Next: Sampling in time Up: Prediction error filters Previous: Scale invariance and filling

## Implementation

There are two steps. The first calculates a PEF and the second calculates missing data values. Both are linear least squares problems, which I solve using conjugate gradients.

The first step we can write
 (18)
where is a vector containing the PEF coefficients, is a filter coefficient selector matrix, and denotes convolution with the input data. The coefficient selector is like an identity matrix, with a zero on the diagonal placed to prevent the fixed 1 in the zero lag of the PEF from changing. The is a vector that holds the initial value of the residual, .If the unknown filter coefficients are given initial values of zero, then contains a copy of the input data. makes up for the fact that the 1 in the zero lag of the filter is not included in the convolution (it is knocked out by ).

The second step is almost the same, except we solve for data values rather than filter coefficients, so knowns and unknowns are reversed. We use the entire filter, not leaving out the fixed 1, so the is gone, but we need a different selector matrix, this time to prevent the originally recorded data from changing. Split the data into known and unknown parts with an unknown data selector matrix and a known data selector matrix .Between the two of them they select every data sample once: .Now to fill in the missing data values, solve
 (19) (20) (21)
now denotes the convolution operator, and the vector holds the data, with the selector preventing any change to the known, originally recorded data values. The in interp is calculated just like the in pefest, except that now the adjustable filter coefficients are presumably not zero. Where before wound up holding a copy of the data (assuming the adjustable filter coefficients start out zero), here it holds the initial prediction error .

There is some wasted effort implied here. There is no sense, for instance, in actually running the filter over all those zero-valued missing traces in the PEF calculation step, so they can be left out.

Next: Sampling in time Up: Prediction error filters Previous: Scale invariance and filling
Stanford Exploration Project
1/18/2001