Finite-Element Ray-Tracing Tomography

Finite-element ray-tracing tomography of a salt dome in the Gulf of Mexico.

Research Objectives

To implement and develop a new ray-tracing method that provides a useful tool to analyze surface-to-surface, surface-to-borehole, and crosshole data. To develop next-generation wave propagation and hydrodynamics codes for computational geophysics.

Computational Approach

A graphic topology method is developed to generate a finite element mesh that not only eliminates the complications posed by complex, unstructured computational grids, but also enhances the efficiency of the ray-tracing method. The rays are accurately traced using the known velocity expression in the current element instead of in-layered or stratified medium. Critical conditions for total refraction and total internal reflection have been developed and applied. Instead of only tracing for one kind of reflection or refraction ray, the method traces the ray according to the critical conditions. Therefore, we obtain not only the first arrival ray, but also the last arrival ray. Obviously, the last arrival ray brings us more information about the deeper subsurface. These calculations are ideally suited to take advantage of developments in parallel computing on the T3E.

Accomplishments

We have tested the ray-tracing method using the SEG/EAGE salt model (in the Gulf of Mexico). The numerical experiment was conducted for the velocity distribution in two dimensions, in which ray tracings with 109 sources and 22 initial angles for each source were used. In the figure, we plotted only the farthest rays that returned to the surface, with the highest velocity representing the salt dome.

The difficulties in ray tracing come from both the geometry and the velocity distribution of the model. The interfaces are complex; some parts are linear, some parts are curved, and some parts are fuzzy; and there are many corners that could result in a singularity when a ray passes them. Velocity distribution is complex with a large contrast (3.16 times), while some parts change continuously.

Significance

The method is applicable regardless of how heterogeneous the medium is, as long as the geological parameters (shape and size) and physical parameters (velocity of propagation and gravity) are known analytically or numerically. The method has been implemented for a two-dimensional case, and it can be easily extended to a three-dimensional case.

Publications

J. Li and R. P. Srivastav, "Computing the singular behavior of solutions of Cauchy singular integral equations with variable coefficients," Appl. Math. Lett. 10, N3, 57-62 (1997).

J. Li and G. Xie, "A new cubic-hole element and its application in resistivity imaging," Proceedings of the International Symposium on Three-Dimensional Electromagnetics (1995), pp. 415-419.

J. Li, K. H. Lee, I. Javandel, and G. Xie, "Nonlinear three-dimensional inverse imaging for direct current data," SEG 65th Annual International Meeting and Exposition, Expanded Abstracts (1995), pp. 250-253.