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| Traveltime sensitivity kernels: Banana-doughnuts
or just
plain bananas? | |
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Next: Rytov traveltime sensitivity
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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 (O-7), 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, .
Cross-correlating the total wavefield, , with , 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 , and and
define a temporal window around the event of interest.
Equation (7) is only valid for small time-shifts,
.
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|
|
| Traveltime sensitivity kernels: Banana-doughnuts
or just
plain bananas? | |
|
Next: Rytov traveltime sensitivity
Up: Theory
Previous: Theory
2013-03-03