Model Variability

What we choose for $\bf A$ can have a significant affect on our model estimate. In theory $\bf A$ should be a matrix of size nm by nm (where nm is the number of model elements). People use numerous approximations for $\bf A$.Some of the more common

• a Laplacian or some type of symmetric operator,
• a stationary Prediction Error Filter Claerbout (1999),
• a steering filter Clapp (2001a), or
• a non-stationary PEF (NSPEF) Crawley (2000).

The first option makes the assumption that our model is smooth and stationary. The second option still assumes stationarity, but allows for much more sophisticated covariance descriptions. The third option allows for non-stationarity but is only valid when a model has a single dip at each location and requires some a priori knowledge about the model. The last is the best representation of the inverse model covariance matrix. Unfortunately, it requires a field with the same properties as the model in order to be constructed. In addition, it faces stability problems Rickett (2001).

At least the first three methods, and possibly all four (depending on the field we use and the filter description we choose to estimate the NSPEF) are limited to describing second order statistics. In addition we normally try to describe the inverse covariance through a filter with only a few coefficients. In terms of the covariance matrix, we are putting non-zero coefficients along only a few diagonals. Finally,/ all of these approaches assume that the main diagonal of the inverse covariance matrix is a constant.

The problems with these approximations are demonstrated in Figure . The left panel shows measurements of the sea depth for a day. A PEF is estimated at the known locations and used as $\bf A$with fitting goals,

$$\begin{eqnarray} \bf 0&\approx&\bf r_{noise} = \bf d- \bf J\bf m \\ \bf 0&\approx&\bf r_{model} = \epsilon \bf A\bf m, \nonumber\end{eqnarray}$$ (6)

where $\bf d$ is shown in the left panel of Figure  and $\bf J$ is a selector matrix (1 at known locations, 0 at unknown). The center panel of Figure  shows the estimated model. Note how the estimated portion of the model doesn't have the right texture. The variance of the model is not the same as the variance of the data. This is due to the combination of two factors.

 

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Figure 1 The left panel shows the result of recording the sea depth for one day. The center panel shows the interpolation result using a filter estimated from the known portion of the data and then solving (6). Note the difference in variance in the known and unknown portions of the model. The right panel is the filter response of the PEF used for interpolation.

• Our covariance description has a limited range. The right panel of Figure  show the result of applying the inverse PEF (or more correctly $(\bf A^{-1})'\bf A^{-1}$) to a spike in the center of the model. Note how the response tends towards zero.
• Our inverse problem will give us a minimum energy solution, which means it is going to want to fill the residual vectors with as small numbers as possible.

In previous papers Clapp (2000, 2001a), I showed how we can change what the minimum energy solution will be by introducing an initial condition to our residual vector filled with random numbers. In terms of our fitting goals (6) we are replacing the zero vector of our model styling fitting goal with with standard normal noise vector $\bf n$, scaled by some scalar $\sigma_{m}$,

$$\begin{eqnarray} \bf 0&\approx&\bf r_{noise} = \bf d- \bf J\bf m \\ \sigma_{m} \bf n &\approx&\bf r_{model} = \epsilon \bf A\bf m. \nonumber\end{eqnarray}$$ (7)

For the special case of missing data problems, where $\bf L$ is simply a masking operator $\bf J$ delineating known and unknown points, Claerbout (1999) showed how $\sigma_{m}$ can be approximated by first estimating the model through the fitting goals in (6), then by solving  

$$\sigma_{m} \quad = \quad { \bold 1' \bold J \bf r_{model}^2 \over \bold 1' \bold J \bold 1 } ,$$ (8)

where $\bf 1$ is a vector composed of 1s. Figure  shows the model using three different $\bf n$ vectors. Note how the variance of the model is now similar to the variance of the data. A good check of this is whether the original recording path can be seen.

 

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Figure 2 Three realizations of the interpolation problem using fitting goals (8).

Using equation (8) to estimate $\sigma_{m}$ makes the assumption that our covariance description is correct. When our assumptions on stationarity are incorrect or filter shape is insufficient to correctly describe the covariance, this approach will be ineffective.