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The next processing step involves tracing geometric coordinate system
rays from the generated PF. The goal here is
to develop an orthogonal, raybased coordinate system related to an
underlying Cartesian mesh through onetoone mappings,
 

 (7) 
 
where is the wavefield extrapolation direction (equivalent
to z in Cartesian), and are the two orthogonal
directions (equivalent to x and y in Cartesian), and J is the
Jacobian of the coordinate system transformation. Recorded wavefield,
, is extrapolated from the acquisition surface
defined by into the subsurface along the rays coordinate system
defined by triplet .
Geometric rays are traced by solving a firstorder ordinary
differential equation through integrating the PF gradient field
along the gradient direction,
 
(8) 
where a(z_{0},x_{0},y_{0}) is a known lower integration bound at
equipotential , and b(z_{1},x_{1},y_{1}) is an unknown upper
integration bound located on equipotential surface, , and
is the L^{2} norm of the gradient function.
The only unknown parameter is b(z_{1},x_{1},y_{1});
hence, Equation (8) is an integral equation with an
unknown integration bound. This approach is similar to phaseray
tracing method described in Shragge and Sava (2004b); however,
in this case the integration step lengths are now unknown. Note also
that the equipotentials of the upper and lower bounding surfaces in
Equation (4) require PF steps of .
The following approach locates unknown integration bound, b, on the
next equipotential:
 1.
 Numerically integrate Equation (8) on the interval
[] where is smaller than the expected step
size, and test to see whether ; if yes goto step 3;
 2.
 Numerically integrate Equation (8) on next interval
and test whether ; if yes goto step 3; if no, repeat step 2 n
times until true;
 3.
 Linearly interpolate between points and
to find the b corresponding to .
A geometric ray is initiated at a particular
on acquisition surface defined by , and computed by
integrating through each successive step until the lower
bounding surface is reached. This procedure is repeated for
all and acquisition points.
Next: Numerical Examples
Up: Theory
Previous: Generating Potential Functions
Stanford Exploration Project
5/3/2005