Summary

In this abstract, we discuss the implementation of a TTI reverse time migration. Also we address several practical issues regarding numerical accuracy, stability and artifact removal related to the numerical implementation. The impulse responses from modeling and migration, in both synthetic and real models and for both strong anisotropy and strong variations of symmetry axis, validate the proposed formulation and algorithm.

Introduction

The presence of geological anisotropy causes complexities in seismic imaging. When the sedimentary layers are not horizontally oriented, imaging anisotropic structures under the VTI (Vertical Transverse Isotropy) assumption may introduce errors both kinetically and dynamically. In a general case, the Tilted Transverse Isotropy (TTI) could be a reasonably good processing basis.

The full-way modeling in TTI media has been proposed by different authors with different formulations. Alkhalifah (2000) derived a 4th-order partial-differential equation (PDE) in time to describe a TTI propagation. To avoid solving a high order PDE, Zhou et al. (2006) split Alkhaifah''s equation into two coupled 2nd-order (in time) PDEs. Therefore any existing numerical method to solve the isotropic acoustic equation can be directly used to solve the VTI equations.

Although it is fairly straightforward to implement a VTI modeling and migration, there are some challenges for modeling and migrating seismic data in a 3D heterogeneous TTI medium. The non-vertical symmetric axis complicates the formulations and the algorithms, and the lateral heterogeneity may cause numerical instabilities. The computational cost also increases dramatically. In this paper, we present some numerical solutions to TTI modeling and migration. The paper is organized as follows: We first propose a different splitting form of TTI wave equation in 3D heterogeneous media. Then we address several issues regarding the implementation of modeling and reverse time migration in 3D TTI media. At the end we present some numerical examples and draw some conclusions.

Formulation of Reverse Time Migration in 3D Heterogeneous TTI media

The equations for wave propagation in 3D TTI media can be derived in a similar way as in Zhou (2006). Start with the P-wave phase velocity expression (in the plane passing through symmetry axis and S-wave velocity is zero) (Tsvankin, 2001).

Numerical Implementation and Stability Issue

As stated above, the equation system (2) represents a set of The 2nd- order hyperbolic PDEs with variable coefficients. Ordinary centered finite-difference schemes and pseudospectral methods with a centered Fourier transform can cause energy leakage during wave propagation, and results in an instability in the media with inhomogeneous azimuth and dip angle, even with constant velocity, e and d. The instabilities usually start at the locations in the azimuth and dip models where sharp contrasts exist. This problem is more visible with strong anisotropic media.

To stabilize the computation process in heterogeneous media, computation of 1st- order spatial derivatives in (3) have to be calculated on a staggered grid (Fornberg, 1990, Corrêa et. al. 2002) so that the differentiation is spatially localized (Özdenvar, 1996) to reduce the ringing in the wavefield. Since the coefficients in (2) are varying, theyshould be relatively smooth before being used for modeling or migration.

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