Elastic 3-D forward modeling by the Fourier method

We present an elastic forward modeling algorithm based on the Fourier method. This modeling is capable of handling arbitrary velocity variation in both the horizontal and vertical direction. Although from a mathematical viewpoint 3-D modeling is not very different from 2-D modeling, from a computational viewpoint, however, 3-D modeling is an order of magnitude more difficult. For example, modeling of a region of size 5 km in all spatial dimensions with a frequency band reaching 40 Hz, requires a grid size of at least 250 × 250 × 250. The total amount of computer storage required for such a problem is over 200 million words. Obviously problems of this magnitude pose a serious challenge to current computer technology. The elastic modeling algorithm is first tested against simple problems for which the results can be readily understood. In a later stage more complicated problems will be introduced. The input for a typical problem includes the density and the P and S velocities at all grid points as well as a specification of the seismic source. The output can include a variety of displays such as displacement or stress time sections at selected receivers, or snapshots of these variables at fixed times.