Optimized implicit finite-difference migration for VTI media

I develop an implicit finite-difference migration method for vertical transversely isotropic (VTI) media with laterally varying anisotropy parameters. I approximate the dispersion relation of VTI media with a rational function series, the coefficients of which are estimated by least-squares optimization. These coefficients are functions of Thomsen anisotropy parameters. They are calculated and stored in a table before the wavefield extrapolation. The implicit finite-difference scheme for VTI media is almost the same as that of the isotropic media, except that the coefficients are derived from the pre-calculated table. In the 3D case, a phase-correction filter is applied after the finite-difference operator to eliminate the numerical-anisotropy error caused by two-way splitting. This finitedifference operator for VTI media is accurate to 60, and its computational cost is almost the same as the isotropic migration. I apply this method to a 2D synthetic dataset and a 2D slice of a real 3D dataset to validate the method. INTRODUCTION Anisotropy is becoming increasingly important in seismic imaging. If anisotropy is not included in migration, reflectors will not be imaged at the right positions, or even worse, they will be defocused. However, imaging in a general anisotropic medium is still a challenging problem. A vertical transversely isotropic (VTI) medium is one of the simplest and most practical approximations for anisotropic media in seismic imaging. Compared to that of isotropic media, the dispersion relation of VTI media is much more complicated. As a result, phaseshift-based methods (Rousseau, 1997; Ferguson and Margrave, 1998) and explicit convolution methods (Uzcategui, 1995; Zhang et al., 2001a,b; Baumstein and Anderson, 2003; Shan and Biondi, 2005; Ren et al., 2005) are usually used in anisotropic migration, because the complex dispersion relation does not increase the difficulty of these algorithms. However, phase shift with interpolation requires a lot of reference wavefields, because there are two Thomsen anisotropy parameters in addition to the vertical velocity. Explicit convolution methods do not guarantee stability, and they also require long convolution filters to achieve good accuracy. The implicit finite-difference method has been one of the most attractive migration methods for isotropic media. It can handle lateral variation naturally and guarantee stability. Traditional finite-difference methods, such as the 15 equation (Claerbout, 1971) and the 45 equation (Claerbout, 1985), approximate the dispersion relation by the truncation of Taylor series. Lee and Suh (1985) approximate the square-root equation with rational functions, and 17