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This section is largely inspired by chapter 7.5 in Claerbout (1992). I
show that the proper way of dealing with amplitude problems with seismic
data for the PEF estimation is to weight the residual and *not* the data
itself.
I will now describe the theory of PEF estimation for a three
coefficients filter from a time series
. Helical boundary conditions
Claerbout (1998) make
this analysis valid for multidimensional PEFs as well.
What we want is to minimize the energy of the residual according to

| |
(1) |

which can be rewritten
| |
(2) |

where is the convolution matrix with , is
a masking operator and the first column of the convolution
matrix . The next step is to find such that
| |
(3) |

is minimum. We can solve this problem with an iterative solver such as
a conjugate-direction method Claerbout and Fomel (2002). The least-squares inverse of
becomes
| |
(4) |

It is common practice to
weight the data before doing any processing in order to correct for
amplitude anomalies. Doing so, the fitting goal in equation
(1)
becomes
| |
(5) |

where are weights for a particular point
of the time series . This is the wrong approach to correct for
amplitude anomalies but it has practical values. First,
equation (5) keeps the Toeplitz
structure of the Hessian in equation (4), allowing
fast computation of . Then, the PEF can be
easily estimated in the Fourier domain. However, inverse theory
teaches us that we should be weighting the residual instead such that
| |
(6) |

with
| |
(7) |

The difference between equations (5) and
(6) is that in the first case, we weight
the data points and that in the second case, we weight the regression
equations. In equation (6), we weight
each row independently and leverage the PEF estimation globally.
With the weighted residual, the Hessian loses its Toeplitz structure
making the estimation of less computationally efficient. In
addition, the estimation of in the Fourier domain becomes
much more difficult.
It might not be clear why one method is better than the other. The
choice of a weighting function can also be tricky. In the next
section, I present guidelines and results on how to choose and what it changes for two signal/noise separation examples.

** Next:** Signal-noise separation with weighted
** Up:** Guitton: Weighted PEF estimation
** Previous:** Introduction
Stanford Exploration Project

7/8/2003