Convolutional model in the frequency domain

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
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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 11
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

An important alternative to processing by time-domain modeling is processing by spectral modeling (Rosa and Ulrych, 1991[1]). The convolutional model for a seismic trace x is , where w is the wavelet and is the reflectivity. We show an example of such a model in Figure 13a, in which the wavelet is minimum phase and the reflectivity is white. In Figure 13b, we see the resulting synthetic trace on the left and the deconvolved trace on the right. On the other hand, Figure 13c shows the minimum-phase wavelet autocorrelation on the left and the reflectivity autocorrelation on the right.

Figure 13.  (a) The two components of the convolutional model: a minimum-phase wavelet and a white reflectivity. (b) The trace obtained by the convolution of the minimum-phase wavelet and the white reflectivity in (a) and the spike-deconvolved trace. Compare the spike-deconvolved trace with the white reflectivity shown in (a). (c) The autocorrelation of the minimum-phase wavelet and the autocorrelation of the white reflectivity shown in (a). (d) The autocorrelation of the trace and the autocorrelation of the deconvolved trace shown in (b). (e) The energy spectrum of the minimum-phase wavelet shown in (a) and the power spectrum of the white reflectivity shown in (a). (f) The power spectra of the original trace shown on the left in (b) and of the deconvolved trace shown on the right in (b).
Figure 13. continued.

Convolution in the time domain is equivalent to multiplication in the frequency domain. The frequency spectrum of the trace is the product of the frequency spectrum of the seismic wavelet and the frequency spectrum of the reflectivity. Figure 13d shows the trace autocorrelation on the left and the autocorrelation of the deconvolved trace on the right. The energy spectrum is the squared magnitude of the amplitude spectrum. Thus, the energy spectrum of the trace equals the product of the energy spectrum of the seismic wavelet and the energy spectrum of the reflectivity. Figure 13e shows the energy spectrum of the seismic wavelet and the power spectrum of the reflectivity. Figure 13f shows the power spectra of the trace and of the deconvolved trace. The similarity in the overall shape between the energy spectrum of the wavelet and the power spectrum of the trace is apparent. In fact, a smoothed version of the power spectrum of the trace is nearly indistinguishable from the energy spectrum of the wavelet. The rapid fluctuations observed in the energy spectrum of the trace result from the high frequencies present in the reflectivity component, whereas the smoother and lower-frequency component is associated primarily with the wavelet.

The similarity between the energy spectrum of the minimum-phase wavelet and the power spectrum of the trace results from the near flatness of the power spectrum of the reflectivity. By definition, any infinitely long signal generated by a white-noise process has a power spectrum that is flat over its entire spectral bandwidth. Of course, in any real-life realization of a white-noise process, the signal is not infinitely long, and therefore the power spectrum is not entirely flat.

Now consider the wavelet, reflectivity, and trace autocorrelations. The autocorrelation of the wavelet (left panel of Figure 13c) is similar to the autocorrelation of the trace (left panel of Figure 13d). In addition, the reflectivity autocorrelation (right panel of Figure 13c) is similar to although not quite as sharp as the autocorrelation of the deconvolved trace (right panel of Figure 13d). Mathematically, the similarity between the trace autocorrelation and the wavelet autocorrelation indicates that the reflectivity has an autocorrelation with small magnitudes at all lags except for lag zero, as Figure 13c shows.

Experimental studies indicate that the reflectivity is never entirely a white-noise process. Spectral properties of reflectivity functions derived from a worldwide selection of sonic logs indicate that the reflectivity is closer to so-called blue noise — that is, to noise richer in the higher frequencies than in the lower frequencies. The autocorrelation of blue noise has a significantly large negative value at lags following lag zero. This is not the case for the autocorrelation of purely white noise. The positive zero-lag peak followed by the smaller negative peak in the autocorrelation of the impulse response thus arises from the “blueness” of the spectrum.


References

  1. Rosa, A. L. R., and T. J. Ulrych, 1991, Processing via spectral modeling: Geophysics, 56, 1244-1251.

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