Elastic wave-vector decomposition in heterogeneous anisotropic media

Introduction

In seismic imaging by elastic reverse-time migration (RTM), different seismic wave modes generally need to be decoupled. Wave-mode separation in isotropic media is relatively simple to accomplish by means of the divergence and curl operators (Aki and Richards, 2002). Dellinger and Etgen (1990) extended this concept to 2D anisotropic media by projecting the vector wavefield onto the polarization vectors found by solving the Christoffel equation. Yan and Sava (2008) applied wave-mode separation for elastic RTM in isotropic media. Yan and Sava (2009) addressed wave-mode separation in the space domain via the use of non-stationary spatial filters, which enable handling of heterogeneity. They later improved this method by frequency-domain phase-shift plus interpolation (PSPI) technique, which increases the computational efficiency (Yan and Sava, 2011), and extended the application to the case of transversely isotropic media with tilted symmetry axis (TTI) (Yan and Sava, 2012). Alternatively, Zhang and McMechan (2010) developed the wave-vector decomposition method originally studied by Dellinger (1991) and successfully appiled this method to transversely isotropic media with vertical symmetry axis (VTI). The wave-vector decomposition is based on the principle of projecting wavefields onto polarization vectors in the wavenumber domain (Dellinger, 1991).

In both wave-mode separation and wave-vector decomposition, the computational cost presents a primary challenge because of the need to solve the Christoffel equation in all phase directions for a given set of stiffness tensor coefficients at each spatial location of the medium. Cheng and Fomel (2014) proposed to reformulate the separation and decomposition operators as Fourier Integral Operators (FIO) and applied the low-rank approximation approach (Fomel et al., 2013), which significantly improved computational efficiency. As shown by Cheng and Kang (2014,2016), partial mode separation during extrapolation using the pseudo-pure-mode wave equations also helps improving the efficiency of the procedure.

In recent years, orthorhombic anisotropy has become an important topic of interest in places where a simpler transverse isotropy model becomes insufficient for characterizing the subsurface (Xu et al., 2005; Bakulin et al., 2000; Thomsen, 2014; Tsvankin, 2012; Sripanich et al., 2016; Grechka, 2009; Vasconcelos and Tsvankin, 2006; Tsvankin, 1997; Sripanich et al., 2015; Sripanich and Fomel, 2015). An example of an orthorhombic medium is a sedimentary basin with parallel fractures in a transversely isotropic background (Thomsen, 2014; Tsvankin and Grechka, 2011). Elastic wave phenomena in such media involve three mutually orthogonal symmetry planes and can be described by nine independent stiffness coefficients as opposed to five in the case of TI media. This behavior leads to a higher degree of complexity of velocity and polarization vector characterization in comparison with TI media (Schoenberg and Helbig, 1997). In orthorhombic media, P-wave mode is, generally, well-separated from the S modes because of its higher phase velocity and therefore, P waves can be straightforwardly extracted from the full elastic wavefield (Dellinger, 1991). The S-wave modes, however, are extricably coupled and more difficult to separate (Schoenberg and Helbig, 1997). Dellinger (1991) showed that if the S waves were separated based on the magnitude of their phase velocities (S1 and S2), the resulting wavefields would be plagued with strong planar artifacts associated with the effects of a polarization discontinuity at the singularity, which in the wavenumber domain, behaved similarly to the delta function. Moreover, orthorhombic symmetry is associated with kiss and point singularities but not the intersection singularity observed in the TI case (Crampin, 1991; Crampin and Yedlin, 1981; Crampin, 1984). The number of kiss and point singularities, and their corresponding phase directions are not fixed and depend on the model parameters. Therefore, a simple global weighting function cannot eliminate artifacts caused by a discontinuity of polarization vectors as in the case of a kiss singularity along the symmetry axis in TI media (Yan and Sava, 2012).

In this study, we first extend the wave-vector decomposition method using low-rank approximation (Cheng and Fomel, 2014) and apply it to separate elastic wavefield in orthorhombic media, as well as, in monoclinic and triclinic media, where only the effects of kiss and point singularities for S waves are possible. Because the velocity of the P-wave mode in models considered in practice are generally larger than those of the S-wave modes, there is no ambiguity in identifying its polarization from the solution of the Christoffel equation. To distinguish between the two S-wave modes, we follow the definitions of S modes proposed by Dellinger (1991) and separate them into S1 and S2 based on their phase velocities. We subsequently propose a constructive method for locating singularities that can be used for defining a smoothing filter (weighting function) to mitigate the artifacts caused by discontinuities of polarization vectors. We recover from amplitude loss caused by the smoothing filter via the process of local signal-noise orthogonalization (Chen and Fomel, 2015). We test the proposed method with a set of synthetic examples of increasing complexity.

Elastic wave-vector decomposition in heterogeneous anisotropic media