Solution steering with space-variant filters

INTRODUCTION

When attempting to do inversion we are constantly confronted with the problem of slow convergence. Claerbout and Nichols (1994) suggested using a preconditioner to speed up convergence. Unfortunately it is often difficult to find an appropriate preconditioner and/or the preconditioner is so computationally expensive that it negates the savings gained by reducing the number of iterations (Claerbout, 1994). Claerbout (1997) proposed designing helicon-style operators to provide a method to find stable inverses, and potentially, appropriate preconditioners (Fomel et al., 1997; Fomel, 1997).

In addition, geophysical problems are often under-determined, requiring some type of regularization. Unfortunately the simplest, and most common, regularization techniques tend to create isotropic features when we would often prefer solutions that follow trends. This problem is especially prevalent in velocity estimation. The result obtained through many inversion schemes produce a velocity structure that geologists (whose insights are hard to encode into the regression equations) find unreasonable (Etgen, 1997). Fortunately, there are often other sources of information that can be encoded into the regularization operator that allow the inversion to be guided towards a more appealing result. For example, in the case of velocity estimation, reflector dips might be appropriate.

We create small, space-variant, steering filters from dip or other a priori information. We use the inverse of these filters to form a preconditioner which acts as our regularization operator. We show this methodology applied to three different types of problems. In the first set of examples we interpolate well-log information using reflector dip as the basis for our steering filters. For the second set of examples we do a more traditional seismic data interpolation problem. Starting from a shot gather with a portion of the data missing. We use a velocity function to create hyperbolic paths, which in turn are used to construct steering filters. In the final example we show some preliminary results of using steering filters in conjunction with a tomographic operator to create velocity models which both satisfy the data and are geologically reasonable.

Solution steering with space-variant filters