Spiking deconvolution - book

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

By its very nature, deconvolution is based on a convolutional model. Without a convolutional model, there can be prediction-error filtering but no deconvolution as such. The convolutional model is our particular representation of the physical structure of the earth. The standard convolutional model states that a seismic trace is the convolution of a minimum-phase wavelet with a white reflectivity. Any generalization of the convolutional model always must begin with ways to relax either the minimum-phase assumption or the white-reflectivity assumption or both.

Let $x=\left(x_{0}{,\ }x_{1}{\ ,\ }x_{2}{,\ .\ .\ .}\right)$ be the seismic trace, let $b=\left(b_{0}{,\ }b_{1}{\ ,\ }b_{2}{,\ }\dots \right)$ be the minimum-phase wavelet, and let $\varepsilon =\left({\varepsilon }_{0}{,\ }{\varepsilon }_{\rm {l}}{,\ }{\varepsilon }_{2}{,\ .\ .\ .}\right)$ be the random white reflectivity. Mathematically, the convolutional model is

${\begin{aligned}x_{n}=b_{0}{\varepsilon }_{n}+b_{1}{\varepsilon }_{n-1}+\dots =\sum _{i={0}}^{\infty }{b_{i}}{\varepsilon }_{n-i}.\end{aligned}}$

(19)

For convenience, the first coefficient $b_{0}$ of the wavelet is taken to be equal to one. In more concise notation, the convolutional model is $x=b*\varepsilon$. The deconvolution problem consists of removal of the wavelet from the trace, leaving behind an estimate of the reflectivity.

The deconvolution process can be outlined as follows. The trace is the given data. The autocorrelation function of this trace is computed next. Because the reflectivity is white, computing the trace’s autocorrelation function averages out the random, uncorrelated reflectivity series. As a result, the trace’s autocorrelation is the same function, within statistical bounds, as the wavelet’s autocorrelation. From the seismic trace, we thereby can estimate the autocorrelation of the wavelet. This is the crucial step in the deconvolution process. However, rough edges on the computed autocorrelation always remain. As a result, the computed autocorrelation must be adjusted or tapered to render it more suitable from the viewpoint of its spectral energy. A typical adjustment is to increase the zero-lag autocorrelation coefficient by a small percentage of it. Because this increase in the zero-lag autocorrelation coefficient boosts the white-noise level of the spectrum, this adjustment is often called prewhitening.

The seismic wavelet is unknown; however, given the assumptions of the convolutional model, the autocorrelation of the wavelet can be found. Note, however, that the autocorrelation function preserves no phase information. It turns out that many wavelet shapes share the same autocorrelation function but have different phase spectra (Chapter 7). However, only one of these wavelets is minimum phase. Because the convolutional model assumes that the seismic wavelet is minimum phase, specification of the minimum-phase wavelet provides us with the necessary information to determine the wavelet from the given autocorrelation function. Once the minimum-phase wavelet is known, it can be removed from the trace by deconvolution, thereby leaving, as a residual, the random reflectivity series.

Whereas prediction operators can be used with various values of the prediction distance, our attention now turns to the prediction operator for unit prediction distance — that is, for $\alpha =1$. The prediction equation for unit prediction distance is

${\begin{aligned}{\hat {x_{n}}}=k_{0}x_{n-1}+k_{\rm {1}}x_{n-2}+...+k_{N-{\rm {1}}}x_{n-N}.\end{aligned}}$

(20)

The given data are the autocorrelation of the trace, which is the same as the autocorrelation of the wavelet. The critical simplification brought about by the convolutional model is that now one needs only to deal with the wavelet instead of with the entire trace when it comes to operator design. Hence, now wavelet b can be regarded as the input to the filter. The autocorrelation of the wavelet is the same as the autocorrelation r of the trace.

The first step is to determine the coefficients of the prediction operator for the unit prediction distance for this wavelet. Setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha = 1} in equation 9, we obtain the normal equations

$\left[{\begin{array}{l}\,r_{0}\;\;\;\;\;r_{1}\;\;\;\;\;\,\ldots \;\;\;\;\;\,r_{N-1}\\r_{1}\;\;\;\;\;\;r_{0}\;\;\;\;\,\,\ldots \;\;\;\;\;r_{N-2}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\,\,\,\,\,\ldots \;\;\;\;\;\;\\r_{N-1}\;\;r_{N-2}\;\;\;\ldots \;\;\;\;r_{0}\,\\\end{array}}\right]\left[{\begin{array}{l}k\\k_{1}\\\ldots \\k_{N-1}\,\\\end{array}}\right]=\left[{\begin{array}{l}r_{1}\\r_{2}\\\ldots \\r_{N}\\\end{array}}\right],$

(21)

The solution of these normal equations gives the coefficients $k_{0},k_{\rm {1}},\ldots ,k_{N-1}$ of the prediction operator for unit prediction distance. By equation 12, the prediction-error operator for a prediction distance of 1 is

${\begin{aligned}\left(a_{0}{\ ,\ }a_{1}{\ ,\ }a_{2}{,\dots ,}a_{N}\right)=\left({1,\ }-k_{0}{,\ }-k_{1}{,\dots ,}-k_{N-1}\right).\end{aligned}}$

(22)

The operator $a=\left({1\ ,\ }a_{\rm {1}}{,\ }a_{2}{,\dots ,\ }a_{N}\right)$ is a spiking operator or spiking filter (normalized so that the leading coefficient is 1). The spiking operator is necessarily minimum phase.

The convolutional model has two components: the minimum-phase wavelet and the white reflectivity. The spiking operator represents the means to obtain both components. The inverse of the spiking operator is the minimum-phase wavelet. Thus, the wavelet can be found from the spiking filter a with the relation $a*b={\delta }_{0}$, which is

${\begin{aligned}a_{0}\ b_{0}={1,\ }\;\;\;\;\;\sum _{i={0}}^{M}{a_{i}}b_{n-i}={0,\ }\;\;\;\;\;n={1,\ 2,\ 3,\dots }.\end{aligned}}$

(23)

This equation says that the minimum-phase wavelet b is the inverse of the spiking filter a — that is, $b=a^{-1}$. The spiking filter finally yields the reflectivity $\varepsilon$ with

${\begin{aligned}y=x*a=b*\varepsilon *b^{-1}=\varepsilon .\end{aligned}}$

(24)

This equation denotes spiking deconvolution. Thus, the spiking filter produces both the wavelet and the reflectivity. Deconvolution of the conventional model of the trace is now complete.

Once the spiking operator has been found, any other linear filter of given length can be obtained easily within given limits of computational accuracy. Suppose that the required filter q has an input b and desired output z. Then, the required filter has impulse response $q=z*a$. We can check this result by computing the output. The output of the filter is

${\begin{aligned}b*q=b*\left(z*a\right)\approx z.\end{aligned}}$

(25)

Once the normal equations for the spiking filter have been solved, no further set of normal equations needs to be solved for the given input. Any other filter can be obtained by merely convolving the required desired output with the spiking filter.

As an example, consider the problem of shaping the wavelet $b=(\ {1},0.5)$ into the desired output $d=(0.3,1)$. If we compute the least-squares shaping filter, we obtain

${\begin{aligned}f=\left({0.3012,0.8469,\ }-{0.4185,0.1993,\ }-{0.07971}\right).\end{aligned}}$

(26)

If we compute the least-squares spiking filter, we obtain

${\begin{aligned}a=\left({0.9971,\ }-{0.4927,0.2346,\ }-{0.09384}\right)\end{aligned}}$

(27)

which, when convolved with the desired output, gives

${\begin{aligned}f=\left({0.2991,0.8493,\ }-{0.4223,0.2065,\ }-{0.09384}\right).\end{aligned}}$

(28)

We see that the filters in equations 26 and 28 are the same, within the bounds of accuracy obtainable for filters of such short length.

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Wavelet Processing	Fine Points
Table of Contents (book)

Also in this chapter

• Model used for deconvolution
• Least-squares prediction and smoothing
• The prediction-error filter
• Gap deconvolution
• Tail shaping and head shaping
• Seismic deconvolution
• Piecemeal convolutional model
• Time-varying convolutional model
• Random-reflection-coefficient model
• Implementing deconvolution
• Canonical representation
• Appendix J: Exercises

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