Fourier finite-difference wave propagation

Introduction

The wavefield extrapolation problem refers to advancement of a wavefield through space or time. Both extrapolation in depth and extrapolation in time can be used in seismic modeling and seismic migration. Reverse time migration, or RTM (McMechan, 1983; Baysal et al., 1983; Levin, 1984; Whitmore, 1983), involves wave extrapolation forward and backward in time. RTM is useful for accurate imaging in complex areas and is drawing more and more attention as the most powerful depth-imaging method (Fletcher et al., 2009; Yoon et al., 2004; Symes, 2007; Fowler et al., 2010).

Reverse-time migration can correctly handle complex velocity models without dip limitations on the image. However, it has large memory requirements and needs a significant amount of computation. The most popular and straightforward way to implement reverse-time migration is the method of explicit finite differences, which is only conditionally stable because of the limit on time-step size. Finite-difference methods also suffer from numerical dispersion problems, which can be overcome either by decreasing the time step or by high-order schemes (Wu et al., 1996; Liu and Sen, 2009). Several alternative algorithms have been developed for seismic wave extrapolation in variable velocity media. Soubaras and Zhang (2008) introduced an algorithm based on a high-order differential operator, which allows a large extrapolation time step by solving a coefficient optimization problem. Zhang and Zhang (2009) proposed one-step extrapolation method by introducing a square-root operator. This method can formulate the two-way wave equation as a first-order partial differential equation in time similar to the one-way wave equation. Etgen and Brandsberg-Dahl (2009) modified the Fourier Transform of the Laplacian operator to compensate exactly for the error resulting from the second-order time marching scheme used in conventional pseudo spectral methods (Reshef et al., 1988a). Fowler et al. (2010) provided an accurate VTI P-wave modeling method with coupled second-order pseudo-acoustic equations. Pestana and Stoffa (2010) presented an application of Rapid Expansion Method (REM) (Tal-Ezer et al., 1987) for forward modeling with one-step time evolution algorithm and RTM with recursive time stepping algorithm.

In this paper, we present a new wave extrapolator derived from the pseudo-analytical approach of Etgen and Brandsberg-Dahl (2009). Our method combines FFT and finite differences. We call it the Fourier Finite Difference method because it is analogous to the concept introduced previously for one-way wave extrapolation by Ristow and Ruhl (1994).

As a chain operator of Fast Fourier Transform and Finite Difference operators, the proposed extrapolator can be as accurate as the parameter interpolation approach employed by Etgen and Brandsberg-Dahl (2009) but at a cost of only one Fast Fourier Transform (FFT) and inverse Fast Fourier Transform (IFFT) operation. The advantages of the FFD operator are even more apparent in the anisotropic case: no need for several interpolations for different parameters with the corresponding computational burden of several FFTs and IFFTs. In addition, the operator can overcome the coupling of qP-waves and qSV-waves (Zhang et al., 2009). We demonstrate the method on synthetic examples and propose to incorporate FFD into reverse-time migration in order to enhance migration accuracy and stability.

Fourier finite-difference wave propagation