Solution steering with space-variant filters

Regularization

In general, geophysical problems are ill-posed. To obtain pleasing results we impose some type of regularization criteria such as diagonal scaling, limiting solutions to large singular values (Clapp and Biondi, 1995), or minimizing different solution norms (Nichols, 1994). The typical SEP approach is to minimize the power out of a regularization operator ($\mathbf A$) applied to the model ($\mathbf m$), described by the fitting goal

$$0 \approx \mathbf A \mathbf m .$$ (1)

Where $\mathbf A$'s spectrum will be the inverse of $\mathbf m$, so to produce a smooth $\mathbf m$, we need a rough $\mathbf A$(Claerbout, 1994)). The regularization operator can take many forms, in order of increasing complexity:

Laplacian operator ($\nabla^2$)

The symmetric nature of the Laplacian leads to isotropic smoothing of the image.

Steering filters

Simple plane wave annihilation filters which tend to orient the data in some preferential direction, chosen a priori. These filters can be simple two point filters, Figure 1, to larger filters that sacrifice compactness for more precise dip annihilation.

steering Figure 1. An example of steering filter. In this case preference is given to slopes at 45 degrees.

Prediction Error Filters (PEF)

Like steering filters apply a preferential smoothing direction, but are not limited to a single dip and determine their smoothing directions from the known data (Schwab, 1997).

Solution steering with space-variant filters