# Vibroseis

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

Should vibroseis records be correlated? The signature of a vibroseis record is the sweep signal, which is recorded separately from the received seismic trace. The Klauder wavelet is the autocorrelation of this signature. In vibroseis processing, it is traditional (as the first step) to correlate the received trace with the signature (i.e., the sweep signal), which replaces each sweep signal on the trace by the Klauder wavelet. The Klauder wavelet is a broadband zero-phase signal, so it is an ideal interpreter wavelet.

During earlier days, the correlated vibroseis traces constituted the final result for interpretation. Today, we no longer consider it prudent merely to correlate the vibroseis trace with the swept-frequency signal because we now realize that this step inserts an interpreter wavelet at the front end of the processing sequence. Most processing is designed to deal with physical wavelets, which are causal. But the zero-phase interpreter wavelet is noncausal and two-sided and thus interferes with standard processing. In other words, the older initial correlation for vibroseis records should not be carried out. It only adds trouble. The signature-free trace described above avoids this trouble. Ideally, signature deconvolution should be carried out by each crew in the field so that processing centers receive clean records that are compatible with those coming from other seismic crews (Robinson and Saggaf, 2001[1]).

Signature deconvolution works for air-gun records at sea as well as for vibroseis records on land (or sea). By removing the source signature, the resulting signature-free trace is the response of the earth and the receiver system to a spike impulse. Signature deconvolution requires knowledge of the signature wavelet to remove it from the trace. By implementing signature deconvolution at the front end of the processing sequence, fewer remnants of the signature will have to be removed at a later processing stage by predictive deconvolution.

We now will deal with the case of correlating field vibroseis records during acquisition. Correlated vibroseis records are still quite common, so it is useful to know how to process them. How can correlated vibroseis records be processed? Combining equation 2 of Chapter 8 and equation 31 above for the trace, we obtain

 {\displaystyle {\begin{aligned}x=s*m*\varepsilon =s*z,\end{aligned}}} (47)

where z is the source-free trace.

Correlation of s with the field trace is the same as convolution of the reverse ${\displaystyle s^{R}}$ with the field trace. The reverse is taken with respect to time zero. Thus, the correlated trace is

 {\displaystyle {\begin{aligned}u=x*s^{R}=s*z*s^{R}=\left(s*s^{R}\right)*z=k*z,\end{aligned}}} (48)

where ${\displaystyle k=s*s^{R}}$ is called the Klauder wavelet. The processing center receives only the correlated trace u and the Klauder wavelet k. The canonical representation is ${\displaystyle s=b*p}$, so the Klauder wavelet is

 {\displaystyle {\begin{aligned}k=s*s^{R}=b*p*{\left(b*p\right)}^{R}=b*b^{R}.\end{aligned}}} (49)

Thus, we see that the correlation process has wiped out the extra phase contained in the all-pass system.

In a common method of vibroseis deconvolution, each Klauder wavelet is replaced by its minimum-phase counterpart. The minimum-phase counterpart is removed later by predictive deconvolution. However, such a procedure is not always prudent because it adds an extra load to the task of predictive deconvolution. It usually is better to remove the entire Klauder wavelet before predictive deconvolution. The following method does this:

1) Observe that the Klauder wavelet is in fact the autocorrelation r that enters into the normal equations for computation of the spiking filter f for the swept-frequency signature. As we showed above, the spiking filter is approximately the inverse of the minimum-delay counterpart b of the swept frequency signature; that is, ${\displaystyle f\approx b^{-{1}}}$. The correlation process has wiped out the all-pass wavelet; now the spiking filter gives us the means to eliminate also this minimum-delay counterpart.

2) Convolve the correlated trace u with the spiking filter f and then convolve the result with the reverse of the spiking filter. The final result is the signature-free trace

${\displaystyle f^{R}*f*u=f*f^{R}*s*s^{R}*z}$

 {\displaystyle {\begin{aligned}=f*f^{R}*b*b^{R}*z=\left(f*b\right)*\left(f^{R}*b^{R}\right)*z=z.\end{aligned}}} (50)

An ideal Klauder wavelet has a zero-phase spectrum and a positive constant-amplitude spectrum over the sweep band and is zero elsewhere. However, because such an ideal is not achievable, we must specify amplitudes for the spectrum outside the sweep. A usual way to do so is by adding white noise in the form of a small constant value (say, ${\displaystyle \varepsilon }$) at each frequency. From the resulting amplitude spectrum, the minimum-phase counterpart of the Klauder wavelet then can be determined. In essence, the amplitude spectrum of the ideal Klauder wavelet is altered by ${\displaystyle \varepsilon }$, so the resulting actual Klauder wavelet depends on the selected value of ${\displaystyle \varepsilon }$.

## References

1. Robinson, E. A., and M. M. Saggaf, 2001, Klauder wavelet removal before vibroseis deconvolution: Gephysical Prospecting, 49, 335-340.