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One approach to building a linear finite-frequency traveltime operator
is to apply the first-order Born approximation, to obtain a linear
relationship between slowness perturbation, , and wavefield
perturbation, ,
| |
(5) |

The Born operator, , is a discrete implementation of
equation (16), which is described in the Appendix.
Traveltime perturbations may then be calculated from the wavefield
perturbation through a (linear) picking operator, , such that

| |
(6) |

where is a (linearized) picking operator, and a function
of the background wavefield, *U*_{0}.
Cross-correlating the total wavefield, *U*(*t*), with *U*_{0}(*t*), provides
a way of measuring their relative
time-shift, . Marquering et al. (1999) uses
this to provide the following explicit linear relationship between
and ,

| |
(7) |

where dots denote differentiation with respect to *t*, and *t*_{1} and
*t*_{2} define a temporal window around the event of interest.
Equation (7) is only valid for small time-shifts,
.

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Stanford Exploration Project

4/27/2000