Three-dimensional wave propagation using boundary integral equation techniques

The Boundary Integral Equation (BIE) approach for simulating wave propagation in three-dimensional, irregular multilayered, viscoelastic media is formulated. The BIE formulation takes advantage of known wave propagation properties within an individual layer, leaving only the interactions at the layer boundaries to be treated numerically. This essentially reduces the problem by one spatial dimension and represents a concise treatment of the pertinent physics involved. The resulting system of singular boundary integral equations is much smaller than the corresponding system of equations using the Finite Difference or Finite Element approach, but the block diagonal matrices are much more dense. Two methods are presented for dealing with these dense matrix equations. First an approximate Kirchhoff technique is derived in which only local values of the wave field are allowed to interact with the layer boundaries and the propagation through multilayered structures is accomplished by cascading up and down through the stack to get higher order reflections. Since the Kirchhoff approximation is not valid for critical reflections and some diffraction effects, a second and more complete BIE solution technique was developed which iteratively deals with the singular matrix equation from a perturbation point of view with respect to known flat layer solutions. While the Kirchoff algorithm is a fully three dimensional code for any number of layers, the current iterative BIE algorithm solves a more specialized class of problems and planned extensions to the general case are outlined.