(26) |

Application of a prediction-error filter removes the predictable information from a dataset, leaving the unpredictable information, that is, the prediction error. A typical use of prediction-error filters is seen in the deconvolution problem, where the predictable parts of a seismic trace, such as the source wavelet and multiples, are removed, leaving the unpredictable reflections.

The condition that contains no predictable information may be expressed in several ways. One method is by minimizing , where is the conjugate transpose of ,by calculating a filter that minimizes .This minimization reduces to have the least energy possible, where the smallest is assumed to contain only unpredictable information.

Another equivalent expression of unpredictability is that the non-zero lags of the normalized autocorrelation are zero, or that

(27) |

The prediction-error filter may also be defined
in the frequency domain using the condition that
the expectation ** E[r_{i} r_{j}] = 0** for .Transforming the autocorrelation into the frequency domain
gives ,where is the complex conjugate of .Since is the convolution of and ,,

(28) |

(29) |

Another way of expressing the unpredictability of is ,that is, the expectation of is the identity matrix. This states that the expectation of the cross-terms of the errors are zero, that is, the errors are uncorrelated. This also states that the variances of the errors have equal weights. To make the matrices factor with an decomposition Strang (1988), the expression needs to be posed as a matrix operation ,with as an upper triangular matrix. To do this, the indices of and are reversed from the usual order in their vector representations. A small example of is

(30) |

Building one small realization of gives

(31) |

Starting from the expression and substituting for gives

(32) |

(33) |

(34) |

(35) |

(36) |

(37) |

Generally, will be invertible and positive definite for real data. There are some special cases where is not invertible and positive definite, for example, when contains a single sine wave. To avoid these problems, the stabilizer is often added to the autocorrelation matrix, where is a small number and is the identity matrix. In the geophysical industry, this is referred to as adding white noise, or whitening, since adding to the autocorrelation matrix is equivalent to adding noise to the data .

The matrix may be obtained from the matrix by Cholesky factorizationStrang (1988), since is symmetric positive definite. Cholesky factorization factors a matrix into ,where looks like

(38) |

Another way of looking at this definition of a prediction-error filter is to consider the filter as a set of weights producing a weighted least-squares solution. Following Strang 1986, the weighted error is , where is to be determined, and is the error of the original system. The best will make

(39) |

(40) |

While I've neglected a number of issues, such as the invertability of ,the finite length of , and the quality of the estimations of the expectations, in practice the explicit solution to () will not be used to calculate a prediction-error filter. Most practical prediction-error filters will be calculated using other, more efficient, methods. A traditional method for calculating a short prediction-error filter using Levinson recursion will be shown in the next section. In this thesis, most of the filters are calculated using a conjugate-gradient technique such as that shown in Claerbout 1992a.

Most prediction-error filters are used as simple filters, and the desired output is just the error from the application of the filter . Another use for these prediction-error filters is to describe some class of information in an inversion, such as signal or noise. In this case, these filters are better described as annihilation filters, since the inversion depends on the condition that the filter applied to some data annihilates, or zeros, a particular class of information to a good approximation. For example, a signal may be characterized by a signal annihilation filter expressed as a matrix , so that which may be expressed as , where is small compared to .A noise may be characterized by a noise annihilation filter , so that .Examples of annihilation filters used to characterize signal and noise will be shown in chapters , , , and .

In this thesis, prediction-error filters will be referred to as annihilation filters when used in an inversion context. While prediction-error filters and annihilation filters are used in different manners, they are calculated in the same way. In spite of the similarities in calculating the filters involved, the use of these filters in simple filtering and in inversion is quite different. Simple filtering, whether one-, two-, or three-dimensional, involves samples that are relatively close to the output point and makes some simplifying assumptions. An important assumption is that the prediction error is not affected by the application of the filter. Inversion requires that the filters describe the data, and the characterization of the data is less local than it is with simple filtering. The assumption that the prediction error is not affected by the filter can be relaxed in inversion, a topic to be further considered in chapters and .

2/9/2001