3D Frequency-domain Full Waveform Inversion Research Articles

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

Efficient frequency-domain full-waveform inversion (FWI) of long-offset node data can be designed with a few discrete frequencies, which lead to modest data volumes to be managed during the inversion process. Moreover, attenuation effects can be straightforwardly implemented in the forward problem without the computational overhead. However, 3D frequency-domain seismic modeling is challenging because it requires solving a large and sparse linear indefinite system for each frequency with multiple right-hand sides (RHSs). This linear system can be solved by direct or iterative methods. The former allows efficient processing of multiple RHSs but may suffer from limited scalability for very large problems. Iterative methods equipped with a domain-decomposition preconditioner provide a suitable alternative to process large computational domains for sparse-node acquisition. We have investigated the domain-decomposition preconditioner based on the optimized restricted additive Schwarz (ORAS) method, in which a Robin or perfectly matched layer condition is implemented at the boundaries between the subdomains. The preconditioned system is solved by a Krylov subspace method, whereas a block low-rank lower-upper decomposition of the local matrices is performed at a preprocessing stage. Multiple sources are processed in groups with a pseudoblock method. The accuracy, the computational cost, and the scalability of the ORAS solver are assessed against several realistic benchmarks. In terms of discretization, we compare a compact wavelength-adaptive 27-point finite-difference stencil on a regular Cartesian grid with a P3 finite-element method on h-adaptive tetrahedral mesh. Although both schemes have comparable accuracy, the former is more computationally efficient, the latter being beneficial to comply with known boundaries such as bathymetry. The scalability of the method, the block processing of multiple RHSs, and the straightforward implementation of attenuation, which further improves the convergence of the iterative solver, make the method a versatile forward engine for large-scale 3D FWI applications from sparse node data sets.

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

Efficient frequency-domain full-waveform inversion (FWI) of long-offset node data can be performed with a few frequencies. The seismic response of these frequencies can be computed with compact finite-difference stencils on regular Cartesian grid with direct or hybrid direct/iterative methods. Compactness, which is necessary to mitigate the fill-in induced by the lower-upper factorization, is implemented with second-order stencils, whereas accuracy is achieved by building a consistent mass matrix and a compound stiffness matrix with optimal weights so that the stencil covers the eight cells surrounding the central point, leading to a 27-point stencil. Classical approaches estimate constant weights by jointly minimizing numerical dispersion in homogeneous media for several numbers of grid points per wavelength ([Formula: see text]). Then, the impedance matrix is built by using these constant weights at each central point covering the heterogeneous subsurface model, leading to nonuniform wavefield accuracy. Instead, we estimate [Formula: see text]-dependent weights once and for all by minimizing dispersion separately for each value of [Formula: see text] in a range found in FWI applications. Then, we build the impedance matrix without computational overhead by selecting at each central point the weights corresponding to the local [Formula: see text], hence leading to a wavelength-adaptive stencil. This separate approach with adaptive weights makes wavefield accuracy uniform. We benchmark wavefields, which are computed in several 3D large-scale subsurface models with a sparse multifrontal direct solver and the nonadaptive/adaptive stencils, against analytical solutions when available and the highly accurate discretization-free convergent Born series method. Each benchmark reveals the higher accuracy of the adaptive stencil relative to the nonadaptive one. In the presence of sharp contrasts, the adaptive stencil also is more accurate than a classical [Formula: see text] finite-different time-domain staggered-grid stencil. The relevance of the adaptive stencil is finally illustrated with a 3D FWI case study in the 3.5–13 Hz frequency band.

Three-dimensional frequency-domain full waveform inversion based on the nearly-analytic discrete method

The nearly analytic discrete (NAD) method is a kind of finite difference method with advantages of high accuracy and stability. Previous studies have investigated the NAD method for simulating wave propagation in the time-domain. This study applies the NAD method to solving three-dimensional (3D) acoustic wave equations in the frequency-domain. This forward modeling approach is then used as the “engine” for implementing 3D frequency-domain full waveform inversion (FWI). In the numerical modeling experiments, synthetic examples are first given to show the superiority of the NAD method in forward modeling compared with traditional finite difference methods. Synthetic 3D frequency-domain FWI experiments are then carried out to examine the effectiveness of the proposed methods. The inversion results show that the NAD method is more suitable than traditional methods, in terms of computational cost and stability, for 3D frequency-domain FWI, and represents an effective approach for inversion of subsurface model structures.

Fast 3D frequency-domain full-waveform inversion with a parallel block low-rank multifrontal direct solver: Application to OBC data from the North Sea

Wide-azimuth long-offset ocean bottom cable (OBC)/ocean bottom node surveys provide a suitable framework to perform computationally efficient frequency-domain full-waveform inversion (FWI) with a few discrete frequencies. Frequency-domain seismic modeling is performed efficiently with moderate computational resources for a large number of sources with a sparse multifrontal direct solver (Gauss-elimination techniques for sparse matrices). Approximate solutions of the time-harmonic wave equation are computed using a block low-rank (BLR) approximation, leading to a significant reduction in the operation count and in the volume of communication during the lower upper (LU) factorization as well as offering great potential for reduction in the memory demand. Moreover, the sparsity of the seismic source vectors is exploited to speed up the forward elimination step during the computation of the monochromatic wavefields. The relevance and the computational efficiency of the frequency-domain FWI performed in the viscoacoustic vertical transverse isotropic (VTI) approximation was tested with a real 3D OBC case study from the North Sea. The FWI subsurface models indicate a dramatic resolution improvement relative to the initial model built by reflection traveltime tomography. The amplitude errors introduced in the modeled wavefields by the BLR approximation for different low-rank thresholds have a negligible footprint in the FWI results. With respect to a standard multifrontal sparse direct factorization, and without compromise of the accuracy of the imaging, the BLR approximation can bring a reduction of the LU factor size by a factor of up to three. This reduction is not yet exploited to reduce the effective memory usage (ongoing work). The flop reduction can be larger than a factor of 10 and can bring a factor of time reduction of around three. Moreover, this reduction factor tends to increase with frequency, namely with the matrix size. Frequency-domain viscoacoustic VTI FWI can be viewed as an efficient tool to build an initial model for elastic FWI of 4C OBC data.