COARSE GRID CALCULATIONS OF WAVES IN INHOMOGENEOUS MEDIA WITH APPLICATION TO DELINEATION OF COMPLICATED SEISMIC STRUCTURE

Abstract

The multidimensional scalar wave equation at a single frequency is split into two equations. One controls the downgoing transmitted wave; the other controls the upcoming reflected wave. The equations are coupled, but in many reflection seismology situations the transmitted wave may be calculated without consideration of the reflected wave. The reflected wave is then calculated from the transmitted wave and the assumed velocity field. The waves are described by a modulation on up‐ or downgoing plane waves. This modulation function is calculated by difference equations on a grid. Despite complicated velocity models (steep faults, buried focus, etc.), the grid may be quite coarse if waves of interest do not propagate at large angles from the vertical. A one‐dimensional grid may be used for a two‐dimensional velocity model. With approximations, a point source emitting waves spreading in three dimensions may be included on the one‐dimensional grid. Calculation time for representative models is a few seconds. Phenomena displayed are interference, spherical spreading, propagation through focus, refraction, and diffraction. Converted waves are neglected. A procedure is suggested for the construction of a depth map of reflectors from observations at the surface. Assuming a velocity model, we may integrate the downgoing wave away from a surface source. Likewise, the upcoming wave may be approximately integrated back down into the earth. Since reflection coefficients are real, the ratio of upcoming to downgoing waves tends to be real at a reflector. An example is given in two dimensions which shows that this ratio over a dipping bed gives the dip correctly independent of source/receiver‐group offset.