Amplitude effect of the free surface in elastic reverse-time extrapolation

Abstract

We evaluate the physical validity of surface boundary conditions of the computational model in reverse-time extrapolation of 3D, three-component (3-C) elastic seismic data acquired at the earth’s free surface by using mathematical derivations and numerical simulations. Reverse-time extrapolation of elastic data assumes that only the incident P- or S-waves are reconstructed during extrapolation into the computational grid. However, superposition of the (upgoing) incident waves and the (downgoing) reflected and converted waves generated at the free surface also is recorded in data acquisition and is input into reverse-time extrapolation. In elastic reverse-time extrapolation, the computational model needs to have an absorbing top boundary. When the 3D, 3-C elastic data are inserted into the computational model during reverse-time extrapolation, the originally incident P- or S-wave is reconstructed. In addition, the free-surface P-to-P reflected and P-to-S converted waves recombine to reconstruct a second incident P-wave, and the free-surface S-to-S reflected and S-to-P converted waves recombine to reconstruct a second incident S-wave. Therefore, 3D elastic reverse-time extrapolation reconstructs the incident waves with displacement amplitudes increased by a fixed factor of exactly two when free-surface reflections and conversions are in the data. In this implementation, reconstructed (virtual) waves propagating upward from the free surface enter an absorbing zone and disappear.