Exact Ray-Theory Solutions for Two-Dimensional Velocity Variation

Abstract

This paper presents exact ray-path and travel-time equations using solutions to the eikonal equation for a general class of sound velocity profiles which vary with range as well as depth. In the past, ray-theory for two-dimensional velocity variation has been treated mainly by numerical methods. Exact solutions have been available for only a few profiles of a very simple form. In this paper the velocity profile is expressed in terms of α and β of a transformed coordinate system in which α(z,r) and β(z,r) are functions of the depth, z, and range, r. This transformation and velocity function are such that the transformed eikonal equation reduces to a partial differential equation separable in α and β. The rotation of a velocity profile which depends on depth only is the simplest example of this approach. There are, however, transformations which lead to new velocity profiles not previously considered in the literature. The exact ray-path solutions and an equation for travel time are presented in terms of the α and β variables.