In order to examine the results of preconditioned inversion, regularized inversion, and my proposed CIPR, I needed a problem that was easier to understand than that shown in Prucha and Biondi (2002). I am concerned with two issues: frequency content and solutions at every model point. To address the first issue, I chose to make my inversion operator a ``smoother'' that causes the model to have a higher frequency content than the data. This can be expressed as:
where is the data, is the model, and is a smoothing operator that maps the average of 5 vertical points in the model to one point in the data. Since the model should be high frequency, the effects of the preconditioned inversion should be quite obvious.
Given such a simple inversion operator, creating a need for regularization or preconditioning requires that I cause the model created by inversion (fitting goal (1)) to have points that do not have solutions defined by the inversion operator. I chose to do this by introducing a masking operator .The combined operator will now have a null space where .This changes my fitting goal to:
To interpolate the model in the areas affected by the null space, I add a second fitting goal to fitting goal (2):
where the new operator, , is a regularization operator. I have chosen to make a steering filter Clapp et al. (1997); Clapp (2001) generated as described in Prucha et al. (2000, 2001). Briefly, a steering filter consists of dip penalty filters at every model point, meaning that it is a non-stationary roughening operator that acts over short distances. To precondition this problem, I perform a change of variables to replace the model with the preconditioned variable :
Applying this to fitting goals (3) results in a new set of fitting goals:
The inverse of the steering filter () is applied using the helix transform. The inverse operator will be a smoothing operator that will act over a much larger distance than .