The prediction-error filter

**Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing**
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

<translate> The prediction-error filter results directly from the prediction filter. We define the prediction-error series as the difference between the true value $x_{n+\alpha }$ and the estimated or predicted value ${\hat {x}}_{n+\alpha }$ . The prediction error at time instant $n+\alpha$ is thus

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{\begin{aligned}{\varepsilon }_{n+\alpha }=x_{n+\alpha }-{\hat {x}}=x_{n+\alpha }-k_{0}x_{n}-k_{1}x_{n-1}-...-k_{N-1}x_{n-N+1}.\end{aligned}}

(10)

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

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\varepsilon }_{n}= x_n-k_0x_{n-\alpha }-k_{1}x_{n-\alpha -1}-... -k_{N-1}x_{n-\alpha -N+1}. \end{align}

(11)

The prediction error Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\varepsilon }_n is a signal that represents the nonpredictable part of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x_n . The above equation shows that the filter is

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align}f = (1,\;0,\;0,\;...\;,\;0,\; - k_0 ,\;...\;, - k_{N - 1}). \end{align}

(12)

This filter is called the prediction-error filter or the prediction-error operator. There are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha -1 zeros in the prediction-error operator that lie between the leading coefficient, namely 1, and the negative prediction-operator coefficients. These Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha -1 zeros constitute the gap. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\delta }_0= \left({ 1\ ,\ 0,\ 0,\ .\ .\ .}\right) 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta _\alpha = (0,\;0,\;0,...\;,\;1) represent the unit spike for delay Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha (where there are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha -1 zeros before the 1). The Z-transform of the unit spike for delay Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z^{\alpha } . 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,

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f= {\delta }_0-{\delta }_{\alpha }*k. \end{align}

(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

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[ \begin{array}{l} r_\alpha \\ r_{\alpha + 1} \\ \ldots \\ r_{\alpha + N} \\ \end{array} \right] - \left[ \begin{array}{l} r_0 \;\;\;\;\;\;r_1 \;\;\;\;\;\;\; \ldots \;\;\;\;\;r_{N - 1} \\ r_1 \;\;\;\;\;\;r_0 \;\;\;\;\;\;\; \ldots \;\;\;\;r_{N - 2} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ldots \\ r_{N - 1\;\;\;} \;r_{N - 2} \;\; \ldots \;\;\;\;r_0 \\ \end{array} \right]\left[ \begin{array}{l} k_0 \\ k_1 \\ \ldots \\ k_{N - 1} \\ \end{array} \right] = \left[ \begin{array}{l} 0 \\ 0 \\ \ldots \\ 0 \\ \end{array} \right],

(14)

which is

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[ \begin{array}{l} r_\alpha \;\;\;\;\;\;\;\;\;r_0 \;\;\;\;\;r_1 \;\;\;\;\; \ldots \;\;\;r_{N - 1} \\ r_{\alpha + 1} \;\;\;\;\;\;\;r_1 \;\;\;\;\;r_0 \;\;\;\;\; \ldots \;\;\;r_{N - 2} \;\; \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ldots \;\;\; \\ r_{\alpha + N - 1\;\;} r_{N - 1} \;\;r_{N - 2} \;\; \ldots \;\;\;r_0 \\ \end{array} \right]\left[ \begin{array}{l} 1 \\ - k_0 \\ - k_1 \\ \ldots \\ - k_{N - 1} \\ \end{array} \right] = \left[ \begin{array}{l} 0 \\ 0 \\ \ldots \\ 0 \\ \end{array} \right],

(15)

and which also can be written as

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[ \begin{array}{l} r_\alpha \;\;\;\;\;\;\;r_{\alpha - 1} \;\;\;\;\;\; \ldots \;\;\;\;r_1 \;\;\;\;\;r_0 \;\;\;\;\;r_1 \;\;\;\;\; \ldots \;\;\;r_{N - 1} \\ r_{\alpha + 1} \;\;\;\;\;r_{\alpha \;} \;\;\;\;\;\;\;\; \ldots \;\;\;r_2 \;\;\;\;\;r_1 \;\;\;\;\;r_0 \;\;\;\;\; \ldots \;\;\;r_{N - 2} \;\; \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ldots \;\;\; \\ r_{\alpha + N - 1} \;\;r_{\alpha + N - 2} \;\; \ldots \;\;\;\;r_{N + 1} \;\;r_N \;\;r_{N - 1} \; \ldots \;\;\;r_0 \\ \end{array} \right]\left[ \begin{array}{l} 1 \\ 0 \\ \ldots \\ 0 \\ - k_0 \\ - k_1 \\ \ldots \\ - k_{N - 1} \\ \end{array} \right] = \left[ \begin{array}{l} 0 \\ 0 \\ \ldots \\ 0 \\ \end{array} \right].

(16)

Next, we define the quantities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rho }_0,{\rho }_{{\rm l}}, . . . , {\rho }_{\alpha -1} with the equation

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[ \begin{array}{l} \,r_0 \;\;\;\;\;\;\;\;r_1 \;\;\;\;\;\,\,\, \ldots \;\;\;\;\;\,r_{\alpha - 1} \;\;\;\;r_\alpha \;\;\;\;\;r_{\alpha + 1} \,\,\,\,\, \ldots \,\,\,\,r_{\alpha + N - 1} \\ r_1 \;\;\;\;\;\;\;\;r_0 \;\;\;\;\,\,\,\,\, \ldots \;\;\;\;\;r_{\alpha - 2} \;\;\;\;r_{\alpha - 1} \;\;\;r_\alpha \;\;\;\;\;\, \ldots \;\;\;r_{\alpha + N - 2} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\,\;\;\,\,\,\,\;\;\; \ldots \;\;\;\;\;\; \\ r_{\alpha - 1} \;\;\,\;\;r_{\alpha - 2} \;\;\;\;\;\; \ldots \;\;\;\;\;r_0 \;\;\;\;\;\;\;r_1 \,\;\;\;\;r_2 \;\;\;\;\;\,\, \ldots \;\;\;\,r_{N + 1} \\ r_\alpha \;\;\;\;\,\;\;r_{\alpha - 1} \;\;\;\;\;\;\, \ldots \;\;\;\;\;r_1 \;\;\;\;\;\;\;r_0 \;\;\;\;r_1 \;\;\;\;\;\,\,\, \ldots \;\;\;\,r_N \; \\ r_{\alpha + 1} \;\;\,\;\;r_\alpha \;\;\;\;\;\;\;\,\, \ldots \;\;\;\;r_2 \;\;\;\;\;\;\;r_1 \;\;\;\;\;r_0 \;\;\;\;\;\;\,\, \ldots \;\;\,\,r_{N - 1} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,\,\,\,\;\;\; \ldots \\ r_{\alpha + N - 1} \;\;r_{\alpha + N - 2} \,\,\,\, \ldots \;\;\;\;r_{N + 1} \,\,\,\,\,r_N \;\;\;\,r_{N - 1} \,\,\,\,\,\,\, \ldots \;\;\;\;r_0 \\ \end{array} \right]\left[ \begin{array}{l} 1 \\ 0 \\ \ldots \\ 0 \\ - k_0 \, \\ - k_1 \\ \ldots \\ - k_{N - 1} \\ \end{array} \right] = \left[ \begin{array}{l} \rho _0 \\ \rho _1 \\ \ldots \\ \rho _{\alpha - 1} \\ 0 \\ 0 \\ \ldots \\ 0 \\ \end{array} \right].

(17)

Then, we combine equations 16 and 17 to yield

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[ \begin{array}{l} \,r_0 \;\;\;\;\;\;\;\;r_1 \;\;\;\;\;\,\,\, \ldots \;\;\;\;\;\,r_{\alpha - 1} \;\;\;\;r_\alpha \;\;\;\;\;r_{\alpha + 1} \,\,\,\,\, \ldots \,\,\,\,r_{\alpha + N - 1} \\ r_1 \;\;\;\;\;\;\;\;r_0 \;\;\;\;\,\,\,\,\, \ldots \;\;\;\;\;r_{\alpha - 2} \;\;\;\;r_{\alpha - 1} \;\;\;r_\alpha \;\;\;\;\;\, \ldots \;\;\;r_{\alpha + N - 2} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\,\;\;\;\;\;\,\,\,\, \ldots \;\;\;\;\;\; \\ r_{\alpha - 1} \;\;\,\;\;r_{\alpha - 2} \;\;\;\;\;\; \ldots \;\;\;\;\;r_0 \;\;\;\;\;\;\;r_1 \,\;\;\;\;r_2 \;\;\;\;\;\,\, \ldots \;\;\;\,r_{N + 1} \\ r_\alpha \;\;\;\;\,\;\;r_{\alpha - 1} \;\;\;\;\;\;\, \ldots \;\;\;\;\;r_1 \;\;\;\;\;\;\;r_0 \;\;\;\;r_1 \;\;\;\;\;\,\,\, \ldots \;\;\;\,r_N \; \\ r_{\alpha + 1} \;\;\,\;\;r_\alpha \;\;\;\;\;\;\;\,\, \ldots \;\;\;\;r_2 \;\;\;\;\;\;\;r_1 \;\;\;\;\;r_0 \;\;\;\;\;\;\,\, \ldots \;\;\,\,r_{N - 1} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,\, \ldots \\ r_{\alpha + N - 1} \;\;r_{\alpha + N - 2} \,\,\,\, \ldots \;\;\;\;r_{N + 1} \,\,\,\,\,r_N \;\;\;\,r_{N - 1} \,\,\,\,\,\,\, \ldots \;\;\;\;r_0 \\ \end{array} \right]\left[ \begin{array}{l} 1 \\ 0 \\ \ldots \\ 0 \\ - k_0 \, \\ - k_1 \\ \ldots \\ - k_{N - 1} \\ \end{array} \right] = \left[ \begin{array}{l} \rho _0 \\ \rho _1 \\ \ldots \\ \rho _{\alpha - 1} \\ 0 \\ 0 \\ \ldots \\ 0 \\ \end{array} \right].

(18)

The column vector on the left-hand side is the prediction-error operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f= {\delta }_0-{\delta }_{\alpha }*k . Thus, matrix equation 17 is a representation for the prediction-error operator with prediction distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \alpha Such an operator performs what is known as gap deconvolution.