A non-periodic two scale asymptotic method to take account of rough topographies for 2D elastic wave propagation

International audienceWe propose a two scale asymptotic method to compute the effective effect of a free surface topography varying much faster than the minimum wavelength for 2-D P-SV elastic wave propagation. The topography variation is assumed to be non-periodic but with a deterministic description and, in this paper, the elastic body below the topography is assumed to be ho- mogeneous. Two asymptotic expansions are used, one in the boundary layer close to the free surface and one in the volume. Both expansions are matched appropriately up to the order 1 to provide an effective topography and an effective boundary condition. We show that the effective topography is not the averaged topography but it is a smooth free surface lying below the fast variations of the real topography. Moreover, the free boundary condition has to be modified to take account of the inertial effects of the fast variations of the topography above the effective topography. In other words, the wave is not propagating in the fast topography but below it and is slowed down by the weight of the fast topography. We present an iterative scheme allowing to find this effective topography for a given minimum wavelength. We do not attempt any mathematical proof of the proposed scheme, nevertheless, numerical tests show good results

1-D non periodic homogenization for the seismic wave equation

International audienceWhen considering numerical acoustic or elastic wave propagation in media containing small heterogeneities with respect to the minimum wavelength of the wavefield, being able to upscale physical properties (or homogenize them) is valuable mainly for two reasons. First, replacing the original discontinuous and very heterogeneous medium by a smooth and more simple one, is a judicious alternative to the necessary fine and difficult meshing of the original medium required by many wave equation solvers. Second, it helps to understand what properties of a medium are really ‘seen' by the wavefield propagating through, which is an important aspect in an inverse problem approach. This paper is an attempt of a pedagogical introduction to non- periodic homogenization in 1-D, allowing to find the effective wave equation and effective physical properties, of the elastodynamics equation in a highly heterogeneous medium. It can be extrapolated from 1-D to a higher space dimensions. This development can be seen as an extension of the classical two-scale homogenization theory applied to the elastic wave equation in periodic media, with this limitation that it does not hold beyond order 1 in the asymptotic expansion involved in the classical theory

2-D non-periodic homogenization of the elastic wave equation: SH case

International audienceIn the Earth, seismic waves propagate through 3-D heterogeneities characterized by a large variety of scales, some of them much smaller than their minimum wavelength. The costs of computing the wavefield in such media using purely numerical methods, are very high. To lower them, and also to obtain a better geodynamical interpretation of tomographic images, we aim at calculating appropriate effective properties of heterogeneous and discontinuous media, by deriving convenient upscaling rules for the material properties and for the wave equation. To progress towards this goal we extend our successful work from 1-D to 2-D. We first apply the so-called homogenization method (based on a two-scale asymptotic expansion of the field variables) to model antiplane wave propagation in 2-D periodic media. These latter are characterized by short-scale variations of elastic properties, compared to the smallest wavelength of the wavefield. Seismograms are obtained using the 0th-order term of this asymptotic expansion, plus a partial first-order correction. Away from boundaries, they are in excellent agreement with solutions calculated at a much higher computational cost, using spectral elements simulations in the reference media. We then extend the homogenization of the wave equation, to 2-D non-periodic, deterministic media

Reliability of mantle tomography models assessed by spectral element simulation

Global tomographic models collected in the Seismic wave Propagation and Imaging in Complex (SPICE media: a European network) model library (http://www.spicertn.org/research/planetaryscale/tomography/) share a similar pattern of long, spatial wavelength heterogeneity, but are not consistent at shorter spatial wavelengths. Here, we assess the performance of global tomographic models by comparing how well they fit seismic waveform observations, in particular Love and Rayleigh wave overtones and fundamental modes. We first used the coupled spectral element method (CSEM) to calculate long-period (>100 s) synthetic seismograms for different global tomography models. The CSEM can incorporate the effect of three-dimensional (3-D) variations in velocity, anisotropy, density and attenuation with very little numerical dispersion. We then compared quantitatively synthetic seismograms and real data. To restrict ourselves to high-quality overtone data, and to minimize the effects of the finite extent of seismic sources and of crustal heterogeneity, we favour deep (>500 km) earthquakes of intermediate magnitude (Mw ∼ 7). Our comparisons reveal that: (1) The 3-D global tomographic models explain the data much better than the one-dimensional (1-D) anisotropic Preliminary Reference Earth Model (PREM). The current 3-D tomographic models have captured the large-scale features of upper-mantle heterogeneities, but there is still some room for the improvement of large-scale features of global tomographic models. (2) The average correlation coefficients for deep events are higher than those for shallow events, because crustal structure is too complex to be completely incorporated into CSEM simulations. (3) The average correlation coefficient (or the time lag) for the major-arc wave trains is lower (or higher) than that for the minor-arc wave trains. Therefore, the current tomographic models could be much improved by including the major-arc wave trains in the inversion. (4) The shallow-layer crustal correction has more effects on the fundamental surface waves than on the overtone

Time-reversal imaging of seismic sources and application to the great Sumatra earthquake

International audienceThe increasing power of computers and numerical methods (like spectral element methods) allows continuously improving modelization of the propagation of seismic waves in heterogeneous media and the development of new applications in particular time reversal in the three-dimensional Earth. The concept of time-reversal (hereafter referred to as TR) was previously successfully applied for acoustic waves in many fields like medical imaging, underwater acoustics and non destructive testing. We present here the first application at the global scale of TR with associated reverse movies of seismic waves propagation by sending back long period time-reversed seismograms. We show that seismic wave energy is refocused at the right location and the right time of the earthquake. When TR is applied to the Sumatra-Andaman earthquake (26 Dec. 2004), the migration of the rupture from the south towards the north is retrieved. Therefore, TR is potentially interesting for constraining the spatio-temporal history of complex earthquakes

Oceanic lithosphere-asthenosphere boundaryfrom surface wave dispersion data

International audienceAbstract According to different types of observations, the nature of lithosphere-asthenosphereboundary (LAB) is controversial. Using a massive data set of surface wave dispersions in a broad periodrange (15–300 s), we have developed a three-dimensional upper mantle tomographic model (first-orderperturbation theory) at the global scale. This is used to derive maps of the LAB from the resolved elasticparameters. The key effects of shallow layers and anisotropy are taken into account in the inversion process.We investigate LAB distribution primarily below the oceans, according to different kinds of proxies thatcorrespond to the base of the lithosphere from the shear velocity variation at depth, the amplituderadial anisotropy, and the changes in azimuthal anisotropy G orientation. The estimations of the LAB depthbased on the shear velocity increase from a thin lithosphere (∼20 km) in the ridges, to a thick old-oceanlithosphere (∼120–130 km). The radial anisotropy proxy shows a very fast increase in the LAB depth fromthe ridges, from ∼50 km to the older ocean where it reaches a remarkable monotonic subhorizontal profile(∼70–80 km). The LAB depths inferred from the azimuthal anisotropy proxy show deeper values for theincreasing oceanic lithosphere (∼130–135 km). The difference between the evolution of the LAB depth withthe age of the oceanic lithosphere computed from the shear velocity and azimuthal anisotropy proxies andfrom the radial anisotropy proxy raises questions about the nature of the LAB in the oceanic regions and ofthe formation of the oceanic plate

On the amplitude of surface waves obtained by noise correlation and the capability to recover the attenuation: a numerical approach

International audienceCross-correlation of ambient seismic noise recorded by a pair of stations is now commonly recognized to contain the Green's function between the stations. Although traveltimes extracted from such data have been extensively used to get images of the Earth interior, very few studies have attempted to exploit the amplitudes. In this work, we investigate the information contained in the amplitudes and we probe the capability of noise correlations to recover anelastic attenuation. To do so, we carry out numerical experiments in which we generate seismic noise at the surface of a 1-D Earth model. One of the advantages of our approach is that both uniform and non-uniform distributions of noise sources can be taken into account. In the case of a uniform distribution, we find that geometrical spreading as well as intrinsic attenuation are retrieved, even after strong non-linear operations such as one-bit normalization and spectral whitening applied to the noise recordings. In the case of a non-uniform distribution of sources, the geometrical spreading of the raw noise correlations depends on the distribution, but intrinsic attenuation is preserved. For the one-bit noise and whitened noise correlations, the interpretation of observed amplitude decays requires further study

Multiscale full waveform inversion

We develop and apply a full waveform inversion method that incorporates seismic data on a wide range of spatio-temporal scales, thereby constraining the details of both crustal and upper-mantle structure. This is intended to further our understanding of crust-mantle interactions that shape the nature of plate tectonics, and to be a step towards improved tomographic models of strongly scale-dependent earth properties, such as attenuation and anisotropy. The inversion for detailed regional earth structure consistently embedded within a large-scale model requires locally refined numerical meshes that allow us to (1) model regional wave propagation at high frequencies, and (2) capture the inferred fine-scale heterogeneities. The smallest local grid spacing sets the upper bound of the largest possible time step used to iteratively advance the seismic wave field. This limitation leads to extreme computational costs in the presence of fine-scale structure, and it inhibits the construction of full waveform tomographic models that describe earth structure on multiple scales. To reduce computational requirements to a feasible level, we design a multigrid approach based on the decomposition of a multiscale earth model with widely varying grid spacings into a family of single-scale models where the grid spacing is approximately uniform. Each of the single-scale models contains a tractable number of grid points, which ensures computational efficiency. The multi-to-single-scale decomposition is the foundation of iterative, gradient-based optimization schemes that simultaneously and consistently invert data on all scales for one multi-scale model. We demonstrate the applicability of our method in a full waveform inversion for Eurasia, with a special focus on Anatolia where coverage is particularly dense. Continental-scale structure is constrained by complete seismic waveforms in the 30-200s period range. In addition to the well-known structural elements of the Eurasian mantle, our model reveals a variety of subtle features, such as the Armorican Massif, the Rhine Graben and the Massif Central. Anatolia is covered by waveforms with 8-200s period, meaning that the details of both crustal and mantle structure are resolved consistently. The final model contains numerous previously undiscovered structures, including the extension-related updoming of lower-crustal material beneath the Menderes Massif in western Anatolia. Furthermore, the final model for the Anatolian region confirms estimates of crustal depth from receiver function analysis, and it accurately explains cross-correlations of ambient seismic noise at 10s period that have not been used in the tomographic inversion. This provides strong independent evidence that detailed 3-D structure is well resolve

2-D non-periodic homogenization to upscale elastic media for P-SV waves

International audienceThe pur pose of this paper is to give an upscaling tool valid for the wave equation in general elastic media. This paper is focused on P–SV wave propagation in 2-D, but the methodology can be extended without any theoretical difficulty to the general 3-D case. No assumption on the heterogeneity spectrum is made and the medium can show rapid variations of its elastic properties in all spatial directions. The method used is based on the two-scale homogenization expansion, but extended to the non-periodic case. The scale separation is made using a spatial low-pass filter. The ratio of the filter wavelength cut-off and the minimum wavelength of the propagating wavefield defines a parameter ε0 with which the wavefield propagating in the homogenized medium converges to the reference wavefield. In the general case, this non- periodic extension of the homogenization technique is only valid up to the leading order and for the so-called first-order cor rector. We apply this non-periodic homogenization procedure to two kinds of heterogeneous media: a randomly generated, highly heterogeneous medium and the Marmousi2 geological model. The method is tested with the Spectral Element Method as a solver to the wave equation. Comparing computations in the homogenized media with those obtained in the original ones shows that convergence with ε0 is even better than expected. The effects of the leading order cor rection to the source and first cor rection at the receivers' location are shown

Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation

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