Summary

This paper presents an implicit finite difference algorithm for 3D TTI media. We first give the detailed derivation of 3-D TTI dispersion relation equation. We approximate the 3-D TTI dispersion equation with four-direction splitting partial fractional expansion to enable the cascading implementation with implicit finite difference. The expansion coefficients are obtained through a nonlinear optimization of the dispersion error cost function. The strong-amplitude evanescent energy is attenuated with a frequency-dependent phase shift at each depth step. We test and verify our method and implementation with few synthetic examples.

Introduction

Anisotropy exists in many physical rocks such as shales due to the intrinsic anisotropic properties of rock mineral components and fine-layered sandstones with preferable principal directions. Anisotropy has been observed in quite different geological regions from North Sea to the Gulf of Mexico, from onshore survey (foothills) to offshore survey. It is important to incorporate anisotropy in seismic depth imaging algorithms to get high quality structurally-correct images. In present seismic exploration practice, anisotropy is either described by a VTI model or by a TTI model. TTI model provides better description to real life anisotropy and TTI image techniques have been increasingly demanded. Due to the amazing advances both in seismic imaging algorithms and high performance computing techniques, TTI anisotropic imaging has been made possible and been widely used in field data processing in industries in the last few years. As imaging in isotropic media, there are three technical classes for TTI imaging: 1. Kirchhoff-type imaging based on the high frequency asymptotic integral solution of wave equation with ray tracing (Jiao et al, 2005; Zhu et al, 2006); 2. Downward extrapolation imaging based on one-way wave equation (Shan, 2005, 2007; Bale et al, 2007; Valenciano et al., 2009a, 2009b); and 3. Reverse time migration (RTM) by directly solving two-way wave equation (Du et al, 2007; Zhang et al, 2008; Fletcher et al, 2008, 2009). On one hand, in many cases, wave equation migration is superior to Kirchhoff migration because it can naturally handle multi-pathing issues for complex velocity models and provides better images than Kirchhoff migration. On the other hand, despite the successes of TTI reverse time migration (RTM), one-way migrations remain important imaging tools in industries: One-way propagator is much faster than RTM,. it is useful in iterative velocity updating (Shen et al, 2005, Fei et al, 2007) and wide bandwidth data imaging where finer grid space must be used and higher frequency has to be migrated. One-way TTI migration is complicated to solve because it involves more model parameters (four in 2-D and five in 3- D) than isotropic migration (only one parameter – velocity) and TTI dispersion relation equation is in implicit form (difficult to be analytically solved, especially for 3-D). Algorithms applied in isotropic migration have been extended to TTI model. Shan and Biondi (2005) developed a 3-D TTI wave field extrapolation method using an implicit isotropic operator with an explicit anisotropic correction. Bale et al (2007) adapted isotropic PSPI and Fourier split-step operators for TTI media.

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