Research topics

The aim of this research project is to solve equations (5) for a general elasticity tensor. Several methods may be used for comparison, including the following two: (1) full numerical solution of at least one test example and (2) the method of characteristics, giving analytical derivation of an eikonal equation, followed by numerical solution of that eikonal equation. Both the standard method of characteristics and a principal-symbol variant should yield the same characteristics. However, another more original aspect of this research will be its application of Fourier-integral operator (FIO) methods to solving equations (5). Such a solution ideally will provide information not present in the high-frequency ansatz-derived eikonal- and transport-equation solution. This information may throw light on hitherto unexplained features in real data, and thus provide an aid to interpretation of other real data that shows such features. The solution should provide information on amplitude, traveltime, and ray trajectory. The next step is to determine whether the amplitude information shows any improvement over that obtained by solution of the transport equation, when both are compared to a reliable numerical solution of a test example of an inhomogeneous fully-elastic medium.

Fourier-integral operators (FIOs) have been known for about half a century. Some of their advantages are that

1. unlike the related theory of pseudo-differential operators, which combined with FIOs form the field of micro-local analysis, they can be applied to the analysis and solution of non-elliptic problems such as equations (5);
2. they handle propagation of singularities very well, and indeed for the 3-D wave equation singularities propagate according to the strong Huygens' principle, according to the applications section of Saint Raymond (1991);
3. they are a powerful tool for the investigation of partial differential equations in terms of symbols, distributions and the Fourier transform. While they also to some extent give a high-frequency approximation, they may yield some new information about the solution of the equations of motion that is not present in ray-theory solutions of the eikonal and transport equations.