Thin-bed vowels and heterolithic consonants

Seismologists see the world differently. Or, rather, they hear the world differently. Sounds become time series, musical notes become Fourier components. The notes we make with our vocal chords come from the so-called sonorants, especially the vowel sounds, and they look like this:

Consontants aren't as pretty, consisting of various obstruents like plosives and fricatives—these depend on turbulence, and don't involve the vocal chords. They look very different:

Geophysicists will recognize these two time series as being signal-dominated and noise-dominated, respectively. The signal in the vowel sound is highly periodic: a small segment about 12 ms long is repeated four times in this plot. There is no repeating signal in the consonant sound: it is more or less white noise.

When quantitative people hear the word periodic, their first thought is usually Fourier transform. Like a prism, the Fourier transform unpacks mixed signals into monotones, making them easier to examine and explain. For instance, the Fourier transform of a set of limestone beds might reveal the Milankovitch cycles of which I am so fond. What about S and E?

The spectrum of the consonant S is not very organized and close to being random. But the E sound has an interesting shape. It's quite smooth and has obvious repetitive notches. Any geophysicist who has worked with spectral decomposition—a technique for investigating thin beds—will recognize these. For example, compare the spectrums for a random set of reflection coefficients (what we might call background geology) and a single thin bed, 10 ms thick:

Notches! The beauty of this, from an interpreter's point of view, is that one can deduce the thickness of the thin-bed giving rise to this notchy spectrum. The thickness is simply 1/n, where n is the notch spacing, 100 Hz in this case. So the thickness is 1/100 = 0.01 s = 10 ms. We can easily compute the spectrum of seismic data, so this is potentially powerful.

While obvious here, in a complicated spectrum the notches might be hard to detect and thus measure. But the notches are periodic. And what do we use to find periodic signals? The Fourier transform! So what happens if we take the spectrum of the spectrum of my voice signal—where we saw a 12 ms repeating pattern?

There's the 12 ms periodic signal from the time series! 

The spectrum of the spectrum is called the cepstrum (pronounced, and sometimes spelled, kepstrum). We have been transported from the frequency domain to a new universe: the quefrency domain. We are back with units of time, but there are other features of the cepstral world that make it quite different from the time domain. I'll discuss those in a future post. 

Based on a poster paper I presented at the 2005 EAGE Conference & Exhibition in Madrid, Spain, and on a follow-up article Hall, M (2006), Predicting bed thickness with cepstral decomposition, The Leading Edge, February 2006, doi:10.1190/1.2172313.