Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media |

Multicomponent seismic data are increasingly acquired on land and at the ocean bottom in an attempt to better understand the geological structure and characterize oil and gas reservoirs. Seismic modeling, reverse-time migration (RTM), and full-waveform inversion (FWI) in areas with complex geology all require high-accuracy numerical algorithms for time extrapolation of waves. Because seismic waves propagate through the earth as a superposition of P- and S-wave modes, an elastic wave equation is usually more accurate for wavefield extrapolation than an acoustic wave equation. Wave mode decoupling can not only help elastic imaging to produce physically interpretable images, which characterize reflectivities of various reflection types (Yan and Sava, 2008; Dellinger and Etgen, 1990; Wapenaar et al., 1987), but also provide more opportunity to mitigate the parameter trade-offs in elastic waveform inversion (Wang and Cheng, 2015).

For isotropic media, far-field P and S waves can be separated by taking the divergence and curl in the extrapolated elastic wavefields (Aki and Richards, 1980; Sun and McMechan, 2001). Alternatively, Ma and Zhu (2003) and Zhang et al. (2007) extrapolated vector P and S modes separately in an elastic wavefield by decomposing the wave equation into P- and S-wave components. In the meantime, decoupling of wave modes yields familiar scalar wave equations for P and S modes (Aki and Richards, 1980). In anisotropic media, one cannot, so simply, derive explicit single-mode time-space-domain differential wave equations. Generally, P and S modes do not respectively polarize parallel and perpendicular to the wave vectors, and thus are called quasi-P (qP) and quasi-S (qS) waves. They cannot be fully decoupled with divergence and curl operations (Dellinger and Etgen, 1990).

Anisotropic wave propagation can be formally decoupled in the wavenumber-domain to yield single-mode pseudo-differential equations (Liu et al., 2009). Unfortunately, these equations in time-space domain cannot be solved with traditional numerical schemes unless further approximation to the dispersion relation or phase velocity is applied (Zhan et al., 2012; Wu and Alkhalifah, 2014; Song and Alkhalifah, 2013; Du et al., 2014; Fowler and King, 2011; Etgen and Brandsberg-Dahl, 2009; Chu et al., 2011). To avoid solving the pseudo-differential equation, Xu and Zhou (2014) proposed a nonlinear wave equation for pseudoacoustic qP-wave with an auxiliary scalar operator depending on the material parameters and the phase direction of the propagation at each spatial location. All these efforts are restricted to pure-mode scalar waves and do not honor the elastic effects such as mode conversion. Cheng and Kang (2014) and Kang and Cheng (2012) have proposed approaches to propagate the partially decoupled wave modes using the so-called pseudo-pure-mode wave equations, and then obtain completely decoupled qP or qS waves by correcting the polarization deviation of the pseudo-pure-mode wavefields. Their approaches honor the kinematics of all wave modes but may distort the reflection/transmission coefficients if high contrasts exist in the velocity fields.

Alternatively, many have developed approaches to decouple qP- and qS-wave modes from the extrapolated elastic wavefields. Dellinger and Etgen (1990) generalized the divergence and curl operations to anisotropic media by constructing separators as polarization projection in the wavenumber-domain. To tackle heterogeneity, these mode separators were rewritten by Yan and Sava (2009) as nonstationary spatial filters determined by the local polarization directions. Zhang and McMechan (2010) proposed a wavefield decomposition method to separate elastic wavefields into vector qP- and qS-wave fields for vertically transverse isotropic (VTI) media. Accordingly, Cheng and Fomel (2014) proposed fast mixed-domain algorithms for mode separation and vector decomposition in heterogeneous anisotropic media by applying low-rank approximation to the involved Fourier integral operators (FIOs) of the general form.

The motivation of this study is to develop an efficient approach to propagate and decouple the elastic waves for general anisotropic media. The primary strategy is to merge the numerical solutions for time extrapolation and vector decomposition into a unified Fourier integral framework and speed up the solutions using the low-rank approximation. This paper is organized as follows. We first demonstrate a pseudo-spectral solution to extrapolate the elastic displacement wavefields in time-domain. Then we propose to merge the spectral operations for time extrapolation into the integral framework for vector decomposition of the wave modes. Applying low-rank approximations to the involved mixed-domain matrices, we obtain an efficient algorithm for simultaneous propagating and decoupling the elastic wavefields. We demonstrate the validity of the proposed method using 2-D and 3-D synthetic examples on the transversely isotropic (TI) and orthorhombic models with increasing complexity.

Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media |

2016-11-21