Although the eigenvalue decomposition is unconditionally stable and precise, the N3-dependence of the computation time makes it prohibitively expensive. To make the method affordable, I have derived a scheme that uses a time propagator operator to find the solution at time ,using only the solution at time t. The basic principle is the same used by Dablain (1986) to derive a fourth order in time, explicit, finite difference scheme. Contrary to the method presented here, Dablain's scheme used two time-step solutions (and ) in order to find .
A recursive solution in the time domain can be obtained with the use of the following relations:
in the Taylor expansion of around
Combining these two equations, we obtain the time-propagation equation
where the forward time-propagation operator has the following form:
Figure shows one frame of the horizontal displacement field, for the same source and media used to generate the synthetic data of Figure (but with a finer grid spacing, and at a different time frame). This algorithm was implemented in the Connection Machine C2 with parellization in both spatial axes.
Stability and energy conservation associated with the truncation of the Taylor expansion in equation (8) are discussed in Appendix B.