Data dependent parameterization

As inversion problems are generally under-determined, they should be stabilized. Stabilization methods can be divided into two groups:

• Regularization of optimization;
• Adjustment of parameterization;

The first method can be performed by both regularization criteria (e.g. damping of the least squares method, or truncated SVD) and model regularization (e.g. including a priori information or conditional covariance matrices in the optimization). A drawback of this method, however, is that the problem of over-parameterization is still not solved, which might cause problems when the inversion problem is regularized. The second method, the adjustment of parameterization, can for example be done by using a global parameterization. However, as mentioned before, this parameterization constrains the possible range of solutions, and incorrect initial parameterization can lead to slow or non-convergence.

In the inversion of focusing operators an attempt is made to combine both stabilizing methods; the adjustment of parameterization is based on the covariance after optimization; the a posteriori covariance. Model parameters that have a high variance are removed, and in regions containing low variance parameters, extra parameters can be added. In this way, the inversion problem is well determined and over-parameterization is avoided. Moreover, all the available information within the data will be translated to the model, as the parameterization adapts to the data.

If the data is used to determine the parameterization, as a consequence the obtained parameterization tells something about the data. A coarse parameterization exposes for example that more focus points should positioned in that region (of course this is only possible if data related to the focus point is available). On the other hand, when updated focus points are positioned very close to each other, it might be wise to remove/shift them, as they will only duplicate information.

 

flowap Figure 5 Strategy flowchart for inversion if focusing operators with data dependent parameterization. The extra steps w.r.t. Figure 3 are underlined.