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I call the convolution or combination operator with a bank of
non-stationary filters. For the non-stationary convolution, the
filters are in the column of (one filter corresponds to one
point in the input space) whereas for the non-stationary
combination, the filters lie in the rows of (one filter corresponds
to one point in the output space). For the convolution matrix,
I define *a*_{i,j} as the *i*^{th} coefficient of the filter for the *j*^{th}
data point in the input space. For the combination matrix, I define
*a*_{i,j} as the *j*^{th} coefficient of the filter for the *i*^{th}
data point in the output space. Therefore, for the non-stationary convolution we have
| |
(13) |

and for the non-stationary combination we have
| |
(14) |

The size of both matrices is where *n* is the size of
an output vector () and *m* the size of an input vector
() if
| |
(15) |

The helical boundary conditions allow to generalize this
1-D convolution to higher dimensions. We can rewrite equation
(15) for the convolution as follows:
| |
(16) |

where *nf* is the number of filter coefficients, and for the
combination
| |
(17) |

In the next section, I show how the non-stationary PEFs are estimated.

** Next:** Filter estimation
** Up:** Estimation of nonstationary PEFs
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Stanford Exploration Project

7/8/2003