Next: LOCAL WAVEFRONT EXTRAPOLATION Up: Zhang: Local paraxial ray Previous: Introduction

# PARAXIAL RAY APPROXIMATION

Ray tracing is a technique for finding the coordinates of a ray. Suppose, by using this technique, that we find the coordinates of the ray. We can then define a new orthogonal coordinate system (s,n), named ray-centered coordinates. As Figure shows, the coordinate s measures the arc-length along the ray, and the coordinate n is the normal distance from the ray at point s on the ray. Let us denote the relations between Cartesian coordinates and ray-centered coordinates as and . Then, the coordinates of the ray are . The traveltime on any point of the ray can be computed by integrating the slowness function along the ray, as follows:
 (1)
where m(s,n) is the slowness function in ray-centered coordinates.

 rccoor Figure 1 Ray-centered coordinates.

The amplitude calculation is more complicated than the traveltime calculation. Cervený et al. (1977) showed that the amplitude function along the ray is related to a Jacobian function determined by
 (2)
where M(s,0) is the second-order partial derivative of the traveltime with respect to n along the ray and can be found by solving the dynamic ray-tracing equation
 (3)
along the ray.

To compute the traveltimes and amplitudes off but near the ray, we use the paraxial ray approximation. The resulted traveltime is
 (4)
The direction of a ray follows the direction of the traveltime gradient that is determined by
 (5)
where and are unit vectors tangential and normal to the ray at (s,0), respectively. Using the paraxial approximation also yields
 (6)
and
 (7)
The next section explains how these equations are used in the local paraxial ray method.

Next: LOCAL WAVEFRONT EXTRAPOLATION Up: Zhang: Local paraxial ray Previous: Introduction
Stanford Exploration Project
12/18/1997