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Introduction

Ultrasonic laboratory studies as well as seismic field investigations have shown that many geological materials and subsurface structures are elastically anisotropic. It is well known that a shear wave passing through an anisotropic material splits into two mutually orthogonal waves, which propagate at different velocities. Therefore, seismic waves propagate through the earth as a superposition of three body wave modes, one P-wave and two S-waves. Generally they are polarized neither parallel to nor perpendicular to the direction of wave travel, thus are called quasi-P (qP) and quasi-S (qS) waves, with quasi- means similar to but not exactly. P- and S-waves were originally named for their arrival times, with P for the first (primary) and S for the second. Today, the indicators P and S are often connected with polarization, i.e., P with compressional (or longitudinal) and S with shear (or transverse), with the specification SH and SV for waves with transverse displacements in the horizontal and vertical planes, respectively (Winterstein, 1990). For vertical transversely isotropic (VTI) media, one often uses this terminology with qP, qSV, and SH, since the first two of these waves are generally not purely longitudinal and transverse, respectively.

Because seismic anisotropy by nature is an elastic phenomenon, the full elastic wave equation is usually more accurate for wavefield extrapolation than the acoustic equation. However, seismic imaging using the elastic wave equation involve high computational cost and many challenges in decoupling wave modes to get physically interpretable images of the subsurface (Zhang and McMechan, 2010; Dellinger and Etgen, 1990; Cheng and Fomel, 2014; Yan and Sava, 2011). For real-sized applications of seismic imaging and inversion, it is necessary to resort to a simplified description of wave propagation in anisotropic media.

Pseudoacoustic wave equations are the most common approximations made to mono-component (mainly pressure) seismic data. They are derived by setting the qS-wave phase velocity along the symmetry axis to zero for VTI or orthorhombic media (Alkhalifah, 2003; Duveneck and Bakker, 2011; Alkhalifah, 2000). Pseudoacoustic wave equations describe the kinematic signatures of qP-waves with sufficient accuracy and are simpler than their elastic counterparts, which leads to computational savings in practice (Fletcher et al., 2009; Zhang and Zhang, 2011; Zhou et al., 2006). They also have fewer parameters, which is important for inversion. However, we note several limitations of the acoustic anisotropic wave equation. First, acoustic approximation does not prevent the propagation of qS-waves in directions other than the symmetry axis (Grechka et al., 2004; Zhang et al., 2005); the residual qS-waves are regarded as artifacts in the framework of acoustic modeling, reverse-time migration (RTM) and full waveform inversion (FWI) (Operto et al., 2009; Zhang et al., 2009; Alkhalifah, 2000). Second, stability analysis based on requiring the stiffness tensor to remain positive definite (Helbig, 1994) shows that wavefield extrapolation in a pseudo-acoustic TI or orthorhombic medium can become unstable (Grechka et al., 2004; Fowler and King, 2011; Alkhalifah, 2000). Alternatively, qP- and qS-wave propagation can be formally decoupled in the wavenumber domain to yield pure-mode pseudodifferential equations (Liu et al., 2009; Du et al., 2014). Unfortunately, these equations in time-space domain cannot be solved with traditional numerical schemes. Through factorizing and approximating the qP-qSV dispersion relations or phase velocities, many authors proposed to simulate propagation of scalar pure-mode waves using mixed-domain recursive integral operators (Liu et al., 2009; Fomel et al., 2013; Crawley et al., 2010; Zhan et al., 2012; Pestana et al., 2011; Zhang and Zhang, 2009; Chu et al., 2011; Etgen and Brandsberg-Dahl, 2009; Fowler et al., 2010) or novel finite-difference operators (Song et al., 2013). To avoid solving the pseudodifferential equation, Xu and Zhou (2014) proposed a nonlinear wave equation for a pseudo-acoustic qP-wave with an auxiliary scalar operator depending on the material parameters and the phase direction of the propagation at each spatial location.

Multicomponent seismic data are increasingly acquired on land and at the ocean bottom to better delineate geological structures and to characterize oil and gas reservoirs (Hardage et al., 2011; Thomsen, 1999; Li, 1997; Stewart et al., 2002; Cary, 2001). The development of unconventional reservoirs and the microseismic monitoring of hydraulic fracturing has led to more interest in shear waves because microseismic sources emit strong shear energy that is routinely recorded by three-component geophones (Maxwell, 2010) and is widely recognized as being useful for locating microseismic events and estimating their focal mechanisms (Baig and Urbancic, 2010; Grechka and Yaskevich, 2014). In fact, anisotropic phenomena are especially noticeable in shear and mode-converted wavefields. Therefore, modeling of anisotropic shear waves may be important both theoretically and practically. As we know, the pseudoacoustic approximation is not appropriate for qS-waves. In addition to amplitude errors, the kinematic accuracy of qS-waves is reduced if we use the existing numerical schemes based on factorizing and approximating the dispersion relations or phase velocities.

In kinematics, there are various forms equivalent to the original elastic wave equations. In our previous paper (part I), we derived the pseudo-pure-mode wave equation for qP-waves by applying a particular similarity transformation to the Christoffel equation and demonstrated its features in describing wave propagation for TI and orthorhombic media. Except for its application to scalar qP-wave RTM (Cheng and Kang, 2014), the pseudo-pure-mode wave equation provides new insight into developing approaches for multicomponent qP-wave inversion (Djebbi and Alkhalifah, 2014). The same theoretical framework described in part I is applied to qS-waves in this paper. First we derive the pseudo-pure-mode wave equations for qS-waves in TI media through new similarity transformations to the original Christoffel equation. Numerical examples demonstrate the features of the proposed qS-wave propagators in 2D and 3D TI media. Then we discuss the dynamic features of the pseudo-pure-mode qS-wave equations and the challenges to extending them to anisotropic media with lower symmetry.


next up previous [pdf]

Next: Phase velocity and polarization Up: Cheng & Kang: Propagate Previous: Cheng & Kang: Propagate

2016-10-14