Solution steering with space-variant filters

Helix transform

The next question is how to choose $\mathbf B$? We have three general requirements:

• it produces relatively smooth (by some criteria) results;
• it spreads information quickly;
• and it is computationally inexpensive.

By defining our operators via the helix method (Claerbout, 1997) we can meet all of these requirements. The helix concept is to transform N-Dimensional operators into 1-D operators to take advantage of the well developed 1-D theory. In this case we utilize our ability to construct stable inverses from simple, causal filters. We can set $\mathbf B$, from equation (4) to

$$\mathbf B = \mathbf A^{-1} ,$$ (5)

where $\mathbf A$ is the roughening operator from fitting goal (1), and $\mathbf B$ is simulated using polynomial division. If $\mathbf A$ is a small roughening operator, $\mathbf B$ is a large smoothing operator without the heavy costs usually associated with larger operators.