Vincent Etienne

Frequency-Domain Numerical Modelling of Visco-Acoustic Waves Based on Finite-Difference and Finite-Element Discontinuous Galerkin Methods

Seismic exploration is one of the main geophysical methods to extract quantitative inferences about the Earth’s interior at different scales from the recording of seismic waves near the surface. Main applications are civil engineering for cavity detection and landslide characterization, site effect modelling for seismic hazard, CO2 sequestration and nuclearwaste storage, oil and gas exploration, and fundamental understanding of geodynamical processes. Acoustic or elastic waves are emitted either by controlled sources or natural sources (i.e., earthquakes). Interactions of seismic waves with the heterogeneities of the subsurface provide indirect measurements of the physical properties of the subsurface which govern the propagation of elastic waves (compressional and shear wave speeds, density, attenuation, anisotropy). Quantitative inference of the physical properties of the subsurface from the recordings of seismic waves at receiver positions is the so-called seismic inverse problem that can be recast in the framework of local numerical optimization. The most complete seismic inversion method, the so-called full waveform inversion (Virieux & Operto (2009) for a review), aims to exploit the full information content of seismic data by minimization of the misfit between the full seismic wavefield and the modelled one. The theoretical resolution of full waveform inversion is half the propagated wavelength. In full waveform inversion, the full seismic wavefield is generally modelled with volumetric methods that rely on the discretization of the wave equation (finite difference, finite element, finite volume methods). In the regime of small deformations associated with seismic wave propagation, the subsurface can be represented by a linear elastic solid parameterized by twenty-one elastic constants and the density in the framework of the constitutive Hooke’s law. If the subsurface is assumed isotropic, the elastic constants reduce to two independent parameters, the Lame parameters, which depend on the compressional (P) and the shear (S) wave speeds. In marine environment, the P wave speed has most of the time a dominant footprint in the seismic wavefield, in particular, on the hydrophone component which records the pressure wavefield. The dominant footprint of the P wave speed on the seismic

Modelling Seismic Wave Propagation for Geophysical Imaging

The Earth is an heterogeneous complex media from the mineral composition scale (10−6m) to the global scale ( 106m). The reconstruction of its structure is a quite challenging problem because sampling methodologies are mainly indirect as potential methods (Gunther et al., 2006; Rucker et al., 2006), diffusive methods (Cognon, 1971; Druskin & Knizhnerman, 1988; Goldman & Stover, 1983; Hohmann, 1988; Kuo & Cho, 1980; Oristaglio & Hohmann, 1984) or propagation methods (Alterman & Karal, 1968; Bolt & Smith, 1976; Dablain, 1986; Kelly et al., 1976; Levander, 1988; Marfurt, 1984; Virieux, 1986). Seismic waves belong to the last category. We shall concentrate in this chapter on the forward problem which will be at the heart of any inverse problem for imaging the Earth. The forward problem is dedicated to the estimation of seismic wavefields when one knows the medium properties while the inverse problem is devoted to the estimation of medium properties from recorded seismic wavefields.

Algorithmic and methodological developments towards full waveform inversion in 3D elastic media

IONFORDIRECT& INVERSE PROBLEMS The discretizations of the forward and inverse problems can be different. Our abstraction concept is based on this principle and mainly consists in forward and backward projections between the modeling and inverse discrete models. The abstraction concept in a parallel environment is depicted in the figure 1 which represents a schematic view in 2D. At each iteration, the incident and back-propagated wavefields are computed via equations (3) and (25). The frequency solutions are extracted with a discrete Fourier transform during the time steps as initially proposed by (Sirgue et al., 2008). Once the modeling is over, the frequency wavefields are projected onto the inverse discretization (arrow 1 in Fig. 1). The computation of the gradient is done through the expression (30), which is totally independent from the numerical method used in the forward modeling. When the new model has been evaluated, the physical properties are then projected onto the forward modeling discretization (arrow 2 in Fig. 1). After this last projection, the search of the new model can be done with the classical sequence of forward modelings in order to retrieve the model that minimizes the objective function. Figure 1: Illustration of the abstraction principle between the forward problem (left) and inverse problem (right). INVERSION INTHE SEG/EAGEOVERTHRUSTMODEL We have applied our FWI algorithm to a limited target of the SEG/EAGE overthrust model in the acoustic approximation. Seismic modeling is performed with an acoustic velocity-stress O(∆t,∆x) finite-difference time-domain staggered grid method. Some preliminary results (Fig. 2) are obtained for a coarse acquisition involving 11 × 10 sources and 75 × 91 receivers located at the top of the model. The spacing between sources is 700 m in the xand y-directions, respectively. The spacing between receivers is 100 m in both directions. We used two groups of frequencies: the first one with 8 frequencies ranging from 3 to 8 Hz and the second with 8 frequencies ranging from 6.5 to 12.5 Hz. Twelve iterations have been performed for each frequency group. The initial model has been obtained with a Gaussian smoothing of the true model. The final results exhibit the main features of the model, especially the channel structure, although the footprint of the coarse acquisition is clearly visible in the FWI model. Comparison between vertical profiles of the true and of the FWI models shows a good quantitative reconstruction of the velocities. The computation time is about 24 h with 128 CPUs. Figure 2: (a-b) Horizontal (a) and vertical (b) sections of the SEG/EAGE Overthrust model; (c-d) Same as (a-b) for the initial model; (e-f) Same as (a-b) for the FWI model. g) Log along z axis in the middle of the model The black, light gray, and drak gray curves are from the initial model, the true model and the FWI model, respectively. Note the footprint of the coarse acquisition in the FWI model. CONCLUSIONS AND PERSPECTIVES We have presented methodological developments specific to 3D FWI in elastic media using standard equations as well as conservative equations. We have shown how to estimate the gradient through the adjoint method and we have illustrated the interest of the conservative approach avoiding model parameter spatial derivatives whatever are values of medium parameters. This gradient has been applied to a toy synthetic example in order to illustrate its efficiency. We are currently applying the proposed FWI tool to more realistic targets.

Long Duration of Ground Motion in the Paradigmatic Valley of Mexico

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.

A Massively Parallel Time-domain Discontinuous Galerkin Method For 3D Elastic Wave Modeling

SUMMARY We present a massively parallel time-domain Discontinuous Galerkin (DG) finite-element formulation with Convolutional Perfectly Matched Layer (CPML) absorbing condition for 3D elastic seismic wave modeling with tetrahedral meshes. Parallelism is implemented by domain decomposition. The modeling is purposely developed for frequency-domain full-waveform inversion (FWI) in 3D elastic media. We designed an efficient and computationally-efficient CPML absorbing condition by combining two interpolation orders in the computational domain: we used the DG P2 scheme in the medium to get the required accuracy while we used the low-order DG P0 scheme in the CPMLs to reduce the whole numerical cost and to obtain a well-balanced workload over processors. The use of the lowinterpolation orders has been driven by the theorical resolution of FWI, which constrains the spatial discretization of the medium, and hence, the required interpolation order. We applied the proposed method to the SEG/EAGE overthrust model. The measured computation times of the modeling provide clear insights on the feasibility of 3D elastic FWI.

Computational issues and strategies related to full waveform inversion in 3D elastic media: method- ological developments

ION BETWEEN THE FORWARD MODELING AND THE INVERSE PROBLEM As stated before, discretizations of the forward and inverse problems can be different. Our abstraction concept is based on this principle and mainly consists in forward and backward projections between the modeling and inverse discrete models. Before going further, we should introduce some aspects related to parallel computing. In order to treat large-scale problems, 3D modeling tools are usually designed with the single program multiple data (SPMD) architecture, which means that there is only one program and each CPU uses the same executable to work on different parts or partitions of the 3D model. Our FDM and DG-FEM tools are based on such architecture with MPI communications between partitions. For an efficient load balancing between CPUs, the cartesian grid or the tetrahedral mesh are divided in such a way that the number of unknowns is balanced among partitions while the number of unknowns shared between partitions is minimized. The abstraction concept in a parallel environment is depicted in Fig. 3 which represents a schematic view in 2D. The top of the figure is dedicated to the forward problem and contains a triangular mesh divided into 8 partitions. On the left, right and bottom parts of the mesh, there are absorbing boundaries such as CPML layers (Komatitsch and Martin, 2007) in order to simulate an infinite medium. These artificial layers should not be included in the inversion. Therefore, we can define an imaging target and wish to perform FWI only in this area. Here, we choose a relative simple example but it is worth to note that the proposed scheme allows a flexible target oriented approach. The bottom of Fig. 3 is dedicated to the inverse problem and represents the imaging target discretized differently from the forward problem (here with a regular cartesian grid). Transition 1052 SEG Denver 2010 Annual Meeting

An overview of the SEISCOPE project on frequency-domain Full Waveform Inversion Multiparameter inversion and efficient 3D full-waveform inversion

We present an overview of the SEISCOPE project on frequency-domain full waveform inversion (FWI). The two main objectives are the reconstruction of multiple classes of parameters and the 3D acoustic and elastic FWI. The optimization relies on a preconditioned L-BFGS algorithm which provided scaled gradients of the misfit function for each classes of parameter. For onshore applications where body waves and surface waves are jointly inverted, P- and S-wave velocities (VP and VS) must be reconstructed simultaneously using a hierarchical inversion algorithm with two nested levels of data preconditioning with respect to frequency and arrival time. Simultaneous inversion of multiple frequencies rather than successive inversions of single frequencies significantly increases the S/N ratio of the models. For offshore applications where VS can have a minor footprint in the data, a hierarchical approach which first reconstructsVP in the acoustic approximation from the hydrophone component followed by the joint reconstruction of VP and VS from the geophone components can be the approach of choice. Among all the possible minimization criteria, we found that the L1 norm provides the most robust and easy-to-tune criterion as expected for this norm. In particular, it allowed us to successfully reconstruct VP and VS on a realistic synthetic offshore case study, when white noise with outliers has been added to the data. The feasibility of 3D FWI is highly dependent on the efficiency of the seismic modelling. Frequency-domain modelling based on direct solver allows one to tackle small-scale problems involving few millions of unknowns at low frequencies. If the seismic modelling engine embeds expensive source-dependent tasks, source encoding can be used to mitigate the computational burden of multiple-source modelling. However, we have shown the sensitivity of the source encoding to noise in the framework of efficient frequency-domain FWI where a limited number of frequencies is inverted sequentially. Simultaneous inversion of multiple frequencies is required to achieve an acceptable S/N ratio with a reasonable number of FWI iterations. Therefore, time-domain modelling for the estimation of harmonic components of the solution can be the approach of choice for 3D frequency-domain FWI because it allows one to extract an arbitrary number of frequencies at a minimum extra cost.

Challenges in the Full Waveform Inversion Regarding Data, Model and Optimisation

Full waveform inversion has been proposed in the early eighties and we now find various illustrations of this high resolution seismic imaging technique on both synthetic and real data. We investigate the different issues one may address regarding the three elements of this technique. The optimisation formulation should move towards more complete Newton-like methods. The hierarchical data sampling strategy should prevent the local optimisation approach to be trapped into a local minimum. The model description should keep the number of degrees of freedom as low as possible while prior information should be integrated into a regularisation term moving the full waveform inversion from a data-driven approach to a more balanced data and model contributions when available.