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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