Computer algebra solutions for seismic models

Abstract

The usage of computer algebra is in its infancy in many fields but is likely to increase as the speed and practicability of computer generated analytical solutions improves. This paper discusses the use of computer algebra to solve ray-tracing problems in simple seismic reflection models. The classical two-point problem, of finding ray travel times as a function of source receiver distances for multi-layer models, does not have a complete explicit solution except in the simplest of cases or where unrealistic assumptions are made. Approximation methods have therefore been adopted to obtain numerical solutions. Such methods, however, have the disadvantage of sometimes requiring remodelling and recalculation if parameters are altered. If analytical solutions can be found, they could incorporate parameters of the model as variables and the expressions only need re-evaluation rather than a complete re-determination. The re-evaluation would be based on exact expressions and not on approximations such as are used now. Exact parametric solutions can be derived for three- dimensional models with multiple layers having different acoustic velocities, with reflections and refraction at interfaces according with Snell's Law. These solutions have been generated using subroutines for the Maple computer algebra system which simulate three-dimensional ray tracing through acoustic media and for the interactions with interfaces. The solutions result in the generation of long expressions, which evidently could not be reasonably found by hand calculation. The symbolic solutions give travel time, ray position and ray direction as parametric equations using initial ray direction as the parameters. When a numerical receiver position is entered, travel time and ray-path solutions are returned by solution of the parametric equations. The subroutines are general and could be used to generate mathematical expressions for other simple earth models. For more complex models the analytical solutions may yield very long expressions. Geophysicists will find further applications for computer algebra packages for problems which are currently solved by numerical and approximate methods.