Regularization of the inversion

In order to compute stable solutions to ill-conditioned systems it is often necessary to apply regularization methods. Claerbout 1997 writes: `` In geophysical fitting we generally have two goals, the first being data ``fitting'' and the second being the ``damping'' or regularization goal''. One way to achieve such objective is to augment the problem with a second regression that adds assumptions about the model (e.g. roughness, smoothness, curvature, energy in one dip, etc...).

For the data-space inverse, one now solves the problem:

$$\begin{eqnarray} 0 \quad\approx\quad \bold r_d &=& \bold A \; {\bold {\hat{d}}} ... ...d}} &=& \lambda \; \bold P \; {\bold {\hat{d}}} \EQNLABEL{reg-dat}\end{eqnarray}$$ (59)

The solution $\bold {\hat{d}}$ represents an equalized data vector that is unevenly sampled. I chose the penalty operator, $\bold P$, to be the identity matrix. This is the standard Tikhonov regularization where the solution $\bold {\hat {d_{\lambda}}}$ solves the problem:

$$\bold {\hat {d_{\lambda}}} = \bold L^T {({\bold {L L^T} + \lambda^{2} \bold I})}^{-1} \bold d \EQNLABEL{reg-mod}$$ (60)

In the solution above, $\bf I$ is the identity operator and the damping parameter ${\lambda}$is an estimate of the smallest resolved singular value of $\bold L$.

Similar to the data-space solution, to regularize the model-space inverse one seeks a solution to the system of regressions:

$$\begin{eqnarray} 0 \quad\approx\quad \bold r_d &=& \bold L \; \bold m \; - \; \b... ...\\ 0 \quad\approx\quad \bold r_m &=& \lambda \; \bold P \; \bold m\end{eqnarray}$$ (61) (62)

Since the model $\bold m$ is regularly sampled, I chose $\bold P$ to be the Laplacian operator, which represents differentiation in the midpoint-space. The parameter ${\lambda}$ controls the smoothness of the solution and is again an estimate of the smallest resolved singular value of $\bold L$.At the time of writing the dissertation, I didn't investigate a robust strategy for estimating ${\lambda}$. However, by processing a single frequency (e.g the dominant frequency of the survey), I was able to iteratively guess a good estimate for ${\lambda}$. Ideally, one might assume a different value for ${\lambda}$ should be used for each frequency inversion. Results showed that a single good estimate of ${\lambda}$ produced a reasonably smooth solution while still preserving the high frequency components of the reflectivity function.