Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

/pdf/an-overview-of-full-waveform-inversion-in-exploration-59sei7fzvh.pdf

An overview of full-waveform inversion in exploration geophysics

SUMMARY

An application of full-waveform tomography to dense onshore wide-aperture seismic data recorded in a complex geological setting (thrust belt) is presented.



The waveform modelling and tomography are implemented in the frequency domain. The modelling part is solved with a finite-difference method applied to the visco-acoustic wave equation. The inversion is based on a local gradient method. Only the P-wave velocity is involved in the inversion. The inversion is applied iteratively to discrete frequency components by proceeding from low to high frequencies. This defines a multiscale imaging in the sense that high wavenumbers are progressively incorporated in images. The linearized waveform tomography requires an accurate starting velocity model that has been developed by first-arrival traveltime tomography.



After specific pre-processing of the data, 16 frequency components ranging between 5.4 and 20 Hz were inverted. Ten iterations were computed per frequency component leading to 160 tomographic models. The waveform tomography has successfully imaged southwest-dipping structures previously identified from other geophysical data as being associated with high-resistivity bodies. The relevance of the tomographic images is locally demonstrated by comparison of a velocity–depth function extracted from the waveform tomography models with a coincident vertical seismic profiling (VSP) log available on the profile. Moreover, comparison between observed and synthetic seismograms computed in the (starting) traveltime and waveform tomography models demonstrates unambiguously that the waveform tomography successfully predicts for wide-angle reflections from southwest-dipping geological structures.



This study demonstrates that the combination of first-arrival traveltime and frequency-domain full-waveform tomographies applied to dense wide-aperture seismic data is a promising approach to quantitative imaging of complex geological structures. Indeed, wide-aperture acquisition geometries offer the opportunity to develop an accurate background velocity model for the subsequent waveform tomography. This is critical, because the building of the macromodel remains an open question when only near-vertical reflection data are considered.

/pdf/multiscale-imaging-of-complex-structures-from-multifold-wide-4qt74jyimb.pdf

Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: application to a thrust belt

We present a finite-difference frequency-domain method for 3D visco-acoustic wave propagation modeling. In the frequency domain, the underlying numerical problem is the resolution of a large sparse system of linear equations whose right-hand side term is the source. This system is solved with a massively parallel direct solver. We first present an optimal 3D finite-difference stencil for frequency-domain modeling. The method is based on a parsimonious staggered-grid method. Differential operators are discretized with second-order accurate staggered-grid stencils on different rotated coordinate systems to mitigate numerical anisotropy. An antilumped mass strategy is implemented to minimize numerical dispersion. The stencil incorporates 27 grid points and spans two grid intervals. Dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3D domain. To assess the feasibility of the method for frequency-domain full-waveform inversion, we computed simulations in the 3D SEG/EAGE overthrust model for frequencies 5, 7, and 10 Hz. Results confirm the huge memory requirement of the factorization (several hundred Figabytes) but also the CPU efficiency of the resolution phase (few seconds per shot). Heuristic scalability analysis suggests that the memory complexity of the factorization is O(35N(4)) for a N-3 grid. Our method may provide a suitable tool to perform frequency-domain full-waveform inversion using a large distributed-memory platform. Further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time- and frequency-domain modeling of wave propagation to perform 3D full-waveform inversion.

/pdf/3d-finite-difference-frequency-domain-modeling-of-visco-1seg9kqkya.pdf

3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study

SUMMARY

We compare different finite-difference schemes for two-dimensional (2-D) acoustic frequency-domain forward modelling. The schemes are based on staggered-grid stencils of various accuracy and grid rotation strategies to discretize the derivatives of the wave equation. A combination of two staggered-grid stencils on the classical Cartesian coordinate system and the 45° rotated grid is the basis of the so-called mixed-grid stencil. This method is compared with a parsimonious staggered-grid method based on a fourth-order approximation of the first derivative operator. Averaging of the mass acceleration can be incorporated in the two stencils. Sponge-like perfectly matched layer absorbing boundary conditions are also examined for each stencil and shown to be effective.



The deduced numerical stencils are examined for both the wavelength content and azimuthal variation. The accuracy of the fourth-order staggered-grid stencil is slightly superior in terms of phase velocity dispersion to that of the mixed-grid stencil when averaging of the mass acceleration term is applied to the staggered-grid stencil.



For fourth-order derivative approximations, the classical staggered-grid geometry leads to a stencil that incorporates 13 grid nodes. The mixed-grid approach combines only nine grid nodes. In both cases, wavefield solutions are computed using a direct matrix solver based on an optimized multifrontal method. For this 2-D geometry, the staggered-grid strategy is significantly less efficient in terms of memory and CPU time requirements because of the enlarged bandwidth of the impedance matrix and increased number of coefficients in the discrete stencil.



Therefore, the mixed-grid approach should be suggested as the routine scheme for 2-D acoustic wave propagation modelling in the frequency domain.

/pdf/mixed-grid-and-staggered-grid-finite-difference-methods-for-47mazeg9mb.pdf

Mixed‐grid and staggered‐grid finite‐difference methods for frequency‐domain acoustic wave modelling

SUMMARY
In this study, we propose a new numerical method, named as Traction Image method, to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography. In this algorithm, the computational domain is discretized by boundary-conforming grids, in which the irregular surface is transformed into a ‘flat’ surface in computational space. Thus, the artefact of staircase approximation to arbitrarily irregular surface can be avoided. Such boundary-conforming gridding is equivalent to a curvilinear coordinate system, in which the first-order partial differential velocity-stress equations are numerically updated by an optimized high-order non-staggered finite difference scheme, that is, DRP/opt MacCormack scheme. To satisfy the free surface boundary conditions, we extend the Stress Image method for planar surface to Traction Image method for arbitrarily irregular surface by antisymmetrically setting the values of normal traction on the grid points above the free surface. This Traction Image method can be efficiently implemented. To validate this new method, we perform numerical tests to several complex models by comparing our results with those computed by other independent accurate methods. Although some of the testing examples have extremely sloped topography, all tested results show an excellent agreement between our results and those from the reference solutions, confirming the validity of our method for modelling seismic waves in the heterogeneous media with arbitrary shape topography. Numerical tests also demonstrate the efficiency of this method. We find about 10 grid points per shortest wavelength is enough to maintain the global accuracy of the simulation. Although the current study is for 2-D P-SV problem, it can be easily extended to 3-D problem.

Traction image method for irregular free surface boundaries in finite difference seismic wave simulation

SUMMARY

An adaptive wavelet-based finite-difference method for 2-D SH-wave propagation modelling is presented. The discrete orthogonal wavelet transform allows the decomposition of spatial wavefield coordinates on to different grids of various resolution. At different times during propagation and locations in the model, the different scales involved in the decomposition give different contributions to the wavefield construction. The orthogonal wavelet basis provides a natural framework to adapt spatial grids to local wavefield properties in time and space. In this paper, the efficiency of this approach is assessed in terms of computational cost and accuracy for different 2-D heterogeneous media.



The classic (Δt2, Δx4) time–space finite-difference method is recast into the time–spatial-wavelet domain. Wavefields, spatial differential and medium convolution operators are decomposed on to spatial orthogonal wavelet bases. These spatial operators may be applied in the wavelet domain or may be applied by going back and forth in the spatial domain. These two strategies provide similar results but differ in efficiency. Then, wavelet coefficients are extrapolated in time through second-order differencing using a constant time step. Recomposition in the original spatial domain may be performed for analysis of the results. Wavelet implementations of source excitation, PML-like absorbing boundary conditions and a free surface have also been implemented and are described for realistic wave propagation using the wavelet approach.



Contrasted and structurally complex heterogeneous models such as the corner–edge and Marmousi models are considered for a comparison with the staggered-grid time–space finite-difference method. The numerical results compared well with those of time–space finite-difference method provided that sharp variations in the medium and the wavefields can be oversampled in the finest grid of the multiresolution analysis. In the smooth parts of the medium, a rule of thumb of five nodes per wavelength is used for the wavelet approach as for the (Δt2, Δx4) standard finite-difference methods. Nevertheless, the discretization is adapted to the local wavelength in the wavelet approach, while the unique discretization involved in the standard finite-difference method is matched to the minimum wavelength.



An accurate presentation of the efficiency of the method is difficult because interpolations are not identical to those used in staggered-grid time–space finite-differences approaches and because the multiresolution analysis provides a complementary discretization of the medium, which may affect the numerical results without any corresponding effect in the standard approach. Actually, the wavelet approach in its different forms requires much greater computer resources than standard approaches. A future implementation of the time adaptivity with the time step being adapted to the local grid resolution of the multiresolution analysis will improve the CPU efficiency and will enhance the accuracy of the method by limiting the grid dispersion.



Despite its present CPU limitations, the wavelet method provides a new flexible numerical tool to adapt wave phenomena discretization to local media properties. This flexibility is invaluable and could be important when complex boundary interactions and non-linear rheology of near-surface models are treated.

https://hal.archives-ouvertes.fr/hal-00406712/document

Adaptive wavelet-based finite-difference modelling of SH-wave propagation

SUMMARY We present a new numerical modelling approach for frequency-domain finite-difference (FDFD) wave simulations. The new approach is developed as an extension to standard FDFD modelling schemes, when wave propagation simulations are performed in large-scale 2-D or 3-D models with complex heterogeneous rheology. Partial differential equations are presented in matrix-type form. Wavefield solutions are computed on different coarse- and fine-discretized numerical grids by a combination of a direct solver with an iterative solver. Two different connection strategies are designed. Both compute a coarse-grid wavefield solution using a direct matrix solver. The obtained solution is projected on a fine-discretized grid, which is used as an initial solution for an iterative solver to compute the desired fine-grid solution. The wavefield projection that combines coarse and fine grids, is either based on a space interpolation scheme, called the direct iterative space solver (DISS) or on a multi-scale wavelet expansion, called the direct iterative wavelet solver (DIWS). The DISS scheme mimics a nested iteration scheme of a full multi-grid method, since numerical grids are prolonged by a simple bilinear interpolation scheme. The simple grid combination leads to wavefield solutions that are affected by spatial phase-shift artefacts (aliasing), which may be suppressed by a large number of iteration steps or a standard V- and W-cycles sequence between grids. The actual DIWS matrix construction implementation is computationally more expensive, though the wavelet iteration scheme guarantees fast and stable iterative convergence. Coarse-grid wavefield solutions are combined with fine-grid solutions through the multi-resolution scaling property of a standard orthogonal wavelet expansion. Since the wavelet transformation accounts for grid interactions, phase-shift artefacts are greatly reduced and significantly fewer iteration steps are required for convergence. We demonstrate the performance and accuracy of the DISS and DIWS strategies for two complex 2-D heterogeneous wave simulation examples.

/pdf/a-multi-level-direct-iterative-solver-for-seismic-wave-uu79cv9knc.pdf

A multi‐level direct‐iterative solver for seismic wave propagation modelling: space and wavelet approaches

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Papers

Mixed‐grid and staggered‐grid finite‐difference methods for frequency‐domain acoustic wave modelling

Adaptive wavelet-based finite-difference modelling of SH-wave propagation

A multi‐level direct‐iterative solver for seismic wave propagation modelling: space and wavelet approaches