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Thursday
Jun092011

## F is for Frequency

Frequency is the number of times an event repeats per unit time. Periodic signals oscillate with a frequency expressed as cycles per second, or hertz: 1 Hz means that an event repeats once every second. The frequency of a light wave determines its color, while the frequency of a sound wave determines its pitch. One of the greatest discoveries of the 18th century is that all signals can be decomposed into a set of simple sines and cosines oscillating at various strengths and frequencies.

I'll use four toy examples to illustrate some key points about frequency and where it rears its head in seismology. Each example has a time-series representation (on the left) and a frequency spectrum representation (right).

### The same signal, served two ways

This sinusoid has a period of 20 ms, which means it oscillates with a frequency of 50 Hz (1/20 ms-1). A sinusoid is composed of a single frequency, and that component displays as a spike in the frequency spectrum. A side note: we won't think about wavelength here, because it is a spatial concept, equal to the product of the period and the velocity of the wave.

In reflection seismology, we don't want things that are of infinitely long duration, like sine curves. We need events to be localized in time, in order for them to be localized in space. For this reason, we like to think of seismic impulses as a wavelet.

The Ricker wavelet is a simple model wavelet, common in geophysics because it has a symmetric shape and it's a relatively easy function to build (it's the second derivative of a Gaussian function). However, the answer to the question "what's the frequency of a Ricker wavelet?" is not straightforward. Wavelets are composed of a range (or band) of frequencies, not one. To put it another way: if you added monotonic sine waves together according to the relative amplitudes in the frequency spectrum on the right, you would produce the time-domain representation on the left. This particular one would be called a 50 Hz Ricker wavelet, because it has the highest spectral magnitude at the 50 Hz mark—the so-called peak frequency

### Bandwidth

For a signal even shorter in duration, the frequency band must increase, not just the dominant frequency. What makes this wavelet shorter in duration is not only that it has a higher dominant frequency, but also that it has a higher number of sine waves at the high end of the frequency spectrum. You can imagine that this shorter duration signal traveling through the earth would be sensitive to more changes than the previous one, and would therefore capture more detail, more resolution.

The extreme end member case of infinite resolution is known mathematically as a delta function. Composing a signal of essentially zero time duration (notwithstanding the sample rate of a digital signal) takes not only high frequencies, but all frequencies. This is the ultimate broadband signal, and although it is impossible to reproduce in real-world experiments, it is a useful mathematical construct.

Real seismic data, which is acquired by sending wavelets into the earth, also has a representation in the frequency domain. Just as we can look at seismic data in time, we can look at seismic data in frequency. As is typical with all seismic data, the example below set lacks low and high frequencies: it has a bandwidth of 8–80 Hz. Many geophysical processes and algorithms have been developed to boost or widen this frequency band (at both the high and low ends), to increase the time domain resolution of the seismic data. Other methods, such as spectral decomposition, analyse local variations in frequency curves that may be otherwise unrecognizable in the time domain.

High resolution signals are short in the time domain and wide or broadband in the frequency domain. Geoscientists often equate high resolution with high frequency, but that it not entirely true. The greater the frequency range, the larger the information carrying capacity of the signal.

In future posts we'll elaborate on Fourier transforms, sampling, and frequency domain treatments of data that are useful for seismic interpreters.

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Nice tutorial - hopefully (?!) your readership has a pretty good handle on that. The missing step I think many do not understand, thati is how we measure signal and the limitations of that step. In 2001 at the Canadian SEG meeting Norm Cooper and I presented a paper "Excuse me sir, will that be one millisecond or two?" Where we looked at the measurement and computational affects on bandwidth.

In short, I was annoyed that seismic contractors were convincing clients to sample at 1ms instead of 2ms (at the time with I/O System II). More samples means better data right? When in fact, for those clients, thanks to Fourier, 1ms was oversampling, but worse, happens in the box actually mathematically reduces the bandwidth.

A side comment too, that old I/O system had a slick little digital whitening filter that worked like a champ, misunderstood and feared though. It applied it's whitening to the data in the A/D converter - as the analog stream was being digitized, in the box. Since our great processors never get their hands on the data at that point, it was a very high quality step that could not be recreated as effectively, ever.

The point is I guess, geophysics is complicated...and there are pitfalls, and you need to know how your data is being sampled. Often us poor interpreters are stuck 'dealing with' what we are given - ie some other schmuck's attempt at data acquisition...

June 11, 2011 | Peter MacKenzie

@Peter: Thanks for the thoughtful comment. I found the abstract for your paper with Norm here (PDF). Thanks for that—I really liked the explanation of Nyquist, and the pitfalls of sample rate selection.

I've only worked on one project where we acquired with a 1 ms interval, at Surmont in the Athabasca, Alberta. We were getting 200+ Hz of signal out of the ground—it's very shallow (< 500 ms), and we were using small charges (125 g). Since a 0.8 × Nyquist field filter is normal (and necessary, because as you approach Nyquist amplitude preservation is a problem), we didn't want to risk missing the high frequencies.

I too hope this is basic stuff for most geophysicists, but I think interpreting geologists are sometimes stumped by the signal processing aspects of seismic (like sampling, aliasing, filtering); I know I was baffled by that stuff when I started out. We're hoping to use this primer as a foundation. I think we'll be referring to your abstract in the sampling edition.

June 12, 2011 | Matt Hall