# Difference between revisions of "Vibroseis deconvolution"

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The vibroseis source is a long-duration sweep signal in the form of a frequency-modulated sinusoid that is tapered on both ends. Just as a convolutional model was proposed for the marine seismogram given by equation (41), a similar convolutional model can be proposed for the vibroseis seismogram

 ${\displaystyle x(t)=s(t)\ast w(t)\ast e(t),}$ (42)

where x(t) is the recorded seismogram, s(t) is the sweep signal, w(t) is the seismic wavelet with the same meaning as in equation (41), and e(t) is the earth’s impulse response. Convolutions in equation (42) become multiplications in the frequency domain:

 ${\displaystyle X(\omega )=S(\omega )W(\omega )E(\omega ).}$ (43)

In terms of amplitude A(ω) and phase ϕ(ω) spectra, equation (43) yields

 ${\displaystyle A_{x}(\omega )=A_{s}(\omega )A_{w}(\omega )A_{e}(\omega )}$ (44a)

and

 ${\displaystyle \phi _{x}(\omega )=\phi _{s}(\omega )+\phi _{w}(\omega )+\phi _{e}(\omega ).}$ (44b)

Crosscorrelation of the recorded seismogram x(t) with the sweep signal s(t) is equivalent to multiplying equation ('44a) by As(ω) and subtracting ϕs(ω) from equation (44b). The correlated vibroseis seismogram x(t) therefore would have the following amplitude and phase spectra:

 ${\displaystyle A'(\omega )=A_{s}^{2}(\omega )A_{w}(\omega )A_{e}(\omega )}$ (45a)

and

 ${\displaystyle \phi \prime (\omega )=\phi _{w}(\omega )+\phi _{e}(\omega ).}$ (45b)
Figure 2.5-13  An outline of vibroseis correlation.

The inverse Fourier transform of ${\displaystyle A_{s}^{2}\left(\omega \right)}$ yields the autocorrelation of the sweep signal, which is called the Klauder wavelet k(t). Returning to the time domain, equations (45a,45b) yield

 ${\displaystyle x\prime (t)=k(t)\ast w(t)\ast e(t).}$ (46)

Figure 2.5-13 outlines the process of vibroseis correlation where the seismic wavelet w(t) has been omitted for convenience. Note that, following vibroseis correlation, the sweep s(t) contained in the recorded seismogram x(t) is replaced with its autocorrelogram — the Klauder wavelet k(t).

Since it is an autocorrelation, the Klauder wavelet is zero-phase. Convolution of k(t) with the assumingly minimum-phase wavelet w(t) yields a mixed-phase wavelet. Because spiking deconvolution is based on the minimum-phase assumption, it cannot recover e(t) properly from vibroseis data.

One approach to deconvolution of vibroseis data is to apply a zero-phase inverse filter to remove k(t), followed by a minimum-phase deconvolution to remove w(t). The amplitude spectrum of the inverse filter is defined as ${\displaystyle {1}/{A_{s}^{2}\left(\omega \right)}}$. In practice, problems arise because of zeroes in the spectrum that are caused by the band-limited nature of the Klauder wavelet. Inversion of an amplitude spectrum, which has zeroes, yields an unstable operator (Optimum wiener filters). To circumvent this problem, a small percent of white noise, say 0.1%, usually is added before inverting the Klauder wavelet spectrum.

Another approach is to design a filter that converts the Klauder wavelet to its minimum-phase equivalent [1]. A technique to compute the minimum-phase spectrum from a given amplitude spectrum is described in Section B.4 and is included in the discussion on frequency-domain deconvolution. If the Klauder wavelet were converted to its minimum-phase equivalent, then equation (45b) would take the form:

 ${\displaystyle \phi \prime (\omega )=\phi _{k}(\omega )+\phi _{w}(\omega )+\phi _{e}(\omega ).}$ (47)

If we assume that w(t) is minimum-phase and if we make k(t) minimum-phase, then the result of their convolution also is minimum-phase. Spiking deconvolution now is applicable since the minimum-phase assumption is satisfied.

There is a 90-degree phase difference in some vibrator systems between the control sweep signal and the baseplate response. As an option, we may want to subtract out this phase difference. Figure 2.5-14 shows the recommended sequence of operations for vibroseis processing.

Figure 2.5-14  A flowchart for vibroseis deconvolution.

Figure 2.5-15 shows how the flowchart in Figure 2.5-14 is used with a synthetic reflectivity series. By including the step to convert the Klauder wavelet into its minimum-phase equivalent before spiking deconvolution, a closer representation of the impulse response is produced as seen by comparing steps (k) and (l) with (m).

Although sound in theory, the above scheme may have problems in practice. Fundamental issues, such as whether the convolutional model given in equation (42) really represents what goes on in the earth, are not resolved.

Vibroseis data often are deconvolved as dynamite data, without converting the Klauder wavelet to its minimum-phase equivalent. An example of deconvolution of a correlated vibroseis record is shown in Figure 2.5-16.

Despite the fact that the basic minimum-phase assumption is violated for vibroseis data as compared to explosive data, spiking deconvolution without conversion of the Klauder wavelet to its minimum-phase equivalent seems to work for most field data. Figure 2.5-17 shows a set of correlated vibroseis records before and after spiking deconvolution. Prominent reflections after deconvolution are enhanced and reverberations are attenuated significantly. Nonetheless, the problem of tying vibrator lines to lines recorded with other sources, say dynamite, is more difficult if the vibrator data have not been phase-corrected. Field systems now exist to do minimum-phase vibroseis correlation in the field.