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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