Using the rapid expansion method for accurate time-stepping in modeling and reverse-time migration

Reverse-time migration is based on seismic forward modeling algorithms, where spatial derivatives usually are calculated by finite differences or by the Fourier method. Time integration in general is done by finite-difference time stepping of low orders. If the spatial derivatives are calculated by high-order methods and time stepping is based on low-order methods, there is an imbalance that might require that the time-step size needs to be very small to avoid numerical dispersion. As a result, computing times increase. Using the rapid expansion method (REM) avoids numerical dispersion if the number of expansion terms is chosen properly. Comparisons with analytical solutions show that the REM is preferable, especially at larger propagation times. For reverse-time migration, the REM needs to be applied in a time-stepping manner. This is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time. This is not appropriate for reverse-time migration where the sources have different time histories. In reverse-time migration, it might be desirable to use the Poynting vector information to estimate opening angles to improve the quality of the image. In the solution of the wave equation, this requires that one calculates not only the pressure wavefield but also its time derivative. The rapid expansion method can be extended easily to provide this time derivative with negligible extra cost.