Multidimensional recursive filter preconditioning in geophysical estimation problems

Conclusions

Regularization is often a necessary part of geophysical estimation. Its goal is to impose additional constraints on the model and to guide the estimation process towards the desired solution.

We have considered two different regularization methods. The first, model-space approach involves a convolution operator that enhances the undesired features in the model. The second, data-space, approach involves inverse filtering (deconvolution) to precondition the model. Although the two approaches lead to the theoretically equivalent results, their behavior in iterative estimation methods is quite different. Using several synthetic and real data examples, we have demonstrated that the second, preconditioning approach is generally preferable because it shows a significantly faster convergence at early iterations.

We suggest a constructive method for preconditioning multidimensional estimation problems using the helix transform. Applying inverse filtering operators constructed this way, we observe a significant (order of magnitude) speed-up in the optimization convergence. Since inverse recursive filtering takes almost the same time as forward convolution, the acceleration translates straightforwardly into computational time savings.

For simple test problems, these savings are hardly noticeable. On the other hand, for large-scale (seismic-exploration-size) problems, the achieved acceleration can have a direct impact on the mere feasibility of iterative least-squares estimation.

Multidimensional recursive filter preconditioning in geophysical estimation problems