Introduction

Implicit finite-difference methods that are adapted to strongly laterally varying media and guarantee stability, have been one of the most attractive methods for isotropic media. Traditional implicit finite-difference migration methods are based on the truncation of the Taylor series of the dispersion relation. For anisotropic media, phase-shift plus interpolation (PSPI) methods Ferguson and Margrave (1998); Rousseau (1997) or explicit finite-difference methods Baumstein and Anderson (2003); Ren et al. (2005); Shan and Biondi (2005); Uzcategui (1995); Zhang et al. (2001a,b) are usually chosen for migration because the dispersion relation of anisotropic media is very complex and it is difficult to derive a Talyor series for the implicit finite-difference scheme with high accuracy. However, TTI (tilted TI) media are not circularly symmetric and a 2D convolution operator is required instead of the McClellan transformations Hale (1991) to implement the explicit finite-difference scheme Shan and Biondi (2005). Although Lloyd's algorithm can be used to reduce the number of reference velocity and anisotropy parameters in PSPI Tang and Clapp (2006), too many reference wavefields are required to achieve decent accuracy in a strongly laterally varying TTI medium.

Lee and Suh (1985) approximate the square-root operator with rational functions and optimize the coefficients by least-squares function fitting. This method improves the accuracy of the finite-difference scheme without increasing the computational cost. Under the weak anisotropy assumption, Ristow and Ruhl (1997) design an implicit finite-difference scheme for VTI (transversely isotropic with a vertical symmetry axis) media. Liu et al. (2005) apply a phase-correction operator Li (1991) in the Fourier domain in addition to the implicit finite-difference operator for VTI media and improve the accuracy. Shan (2006b) approximates the dispersion relation of VTI media with rational functions and obtains the coefficients for the finite-difference scheme by using the weighted least-squares optimization. Similarly, Shan (2006a) designs implicit-finite difference scheme for TTI media by fitting the dispersion relation with rational functions and shows impulse responses in a homogeneous medium.

In this paper, I review the optimized implicit finite-difference method for TTI media and apply it to a 2D synthetic dataset to verify the methodology in laterally varying media.