Surface Boundary Conditions in Reverse-Time Extrapolations of Three-Dimensional, Three-Component Elastic Seismic Data

We evaluate the physical validity of surface boundary conditions of the computational model in reverse-time extrapolations of three-dimensional (3-D), three-component (3-C) elastic seismic data acquired at the earth’s free surface, using mathematical derivations and numerical simulations. Reverse-time extrapolation of elastic data assumes that only the incident (P or S) waves are reconstructed and downward extrapolated into the computational grid. In fact, the superposition of the (up-going) incident waves and the (down-going) reflected and converted waves is recorded in data acquisition on the free surface, and is input in reverse-time extrapolation. When inserted as boundary conditions, the input incident waves propagate both downward and upward. Using an absorbing top boundary makes the upward propagating incident, reflected and converted waves disappear, the downward propagating waves enter the computational grid as signal, and the downward propagating reflected and converted waves (with small amplitudes) as artifacts. Using a free-surface top boundary causes all the upward propagating waves to generate secondary reflections and conversions that propagate downward into the computational grid as additional artifacts.