# White convolutional model

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 |

What is the white convolutional model? The white convolutional model for the seismic trace *x* is

**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/":): {\displaystyle \begin{align} x= w*\varepsilon . \end{align}}****(**)

In this model, *w* is the field wavelet and **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/":): {\displaystyle \varepsilon }**
is a white-reflection-coefficient series. The *field wavelet* can be represented by the triple convolution

**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/":): {\displaystyle \begin{align} w = s*i*a*m, \end{align}}****(**)

where *s* is the *source wavelet*, *a* is the *absorption filter*, *m* is the overall signal representing the *multiples*, and *i* is the response of the receiving *instruments*. These instruments include the geophones or hydrophones, the amplifiers, and the modulation equipment. Often the actual source wavelet, the instrumentation response, and certain near-surface effects all can be lumped together under the rubric of *signature wavelet*. The field wavelet *w *is necessarily causal, but it will not necessarily be minimum phase unless *all* its components are minimum phase.

As seismic waves propagate through the earth, part of the wave energy is transformed into heat. This inelastic absorption phenomenon has been an active research topic for many years. Experimental results suggest that as a seismic wavelet travels through homogeneous rock, it systematically loses high frequencies to heat according to a process that can be represented by the action of a linear convolutional filter, the so-called absorption filter (see above). This absorption filter *a *simulates the real earth’s absorption response and has a magnitude spectrum approximately given 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/":): {\displaystyle {\rm \ exp\ }\left(-\alpha d\right)}**
, where **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/":): {\displaystyle \alpha }**
is the distance attenuation coefficient in units of reciprocal distance and *d* is the distance that the pulse has traveled. For most dry rocks, **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/":): {\displaystyle \alpha }**
is roughly proportional to the first power of frequency at frequencies used in the field and is given 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/":): {\displaystyle \alpha = \omega {\rm /2}vQ}**
, where **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/":): {\displaystyle \omega }**
is angular frequency, *v* is velocity, and *Q* is the seismic quality factor, which for most dry rocks is independent of **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/":): {\displaystyle \omega }**
in the seismic frequency band (Aki and Richards, 1980^{[1]}). In addition, it can be shown that the phase spectrum of the earth filter is minimum phase. This absorption response is approximately removable by a process we can call *absorption deconvolution*. It consists of the convolution of a given trace *x* with the inverse **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/":): {\displaystyle a^{-1}}**
of the minimum-phase absorption filter *a*.

## References

- ↑ Aki, K., and P. G. Richards, 1980, Quantitative seismology: Freeman and Company.

## Continue reading

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Spiking filter | Wavelet processing |

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Synthetics | Deconvolution |

## Also in this chapter

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