Although the eigenvalue decomposition is unconditionally stable
and precise, the *N ^{3}*-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

A recursive solution in the time domain can be obtained with the use of the following relations:

in the Taylor expansion of around(7) | ||

Combining these two equations, we obtain the time-propagation equation

where the forward time-propagation operator has the following form:(8) |

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.

Figure 7

Stability and energy conservation associated with the truncation of the Taylor expansion in equation (8) are discussed in Appendix B.

12/18/1997