Time Evolution of the Wave Equation using the Rapid Expansion Method for Highly Accurate RTM

Starting from the spatially variable acoustic wave equation we show how to produce the future time response of the seismic wave field in "one step". This method is called the Rapid Expansion Method, REM, and it uses Chebyshev polynomials to compute the time response. We use the pseudo spectral method for the spatial derivatives and the REM method to get highly accurate wave field snap shots in space and time. We then develop a recursive solution for arbitrary time steps for seismic modelling and RTM. We develop each method and show their relation to previous work and illustrate them with numerical examples of subsalt modeling and imaging.