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The elements of matrix are the derivatives of the traveltimes
with respect to the slowness model parameters. Let us define

| |
(8) |

Taking the derivatives of both sides of the eikonal equation with respect
to *m*_{j} gives
| |
(9) |

If we assume that has been
computed by solving the eikonal equation, then equation (9)
is a first-order
linear partial differential equation (PDE). Because the traveltime
at the source location is always zero no matter how the slowness model
changes, equation (9) should be solved with the initial condition
that *p*_{ij}(*x*,*z*)=0 at the source location.
Evaluating the function *p*_{ij}(*x*,*z*)
at receiver locations gives the elements of matrix .

** Next:** Computation of the gradient
** Up:** FINITE DIFFERENCE METHODS
** Previous:** The computation of traveltimes
Stanford Exploration Project

12/18/1997