Summary
The dynamic form of the double square-root (DSR) equation provides a mechanism to extrapolate prestack wavefields by moving the sources and receivers in space with respect to a potential image (scatterer) point. Its classical implementation for imaging often assumes sources and receivers are at the same horizontal surface. Reverse time migration (RTM), as well as common shot migrations in general, through its separate treatment of the sources and receivers, allows for more flexibility in source and receiver configurations. A simple modification to the classical DSR, equation provides such flexibility. Specifically, we define a 7-dimensional prestack wavefield for 3-D media that includes the vertical source and receiver offset. The corresponding dispersion relation can be used to extrapolate such wavefields. However, the cost for such a definition and extrapolation can be prohibitive, considering the high dimensionality of the problem. We reduce the dimensionality by recognizing that the sources and receivers often share the same horizontal plane, and thus, obtain the conventional DSR, formulation. An efficient implementation of DSR in time yields extrapolation speeds that nominally exceed those obtained from reverse time migration. We can also reduce the dimensionality by setting the horizontal offset between the source and receiver at the image point to zero, or using a DSR-like formulation to correct for source-receiver vertical offset or topography.
