Adaptive seismic wave modeling in wavelet bases

Wavelets and multiresolution analysis are efficient tools to represent a signal at different scales [Meyer, 1990, Mallat, 1989, Daubechies, 1992]. Seismic wave propagation in the Earth samples the multiple scales of the Earth heterogeneity. Classic finite-difference methods for wave modeling require the discretization of the Earth model and the computed wavefield into an uniform grid in space and time e.g, [Virieux, 1986]. These methods may oversample some regions of the heterogeneous medium such as smooth parts of the model using an unecessary small numerical step for time extrapolation. Especially in 3D, this leads to heavy computational costs. Different strategies based on multigrids have been proposed recently (see the review by [Moczo et al, 1999]) where the user defines by himself how to describe the 3D model. This task could be very time-consuming and may lead to some unexpected inexact sampling of some regions of the model. The multiresolution structure of orthogonal wavelet basis will allow an automatic distribution of the medium properties and the wavefield onto different grids. If adaptive spatial and temporal discretisations of the wavefield based on its local properties are possible, fast algorithms are expected. We propose an adaptation of the time-space second-order finite- difference method [Virieux, 1986] in the time-wavelet domain. The wavefields and the Earth properties are decomposed onto spatial orthogonal wavelet basis. The wavelet coefficients are extrapolated in time through a finite difference scheme. The recomposition in the space domain is performed only when wave propagation results are analysed. Since time extrapolation is performed in the wavelet domain, one may say that the proposed method belongs to the class of spectral methods. The wave differential equation is explicitely formulated in the wavelet basis. The so-called spatial adaptivity can be implemented in the wavelet domain by propagating only the wavelet coefficients from time t to t + Deltat which are greater than a given threshold. The other possible adaptivity is the so-called time adaptivity. Its purpose is to match the time step to each scale of the multiresolution analysis. Until now, we were unable to implement this time adaptivity although potential strategies can be proposed. The coupling between scales precludes an immediate simple solution. We would like to present the time-wavelet method and few synthetic examples in order to show that the computation is indeed possible with good precision. We certainly do not address yet the computational performances of this approach. As an illustration, we present a simulation of elastic wave propagation in the time-wavelet domain for a 2-D canonical heterogeneous wedge model.