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Regardless of how the model is described, the problem of ray theoretic
traveltime tomography always reduces to the solution of a system of
equations of the form:
| ![\begin{displaymath}
\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}\Delta \mbox{\boldmath$m$} \ =\ \Delta \mbox{\boldmath$t$},\end{displaymath}](img8.gif) |
(2) |
where
is a matrix whose nature depends on how the
model is described,
is a vector that
contains the variations in model parameters with
respect to a reference model, and
is the misfit between real and calculated
traveltimes.
The model can be described in many different ways. However, in
this paper I will only focus on three of them .
The first one will be the conventional
discretization of the medium in cells of constant slowness Sj (McMechan,
1983):
| ![\begin{displaymath}
\Delta \mbox{\boldmath$m$} \ =\
(\Delta S_1, \Delta S_2, ... , \Delta S_N )^T,\end{displaymath}](img11.gif) |
(3) |
where N is the total number of cells.
The second discretization I will consider will be the one based
on natural pixels (Michelena and Harris, 1991):
| ![\begin{displaymath}
\Delta \mbox{\boldmath$m$} \ =\
(a_1, a_2, ... , a_M)^T ,\end{displaymath}](img12.gif) |
(4) |
where the coefficients ai are used to calculate the slowness
perturbation
as a superposition of natural pixels
![\begin{displaymath}
\Delta S \ =\ \sum_{i=1}^{M} a_i \phi_i (x,z).\end{displaymath}](img15.gif)
In the previous expressions, M is the total number of traveltimes, which
are calculated also along natural pixels.
Finally, I will consider the discretization of the model in homogeneous
orthogonal regions with elliptical velocity dependencies
(Michelena and Muir, 1991):
| ![\begin{displaymath}
\Delta \mbox{\boldmath$m$} \ =\
(\Delta S_{x_{1}}, \Delta S...
... \Delta S_{z_{1}}, \Delta S_{z_{2}}, ... , \Delta S_{z_{N}})^T.\end{displaymath}](img16.gif) |
(5) |
The reasons why I will perform the SVD only for these three particular
parametrizations are the following:
- Compare the matrices obtained by discretizing the model
in orthogonal
homogeneous cells with those obtained by discretizing the model in
natural pixels (for the isotropic case).
- Compare the matrices obtained by allowing the model to be either
isotropic or anisotropic (when the model is discretized in
orthogonal homogeneous cells).
Next: SINGULAR VALUE DECOMPOSITION: APPLICATION
Up: Michelena: SVD
Previous: SINGULAR VALUE DECOMPOSITION: SHORT
Stanford Exploration Project
12/18/1997