Recursive integral time extrapolation of elastic waves using low-rank symbol approximation |

To mitigate cross-talk between P- and S-waves, it is often necessary to decouple different wave modes prior to imaging. In isotropic media, wave-mode separation can be achieved using the divergence and curl operators (Aki and Richards, 1980). Dellinger and Etgen (1990) implement wave-mode separation in anisotropic media by projecting the vector wavefield onto the polarization directions defined by the Christoffel equation. Yan and Sava (2009,2012) implement wave-mode separation in vertical transversely isotropic (VTI) and tilted transverse isotropic (TTI) media by introducing space-domain non-stationary filters to handle spatial heterogeneity, and improve the efficiency using the idea of phase-shift plus interpolation (Yan and Sava, 2011). Zhang and McMechan (2010) further investigate wavefield vector decomposition method in the wavenumber domain and apply it to VTI media. Cheng and Fomel (2014) formulate the wave-mode separation and decomposition operators in heterogeneous media as Fourier Integral Operators (FIOs) and efficiently apply them using the low-rank approximation (Fomel et al., 2013). Sripanich et al. (2015) extend the low-rank decomposition operator to wave-mode decomposition in orthorhombic media.

Conventionally, wave-mode decomposition and wave extrapolation are considered as two separated steps. Hou et al. (2014) and Cheng et al. (2014,2016) combine these two steps into a single FIO, which can be implemented by low-rank approximation. These methods are based on the assumption that the medium properties are sufficiently smooth so that their spatial derivatives can be neglected. However, the Earth model can be strongly heterogeneous and contain discontinuities, e.g., at salt/sediment boundaries. In such cases, the assumption about the smoothness of the Earth model is no longer valid and could lead to inaccurate calculation of polarization directions. More importantly, simultaneous wave extrapolation and wave-mode separation based on such an assumption may fail to provide reliable phase and amplitude information.

In this paper, we introduce a general framework for elastic wave extrapolation in strongly heterogeneous and anisotropic media without the assumption of the smoothness of the medium. The proposed method uses FIOs which allow accurate and stable wave extrapolation to be performed without explicit wave-mode separation. The proposed formulation reveals a simple connection between wave-mode decomposition and wave extrapolation through matrix exponentials. We also show that it is not necessary to explicitly decompose the wavefield into separate wave modes in order to apply recursive integral operators. We first derive one-step elastic wave extrapolation in homogeneous and smooth media motivated by one-step acoustic wave extrapolation (Zhang and Zhang, 2009; Sun et al., 2016a). In one-step extrapolation of elastic waves, only positive or negative frequency components are propagated, naturally providing the direction information that is useful for efficient wavefield up-down separation and angle gathers computation during imaging (Hu et al., 2016; Shen and Albertin, 2015; Sun et al., 2016a). We also construct the corresponding two-step scheme which uses only a real-valued vector wavefield. We draw connections of the proposed method to simultaneous propagation of decoupled elastic wave modes. Next, we show that to accurately model wave propagation in strongly heterogeneous media, spatial gradients of stiffnesses need to be included in the Christoffel matrix, leading to complex eigenvalues and polarization directions. Efficient calculation of the proposed FIOs in heterogeneous media is enabled by approximating the wave extrapolation matrix symbol with a low-rank decomposition. Our numerical examples demonstrate that the proposed method is stable and free of dispersion artifacts and therefore is suitable for accurate elastic RTM and FWI in (strongly) heterogeneous anisotropic media.

Recursive integral time extrapolation of elastic waves using low-rank symbol approximation |

2018-11-16