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A standard formulation for calculating PEFs from known data is
to solve a linear least-squares problem like

| |
(1) |

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 ).
When there are many coefficients, as when PEFs are spread
densely on the data grid, it makes sense to add damping
equations and/or precondition the problem.
Inserting the preconditioned variable (where
is a somewhat arbitrary smoother) for and
adding the also somewhat arbitrary roughener to regularize the model, gives a formulation like

| |
(2) |

| (3) |

In many cases we can set and just use equation goodleak2,
being careful not to let it go for too many iterations.
We still have to define (or ).

** Next:** Radial smoothing
** Up:** Crawley: Nonstationary filtering
** Previous:** INTRODUCTION
Stanford Exploration Project

4/27/2000