The prediction-error filter

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 10
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

The prediction-error filter results directly from the prediction filter. We define the prediction-error series as the difference between the true value and the estimated or predicted value . The prediction error at time instant is thus


(10)

If is replaced by n in the above equation, the result is the expression for the prediction error, at the present time n, given by


(11)

The prediction error is a signal that represents the nonpredictable part of . The above equation shows that the filter is


(12)

This filter is called the prediction-error filter or the prediction-error operator. There are zeros in the prediction-error operator that lie between the leading coefficient, namely 1, and the negative prediction-operator coefficients. These zeros constitute the gap. Let represent the zero-delay unit spike (where the 1 occurs at time instant 0). The Z-transform of the zero-delay unit spike is 1. Let represent the unit spike for delay (where there are zeros before the 1). The Z-transform of the unit spike for delay is . The prediction-error operator is now the difference between the zero-delay unit spike and the prediction operator delayed by the prediction distance; that is,


(13)

A matrix equation can be derived for the prediction-error operator. Normal equations 9 for the prediction filter can be augmented in such a way that the prediction operator is converted into the prediction-error operator. First, in equations 9, we subtract the left-hand side from the right-hand side. The result is


(14)

which is


(15)

and which also can be written as


(16)

Next, we define the quantities with the equation


(17)

Then, we combine equations 16 and 17 to yield


(18)

The column vector on the left-hand side is the prediction-error operator . Thus, matrix equation 17 is a representation for the prediction-error operator with prediction distance Such an operator performs what is known as gap deconvolution.

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