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Estimation of a stationary PEF can be phrased as a least-squares problem, where the
following fitting goal is minimized with respect to an unknown filter **f**:
| |
(1) |

in which **W** is a weight to exclude equations with missing data,
**D** is convolution with the data, **K** constrains the
first filter coefficient to 1, **f** is the unknown filter,
and **d** is a copy of the data.
Once the PEF has been estimated, it can then be used to interpolate missing data
by solving another inverse problem:

| |
(2) |

| |
(3) |

where **L** is a selecting operator that selects the known data within the
interpolated model **m**, **d** is the known data, is a scaling
factor, and **F** represents
convolution with the newly-found PEF. The output model **m** is
referred to as the restored data.
Instead of a restored version of the data, multiple equiprobable
realizations Clapp (2000) of
the missing data can be generated by changing fitting goal (3)
to

| |
(4) |

where instead of desiring the output of the filter convolved with the model to
be zero, we now choose for it to be equal to random noise **n** scaled by a
factor . Multiple realizations of the interpolation can be generated by
using different random numbers for **n** that are identically distributed.
This noise is only introduced where the data are missing, as the
residual of fitting goal (3) will already look like random noise where data
are present.
In geostatistical language, the restored data is very similar to an
E-type, which is the same as an average of multiple realizations (or
simulations) of the result generated by fitting goals
(2) and (4). Dividing random
noise by the PEF, which is the same as solving fitting goal
(4) is equivalent to an unconditional simulation,
where no data constrain the output.

All of the previous theory has been utilized for stationary PEFs. A non-stationary
PEF can be estimated by a fitting goal similar to fitting goal (1),
except that instead of the PEF having a single set of coefficients, the PEF now
has a separate set of coefficients for each data point. Since there are now many
more unknown filter coefficients than known data, a regularization term needs to
be added to constrain the problem:

| |
(5) |

where **f** is the unknown non-stationary PEF and **A** is a regularization
operator that acts over space for each filter coefficient independently, and is typically either a
Laplacian or helix derivative Claerbout (1999). Fitting goals (3) and (4) can be used with
a non-stationary PEF **f** in the place of a stationary PEF.

** Next:** Application to Herringbone data
** Up:** Curry: Non-stationary PEFs and
** Previous:** INTRODUCTION
Stanford Exploration Project

5/3/2005