MULTIPLE REALIZATIONS REVIEW

Inverse problems obtain an estimate of a model $\bf m$, given some data $\bf d$ and an operator $\bf L$ relating the two. We can write our estimate of the model as minimizing the objective function in a least-squares sense,

$$f(\bf m) = \Vert\bf d- \bf L\bf m\Vert^2 .$$ (1)

We can think of this same minimization in terms of fitting goals as

$$\bf 0\approx \bf r_{} = \bf d- \bf L\bf m,$$ (2)

where $\bf r_{}$ is a residual vector.

Bayesian theory tells us Tarantola (1987) that convergence rate and the final quality of the model is improved the closer $\bf r_{}$ is to being Independent Identically Distributed (IID). If we include the inverse noise covariance $\bf N$ in our inversion our data residual becomes IID,  

$$\bf 0\approx \bf r_{} = \bf N( \bf d- \bf L\bf m) .$$ (3)

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,

$$\begin{eqnarray} \bf 0&\approx&\bf r_{d} = \bf N_{noise} ( \bf d- \bf L\bf m) \n... ...0&\approx&\bf r_{m} =\epsilon \bf N_{model} ( \bf 0- \bf I\bf m) .\end{eqnarray}$$ (4)

In this new formulation, the first expression is the ``data fitting goal'' and the second is the ``model styling goal'', $\bf r_{d}$ is the residual from the data fitting goal, $\bf r_{m}$ is the residual from the model styling goal, $\bf N_{noise}$ is the inverse noise covariance, $\bf N_{model}$ is the inverse model covariance, $\bf I$ is the identity matrix, and $\epsilon$ is a scalar that balances the fitting goals against each other. Normally we think of $\bf N_{{\rm model}}$ as the regularization operator $\bf A$. Simple linear algebra leads to a more standard set of fitting goals:

$$\begin{eqnarray} \bf 0&\approx&\bf r_{d} = \bf N_{{\rm noise}} ( \bf d- \bf L\bf m) \nonumber \\ \bf 0&\approx&\bf r_{m} = \epsilon \bf A\bf m .\end{eqnarray}$$ (5)

A problem with this approach is that we never know the true inverse noise or model covariance and therefore are only capable of applying approximate forms of these matrices.

One way to approximate $\bf \bf N_{{\rm noise}}$ is to think of it as a chain of two operators Clapp (2003a). One operator will describe the two-point covariance of the matrix $\bf \bf N_{{n,c}}$ and one operator that describes the variance $\bf \bf N_{{n,v}}$.We have a number of options in designing $\bf \bf N_{{n,c}}$.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 $\bf \bf N_{{n,v}}$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 $\bf \bf N_{{n,v}}$ 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 $\bf r_{m}$ with a random noise vector $\bf n$.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 $\bf r_{d}$ 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 $\bf \bf N_{{n,c}}$ was a derivative operator. The $\bf \bf N_{{n,v}}$ 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.