# The prediction-error filter

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 10 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 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 ${\displaystyle x_{n+\alpha }}$ and the estimated or predicted value ${\displaystyle {\hat {x}}_{n+\alpha }}$. The prediction error at time instant ${\displaystyle n+\alpha }$ is thus

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

 {\displaystyle {\begin{aligned}{\varepsilon }_{n}=x_{n}-k_{0}x_{n-\alpha }-k_{1}x_{n-\alpha -1}-...-k_{N-1}x_{n-\alpha -N+1}.\end{aligned}}} (11)

The prediction error ${\displaystyle {\varepsilon }_{n}}$ is a signal that represents the nonpredictable part of ${\displaystyle x_{n}}$. The above equation shows that the filter is

 {\displaystyle {\begin{aligned}f=(1,\;0,\;0,\;...\;,\;0,\;-k_{0},\;...\;,-k_{N-1}).\end{aligned}}} (12)

This filter is called the prediction-error filter or the prediction-error operator. There are ${\displaystyle \alpha -1}$ zeros in the prediction-error operator that lie between the leading coefficient, namely 1, and the negative prediction-operator coefficients. These ${\displaystyle \alpha -1}$ zeros constitute the gap. Let ${\displaystyle {\delta }_{0}=\left({1\ ,\ 0,\ 0,\ .\ .\ .}\right)}$ 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 ${\displaystyle \delta _{\alpha }=(0,\;0,\;0,...\;,\;1)}$ represent the unit spike for delay ${\displaystyle \alpha }$ (where there are ${\displaystyle \alpha -1}$ zeros before the 1). The Z-transform of the unit spike for delay ${\displaystyle \alpha }$ is ${\displaystyle Z^{\alpha }}$. 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,

 {\displaystyle {\begin{aligned}f={\delta }_{0}-{\delta }_{\alpha }*k.\end{aligned}}} (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

 ${\displaystyle \left[{\begin{array}{l}r_{\alpha }\\r_{\alpha +1}\\\ldots \\r_{\alpha +N}\\\end{array}}\right]-\left[{\begin{array}{l}r_{0}\;\;\;\;\;\;r_{1}\;\;\;\;\;\;\;\ldots \;\;\;\;\;r_{N-1}\\r_{1}\;\;\;\;\;\;r_{0}\;\;\;\;\;\;\;\ldots \;\;\;\;r_{N-2}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots \\r_{N-1\;\;\;}\;r_{N-2}\;\;\ldots \;\;\;\;r_{0}\\\end{array}}\right]\left[{\begin{array}{l}k_{0}\\k_{1}\\\ldots \\k_{N-1}\\\end{array}}\right]=\left[{\begin{array}{l}0\\0\\\ldots \\0\\\end{array}}\right],}$ (14)

which is

 ${\displaystyle \left[{\begin{array}{l}r_{\alpha }\;\;\;\;\;\;\;\;\;r_{0}\;\;\;\;\;r_{1}\;\;\;\;\;\ldots \;\;\;r_{N-1}\\r_{\alpha +1}\;\;\;\;\;\;\;r_{1}\;\;\;\;\;r_{0}\;\;\;\;\;\ldots \;\;\;r_{N-2}\;\;\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots \;\;\;\\r_{\alpha +N-1\;\;}r_{N-1}\;\;r_{N-2}\;\;\ldots \;\;\;r_{0}\\\end{array}}\right]\left[{\begin{array}{l}1\\-k_{0}\\-k_{1}\\\ldots \\-k_{N-1}\\\end{array}}\right]=\left[{\begin{array}{l}0\\0\\\ldots \\0\\\end{array}}\right],}$ (15)

and which also can be written as

 ${\displaystyle \left[{\begin{array}{l}r_{\alpha }\;\;\;\;\;\;\;r_{\alpha -1}\;\;\;\;\;\;\ldots \;\;\;\;r_{1}\;\;\;\;\;r_{0}\;\;\;\;\;r_{1}\;\;\;\;\;\ldots \;\;\;r_{N-1}\\r_{\alpha +1}\;\;\;\;\;r_{\alpha \;}\;\;\;\;\;\;\;\;\ldots \;\;\;r_{2}\;\;\;\;\;r_{1}\;\;\;\;\;r_{0}\;\;\;\;\;\ldots \;\;\;r_{N-2}\;\;\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots \;\;\;\\r_{\alpha +N-1}\;\;r_{\alpha +N-2}\;\;\ldots \;\;\;\;r_{N+1}\;\;r_{N}\;\;r_{N-1}\;\ldots \;\;\;r_{0}\\\end{array}}\right]\left[{\begin{array}{l}1\\0\\\ldots \\0\\-k_{0}\\-k_{1}\\\ldots \\-k_{N-1}\\\end{array}}\right]=\left[{\begin{array}{l}0\\0\\\ldots \\0\\\end{array}}\right].}$ (16)

Next, we define the quantities ${\displaystyle {\rho }_{0},{\rho }_{\rm {l}},...,{\rho }_{\alpha -1}}$ with the equation

 ${\displaystyle \left[{\begin{array}{l}\,r_{0}\;\;\;\;\;\;\;\;r_{1}\;\;\;\;\;\,\,\,\ldots \;\;\;\;\;\,r_{\alpha -1}\;\;\;\;r_{\alpha }\;\;\;\;\;r_{\alpha +1}\,\,\,\,\,\ldots \,\,\,\,r_{\alpha +N-1}\\r_{1}\;\;\;\;\;\;\;\;r_{0}\;\;\;\;\,\,\,\,\,\ldots \;\;\;\;\;r_{\alpha -2}\;\;\;\;r_{\alpha -1}\;\;\;r_{\alpha }\;\;\;\;\;\,\ldots \;\;\;r_{\alpha +N-2}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\,\;\;\,\,\,\,\;\;\;\ldots \;\;\;\;\;\;\\r_{\alpha -1}\;\;\,\;\;r_{\alpha -2}\;\;\;\;\;\;\ldots \;\;\;\;\;r_{0}\;\;\;\;\;\;\;r_{1}\,\;\;\;\;r_{2}\;\;\;\;\;\,\,\ldots \;\;\;\,r_{N+1}\\r_{\alpha }\;\;\;\;\,\;\;r_{\alpha -1}\;\;\;\;\;\;\,\ldots \;\;\;\;\;r_{1}\;\;\;\;\;\;\;r_{0}\;\;\;\;r_{1}\;\;\;\;\;\,\,\,\ldots \;\;\;\,r_{N}\;\\r_{\alpha +1}\;\;\,\;\;r_{\alpha }\;\;\;\;\;\;\;\,\,\ldots \;\;\;\;r_{2}\;\;\;\;\;\;\;r_{1}\;\;\;\;\;r_{0}\;\;\;\;\;\;\,\,\ldots \;\;\,\,r_{N-1}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,\,\,\,\;\;\;\ldots \\r_{\alpha +N-1}\;\;r_{\alpha +N-2}\,\,\,\,\ldots \;\;\;\;r_{N+1}\,\,\,\,\,r_{N}\;\;\;\,r_{N-1}\,\,\,\,\,\,\,\ldots \;\;\;\;r_{0}\\\end{array}}\right]\left[{\begin{array}{l}1\\0\\\ldots \\0\\-k_{0}\,\\-k_{1}\\\ldots \\-k_{N-1}\\\end{array}}\right]=\left[{\begin{array}{l}\rho _{0}\\\rho _{1}\\\ldots \\\rho _{\alpha -1}\\0\\0\\\ldots \\0\\\end{array}}\right].}$ (17)

Then, we combine equations 16 and 17 to yield

 ${\displaystyle \left[{\begin{array}{l}\,r_{0}\;\;\;\;\;\;\;\;r_{1}\;\;\;\;\;\,\,\,\ldots \;\;\;\;\;\,r_{\alpha -1}\;\;\;\;r_{\alpha }\;\;\;\;\;r_{\alpha +1}\,\,\,\,\,\ldots \,\,\,\,r_{\alpha +N-1}\\r_{1}\;\;\;\;\;\;\;\;r_{0}\;\;\;\;\,\,\,\,\,\ldots \;\;\;\;\;r_{\alpha -2}\;\;\;\;r_{\alpha -1}\;\;\;r_{\alpha }\;\;\;\;\;\,\ldots \;\;\;r_{\alpha +N-2}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\,\;\;\;\;\;\,\,\,\,\ldots \;\;\;\;\;\;\\r_{\alpha -1}\;\;\,\;\;r_{\alpha -2}\;\;\;\;\;\;\ldots \;\;\;\;\;r_{0}\;\;\;\;\;\;\;r_{1}\,\;\;\;\;r_{2}\;\;\;\;\;\,\,\ldots \;\;\;\,r_{N+1}\\r_{\alpha }\;\;\;\;\,\;\;r_{\alpha -1}\;\;\;\;\;\;\,\ldots \;\;\;\;\;r_{1}\;\;\;\;\;\;\;r_{0}\;\;\;\;r_{1}\;\;\;\;\;\,\,\,\ldots \;\;\;\,r_{N}\;\\r_{\alpha +1}\;\;\,\;\;r_{\alpha }\;\;\;\;\;\;\;\,\,\ldots \;\;\;\;r_{2}\;\;\;\;\;\;\;r_{1}\;\;\;\;\;r_{0}\;\;\;\;\;\;\,\,\ldots \;\;\,\,r_{N-1}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,\,\ldots \\r_{\alpha +N-1}\;\;r_{\alpha +N-2}\,\,\,\,\ldots \;\;\;\;r_{N+1}\,\,\,\,\,r_{N}\;\;\;\,r_{N-1}\,\,\,\,\,\,\,\ldots \;\;\;\;r_{0}\\\end{array}}\right]\left[{\begin{array}{l}1\\0\\\ldots \\0\\-k_{0}\,\\-k_{1}\\\ldots \\-k_{N-1}\\\end{array}}\right]=\left[{\begin{array}{l}\rho _{0}\\\rho _{1}\\\ldots \\\rho _{\alpha -1}\\0\\0\\\ldots \\0\\\end{array}}\right].}$ (18)

The column vector on the left-hand side is the prediction-error operator ${\displaystyle f={\delta }_{0}-{\delta }_{\alpha }*k}$. Thus, matrix equation 17 is a representation for the prediction-error operator with prediction distance ${\displaystyle \alpha }$ Such an operator performs what is known as gap deconvolution.