# Signature deconvolution - book

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 filterf.2) Convolve the spiking filter

fwith the signaturesto obtain the all-pass filterp, as shown above; that is, compute .3) Convolve the field trace

xwith the reverse of the all-pass filter to obtain the dephased tracey; that is, compute .4) Convolve the dephased trace

ywith the spiking filterfto obtain the signature-free tracez; 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 .

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Nonminimum-delay wavelet | Vibroseis |

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Synthetics | Deconvolution |

## Also in this chapter

- Wavelets
- The shaping filter
- Spiking filter
- White convolutional model
- Wavelet processing
- All-pass filter
- Convolutional model
- Nonminimum-delay wavelet
- Vibroseis
- Dual-sensor wavelet estimation
- Deconvolution: Einstein or predictive?
- Summary
- Appendix I: Exercises