Hooray for Fourier!
The theory of truth is a series of truisms - J.L Austin
The mathematical notion that any periodic function, no matter how jagged or irregular, can be represented as a sum of sines — called a Fourier series — is one of the most extraordinarily useful ideas ever. Ever! It is responsible for the theory of transmitting and recovering information, and yes, is ubiquitious in geophysics. Strikingly, any signal can be decomposed into an ensemble of sine waves. They are two different representations of the same object, two equal representations of the same information. Fourier analysis is this act of sending waves through a mathematical prism, breaking up a function into the frequencies that compose it.
To build an arbitrary signal, the trick is to mulitply each of the sines by a coefficient (to change thier amplitude) and to shift them so that they either add together or cancel (changing the phase). From the respective coefficients and phases of the composite sinusoids, one can reconstruct the original curve: no information is lost in translating from one state to the other.
So the wiggle trace we plot of the seismic waveform has bits of information partitioned across each of its individual sinusoids. The more frequencies it has, the more information carrying capacity it has. Think of it as being able to paint with a full color palette. The degree of richess or range is known as bandwidth. In making this example, I was surprised how few sine waves (only ten) it took to make a signal that actually looks like a bonafide seismic trace.
In 52 Things You Should Know About Geophysics, Mostafa Nagizadeh wrote an essay on the magic of Fourier; it's applications for geophysics data analysis. And he should know. In tomorrow's post, I will elaborate on the practical and economical issues we encounter making discrete measurements of continuous (analog) phenomena.
Evan Bianco
Here is a Google spreadsheet where I contructed this signal. You can play with the amplitudes, frequencies, and phases to make your very own signal. Try it out. Change a few values in green and watch what happens. Make some small changes, makes some big changes and see how sensitive the summation is on individual components.
Reader Comments (4)
You know, I recently had a discussion here at the office about the problems of the Fourier method. In particular the concern was whether gibbs oscillations were a problem. And should we therefore use a more robust transformation when decomposing frequency ranges.
@Toastar,
Ah yes, Gibbs. A subtle but important point when we are trying to reconstruct sharp edges. Indeed the earth's impedance has sharp edges, but as you know the sources in the seismic experiment seldom have such high frequencies. Does that mean that we are out of the woods? You pose an interesting question, whether Gibbs phenomenon hinders seismic inversion might in fact be a real concern. I'd like to point out that many geologists and geophysicists that I talk to find it a bit counter-intuitive that low frequencies are perhaps even more important than high frequencies for constructing the signal. I suppose that is why I am motivated to share figures like this one to reiterate this fact. The lowest frequency sine wave shown here is 10% of the signal. What this says is "high resolution" does not simply mean "high frequency", it means "full bandwidth", I find some people forget about the importance of low frequencies. If you don't record them, you are missing important information. Mauricio Sacchi helped me see that for the first time.
Thanks for your considerate comments as usual. If you come across any good examples of Gibbs causing heck in earth systems, please share. After writing this post, I wondered why I haven't done Fourier Analysis on logs yet. Certainly within a blocky channel sand (not unlike the square wave in the text book Gibbs example), you'd run into this problem. I'll put that on my to do list (a.k.a to putter with list).
Great didactical explanation of Fourier synthesis Evan!
@Mauricio,
Thanks for the compliment, I am glad you enjoyed it. The exercise I went through to make this figure was a good reminder about the synthesis of signals. What surprised me is the impact caused by changing the phase of only one or two sinusoids. I did this somewhat randomly, and I have no idea if this kind of notched phase shift is even possible with natural signals. So, I feel like there is a real importance in phase, and I for one, don't really know what phase means on a synthesized signal, phase wrapping and so on. I mean, I know the basics of what zero phase, and minimum phase, and phase rotation mean, but I feel like it is too seldom reported alongside power spectra, especially in industry geophysics software. I know Sam Kaplan, Tad, and yourself, are all wrapped up in phase, and I need to explore a few more exercises like this one to see what the fuzz is about. Spectral decomposition is all the rage, and it makes me wonder if phase can be brought into the spotlight.