Rapid expansion method (REM) for time-stepping in reverse time migration (RTM)

We show that the wave equation solution using a conventional finite-difference scheme, derived commonly by the Taylor series approach, can be derived directly from the rapid expansion method (REM). After some mathematical manipulation we consider an analytical approximation for the Bessel function where we assume that the time step is sufficiently small. From this derivation we find that if we consider only the first two Chebyshev polynomials terms in the rapid expansion method we can obtain the second order time finite-difference scheme that is frequently used in more conventional finite-difference implementations. We then show that if we use more terms from the REM we can obtain a more accurate time integration of the wave field. Consequently, we have demonstrated that the REM is more accurate than the usual finite-difference schemes and it provides a wave equation solution which allows us to march in large time steps without numerical dispersion and is numerically stable. We illustrate the method with post and pre stack migration results.