Introduction

Anisotropy is becoming increasingly important in seismic imaging. A vertical transversely isotropic (VTI) medium is one of the simplest and most practical approximations for anisotropic media. However, the VTI approximation is only valid for simple geologic formations, where the bedding plane is horizontal. In an area where the sediments are steeply dipping, such as anticline structures and thrust sheets, the symmetry axis of the medium is not vertical and the medium cannot be simply approximated as VTI medium. In these area, it is usually better to consider them as tilted transversely isotropic (TTI) media. For VTI media, to image steeply dipping reflector using one-way wave equation, Shan and Biondi (2004) rotate the coordinates. In the new coordinates, the medium becomes TTI media. For both cases, we need to design wavefield-extrapolation operators for TTI media.

Compared to those of isotropic and VTI media, the dispersion relation of TTI media is much more complicated. The dispersion relation of an isotropic medium is very simple, and we have an explicit expression for it. For a VTI medium, under the assumption that the S-wave velocity equals zero, we can still derive an explicit formula for its dispersion relation. The dispersion relation of TTI media is a quartic equation, and we have to solve it numerically. Conventional implicit finite-difference methods rely on the Taylor series approximation of the explicit dispersion relation. It is very hard to derive a Taylor series for the dispersion relation of TTI media. As a result, most wavefield extrapolation algorithms for anisotropic media are based on either explicit finite-difference Baumstein and Anderson (2003); Ren et al. (2005); Shan and Biondi (2005); Uzcategui (1995); Zhang et al. (2001a,b) or phase-shift plus interpolation method Ferguson and Margrave (1998); Rousseau (1997). For both explicit finite-difference methods and phase-shift plus interpolation (PSPI), the complex dispersion relation does not increase the complexity of the algorithm. However both of them are very expensive; explicit finite-difference methods for TTI media require running 2D convolutions in 3D and PSPI requires extrapolating many reference wavefields.

Implicit finite-difference method has been one of the most attractive methods for isotropic media. It can handle lateral variation of velocity naturally and guarantee stability. Traditional finite-difference methods, such as the $15^\circ$ equation Claerbout (1971) and the $45^\circ$ 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 optimize the coefficient with least-squares. This method improves the accuracy with the same computational cost. Under the weak anisotropy assumption, Ristow and Ruhl (1997) design an implicit scheme for VTI media. Liu et al. (2005) apply a phase-correction operator Li (1991) after the finite-difference operator for VTI media and improve the accuracy. Shan (2006) approximates the VTI dispersion relation with rational functions and obtains the coefficients using weighted least-squares optimization.

In this paper, I present an optimized one-way wave equation for TTI media and use a table-driven implicit finite-difference method Shan (2006) for laterally varying media. I compared the impulse responses of this algorithm with those of phase-shift methods.