Separating P- and S-waves in prestack elastic seismograms using the equivalent form of elastic wave equation

Based on the Helmholtz decomposition, P- and S-waves in isotropic media can be separated by using divergence and curl operators. However, divergence and curl operators cannot be used directly for the wavefield separation of surface seismograms, because their first vertical derivatives are incalculable. An approach to solve this problem is combining elastic full wavefield extrapolation and wave-type separation using divergence and curl operators. It includes three steps: (1) downward extrapolate the elastic data from the surface to a reference depth, (2) separate the P- and S-waves by calculating the divergence and curl of the full elastic wavefield at the reference depth, and (3) upward extrapolate the separated P- and S-waves back to the surface, respectively. However, the phase and the P-S amplitude ratio are changed, after the divergence and curl calculations. So the phase correction and amplitude balancing need to be done after the separation. In this paper, after the above step (1), the P- and S-wave separated elastic equation, which is the equivalent form of the conventional elastic wave equation, are used to upward extrapolate the downward extrapolated wavefield back to the surface. The P- and S-waves are separated naturally during the extrapolation, and their phase and amplitude are the same as the preseparation.