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:

$$\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}\Delta \mbox{\boldmath$m$} \ =\ \Delta \mbox{\boldmath$t$},$$ (2)

where $$\mathop{\mbox{\bf J}}_{\mbox{$\sim$}}$$ 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 S_j (McMechan, 1983):

$$\Delta \mbox{\boldmath$m$} \ =\ (\Delta S_1, \Delta S_2, ... , \Delta S_N )^T,$$ (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):

$$\Delta \mbox{\boldmath$m$} \ =\ (a_1, a_2, ... , a_M)^T ,$$ (4)

where the coefficients a_i are used to calculate the slowness perturbation $\Delta S$ as a superposition of natural pixels $\phi_i (x,z)$

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

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):

$$\Delta \mbox{\boldmath$m$} \ =\ (\Delta S_{x_{1}}, \Delta S... ... \Delta S_{z_{1}}, \Delta S_{z_{2}}, ... , \Delta S_{z_{N}})^T.$$ (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).