** Next:** Regularization of the filter
** Up:** Estimation of nonstationary PEFs
** Previous:** Definitions

When PEFs are estimated, the matrix is unknown. If is the data vector from which we want to estimate the filters, we
minimize the vector as follows:
| |
(18) |

which can be rewritten
| |
(19) |

where is the matrix representation of the non-stationary
convolution or combination with the input vector .The transition between equations (18) and (19)
is not simple. In particular, the shape of the matrix is
quite different if we are doing non-stationary convolution or
combination. For the convolution, we have
| |
(20) |

We see that for the convolution case, the matrices
are diagonal operators, translating the need for one filter to be
applied to one input point. The size of the matrix is
where *nf* is the number of filter
coefficients.
Now, for the combination, we have
| |
(21) |

We see that for the combination case, the matrices
are row operators, translating the need for one filter to be
constant for one output point. The size of is equal to
the size of .For the vector in equation (19) we have
| |
(22) |

where *nf* is the number of coefficients per filter. This definition
of is independent of .We might want to have one filter common to different input or
output points instead of one filter per point. In that case, the
matrix is obtained by adding successive matrices
depending on how many points have a similar filter.
Note that in the stationary case, for both the convolution and the
combination case we have
and
| |
(23) |

Therefore, for the matrix , we have to add all the matrices together. If we take advantage of the
special structure of for the convolution and the
combination, we obtain for the stationary case
| |
(24) |

which is the matrix formulation of the stationary convolution.
With the definitions given in equations
(20), (21) and (22), the fitting goal in
equation (19) can be rewritten

| |
(25) |

or
| |
(26) |

Each vector has one constrained coefficient. We can then
rewrite equations (25) and (26) as follows:
| |
(27) |

and
| |
(28) |

with
| |
(29) |

The definition of assumes that the first coefficient of each
filter is known. Note that is equal for both convolution and
combination methods.
Having defined the matrix , we can now rewrite
equation (19) as follows:
| |
(30) |

where the square matrix is
| |
(31) |

The next step consists of estimating the filter coefficients in a
least-squares sense.

** Next:** Regularization of the filter
** Up:** Estimation of nonstationary PEFs
** Previous:** Definitions
Stanford Exploration Project

7/8/2003