Abstract

My research will focus on the mathematics of seismic wave theory in elastic media. In particular, dealing with a three-dimensional medium, I will study the three coupled Cauchy's equations of motion in anisotropic inhomogeneous media using: (1) the standard method of characteristics, (2) the principal-symbol analysis, (3) Fourier-integral operator (FIO) theory, and (4) numerical analysis. I have already begun and will continue to apply the method of characteristics to the equations of motion. I have also begun to apply principal-symbol theory (related to Fourier-integral operators) to the equations of motion. From that I should also derive the same characteristics. Symbols should put the equations in a framework amenable to application of Fourier-integral operators. I will also try to apply Fourier-integral operator theory to simple cases so that FIO results can be tested against other known solutions. Then, I will extend that theory to the inhomogeneous anisotropic case and ideally show that it improves on the results of ray theory. In that general case I expect the FIO theory will reduce the complexity of the equations somewhat. Thus less numerical work will be required to complete the solution than for full numerical analysis of the original equations. That will involve some computer implementation, and comparison to real data over known geology. The FIO modelling results for the inhomogeneous anisotropic case can be checked by various methods, for example, inserting the isotropic elasticity matrix into the algorithm and checking that the solution reduces to the isotropic solution. For at least one test case I will also check the anisotropic inhomogeneous FIO solution against a full numerical solution.