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## Computation of the gradient vector

The gradient vector defined in equation (5) is the sum of traveltime residual term and condition term . The computation of the condition term is straightforward because matrix is given. In principle, we can compute the traveltime residual term by first computing matrix , and then performing the matrix multiplication. However, matrix has elements that are functions of r; hence, both its calculation and storage are of high cost. If we can compute the vector directly using a finite difference method, we can considerably reduce the cost in computation time and storage.

If we solve the first-order linear PDE (9) using the method of characteristics, we find that
 (10)
where and are the coordinates of the ray that starts from the number i source and ends at the receiver at depth r, and s is the arc-length of the ray. Clearly, the elements of matrix are the integrations of the basis functions along rays. The component of the gradient vector can be computed as follows:
 (11)
where J is the Jacobian of the coordinate transformation:
 (12)
The last step of equation (11) is an area integration that is easy to implement. Appendix A shows that function satisfies the first-order linear PDE, as follows:
 (13)
with the initial condition that function is equal to the receiver depth at each receiver location. We can use a finite difference algorithm, similar to the one used for traveltime calculation, to solve equation (13). The algorithm should start from receiver locations, and extrapolate function in the opposite directions of the wave propagations, towards the source.

Next: Computation of the conjugate Up: FINITE DIFFERENCE METHODS Previous: Computation of matrix
Stanford Exploration Project
12/18/1997