An improved elastic wavefield separation method based on the Helmholtz decomposition

Abstract

Elastic wave-mode separation plays an important role in elastic reverse-time migration (RTM) and elastic full-waveform inversion. It helps to remove crosstalk artifacts and improve imaging quality. A class of efficient methods for elastic wave-mode separation in isotropic elastic media is the Helmholtz decomposition technique. Although this kind of approach produces pure-mode vector wavefields with correct amplitudes, phases, and physical units, their computational costs are still high especially for 3D large-scale problems. By making use of the relationships among the divergence, curl, gradient, and exterior derivative operations, we develop an improved elastic wave-mode separation based on the Helmholtz decomposition. We also need to solve a Poisson equation, but the Laplace operator operates on a scalar function rather than a vector function. Thus, for multidimensional (2D or 3D) problems, the Poisson equation only needs to be solved once for the vector P- and S-wavefields. This allows us to reduce the computational cost of the conventional Helmholtz decomposition method by a factor of 2 for solving 2D problems. For a 3D problem, the computational cost can be reduced by a factor of 3. To further reduce the computational cost, by introducing a smooth extension technique, we transform the problem into the wavenumber domain via Fourier transform and propose a fast solver for the Poisson equation. The resulting wavefield separation method not only produces P-wave and S-wave with the same phases and amplitudes as the input coupled wavefields but also significantly reduces the computational cost. Numerical tests show the efficiency of the proposed method and confirm the theoretical results.

qS‐wave decoupling equation for wavefield separation in transversely isotropic media

“…(2018), in recent years, reverse‐time migration (RTM) (Baysal et al., 1983;McMechan, 1983; Whitemore, 1983) has become a powerful practical tool for prestack imaging in the exploration and production of oil and gas reservoirs. Currently, the wavefield decoupling techniques in isotropic media (Ma & Zhu, 2003; Li et al., 2007; Zhang et al., 2007; Xiao & Leaney, 2010; Wang & McMechan, 2015; Zheng & Yao, 2023; Zhu, 2017) have become relatively mature and are widely used in elastic RTM (ERTM) (Du et al., 2017, 2020; Zhang & Lu., 2022; Zhou et al., 2019; Zhu, 2017) and full waveform inversion (Qu et al., 2018; Ren & Liu, 2016; Sears et al., 2008; Wang & Cheng, 2017). However, due to the fact that the polarization direction of waves in anisotropic media is neither parallel nor perpendicular to the direction of wave propagation, there are coupling or interference phenomena between the separated P‐ and S‐wave.…”

“…In order to tackle this issue, a new pseudo‐Helmholtz operator based on the wavefront phase direction was proposed, and the generalized minimum residual algorithm was utilized to iteratively solve the Poisson equation for the acquisition of separated vector P‐ and S‐wavefields. Zheng and Yao (2023) introduced a scalar Poisson equation and developed a fast solver for it, enabling wavefield separation in isotropic media. They suggested that this method can be extended to anisotropic media.…”