Shaping regularization in geophysical estimation problems

Conclusions

Shaping regularization is a new general method for imposing regularization constraints in estimation problems. The main idea of shaping regularization is to recognize shaping (mapping to the space of acceptable functions) as a fundamental operation and to incorporate it into iterative inversion.

There is an important difference between shaping regularization and conventional (Tikhonov's) regularization from the user prospective. Instead of trying to find and specify an appropriate regularization operator, the user of the shaping regularization algorithm specifies a shaping operator, which is often easier to design. Shaping operators can be defined following a triangle construction from local predictions or by combining elementary shapers.

I have shown two simple illustrations of shaping applications. The examples demonstrate a typical behavior of the method: enforced model compliance to the specified shape at each iteration and, in many cases, fast iterative convergence of the conjugate gradient iteration. The model compliance behavior follows from the fact that shaping enters directly into the iterative process and provides an explicit control on the shape of the estimated model.

Shaping regularization in geophysical estimation problems