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THE LINEAR SYSTEM

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} (2)
where $\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$ is a matrix whose nature depends on how the model is described, $\Delta 
\mbox{\boldmath$m$}$ is a vector that contains the variations in model parameters with respect to a reference model, and $\Delta \mbox{\boldmath$t$} $ 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} (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} (4)
where the coefficients ai are used to calculate the slowness perturbation $\Delta S$ as a superposition of natural pixels $\phi_i (x,z)$

\begin{displaymath}
\Delta S \ =\ \sum_{i=1}^{M} a_i \phi_i (x,z).\end{displaymath}

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} (5)

The reasons why I will perform the SVD only for these three particular parametrizations are the following:


previous up next print clean
Next: SINGULAR VALUE DECOMPOSITION: APPLICATION Up: Michelena: SVD Previous: SINGULAR VALUE DECOMPOSITION: SHORT
Stanford Exploration Project
12/18/1997