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Introduction

Seismic waves are described by the elastic wave equation with P- and S-waves intrinsically coupled. An elastic migration or inversion program should be able to handle both wave modes. Normally the P and S modes are separated and each is treated independently. Otherwise, the two modes are mixed on all wavefield components and cause crosstalk and image artifacts (Yan and Sava, 2009b). In isotropic media, far-field P- and S-wave modes can be separated by taking the divergence and curl in the extrapolated elastic wavefield. It is well known that a shear wave passing through an anisotropic medium can split into two mutually orthogonal waves (Crampin, 1984). Generally the P-wave and the two S-waves in anisotropic materials are not polarized parallel and perpendicular to the wave vectors and can not be fully separated with divergence and curl operations.

To account for seismic anisotropy, wave mode separation concept and approach have been extendedin the past two decades. Dellinger and Etgen (1990) generalize divergence and curl to anisotropic media by constructing the separators in the wavenumber domain, and independently solving the Christoffel equation in each wave propagation direction. For heterogeneous media, these divergence-like and curl-like separators are rewritten by Yan and Sava (2009b) as nonstationary spatial filters determined by the local polarization vectors. Zhang and McMechan (2010) develop a wavefield decomposition method to separate elastic wavefields into vector P- and S-wave fields for vertically transverse isotropic (VTI) media. Alternatively, we may implicitly achieve partial mode separation during wavefield extrapolation using the so-called pseudo-pure-mode wave equations, and then obtain completely separated wave modes by correcting the polarization projection deviation of the pseudo-pure-mode wavefields from the isotropic reference (Cheng and Kang, 2013,2012). Although these studies provide significant insights into wave mode separation in anisotropic media, many challenges remain, especially in the computational implementation if the proposed approaches are directly used in practice. For example, mode separation using nonstationary filtering is computationally expensive, especially in 3D. To improve efficiency, Yan and Sava (2011) present a mixed-domain algorithm that resembles the phase-shift plus interpolation (PSPI) scheme from one-way wave equation migration (Gazdag and Sguazzero, 1984). The compromise between accuracy and cost requires to determine the minimal reference models that best represent the true model space, and the choice of the models is case dependent. On the other hand, Zhang and McMechan (2010)'s wavenumber-domain vector decomposition approach is effective when the model can be separated into distinct geologic units. In addition, spectral methods were proposed to provide solutions which can completely avoid the crosstalk between the qP and qS modes in wavefield modeling and reverse-time migration (RTM) (Liu et al., 2009; Pestana et al., 2011; Song et al., 2013; Etgen and Brandsberg-Dahl, 2009; Chu et al., 2011; Fomel et al., 2013). However, these pure-mode solutions fail to provide accurate amplitudes for qP- and qS-waves. For true-amplitude ERTM in anisotropic media, effective mode separation and decomposition are highly required before applying the imaging condition to the extrapolated elastic wavefields (Zhang and McMechan, 2011).

In this paper, we respectively propose fast algorithms for elastic wave mode separation and vector decomposition in 3D heterogeneous transverse isotropic (TI) media. First, we give a brief review of the underlying principles. Then we present space-wavenumber-domain operations for mode separation and vector decomposition in the form of Fourier integral operators, and discuss how to construct efficient algorithms using low-rank approximation (Engquist and Ying, 2008). At the end, we test efficiency and accuracy of the proposed method using synthetic models of increasing complexity.


next up previous [pdf]

Next: Elastic wave mode separation Up: Cheng & Fomel: Anisotropic Previous: Cheng & Fomel: Anisotropic

2014-06-24