Open AccessA multi‐level direct‐iterative solver for seismic wave propagation modelling: space and wavelet approaches
Author(s) -
Hustedt Bernhard,
Operto Stéphane,
Virieux Jean
Publication year - 2003
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.2003.02098.x
Subject(s) - solver , discretization , interpolation (computer graphics) , wavelet , grid , iterative method , algorithm , bilinear interpolation , computer science , mathematics , mathematical optimization , mathematical analysis , geometry , artificial intelligence , animation , computer graphics (images) , computer vision
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.