View source for The prediction-error filter

<languages/>
<translate>
</translate>
{{publication
 | name    = Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
 | image   = [[File:DigitalImaging.png|frameless|180px]]
 | width   = 120px
 | series  = Geophysical References Series
 | title   = Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
 | part    = 
 | chapter = 10
 | frompg  = 253
 | topg    = 288
 | author  = [[Special:MyLanguage/Enders A. Robinson|Enders A. Robinson]] and [[Special:MyLanguage/Sven Treitel|Sven Treitel]]
 | doi     = 10.1190/1.9781560801610
 | pdf     = 
 | store   = https://seg.org/shop/products/detail/898
 | isbn    = 9781560801481
}}
<translate>
<!--T:1-->
The prediction-error filter results directly from the prediction filter. We define the prediction-error series as the difference between the true value <math>x_{n+\alpha }</math> and the estimated or predicted value <math>\hat{x}_{n+\alpha}</math>. The prediction error at time instant <math>n+\alpha</math> is thus

</translate>
<translate><!--T:2-->
{{NumBlk|:|<math>\begin{align}
{\varepsilon }_{n+\alpha }= x_{n+\alpha }-\hat{x}= x_{n+\alpha }-k_0x_n-k_{1}x_{n-1}-... -k_{N-1}x_{n-N+1}.
\end{align}</math>|{{EquationRef|10}}}}

<!--T:3-->
If <math>n+\alpha</math> is replaced by ''n'' in the above equation, the result is the expression for the prediction error, at the present time ''n'', given by

</translate>
<translate><!--T:4-->
{{NumBlk|:|<math>\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}</math>|{{EquationRef|11}}}}

<!--T:5-->
The prediction error <math>{\varepsilon }_n</math> is a signal that represents the nonpredictable part of <math>x_n</math>. The above equation shows that the filter is

</translate>
<translate><!--T:6-->
{{NumBlk|:|<math>\begin{align}f = (1,\;0,\;0,\;...\;,\;0,\; - k_0 ,\;...\;, - k_{N - 1}).
\end{align}</math>|{{EquationRef|12}}}}

<!--T:7-->
This filter is called the ''prediction-error filter'' or the ''prediction-error operator''. There are <math>\alpha -1</math> zeros in the prediction-error operator that lie between the leading coefficient, namely 1, and the negative prediction-operator coefficients. These <math>\alpha -1</math> zeros constitute the gap. Let <math>{\delta }_0= \left({ 1\ ,\ 0,\ 0,\ .\ .\ .}\right)</math> 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 <math>\delta _\alpha   = (0,\;0,\;0,...\;,\;1)</math> represent the unit spike for delay <math>\alpha</math> (where there are <math>\alpha -1</math> zeros before the 1). The ''Z''-transform of the unit spike for delay <math>\alpha</math> is <math>Z^{\alpha }</math>. 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,

</translate>
<translate><!--T:8-->
{{NumBlk|:|<math>\begin{align}
f= {\delta }_0-{\delta }_{\alpha }*k.
\end{align}</math>|{{EquationRef|13}}}}

<!--T:9-->
A matrix equation can be derived for the prediction-error operator. Normal equations {{EquationNote|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 {{EquationNote|9}}, we subtract the left-hand side from the right-hand side. The result is

</translate>
<translate><!--T:10-->
{{NumBlk|:|<math>\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],</math>|{{EquationRef|14}}}}

<!--T:11-->
which is

</translate>
<translate><!--T:12-->
{{NumBlk|:|<math>\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],</math>|{{EquationRef|15}}}}

<!--T:13-->
and which also can be written as

</translate>
<translate><!--T:14-->
{{NumBlk|:|<math>\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].</math>|{{EquationRef|16}}}}

<!--T:15-->
Next, we define the quantities <math>{\rho }_0,{\rho }_{{\rm l}}, . . . , {\rho }_{\alpha -1}</math> with the equation

</translate>
<translate><!--T:16-->
{{NumBlk|:|<math>\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].</math>|{{EquationRef|17}}}}

<!--T:17-->
Then, we combine equations {{EquationNote|16}} and {{EquationNote|17}} to yield

</translate>
<translate><!--T:18-->
{{NumBlk|:|<math>\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].</math>|{{EquationRef|18}}}}

<!--T:19-->
The column vector on the left-hand side is the prediction-error operator <math>f= {\delta }_0-{\delta }_{\alpha }*k</math>. Thus, matrix equation {{EquationNote|17}} is a representation for the prediction-error operator with prediction distance <math>\alpha</math> Such an operator performs what is known as [[Special:MyLanguage/gap deconvolution|gap deconvolution]].
__NOTOC__

== Continue reading == <!--T:20-->

<!--T:21-->
{| border="1" cellpadding="5" cellspacing="0" align="center"
|-
! style="background: #426580; color: white;" | Previous section
! style="background: #E5D75F;" | Next section
|-
| align="center" | [[Special:MyLanguage/Least-squares prediction and smoothing|Least-squares prediction and smoothing]]
| align="center" | [[Special:MyLanguage/Spiking deconvolution - book|Spiking deconvolution]]
|-
! style="background: #426580; color: white;" | Previous chapter
! style="background: #E5D75F;" | Next chapter
|-
| align="center" | [[:Category:Wavelet Processing - Chapter 9|Wavelet Processing]]
| align="center" | [[:Category:Fine Points|Fine Points]]
|-
| align="center" colspan="2" |
[[Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing|Table of Contents (book)]]
|}


== Also in this chapter == <!--T:22-->

 
<!--T:23-->
* [[Special:MyLanguage/Model used for deconvolution|Model used for deconvolution]]
* [[Special:MyLanguage/Least-squares prediction and smoothing|Least-squares prediction and smoothing]]
* [[Special:MyLanguage/Spiking deconvolution - book|Spiking deconvolution]]
* [[Special:MyLanguage/Gap deconvolution|Gap deconvolution]]
* [[Special:MyLanguage/Tail shaping and head shaping|Tail shaping and head shaping]]
* [[Special:MyLanguage/Seismic deconvolution|Seismic deconvolution]]
* [[Special:MyLanguage/Piecemeal convolutional model|Piecemeal convolutional model]]
* [[Special:MyLanguage/Time-varying convolutional model|Time-varying convolutional model]]
* [[Special:MyLanguage/Random-reflection-coefficient model|Random-reflection-coefficient model]]
* [[Special:MyLanguage/Implementing deconvolution|Implementing deconvolution]]
* [[Special:MyLanguage/Canonical representation – Chapter 10|Canonical representation]]
* [[Special:MyLanguage/Appendix J: Exercises|Appendix J: Exercises]]


== External links == <!--T:24-->

</translate>
{{search}}
<translate> 

<!--T:25-->
[[Category:Digital Imaging and Deconvolution]]
[[Category:Publications]] 
[[Category:Deconvolution - Chapter 10|Deconvolution]]
</translate>