A compact three‐dimensional fourth‐order scheme for elasticity using the first‐order formulation (original) (raw)

Elastic wave modelling by an integrated finite difference method

We have developed a finite difference method for modelling the elastic wave equation in the time domain, based on integrating the elastic parameters. In this method, we adopt the strategy of integrating the elastic parameters over a limited space; so, it is suitable for wave propagation modelling in fractured media, for which we use an equivalent media with the elastic coefficients averaged over a fractured space. This elastic parameters integration allows us to reduce the five simultaneous equations usually used to describe the velocity and stress propagation to just two, in terms of velocity alone, providing a significant saving in computational memory. In this paper, we discuss the derivation and computational implementation of the method for 2-D media, including the seismic source and both reflecting and absorbing boundary conditions , and illustrate it with some synthetic models of heterogeneous, anisotropic and fractured media. In the forward modelling of seismic waveforms, there has been steady improvement in the formulation of finite difference implementations. Explicit second-order time-domain schemes for modelling wave propagation in isotropic homogeneous and heterogeneous cases were proposed by Kelly et al. (1976). A second-order staggered grid scheme, based upon the five-equation velocity–stress formulation of the elastic wave equation was proposed by Virieux (1986) to deal with more complicated problems and also to model a boundary between acoustic and elastic media in a stable manner. This was extended to a fourth-order scheme for greater accuracy by Levander (1988). The second-order scheme of Kelly et al. (1976) was applied in the frequency domain by Pratt (1990) to allow for more efficient studies of multiple sources and the effects of attenu-ation. In this paper, we present a finite difference scheme, based upon the method of integrating elastic parameters over a limited space (Tikhonov & Samarskii 1961). This method works for fully heterogeneous and anisotropic 2-D media. It is implemented using the elastic wave equation written in terms of velocity only, which is an improvement in computational efficiency over the commonly used velocity–stress formulation, because the stress does not need to be calculated or saved. It can be implemented in the time domain, as presented here, or in the frequency domain. In the implementation, we also propose a scheme for how to properly set up a seismic source in the waveform simulation. We illustrate our method by applying it to some synthetic velocity models including heterogeneous , anisotropic and fractured media.