# Geophysics

- GeoRef, Copyright 2005, American Geological Institute. Reference includes data supplied by Society of Exploration Geophysicists, Tulsa, OK, United States

## Abstract

The traveltimes for reflected waves from plane, dipping layer interfaces for split-spread arrays and CGP gathers are determined by a computationally efficient method. The computational efficiency is obtained by (1) interpolating the traveltimes for particular source-receiver distances from least-squares curves fitted to traveltime data rather than by using iterative ray-tracing techniques, and (2) using the traveltime curves for a fixed source to determine those for other source locations and for CGP gathers over the same dipping layer interfaces. The additional computations necessary in this latter case are minimal when this approach is used. The traveltime curves for other source locations and CGP gathers are obtained by taking advantage of the fact that the traveltimes and travel distances for parallel rays are simply related for plane, dipping layer interfaces.For the fixed source array, standard deviations between 0.95 msec for the fourth interface (in a four layer model) and 9.5 msec for the first interface were obtained when the traveltimes t(x) were fitted to a second order curve: t(x) = a _{0} + a _{1} x + a _{2} x ^{2} . These standard deviations decreased to 0.81 and 3.5 msec, respectively, when a third order curve t(x) = a _{0} + a _{1} x + a _{2} x ^{2} + a _{3} x ^{3} was used. The standard deviations became 2.97 and 0 msec, respectively, for a second order curve in t ^{2} (x): t ^{2} (x) = a _{0} + a _{1} x + a _{2} x ^{2} ; and 0.95 and 0 msec, respectively, for the third order curve: t ^{2} (x) = a _{0} + a _{1} x + a _{2} x ^{2} + a _{3} x ^{3} . For a CGP gather over the same layers, the standard deviations were 0.84 and 0 msec for the fourth and first interfaces, respectively, when the traveltime data were fitted to a first order curve in t ^{2} (y ^{2} ): t ^{2} (y ^{2} ) = a _{0} + a _{1} y ^{2} . These standard deviations became 0.10 and 0 msec, respectively, when the curve fitted was: t ^{2} (y ^{2} ) = a _{0} + a _{1} y ^{2} + a _{2} y ^{4} .Errors in the traveltimes for a fitted curve, t ^{2} (y ^{2} ) = a _{0} + a _{1} y ^{2} , in a CGP gather for the third interface at a depth of about 5000 ft were less than 0.5 msec for source-receiver separations up to 5000 ft. For the same interface, errors were much less than 0.1 msec over the same separations for the fitted curve: t ^{2} (y ^{2} ) = a _{0} + a _{1} y ^{2} + a _{2} y ^{4} .