Analog prediction error

**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	15
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

<translate> Because the same situation is obtained in the analog case as in the digital case, let us merely give examples.

The minimum-delay signal

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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\left(t\right)={2}^{{\rm 1/2}} e^{-{\rm t}} \mathrm \;\;{\rm for} \;\; t\ge 0 \end{align}

(191)

and the causal nonminimum-delay signal

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{\begin{aligned}g\left(t\right)={2}^{\rm {l/2}}e^{-t}\left(1-2t\right)\mathrm {\;} \;{\rm {for}}\;t\geq 0\end{aligned}}

(192)

have the same magnitude spectrum. The prediction filter for 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 =1 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} E\left(s\right)=\frac {{\rm L} \left\{f\left(t{\rm +l}\right)\right\}} {{\rm L} \left\{f\left(t\right)\right\}}=e^{-1}. \end{align}

(193)

We recall that L denotes the one-sided Laplace transform (equation 39). First let us consider the minimum-delay signal 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\left(t\right) . The prediction is obtained by multiplying the signal by 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/":): e^{ - 1} ; 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} \hat{f}\left(t|\varepsilon \right)=\hat{f}\left(t{\rm |1}\right)=f\left(t\right)e^{-1}={2}^{{\rm l/2}} e^{-t}e^{-1}={2}^{{\rm 1/2}} e^{-\left(t+1\right)} \mathrm \;\; {\rm for} \;\; t\ge 0, \end{align}

(194)

which exactly reproduces the tailgate of the advanced signal 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\left(t{\rm +\ 1}\right) . Thus, the prediction error is given by the front end of the advanced signal; 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} \tilde{f}\left(t|\varepsilon \right)=\tilde{f}\left(t{\rm |1}\right)=f\left(t{\rm +l}\right)={2}^{{\rm 1/}2}e^{-\left(t+1\right)} \mathrm \;\; {for} \; -1\le t<0. \end{align}

(195)

The prediction-error energy is the front-end energy

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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} \int\limits^0_{-1}{f^{2}} \left(t+1\right)dt=\int\limits^{1}_0{f^{2}}\left(t\right)dt=\int\limits^{1}_0{2}e^{-2t}dt=\int\limits^{2}_0{e^{-{\rm u}}}du=1-e^{-2}. \end{align}

(196)

Next let us consider the nonminimum-delay signal 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/":): g\left(t\right) . The prediction is again obtained by multiplying the signal by 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/":): e^{ - 1} ; 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} \hat{g}\left(t|\varepsilon \right)=\hat{g}\left(t{\rm |1}\right)=g\left(t\right)e^{-1}={2}^{{\rm 1/2}} e^{-\left(t+1\right)}\left(1-2t\right) \mathrm \;\; {\rm for} \; t\ge 0 . \end{align}

(197)

This prediction does not exactly reproduce the tailgate of the advanced signal 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(t{\rm +\ 1}\right) . Thus, the prediction error ${\tilde {g}}\left(t{\rm {|1}}\right)$ is made up of two components. One component is the front-end error

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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} \tilde{g}\left(t{\rm |1}\right)=g\left(t{\rm +l}\right)={2}^{{\rm 1/2}} e^{-\left({\rm t+1}\right)}\left[1-2\left(t{\rm +l}\right)\right] \mathrm \;\; {\rm for} \; -1\le t<0, \end{align}

(198)

and the other component is the tailgate error

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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} \tilde g(t|1) = g(t + 1) - g(t)e^{ - 1} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ \;\; = 2^{1/2} e^{ - (t + 1)} [1 - 2(t + 1)] - 2^{1/2} e^{ - (t + 1)} [1 - 2t] \;\;\;\; \\ \;\; = - 2^{3/2} e^{ - (t + 1)} \;\;\;\;{\rm for}\;\;t \ge 0. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ \end{align}

(199)

The front-end-error energy 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} \int\limits^0_{-1}{{\left[\tilde{g}\left(t{\rm |1}\right)\right]}^{2}} dt=\int\limits^0_{-1}{{\left[g\left(t+1\right)\right]}^{2}}dt=\int^{1}_0{{\left[g\left(t\right)\right]}^{2}}dt \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ =\int\limits^{1}_0{2}e^{-2t}{\left(1-2t\right)}^{2}dt=\int\limits^{2}_0{e^{-{\rm u}}}{\left(1-u\right)}^{2}du. \end{align}

(200)

Using tables of integrals, we obtain the front-end-error energy 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/":): \begin{align} \int\limits^0_{-1}{{\left[\tilde{g}\left(t{\rm |1}\right)\right]}^{2}} dt=1-5e^{-2}. \end{align}

(201)

The tailgate-error energy 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} \int\limits^{\infty }_0{{\left[\tilde{g}\left(t{\rm |1}\right)\right]}^{2}} dt=\int\limits^{\infty }_0{{\rm 8}}e^{-2\left({\rm t+1}\right)dt}{\rm =4}\int\limits^{\infty }_{2}{e^{-u}}du{\rm =4}e^{-2} . \end{align}

(202)

Thus, the total prediction-error energy 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} \int\limits^{\infty }_{-1}{{\left[\tilde{g}\left(t{\rm |1}\right)\right]}^{2}} dt=1-5e^{-2}{\rm +4}e^{-2}=1-e^{-2} , \end{align}

(203)

which is the same as equation 196 for the case of the minimum-delay signal, as we would expect.

Alternatively, 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/":): \hat{g}\left({\rm t|1}\right) for 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/":): t\ge 0 can be obtained by passing 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/":): \hat{f}\left({\rm t|1}\right) through the all-pass system. Because

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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\left(s\right)={\rm L}\left\{{2}^{{\rm 1/2}} e^{-t}\right\}={2}^{{\rm 1/2}}{\left(s+1\right)}^{-1} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ G\left(s\right)={\rm L}\left\{{2}^{{\rm 1/2}}e^{-t}\left(1-2t\right)\right\}={2}^{{\rm l/2}}\left(s-1\right){\left(s+1\right)}^{-2} , \end{align}

(204)

the all-pass system 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} P\left(s\right)=G\left(s\right)F^{-1}\left(s\right)=\left(s-1\right){\left(s+1\right)}^{-1}. \end{align}

(205)

If we expand in partial fractions, we have

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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} P\left(s\right)={\rm l}-2{\left(s{\rm +l}\right)}^{-1} , \end{align}

(206)

so the all-pass impulse response is the causal function

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{\begin{aligned}p\left(t\right)=\delta \left(t\right)-2e^{-t}\mathrm {\;} \;{\rm {for}}\;t\geq 0.\end{aligned}}

(207)

Thus, the prediction of the nonminimum-delay signal 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} \hat{g}\left(t{\rm |1}\right)=p\left(t\right)*\hat{f}\left(t{\rm |1}\right) \mathrm \;\; {\rm for} \; t\ge 0, \end{align}

(208)

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/":): \begin{align} \hat{g}\left(t{\rm |1}\right)=\left[\delta \left(t\right)-2e^{-t}\right]*\left[{2}^{{\rm 1/2}} e^{-\left(f+1\right)}\right] \;\;\;\;\;\;\;\;\;\; \\ ={2}^{{\rm l/2}}e^{-\left(t{\rm +l}\right)}-{2}^{{\rm 3/2}}{\rm \ }\int^t_0{e^{-\tau }}e^{-t+\tau -1}d\tau \;\;\;\\ ={2}^{{\rm 1/2}}e^{-\left(t{\rm +l}\right)}\left(1-2t\right) \mathrm \; {\rm for} \; t\ge 0. \;\;\;\;\;\;\;\;\;\;\;\;\; \end{align}

(209)

This is the same result we obtained before (equation 197).

Likewise, 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/":): \tilde{g}\left(t{\rm |1}\right) can be obtained by passing 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/":): \tilde{f}\left(t{\rm |1}\right) through 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/":): p\left(t\right) . The 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/":): \tilde{f}\left(t{\rm |1}\right) is nonzero only in the range 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 \le t<0 , whereas the 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/":): \tilde{g}\left(t{\rm |1}\right) is nonzero in the range 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 \le t<\infty . First we will evaluate 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/":): \tilde{g}\left(t{\rm |1}\right) in the subrange 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 \le t<0 and then in the subrange 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/":): 0 \le t< \infty . We have

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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} \tilde{g}\left(t{\rm |1}\right)=\tilde{f}\left(t{\rm |l}\right)*p\left(t\right){\rm =\ }\int\limits^t_{-1}{{2}^{{\rm 1/2}} }{\rm \ }e^{-\left(\tau {\rm +l}\right)}\left[\delta \left(t-\tau \right)-2e^{-\left(t\tau \right)}\right]d\tau \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ ={2}^{{\rm 1/2}}e^{-\left(t+1\right)}\left(1-2\right)=-{2}^{{\rm l/2}}e^{-\left(t{\rm +l}\right)} \mathrm \;\; {\rm for} \; -\varepsilon \le t<0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{align}

(210)

whereas

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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} \tilde g(t|1) = \int\limits_{ - 1}^0 {2^{1/2} e^{ - (\tau + 1)} [\delta (t - \tau ) - 2e^{ - (t - \tau )} ]\;d\tau \;\;\;{\rm for}\;\;{\rm 0}\; \le t < \infty .} \end{align}

(211)

In this last integral, the spike of the delta function does not lie within the integration range 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/":): - 1 \le \tau < 0 , so the contribution resulting from the delta function is zero. Thus,

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{\begin{aligned}{\tilde {g}}(t|1)=\int \limits _{-1}^{0}{2^{1/2}e^{-(\tau +1)}(-2)e^{-(t{\rm {-}}\tau )}}d\tau =-2^{3/2}e^{-(t+1)}\;\;\;{\rm {for}}\;{\rm {0}}\leq {\rm {t<}}\infty {\rm {.}}\end{aligned}}

(212)

We see that we have obtained the same prediction error that we did before (equation 199).

The examples we have just treated are illustrated in Figure 12, which is similar to Figures 9.81 and 9.82 in Robinson (1962)^[1]. The minimum-delay case is on the left. Note how the prediction system clearly splits the desired output (i.e., the advanced input) into two parts so that one part is the prediction error (anticausal) and the other part is the prediction (causal). The nonminimum-delay case is on the right. Each curve on the right can be obtained from the corresponding curve on the left by means of the all-pass filter.