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International audienceWe present a new method for wave propagation in global earth models based upon the coupling between the spectral element method and a modal solution method. The Earth is decomposed into two parts, an outer shell with 3-D lateral heterogeneities and an inner sphere with only spherically symmetric heterogeneities. Depending on the problem, the outer heterogeneous shell can be mapped as the whole mantle or restricted only to the upper mantle or the crust. In the outer shell, the solution is sought in terms of the spectral element method, which stem from a high order variational formulation in space and a second-order explicit scheme in time. In the inner sphere, the solution is sought in terms of a modal solution in frequency after expansion on the spherical harmonics basis. The spectral element method combines the geometrical flexibility of finite element methods with the exponential convergence rate of spectral methods. It avoids the pole problems and allows for local mesh refinement, using a non-conforming discretization, for the resolution of sharp variations and topography along interfaces. The modal solution allows for an accurate isotropic representation in the inner sphere. The coupling is introduced within the spectral element method via a Dirichlet-to-Neumann (DtN) operator. The operator is explicitly constructed in frequency and in generalized spherical harmonics. The inverse transform in space and time requires special attention and an asymptotic regularization. The coupled method allows a significant speed-up in the simulation of the wave propagation in earth models. For spherically symmetric earth model, the method is shown to have the accuracy of spectral transform methods and allow the resolution of wavefield propagation, in 3-D laterally heterogeneous models, without any perturbation hypothesis

International audienceWe present a discontinuous Galerkin finite-element method (DG-FEM) formulation with Convolutional Perfectly Matched Layer (CPML) absorbing boundary condition for 3-D elastic seismic wave modelling. This method makes use of unstructured tetrahedral meshes locally refined according to the medium properties (h-adaptivity), and of approximation orders that can change from one element to another according to an adequate criterion (p-adaptivity). These two features allow us to significantly reduce the computational cost of the simulations. Moreover, we have designed an efficient CPML absorbing boundary condition, both in terms of absorption and computational cost, by combining approximation orders in the numerical domain. A quadratic interpolation is typically used in the medium to obtain the required accuracy, while lower approximation orders are used in the CPMLs to reduce the total computational cost and to obtain a well-balanced workload over the processors. While the efficiency of DG-FEMs have been largely demonstrated for high approximation orders, we favour the use of low approximation orders as they are more appropriate to the applications we are interested in. In particular, we address the issues of seismic modelling and seismic imaging in cases of complex geological structures that require a fine discretization of the medium. We illustrate the efficiency of our approach within the framework of the EUROSEISTEST verification and validation project, which is designed to compare high-frequency (up to 4 Hz) numerical predictions of ground motion in the Volvi basin (Greece). Through the tetrahedral meshing, we have achieved fine discretization of the basin, which appears to be a sine qua non condition for accurate computation of surface waves diffracted at the basin edges. We compare our results with predictions computed with the spectral element method (SEM), and demonstrate that our method yields the same level of accuracy with computation times of the same order of magnitude

S U M M A R YThis paper deals with the spectral element modelling of seismic wave propagation on a global scale. Two aspects relevant to low-frequency studies are particularly emphasized. First, the method is generalized beyond the Cowling approximation in order to fully account for the effects of self-gravitation. In particular, the perturbation of the gravity field outside the Earth is handled by a projection of the spectral element solution onto the basis of spherical harmonics. Second, we propose a new formulation inside the fluid which allows us to account for an arbitrary density stratification. It is based upon a decomposition of the displacement into two scalar potentials, and results in a fully explicit fluid-solid coupling strategy. The implementation of the method is carefully detailed and its accuracy is demonstrated through a series of benchmark tests.It has been recently established by several authors (Chaljub 2000; Komatitsch & Tromp 2002a,b;Chaljub et al. 2003) that the spectral element method (SEM) provides an efficient solution to the issue of computing synthetic seismograms in three-dimensional (3-D) models of the Earth. Whereas most of current spectral element studies aim to push calculations toward high frequencies, where the methods traditionally used on a global scale reach their limits, this paper focuses on some physical effects that are critical for the lower part of the seismic frequency band: (i) the full treatment of self-gravitation and (ii) the ability to take into account any density stratification in the fluid regions of the Earth.The first novelty of this paper stands in the incorporation of self-gravitation, the effect of which is important for seismic and gravimetric observations with periods larger than 100 s. All the previously mentioned studies based upon the SEM accounted for the effects of gravity within the Cowling approximation (Cowling 1941), i.e. by neglecting the perturbation of the gravity field by seismic waves. The main reason for making this assumption lies in the intrinsic difficulty of the problem. Considering the full effects of self-gravitation requires solving Poisson's equation for the perturbed gravitational potential which is defined over the whole space. Unlike spherical harmonics approaches, the use of a grid-based method such as the SEM does not provide a natural framework for the resolution of the exterior problem. Grid-based approximations in unbounded domains proceed first by restricting the computational domain, then by imposing an appropriate condition on the truncating boundary. Different methods arise depending on whether the artificial boundary condition (ABC) is local or not. Methods based upon a local ABC have the advantage of being computationally inexpensive and valid for arbitrary geometries. An example is the infinite-element method (e.g. Bettess 1992; Gerdes & Demkowicz 1996), in which the behaviour of the exterior solution is enforced in the radial direction. Methods of the second class, based upon a non-local ABC, are not as general sin...

International audienceThis article documents a comparative exercise for numerical simulation of ground motion, addressing the seismic response of the Grenoble site, a typical Alpine valley with complex 3D geometry and large velocity contrasts. Predictions up to 2 Hz were asked for four different structure wave-field configurations (point source and extended source, with and without surface topography). This effort is part of a larger exercise organized for the third international symposium on the effects of surface geology (ESG 2006), the complete results of which are reported elsewhere (Tsuno et al., 2009). While initial, blind computations significantly differed from one another, a remarkable fit was obtained after correcting for some nonmethodological errors for four 3D methods: the arbitrary high-order derivative discontinuous Galerkin method (ADER-DGM), the velocity-stress finite-difference scheme on an arbitrary discontinuous staggered grid (FDM), and two implementations of the spectral-element method (SEM1 and SEM2). Their basic formulation is briefly recalled, and their implementation for the Grenoble Valley and the corresponding requirements in terms of computer resources are detailed. Besides a visual inspection of PGV maps, more refined, quantitative comparisons based on time-frequency analysis greatly help in understanding the origin of differences, with a special emphasis on phase misfit. The match is found excellent below 1 Hz, and gradually deteriorates for increasing frequency, reflecting differences in meshing strategy, numerical dispersion, and implementation of damping properties. While the numerical prediction of ground motion cannot yet be considered a mature, push-button approach, the good agreement reached by four participants indicates that, when used properly, numerical simulation is actually able to handle correctly wave radiation from extended sources in complex 3D media. The main recommendation to obtain reliable numerical predictions of earthquake ground motion is to use at least two different but comparably accurate methods, for instance the present formulations and implementations of the FDM, SEM, and ADER-DGM

Abstract. The sensitivity of SS precursors to the presence of topography on the 660-kin discontinuity is addressed using an axisymmetric finite difference approximation to the $H wave propagation in the Earth mantle. Numerical experiments are lead to quantify the bias in both wavelength and amplitude committed in estimating depth information on the upper-mantle discontinuities from ray inversion of .SdS arrival times. It is further shown that long period $660S arrival-times do not provide information on the deepening of the '660' in presence of a velocity increase expected as the thermal signature of a subducting slab.

Built-up on top of ancient lake deposits, Mexico City experiences some of the largest seismic site effects worldwide. Besides the extreme amplification of seismic waves, duration of intense ground motion from large subduction earthquakes exceeds three minutes in the lake-bed zone of the basin, where hundreds of buildings collapsed or were seriously damaged during the magnitude 8.0 Michoacán earthquake in 1985. Different mechanisms contribute to the long lasting motions, such as the regional dispersion and multiple-scattering of the incoming wavefield from the coast, more than 300 km away the city. By means of high performance computational modeling we show that, despite the highly dissipative basin deposits, seismic energy can propagate long distances in the deep structure of the valley, promoting also a large elongation of motion. Our simulations reveal that the seismic response of the basin is dominated by surface-waves overtones, and that this mechanism increases the duration of ground motion by more than 170% and 290% of the incoming wavefield duration at 0.5 and 0.3 Hz, respectively, which are two frequencies with the largest observed amplification. This conclusion contradicts what has been previously stated from observational and modeling investigations, where the basin itself has been discarded as a preponderant factor promoting long and devastating shaking in Mexico City.