Examples
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| Series | Geophysical References Series |
|---|---|
| Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |
| Author | Enders A. Robinson and Sven Treitel |
| Chapter | 8 |
| DOI | http://dx.doi.org/10.1190/1.9781560801610 |
| ISBN | 9781560801481 |
| Store | SEG Online Store |
Let us now discuss a model consisting of a surface interface overlying two buried interfaces, with the three interfaces separated by arbitrary two-way layer traveltimes. Let the reflection coefficients be given by a, b, c. Let the two-way traveltime between the surface and the first buried interface be S, and let the two-way traveltime between the first buried interface and the second buried interface be T. In other words, the surface reflection coefficient is $ {\varepsilon }_{0}=a $, the reflection coefficient for the first buried interface is $ {\varepsilon }_{S}=b $, and the reflection coefficient for the second buried interface is $ {\varepsilon }_{T+S}=c $. The Z-transform of the reflectivity is
$ {\begin{aligned}a+bZ^{S}+cZ^{S+T}.\end{aligned}} $ ()
The right-hand side of the autocorrelation of the reflectivity is
$ {\begin{aligned}g_{0}+g_{S}Z^{S}+g_{T}Z^{T}+g_{S+T}Z^{S+T}=g_{0}+abZ^{S}+bcZ^{T}+acZ^{S+T}.\end{aligned}} $ ()
If we replace $ g_{0} $ by 1, we obtain the feedback loop
$ {\begin{aligned}{1}+abZ^{S}+acZ^{T}+acZ^{S+T}.\end{aligned}} $ ()
There are three reverberations. The reverberation between the surface and the first buried interface contributes $ g_{S}=a\ b\ Z^{s} $, the reverberation between the second and third buried interfaces contributes $ g_{T}=b\ c\ Z^{T} $, and the reverberation between the surface and third buried interface contributes $ g_{T+S}=a\ c\ Z^{S+T} $. The synthetic trace now is given by the feedforward-feedback filter (Figure 11):
$ {\begin{aligned}{\frac {bZ^{S}+cZ^{S+T}}{{1}+abZ^{S}+bcZ^{T}+acZ^{S+T}}}.\end{aligned}} $ ()
Suppose that $ S={2,}T={=5} $. Then the synthetic trace is given by
$ {\begin{aligned}{\frac {bZ^{2}+cZ^{7}}{{1}+abZ^{2}+bcZ^{5}+acZ^{7}}},\end{aligned}} $ ()
which is
$ {\rm {shot}},0,b,0,-\left(ab^{2}\right),{\rm {0,}}a^{2}b^{3},c-b^{2}c,-\left(a^{3}b^{4}\right),2ab\left(-{1}+b^{2}\right)c,a^{4}b^{5}, $
$ 3a^{2}b^{2}\left(1-b^{2}\right)c{,\ }-\left(a^{5}b^{6}\right)-bc^{2}+b^{3}c^{2}{\ ,\ 4}a^{3}b^{3}\left(-{1+}b^{2}\right)c, $
$ a\left(a^{5}b^{7}-c^{2}+{4}b^{2}c^{2}-2b^{2}c^{2}\right){\ ,\ 5}a^{4}b^{4}\left(1-b^{2}\right)c, $
$ {\begin{aligned}a^{2}b\left(-\left(a^{5}b^{7}\right)+{3}c^{2}-9b^{2}c^{2}+{6}b^{4}c^{2}\right){\ ,\ }b^{2}\left(-{1+}b^{2}\right)c\left(6a^{5}b^{3}-c^{2}\right),\dots .\end{aligned}} $ ()
The shot occurs at time 0, the first primary b occurs at time 2, the first-surface- first-interface multiple $ {\text{ – }}ab^{2} $ occurs at time 4, the second-surface–first-interface multiple $ a^{2}b^{3} $ occurs at time 6, the second primary occurs at time 7, the third-surface–first-interface multiple $ -\left(a^{3}b^{4}\right) $ occurs at time 8, the peg-leg multiple $ 2ab\left(-{1\ +}b^{2}\right)c $ occurs at time 9, and so on. Figure 11a shows the reflectivity for the case $ a\;=0.8;\;b=-0.4;\;c=0.7 $. The leading portion of the corresponding impulsive synthetic trace is shown in Figure 11b.
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| Synthetic seismogram with multiples | Small and white reflection coefficients |
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| Wavelets | Wavelet Processing |
Also in this chapter
- Introduction
- Polarity
- Reflection coefficients and transmission coefficients
- Ghost reflection
- Layer-cake model
- Synthetic seismogram without multiples
- Water reverberations
- Synthetic seismogram with multiples
- Small and white reflection coefficients
- Appendix H: Exercises
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