Sergio Del Gaudio
email: sergio.delgaudio@irsn.fr

Sebastien Hok

Gaetano Festa

Mathieu Causse

Maria Lancieri

Near fault broadband ground motion simulations using empirical Green's functions: application to the Upper Rhine Graben (France-Germany) case study

Seismic hazard estimation relies classically on data based ground motion prediction equations (GMPEs) giving the expected motion level as a function of several parameters characterizing the source and the sites of interest. However, records of moderate to large earthquakes at short distances from the faults are still rare. For this reason, it is difficult to obtain a reliable ground motion prediction for such a class of events and distances where also the largest amount of damage is usually observed. A possible strategy to fill this lack of information is to generate synthetic accelerograms based on an accurate modeling of both extended fault rupture and wave propagation process.

The development of such modelling strategies is essential for estimating seismic hazard close to faults in moderate seismic activity zones, where data are even scarcer. For that we reason we selected a target site in Upper Rhine Graben (URG), at the French-German border. URG is a region where faults producing micro-seismic activity are very close to the sites of interest (e.g. critical infrastructures like supply lines, nuclear power plants, etc.) needing a careful investigation of seismic hazard.

In this work we demonstrate the feasibility of performing near-fault broadband ground motion numerical simulations in a moderate seismic activity region such as Upper Rhine Graben (URG) and discuss some of the challenges related to such an application. The modelling strategy is to couple the multi-Empirical Green's Function technique (multi-EGFt) with a k -2 kinematic source model. One of the advantages of the multi-EGFt is that it does not require a detailed knowledge of the propagation medium since the records of small events are used as the medium transfer function, if, at the target site, records of small earthquakes located on the target fault are available.

The selection of suitable events to be used as multi-EGF is detailed and discussed in our specific situation where little number of events are available. We then showed the impact that each source parameter characterizing the k -2 model has on ground motion amplitude. Finally we performed ground motion simulations showing results for different probable earthquake scenarios in the URG.

Dependency of ground motions and of their variability are analyzed at different frequencies in respect of rupture velocity, roughness degree of slip distribution (stress drop), and hypocenter location. In near source conditions, ground motion variability is shown to be mostly governed by the uncertainty on source parameters. In our specific configuration (magnitude, distance), the directivity effect is only observed in a limited frequency range. Rather, broad-band ground motions are shown to be sensitive to both average rupture velocity and its possible variability, and to slip roughness. Ending up with a comparison of simulation results and GMPEs, we conclude that source parameters and their variability should be set up carefully in order to obtain reliable broadband ground motion estimations. In particular, our study shows that slip roughness should be set up in respect to the target stress drop. This entails the need for a better understanding of the physics of earthquake source and its incorporation in the ground motion modelling.

Introduction

To estimate the seismic hazard assessment (SHA), a proper modeling of the ground motion is a fundamental step. This is commonly done through the use of ground motion prediction equations (GMPEs), which are empirical relations obtained by the regression of recorded data and metadata. The GMPEs estimate the ground motion level as a function of several parameters concerning either the source, the wave path and the site such as distance, magnitude, fault type, soil conditions, etc. The increase of the number and quality of strong motion records associated to moderate to large earthquakes worldwide led to a significant improvement in the GMPEs [START_REF] Douglas | Recent and future developments in earthquake ground motion estimation[END_REF]. A variety of GMPEs has been proposed for several engineering parameters (PGA, spectral ordinates, PGV, Intensity measurement) accounting for different source mechanisms, tectonic environment, geographical areas and specific sites effects (e.g. [START_REF] Campbell | NGA-West2 ground motion model for the average horizontal components of PGA, PGV, and 5% damped linear acceleration response spectra[END_REF]. A comprehensive collection of GMPEs can be found in Douglas (2016). GMPEs are widely used because they are computationally inexpensive and they have associated uncertainties that can be integrated in the hazard evaluation.

Nevertheless, GMPEs do not represent the variety of source and propagation, indeed the source is mainly described by the magnitude and, for the recent GMPEs, also by the fault geometry and focal mechanism, but the rupture process is not taken into account. In addition, one could question of the confidence we can have in the extrapolation beyond their range of validity. This is especially true in near fault conditions and for large crustal events, where the scarcity of available data can lead to incorrect estimates when applied to some specific case studies [START_REF] Boore | Comparisons of ground motions from the 1999 Chi-Chi earthquake with empirical predictions largely based on data from California[END_REF]. These specific cases include for example, interaction between the earthquake rupture and basins [START_REF] Koketsu | A seismological overview of long period ground motion[END_REF] or surface rupture for normal or thrust faults [START_REF] Donahue | Simulation-Based Hanging Wall Effects[END_REF]) and all the cases where the time history content is dominated by the source characteristics (e.g. see example of Tocopilla earthquake studied by [START_REF] Lancieri | Spectral scaling of the aftershocks of the Tocopilla 2007 earthquake in northern Chile[END_REF]. In such cases, ground motion simulations will likely be a better modelling than GMPEs, provided that both source and propagation are accurately described and uncertainties properly addressed. Moreover, we simulate complete waveforms having amplitudes and phases information, while GMPEs only gives us information about the amplitude of the ground motion.

The knowledge of the seismic source is generally based on different parameters: some of them can be investigated or inferred from local geology (e.g. position and geometry of the faults), or geophysical studies (e.g. the depth of the faults); other parameters, related to the kinematic and dynamic properties of the source, can be either assumed a priori making some hypothesis (e.g. magnitude of the event we want to simulate) or chosen from a range of possible values. This latter strategy leads to different resulting scenarios by defining an a priori probability distribution for each parameter: e.g. rupture velocity, slip distribution, hypocenter position on the fault plane, etc. In this paper, we will show how to produce a population of ground motion records representing the possible variability of the source process for a target earthquake (e.g. [START_REF] Causse | Calibrating median and uncertainty estimates for a practical use of empirical Green's functions technique[END_REF][START_REF] Cultrera | Variability of kinematic parameters and its implication on the choice of the design scenario[END_REF].

When the target scenario refers to historical or paleoseismic events, and no instrumental record is available, the ground motion simulations are referred to as "blind". Performing blind simulations is helpful not only to deduce an expected ground motion level, but its variability as well.

We can separate this variability in two different parts: the aleatory and the epistemic contributions. The aleatory part cannot be reduced because it comes from features that are essentially or apparently random in nature. The aleatory variability comes from any parameter that is intrinsically variable during the rupture process. The epistemic part though, since it is related to the variable modelling strategy and parameterization can be potentially reduced investigating the physics and improving our knowledge of the rupture process. For instance, physics-based rupture models showed correlations between different kinematic source parameters and the influence that they have on the ground motion [START_REF] Song | Pseudo-dynamic source modeling with 1point and 2-point staistcs of earthquake source parameters[END_REF][START_REF] Schmedes | Correlation of earthquake source parameters inferred from dynamic rupture simulations[END_REF].

Simulations of ground motions accounting for source variability can be performed through the use of numerical codes that couple the source dynamics or kinematics with the complete wavefield propagation in 3D complex media. Several techniques are commonly used to provide reliable ground motion accelerograms, such as the Spectral Element Method (e.g. Komatitcsh et al., 2004;Cupillard et al., 2012), the Finite Differences method (e.g. [START_REF] Day | Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture[END_REF][START_REF] Dalguer | Staggered-grid split-node method for spontaneous rupture simulation[END_REF][START_REF] Aochi | Influence of super-shear earthquake rupture models on simulated near-source ground motion from the 1999 Izmit (Turkey) earthquake Bull[END_REF] and the Discontinuous Galerkin Method (e.g. Dumbser and Kaeser 2007;[START_REF] Mazzieri | Speed: Spectral elements in elastodynamics with discontinuous galerkin: a non-conforming approach for 3d multiscale problems[END_REF]. Accuracy in the strong motion modelling depends on the knowledge of the structure, which is, in the best cases, limited at space scales larger than hundreds of meters. The main problem of these approaches is that for many sites crustal models are barely known. In addition, for broadband ground motion estimations (above 10Hz) those numerical schemes need to be completed by stochastic high frequency methods.

For these reasons, the use of Empirical Green's Functions (EGF) may help in correctly model the wave propagation up to short wavelengths [START_REF] Hartzell | Earthquakes aftershocks as Green's functions[END_REF][START_REF] Irikura | Semi-empirical estimation of strong ground motions during large earthquakes[END_REF], providing an elastic response of the medium and a compatibility of the small earthquakes with the target event to simulate in terms of location on the fault, focal mechanism and stress drop [START_REF] Irikura | Prediction of strong ground motions using observed seismograms from small events[END_REF]. Although this approach was developed and validated for far-field broadband simulations, it has been recently extended by Del [START_REF] Del Gaudio | Broad-band strong motion simulations coupling k-square kinematic source models with empirical Green's functions: the 2009 L'Aquila earthquake[END_REF] to the near fault case through the introduction of the multi-EGF concept. In this concept, different EGFs are attributed to different part of the target fault in order to cover the whole range of different orientations between the source and a receiver that is close to it, so that the approximation of EGF remains valid locally. Regarding the source process, the multi-EGF technique is coupled with a k -2 kinematic model [START_REF] Herrero | Modeling directivity of heterogeneous earthquake ruptures[END_REF] that will be described in the methodology section of the paper. This methodology allows for rapid estimations of broadband ground motions in the near fault area together with a detailed extended source description, without needing either a detailed knowledge of propagation medium nor additional methodologies for modeling different frequency bands. This methodology of broadband ground motion simulations is thus adapted to the need for challenging GMPEs on their limited range of validity close to faults, especially due to their lack of precision regarding source description.

Finally, we face an additional challenge regarding SHA and GMPEs in moderate to low seismic activity context. In such areas, small earthquakes are recorded and estimation of seismic hazard is needed. However, GMPEs are criticizable in such places because of the possible inconsistency between the local context and the data used in the GMPE. Therefore, simulations, and in particular EGFt, can improve the ground motion prediction. Thanks to the presence of multiple small earthquakes, EGFt can be applied directly to moderate seismicity areas. For that reason, we applied our methodology to the Upper Rhine Graben (URG) area, where minimal experimental conditions are found: active faults, recorded moderate seismicity, and need for a near-fault SHA. As a first step toward near-fault SHA, we perform broadband blind simulations for two scenarios (Mw 6 and Mw 6.5) on an active fault in the URG. Along the text we point out the difficulty in doing such an exercise in an area scarcely instrumented where information on small recorded events are imprecise.

After a description of the URG, we describe the implemented strategy to select and characterize the EGFs, to explore the source parameters variability and its influence on the resulting ground motion simulations.

We selected a target site which is located in very near fault conditions, but also close to seismic activity in the center of the Graben, where most of the infrastructures (e.g. buildings, roads, industries, etc.) and also some critical facilities (e.g. supply lines, nuclear power plants, etc.) are located.

Simulations method

Simulations method: EGF technique

Using the representation integral [START_REF] Aki | Quantitative Seismology[END_REF], the ground displacement recorded at a position 𝑥̅ on the Earth's surface at a time 𝑡 can be expressed as:

𝒖(𝑥̅ , 𝑡) = ∫ 𝑠 𝑖 (𝜉 ̅ , 𝜏) * [𝑐 𝑖𝑗𝑘𝑙 (𝜉 ̅ )𝝂 𝑗 (𝜉 ̅ )𝑮 𝑘,𝑙 (𝑥̅ , 𝜉 ̅ ; 𝑡, 𝜏)]𝑑Σ(𝜉 ̅ ) 𝛴 (1)
where s is the slip vector, c is the elastic coefficient matrix, 𝝂 is the normal to the fault, G is the Green's function and ∑ is the fault surface. Summation over repeated indices is assumed and the symbol * denotes time convolution. The Green's function represents the crustal medium response to an impulsive wave and it can be computed knowing the crustal structure (i.e. geometrical and rheological properties). In this work we adopted a simulation method based on the empirical Green's functions (EGF) approach. Many applications have made use of the EGFs for ground motion simulation [START_REF] Hutchings | Kinematic earthquake models and synthesized ground motion using empirical Green's functions[END_REF][START_REF] Irikura | Estimation of strong ground motion in broad-frequency band based on a seismic source scaling model and an empirical Green's function technique[END_REF][START_REF] Hartzell | Calculation of broadband time histories of ground motion: Comparison of methods and validation using strong-ground motion from the 1994 Northridge earthquake[END_REF][START_REF] Causse | New approach for coupling k-2 and empirical Green's functions: Application to the blind prediction of broadband ground-motion in the Grenoble basin[END_REF][START_REF] Del Gaudio | Broad-band strong motion simulations coupling k-square kinematic source models with empirical Green's functions: the 2009 L'Aquila earthquake[END_REF] since the pioneering works of [START_REF] Hartzell | Earthquakes aftershocks as Green's functions[END_REF]

The record of small earthquake used to describe the medium response is generally referred to as an EGF. The index i of the slip function has been removed because we assumed a constant rake direction, so S now only represents the amplitude of the slip.

For broadband simulations, the main advantage of using EGFs is that we do not need a detailed knowledge of the crustal velocity structure; we indeed need a good quality location catalog of small events and some reliable indication of the fault geometry, moreover the earthquakes used as EGF should satisfy specific criteria: first of all, the signal-to-noise ratio has to be sufficiently large in all the frequency range of interest, its location and focal mechanism has to be compatible with the fault plane considered for the target event and its magnitude must be at least 2 point smaller than the one of the target event.

Under this latter condition an EGF can be considered as the medium response to a Dirac delta function because it is a good approximation of a point source with respect to the large event to be simulated.

The space discretization on the fault plane is controlled by the size of the event used as EGF, according to earthquake scaling laws. Assuming self-similarity between the rupture of small and large events [START_REF] Brune | Tectonic stress and the spectra of seismic shear waves from earthquakes[END_REF], the number of subfaults associated to small events to be considered is

(3) N is the spatial scale factor between the mainshock and the associated EGF. In this equation L, W and M0 are the length, width, and seismic moment of the target event, respectively, while l, w and 𝑀 𝐺 ̅ represent the same quantities for the small event. In this study we used squared subfaults with size Δx=w to discretely solve equation (2).

Simulations method: the Multi-EGF approach

The classic EGF approach uses a single event as EGF translated along the fault plane. This is equivalent to assume that the propagation term (path, radiation) is the same for all the points of the crustal volume surrounding the fault plane. This is a strong approximation, especially for large faults and in near source conditions, where the wave path from fault to site varies strongly.

To overcome this limit, Del [START_REF] Del Gaudio | Broad-band strong motion simulations coupling k-square kinematic source models with empirical Green's functions: the 2009 L'Aquila earthquake[END_REF] proposed to use several EGFs, each EGF sampling the closest portion of the fault plane, allowing to better take into account the wave path and source radiation variability. This technique is called multi-EGF. Del Gaudio et al.

1/3 0 G M LW N l w M      
(2015) demonstrated the validity of the methodology on the 2009 L'Aquila earthquake using several aftershocks as multi-EGF to reproduce the main event accurately.

In practical, the difference with the single EGF technique is that the Green's function is variable along the fault plane. Each portion of the fault plane gets its own EGF as being the closest one, such as illustrated on Figure 1. In addition, a correction in time and amplitude is also applied to each EGF to take into account the difference between original hypocentral position of the event and applied subfault (see [START_REF] Causse | New approach for coupling k-2 and empirical Green's functions: Application to the blind prediction of broadband ground-motion in the Grenoble basin[END_REF]). After a discretization of equation ( 2), the simulated displacement can be written as a summation of EGF

𝒖(𝑥̅ , 𝑡) = ∑ 𝜇 (𝑖𝑗) ∆𝑥 2 𝑀 𝐺 ̅ (𝑖𝑗) 𝑆 (𝑖𝑗) * 𝑮 (𝑖𝑗) (𝑥̅ , 𝑡) 𝑖𝑗 (4)
The indices i and j are related to subfault (i,j) and run over strike and dip directions respectively.

Simulations method: source model

To describe the source process we adopted the k -2 kinematic source model [START_REF] Herrero | Modeling directivity of heterogeneous earthquake ruptures[END_REF]Gallovič and Brokešová, 2004). In the wavenumber domain k, this model is characterized by an asymptotic k -2 decay of the slip amplitude beyond the corner wavenumber

𝑘 𝑐 = K/𝐿,
where 𝐿 is the characteristic fault dimension and K is a non-dimensional parameter which represents the roughness degree of the slip distribution (in Figure S1 of the electronic supplement are shown two example of slip distributions for two different values of K). This particular asymptotic behavior is able to reproduce the high frequency 𝑓 -2 decay observed in far-field displacement spectra. While the slip amplitude spectrum in the wavenumber domain follows this model, the phase distribution 𝜑 is randomly chosen and it represents a source of variability for k -2 models. The slip distribution in the wavenumber domain can be summarized as

𝑆 𝑘 (𝑘 𝑥 , 𝑘 𝑦 ) = 𝑆 ̅ 𝐿𝑊 √ 1+[( 𝑘 𝑥 𝐿 K ) 2 +( 𝑘 𝑦 𝑊 K ) 2 ] 2 𝑒 𝑖𝜑(𝑘 𝑥 ,𝑘 𝑦 ) (5)
where 𝑆 ̅ is the average slip value, 𝐿, 𝑊, 𝑘 𝑥 and 𝑘 𝑦 are the fault dimensions and wavenumbers respectively along strike and dip directions.

Rise time is assumed to be k-dependent. In particular it is constant for 𝑘 < 𝑘 𝑐 and it linearly decreases with k for larger wavenumbers. Rupture velocity is assumed to be constant over the fault plane.

All the source parameters such as the roughness degree of the slip K, the rupture velocity 𝑣 𝑟 , the rise time 𝜏 and the position of the hypocenter on the fault plane, can be deterministically defined (i.e. from a kinematic source model obtained from low frequency inversion) or randomly chosen from a priori distribution [START_REF] Causse | New approach for coupling k-2 and empirical Green's functions: Application to the blind prediction of broadband ground-motion in the Grenoble basin[END_REF][START_REF] Del Gaudio | Broad-band strong motion simulations coupling k-square kinematic source models with empirical Green's functions: the 2009 L'Aquila earthquake[END_REF].

Upper Rhine Graben case study

To better investigate the advantages and limitation of the technique, and to test the influence that each considered source parameter has on the simulated ground motion and its variability, we set up an application to the Upper Rhine Graben (URG). The URG is a region located along the border line between French and Germany, characterized by a moderate level of seismicity.

Historically, one of the major West-European earthquakes happened in 1356, near Basel (Switzerland) at the Southern termination of the Graben. MKS maximum intensity of this event was estimated between VIII and IX (Mayer-Rosa and [START_REF]A review of the 1356 Basel earthquake: Basic data[END_REF], but the fault source has been not clearly identified (Mayer et al., 1994;[START_REF] Lambert | New hypotheses on the maximum damage area of the 1356 Basel earthquake (Switzerland)[END_REF][START_REF] Meghraoui | Active normal faulting in the Upper Rhine Graben and Paleoseismic identification of the 1356 Basel earthquake[END_REF][START_REF] Ferry | Evidence for Holocene palaeoseismicity along the Basel-Reinach active normal fault (Switzerland): a seismic source for the 1356 earthquake in the Upper Rhine graben[END_REF].

The Rhine Graben is characterized by the presence of several fault systems. In Figure 2 all the segments that have shown evidence of activity are represented. Specifically we can recognize the Western Rhine Fault, at the western border of the Graben; the Black Forest Fault, at the eastern border of the Graben and the Rhine River Fault [START_REF] Bertrand | Quaternary faulting in the Upper Rhine Graben revealed by high-resolution multi-channel reflection seismic[END_REF][START_REF] Nivière | Active tectonics of the southeastern Upper Rhine Graben, Freiburg area (Germany)[END_REF][START_REF] Palumbo | Devising BDFA: a new active fault database conceived behind nuclear safety assessment in France[END_REF], following the river path.

Considering historical seismicity and the large number of active faults in the area, people and infrastructures (e.g. buildings, roads, industries, etc.) and also some critical facilities (e.g.

supply lines, nuclear power plants, etc.) are potentially exposed to the risk that comes from earthquakes generated along the URG. The size of the fault segments in the URG suggests that earthquakes with magnitude larger than 6 are likely to happen. Their return periods are estimated as a function of the magnitude [START_REF] Nivière | Active tectonics of the southeastern Upper Rhine Graben, Freiburg area (Germany)[END_REF] and, for instance, a return interval of about 8000 years was estimated for the Rhine River Fault for an event of magnitude of 6.5.

In addition, such faults were also considered in long return period probabilistic seismic hazard assessment (PSHA) by [START_REF] Chartier | Étude probabiliste de l'aléa sismique pour un site du Fossé Rhénan Supérieur[END_REF]. Such studies use GMPEs to evaluate the ground motions close to the fault, where they are likely to be biased. As a consequence, ground motion simulations can provide a useful comparison with GMPEs and eventually an alternative choice for PSHA. For these reasons the URG is an interesting test case for applying our technique and performing blind ground motion simulations.

For the simulations, we decided to focus on the Rhine River fault. We considered this fault as divided in two segments. From surface evidence, we set their strikes equal to 172° for the northern segment, and 203° for the southern segment. From geological and geophysical investigations [START_REF] Nivière | Active tectonics of the southeastern Upper Rhine Graben, Freiburg area (Germany)[END_REF] the faults are west dipping with an angle of approximately 60°-70°. We decided to adopt a dip angle of 70° but, for future applications, an exploration of different values should be performed in order to investigate the influence that it could have on simulation results. Regarding the depth of the faults, we prolonged the surface trace up to a maximum depth of 15 km. In Figure 3 the two selected segments of the Rhine River Fault are schematically represented.

To perform ground motion simulation in the URG we faced several problems. Due to the fact that we are in "blind" conditions, kinematic parameters (e.g. rupture velocity, slip distribution) and dynamic parameters (e.g. stress drop, interaction between different fault segments) are unknown, as well as details regarding the fault geometry (e.g. hypocenter location, depth of the fault). Sensitivity studies for all the different parameters are, hence, required to characterize the aleatory and epistemic variability of the simulated ground motion. Moreover, the low rate of seismicity and the consequent small amount of records requires a careful investigation of the different events to be used as EGF, whose selection depends on the hypothesis for the target event (fault geometry, magnitude, etc.).

EGF Selection and Analysis

After selecting the fault geometry, we started screening data of small earthquakes to be used as an empirical approximation of the propagation medium response function. As described in the methodology section, to be used as an EGF, an event must satisfy specific criteria.

We screened out the French instrumental catalog Si-Hex (Sismicité instrumentale de l'Hexagone), the Renass (Réseau National de Surveillance Sismique) catalog, and the seismic catalog of the LGRB (Landesamt für Geologie, Rohstoffe und Bergbau) Institute of Freiburg in a time period ranging between January 2000 and December 2014. The seismological institutes responsible for the location of earthquakes indeed share the data across the border. Twentyseven events were identified with a local magnitude ranging between 1.5 and 3.2 (Figure 3).

We decided to adopt the locations proposed by the LGRB catalog, because they are done without fixing the hypocentral depth. This choice will reduce possible biases in the correction for the geometrical spreading.

As target site we chose the BREM station (see Figure 3) which is close to both the considered faults (it's located exactly on the surface projection of both the faults planes and at about 5 km of geometrical distance from them).

We selected five events, three having the largest magnitude in the catalog, to have a large signal to noise ratio in a wide frequency range, and the two whose location is the closest to the selected fault system. In Figure 3, we plot all the considered events (black dots) and the selected EGF (red dots). From the figure results clear that two of the five selected EGF can be associated to the northern fault segment and three to the southern segment. Therefore our selection covers spatially the target faults, which is needed in the multi-EGF approach for near field computation. On the map the seismic stations located in the area belonging to both the French and the German national networks are also showed.

To verify the hypothesis of self-similarity of the EGF, we computed seismic moment and corner frequency and we inferred the apparent stress drop by inverting the S phase Fourier amplitude spectrum. The S wave has been manually picked on the records at the available stations. A time window of 5s (0.5s before and 4.5s after the S wave arrival time) containing the whole direct S-wave train has been isolated. We removed baseline from signals and computed displacement from acceleration by double integration. After computing Fourier spectra of the three components we combined them by vector summation. For the spectral inversion, we assumed in the angular frequency domain ω a ω -2 decay of the displacement spectrum beyond its corner frequency fc [START_REF] Brune | Tectonic stress and the spectra of seismic shear waves from earthquakes[END_REF]. We also estimated the focal mechanisms for eligible EGF computing long-period spectral level ratios at each station [START_REF] De Matteis | BISTROP: Bayesian Inversion of Spectral-Level Ratios and P-Wave Polarities for Focal Mechanism Determination[END_REF]. Because of the sparse network geometry, azimuthal gap (>300°), paucity of records and low signal to noise ratio for P wave, we fixed the strike and the dip angle according to the geometry of the faults to which the event is related by proximity, and we computed the rake angle using single station spectral ratios between S and P displacement levels at low frequency. Assuming a Brune's source model, the ratio between the low frequency plateaus, can be expressed as:

Ω 0,𝑠 Ω 0,𝑝 = 𝑅 𝜆𝜑 𝑃 𝑅 𝜆𝜑 𝑆 ( 𝑣 𝑠 𝑣 𝑝 ) 3 (6)
where 𝑣 𝑠 and 𝑣 𝑝 are the mean values of P and S wave velocities and 𝑅 𝜆𝜑 𝑃,𝑆 are the radiation pattern coefficients which depend on the rake angle (a detailed formulation of this coefficients is given in [START_REF] Aki | Quantitative Seismology[END_REF]. After measuring the ratio, a grid search was performed to determine the value of the rake angle which at best approximates the equation ( 5). Two symmetric solutions are always available for the rake angle and, to remove this ambiguity, we have to measure P-wave polarities for each station. Obtained results show mainly a normal fault behavior with a strike slip component in the northern part, as showed also in previous studies performed for the same region [START_REF] Homuth | Seismicity of the northern Upper Rhine Graben -Constraints on the present-day stress field from focal mechanisms[END_REF]. For this reason, we did not apply any correction for the focal mechanism differences of the EGF.

Location and computed parameters of the selected EGFs are reported in TABLE 1. Figure 4 shows the retrieved corner frequency as a function of the seismic moment in a logarithmic scale respectively, for all the selected EGFs. In Figure S2 of supplementary material are also shown the spectral inversions for all the EGFs. As shown in Figure S2, not all the signals are characterized by a good signal to noise ratio in all the considered frequency range (from 1 to 20 Hz). In particular this ratio becomes lower than 1 at low frequencies for stations END, LIBD and STAU and EGF1, EGF3 and EGF4. We decided to keep these data and source parameters were obtained by making a weighted average of values computed for each station, with weights proportional to signal-to-noise ratios.

For the selected events, we found that the average stress drop ranges between 3MPa and 35

MPa

Ground motion simulations

Sensitivity analysis

We recall that, in blind conditions, we do not have a benchmark time history, so our aim is not to simulate an observed motion, but all the possible ground motions produced by all the possible source configurations. In fact, while the knowledge on fault geometry can be improved through geophysical investigations), dynamic and kinematic source parameters are unknown until the occurrence of a big earthquake. Moreover, they can be unique for each event and then, for this reason, it is necessary to make several simulations with variable source parameters to investigate all the possible scenarios and the variability associated to them.

The k -2 model is characterized by three parameters that mainly affect the ground motion: the roughness parameter K (eq. 4), the rupture velocity 𝑣 𝑟 and the nucleation position on the fault plane. We did sensitivity analysis for all of them. Figure 5, Figure 6 and Figure 8 show simulation results for a magnitude 6 event occurring on the southern segment of the Rhine River We considered a large range of variation for each parameter to explore the effect on the ground motion. Several authors (e.g. [START_REF] Gallovic | On strong ground motion synthesis with the k-2 slip distribution[END_REF][START_REF] Causse | New approach for coupling k-2 and empirical Green's functions: Application to the blind prediction of broadband ground-motion in the Grenoble basin[END_REF]2010) have

given different probable values of roughness for the slip distribution ranging it between 0.7 and 1.6, we then let K vary uniformly between 0.3 and 2 to explore a wider range.

For 𝑣 𝑟 as well we selected a wide range letting it vary between 2100 m/s and 3400 m/s. For this latter parameter we assumed subshear rupture propagation on average with the maximum rupture velocity smaller than the average crustal shear wave speed of about 3500 m/s.

For the nucleation position, any location on the fault plane has been considered.

Simulations were first performed looking at the influence of one parameter at a time. In Figure 5, the influence of K on ground motion was investigated performing 100 simulations with a fixed 𝑣 𝑟 = 2800𝑚/𝑠 and the hypocenter location at the center of the fault. In Figure 6, we performed 100 simulations fixing K=1 and the hypocenter location again at the center of the fault to check the impact of 𝑣 𝑟 . We observe a linear trend at low frequency and an exponential trend at high frequencies. Variability for specific values of the source parameters is inherited by the aleatory variability of the slip phases, which are randomly chosen in the k -2 model as discussed in the first section. We can also observe that, for high frequencies (above 5Hz), ground motion is not influenced by rupture velocity (the trend is linear with a slope close to zero) up to a threshold above that we observe an exponential increase. This threshold is between 3000m/s and 3200m/s.

To simultaneously investigate the influence that these two parameters have on ground motion, in Figure 7 results of simulations obtained combining the variability of K and 𝑣 𝑟 are shown, while the nucleation position is always fixed at the center of the fault. The six subplots refer to six frequencies and, for each frequency, a clear trend of the ground motion parameters as a function of both the roughness and rupture velocity is observed. Specifically the dependency on the roughness looks quite linear at all the considered frequencies. From Figure 7 we observe how roughness and rupture velocity jointly affect the ground motion. We can see how amplitude variation for low frequencies (up to 2Hz) is of about two order of magnitude: we identified three different color areas (each color area corresponds to half order of magnitude), separated by black lines, for which the logarithm of the spectral acceleration and PGA is almost constant. If we move to higher frequencies, we observe a wider range of variation (with five different areas highlighted for the 10Hz plot). From this figure we can also conclude that for small value of K (below 0.5 for spectral acceleration and between 0.4 and 0.9 for PGA) ground motion amplitudes is less sensitive to rupture velocity. For larger values of K, variations of 𝑣 𝑟 become really important to control the simulated ground motion values.

Concerning the variability associated with the nucleation position on the fault plane, we separated its variation along the strike and the dip directions, here represented by the variables 𝑥 𝑛 and 𝑦 𝑛 respectively. Figure 8 shows the ground motion parameters as extracted from the simulations as functions of the nucleation position on the fault plane. In this subset of simulations we fixed 𝑣 𝑟 = 2800𝑚/𝑠 and K=1. A correlation at very low frequencies (0.5 Hz) can be observed. Specifically we computed the corner frequency for the simulated events. Since they range between 0.1 and 0.5 Hz, we expect that the directivity effect is more important around this interval while it starts to disappear at higher frequencies (>=1 Hz). To investigate this aspect we performed another set of simulations for a magnitude Mw 5.0 event. To be consistent with the new magnitude, also the size of the fault was reduced while the corner frequency of the synthetics increased at around 1.5 Hz. It is well known that the directivity effect is mainly observed in a range below the corner frequency (e.g. [START_REF] Causse | New approach for coupling k-2 and empirical Green's functions: Application to the blind prediction of broadband ground-motion in the Grenoble basin[END_REF]. For this reason, in Figure S3 of the supplementary material, the results of the test for a Mw 5 event clearly show the rupture directivity effect up to 2 Hz.

This sensitivity analysis shows that both the rupture velocity and the roughness degree of the slip have a strong impact on the high frequency strong motion parameters while they generally control the amplitude of the ground motion. For this reason, constraining their range of variation could significantly reduce the epistemic variability of the simulated ground motion. We should take into account these results to build up a proper strategy of selection of realistic values for these parameters to not introduce strong biases in the calculation.

Fault geometry and Stress Drop

Another important aspect consists in evaluating the static stress drop associated to our simulations. This stress drop is related to the slip distributions and hence to the roughness parameter. It is independent from seismological data (but depends on geological information)

and we can compare it with the estimates from the EGF reported in TABLE 1. According to [START_REF] Kanamori | Theoretical basis of some empirical relations in seismology[END_REF], for an extended fault, the static stress drop associated to a particular slip distribution can be defined as:

Δ𝜎 = 𝐶𝜇 𝑆 ̅ 𝐿 ̃ (7)
where C is a non-dimensional shape factor (≈1 in many cases), μ is the rigidity, 𝑆 ̅ is the mean slip value over the fault plane and 𝐿 ̃ represents the length of the rupture area defined as the square root of the surface where the slip is over 20% of the maximum slip. The size of the just one order of magnitude of variation for Δ𝜎), it is important to note that stress drop values obtained using the entire fault surface inferred from geological knowledge are really small compared to the average stress drop computed from the EGF (~8𝑀𝑃𝑎). Since the EGF technique is based on the self-similarity assumption, we concluded that using a smaller fault surface could be more appropriate for our simulations. Geological faults are generally big structures, while rupture surface should be related to the characteristics of the event we want to simulate and can be just a portion of the geological fault plane. For the rest of the paper, all the simulations relative to a Mw 6 event will be performed for a smaller fault plane (the red rectangle in Figure 9b).

Finally, we can also observe that high values of rupture velocity result in very large peak amplitudes on the simulations.

Blind simulations

To perform blind simulations a useful strategy consists in defining the probability distribution of the source parameters to control their variability. An example is given in Figure 10 for the rupture velocity. For a uniformly distributed probability (between 2200 m/s and 3400 m/s), resulting values of PGA are between 10% and 60% of g. For a normal distribution centered in 𝑣 𝑟 = 0.8𝑣 𝑠 and truncated in the same interval as above, the resulting PGA variability is significantly reduced (from 10% to 38% of g). From this simple example we can easily deduce that constraining the variability in the source parameters with a proper probability distribution will reduce ground motion variability of our simulations.

For the roughness parameter, [START_REF] Causse | Constraining the roughness degree of slip heterogeneity[END_REF] showed a correlation between its logarithm and the logarithm of the PGA. Under the far-field approximation, they show that PGA∝(K/L) 3/2 proposing a lognormal distribution for this parameter with a magnitude dependent mean value and a standard deviation 𝜎 = 0.12. Our results show an almost linear increase of the PGA with K (Figure 5) probably due to the very near fault conditions. In any case, the dependence of the PGA from the roughness of the slip is confirmed and it is still reasonable to adopt a lognormal distribution. Nevertheless, mean value of this distribution is not clearly identified in literature. According to [START_REF] Gallovic | On strong ground motion synthesis with the k-2 slip distribution[END_REF], a reasonable value of K must be chosen around 1, while using the hypothesis of self-similarity between the target event and the EGFs, [START_REF] Causse | New approach for coupling k-2 and empirical Green's functions: Application to the blind prediction of broadband ground-motion in the Grenoble basin[END_REF] found K ̅ = 0.74. The empirical relation derived by [START_REF] Causse | Constraining the roughness degree of slip heterogeneity[END_REF] shows, for a Mw 6 event, K ̅ = 1.6. For this reason, we decided to perform three different sets of simulations changing the mean value for the distribution of the roughness parameter.

For the rupture velocity we selected a range of variation according to observations from source studies where 𝑣 𝑟 is generally found between 60% and 90% of 𝑣 𝑠 (e.g. [START_REF] Heaton | Evidence for and implications of self-healing pulses of slip in earthquake rupture[END_REF]). So we decided to assume a normal distribution with a mean value 𝑣 𝑟 ̅ = 0.8𝑣 𝑠 = 2800𝑚/𝑠 and 𝜎 = 400𝑚/𝑠, truncated between 2100m/s and 3400m/s to avoid super shear propagation.

According with results obtained by [START_REF] Mai | Hypocenter locations in finite-source rupture models[END_REF], nucleation position should be chosen close to zones with significant slip values. Since our explored slip distributions mostly have large slip patches far from the fault edges, we chose normal distributions centered in the middle of the fault plane for 𝑥 𝑛 and 𝑦 𝑛 .

In Figure 11 are summarized all the chosen source parameters probability density functions (Figure 11a) and the choices we made for the roughness mean value, the fault size and its position with respect to the main fault plane (Figure 11b).

Finally we let vary all the source parameters together, according to the probability distributions described above, and performed three sets of 500 blind simulations for a magnitude 6 event located on the southern fault segment and we compared results with prediction of different GMPEs. Each set has a different mean roughness value: 0.74 (from self-similarity assumption), 1.0 (from Gallovic & Brokesova 2004) and 1.6 (from empirical relations).

We chose four different prediction equations that were compatible with the considered case study: [START_REF] Boore | Ground-Motion Prediction Equations for the Average Horizontal Component of PGA, PGV, and 5%-Damped PSA at Spectral Periods between 0.01 s and 10.0 s[END_REF], [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF], [START_REF] Cauzzi | Updated predictive equations for broadband (0.01-10 s) horizontal response spectra and peak ground motions, based on a global dataset of digital acceleration records[END_REF] and [START_REF] Akkar | Empirical equations for the prediction of PGA, PGV and spectral accelerations in Europe, the Mediterranean and the Middle East[END_REF]. Some details about the considered GMPEs are shown in TABLE 2. All these GMPEs take into account the soil site condition through the VS30 parameter (mean shear wave velocity in the first 30 m from the surface). For the selected site we assumed VS30 = 580 m/s [START_REF] Chartier | Étude probabiliste de l'aléa sismique pour un site du Fossé Rhénan Supérieur[END_REF].

As an example, in Figure 12 PGA and Sa (computed at 0.5, 1, 2, 5 and 10 Hz) values from simulations are compared with GMPEs predictions for a Mw 6 event recorded at BREM station.

As observed also by other authors (e.g. [START_REF] Baumann | Evaluating the compatibility of dynamic rupturebased synthetic ground motion with empirical ground-motion prediction equation[END_REF][START_REF] Dalguer | Prediction of near-source ground motion exceeding 1g at low frequencies (<2 Hz) from Mw∼6.5 deterministic physics-based dynamic rupture simulations[END_REF], empirical models predictions generally produce higher values than simulation results in near source conditions (distances lower than 10 km). Here we observe the same discrepancy at all frequencies, except for the simulations obtained with a mean roughness value of K ̅ = 1.6. This is clearly visible between 5 and 10 Hz, where we can observe a strong increase of the simulated ground motion values.

Figure 12 also shows simulation results for different ranges of rupture velocities: 𝑣 𝑟 ≤ 2400𝑚/𝑠, 2400 < 𝑣 𝑟 < 3000𝑚/𝑠 and 𝑣 𝑟 ≥ 3000𝑚/𝑠. As expected, due to the choice of a normal distribution for rupture velocity centered on 2800m/s, most of the data are concentrated in the range between 2400 and 3000m/s.

Finally, we decided to adopt the roughness mean value that comes from the self-similarity assumption (0.74) in order to fix the stress drop of the target event as the average of the values computed for the EGFs. Under this assumption, the Figure 13 shows the results of three sets of 500 blind simulations obtained for three different scenarios: the first one considering only the southern segment of the Rhine River Fault; the second one considering only the northern segment; the third one considering the two segments together.

In Figure 13 the spectral accelerations are shown as functions of the period 𝑇 (obtained as the geometrical mean of the two horizontal components) in terms of mean and standard deviation for each scenario. Each single fault has been associated to a magnitude 6 event, such that a rupture scenario propagating through both the segments generates a Mw 6.5 event.

As we can see, if we consider the two faults separately, results are very similar with the northern segment producing larger amplitudes especially for periods between 0.05 and 0.1 s (10 to 20 Hz). Results obtained involving both the two fault segments are giving, as expected, larger amplitudes with values of response spectra over 1 g for periods between 0.1 s and 0.2 s (5 to 10 Hz).

The possibility to investigate different scenarios results extremely important in the case of a probabilistic hazard analysis. Including faulting models in PSHA let us to identify which could be the source contributing the most to the hazard [START_REF] Chartier | Étude probabiliste de l'aléa sismique pour un site du Fossé Rhénan Supérieur[END_REF].

Discussion and conclusions

In the framework of the BestPSHANI workshop (Best Practices in Physics-Based Fault Rupture Models for Seismic Hazard Assessment of Nuclear Installations, Vienna, Austria, 18-20 November 2015), whose purpose was to discuss the applicability of different ground motion simulation techniques for hazard assessment of nuclear installations, we propose a near fault application to the Upper Rhine Graben based on the multi empirical Green's function approach coupled to a k -2 kinematic source description. The importance of performing simulations relies on the fact that GMPEs are often not robust in near fault conditions. The lack in near fault records makes them not able to cover all the possible earthquake scenarios likely leading to incorrect ground motion estimation.

Numerical simulations can be a way to overcome this problem, although they need to be properly calibrated with the weight of each parameter investigated. One of the advantages of the approach we use for modeling the seismic wave propagation is that a very detailed description of the crustal medium is not required because it is replaced by a significant number of records of small earthquakes. In the case of the application shown here, the Upper Rhine Graben, a region located at the border between France and Germany which is characterized by a low to moderate seismic activity, finding enough records was part of the challenge. From a database of about 30 events recorded in the last 15 years, we were able to extract 5 events which satisfy strict quality criteria to be used as EGFs. Even if it worked out in this case, this is one limitation of our application. Indeed improvement of seismic networks for regions demanding seismic hazard assessment is required to increase the robustness of ground motion prediction.

After choosing the BREM station as the target site, we tested the influence of source parameters on the simulations. In particular we showed the correlations of the ground motion parameters with the roughness of the slip, the rupture velocity and the nucleation position. The roughness parameter mostly shows a linear correlation with ground motion amplitudes, while for the rupture velocity we observe a faster growth at high frequency (mimicking an exponential behavior). Nucleation position is related to rupture directivity effect and it has a strong influence only at low frequencies below the corner frequency of the simulated event. To better understand the physical meaning of the roughness parameter and try to reduce its range of variability, we studied its correlation with the static stress drop. All these parameters have shown to play a key role and we studied in details this effect to better constrain the variability of the ground motion and properly calibrate simulations.

While for rupture velocity and the hypocentral position we can make some assumptions on the most probable values and the associated variability, the roughness parameter has no a priori constraints because it is directly related to the slip distribution, which can be quite different from earthquake to earthquake. So we decided to adopt a lognormal probability distribution for it according to [START_REF] Causse | Constraining the roughness degree of slip heterogeneity[END_REF] and to explore three (very) different mean values as proposed in several previous studies (e.g. [START_REF] Causse | New approach for coupling k-2 and empirical Green's functions: Application to the blind prediction of broadband ground-motion in the Grenoble basin[END_REF][START_REF] Gallovic | On strong ground motion synthesis with the k-2 slip distribution[END_REF]. Without any a priori assumption, as discussed also by Del [START_REF] Del Gaudio | Broad-band strong motion simulations coupling k-square kinematic source models with empirical Green's functions: the 2009 L'Aquila earthquake[END_REF], using the EGF approach, K ̅ = 0.74 should be preferred according to the self-similarity hypothesis.

We then performed blind simulations for a magnitude 6 event occurring on the southern segment of the Rhine River Fault system assuming a priori probability distributions of source parameters and comparing them with four different GMPEs. We found that generally our simulations lead to lower ground motion estimations (and variability) than the ones obtained from GMPEs, with a strong dependence on the input source parameters. This comparison is not straightforward though, because in both cases we are not comparing the same ground motions estimators. Regarding simulations, the level and the variability of the results are reflecting the large uncertainties on the controlling source parameters, exhibiting the lack of physical constrains on them. In the future, physics-based rupture models should help to reduce those uncertainties. Regarding the GMPEs, the results are reflecting the lack of near fault data and the summation of uncertainties due to both extrapolation and ergodicity. Large epistemic uncertainties remain in both methodologies, although they are essentially different for each.

The relative consistency between GMPEs with respect to simulations may just be symptomatic of the common methodology of GMPEs which are essentially lacking of data in near source conditions, while the biases of simulations are essentially different and independent ones originating from the lack of physics.

With this work we confirm that, for a defined earthquake scenario, distance is not sufficient to explain the variability of the ground motion. In this synthetic case, it is dominated by the uncertainty on sources parameters (roughness degree, average rupture velocity, etc.) rather than on aleatory parameters such as slip distribution or hypocenter location. Our work clearly identifies parameters controlling the ground motion, and subsequently point out which uncertainties should future studies aim at reducing. Meanwhile, those uncertainties should be taken into account in ground motion computations. This is an important point if the simulations carried out have to be included into a probabilistic seismic hazard model to complete information that comes from GMPEs at short distances from the fault. For a PSHA, it is necessary not only to retrieve the average ground motion produced by a given earthquake, but the distribution of all the possible scenarios that can be generated.

Finally, for the specific case study we also showed results for three different sets of scenarios considering the two selected segments of the Rhine River Fault. For each of them we showed average simulated ground motion and its variability in terms of acceleration response spectra as functions of the period of the oscillation. For these simulations we fixed the magnitude to Mw 6 for the case of a single fault involved in the rupture process, and Mw 6.5 when considering both the fault segments. Our results show that the northern segment produces on average larger amplitudes than the southern one, and that the worst case of a bigger earthquake producing over the two fault segments can reach high ground motion level for a target site located in very near fault conditions (more than 1 g for response spectra ~5 Hz with a PGA of 0.4 g). Again, this result is strongly dependent on the uncertainties we have on source parameters. In blind conditions, where no benchmark are available, source parameters should be accurately investigated and their natural variability properly reproduced. A strategy which combines physics-based simulation methods with the knowledge that comes from real observations could be in future the right choice to correctly calibrate source parameters and hence to obtain realistic strong ground motion predictions.

.

  Nevertheless uncertainties are very large owing to the difficulty in well constraining the corner frequencies with a small number of records. Different values in the stress drop may lead to differences in simulated ground motion amplitude, increasing the epistemic variability associated with the EGFs. According to Del Gaudio et al. (2015), we applied a correction to each EGF. The EGF amplitude spectrums were first divided by their corresponding Brune spectrums, considering the seismic moment and stress drop values of TABLE 1. The spectra were next multiplied by a Brune spectrum associated to a reference seismic moment of a Mw 3 event with a stress drop =8 MPa, this latter value being the geometric mean of the EGF stress drops.

Fault

  in terms of PGA and spectral acceleration (Sa) at 0.5 Hz, 1Hz, 2Hz, 5 Hz and 10 Hz as a function of the roughness parameter, rupture velocity and nucleation position respectively. On each of these figures, an example of simulated waveforms (East-West component of acceleration) is also shown to see clearly the effect of each considered source parameters. PGA and Sa are obtained as the geometrical mean of two quantities measured on the horizontal components. This choice is consistent with what is generally done for GMPEs and give us the possibility of a comparison with them.

Figure 1

 1 Figure 1 Schematic view of multi-EGF technique. Stars locate n different small earthquakes recorded at station x, that are candidate for being the EGF used at each point of the gridded fault. To determine which of the n different EGF is applied at each grid point, distance D1, D2, … Dn are calculated. At the red spot, D2 is found to be the shortest distance. The EGF for this point is 𝑮 𝟐 (𝒙 ̅, 𝒕).

Figure 2

 2 Figure 2 Seismicity and faults map of the Upper Rhine Graben; in particular the Western Rhine Fault and the Black Forest Fault systems are highlighted, which follow the borders of the Graben, and the Rhine River Fault, which follows the river path. As indicated in the upper legend, white circles represents recorded earthquakes in the area (from 1964 to 2009), while colored circles represents historical earthquakes in the area (different colors refers to different estimated intensities).

Figure 3

 3 Figure3Simplified map of the area, showing the location of the considered events (black dots with a magnitude dependent size) and the surface projection of the modelled faults. Red dots represent the selected EGFs (identification numbers are the same as Table1) for which focal mechanism solutions (computed fixing the strike and the dip angle, according to the hypothesis made for the considered fault geometry, and retrieving the rake angle from eq. 5) are also plotted on the map. The French and German stations located in the area are represented with blue triangles. Bottom and right panels are vertical sections in the EW and NS directions respectively.

Figure 3

 3 Figure 3 Spectral acceleration at 0.5 Hz, 1 Hz, 2 Hz, 5 Hz, 10 Hz and PGA as functions of the roughness parameter for a Mw 6 event occurring on the southern segment of the Rhine River fault. The nucleation position is fixed at the center of the fault and the rupture velocity is fixed to 2800 m/s. Red and blue lines on the plots corresponds to two values of the roughness parameter, 0.74 and 1.6, for which an example of simulated waveforms is shown.

Figure 4

 4 Figure 4Spectral acceleration at the same frequencies as in Figure5and PGA as functions of the rupture velocity for a Mw 6 event occurring on the southern segment of the Rhine River fault. The roughness parameter is fixed to 1 and the nucleation position is fixed at the center of the fault. Red and blue lines on the plots corresponds to two values of rupture velocity, 2400 and 3200 m/s, for which an example of simulated waveforms is shown.

Figure 5

 5 Figure 5 Spectral acceleration in logarithmic scale at same frequency as in Figure 5 and PGA as functions of the rupture velocity and the roughness parameter for a Mw 6 event occurring on the southern segment of the Rhine River fault. Only the nucleation position is fixed at the center of the fault. Paired color scale is used to highlight ground motion variations. Black lines separate qualitatively zones in which amplitudes are almost constant.

Figure 6

 6 Figure 6 Normalized spectral acceleration and PGA as functions of the nucleation position for a Mw 6 event occurring on the southern segment of the Rhine River fault. Each point in a plot corresponds to a specific location for the rupture initiation points. The roughness parameter is fixed to 1 and the rupture velocity to 2800 m/s. The location of the target station with respect to the fault surface trace is also indicated. Red and blue spots on the plots corresponds respectively to an anti-directive and directive positions for which an example of simulated waveforms is shown.

Figure 9 Figure 10

 910 Figure 9 Top panels : Stress drop as a function of the roughness parameter (red crosses represent average values for fixed K) for the three different fault geometries a) Southern segment of the Rhine River fault; b) smaller fault surface with the same shape ratio of a); c) same fault of b) but placed at a shallow depth. In the bottom panels PGA and PGV distributions are shown as functions of mean Δ𝜎 and 𝑣 𝑟 values for the same cases.

Figure 11

 11 Figure 11 Considered probability density functions for the roughness parameter, the rupture velocity and the hypocentral position on the fault (a). Hypothesis made for blind simulations are also shown (b): mean values for the roughness parameter, considered fault size and its relative position with respect to the main fault plane.

Figure 12

 12 Figure 12 Comparison between simulations and GMPEs in terms of PGA and Sa (0.5, 1, 2, 5 and 10 Hz) for a Mw 6 event recorded at BREM station. Red dots represent the mean values for simulations (with different mean values of roughness parameter distribution) and GMPEs; error bars represent one σ interval. For simulations we also show density of values (in grey scale), mean and standard deviation for different ranges of rupture velocity.

TABLE 1

 1 Source parameters for the selected EGFs: location from LGRB; moment magnitude, corner frequency and stress drop calculated as the average of the estimations at each station. Standard deviations for magnitude, corner frequency and stress drop are also indicated. For EGF 2 standard deviations are not indicated because only one record is available.

	EGF #	Date	Time	Mw σMw	Lat (°)	Lon (°)	Depth (km)	fc (Hz)	σfc	Δσ (MPa)	σΔσ
	1	26/09/2002 21:01:40.30	3.0	0.2 47.67 7.47	15	8	2	3	1
	2	18/05/2005 15:34:10.10	2.3	-	47.91 7.61	5	25	-	15	-
	3	25/07/2005 01:36:32.00	3.1	0.2 47.99 7.66	6	11	4	35	40
	4	15/12/2009 21:12:35.00	2.3	0.1 47.89 7.60	6	14	3	3	2
	5	01/11/2011 19:22:16.00	3.8	0.7 48.18 7.69	10	3	1	7	7

Acknowledgments

We thank LBRG and Stefan Stange in particular for providing the BREM station data that we used in the simulations, and the additional stations data needed to invert for moment tensors.

We thank Sophie Lambotte and Cécile Doubre at EOST/Renass for their help getting the data.

We warmly thank Hervé Jomard for his help plotting the BDFA map. This research activity was supported by the PIA ANR SINAPS@ project, grant no ANR-11-RSNR-0022.

rupture area is correlated to the value of the roughness degree of the slip: for large values of K the slip is concentrated in small areas characterized by large amplitudes, inversely, for small values of K, we will have large rupture area with lower amplitude.

The stress drop depends on selection for the fault geometry and different fault surfaces can lead to different stress drop values. Moreover, for the same fault surface, different rupture areas (different values of K) corresponds to different stress drops. To better understand this aspect, in Figure 9 we show Δ𝜎 as a function of K for three different fault geometries. For each case we performed 450 simulations linearly varying K and 𝑣 𝑟 with the same procedure as for the tests showed in Figure 7. The first test corresponds to the entire extension of the segment identified for the southern part of the Rhine River Fault, (27 km along strike and 16 km along the dip) (Figure 9a). For the other two cases, we reduced the fault area to 1/3 of the original one keeping the same shape ratio between L and W and considered two different fault position with respect to the surface: one is placed in depth while the other reaches the surface. As discussed by [START_REF] Causse | Variability of dynamic source parameters inferred from kinematic models of past earthquakes[END_REF], stress drop should increase with K for a fixed fault area. We observe in all the cases an almost linear increase of mean Δ𝜎 value with the roughness. Also stress drop variability at fixed K comes from randomness of slip phases. Another intuitive result is that reducing the fault area the mean slip value increases producing larger stress drops (Figure 9b andc).

Strong motion peak parameters (PGA and PGV) are also shown in Figure 9 as functions of the mean stress drop value (computed for each K) and rupture velocity. They are well correlated with the position of the fault with respect to the target site. Larger values of peak parameters are observed when reducing the fault size and increasing the stress drop, with a significant increase when the small fault is placed at shallow depth (Figure 9c).

Even if these analyses shown that stress drop variability associated to K is not large enough to better constrain roughness values (one order of magnitude of variation for K corresponds in


Courtois, R Claude Ferrand

Robert Courtois

Guillaume Martinent

Michèle Rivière

Rivière, M.* Emmanuel Rusch

Relationships between Work-related Characteristics, Needs Satisfaction, Motivation and Mental Health in Midwifery Students

Keywords: self-determination, midwifery students, psychological needs satisfaction, motivation, mental health

Relationships between Work-related Characteristics, Needs Satisfaction, Motivation and Mental Health in Midwifery Students

In midwifery studies, internships in hospitals are a vital component of the student experience. Some studies have pointed out that internships were often recognized as work-stressful places (e.g., [START_REF] Carolan-Olah | Final year students' learning experiences of the bachelor of midwifery course[END_REF], and too much work-related stress might have negative effects on students' learning and success (e.g., [START_REF] Sarikaya | The anxieties of medical students related to clinical training[END_REF], and can cause mental health problems (e.g., [START_REF] Gammon | A study to ascertain the effect of structured student tutorial support on student stress, self-esteem and coping[END_REF]. Indeed, learning occurs though interaction in the environment and the conflicting demands of being both a learner and an emerging professional can be particularly a difficult experience physically and stressful, for students when the environment is not conductive to learning (Flenghi, 2012;[START_REF] Pearcey | Student impressions of clinical nursing[END_REF]). Thus, it seems necessary to gain insight into the psychological processes underlying the relationships between work-related characteristics and mental health.

Few studies have focused on the interplay between work-related characteristics in internships, psychological needs satisfaction, motivation and mental health. Consequently, the present study examined these relationships using a partial least squares path modeling approach (PLS-PM; [START_REF] Sanchez | PLS Path Modelling with R[END_REF].

One theory on which we can rely for this study is the self-determination theory (SDT, [START_REF] Deci | Intrinsic motivation and self-determination in human behavior[END_REF]. SDT has shown the existence of three innate basic needs: autonomy need (the need to experience volition, freedom in one's actions and choices), competence need (the need to feel competent in doing optimally challenging activities and in mastering one's environment) and relatedness need (the need to have a sense of belonging and mutual respect) (e.g., Li, Wang, Pyun, & Kee, 2013, p.327). The satisfaction of these needs plays a crucial part in the process of maintaining autonomous forms of motivation in individuals.

Indeed, the distinction between autonomous forms of motivation (intrinsic motivation, identified and integrated regulations), controlled regulations (introjected, external regulations) and amotivation allows us to understand that the extent to which needs are satisfied will determine the types of motivation endorsed by individuals and their effects on mental health [START_REF] Ryan | An overview of self-determination theory[END_REF]. High levels of needs satisfaction would be positively related to autonomous regulations, and negatively associated to controlled regulations and amotivation.

Moreover, SDT has shown that greater mental health was associated with needs satisfaction and autonomous regulations (e.g., [START_REF] Ryan | An overview of self-determination theory[END_REF], and considers mental health as a state of complete physical, mental and social well-being and not merely the absence of disease. For these reasons two indexes such as quality of life and psychological distress will be taken into account to measure mental health.

The Demand-Control-Support model (DCS, [START_REF] Karasek | Healthy work: stress, productivity and the reconstruction of working life[END_REF]) is one of the most used models of stress in the workplace and indicated that individuals who were exposed to high levels of psychological work demands and low work control and social support levels were likely to be subjected to a lot of stress. In contrast, the work resources (e.g., work control and social support), when they are high, can reduce psychological work demands and contribute to the good mental health of individuals. SDT has shown some positive relationships between work resources and psychological needs. Work control was positively related to autonomy and competence needs, and social support was positively linked to relatedness need (e.g., Van den Broeck, Vanteenkiste, De Witte, Soenens, & Lens, 2010) or to relatedness and autonomy needs (e.g., Ryan & Solby, 1996). 312). Table 1 shows the students' characteristics.

Measures

Questionnaires. Participants will be asked to complete several validated questionnaires relating to ( 1

Data analysis

PLS-PM was used. To retain a reasonable number of manifest variables in the model, the different constructs were measured by three or four aggregates of items (e.g., [START_REF] Little | To parcel or not to parcel: Exploring the question, weighing the merits[END_REF]. See Table 3.

Results

Descriptive statistics and correlations for all study variables are shown in Table 2.

Results from the PLS-PM measurement model (outer model) provided evidence for the reliability and validity of all the constructs. Loadings, ρ values, AVE values, and first and second eigenvalues are reported in Table 3.

Results from the PLS-PM structural model (inner model) are presented in Figure 1.

Discussion

Results showed that work control and social support were positively related to the three needs satisfaction. Individuals who are surrounded by resourceful work characteristics would be more likely to experience a general feeling of psychological freedom (i.e., autonomy), interpersonal connectedness (i.e., relatedness), and effectiveness (i.e., competence). So, our findings highlight the importance of these factors for improving psychological needs satisfaction. Contrary to our expectation, psychological work demands were unrelated to the three needs satisfaction. This result suggests that the level of demands has not functional significance in terms of needs satisfaction [START_REF] Deci | Need satisfaction, motivation, and well-being in the work organizations of a former Eastern Bloc country[END_REF].

Results also showed that only competence need satisfaction contributed positively and significantly to the prediction of autonomous motivation and negatively and significantly to controlled regulations and amotivation. Autonomous motivation was a significant positive predictor of quality of life, and controlled regulations and amotivation were significant positive predictors of psychological distress. The need for competence leads to the perception of mastery, achievement and control, and allows individuals to efficiently adapt to complex and changing environments ( [START_REF] Van Den Broeck | Capturing autonomy, competence, and relatedness at work: Construction and initial[END_REF]. Findings highlight the link between competence need satisfaction and motivational mechanisms leading to a good perceived mental health (e.g., low psychological distress and high quality of life). Some limitations of the present study should be considered. First, all measures were self-reported. Caution with regard to over-interpretation should be considered. Second, participants are in different levels of training and internships (1st through 4th year). Warr (1990) suggested that for testing the relationships between work-related characteristics and outcomes, it was important to include a large range of work increasing the variability in workrelated characteristics. Lastly, midwifery student volunteers are a geographically-select sample, and expanding the study into a broader sample of midwifery students is required.

Conclusion and implications for practice

Our results underscore the positive predictive relationships of not only social support, but also work control characteristics on needs satisfaction. If work resources are recognized as stimulating personal growth, development and learning [START_REF] Schaufeli | Job demands, job resources, and their relationship with burnout and engagement: a multi-sample study[END_REF], our findings highlight the importance in training of boosting psychological needs satisfaction to improve the students' optimal functioning and work-related well-being. The findings also provided evidence that competence need satisfaction though motivation played a crucial role in the relationships between work resources and mental health. This suggests that participants are very focused on skills acquisition to better prepare themselves for practice and gain confidence (e.g., [START_REF] Mcintosh | Final year students' perceptions of learning to be a midwife in six British universities[END_REF], are motivated, and it seems possible because the work-related environment is perceived as sufficiently supportive. The overall results suggest the importance of interpersonal relationships and the focal role of clinical preceptor in training [START_REF] Maxwell | The role of the practice educator in supporting nursing and midwifery students' clinical practice learning: An appreciative inquiry[END_REF]. Feedback in general, and debriefing after acute situations in particular, are important in enhancing their learning.

Midwifery schools should continue to develop appropriate strategies such as a personalized follow up or the use of preceptorship programs [START_REF] Licqurish | Bachelor of midwifery students' experiences of achieving competencies: the role of the midwife preceptor[END_REF] to better accompany the transition between the student status and that of professional. .29 [.17, . 39]

A

  model was tested in which psychological needs act as mediators indirectly through motivation of the relationship between psychological work demands, work resources (i.e., work control and social support) and mental health. It was predicted (1) a negative relationship between psychological work demands and psychological needs satisfaction; (2) a positive relationship between work resources and psychological needs satisfaction; (3) that psychological needs satisfaction would be associated positively with autonomous motivation and negatively with controlled regulations and amotivation ; and (4) that autonomous motivation would be associated positively with quality of life and controlled regulations and amotivation would be associated positively with psychological distress. Methods Participants 214 French midwifery students (Mage = 22.8 years, age range: 19-41 years) were recruited from three midwifery schools in the regions "Centre" and 'Pays de la Loire" to participate in this study. Participants represented 68.6% of students of the three schools (n =

  ) psychological needs satisfaction (three different questionnaires assessing autonomy need satisfaction [5 items; Standage, Duda, & Ntoumanis, 2003); competence need satisfaction (IMI; 5 items, Mc Auley, Duncan, & Tammen, 1989) and relatedness need satisfaction (6 items, NRS-10; Richer & Vallerand, 1998)]; (2) behavioural regulations [24 items with six 4-item subscales representing the motivational regulations; Van Hoye, Ramanoel, Heuze, & Sarrazin, 2010)] (3) mental health [26 items; the WHOQOL-Bref, Leplege, Reveillere, Ecosse, Caria, & Riviere (2000) and the General Health Questionnaire-12 items, Lesage, Martens-Resende, Deschamps,& Berjot (2011]; and (4) the Demand-Control-Support model [DCS, 26-item with psychological work demands (9 items), decisional latitude (9 items) and social support at work (8 items), Niedhammer, Chastaing, Gendrey, & David (2006)]. Cronbach's alphas for psychological needs satisfaction range between .70 and .87, for behavioural regulations they range between .67 and .95, for Whoquol-Bref and psychological distress they are .85 and .84, and for DCS they range between .70 and .80, respectively, showing acceptable internal consistency (Nunnally, 1978) Procedure Permission to conduct this study was granted by the University of Human Research Ethics. Written informed consent was obtained from students. Participation was voluntary, and students responded individually and anonymously to an online set of questionnaires (approximately 35 minutes).

  validation of the work related basic need satisfaction scale. Journal of Occupational and Organizational Psychology, 83(4), 981-1002. Van Hoye, A., Ramanoel, S., Heuze, J.-P., & Sarrazin, P. (2010). French validation of the Behavioral Regulation Questionnaire (BQ) by adolescents. Proceedings of the Fourth International Conference on Self-Determination Theory. Ghent, pp. 85. Warr, P. (1990). Decision latitude, job demands, and employee well-being. Work and Stress, 4 (4), 285-294.

Table 1

 1

	French midwifery Students' Characteristics
		N	%
	Less than 25 years	210	98
	Female students	203	94.8
	Distribution of participants in each midwifery school
	First school	85/120	70.8
	Second school	72/98	73.5
	Third school	57/94	60.6
	Distribution by year of specialized training
	First year students	66	31
	Second years students	47	22
	Third years students	54	25
	Four years students	47	22
	__________________________________________________________________________
	NB: In France, specialized training for four years of midwifery students begins after
	examination at the end of the Common First Year of Health Studies -PACES

a Autonomous motivation represented the mean of intrinsic motivation, integrated regulation and identified regulation;

Table 3

 3 Composite Reliability Values, Average Variance Extracted and Eigenvalue-Analysis of Latent Variables and Standardized Factor Loadings for the Measurement Model

	a Self-determined motivation represented the mean of intrinsic motivation, integrated
	regulation and identified regulation;
	b Controlled motivation represented the mean of introjected regulation and external
	regulation;
	*** p < .001