INVERSE MODELING

When solving the inverse problem, the goal is to estimate two different sets of coupled unknowns: the model parameters and the ray paths. The usual way to decouple them is by invoking Fermat's principle, which ``justifies'' the trick of assuming one to estimate the other in an iterative fashion, as long as the magnitude of the changes from one step to the next are kept small.

Once the ray paths have been estimated, the system of nonlinear equations (5) needs to be solved in order to find a new model where rays are going to be traced again. One way to do this is as a sequence of linearized steps starting from a given initial model $\mbox{\boldmath$m_0$}$. The first step is to approximate (5) by its first order Taylor series expansion centered about a given model $\mbox{\boldmath$m_0$}$:

$$\begin{eqnarray} t_i( \mbox{\boldmath$m$}) & \approx & t_i( \mbox{\boldmath$m_0$... ...ystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}_{in} (m_n - m_{0n}),\end{eqnarray}$$ (7)

where the elements of the Jacobian ${\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}_{in}$ are

$${\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}}_{in} \... ...n} \right\vert _{\mbox{\boldmath$m$} = \mbox{\boldmath$m_0$}}.$$

The explicit form of these derivatives is given in Appendix B.

If we assume that $t_i ( \mbox {\boldmath$m$})$represents one component of the vector $\mbox{\boldmath$t$}$ of measured traveltimes, we can compute the perturbations $\Delta \mbox{\boldmath$m$}_n = (m_n - m_{0n})$ once the traveltimes in the reference model $\mbox{\boldmath$m_0$}$has been calculated. The perturbation $\Delta \mbox{\boldmath$m$} = ( \mbox {\boldmath$m$} - \mbox{\boldmath$m_0$})$is the solution of the following system of equations

$$\displaystyle \mathop{\mbox{\bf J}}_{\mbox{$\sim$}}\Delta \mbox{\boldmath$m$} \ =\ \Delta \mbox{\boldmath$t$}$$ (8)

where $\Delta \mbox{\boldmath$t$}_i = t_i ( \mbox{\boldmath$m$}) - t_i ( \mbox{\boldmath$m_0$})$.

In practice, only a fraction r of the correction $\Delta \mbox{\boldmath$m$}$ is added to the given model

$$\mbox{\boldmath$m_1$} = ( \mbox {\boldmath$m_0$} + r \Delta \mbox{\boldmath$m$})$$

where r (the step length) is usually small to avoid large changes in the ray paths from one iteration to the next.

The system of linear equations (8) will be solved using the LSQR variant of the conjugate gradients algorithm (Nolet, 1987)