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A PEF can be estimated by minimizing (in a least-squares sense) the
output of the known data (**d**) convolved (**D**) with a filter
**f** that is unknown except for the first coefficient, that is
constrained to 1 by **K**. This is expressed below, where
**W** is a diagonal weighting operator that is equal to 1
when all filter coefficients lie on known data, and is otherwise.
This is written as:
| |
(1) |

When the data are sparsely sampled, **W** will be zero everywhere,
since there are not enough contiguous data to estimate a PEF. An
example of this is shown elsewhere in this report
Curry (2003). More fitting equations can be added to
this regression by convolving a single filter on multiple copies of
the data that have been rescaled to different grid sizes
Curry and Brown (2001). This can be written as
| |
(2) |

where **W** is now a much larger diagonal weight for all of the
copies of data (**d_i**),
**D_i** represents convolution with **d_i**, and **K** and **f** are the same as in
fitting goal (1).
Often the covariance of the data is
not stationary, so it cannot be adequately described by a single
filter. Non-stationary PEFs may be used to overcome this
limitation. These filters vary with position, so that a filter that
looked like `f(ia)` now looks like `f(ia,id)`. These
filters can be estimated in an fitting goal that looks identical to fitting goal
(1), except that **W**, **K**, **D**, and **f** are all now
non-stationary counterparts to those in fitting goal (1).
The details of these changes are documented elsewhere
Guitton (2003).

Since this non-stationary filter is now likely under-determined due to
a large increase in the number of filter coefficients, a
regularization fitting goal,

| |
(3) |

must also be added, where is a regularization operator that
roughens common filter coefficients in space. This improves the
stability of the PEF, and insures that it will vary smoothly
in space. In order to reduce the number of filter coefficients from
to something more manageable, the filter
coefficients are taken to be constant over a small spatial area, so
that the number of coefficients reduces to ,where *np* is the size of this small area, known as a micropatch
Crawley (2000).
Non-stationary PEF estimation can also be performed on sparse data
Curry (2002), in a multi-scale fashion similar to that
used in fitting goal (2). The only changes to
fitting goals (1) and (3)
are that fitting goal (1) changes to
| |
(4) |

where **W**, **K**, **D**, and **f** are all
now nonstationary versions of those in fitting goal (2). The only new term
in this equation is **P_i**, that sub-samples the non-stationary PEF from to .

** Next:** Size of the parameter
** Up:** Curry: Parameter optimization for
** Previous:** INTRODUCTION
Stanford Exploration Project

7/8/2003