In this work, 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). Then, we show that if we use more terms from the REM one 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 also analyze the behavior of the Chebyshev and Taylor series coefficients to approximate the cosine function which appears in the analytical solution of the wave equation in time. The method is illustrated with pos and pre-stack migration results and it is shown that the REM for the time stepping combined with pseudo spectral operators for the spatial derivatives can be used to obtain numerically stable results with less computational effort than a conventional finite difference time stepping approach for the same level of accuracy.
reverse time migration; finite difference; rapid expansion method
