3-D elastic wave propagation for a VSP geometry

We present a method for a solution of the three dimensional elastic wave equation for VSP geometry. This solution is operated on a three dimensional cylindrical grid using the multi-domain approach. Discretization of the wave-field is carried out on a grid of r 6 and z, where r is the distance from the center, 8 is the angular angle and z is depth. Spatial derivatives are performed using the Chebychev expansion along the radial direction, and using the Fourier expansion along the angular and the vertical direction. We combine the equations of conservation of momentum with the stress-strain relation to yield a system of nine equations for the displacements and for the stresses. This system is resorted to a first order system which includes the variables that are needed for boundary conditions construction and for domain decomposition. Boundary conditions and patching of grids are constructed by the use of the characteristic variables of the wave equation. The numerical algorithm is tested on the Cray YMP super computer.