Signature deconvolution - book

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 9
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

How is signature deconvolution implemented? A first empirical rule for seismic processing is that known things should be removed first. Once the known signature has been removed, the task of removing the unknown things becomes easier. A second (and obvious) empirical rule says that if an operation can be carried out simply, then a more complicated way to achieve the same result should not be attempted. If these rules are followed, removal of a known air-gun ringing wavelet or a known instrument response or a known vibroseis sweep signal becomes simple and easy to understand. All these methods have in common the removal of a known signature wavelet. The method is the same, no matter whether the known waveform is an air-gun signature, an instrument response, or a vibroseis sweep. The known quantities are the trace x and the signature s. The desired quantity is the signature-free trace z. The model is . Signature deconvolution involves the following steps:

1) Given the signature s, compute its least-squares spiking filter f.

2) Convolve the spiking filter f with the signature s to obtain the all-pass filter p, as shown above; that is, compute .

3) Convolve the field trace x with the reverse of the all-pass filter to obtain the dephased trace y; that is, compute .

4) Convolve the dephased trace y with the spiking filter f to obtain the signature-free trace z; that is, compute .

Alternatively, steps 3 and 4 can be combined by first computing the two-sided inverse signature and then computing the signature-free trace .

Figures 11 though 15 describe a simplified example of vibroseis deconvolution. Figure 11a shows the signature s, which is a swept-frequency signal. Figure 11b shows the field trace x, which is the convolution of a reflectivity with the signature s. Figure 12a shows the corresponding Klauder wavelet, which is the autocorrelation of the signature s. This autocorrelation is used to compute the spiking filter f. Figure 12b shows the spiking filter f. Figure 13a shows the all-pass wavelet p, which we obtain by convolving f with s. Figure 13b shows the minimum-delay counterpart b of the signature s. The minimum-delay counterpart b of the signature s, which is the inverse of the filter f, is the wavelet remaining on the dephased trace y in place of the original signature s. Figure 14a shows the dephased trace y. Figure 14b shows the signature-free trace . However, reverberating energy still remains on the signature-free trace, so we introduce predictive (spiking) deconvolution. Figure 15a shows the predictive (spiking) deconvolution filter computed from the signature-free trace. Figure 15b shows the trace we obtain after predictive (spiking) deconvolution, and that trace is an estimate of the approximate reflectivity .

Figure 11.  (a) Swept-frequency signature s. (b) Field trace x.
Figure 12.  (a) Autocorrelation r of the signature. (b) Spiking filter f.
Figure 13.  (a) All-pass wavelet p. (b) Minimum-delay counterpart b.
Figure 14.  (a) Dephased trace y. (b) Signature-free trace z.
Figure 15.  (a) Spiking deconvolution filter. (b) The approximate reflectivity .


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