Investigating equations of motion

So far our research group has focused largely on traveltime and ray trajectories. In my research I will solve for traveltime, ray trajectories, and displacement amplitudes, but the emphasis will be on obtaining new results related to the amplitudes. According to Slawinski (2003), the transport equation for inhomogeneous elastic media is derived from equation (5); it reads thus:

$$\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3\left[\frac{\partial} {\parti... ... \frac{\partial A_k}{\partial x_l}\frac{\partial\psi}{\partial x_j}\right]=0 .$$ (6)

I will review the recent literature for existing methods of solving the transport equation so that amplitude information from such methods can be compared to my Fourier-integral operator solutions and, in at least one test case, to a full numerical solution of equations (5). Only if such existing methods seem inadequate will I devise new methods to solve the transport equation or to improve on existing methods. Otherwise I will focus on solving equations (5).

A new method will involve Fourier-integral operators, which I plan to apply starting in Spring 2004. This procedure should prove fruitful since Fourier-integral operators are applicable to non-elliptic problems such as equations (5) and gracefully handle propagation of singularities, and are a powerful modern mathematical tool for analyzing PDEs. I shall apply Fourier-integral operators directly to the analysis of equations of motion (5) rather than to the transport equation. I hope to show that this analysis extends the results of past analyses of the transport equation with new amplitude (and perhaps other) information that better matches subtleties in real data over known geology, and in a finite-difference or finite-element numerical solution of at least one test model.