ADAPTIVE WAVELET-BASED FINITE-DIFFERENCE MODELLING OF SH -WAVE PROPAGATION

Abstract

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 O(Δt 2, Δx 4 ) 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 O(Δt 2, Δx 4 ) 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.