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1 Seismological
Laboratory
California Institute of Technology
Pasadena, California
91125
We derive the adjoint equations associated with the calculation of
Fréchet derivatives for tomographic inversions based upon a Lagrange multiplier
method. The Fréchet derivative of an objective function
(m), where m denotes the
Earth model, may be written in the generic form 
= 
Km(x) ln m(x)
d3x, where
ln m = m/m denotes the relative model
perturbation and Km the associated 3D
sensitivity or Fréchet kernel. Complications due to artificial absorbing
boundaries
for regional simulations as well as finite sources are accommodated. We
construct
the 3D finite-frequency "banana-doughnut" kernel Km by
simultaneously computing
the so-called "adjoint" wave field forward in time and reconstructing the
regular
wave field backward in time. The adjoint wave field is produced by using time-
reversed signals at the receivers as fictitious, simultaneous sources, while the
regular
wave field is reconstructed on the fly by propagating the last frame of the wave
field,
saved by a previous forward simulation, backward in time. The approach is based
on the spectral-element method, and only two simulations are needed to produce
the
3D finite-frequency sensitivity kernels. The method is applied to 1D and 3D
regional
models. Various 3D shear- and compressional-wave sensitivity kernels are
presented
for different regional body- and surface-wave arrivals in the seismograms. These
kernels illustrate the sensitivity of the observations to the structural
parameters and
form the basis of fully 3D tomographic inversions.
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