyze this effect. Moreover, the incident wavefield is considered to be a plane wave. It would be important for a future work to consider other incoming wavefields such as point or finite fault sources.

Another direction to the current work would be to study the contribution of anelastic attenuation and scattering attenuation due to random heterogeneities. In this study, we explicitly include the anelastic attenuation. However, we also find that random heterogeneities do not always attenuate the ground motion, they even amplify it at some sites locally. This is not well understood for a basin context.

This study considers only spatial variations of the shear wave velocity and assumes all other soil properties constant. For a better understanding of the interaction of the other elastic parameters (i.e. density, Poisson ratio), future work should consider their variability.

In this work, we use a Von Karman and a Gaussian distributions for the 2D random properties and their perturbation, respectively. Yet, we consider that more work should be done both experimentally and numerically to better describe the distribution of heterogeneities in materials and their quantification.

In particular, for seismic hazard assessment we need to include other distributions to capture the epistemic uncertainty.

In this sense, stochastic studies perturb the medium velocity, but we should also consider to link the variability of observed travel times between two stations (i.e. seismic interferometry) to these velocity perturbations. This is possible with borehole data for example.

The results of this dissertation come from 2D simulations. We know, however, that the Earth is 3D, and there is a need to further understand these phenomena at higher dimensions. Under strong ground motion excitation, the soil behavior can behave nonlinearly, and excess pore-pressure may be developed (liquefaction). The effect of nonlinearity and liquefaction of materials was not considered in this study.

It would be important to investigate the combined effect of random heterogeneities and nonlinear soil behavior on seismic wave propagation.

Code verification

Using the TFEM and TFPM, the frequency dependent and the time dependent envelop (FEM and TEM) and phase (FPM and TPM) misfits can be naturally obtained by integrating the time-frequency misfits on the frequency domain, respectively the time domain. In addition, the a single-value envelope (EM) or phase (PM) misfit between the two compared signals can be obtained by integrating the TFEM or TFPM on both the time and frequency domain.

We used all these misfit criteria to quantify the differences between the seismogram of the SEM solver with that of the FD solver. and phase misfit of both signals are 0.14 and 0.09 respectively. Since the envelope misfit can take any value within the range (-∞,∞), and the phase misfit takes values between (-1,1), [START_REF] Kristekova | Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals[END_REF] defined a time-frequency goodness-of-fit criteria to quantify the level of agreement between two signals.

3. 1

 1 Digital elevation model of the Nice (France) region displaying the reliefs and valleys. The black line shows the position along the central valley where the vertical 2D cross-section of the velocity model used in this study was obtained. (After Peyrusse et al., 2013) . . . . 59 3.2 2D velocity model (M) of the Nice sedimentary basin (from 500 m to 1600 m) showing the subsurface geometry of the layers beneath the 2D cross-section (black line in Figure 3.1). Labels c1 -c8 correspond to the layers code listed in Table 3.1. The bedrock is laterally extended on both sides of the basin (from 0 -500 m and 1600 -2100 m) to enable the inclusion of periodic boundaries in the numerical simulations' computation domain. . . . 59 3.3 Left: Source time function (a truncated Gaussian wavelet). Right: source spectrum showing a dominant frequency of 6 Hz (after Peyrusse et al., 2013). . . . . . . . . . . . . . . . . 63 3.4 One realization of the random media in Table 3.2 after interpolation with the basin's spectral element grid for: (a) model R1, (b) model R2, (c) model R3, (d) model R4, (e) model R5, and (f) model R6. See Table 3.2 for the characteristics of each model. . . . . . . 64 3.5 Sythentic seismograms showing the viscoelastic SH seismic wave propagation in: (a) the deterministic velocity model; and one realization of the random models (b) MR1, (c) MR2, (d) MR3, and (e) MR4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6 Envelope (blue line) and phase (red line) misfit between the velocity signals of the the surface receivers for a single realisation of random media and the deterministic model for (a) MR1, (b) MR2, (c) MR3 and (d) MR4. The corresponding 2D velocity profiles for each model are plotted on the bottom. The misfits characterize the difference in amplitude and phase between the random medium signal and the deterministic model. The differences increase with an increase in the coefficient of variation. . . . . . . . . . . . . . . . . . . . . 67 3.7 Average envelope (top) and phase (bottom) misfit curves between the signals of 10 realizations of each random model and the deterministic model. . . . . . . . . . . . . . . . . . 68 3.8 Top: PGV at the surface receivers of the reference medium (black curve) and a single realization of the random media MR1, MR2, MR3, and MR4 (red, blue, magenta, and green curves, respectively); bottom: the ratio between each random medium PGV curve and the deterministic medium PGV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 xii LIST OF FIGURES 3.9 (a) Arrival time of the observed PGV for the deterministic medium (black curve) and one realization of models MR1 to MR4; (b) Seismogram of a surface receiver located at 1210 m on the horizontal profile for the deterministic model (black curve) and a single realization of medium MR4 (red curve). The black and red dots highlight the PGV on both seismograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.10 Coefficient of variation of the PGV curves between the 10 realizations of each random medium. The curves show an increase in coefficient of varaition between the random realizations with the coeffiecient of variation of the random model. . . . . . . . . . . . . . 69 3.11 2D TF computed in (a) the deterministic velocity model; and one realization of the random models (b) MR1, (c) MR2, (d) MR3, and (e) MR4. The small amplifications at some frequency bands in the bedrock are due to the leaking waves from the sediments to the bedrock. The figures displays the degree of change in the 2D amplification patterns and values compared to the deterministic TF (a). . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.12 Top: Coefficient of variation of the PSA between the 10 realizations of each random random model at the periods (a) 0.5 s, and (b) 0.17 s; Bottom: The ratio between median PSA of the random models and the detertiministc model's PSA. . . . . . . . . . . . . . . . . . . 71 3.13 TF Geometric mean between the 10 realizations of the random models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. Notice the smoothening effect on the amplifications due to averaging, which leads lower amplifications (especially for media MR3 and MR4) compared to the reference case (Figure 3.11a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.14 Coefficient of variation (Cov) between the TFs of the 10 realizations of the random models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. A higher variance in the TFs is observed for media MR3 and MR4 with a c v of 30 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.15 The 16th percentile (a) and 84th percentile (b) between the TFs of the 10 realizations of model MR3. (c) Plot showing 16th percentile, 84th percentile, median, and mean for at the point x = 1215 m on the horizontal profile. . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.16 Seismograms for one realization of random model MR5. . . . . . . . . . . . . . . . . . . . 76 3.17 Average enveloppe (top) and phase (bottom) misfit between the seismograms of the random models MR1 (red curve), MR5 (cyan curve), and MR3 (magenta curve) and the deterministic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.18 (a) TF in one realization of MR5. The high-frequency amplifications are locally modified by the velocity heterogeneities compared to the deterministic case. . . . . . . . . . . . . . 77 3.19 (a) Mean TF of the 10 realizations of random model MR5 (b) corresponding coefficient of variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.20 2D simplified velocity model of the basin: (a) Model M1 with a homogeneous filling obtained by performing a harmonic mean of the velocity of the layers in M (Figure 3.2); (b) Model M2 with tabular layers obtained by a laterally extending a 1D cross-section at x = 1200 m of M. The color bars show the velocity of each unit. . . . . . . . . . . . . . . . . 79 3.21 Seismograms of the surface receivers for velocity models (a) M1 and (b) M2 . . . . . . . . 79 3.22 Envelope and phase misfits between the time traces of model M, M1 and M2 in the frequency bands (a) 0.5 -3Hz, and (b) 3 -10 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.23 2D transfer function of (a) M1 and (b) M2. The TFs display similar amplification patterns at both high and low frequencies implying that the basin's layering has only mild effects on the response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.24 2D transfer function for one realization of media (a) M1R3 and (b) M2R3. The TFs display different amplification patterns due to the presence of lateral heterogeneities. . . . . . . . 81 xiii LIST OF FIGURES 3.25 2D transfer function for one realization of media (a) M1R3 and (b) M2R3. The TFs display different amplification patterns due to the presence of lateral heterogeneities. . . . . . . . 81 3.26 2D transfer function for one realization of media (a) M1R3 and (b) M2R3. The TFs display different amplification patterns due to the presence of lateral heterogeneities. . . . . . . . 81 3.27 (a) Arias based intensity (A b I) in the random models R1 to R4 (top to bottom plots). The black curve in each plot is the A b I in the deterministic model. The blue curve in each plot is the mean of 10 realizations of the respective random model. The red curve is the median, while the red shaded area is the inner-68th percentile interval; (b) The ratio between the Arias intensity in the random models (colored curves) and the deterministic model (black curve); (c) The respective Arias based duration (see text for the definition). . 83 3.28 Surface seismograms (top) of the P-SV wave propagation computed in the deterministic model (bottom) for (a) the horizontal component, and (b) the vertical (z) component. . . . 84 3.29 P-SV surface velocity wavefields of the horizontal component (top) and random velocity profiles (bottom) for one realization of models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. As in the SH case (Figure 3.5a), the wavefield is strongly modified as the c v increases. . . 85 3.30 P-SV surface velocity wavefields of the vertical component (top) and random velocity profiles (bottom) for one realization of models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. . 86 3.31 Top: P-SV PGV at the surface receivers for the reference medium (black curve) and for a single realization of the random media MR1, MR2, MR3 and MR4 (red, blue, magenta and green curves respectively); bottom: ratio between each random medium PGV curve and the deterministic medium PGV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.32 Coefficient of variation of the P-SV PGV curves between the 10 realizations of each random medium. For the low intensity of the velocity perturbations (red and blue curve) the coefficient of variation follows the basin geometry. At larger perturbations, the resulting coefficient of variation is more significant and shows no correlation with the basin . . . . 87 3.33 2-D P-SV TF in the deterministic model (a), and in one realization of the random models (b) MR1, (c) MR2, (d) MR3, and (e) MR4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.34 Geometric mean of the P-SV TFs in the 10 realizations of the random models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. The coefficient of variation associated to the mean is represent in Figure 3.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.35 Coefficient of variation between the 10 realization of the random models (a) MR1, (b) MR2, (c) MR3, and (d) MR4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.1 (a) Map showing the location of the 8 earthquakes (coloured dots) used to compute the empirical transfer function in this study. The characteristics of these earthquakes are summarized in Table 4.2; (b) A zoom on the Alps-Ligurian area. The black dot is the February 25th, 2001 Mw 4.5 earthquake with its focal mechanism represented by the beach ball. The dotted black rectangle shows the Nice area (after Salichon et al., 2010); (c) The geographic map of the city of Nice. The black horizontal line shows the 2D crosssection of the basin velocity model we use in this study. The red dots are the borehole locations, and the green stars are the RAP stations considered in this study. . . . . . . . . 93 4.2 The East, North, and Vertical component (top, middle, and bottom plots, respectively) of the February 25, 2001, Nice earthquake recorded at (a) the station NLIB located on soft alluvial deposits; (b) the station NBOR located on an outcropping bedrock. Notice the amplification of the signal due to site effects at NLIB; (c) Map showing the location and magnitude (colored dots) of the earthquakes used to compute the empirical transfer function at NLIB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 xiv LIST OF FIGURES 4.3 The horizontal-to-vertical spectral ratio (HVSR) at NBOR (blue curves) and NLIB (red curves) computed on a 10 seconds S-wave window of East-West component from 8 earthquakes (see the colored dots in Figure 4.1c). The thick curves are the median while the shaded areas are the inner-68th percentile intervals. . . . . . . . . . . . . . . . . . . . . . . 95 4.4 (a) 2D velocity model (M) of the Nice sedimentary basin, labels c1 -c8 correspond to the layers code listed in Table 4.3. The green triangle shows the position of the NLIB station. (b) 1D velocity profile beneath the station NLIB. . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5 (a) Number of boreholes that dissect each sedimentary layer; (b) total number of SPT measurements available in each layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.6 Fitted Gaussian probability density for the distribution of velocities in each layer. The dotted vertical line shows the reference velocity of the layer in the 3D velocity model. The annotations on each subplot report the mean (mu), standard deviation (std), and the coefficient of variation (cov) for each distribution. . . . . . . . . . . . . . . . . . . . . . . . 101 4.7 The gray stars represent the sample ACF of each borehole as a function of the separation distance. The red dots show the average sample ACF of all the boreholes at each separation distance. The green, blue and cyan curves represent the 1D Von Karman ACF computed for correlation lengths of 1, 2 and 5 meters respectively. The Von Karman ACF were computed assuming a Hurst exponent of 0.3. . . . . . . . . . . . . . . . . . . . . . . . 101 4.8 (a) East-West component of the motion recorded at the NBOR station for the 2001 Nice earthquake. The time series is used as the source time function for our simulations; (b) Fourier broadband spectrum of the record in (a) . . . . . . . . . . . . . . . . . . . . . . . . 103 4.9 Velocity fluctuations in the basin for media (a) R1, (b) R2, (c) R3, and (d) R4 . . . . . . . . 103 4.10 (a -e) Horizontal synthetic seismograms (top plots) recorded by the surface receivers in the deterministic velocity model and one realization of the models R1 -R4 (bottom plots), respectively. The vertical green line at 1790 m shows the position of the RAP station NLIB; (f) The synthetic (blue) signal recorded at NLIB in each model in (a) -(e) is plotted against the observed seismic (black) signal at NLIB. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.11 (a) Arias based intensity of the surface receivers along the basin profile for the deterministic medium (black curve) and random media R1 to R4 (top to bottom plots). The red curve is average of the 20 realizations of each random media, while the red shaded zone corresponds to the inner-68 percentile interval. See Table 4.5 for the characteristics of each random media. The blue horizontal dashed line shows the real A b I observed at NLIB, while the dashed vertical shows the position of NLIB on the profile. (b) The ratio between the Arias based intensity of the random media and deterministic medium. The horizontal black dashed line is the amplification/de-amplification limit . . . . . . . . . . 105 4.12 TF along the basin surface for (a) the deterministic veloctity model, and one realization of random models (b) R1, (b) R2, (c) R3, and (d) R4. . . . . . . . . . . . . . . . . . . . . . . 106 4.13 TF geometric mean of the 20 realizations of the random models (a) R1, (b) R2, (c) R3, and (d) R4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.14 Coefficient of variation between the TFs of the 20 realizations of the random models (a) R1, (b) R2, (c) R3, and (d) R4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.15 1D velocity distibution (blue dots) extracted beneath NLIB in one realization of (a) R1, (b) R2, (c) R3, and (d) R4. The deterministic velocity profile beneath NLIB is represented by the black lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 xv LIST OF FIGURES 4.16 In each subplot, we plot the median 1D and 2D TF (blue and red curves, respectively) of the 20 realizations of the random models R1 to R4 (top to bottom subplot, respectively). The shade blue and red areas correspond to the 84th -16th percentile interval. The black curve in each subplot corresponds to the 1D TF in the 1D deterministic velocity model at NLIB (Figure 4.4b), while the dashed black curve corresponds to the 2D TF at NLIB in the 2D deterministic velocity model of the basin (Figure 4.4a). . . . . . . . . . . . . . . . . 110 4.17 Effective shear wave velocity beneath NLIB in the 20 realizations of the stochastic model (a) R1, (b) R2, (c) R3, and (d) R4. The blue vertical line is the median of the 20 realizations. The dotted black vertical line is the effect shear wave velocity beneath NLIB in the deterministic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.18 Empirical transfer function at NLIB computed using 8 earthquakes. The green curve represents the geometric mean of the earthquakes, and the green shaded zone shows the inner-68 percentile interval. The blue curve is the TF for the Nice 2001 earthquake. . . . . 111 4.19 Bottom plots in each subplot: The TF at NLIB of the 2D simulations in the deterministic velocity model (dashed black curve, NTF hm ), and random velocity models (red curves and red shaded area, NTF ht ) (a) R1, (b) R2, (c) R3, (d) R4 are plotted against the observed empirical TF at NLIB (ETF). The top plots in each subplot show the variation of the 84th -16th percentile interval of the NTF ht and ETF as a function of frequency. . . . . . . . . . 113 4.20 Pseudo-spectral acceleration along the basin surface at four frequencies (a) 2 Hz which corresponds to the resonance frequency of the basin, (b) 4 Hz, (c) 6 Hz, and (d) 8 Hz. The top to bottom subplots show the PSA (red curves and shaded area) for the random models R1 to R4, respectively. The black curve in each subplot corresponds to the computed PSA in the deterministic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.21 Velocity distribution in layer 5 of the sedimentary basin computed using the empirical correlation equations in Table 4.7. The dashed vertical line denotes the Vs in the deterministic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.22 Velocity distribution in the layers of the sedimentary basin computed using the average of the velocities obtained by all the empirical equations in Table 4.7. The dashed vertical line denotes the Vs in the deterministic model. . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.1 Deterministic velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

  v of 5 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4 (a) Truncated Gaussian wavelet used a the source time function in the numerical simulations, (b) Fourier spectrum of the wavelet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 Top: seismograms of the surface receivers in the deterministic model. Only the first ten seconds of the signal are shown here. Bottom: the corresponding velocity profile . . . . . 128 5.6 Seismogram for one realization of (a) model M1 and (b) model M3. The wavefield is more scattered by the velocity heterogeneities in model 2. . . . . . . . . . . . . . . . . . . . . . . 128 5.7 Average envelope (EM ) and phase (P M ) differences the deterministic and random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. See Table 5.1 for the describtion of the different random models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Top: 2D TF for the deterministic model. Bottom: the corresponding velocity profile . . . . 131 5.9 (a) and (b) TF for one realization of random models M1 and M3; (c) and (d) are the geometric mean of the 10 realization of the models; (e) and (f) are the coefficients of variaton between the 10 realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 xvi LIST OF FIGURES 5.10 (a) -(c) TFs for one realization of the random models M5, M11, and M12, respectively; (d) -(e) The geometric mean of the 10 realizations of each random model; (g) -(i) The corresponding coefficient of variation. The random models plotted here belong to set 4, where the Hurst exponent varies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.11 TF at the basin center for the random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. The black curve in each subplot is the TF for the deterministic velocity model. The red curve is average TF over the 10 realizations of each random model. The shaded zone denotes the inner-68th percentile interval between the realizations. The random models in each set are organized such that the parameter varied increases from top to bottom. The characteristics of the random random models are annotated in the legend of each plot. 134 5.12 PSA at the basin center for the random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. The black curve in each subplot is the PSA for the deterministic velocity model. The red curve is average PSA over the 10 realizations of each random model. The shaded zone denotes the inner-68th percentile interval between the realizations. The random models in each set are organized such that the parameter varied increases from top to bottom. The characteristics of the random random models are annotated in the legend of each plot. 135 5.13 Arias based intensity A b I computed at 3 surface receivers located at 700, 1000 and 1500 m, respectively for the random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. The black stars are the A b Is for the deterministic simulation while the blue stars represent the average A b I in each random model. The error bars correspond to the inner-68th percentile interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 A.1 Comparison between the solution of the spectral element solver used in this thesis (black signal) and the finite difference solver (red signal) for a receiver located at 1200 m on the Nice 2D velocity (see Figure 3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 A.2 Illustration of the different steps to integrate the velocity heterogeneities in SEM2DPACK for numerical simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 B.1 TF of one realization of the random models in Table 5.1 not shown in the text . . . . . . . 170 B.2 Surface seismograms of one realization of the random models in Table 5.1 not shown in the text. See the text of chapter 5 for their interpretation. . . . . . . . . . . . . . . . . . . . 171 B.3 Arias intensity AI computed at 3 surface receivers located at 700, 1000 and 1500 m, respectively for the random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. The black stars are the AI for the deterministic simulation while the blue stars represent the average AI in each random model. The error bars correspond to the inner-68th percentile interval. 172

Figure 2 . 1 -

 21 Figure 2.1 -Illustration of the different terms that constitute the recorded surface ground motion (G[t]): source (S[t]), path (P[t]), site Si[t], and instrument effects I[t] (edited after[START_REF] Perron | Robustness of kappa (κ) measurement in low-to-moderate seismicity areas: Insight from a site-specific study in Provence, France[END_REF]

Figure 2 . 3 -

 23 Figure 2.3 -This figure illustrates the ground motion parameters defined in the text based on the East-component of the recorded ground motion at the station NLIB of the RAP network during the February 25, 2001, Nice earthquake. The signal has been filtered between 0 and 25 Hz. (a) The red circle denotes the PGV of the ground motion; (b) The green curve represents the cumulated A b I as a function of time. The A b I is denoted by the green star, and the duration (T d ) of the motion corresponds to the time difference between the 95 % and 5 % cumulated A b I.

Figure 2 . 4 -

 24 Figure 2.4 -The figures shows the amplification of seismic waves, due to site effects, at ≈ 400 km (station SCT) away from the epicenter of the Mexico-Michoacan earthquake. The PGA is annotated on each seismogram Celebi et al. (1987).

  Figure 2.5 -A material volume V of the continuum, with surface S. (taken from[START_REF] Aki | Quantitative seismology, theory and methods[END_REF]

Figure 2 . 6 -

 26 Figure 2.6 -Diagram illustrating the propagation (in the direction shown by the blue arrows) of elastic body waves: (a) compressional (P-) wave and (c) transversely polarized shear (S-) waves; and surface waves: (b) Love waves and (d) Rayleigh waves (edited after Sei).

  , who used a numerical code based on the DGM to compute the site response in the Nice (France) sedimentary Theoretical background basin.

  Figure 2.7 -Sketch of the problem to be solved for estimating the solution of the elastic wave equation. The sketch shows a finite earth model with volume Ω and free surface δΩ. The computational domain has to be artificially bounded by absorbing boundaries Γ. Grid-based methods approximate the solution of the wave equation at discrete positions x j and t n .

  Figure 2.9 -Example of a 2D mapping on the reference square based upon 9 control or anchor points (taken from[START_REF] Chaljub | Spectral-element analysis in seismology[END_REF]

  Figure 2.10 -Gauss-Lobatto-Legendre integration points on = [-1, 1] as a function of the polynomial order N

  Figure 2.11 -Lagrange interpolants of degree N=4 on the reference interval = [-1, 1]. The corresponding N + 1 = 5 GLL points can be seen along the horizontal axis. All Lagrange polynomials are, by definition, equal to 1 or 0 at each of these points. Note that the first and last points are exactly -1 and 1 (taken from[START_REF] Chaljub | Spectral-element analysis in seismology[END_REF]

Figure 2 .

 2 Figure 2.12 -Well-logs showing P-and S-wave velocities and mass density vs. depth for well YT2 in Kyushu, Japan.(Edited after[START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]

  Figure 2.13 (right) shows the North component of motion, of the July 1st, 1992 Landers Aftershock, recorded at two soil sites separated by 100 m. The records show evidence of the variability in the ground motion induced by the variability in the soil properties.

  Figure 2.13 -Left: Layout of the soil site dense array of Kinemetrics SSA-2 accelerographs; Right: North component of motion of two soil sites separated by 100 meters (Edited after Steidl and Eeri, 1993).

  Figure 2.14 (from Uzielli et al., 2006) shows the comparison of two 2D spatial distribution of a generic parameter ξ having similar second-moment statistics and distributions, but different magnitude of spatial correlation. The top right plot shows a weak spatial correlation of ξ, whereas the bottom right plot shows a strong spatial correlation 9 .

Figure 2 . 14 -

 214 Figure 2.14 -Comparative representation of spatial data with similar statistical distributions (top and bottom left) but different magnitudes of spatial correlation: weak correlation (top right) and strong correlation (bottom right) (from El-Ramly et al. (2002) ). Edited after[START_REF] Uzielli | Soil variability analysis for geotechnical practice[END_REF]

  Figure 2.15 -Density plots of 2-D random medium samples, where a = 5 km and = 0.05: (a) Gaussian ACF. (b) -(d) von Kármán type ACFs with different ν-values[START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF]

Figure 2 . 16 -

 216 Figure 2.16 -Surface images (left), PSDF (middle) and ACF (right) of the fractional fluctuation of Pwave velocity for (a) Westerly, (b) Oshima, and (c) Inada granites. A dashed curve shown on the ACF indicates the best-fit exponential function. (Sivaji et al. (2002), obtained from Sato and Fehler (2012))

  shows two examples of 2D Von Karman random media generated using RANDOM2D. Both random models have the same characteristics but have different sizes. The vertical and horizontal correlation lengths are 50 m, while Hurst exponent is 0.3. The first random media (Model 1, Figure 2.17a) has a larger size (2000 m × 2000 m), while the second (Model 1, Figure 2.17b) is smaller in size (100 m × 100 m) and violates the first conditions in step 1 above. In Figure 2.17c, we compare the autocorrelation of both models to the theoretical Von Karman autocorrelation function given by equation 4.3. To calculate the models' autocorrelations, we extracted several 1D cross-sections from the 2D media, calculated the autocorrelation for each cross-section and then taking the average. It can be observed that the correlation curve from the small model deviates severely from the theoretical curve, implying that this model does not have the desired characteristics. The larger model fits fairly well the theoretical curve.

Figure 2 .

 2 Figure 2.17 -(a) 2D Von Karman random media with horizontal and vertical correlation length of 50m and a Hurst exponent of 0.3. The model has dimension of 2000 m by 2000 m; (b) same as (a) but has small size of 100 m by 100 m; (c) The dashed blue and orange lines are the autocorrelations of the random media in (a) and (b), respectively, while the solid black line is the theoretical autocorrelation function.

  Figure A.2 of Appendix A illustrates the scheme to integrate the random media generated by RAN2OM2D in SEM2DPACK. In the subsequent chapters, we shall show examples of random media interpolated over the sedimentary basins used in this study.

  the Nice sedimentary basin chapter 3. The methods presented in section 2.3 of this chapter are going to be used to obtain the statistical distribution of the spatial variability of V s in the Nice sedimentary basin in chapter 4. modeling of random heterogeneities . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.2 2D random velocity models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.3 Numerical simulations of seismic wave propagation . . . . . . . . . . . . . . . . . . . . propagation in random heterogeneous media . . . . . . . . . . . . . . . . . . . . . . . 64 3.5.2 Time-frequency analysis: envelope and phase changes . . . . . . . . . . . . . . . . . . . . . 65 3.5.3 Spatial distribution of the peak ground velocity . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5.4 Spectral ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5.5 Pseudo-spectral acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of averaging realizations of random media for site effect analysis . . . . . . . . 72 3.6.2 Link between the coefficient of variation of the random media and velocity contrast . . for model MR5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The P-SV case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.12.1 Velocity wavefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.12.2 Peak ground velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.12.3 Spectral ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.13 General conclusions to the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin This chapter investigates how the selected autocorrelation function parameters correlate with the surface ground motion indicators and site response over a sedimentary basin, using 2D numerical simulations. To this aim, we perform a parametric analysis in which we vary the correlation length and the coefficient of variation of the autocorrelation function and analyze their role on the resulting seismic wavefield. The ACF is chosen to be the Von Karman ACF, and the case study is the sedimentary basin of Nice, France. The chapter is presented in the form of an article submitted to the scientific journal Geophysical Journal International. Complementary work performed and not presented in the article shall also be shown after the article draft. Effects of 2D random velocity perturbations on short-period ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin F.N. Tchawe, C. Gélis, L.F. Bonilla and F. Lopez-Caballero article submitted for publication in Geophysical Journal International

  [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF] observed that for wave propagation at shallow depths (≤ 200 m ), random velocity heterogeneities must exhibit large fluctuations (approximately 25 % of the mean or greater) to reproduce the observed spectral amplification of the ground motion.[START_REF] Pagliaroli | Seismic microzonation of the centeral archaelogical area of Rome: results and uncertainties[END_REF] showed that velocity heterogeneities could locally double the Housner intensity of ground motion computed at Rome's Central Archaeological Area.[START_REF] Stripajova | Extensive Numerical Study on identification of key structural parameters responsible for site effects[END_REF] showed that considering random heterogeneities in 2D ground motion simulations increases the cumulative absolute velocity's aggravation factor. El[START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] investigated the sensitivity of different ground motion indicators to the characteristics of random velocity heterogeneities. They observed that the ground motion indicators are mostly sensitive to the coefficient of variation of the velocity heterogeneities., and that the velocity heterogeneities produces large spatial variability.

Figure 3 . 1 -Figure 3 . 2 -

 3132 Figure 3.2 and is denoted as model M.

  Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, Characteristics of the random media used to model velocity heterogeneities in the sedimentary basin. a x and a z are the horizontal and vertical correlation lengths respectively, ν is the Hurst exponent, and c v is the coefficient of variation.than or equal to the depth of the sedimentary basin. The minimum and maximum V S in the deterministic velocity model are 180 and 300 ms -1 , respectively. The source time function's dominant frequency is 6 Hz (Figure3.3), which gives dominant wavelengths between 30 and 50 m. We chose two values of the correlation length, 10 and 50 m. The correlation length of 10 m is smaller than the dominant wavelengths, while the correlation length of 50 m falls in the range of the dominant wavelength. We also choose three values for the c v , 5, 10, and 30 % to reflect low, intermediate, and high velocity fluctuations in the sedimentary basin. These three values of c v fall in the interval of values commonly reported in

  Figure 3.3 -Left: Source time function (a truncated Gaussian wavelet). Right: source spectrum showing a dominant frequency of 6 Hz (after Peyrusse et al., 2013).

Figure 3 .

 3 Figure 3.4 shows examples of one realization of each random model. In models MR1 and MR2 (Figure 3.4a and b, respectively), with c v = 5%, the initial layering of the sedimentary layers is still preserved, whereas in models MR3 and MR4 (Figure 3.4c and d, respectively) with c v = 30%, the layering structure becomes almost invisible due to the high intensity of the velocity perturbations. Although a c v of 30 % falls in the range of commonly observed values for sediments, this value seems to be high in this case study because the layers have close velocities. Hence, a c v of 30 % results in velocity fluctuations that modifies the basin's background deterministic structure. For this reason, we chose model MR5 (Figure 3.4e) with a c v of 10 % to be an intermediate case. Even though models MR3 and MR4

Figure 3 . 4 -

 34 Figure 3.4 -One realization of the random media in Table 3.2 after interpolation with the basin's spectral element grid for: (a) model R1, (b) model R2, (c) model R3, (d) model R4, (e) model R5, and (f) model R6. See Table3.2 for the characteristics of each model.

Figure 3 .

 3 Figure 3.5a displays the velocity wavefield recorded at the surface of the deterministic model M.The horizontal axis denotes the receivers' position, which can be directly mapped to the basin structure shown in the bottom plot. We show the first 5 seconds of propagation, where the strong ground motion

Figures 3 .

 3 Figures 3.5b -e show wave propagation in one realization of the random models MR1, MR2, MR3, and MR4, respectively. Models MR1 and MR2, with a c v of 5 %, show minor differences in their time histories compared to the deterministic case. The direct or first arrivals are almost unaffected by the velocity heterogeneities, while the later arrivals are slightly scattered. The scattering appears to be more pronounced as the correlation length of the velocity heterogeneities increases. That is, the wavefield in model MR2 is slightly more scattered than the wavefield in model MR1. Increasing the c v to 30 % (models MR3 and MR4) results in a higher scattering of the wavefield. Both direct and latter arrivals are affected by the scattering. The reflected waves by the basin edges are hardly distinguishable in MR3 and are totally scattered in MR4.

Figure 3 .Figure 3 . 5 -

 335 Figure3.6 shows the envelope (blue curve) and phase (red curve) misfit, expressed in percentage, between the signals for one realization of each random model with respect to the deterministic case. The values of EM and PM are approximately zero at the bedrock for all four cases because the medium is homogeneous. The minor differences at the bedrock receivers close to the basin arise due to the diffraction of waves from the basin to the bedrock -which is modeled as an elastic medium; thus, there is no attenuation of the refracted waves inside the bedrock. Larger differences are observed within the basin due to the presence of random velocity heterogeneities. For modelsMR1 and MR2 (Figures 3.6a and b, respectively), EM and PM values vary between 5 % and 30 %. The differences are larger between 1000 m and 1450 m, where the basin structure has lower velocities and more complex geometry. Models MR3 and MR4 (Figures 3.6c and d, respectively) have higher EM and PM values, up to 70%, due to more pronounced local reflections and diffractions of the wavefield by the velocity heterogeneities. The curves show a strong spatial variability with local maxima around the low-velocity zones (blue patches) in the

Figure 3 .Figure 3 . 6 -

 336 Figure3.8 shows the PGV from records at the surface across the basin for the deterministic model (black curve) and one realization of random models MR1, MR2, MR3, and MR4. As before, the horizontal axis is limited between 500 m and 1500 m to show the values for the receivers located within the basin only. Models MR1 and MR2 (red and blue curves) have values close to those of the deterministic case (black curve). In contrast, MR3 and MR4 (magenta and green curves) are spatially variable and

Figure 3 . 7 -

 37 Figure 3.7 -Average envelope (top) and phase (bottom) misfit curves between the signals of 10 realizations of each random model and the deterministic model.

Figure 3 . 8 -

 38 Figure3.8 -Top: PGV at the surface receivers of the reference medium (black curve) and a single realization of the random media MR1, MR2, MR3, and MR4 (red, blue, magenta, and green curves, respectively); bottom: the ratio between each random medium PGV curve and the deterministic medium PGV.

Figure 3 .Figure 3 . 10 -

 3310 Figure 3.11a shows the 2D transfer function of the reference medium. The ordinate axis stands for

Figure 3 .

 3 Figure 3.11 -2D TF computed in (a) the deterministic velocity model; and one realization of the random models (b) MR1, (c) MR2, (d) MR3, and (e) MR4. The small amplifications at some frequency bands in the bedrock are due to the leaking waves from the sediments to the bedrock. The figures displays the degree of change in the 2D amplification patterns and values compared to the deterministic TF (a).

(

  Figures 3.12a and b, lower panels) present values close to one for small velocity fluctuations (MR1 and MR2, red and blue curves, respectively), while they display high spatial variability for larger perturbations (MR3 and MR4), either amplifying or de-amplifying the spectral accelerations, by factors between 0.5 and 1.5. The spatial variability and amplification factors are higher for the shorter period of the oscillator, which coincides with the input motion's dominant frequency.

Figure 3 .

 3 Figure 3.12 -Top: Coefficient of variation of the PSA between the 10 realizations of each random random model at the periods (a) 0.5 s, and (b) 0.17 s; Bottom: The ratio between median PSA of the random models and the detertiministc model's PSA.

Figure 3 .

 3 Figure 3.13 -TF Geometric mean between the 10 realizations of the random models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. Notice the smoothening effect on the amplifications due to averaging, which leads lower amplifications (especially for media MR3 and MR4) compared to the reference case (Figure 3.11a)

Figure 3 . 14 -

 314 Figure 3.14 -Coefficient of variation (Cov) between the TFs of the 10 realizations of the random models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. A higher variance in the TFs is observed for media MR3 and MR4 with a c v of 30 %.

Figure 3 . 15 -

 315 Figure 3.15 -The 16th percentile (a) and 84th percentile (b) between the TFs of the 10 realizations of model MR3. (c) Plot showing 16th percentile, 84th percentile, median, and mean for at the point x = 1215 m on the horizontal profile.

Figure 3 .

 3 Figure 3.16 shows the surface seismograms for one realization of random model MR5 (see Table 3.2 in the main text for its characteristics). The global wave propagation in the basin remains similar to that of the deterministic model (Figure 3.5a) since the deterministic structure of the velocity model is still preserved after perturbation. Compared to model MR1 (Figure 3.5b), the latter arrivals are slightly more scattered due to the higher intensity of velocity perturbation in MR5.

Figure 3 . 16 -

 316 Figure 3.16 -Seismograms for one realization of random model MR5.

Figure 3 . 17 -

 317 Figure 3.17 -Average enveloppe (top) and phase (bottom) misfit between the seismograms of the random models MR1 (red curve), MR5 (cyan curve), and MR3 (magenta curve) and the deterministic model.

Figure 3 .

 3 Figure 3.18 -(a) TF in one realization of MR5. The high-frequency amplifications are locally modified by the velocity heterogeneities compared to the deterministic case.

Figure 3 .

 3 Figure 3.20 -2D simplified velocity model of the basin: (a) Model M1 with a homogeneous filling obtained by performing a harmonic mean of the velocity of the layers in M (Figure 3.2); (b) Model M2 with tabular layers obtained by a laterally extending a 1D cross-section at x = 1200 m of M. The color bars show the velocity of each unit.

Figure 3 . 21 -Figure 3 . 22 -

 321322 Figure 3.21 -Seismograms of the surface receivers for velocity models (a) M1 and (b) M2

  and b the TF of one realization of models M1R3 and M2R3, respectively. It can clearly be observed that the amplification patterns differ from the reference cases at both low and high frequencies, highlighting the importance of lateral heterogeneities compared to the detailed layering of the basin. The geometric mean between the TFs of the 10 realizations of both models is plotted in Figures 3.25. Their corresponding coefficient of variations are plotted in Figure 3.26. The mean TFs of both models are similar amplification values and patterns. The coefficients of variation between the realizations in both random models have high values (greater than 40 %). As a result, the average amplifications are smoother compared to the deterministic models and individual realizations. We also observe that the resonance amplification in both TFs is preserved even after averaging.

Figure 3 .

 3 Figure 3.23 -2D transfer function of (a) M1 and (b) M2. The TFs display similar amplification patterns at both high and low frequencies implying that the basin's layering has only mild effects on the response.

Figure 3 . 25 -

 325 Figure 3.25 -2D transfer function for one realization of media (a) M1R3 and (b) M2R3. The TFs display different amplification patterns due to the presence of lateral heterogeneities.

Figure 3 .

 3 Figure 3.26 -2D transfer function for one realization of media (a) M1R3 and (b) M2R3. The TFs display different amplification patterns due to the presence of lateral heterogeneities.

Figure 3 .

 3 Figure 3.27a shows the plot of the A b I for the four random models MR1 to MR4 (top to bottom plots, respectively). The black curve is the A b I in the deterministic model. The blue curve is the mean of the 10 realizations of the random models. The red shaded zone is the inner-68th percentile interval, while the red curve is the median. The corresponding ratios between the A b I in the random models with respect to the deterministic model are plotted in Figure 3.27b. The plots are limitted between 500 and 1570 m, to show only variations in the basin.

  ity of the A b I. The mean and median A b I along the profile are increased compared to the deterministic case. The mean A b I (blue curve) is higher compared to the median, implying that the A b I distribution between the different realizations is positively skewed. The spread between the realizations of the random models is high compared to models MR1 and MR2. At the right edge of the basin -between 1450 and 1550 m, where the basin displays a hill-like surface geometry and the thin low-velocity layer c1the 84th percentile is particularly high. The median value in the area remains close to the deterministic model, whereas the mean value is high -biased by the few realizations with high A b I values. This

Figure 3 .

 3 Figure 3.27 -(a) Arias based intensity (A b I) in the random models R1 to R4 (top to bottom plots). The black curve in each plot is the A b I in the deterministic model. The blue curve in each plot is the mean of 10 realizations of the respective random model. The red curve is the median, while the red shaded area is the inner-68th percentile interval; (b) The ratio between the Arias intensity in the random models (colored curves) and the deterministic model (black curve); (c) The respective Arias based duration (see text for the definition).

  28a and b the surface seismograms of the horizontal and vertical components, respectively. For an SV incident wave, the vertical component's wavefield is induced from the horizon-Influence of velocity fluctuations on seismic site response : sensitivity study in the Nice, France, sedimentary basin tal component due to the reflection and scattering of the incident wavefield. Notice that the motion is approximately null at the surface bedrock receivers on the vertical component (Figure 3.28b) since it is homogeneous. In contrast, phase conversions occur in the basin, leading to a non-zero motion on the vertical component. The wavefield on the horizontal component (Figure 3.28a) differs from the SH case (Figure 3.5a). In the P-SV case, the reflected waves at the basin edges have lower amplitudes than in the SH case. We also plot the seismograms of the horizontal and vertical components for one realization of each random model in Figures 3.29 and 3.30, respectively. Similar conclusion as in the SH case regarding the effect of the velocity heterogeneities on the wavefield. As the coefficient of variation of the random models increases, the wavefield in more scattered by the velocity heterogeneities and there are more wave conversions from the horizontal component to the vertical component of the motion (Figure 3.30).

Figure 3 . 28 -

 328 Figure 3.28 -Surface seismograms (top) of the P-SV wave propagation computed in the deterministic model (bottom) for (a) the horizontal component, and (b) the vertical (z) component.

Figure 3 .

 3 Figure 3.29 -P-SV surface velocity wavefields of the horizontal component (top) and random velocity profiles (bottom) for one realization of models (a) MR1, (b) MR2, (c) MR3, and (d) MR4. As in the SH case (Figure 3.5a), the wavefield is strongly modified as the c v increases.

3

  Figure 3.30 -P-SV surface velocity wavefields of the vertical component (top) and random velocity profiles (bottom) for one realization of models (a) MR1, (b) MR2, (c) MR3, and (d) MR4.

Figure 3 . 32 -Figure 3 . 34 -

 332334 Figure 3.31 -Top: P-SV PGV at the surface receivers for the reference medium (black curve)and for a single realization of the random media MR1, MR2, MR3 and MR4 (red, blue, magenta and green curves respectively); bottom: ratio between each random medium PGV curve and the deterministic medium PGV.

  Figure 4.1c (green triangles) shows the locations of the stations NLIB and NBOR (green triangles) used in this study. NLIB is located on soft alluvial deposits, whereas NBOR is located on an outcropping rock. The earthquake's epicenter is at ≈ 30 km from NLIB, and the distance between NLIB and NBOR is ≈ 4.5 km. The three-component motions of the earthquake recorded at NLIB and NBOR, filtered between 0.1 and 10 Hz, are plotted in Figure 4.2. An amplification of the signal recorded at NLIB due to site effects can be clearly observed, notably on the horizontal components.

  Figure 4.1 -(a) Map showing the location of the 8 earthquakes (coloured dots) used to compute the empirical transfer function in this study. The characteristics of these earthquakes are summarized in Table 4.2; (b) A zoom on the Alps-Ligurian area. The black dot is the February 25th, 2001 Mw 4.5 earthquake with its focal mechanism represented by the beach ball. The dotted black rectangle shows the Nice area (after Salichon et al., 2010); (c) The geographic map of the city of Nice. The black horizontal line shows the 2D cross-section of the basin velocity model we use in this study. The red dots are the borehole locations, and the green stars are the RAP stations considered in this study.

  Figure 4.2 -The East, North, and Vertical component (top, middle, and bottom plots, respectively) of the February 25, 2001, Nice earthquake recorded at (a) the station NLIB located on soft alluvial deposits; (b) the station NBOR located on an outcropping bedrock. Notice the amplification of the signal due to site effects at NLIB; (c) Map showing the location and magnitude (colored dots) of the earthquakes used to compute the empirical transfer function at NLIB.

For

  our numerical simulations, we use a 2D cross-section of this 3D model represented by the black line in Figure 4.1b. The cross-section (Figure 4.4) is approximately 1.1 km wide and 55 m deep and displays a complex layering structure of 7 layers. The S-wave velocity (V s ) of the layers ranges from 180 -330 ms -1 , and the engineering bedrock has a V s of 1000 ms -1 . The mechanical characteristics and litholog-Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin

  Figure 4.1b. All the boreholes do not necessarily sample each layer in the final velocity model and have different maximum depths. Figures 4.5a and b show the number of boreholes crossing Contribution of heterogeneities characterized from borehole data in site response : case study of Coefficient of variation retained for each layer in the 2D velocity model. each layer of the velocity model and the total number of SPTs performed in those layers, respectively.

Figures 4 .

 4 Figures 4.10a -e show synthetic seismograms recorded along the surface of the deterministic velocity model and one realization of models R1 -R4, respectively. Note that all the seismograms correspond to the horizontal component of the ground velocity. The wavefields show the complexity of the wave propagation and the amplification of the waves in the basin. The velocity heterogeneities tend to increase the wavefields' scattering (Figures 4.10b -e). The amplitude of the later arrival (≥ 8 s) in the

Figure 4 .

 4 Figure 4.12 -TF along the basin surface for (a) the deterministic veloctity model, and one realization of random models (b) R1, (b) R2, (c) R3, and (d) R4.

Figure 4 . 16 -

 416 Figure 4.16 -In each subplot, we plot the median 1D and 2D TF (blue and red curves, respectively) of the 20 realizations of the random models R1 to R4 (top to bottom subplot, respectively). The shade blue and red areas correspond to the 84th -16th percentile interval. The black curve in each subplot corresponds to the 1D TF in the 1D deterministic velocity model at NLIB (Figure 4.4b), while the dashed black curve corresponds to the 2D TF at NLIB in the 2D deterministic velocity model of the basin (Figure 4.4a).

Figure 4 . 17 -

 417 Figure 4.17 -Effective shear wave velocity beneath NLIB in the 20 realizations of the stochastic model (a) R1, (b) R2, (c) R3, and (d) R4. The blue vertical line is the median of the 20 realizations. The dotted black vertical line is the effect shear wave velocity beneath NLIB in the deterministic model.

Figure 4 . 18 -

 418 Figure 4.18 -Empirical transfer function at NLIB computed using 8 earthquakes. The green curve represents the geometric mean of the earthquakes, and the green shaded zone shows the inner-68 percentile interval. The blue curve is the TF for the Nice 2001 earthquake.

Figure 4 . 19 -

 419 Figure 4.19 -Bottom plots in each subplot: The TF at NLIB of the 2D simulations in the deterministic velocity model (dashed black curve, NTF hm ), and random velocity models (red curves and red shaded area, NTF ht ) (a) R1, (b) R2, (c) R3, (d) R4 are plotted against the observed empirical TF at NLIB (ETF). The top plots in each subplot show the variation of the 84th -16th percentile interval of the NTF ht and ETF as a function of frequency.

Figures 4 .

 4 Figures 4.20a, b, c, and d, respectively, for all four random models (red curve and shaded area) and the deterministic model (black curve). As before, the reference case computed using the 2001 Nice earthquake will serve as a reference for comparison. The dash blue horizontal line in each plot shows the response at NLIB to the selected frequency, while the dashed green vertical line at 1790 m shows the position of NLIB along the profile. The plot is limited between 1000 and 2000 m to show only variations in

Figure 5 . 3 -

 53 Figure 5.3 -One realization of each random model defined in Table5.1, all the random models have a c v of 5 %.

Figure 5 . 4 -

 54 Figure 5.4 -(a) Truncated Gaussian wavelet used a the source time function in the numerical simulations, (b) Fourier spectrum of the wavelet.

  Figure 5.5 shows the first 10 seconds of wave propagation recorded by the surface receivers. The direct incident wavefield arrives at bedrock's surface at about 1.2 secs, while it arrives after later in the basin due to the lower velocity. The wavefield is reflected at both edges of the basin, and it propagates across the basin. These reflected waves interfere at the center of the basin after 5 secs of wave propagation. Beyond 5 secs, multiply reflected waves are observed. Due to the large number of simulations, we plot only the velocity wavefield for one realization of models M1 and M3 (Figure 5.6a and b, respectively), which are in the Rayleigh and Mie regimes, respectively, based on our classification. The wavefields for one realization of the remaining random models are shown in Figure B.2 Appendix B. It can be observed that the wavefield in M3, with a x=z = 50 m, is more scattered than the wavefield in M1, with a x=z = 5 m. Indeed, since M3 has a correlation length (50 m) of the order of the incident wavefield's dominant wavelength (45 m), strong scattering is expected. In contrast, the correlation length of M1 is much smaller than the dominant wavelength of the incident wavefield, and hence the wavefield is only weakly scattered. The velocity heterogeneities in M1 only scatter the late arrivals after the direct waves, whereas the velocity heterogneities in M3 scatters both the direct and late arrivals. A similar interpretation can be made for the remaining models plotted in Figure B.2. The wavefield for models M4 and M10 (Figure B.2b and h, respectively), which are in the in the Mie regime category show a stronger scattering than the models in the Rayleigh regime category.

Figure 5 . 5 -

 55 Figure 5.5 -Top: seismograms of the surface receivers in the deterministic model. Only the first ten seconds of the signal are shown here. Bottom: the corresponding velocity profile

  amplitude of the signals recorded at the basin surface and the average Fourier amplitude of the signals recorded at the outcropping bedrock. The reference is taken to be the average Fourier amplitude of the bedrock receivers in the reference model. The signals are first filtered between 0.05 and 10 Hz and their

Figure 5 . 7 -Figure 5 . 8 -

 5758 Figure 5.7 -Average envelope (EM ) and phase (P M ) differences the deterministic and random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. See Table5.1 for the describtion of the different random models

Figure 5 . 9 -

 59 Figure 5.9 -(a) and (b) TF for one realization of random models M1 and M3; (c) and (d) are the geometric mean of the 10 realization of the models; (e) and (f) are the coefficients of variaton between the 10 realization.

Finally

  Figure 5.11 -TF at the basin center for the random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. The black curve in each subplot is the TF for the deterministic velocity model. The red curve is average TF over the 10 realizations of each random model. The shaded zone denotes the inner-68th percentile interval between the realizations. The random models in each set are organized such that the parameter varied increases from top to bottom. The characteristics of the random random models are annotated in the legend of each plot.

  Figure 5.12 -PSA at the basin center for the random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. The black curve in each subplot is the PSA for the deterministic velocity model. The red curve is average PSA over the 10 realizations of each random model. The shaded zone denotes the inner-68th percentile interval between the realizations. The random models in each set are organized such that the parameter varied increases from top to bottom. The characteristics of the random random models are annotated in the legend of each plot.

  Figure 5.13 -Arias based intensity A b I computed at 3 surface receivers located at 700, 1000 and 1500 m, respectively for the random models in (a) Set 1, (b) Set 2, (c) Set 3, and (d) Set 4. The black stars are the A b Is for the deterministic simulation while the blue stars represent the average A b I in each random model. The error bars correspond to the inner-68th percentile interval.

  Figure A.1 -Comparison between the solution of the spectral element solver used in this thesis (black signal) and the finite difference solver (red signal) for a receiver located at 1200 m on the Nice 2D velocity (see Figure 3.2).

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Characteristics of the material properties of the basin and of the random models used to perturb the velocity in the sedimentary basin. a x and a z are the horizontal and vertical correlation lengths, respectively, ν is the Hurst exponent, c v is the coefficient of variation, ar is the aspect ratio (a x /a z ), and ka z is the adimensional product of the wavenumber and a

3.1 Lithology and mechanical characteristics of the nine layers of the 3D geotechnical model of Nice. The layers are denoted by codes 1 to 9. Code R represents the seismic bedrock. . 59 3.2 Characteristics of the random media used to model velocity heterogeneities in the sedimentary basin. a x and a z are the horizontal and vertical correlation lengths respectively, ν is the Hurst exponent, and c v is the coefficient of variation. . . . . . . . . . . . . . . . . . 62 4.1 February 25th, 2001 Nice earthquake hypocenter location and focal mechanism (after Courboulex et al., 2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Characteristics of the 8 events shown in Figure 4.1 (coloured dots). These events are used in this study to compute the empirical transfer function and HVSR. . . . . . . . . . . . . . 94 4.3 Lithology and mechanical characteristics of the nine layers of the 3D geotechnical model of Nice. The layers are denoted by codes 1 to 8. Code R represents the seismic bedrock . . 95 4.4 Coefficient of variation retained for each layer in the 2D velocity model. . . . . . . . . . . 98 4.5 Characteristics of the random media used to model the velocity heterogeneity. a z is obtained by fit a Von Karman ACF to the sample ACF of the velocity heterogeneity. The values of a x are chosen assuming different aspect ratios (see text). ν is the Hurst exponent 100 4.6 Variance reduction between the NTFs (deterministic and random velocity models) and the mean ETF for (top) 2D simulations, and (bottom) theoretical 1D TF curves. We report the VR range (minimum VR -maximum VR) for the different realizations of each random velocity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.7 List of empirical correlation equations most commonly used to convert the standard penetration resistance (N) to the shear wave velocity of the soil (after Bertrand et al., 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1 z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.2 The random models in

Table 5.1 are categorized into four sets. See text for the description of each set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Chapter 1 Introduction Contents 1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Objectives and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Articles / Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

20 2.2 Numerical modeling of seismic wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2

  Tchawe, C. Gélis, L.F. Bonilla. 2D Numerical Investigation on the Effect of Random Velocity Perturbations on Seismic Ground Motion: Application to Site Effect Assessment in the Nice (France) Numerical simulations of seismic wave propagation in complex media : Impact of the spatial variability of soil properties on seismic ground motion. Journées des doctorant SRO-Université Gustave Eiffel, 2020, Champs-sur-Marne, FRANCE. The software implementations, together with the datasets can be found either online at: https://github.com/flominT or available upon request. Methods of site effects estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1 The Elastic wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 The 2D approximation of the elastic wave propagation . . . . . . . . . . . . . . . . . . . . . 25 2.2.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Chapter 2
	Theoretical background
	. . . . . . . . . . . 16
	2.1.3
	Sedimentary Basin. Seismological society of America, 2019, New Mexico,USA. Canceled due to
	COVID-19.

• Two oral presentations: -N.F. Tchawe, C. Gélis, L.F. Bonilla, and F. Lopez-Caballero. Effects of 2D soil heterogeneities on ground motion simulations : Application to the Nice sedimentary basin. Journées des thèses IRSN, 2019, La Grande Motte, FRANCE. Introduction -N.F. Tchawe, C. Gélis, L.F. Bonilla. Contents 2.1 Seismic ground motion and site effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Ground motion parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Site effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 2.3 Modeling and characterizing soil heterogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

	2.3.1 Evidence of soil heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
	2.3.2 Effects of soil heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
	2.3.3 Modeling soil heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
	2.3.4 Random media generation code and coupling with SEM2DPACK . . . . . . . . . . . . . . . 51

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Table 2 .1 -

 2 Characteristics of the ground motion and the ground parameters they reflect (modified after

	Theoretical background
	Ground Motion Characteristic

Table 3

 3

	.1. The bedrock is laterally extended on both sides of the

  in Table 4.7. We choose this correlation equation because it best reproduces (personal communication with the CEREMA members) the average velocities in each layer of the deterministic model. Moreover, the distribution of velocity (Figure 4.6) is very close to the distribution of velocity computed by considering the average of all the equations in Table 4.7 (see Figure 4.22).

Table 4 . 6 -

 46 Variance reduction between the NTFs (deterministic and random velocity models) and the mean ETF for (top) 2D simulations, and (bottom) theoretical 1D TF curves. We report the VR range (minimum VR -maximum VR) for the different realizations of each random velocity model

	Contribution of heterogeneities characterized from borehole data in site response : case study of the
						Nice basin
				2D Velocity model
		NTF hm	R1	R2	R3	R4
	mETF	0.77	0.63 -0.85 0.63 -0.87 0.65 -0.81 0.62 -0.85
				1D Velocity model
		NTF hm	R1	R2	R3	R4
	mETF	0.76	0.33 -0.67 0.44 -0.68 0.44 -0.69 0.43 -0.74

  Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin Vs or time-average velocities in the random media compared to the deterministic medium. Following El Haber et al. (2019) and Rodriguez-Marek et al. (2014) among others, we choose Vs as the parameter to be discretized as it can be measured in the field. An like the study by El Haber et al. (2019), we find a decrease of the fundamental frequency in stochastic media. El Haber et al. (2019) proposes that

The apparent S-wave radiation pattern denotes the spatial distribution of observed maximum S-wave amplitudes during earthquakes according to the fault orientation and style.

seismometers are used to measure velocity wavefield while accelerometers are used to measure the acceleration wavefield.

Cycles are the number of load and stress reversal in an earthquake signal.

Eistein summation convention is a notation that implies summation the indices that appear twice in a term, e.g. kk = 11 + 22 + 33

A property of pseudospectral methods which is related to how fast the approximation converges to the exact solution.

We understand by micro-states, the spatial distribution of the heterogeneities in a random medium realization

This is because small-scale variations of the velocity field represent the variations that are not captured in a deterministic model

Wavelets are small oscillations that are highly localized in time.

Acknowledgements

In the previous chapter, the statistical characteristics (correlation length and coefficient of variation) of the velocity heterogeneities used to parametrize the ACF were chosen based on reported values in the literature. The results of the parametric study showed that the coefficient of variation of the ACF could have a strong influence on the basin's response. Motivated by these results, we estimate the parameters of the velocity fluctuations in the Nice basin from available borehole data collected by CEREMA during the past years. We then use them to compute the basin's response and compare it with the empirical response estimated from earthquake recordings at the NLIB RAP station. This is a follow-up of the previous chapter, where we constrain the basin properties from in-situ data.

Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin ical description of the different layers, which comprise the 2D velocity model, are presented in Table 4.3. The P-wave velocities (V p ) in Table 4.3 were calculated from the V s by assuming a constant Poisson ratio of 0.4. This 2D cross-section has been used in previous research studies, e.g., (1) for comparing two numerical methods to compute site effects in [START_REF] Peyrusse | A nodal discontinuous galerkin method for site effect assessment in viscoelastic media -verification and validation in the nice basin[END_REF] and ( 2) for a parametric analysis on the effect of random velocity heterogeneities on the surface ground motion in [START_REF] Tchawe | Effects of 2D random velocity perturbations on short-period (≤ 1s) ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin[END_REF].

Code

Soil lithology ρ (Kg/m 3 ) V s (m/s) V p (m/s) Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin borehole locations at the surface do not have regular separation intervals. We tried several binnings of boreholes to compute a x . However, we found its value to be very sensitive to the binning interval due to the insufficient spatial distributions of boreholes. For this reason, we could not apply equation 4.2 to estimate the sample ACF along the horizontal direction (and thus, a x ) of the profile.

Nevertheless, it does not restrict us from evaluating the sample ACF in the vertical direction. Since the 2D velocity profile is a cross-section of the 3D velocity model of the basin, the vertical sample ACF estimated at a different location of the 3D velocity model can be conveniently extrapolated to the 2D velocity profile.

As shown in Figure 4.5, only layer 5 of our data set has a sufficient number of data points to reliably estimate the sample ACF. Consequently, we computed the vertical sample ACF (equation 4.2) using the velocity estimates in layer 5 only. We assumed that the velocity estimates in the other layers follow the same vertical sample ACF. The velocity estimates in each borehole were obtained at depth intervals of 1.5 m; hence the sample ACF is calculated for separation distances, which are multiples of the depth interval. Note that this measurement interval of 1.5 m provides a lower bound for the correlation distance. The maximum depth of the velocity estimate in layer 5 is 25.5 m. The procedure we used to compute the sample ACF is as follows:

1. First, we select the boreholes with more than 5 data points.

2. For each borehole, we compute the sample ACF as a function of lag distance. The mean m s in equation 4.2 is taken to be the layer mean velocity from Figure 4.6 to have a common reference for all the boreholes.

3. We then average the sample ACF of all the boreholes at each lag distance.

Regarding step 2, we note that another approach would be to process each borehole data separately and to access the correlation length that best fits most boreholes data. Such a processing would allow to assess velocity fluctuations at a local scale for each borehole, which we plan to do in the future. [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF], assuming a Hurst exponent (ν) of 0.3 as in [START_REF] Tchawe | Effects of 2D random velocity perturbations on short-period (≤ 1s) ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin[END_REF]. Visually, the ACF model with a correlation length of 2 m best fits the sample ACF. We thus retained the correlation length of 2 m to describe the vertical correlation length of the basin's velocity fluctuations. We notice that other sets of parameters could be tested using the Von Karman ACF model. Indeed, when fitting the same borehole data, the choice of a higher ν (e.g. 0.5 for an exponential ACF model) would lower the best fitted correlation length because ν controls the relative amount of high spatial frequency fluctuations (see Figure 2.15 of Chapter 2).

Therefore, several sets of ν and correlation length parameters could be chosen that almost correspond to the same physical size of the heterogeneities. On the other hand, in Chapter 3 [START_REF] Tchawe | Effects of 2D random velocity perturbations on short-period (≤ 1s) ground motion simulations; Application to site effect assessment in the Nice (France) sedimentary basin[END_REF]), we find that c v is the main parameter governing the ground motion variability, and the correlation length is of second order. The observation was made by El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] with c v values of 20 % and 40 %. Consequently, we do not conduct a sensitivity on the influence of the choice of the Von Karman ACF parameter in this study..

As stated before, the borehole configuration is not adequate for the proper estimation of the hor- properties in sediments generally vary differently along different directions of space. Therefore, the spatial variability is generally remarkably anisotropic, with a larger degree of correlation in the horizontal direction. This is mainly because the sedimentation or depositional processes mostly result in stratigraphic geometries. As a result, it is expected that the horizontal correlation length be greater than the vertical correlation length. For example, [START_REF] Nakata | Stochastic characterization of mesoscale seismic velocity heterogeneity in long beach, california[END_REF] found that the velocity heterogeneity at Long Beach, California is highly anisotropic with an aspect ratio (a x /a z ) of 5.1. Salloum (2015) estimated a x in alluvial plain of the river Nahr Beirut (Beirut, Lebanon) using electrical resistivity tomography profiles, and found the value to vary between 3.8 and 10.6 m. In contrast, a z estimated from shear wave velocity profiles was found to vary between 0.5 -2 m. Implying an aspect ratio in the range of 1.9 to 21.2 m, albeit a x and a z were estimated from two different geophysical parameters (shear wave velocity and electrical resistivity). In this study, we consider 4 different correlation lengths of 10, 20, 50, and 100 m, corresponding to aspect ratios of 5, 10, 25, and 50, respectively, for a vertical correlation length of 2 m. We summarize the characteristics of the 4 random models in Table 4.5.

Numerical experiment

We simulate 2D P-SV seismic wave propagation in the sedimentary basin using the spectral element code SEM2DPACK [START_REF] Ampuero | Etude physique et numérique de la nucléation des séismes[END_REF]. The bedrock in Figure 4.4a has been artificially extended in both Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin The green, blue and cyan curves represent the 1D Von Karman ACF computed for correlation lengths of 1, 2 and 5 meters respectively. The Von Karman ACF were computed assuming a Hurst exponent of 0.3.

the lateral and vertical directions to form the computational domain, which is 3100 m wide and 100 m deep. An unstructured mesh containing 5498 elements with a maximum grid resolution of 10 Hz is generated, taking into account all model interfaces using an external mesh generator software Trelis 16.4 [START_REF] Trelis | american fork, ut: csimsoft[END_REF]. The seismic wavefield is integrated on the mesh domain using 5 Gauss-Lobatto-Legendre (GLL) integration points. The minimum GLL separation distance of the spectral element grid of the basin mesh is ≈ 0.054 m. We mimicked an infinite lateral domain using periodic boundary conditions.

Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin

Arias-based intensity of the wavefield

We analyze how the Arias based intensity (A b I) measure, of the wavefield varies with the presence of velocity heterogeneities in the basin. The horizontal black dashed line is the amplification/de-amplification limit .

2D Spectral ratio

We plot in Figure 4.12 the spectral ratio (TF) across the basin in the deterministic medium (Figure

spectral ratio at NLIB

We now focus on the receiver located at NLIB (x=1790 m) on the profile. The velocity profile of the soil column beneath NLIB is shown in Figure 4.4b. The soil column beneath NLIB dissects 5 layers (c1, c3, c5, c6, and c7) and has a total thickness of 39.5 m. In this section, we first compare the 1D and 2D

TFs at NLIB for both the deterministic and random media simulations. We then analyze whether the numerical TFs computed in this study can help explain the observed empirical TF at NLIB.

1D vs 2D TF

In order to assess the importance of accounting for 2D geometry and the related wave propagation, we simplify 2D models into 1D models and compute the 1D theoretical TF in both the deterministic and random models (R1-R4) using the Haskell-Thompson method [START_REF] Haskell | The dispersion of surfacewaves on multilayered media[END_REF]. The 1D TF of the deterministic model (black curve) displays a fundamental resonance frequency at ≈ 1.6 Hz, and has an amplification of 4. The second mode occurs at 4 Hz and is slightly more amplified than the fundamental frequency, with an amplification of 4.3. A third mode is also observed at ≈ 7.9 Hz with an amplification of 3. In the 2D TF of the deterministic model (dashed black curve), the fundamental resonance frequency occurs at 1.8 Hz, with a higher amplification of 4.8. The second resonance mode also occurs at a higher frequency of 4.7 Hz but with a lower amplification of 3.7. The third higher mode also occurs at a higher frequency of 8.3 with amplification of 2.2. This result shows that the 2D wave propagation results in a higher amplification of the fundamental resonance frequency than 1D wave propagation at the site. The is probably due to the lateral propagation of the waves in the 2D case.

Adding velocity heterogeneities in the 1D velocity model shifts the resonance frequencies to lower frequencies in the four random models (blue thick curves and shaded area). To investigate the cause of the fundamental resonance frequency shift, we compute the time-average Vs in 1D stochastic media.

Figure 4.17 shows that the median time-average velocity (blue vertical curve) is, in general, lower than the reference case (black dashed line); hence leading to a lower fundamenetal frequency. As proposed by El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF], this could be related to the choice of the Vs parameter as the random field.

The amplification of the fundamental mode is, on average (blue curve), not changed compared to the deterministic 1D curve. This is interpreted as the smoothing effect due to averaging individual TFs. The amplification of the higher modes (second and third modes) are, on average, decreased by the presence of velocity heterogeneities in all four random models. The variability (shaded blue area) between the Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin 1D TF realizations of the four random models is relatively higher for the higher modes, as shown by the large inner-68 percentile interval.

For the 2D case, the velocity heterogeneities also shift the resonance frequencies to lower frequencies in the four random models (red thick curves and shaded area). There is no major difference between the fundamental resonance mode's amplification in the random and deterministic model. In contrast, the higher modes' amplification is decreased by the velocity heterogeneities in all four random models.

The variability in the amplification around the fundamental mode is higher for models R3 and R4 (with higher a x of 50 and 100 m, respectively) compared to models R1 and R2 (with lower a x , 10 and 20 m, respectively).

The shift in the fundamental resonance frequency in both the 1D and 2D TF observed in this study was also found by [START_REF] Nour | Finite element model for the probabilistic seismic response of heterogeneous soil profile[END_REF]; [START_REF] Salloum | Evaluation de la variabilité spatiale des paramètres géotechniques du sol à partir de mesures géophysiques: application à la plaine alluviale de Nahr-Beyrouth (Liban)[END_REF]; El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF]. El [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] explains the shift as being due to longer travel times of the wavefield associated to a log-normal distribution of velocities in highly perturbed stochastic media. As result, since the effective velocity of the stochastic media is lower than in the deterministic case, there is a shift in the fundamental resonance frequency because of a higher impedance contrast of the medium with the bedrock. In this study, we observe that this shift is also applicable to the higher modes of the 1D and 2D TF and is accompanied by an attenuation of their amplification. where the maximum is taken along the the frequencies.

It can be observed that the variability in the ETF's amplification is very large compared to the NTF in all four random models. Model R3 (4.19c) has a higher variability in the its TF compared to the models R1, R2, and R4, but still low compared to the ETF's variability. These result shows that the earthquake-to-earthquake variability in the ETF at NLIB can't be totally explained by modeling velocity heterogeneities as expected. This is consistent with the influence of the earthquakes location on SSR variability as found by [START_REF] Perron | Robustness of kappa (κ) measurement in low-to-moderate seismicity areas: Insight from a site-specific study in Provence, France[END_REF].

We now compare the fit between the NTFs and the mean ETF using the variance reduction (VR) as in [START_REF] Thompson | Impediments to Predicting Site Response: Seismic Property Estimation and Modeling Simplification[END_REF]. The VR is defined as follows:

where f is frequency and n is the number of frequencies at which the NTF(f ) is computed. The VR takes values in the range (-∞,1), with a VR = 1 implying a perfect fit between the prediction and the observations, whereas smaller values indicate a poorer fit.

We compute the VR only for amplification in the linear frequency range of 1 -10 Hz, which corresponds to the frequency range where NBOR is considered to exhibit no site effects. We first resampled the NTF curves such that their sampling frequency matches that of the ETF before computing the VR. The VR values for the 1D and 2D NTFs are reported in Table 4.6. For the NTF ht , we report the spread or range (minimum and maximum) of the 20 realizations of each random model.

For the 2D case (Table 4.6 top), the VR values for the 2D NTFs in both the deterministic and random models are all close to 1, implying that there is a good fit to the mean ETF. Indeed, the 2D TF curves (Figure 4.19 ) show a good fit with the mean ETF up till 5 Hz. After 5 Hz, the mean ETF has a major peak, which is absent in the NTFs curves. The VR for the deterministic 2D model (NTF hm ) falls in the range of the VRs for the 2D random models. All four random models have similar VR ranges. In the 1D case, the TF in the deterministic velocity model has an overall better fit to the mean ETF compared to the random 1D models. Compared to the 2D case, the four random models have lower minimum and maximum VRs, which shows a better ability of 2D models to reproduce the empirical mean amplification. with the presence of velocity heterogeneities in both the 1D and 2D case. In our study, we find that shift is also applicable to the higher modes of the TF. We link this shift to the decrease in the effective Contribution of heterogeneities characterized from borehole data in site response : case study of the Nice basin would be to consider other medium properties (e.g., Poisson ratio, anelastic attenuation) as spatially correlated random parameters, consider different ACF choices, and different constitutive models.

Appendix

Uncertainty due to the data transformation method

The insitu Vs estimates in the boreholes were obtained by converting the N-value using the empirical correlation by [START_REF] Ohta | Empirical shear wave velocity equations in terms of characteristics soil indexes[END_REF]. Different empirical correlation equations, adapted for different soil types, exist in the geotechnical literature to convert the N-value to Vs. Table 4.7 lists the different equations used during the construction of the 3D deterministic velocity model of the Nice basin.

In this section, we explore the uncertainty due to the choice of one correlation equation over the others. We first admit that the choice of the equation by [START_REF] Ohta | Empirical shear wave velocity equations in terms of characteristics soil indexes[END_REF] in this study was made after personal communication with Etienne Bertrand, member of the GEMGEP project, who observed that this equation better generalized the velocities in the 3D velocity model.

Using the equations suitable for all soil types in Table 4.7, we convert the N-values measured in layer 5 of the sedimentary basin to Vs. We plot in Figure 4.21 the Vs distributions for each equation. The mean and coefficient of variation is also reported on each subplot. The dashed vertical line in each subplot is the layer velocity in the deterministic model, which is 250 ms -1 . The different equations result in different velocity distributions. The mean velocities and coefficient of variation of the different equations equations vary between 196.4 -352.4 ms -1 , and 12.9 -32.4 %, respectively. The average coefficient of variation between the different equations is 23.5 % which is close to the cov of the equation by [START_REF] Ohta | Empirical shear wave velocity equations in terms of characteristics soil indexes[END_REF]. The coefficient of variation of the cov values is 26 %, implying that using a different equation could lead to a change in the coefficient of variation considered as much as 6 %.

Introduction

As we saw in chapter 3, the Von Karman ACF parameter that strongly influences the seismic wavefield and the ground motion intensity measures is the coefficient of variation. The effect of the correlation length (10 and 50 m in both the horizontal and vertical directions) appeared to be less important, while the effect of the Hurst exponent was not investigated in the parametric study.

In chapter 4, we computed the coefficient of variation and the vertical correlation length from borehole data in the Nice sedimentary basin. The coefficient of variation values ranged from 22 to 32 % depending on the geological layer. The vertical correlation length (a z ) was 2 m, and different values of the horizontal correlation length were tested. The Hurst exponent was chosen to be 0.3.

In this chapter, we further investigate the influence of the correlation lengths and the Hurst component (ν) on the surface ground motion. For this, we consider a deeper and larger basin in order to let

Influence of the correlation length and Hurst exponent for Von Karman distribution on the variability basin response the seismic waves interact with the velocity heterogeneities for a longer time and over a larger distance.

We also assume that the basin deterministic velocity model has no layers and has a constant velocity to focus solely on the effect of the velocity heterogeneities.

The work presented in this chapter is exploratory, hence the conclusions made here are preliminary.

Sedimentary basin and random model characteristics

Model geometry

The computational domain is 3000 m wide and 1000 m deep, as shown in Figure 5.1. The sedimentary basin is located between 500 m and 2500 m and is 600 m deep. The basin-bedrock interface has a convex symmetric geometry. We consider the sedimentary filling to be homogeneous with a shear wave velocity (Vs) of 266 ms -1 and a density (ρ) of 1900 kgm -3 , while the underlying bedrock has a Vs of 1000 ms -1 and density of 2100 kgm -3 . [START_REF] Bard | The two-dimensional resonance of sediment-filled valleys[END_REF] used an extensive parameter investigation to categorize the resonance pattern in sine-shaped basins based on the shape ratio and velocity contrast. The relation is presented in Figure 5.2 for an SH wave propagation. For a basin whose shape ratio to velocity contrast dependency falls above the curve in Figure 5.2, 2D resonance modes are expected to occur. Whereas if the dependency falls below the curve, 1D resonance and lateral propagation are expected to dominate. In their study, they found that 2D resonance results in considerably large amplifications and longer propagation duration compared to the 1D case. The bedrock-basin velocity contrast in this study is ≈ 3.76. The shape ratio (the ratio between the sediments thickness, h, and the basin half-width, l) is 0.6. This implies that, for our basin model, the point falls above the curve. Hence, it is expected that the SH wave propagation is dominated by the 2D resonance modes of the basin.

Random models

We perturb the velocity inside the basin using 12 random models with a Von Karman ACF. The characteristics of these 12 random models are summarized in Table 5.1 -Characteristics of the material properties of the basin and of the random models used to perturb the velocity in the sedimentary basin. a x and a z are the horizontal and vertical correlation lengths, respectively, ν is the Hurst exponent, c v is the coefficient of variation, ar is the aspect ratio (a x /a z ), and ka z is the adimensional product of the wavenumber and a z .

Set 1 Set 2 Set 3 Set 4

Table 5.2 -The random models in Table 5.1 are categorized into four sets. See text for the description of each set.

Legendre (GLL) integration points.

To consider the random variations of the velocity in the basin, the random models were interpolated over the 2D spectral element (SEM) mesh of the basin model using a nearest-neighbor interpolation algorithm. The GLL separation distance in the SEM mesh of basin is ≈ 0.2 m (which is five times smaller than the grid step, 1 m, of the random media). This means that we have at least 25 points per minimum correlation distance (5 m) in the random models; hence respecting the condition of at least 2 points per minimum correlation distance [START_REF] Frenje | Scattering attenuation: 2d and 3d finite difference simulations vs. theory[END_REF]. Figure 5.3 shows a plot of one realization of each random model. It can be observed that the spatial distributions of the velocity heterogeneities strongly depends on the model. For models M11 and M12 (Figure 5.3k and l, respectively), in which we vary only ν, we can observe that M11 is rougher than M12. This is because, for a lower ν, the medium contains more short-wavelength heterogeneities and inversely. Hence, model M12 appears smoother than M11.

We consider periodic boundary conditions on the lateral boundaries of the computational domain.

The upper boundary is the free surface, and we use an absorbing boundary condition at the bottom boundary. Intrinsic attenuation is taken into account following the implementation of the rheological viscoelastic model of the Generalized Maxwell body [START_REF] Liu | Efficient modeling of q for 3d numerical simulation of wave propagation[END_REF], and is quantified by the

General conclusions and perspectives

This thesis explores the effects of velocity heterogeneities at the sedimentary scale on seismic ground motion. For this, seismic wave propagation is numerically simulated in 2D sedimentary basins in which the velocity heterogeneities are modeled as spatially correlated random fields. We analyze these effects on several ground motion intensity measures such as the Arias intensity, duration, response spectra, PGV, and the transfer function. This work lies in the boundary of Seismology and Earthquake engineering. Indeed, the dimensions of the studied region are large for classical engineering studies, yet relatively small for traditional research of wave propagation in Seismology. We hope that these results will add some discussion for both communities and help to better understand wave propagation in stochastic media. In what follows, we present the conclusions of this work and propose some directions for future research on this topic.

General discussion and conclusions

We use spatial autocorrelation functions (ACF) to generate correlated random fields describing velocity heterogeneities in a 2D media. The particular choice of the Von Karman ACF is made in this thesis based on the seismological literature. This distribution is implemented and coupled to the spectral element method code we use to simulate seismic wave propagation. The Von Karman distribution has three tunable parameters; the horizontal and vertical correlation length (a x and a z ), the coefficient of variation (c v ), and the Hurst exponent (ν), respectively.

In chapters 3 and 5 of this dissertation, we investigate the contribution of each of these parameters on the simulated ground motion through a parametric analysis. In chapter 3, we use a 2D velocity model of the Nice sedimentary basin, whereas in chapter 5, we use a bigger and deeper canonical model. The choice of this basin is made because (1) the small size of the Nice sedimentary basin limited the investigation of the influence of the vertical correlation length in the sensitivity study, and (2) the complex geometry of the sedimentary layers made it difficult to separate the scattering produced by the velocity heterogeneities and the impedance contrast between layers.

In chapter 4, we model the distribution of velocity heterogeneities in Nice from the geostatistical analysis of borehole data and examine their contribution to the basin site response.

The results show that the effects of velocity heterogeneities, on the seismic ground motion, are complex.

Indeed, their influence can not be explained in terms of their random properties only (i.e. coefficient of variation, correlation length). There is also a combination of other parameters such as the frequency content of the incident wavefield, the size of the basin, the impedance contrast of the sedimentary layers, the complexity in the geometry of the sedimentary filling, and the choice of the ground motion intensity measure.

We enumerate here the conclusions that can be drawn from our study on these different factors. Further-

The choice of the ground motion intensity measures

It is clear from this study that the ground motion may strongly depend on the distribution of the random heterogeneities in the media. Certain ground motion intensity measures are more sensitive than others depending if they represent time or frequency domain measures.

In this study, the envelope misfit (EM) and phase misfit (PM) are not directly used as a ground motion intensity measure but used to compare time series between the stochastic and reference simulations. In Seismology, these quantities, introduced by [START_REF] Kristekova | Misfit criteria for quantitative comparison of seismograms[END_REF], are mostly used to quantify the fit between simulated and observed waveforms. Here, we use them as a relative difference measure that enabled us to evaluate the degree of scattering of each random medium with respect to the deterministic one. We find that for low values of c v these quantities are much lower than those for stronger c v values. This numerical tool allowed to evaluate the spatial variability along the basin. In particular, the presence of local complexities in the eastern part of the Nice basin and random heterogeneities show a large difference between the reference model and the stochastic ones (chapter 3).

We found that the PGV variability along the 2D profile depends on the distribution of local random heterogeneities. In Nice, the PGV is carried by the direct S-waves and reflected waves coming from the borders (edge-generated waves). The presence of the random velocities can produce that sometimes the PGV is carried by late phases.

The A b I is an important time domain parameter that reflects the ground-motion energy density. We found that its amplitude and spatial variability generally increases when the c v of the velocity heterogeneities is high (chapters 3 and 4). Conversely, when the dimensions of the basin increase, therefore having longer propagation times, the resulting ground motion at surface becomes attenuated (for a c v of 5%) due to the scattering and as a consequence, the A b I is also reduced. This also holds for the response spectra.

When looking at the transfer function, which is a frequency domain representation of the site response, it is not sensitive to the frequency content of the incident wavefield. However, we observe a broadening of the resonance peaks and decrease/increase of the amplification for large values of the coefficient of variation (chapter 3). This observation corroborates the study by [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF]. In addition, when averaging the transfer function for all random models we observe a smoothing of the amplification values in both frequency and space distribution. This is important for seismic hazard assessment because a mean basin response from stochastic models may underestimate the amplification.

We propose to present the results in terms of percentiles or mean ± standard deviation to assess the dispersion of such computations. In chapter 4, where 1D and 2D computations of the transfer function were made for station RAP-NLIB, we find that the presence of random heterogeneities shift the response to lower frequencies in both cases. Such effects have also been reported by [START_REF] Haber | Influence of 2D heterogeneous elastic soil properties on surface ground motion spatial variability[END_REF] and [START_REF] Nour | Finite element model for the probabilistic seismic response of heterogeneous soil profile[END_REF]. In chapter 5, where we keep a c v of 5 %, the transfer function is sensitive at frequencies higher than the fundamental resonance frequency of the basin, and generally shows some attenuation. In Chapter 4, we also compared 2D TF with the empirical transfer functions computed from earthquake recordings. Although we only have 8 earthquakes, we compare the earthquake-toearthquake varability of the empirical tranfer function with the numerical one assessed from stochastic models -interpreted as an aleatory variaiblity. As expected, the earthquake-to-earthquake varability of the empirical tranfer function is much higher than the inter-model variability. This is interpreted as being due to the influence of earthquake source location (back-azimuth, distance, depth) on empirical amplification, as shown by [START_REF] Perron | Robustness of kappa (κ) measurement in low-to-moderate seismicity areas: Insight from a site-specific study in Provence, France[END_REF] and [START_REF] Maufroy | 3d numerical simulation and ground motion prediction: Verification, validation and beyond -lessons from the e2vp project[END_REF]. We believe, however, that such kind of comparison could greatly help to understand the variability of the basin response.

Appendix A

Code verification A.1 SEM2DPACK package

In this thesis, we use the 2D spectral element package SEM2DPACK (Ampuero, J.-P., 2012) to perform numerical simulations of seismic wave propagation. At the core of the package, is a 2D solver (SEM2D) for the 2D elastic wave equations and dynamic earthquake rupture based on the Spectral Element Method with explicit time stepping. The core development of the package started during the PhD thesis of [START_REF] Komatitsch | Méthodes spectrales et éléments spectraux pour l'équation de l'élastodynamique 2D et 3D en milieu hétérogène[END_REF] and [START_REF] Ampuero | Etude physique et numérique de la nucléation des séismes[END_REF].

A.1.1 Modifications performed on SEM2DPACK

Our usage of SEM2DPACK in this study is mainly for the modeling of 2D P-SV and SH wave propagation for ground motion assessment at the scale of sedimentary basins, with heterogeneous soil properties. For this, we use the version of the code by [START_REF] Oral | 2-D P-SV and SH spectral element modelling of seismic wave propagation in non-linear media with pore-pressure effects[END_REF] which implements: 1) viscoelastic material rheology following [START_REF] Liu | Efficient modeling of q for 3d numerical simulation of wave propagation[END_REF], 2) material nonlinearity following the MPII model of [START_REF] Iwan | On a class of models for the yielding behavior of continuous and composite systems[END_REF], and 3) a liquefaction front model for excess pore pressure generation under cyclic loading, following the study by [START_REF] Bibliography | Strain space plasticity model for cyclic mobility[END_REF].

This version of the code also allows the introduction of an incident plane wavefield with absorbing layers (classical perfectly matched layers). The interested reader is referred to the PhD thesis [START_REF]Multi-dimensional modeling of seismic wave propagation in linear and nonlinear media[END_REF] by Elif Oral for more details regarding the different implementations she added in the software.

To achieve our goal in simulating wave propagation in random media, certain adjustments on the implementation design had to be made on the code, to allow for the introduction of random material properties. This modifications were done in the viscoelastic module of the package. The initial implementation of the viscoelastic module only allowed the attribution of the mechanical parameters per domain 1 when defining the material properties in the parameter input file. This implies that a soil unit can only be attributed constant mechanical parameter at the nodes of the spectral element mesh. With this implementation design, it is impossible to define random material properties in a domain when using a viscoelastic material rheology. To solve this problem, we redefined the variables of the viscoelatic 1. A domain soil unit in the mesh which is distinguish to other soil units in the mesh by its mechanical properties and labeled by tag Code verification implementation to enable the attribution of the mechanical properties at the GLL nodes of the mesh.

A.1.2 Verification

During her PhD thesis, [START_REF]Multi-dimensional modeling of seismic wave propagation in linear and nonlinear media[END_REF] verified her implementations of the viscoelastic module in the code by comparing the solution of 2D P-SV and SH modeling of ground motion in the Nice (France) sedimentary with those of the Finite difference (FD) solver by [START_REF] Gélis | 2-d p-sv numerical study of soil-source interaction in a non-linear basin[END_REF]. The viscoelasticity in the FD solver was also introduced following the approach of [START_REF] Liu | Efficient modeling of q for 3d numerical simulation of wave propagation[END_REF].

In this thesis, we redo this comparison to check the further modifications we performed on her implementation code. We compare the solution of the SH wave propagation in the deterministic velocity model of Nice (Figure 3.2) presented in chapter 3 to the solution of the FD solution. For this we use the time-frequency envelope and phase misfit analysis as in [START_REF] Kristekova | Misfit criteria for quantitative comparison of seismograms[END_REF][START_REF] Kristekova | Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals[END_REF]. [START_REF] Kristekova | Misfit criteria for quantitative comparison of seismograms[END_REF] where t is time, a is the scale parameter, b is the translational parameter, and ψ is the analyzing wavelet.

The * symbolizes the complex conjugate. The most commonly used CWT wavelet is the Morlet wavelet [START_REF] Bernadino | A real-time gabor primal sketch for visual attention[END_REF], a Gaussian-windowed complex sinusoid that is defined as follows in the domain: ψ(t) = π -1/4 exp(ikt) exp(-t 2 /2)

where t is a non-dimensional time parameter, k is the wavenumber which is generally chosen to be 6.

Using the TFR of two signals (a target signal S1 and a reference signal Sref ), [START_REF] Kristekova | Misfit criteria for quantitative comparison of seismograms[END_REF] defines the local time-frequency evelope difference (∆E(t, f )) and the local time-frequency phase difference (∆P (t, f )) of the two signals as: implies that there is a good agreement between the solution of SEM solver with that of the FD solver.

A.2 Scheme to integrate random heterogeneities in SEM2DPACK