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Collaborating Authors
ABSTRACT We present a study on using a multigrid scheme for a preconditioner for acoustic frequency-domain finite-difference (FDFD) forward modeling. To achieve fourth-order accuracy, we employ the perfectly matched layer (PML) approach. The multi-grid preconditioner combined with the bi-conjugate gradient stabilized (BiCGStab) iterative solver has been successfully applied to solve acoustic forward problems by using the radiation boundary condition, the sponge layer absorbing boundary condition, and the first-order PML boundary condition. The condition number of the discrete Helmholtz equation is highly dependent on the boundary conditions. The high-order PML enables us to match the accuracy of the boundary condition with the accuracy of the finite-difference (FD) scheme in the interior domain, especially for frequency-domain approaches. The geometric multigrid is employed to construct a preconditioner for a fourth-order FDFD forward solver equipped with the PML boundary condition. For efficiency this preconditioner is constructed using a second-order FD scheme with a negligible attenuation function inside the PML domain. The preconditioner is used for accelerating the convergence of the FDFD forward solver for cases where the discretization grids are over-sampled (the number of discretization points per minimum wavelength is greater than 10). The number of multi-grid levels is also chosen adaptively depending on the number of discretization points. Our study shows that the multigrid preconditioner can speed-up the convergence the BiCGStab solver total computational time by a factor of three for cases with over-sampled discretization grids. We also observe that the BiCGStab solver using an accurate PML boundary condition does not have convergence problems as one may encounter with iterative solvers using the radiation boundary condition.
ABSTRACT We tested a biconjugate gradient stabilized (BiCGSTAB) solver using a multigrid-based preconditioner for solving the acoustic wave (Helmholtz) equation in the frequency domain. The perfectly matched layer (PML) method was used as the radiation boundary condition (RBC). The equation was discretized using either a second- or fourth-order finite-difference (FD) scheme. The convergence of an iterative solver depended strongly on the RBC used because the spectrum of the discretized equation also depends on it. We used a geometric multigrid approach to construct a preconditioner for our FD frequency-domain (FDFD) forward solver equipped with the PML boundary condition. For efficiency, this preconditioner was only constructed using a second-order FD scheme with negligible attenuation inside the PML domain. The preconditioner was used for accelerating the convergence rate of the FDFD forward solver for cases when the discretization grids were oversampled (i.e., when the number of discretization points per minimum wavelength was greater than 10). The number of multigrid levels was also chosen adaptively depending on the number of discretization grids. We found that the multigrid preconditioner can speed up the total computational time of the BiCGSTAB solver for oversampled cases or at low frequencies. We also observed that the BiCGSTAB solver using an accurate PML boundary condition converged for realistic SEG benchmark models at high frequencies.
2D frequency-domain finite-difference acoustic wave modeling using optimized perfectly matched layers
Lei, Wen (Zhejiang University) | Liu, Yutao (Zhejiang University) | Li, Gang (Zhejiang University, Zhejiang University) | Zhu, Shuang (Zhejiang University) | Chen, Guoxin (Zhejiang University) | Li, Chunfeng (Zhejiang University, Zhejiang University)
ABSTRACT Seismic forward modeling is fundamental in sensitivity analysis and full-waveform inversions. In finite-difference acoustic wavefield simulation, the absorption boundary, especially the perfectly matched layer (PML), is widely used, but the setting of PML parameters is empirical. An optimized complex frequency-shift PML (CFS-PML) for the modeling of acoustic fields has been developed. It refines the selection of parameters for improving the artificial attenuation. The improved CFS-PML boundary condition is applied to a 2D frequency-domain acoustic wave simulation using an optimal 17-point finite-difference scheme. Numerical results are compared with a conventional nine-point scheme in terms of computational time and physical memory consumption. These tests indicate that our CFS-PML absorption boundary can effectively improve numerical accuracy without increasing the computational burden remarkably.
- Geophysics > Seismic Surveying > Seismic Processing (0.68)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling > Seismic Inversion (0.48)
The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.
Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method
Ma, Xiao (Northwestern Polytechnical University) | Yang, Dinghui (Tsinghua University) | Huang, Xueyuan (Beijing Technology and Business University) | Zhou, Yanjie (Beijing Technology and Business University)
ABSTRACT The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.