Optimization and geophysical inverse problems Page: 9 of 37
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The boundary conditions are that
uz and u, are continuous everywhere,
Tzz and rz, are continuous everywhere except at the depth z, where f is non-zero,
zz = zz=0 at z =0,
v consists of downward propagating waves as z -> oo.
" A(z), y(z), and p(z) at all depths z > 0,
" f(w, p) at z8,
* v(w,p, z) at arbitrary depths z either within or on the surface of the half space and for
arbitrary frequency w and arbitrary slowness p.
* Uz(wn, pm, 0) and ux(Wn, pm, 0) at z = 0 for the specified frequencies w,, (n = 1, .. ., N)
and for the specified slownesses pm (m = 1,..., M),
" f(p,w) at the specified source depth z,,
" A(z), y(z) and p(z) for all depths z > 0.
The observational data are actually acquired in the time-space domain and are of the form
uz(t, x, z) and ux(t, x, z). A Legendre transformation of the form T= t - px is then applied to
obtain data in the tau-slowness domain, uZ(r,p, z) and ux(r,p, z). Finally, a Fourier transfor-
mation of the data leads to uz(W, p, z) and ux(w, p, z).
Problem 4 - Travel Time Tomography
The fourth problem represents the general class of tomography inversions that have become the
preferred approach to the study of three-dimensional earth structure using seismic body waves.
The problem contains a nonlinearity that enters indirectly through the dependence upon the
ray path. A general discussion of one version of this problem can be found in Trampert (1998).
The travel time along a ray between points xs and x, is given by
t(x,, xs) / (7)
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Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F. et al. Optimization and geophysical inverse problems, report, October 1, 2000; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc901685/m1/9/: accessed January 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.