Gap deconvolution

**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> In gap deconvolution, one more step must be added to spiking deconvolution. This step amounts to reaveraging the prediction errors (namely, the reflectivity series). We now describe the head filter and the tail filter and explain their relationship to gap deconvolution.

The first step in understanding gap deconvolution requires splitting the minimum-phase wavelet b into two parts. One part is the head,

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{\begin{aligned}h=(b_{0},\;b_{1},\;b_{2},\;\ldots ,b_{\alpha -1}).\end{aligned}}

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The head consists of the first 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 coefficients, with 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/":): b_0 in time spot 0. The second part is the tail,

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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} t = (b_\alpha ,\;b_{\alpha + 1} ,\;b_{\alpha + 2} , \ldots ). \end{align}

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The tail consists of the remaining coefficients, advanced in time so that 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/":): b_{\alpha } occurs in time spot 0. Hence, the tail must be delayed in the reconstruction of the wavelet 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} b = \left(b_0{,\ }b_{1}{,\ }b_{2}{,\dots,\ }b_{\alpha -1}\right)+\left({ 0,0,0,\ }\dots { \ ,\ 0,\ }b_{\alpha }{ \ ,\ }b_{\alpha +1}{ \ ,\ }b_{\alpha { +2}} { ,\ }\dots \right)= h+{\delta }_{\alpha }*t. \end{align}

(31)

Given the minimum-phase wavelet (which is obtained as the inverse of the spiking-deconvolution filter), we have three ways to compute the gap-deconvolution operator. They are the head-filtering method, the tail-shaping method, and the head-shaping method.

Let the minimum-phase wavelet be the input to a prediction filter 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 . At each time index, the filter tries to predict the value of the input, 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 time units ahead. Thus, at time instant 0, the filter tries to predict 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/":): b_{\alpha } . At time instant 1, the filter tries to predict 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/":): b_{\alpha +1} . At time instant 2, the filter tries to predict $b_{\alpha {+2}}$ , and so on. However, these values make up the tail. Thus, the desired output of the prediction filter is the tail of the minimum-phase wavelet. We therefore can find the prediction filter by merely convolving the spiking filter with the tail; 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} k= a*t \mathrm{\;\;\; or\;\;\;\; } k_i= \sum^{\infty }_{s={0}} {b_{\alpha +}},a_{i-s}, \mathrm{where}\;\;\;\; i={0}, 1, 2,\ldots \end{align}

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For further discussion, see Robinson (1954^[1]), in which his equation 5.332 is the same as our equation 32 here. The corresponding prediction-error filter given by equation 13 in the present case becomes

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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 }*a*t. \end{align}

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The head of the wavelet gives us what we might call the unreachable prediction error for the prediction-error filter.

Let us now find another expression for the prediction-error filter. Because 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= a*b , the prediction-error filter as given by equation 33 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/":): \begin{align} f= a*b-{\delta }_{\alpha }*a*t= a*\left(b-{\delta }_{\alpha }*t\right) . \end{align}

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However, equation 31 gives

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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} h= b-{\delta }_{\alpha }*t. \end{align}

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Thus, the prediction-error filter, as given by equation 34, becomes

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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= a*h. \end{align}

(36)

References

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↑ Robinson, E. A., 1954, Predictive decomposition of time series with applications to seismic exploration: Ph.D. thesis, Massachusetts Institute of Technology. (Reprinted in Geophysics, 32, 418-484, 1967.)