Inverse problems obtain an estimate of a model , given some data and an operator relating the two. We can write our estimate of the model as minimizing the objective function in a least-squares sense,
Bayesian theory tells us Tarantola (1987) that convergence rate and the final quality of the model is improved the closer is to being Independent Identically Distributed (IID). If we include the inverse noise covariance in our inversion our data residual becomes IID,
A regularized inversion problem can be thought of as a more complicated version of (3) with an expanded data vector and an additional covariance operator,
One way to approximate is to think of it as a chain of two operators Clapp (2003a). One operator will describe the two-point covariance of the matrix and one operator that describes the variance .We have a number of options in designing .We can use a Laplacian or some type of symmetric operator, a stationary Prediction Error Filter Claerbout (1999), a steering filter Clapp (2001b), or a non-stationary PEF (NSPEF) Crawley (2000). What we use is based on our a priori knowledge of our noise. The variance component can be thought of simply in terms on how reliable we consider a given component of our data. A simple example of this is the Super Dix Clapp and Biondi (1999) problem where is constructed from stack power.
Another problem with fitting goals (5) is that we produce a single answer, with no information on the variability of different components in the model space. Our single answer is the minimum energy solution. In Clapp (2001b) I showed how, for the interpolation problem, we can produce a range of equi-probable solutions by replacing with a random noise vector .The resulting models all had a more realistic texture than the minimum energy solution because the regularization operator did not fully describe the inverse model covariance.
In Clapp (2003a) I showed, how by replacing with a random noise vector, we could produce a range of equi-probably interval velocity estimates for the Super Dix problem. The Super Dix example was a 1-D problem. The was a derivative operator. The operator was constructed based on the semblance scan. The variance was based on how quickly the semblance fell off from the peak value used to construct the data. The resulting models provided a fuller description of the potential interval velocity models but the 1-D nature of the problem limited its usefulness. A more interesting implementation of the methodology is the migration velocity analysis problem.