The P- and S-wave decomposition in a multicomponent elastic wavefield based on the divergence and curl operators and their applications in elastic reverse time migration

The P- and S-wave separation is an important step in the elastic reverse-time migration (ERTM). It not only removes crosstalk artifacts in the image but also provides additional constrains to subsurface structures by providing images between different wave types. Traditional Helmholtz decomposition based on divergence and curl operations can separate the P and S wave modes but also modify the amplitude, phase and physical dimension of the original coupled wavefields. To recover the separated P- and S-waves to their original forms, we propose a method, in which the corrections to the distorted separated wavefields, including their phase, amplitude and vector polarizations, are organized into two spatial-time domain partial differential equations. Solving these equations can generate the separated P and S wavefields. The method can be conveniently applied in the time-space domain and consistent with most time-domain finite-difference based elastic reverse-time migration method. To verify the quality of the decomposed P- and S-wave components, we apply them to the ERTM using different elastic image conditions. For the PP image, we use the scalar imaging condition by crosscorrelating the corrected scalar Helmholtz potential wavefield from both source- and receiver-sides. For the PS image, we use the magnitude- and sign-based vector imaging condition by crosscorrelating the magnitudes of the source-side vector P wave and the receiver-side vector S wave, whereas the sign is determined through their dot-product results. Thus, accurate PP- and PS-reflectivity images can be obtained. Several numerical examples are used to verify our elastic wavefield separation method and the ERTM workflow.