Shaping regularization in geophysical estimation problems

Introduction

A great number of geophysical estimation problems are mathematically ill-posed because they operate with insufficient data (Jackson, 1972). Regularization is a technique for making the estimation problems well-posed by adding indirect constraints on the estimated model (Zhdanov, 2002; Engl et al., 1996). Developed originally by Tikhonov (1963) and others, the method of regularization has become an indispensable part of the inverse problem theory and has found many applications in geophysical problems: traveltime tomography (Clapp et al., 2004; Bube and Langan, 1999), migration velocity analysis (Zhou et al., 2003; Woodward et al., 1998), high-resolution Radon transform (Trad et al., 2003), spectral decomposition (Portniaguine and Castagna, 2004), etc.

While the main goal of inversion is to fit the observed data, Tikhonov's regularization adds another goal of fitting the estimated model to a priorly assumed behavior. The contradiction between the two goals often leads to a slow convergence of iterative estimation algorithms (Harlan, 1995). The speed can be improved considerably by an appropriate model reparameterization or preconditioning (Fomel and Claerbout, 2003). However, the difficult situation of trying to satisfy two contradictory goals simultaneously leads sometimes to an undesirable behavior of the solution at the early iterations of an iterative optimization scheme.

In this paper, I introduce shaping regularization, a new general method of imposing regularization constraints. A shaping operator provides an explicit mapping of the model to the space of acceptable models. The operator is embedded in an iterative optimization scheme (the conjugate-gradient algorithm) and allows for a better control on the estimation result. Shaping into the space of smooth functions can be accomplished with efficient lowpass filtering. Depending on the desirable result, it is also possible to shape the model into a piecewise-smooth function, a function following geological structure, or a function representable in a predefined basis. I illustrate the shaping concept with simple numerical experiments of data interpolation and seismic velocity estimation.

Shaping regularization in geophysical estimation problems