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 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 two for solving 2D problems. For a 3D problem, the computational cost can be reduced by a factor of three. To further reduce the computational cost, by introducing a smooth extension technique, we transform the problem into the wavenumber domain via the Fourier transform and use a fast solver for the Poisson equation. The resulting wavefield separation method not only produces P and S waves with the same phases and amplitudes as the input-coupled wavefields but also significantly reduces the computational cost. Numerical tests indicate the efficiency of the proposed method and confirm the theoretical results.