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A system of ray tracing equations

The ray tracing method I have implemented is based on the ray tracing system of first-order partial-differential equations, derived from the Eikonal equation by the method of characteristics (see Cerveny1987 for more details):

 (37)
Here w is the slowness, xi=xi(s) (i=1,2,3) are the coordinates of the ray as a function of s, the arclength along the ray, and pi(s) are the components of the slowness vector .The traveltime t(s) along the ray is given by
 (38)

The slowness vector is perpendicular to the wavefront (), and must satisfy
 (39)

The system of ray equations together with equation  () can be solved by a standard numerical integration method. I use a fourth-order Rungge-Kutta method. It propagates the properties of the ray (xi,pi, and t) over an increment in arclength by combining the information from several Euler-style steps (each involving one evaluation of the right-hand side of equations () and ()), and then uses the information obtained to match a fourth-order Taylor series expansion of the ray variables at the current position.

Figure  shows some of rays traced through the reference model shown in Figure .

ray0
Figure 15
Some of the ray trajectories traced through the reference medium to calculate the operator L.

Next: Model parameterization by spline Up: Calculating the slowness error Previous: Calculating the slowness error
Stanford Exploration Project
2/5/2001