Introduction

Summary

In this paper, we decouple the P and SV wave components in an acoustic transversely isotropic media with vertical symmetric axis (VTI), and construct an independent pseudo-differential equation for each wave mode. The resulted wave equation for P-wave is unconditionally stable. A scheme based on the optimized separable approximation is proposed for their numerical implementation. We demonstrate the theory with some simple experiments.

The importance of anisotropy has been identified several decades ago (Backus, 1963). Indeed, ignoring the effect of anisotropy in imaging may result in significant mispositioning of steeply dipping structures. Fig 1a) shows the anisotropic reverse time migration of a 2D synthetic dataset using the correct model parameters. The input data is modeled in a VTI media using the vertical P-wave velocity shown in Fig 1c), the e in Fig. 1d) and the d in Fig. 1e). For comparison, we show the isotropic reverse time migration in Fig 1b) which is imaged using the vertical P-wave velocity. Anisotropic migration has clearly produced much superior result compared to the isotropic one, especially for the steeply dipping fault bed, the salt body and the structures underneath. In addition, the anisotropic migration has correctly imaged the interface that is caused only by the anisotropic parameter variations in the center of the section, it nearly completely disappears in the isotropic image and generates a strong noisy artifact.

Anisotropic migration has gained more and more attention and applications in the industry, especially the acoustic transversely isotropic (TI) case introduced by Alkhalifah (1998). Though many people have started investigating the more complicated and relatively more accurate tilted transverse isotropy (TTI) (Zhou et al., 2006 b, Zhang and Zhang, 2008), the simpler case with vertical symmetric axis (VTI) still has great demand in many geological situations.

Different migration equations have been proposed for an acoustic VTI media following the pioneered work of Alkhalifah (2000). However, those wave equations are not really free of shear waves. In fact, its presence has been observed in full waveform modeling, but it was not fully understood and was simply classified as numerical artifacts. Furthermore, these wave equations are stable and numerically solvable only when the Thomsen’s parameters satisfye =d . Though this is true for a large portion of our exploration targets, there are many rocks that show the opposite relation (Thomsen, 1986). In this paper, we decouple the P and SV wave components from the standard wave equation in an acoustic VTI media, and construct a separate wave equation for each of them. The resulting wave equation for P-wave is still stable even when e <d , but the SV-component becomes evanescent in such case. We will demonstrate the theory using some simple examples.

Wave Equations in Acoustic VTI Media

Acoustic anisotropy is introduced by setting the shear wave velocity to 0, i.e., V s = 0 , along the symmetric axis (Alkhalifah, 1998). Under this definition, Alkhalifah (2000) derived a 4th order wave equation for acoustic VTI media, whose dispersion relation in a 3D case is

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