Accurate 3D frequency-domain seismic wave modeling with the wavelength-adaptive 27-point finite-difference stencil: a tool for full waveform inversion

Abstract

Efficient frequency-domain Full Waveform Inversion (FWI) of long-offset/wide-azimuth node data can be designed with a few discrete frequencies. However, 3D frequency-domain seismic modeling remains challenging since it requires solving a large and sparse linear indefinite system per frequency. When such systems are solved with direct methods or hybrid direct/iterative solvers, based upon domain decomposition preconditioner, finite-difference stencils on regular Cartesian grids should be designed to conciliate compactness and accuracy, the former being necessary to mitigate the fill-in induced by the Lower-Upper (LU) factorization. Compactness is classically implemented by combining several second-order accurate stencils covering the eight cells surrounding the collocation point, leading to the so-called 27-point stencil. Accuracy is obtained by applying optimal weights on the different stiffness and consistent mass matrices such that numerical dispersion is jointly minimized for several number of grid points per wavelength (G). However, with this approach, the same weights are used at each collocation point, leading to suboptimal accuracy in heterogeneous media. In this study, we propose a straightforward recipe to improve the accuracy of the 27-point stencil. First, we finely tabulate the values of G covering the range of wavelengths spanned by the subsurface model and the frequency. Then, we estimate with a classical dispersion analysis in homogeneous media the corresponding table of optimal weights that minimize dispersion for each G treated separately. We however apply a Tikhonov regularization to guarantee smooth variation of the weights with G. Finally, we build the impedance matrix by selecting the optimal weights at each collocation point according to the local wavelength, hence leading to a wavelength-adaptive stencil. We validate our method against analytical solutions in homogeneous and velocity gradient models and 3D heterogeneous benchmarks with sharp contrasts and complex tectonic pattern. In the latter case, the accuracy of the adaptive stencil is checked with the highly accurate solution of the convergent Born series (CBS). Each benchmark reveals the higher accuracy of the adaptive stencil relative to the non-adaptive one. We also show that, in the presence of sharp contrasts, the adaptive stencil is more accurate than a O(∆t 2 , ∆h 8 ) finite-different time-domain method.

Three-dimensional finite-difference finite-element frequency-domain wave simulation with multi-level optimized additive Schwarz domain-decomposition preconditioner: A tool for FWI of sparse node datasets

“…The second discretization scheme relies on the 27-point finite-difference stencil on regular Cartesian grid (constant grid interval h) in which compact 2 nd -order accurate stencils minimize the numerical bandwidth and maximizes the sparsity of A while reaching a high-order accuracy by mixing consistent mass and stiffness matrices on different (rotated) coordinate systems (Aghamiry et al, 2021;Brossier et al, 2010;Chen et al, 2012;Operto et al, 2014Operto et al, , 2007. These sparsity and compactness properties are useful to minimize the matrix fill-in induced by a sparse direct solver, which is used to solve the local problems in each subdomain of the preconditioner.…”

“…Alternatively, G can be first tabulated such that it covers the range of values found in the earth and the corresponding table of weighting coefficients are estimated once and for all by minimizing the phase-velocity numerical dispersion for each value of G treated separately (the weights become functions of G). Then, for a simulation at a given frequency f , each row of the matrix A (which is tied to a given grid point) is built by picking in the table the weights corresponding to the local G = λ/h hence leading to the λ-adaptive 27-point stencil (Aghamiry et al, 2021). We will use the latter in this study since Aghamiry et al (2021) showed that the λ-adaptivity increases the accuracy of the 27-point stencil quite significantly in heterogeneous media, in particular in presence of sharp contrast.…”

“…Then, for a simulation at a given frequency f , each row of the matrix A (which is tied to a given grid point) is built by picking in the table the weights corresponding to the local G = λ/h hence leading to the λ-adaptive 27-point stencil (Aghamiry et al, 2021). We will use the latter in this study since Aghamiry et al (2021) showed that the λ-adaptivity increases the accuracy of the 27-point stencil quite significantly in heterogeneous media, in particular in presence of sharp contrast. Since the grid is regular, we will set h such that h = λ min /4 according to the FWI resolution criterion.…”

“…Our rationale behind the choice of the polynomial order Frequency-domain seismic wave solvers, Tournier et al A PREPRINT is to choose the coarsest discretization providing the desired accuracy while being able to capture heterogeneities during FWI whose size is of the order of half the wavelength, namely the theoretical resolution of FWI . The second discretization scheme relies on the recently proposed compact wavelength-adaptive 27point finite-difference method on uniform Cartesian grid (Aghamiry et al, 2021). The governing idea of this method compared to the classical 27-point stencil (Operto et al, 2007) is to make its accuracy uniform with respect to the local wavelength through a local adaptation of the stencil coefficients.…”

Three-dimensional finite-difference & finite-element frequency-domain wave simulation with multi-level optimized additive Schwarz domain-decomposition preconditioner: A tool for FWI of sparse node datasets

“…Our rationale behind the choice of the polynomial order is to choose the coarsest discretization providing the desired accuracy while being able to capture heterogeneities during FWI whose size is of the order of half the wavelength, namely the theoretical resolution of FWI . The second discretization scheme relies on the recently proposed compact wavelength-adaptive 27point finite-difference method on uniform Cartesian grid (Aghamiry et al, 2021). The governing idea of this method compared to the classical 27-point stencil (Operto et al, 2007) is to make its accuracy uniform with respect to the local wavelength through a local adaptation of the stencil coefficients.…”