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

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 with fitting goals,

(6) | ||

Figure 1

- Our covariance description has a limited range. The right panel of Figure show the result of applying the inverse PEF (or more correctly ) 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 , scaled by some scalar ,

(7) | ||

(8) |

Figure 2

Using equation (8) to estimate 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.

7/8/2003