Rapid travel time calculations for simple earth models using computer algebra

Abstract

This paper describes the use of computer algebra in the derivation of analytical expressions for travel times of rays at finite offsets for a 3D earth model comprising a few layers where layer thickness, velocity and dip angles are represented symbolically. Expressions for offset and travel times are derived symbolically in terms of the initial ray directions and parameters of the earth model. These expressions and the derivatives of the offsets with respect to ray direction are converted to C program subroutines. Numerical techniques are then used to find travel times for given values of offset, limited only by computer precision. Since derivatives of offsets with respect to initial ray directions can also be calculated directly, a modified Newton's method was used to give rapid convergence to initial ray directions required for a given value of offset. The travel time is then calculated from the ray directions so derived. This is an alternative to previously used techniques based either on approximations or on purely numerical calculations. Numerical values of model parameters may be readily changed and the calculations repeated for the changed model. Analytical solutions have been derived for a number of multiple and inter-bed ray paths as well as for primary events. Travel time calculations are fast and accurate since they are based on analytical expressions. Shot records, CMP gathers or other geometry can be readily defined. This paper discusses the potential use of these travel time calculations for improving numerical modelling, identifying multiple reflection events, identifying anisotropic effects and determining the variation of reflection point locations in CMP gathers. The potential for, and possible limitation of, extensions to include multi-layer earth models, converted waves and calculation of amplitudes is also discussed.