Exact travel time calculations for simple three-dimensional Earth
models in seismic exploration using computer algebra
Computers & Geosciences, 28 (3):
327--336 (April 2002)DOI: 10.1016/S0098-3004(01)00042-5
Abstract
In this paper we have used computer algebra to trace rays through
relatively simple three-dimensional models applicable to seismic
exploration and have calculated travel times of the rays for given
values of ray source to receiver spacing. Using standard techniques,
analytical expressions which give travel time in terms of source
receiver spacing or offset can only be derived for a single layer
or where non-physical assumptions are made. The symbolic mathematics
system, Maple, is used to generate parametric equations which give
offset and travel time as analytical expressions with the parametric
variable being the initial direction cosines of the ray as it leaves
the source. These equations contain symbolic constants such as interval
velocities and layer dips and depths which define the layering of
the model. The parametric equations may be solved after assigning
numerical values to the model parameters, using the internal functionality
of Maple. Faster solutions can be obtained by exporting the parametric
equations as C subroutines and using a fast numerical method. In
order to make the solution to the equations rapid, some elements
of the Jacobian matrix, being the derivatives of the ray position
with respect to the initial direction cosines are also calculated
as analytical expressions and output as C subroutines by Maple. These
are used in a 3D Newton's method to converge rapidly to a solution
which gives the initial direction cosines for a given value of offset.
The direction cosines are substituted into the subroutine derived
from the analytical expression for travel time. The subroutines for
each ray path are used in separate programs and the solutions for
each ray path and the construction of shot records and CMP gathers
are managed through a simple graphics interface. Solutions for the
travel times are fast and inherently accurate, limited only by computer
precision. Current limitations of computer power confine the solutions
to a model comprising three layers with different velocities and
dips but a number of complex ray paths can be traced including surface
multiple reflections, and peg-leg and inter-bed multiples.