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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