Cepstral decomposition

Conventional seismic data do not resolve thin beds. Below a thickness of one quarter of a wavelength, the top and base of a thin bed cannot be interpreted accurately. Even at thicknesses greater than but close to this limit, seismic resolution is compromised.

Cepstral decomposition is a new approach to measuring bed thickness even when the bed itself cannot be interpreted. The technique builds on the widely used process of spectral decomposition. Spectral decomposition has emerged recently as an enlightening seismic attribute, producing very informative maps of thin beds, especially in clastic successions with sharp impedance contrasts. These maps are typically interpreted qualitatively, using geomorphologic pattern-recognition, or semi-quantitatively, to infer relative thickness variation. A number of commercial and noncommercial implementations of spectral decomposition are now in use, and the technique is de riguer in the analysis of subtle stratigraphic plays and fractured reservoirs.

There is also considerable scope for applying spectral decomposition quantitatively. As shown in Figure 1, the spectrum of a thin bed has characteristic periodic notches resulting from interference between the signals reflected from the top and base of the thin bed. The spacing of the notches is the reciprocal of the two-way time thickness of the thin bed. Unfortunately, estimating the notch spacing in real data is difficult, because the notches are commonly somewhat cryptic. To solve this problem, I propose using Fourier analysis to elicit the notch spacing. This transforms the spectrum into the cepstrum, and the seismic trace itself beyond the frequency domain into the realm of quefrency. The cepstrum is unfamiliar to most seismic interpreters but well known in several other branches of signal processing.

History
During the Cold War, the United States government was quite concerned with knowing about when and where nuclear test detonations were happening. One technique they employed was seismic monitoring. In order to discriminate between nuclear detonations and earthquakes, a group of mathematicians from Bell Telephone Laboratories proposed detecting and timing echoes in the seismic recordings. These echoes gave rise to periodic but cryptic notches in the spectrum, the spacing of which was inversely proportional to the timing of the echoes. This is exactly analogous to the seismic response of a thin-bed. To measure notch spacing, Bogert, Healy and Tukey (1963 ) invented the cepstrum (an anagram of spectrum and therefore usually pronounced kepstrum). The cepstrum is defined as the Fourier transform of the natural logarithm of the Fourier transform of the signal: in essence, the spectrum of the spectrum. To distinguish this new domain from time, to which is it dimensionally equivalent, they coined several new terms. For example, frequency is transformed to quefrency, magnitude to gamnitude, phase to saphe, filtering to liftering, even analysis to alanysis. Only cepstrum and quefrency are widespread today, though liftering is popular in some fields.

Application
Today, cepstral analysis is employed extensively in linguistic analysis, especially in connection with voice synthesis. This is because, as shown in Figure 2, voiced human speech (consisting of vowel-type sounds that use the vocal chords) has a very different signature in time and frequency from unvoiced speech. These differences are most easily quantified in the cepstrum, in which prominent peaks appear at characteristic quefrencies for voiced speech. These peaks are completely absent for unvoiced speech, which is noise- based and contains no resonant components. Cepstral methods help speech synthesizers must switch seamlessly between two different filters to make convincing human speech.

What is the cepstrum?
To describe the key properties of the cepstrum, we must look at two fundamental consequences of Fourier theory. Firstly, convolution in time is equivalent to multiplication in frequency. Secondly, the spectrum of an echo contains periodic peaks. Let us look at these in turn. A noise-free seismic trace s can be represented in the time t domain by the convolution of a wavelet w and reflectivity series r


 * $$ s(t) = w(t) \ast r(t) \ $$

Then, in the frequency f domain


 * $$ S(f) = W(f) \times R(f) \ $$

In other words, convolution in time becomes multiplication in frequency. The cepstrum is defined as the Fourier transform of the log of the spectrum. Thus, taking logs of the complex moduli


 * $$ \ln|S| = \ln|W| + \ln|R| \ $$

Since the Fourier transform F is a linear operation, the cepstrum is


 * $$ F[\ln|S|] = F[\ln|W|] + F[\ln|R|] \ $$

We can see that the spectra of the wavelet and reflectivity series are additively combined in the cepstrum. This property has given rise to its use in deconvolution and signal estimation. It could also allow the removal of wavelet effects from the cepstrum for more accurate bed thickness estimation. I am currently working on this possibility.

How does the cepstrum respond to a thin bed? To study this, the reflectivity series r can be represented as the sum of two equal and opposite signals, one of which is delayed by time &tau;:


 * $$ r(t) = x(t) - x(t - \tau) \ $$

The Fourier transform of a delayed signal is obtained by multiplication by 2cos(2&pi;f&tau;), so


 * $$ R(f) = X(f) \left \{ 2 - 2 \cos (2\pi f \tau) \right \} \ $$

Calculating the cepstrum as before gives


 * $$ F[\ln|R|] = F[\ln|X|] + F[\ln \left \{ 2 - 2 \cos (2\pi f \tau) \right \}] \ $$

So adding an echo to the original signal has the effect of adding a cosinusoidal pulse with period &tau; to the cepstrum (Bogert et al. 1963). Cepstral analysis is concerned with detecting this pulse and measuring its period. In practice, I have found that taking the log of the spectrum sometimes has an adverse effect on the results. It is not a necessary step for simply finding a periodic 'signal' in the spectrum. Also, there is no reason why any other transformation into the frequency domain could not be used—for example, the wavelet transform, S-transform, or Wigner-Ville transform may offer better temporal resolution than the Fourier transform (FT). In this paper, however, I have used the 'true' cepstrum, that is FT(ln(FT(signal))), taking the complex modulus of the spectrum at each step. Variants, where another transform is used, or where logs are not taken, could be called pseudo-cepstra.