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Using the method of characteristics, we can derive
a system of ordinary differential equations that define the
ray trajectories. To do so, we need to
transform equation (9) to the
following form:
| |
(14) |
or
| |
(15) |
where and . According to
the classic rules of mathematical physics (Courant, 1966), the solutions of this
kinematic equation can be obtained from the system of ordinary differential equations
| |
|
| (16) |
where s is a running parameter along the rays, related to the traveltime t as follows:
with
| |
|
| (17) |
Using equation (9), we obtain
| |
(18) |
| |
(19) |
| |
|
| (20) |
| |
|
| (21) |
and
where
and and , and
the same holds for and .
To trace rays, we must first
identify the initial values x0, , px0, and
. The variables x0 and
describe the source position, and px0 and are extracted from the initial
angle of propagation. Note that, from equation (9),
because =0 at the source position (z=0).
The raytracing system of equations (18-21)
describes the ray-theoretical aspect of wave propagation in the
-domain, and can
be used as an alternative to the eikonal equation. Numerical solutions of the raytracing equations,
as opposed to the eikonal equation, provide multi-arrival traveltimes and amplitudes.
In the numerical examples,
we use raytracing to highlight some of the features of the -domain coordinate system.
Next: The X-TAU acoustic wave
Up: VTI processing in inhomogeneous
Previous: Representing depth with VERTICAL
Stanford Exploration Project
9/12/2000