Fast ray tracing method in 3-D structure and its proof of positive definiteness

Based on Fermat’s principle, two-point ray tracing method was studied in three-dimensional structure. By means of first order Taylor’s incomplete series expansion (i.e. no expansion to the length of the ray), a symmetry block tridiagonal matrix equation set was deduced. Further, the positive definiteness of coefficient matrix was discussed, and the positive definiteness was accurately proved in a mathematical way. It assured that the algorithm was well-posed. Associated with iterative method, the solution to ray tracing can be got through step-by-step linearized iteration of the nonlinear problem. An algorithm of the whole path iterative ray tracing method in three-dimensional velocity structure was obtained. This method shows a clear and simple as well as explicit computation formula, which makes ray tracing computation easily applicable in practice. The correction vector is obtained through finding the solution to the positive definite block tridiagonal equation set, which ensures the method is robust convergence. This study offers a new kind of feasible and efficient ray tracing method for three dimensional seismic migration and tomography. Meanwhile, it also provides the prerequisite guarantee to design a fast algorithm.

Motion of photons in a gravitational wave background

Photon motion in a Michelson interferometer is re-analyzed in terms of both geometrical optics and wave optics. The classical paths of the photons in the background of a gravitational wave are derived from the Fermat principle, which is the same as the null geodesics in general relativity. The deformed Maxwell equations and the wave equations of electric fields in the background of a gravitational wave are presented in a fiat-space approximation. Both methods show that even the envelope of the response of an interferometer depends on the frequency of a gravitational wave, but it is almost independent of the frequency of the mirror＇s vibrations.