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Potential function ray tracing

The next processing step involves tracing geometric coordinate system rays from the generated PF. The goal here is to develop an orthogonal, ray-based coordinate system related to an underlying Cartesian mesh through one-to-one 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 first-order ordinary differential equation through integrating the PF gradient field along the gradient direction,
 (8)
where a(z0,x0,y0) is a known lower integration bound at equipotential , and b(z1,x1,y1) is an unknown upper integration bound located on equipotential surface, , and is the L2 norm of the gradient function. The only unknown parameter is b(z1,x1,y1); hence, Equation (8) is an integral equation with an unknown integration bound. This approach is similar to phase-ray 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