Open AccessThe perturbed traveltime equation and the adjoint‐states gradient of the traveltime error
Author(s) -
Mulder W. A.
Publication year - 2006
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.2006.02932.x
Subject(s) - adjoint equation , derivative (finance) , tracing , mathematical analysis , inverse theory , mathematics , gradient method , fréchet derivative , geology , algorithm , computer science , partial differential equation , oceanography , deformation (meteorology) , financial economics , banach space , economics , operating system
SUMMARY Traveltime tomography based on ray tracing requires the gradient or derivative of the traveltime error with respect to the velocity model parameters. This gradient can be computed by determining the Fréchet derivatives and solving a large sparse linear system. The perturbed traveltime equation offers an alternative way. A third, formal approach is the adjoint‐state method. Here it is shown that the gradient obtained by the adjoint‐state method is the same as the perturbed traveltime equation.