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Wednesday
Oct242012

## N is for Nyquist

In yesterday's post, I covered a few ideas from Fourier analysis for synthesizing and processing information. It serves as a primer for the next letter in our A to Z blog series: N is for Nyquist.

In seismology, the goal is to propagate a broadband impulse into the subsurface, and measure the reflected wavetrain that returns from the series of rock boundaries. A question that concerns the seismic experiment is: What sample rate should I choose to adequately capture the information from all the sinusoids that comprise the waveform? Sampling is the capturing of discrete data points from the continuous analog signal — a necessary step in recording digital data. Oversample it, using too high a sample rate, and you might run out of disk space. Undersample it and your recording will suffer from aliasing.

### What is aliasing?

Alaising is a phenomenon observed when the sample interval is not sufficiently brief to capture the higher range of frequencies in a signal. In order to avoid aliasing, each constituent frequency has to be sampled at least two times per wavelength. So the term Nyquist frequency is defined as half of the sampling frequency of a digital recording system. Nyquist has to be higher than all of the frequencies in the observed signal to allow perfect recontstruction of the signal from the samples.

Above Nyquist, the signal frequencies are not sampled twice per wavelength, and will experience a folding about Nyquist to low frequencies. So not obeying Nyquist gives a double blow, not only does it fail to record all the frequencies, the frequencies that you leave out actually destroy part of the frequencies you do record. Can you see this happening in the seismic reflection trace shown below? You may need to traverse back and forth between the time domain and frequency domain representation of this signal.

Seismic data is usually acquired with either a 4 millisecond sample interval (250 Hz sample rate) if you are offshore, or 2 millisecond sample interval (500 Hz) if you are on land. A recording system with a 250 Hz sample rate has a Nyquist frequency of 125 Hz. So information coming in above 150 Hz will wrap around or fold to 100 Hz, and so on.

It's important to note that the sampling rate of the recording system has nothing to do the native frequencies being observed. It turns out that most seismic acquisition systems are safe with Nyquist at 125 Hz, because seismic sources such as Vibroseis and dynamite don't send high frequencies very far; the earth filters and attenuates them out before they arrive at the receiver.

### Space alias

Aliasing can happen in space, as well as in time. When the pixels in this image are larger than half the width of the bricks, we see these beautiful curved artifacts. In this case, the aliasing patterns are created by the very subtle perspective warping of the curved bricks across a regularly sampled grid of pixels. It creates a powerful illusion, a wonderful distortion of reality. The observations were not sampled at a high enough rate to adequately capture the nature of reality. Watch for this kind of thing on seismic records and sections. Spatial alaising.

Click for the full demonstration (or adjust your screen resolution).You may also have seen this dizzying illusion of an accelerating wheel that suddenly appears to change direction after it rotates faster than the sample rate of the video frames captured. The classic example is the wagon whel effect in old Western movies.

Aliasing is just one phenomenon to worry about when transmitting and processing geophysical signals. After-the-fact tricks like anti-aliasing filters are sometimes employed, but if you really care about recovering all the information that the earth is spitting out at you, you probably need to oversample. At least two times for the shortest wavelengths.

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Nicely written posts. Two enjoyable pieces on Fourier analysis.

October 24, 2012 | Henry

@Henry, Cheers! Make sure to check out the spreadsheet I posted as an update on the previous post. You'll probably laugh at the low tech display. But it's interactive and public, so that is a +1. Switching to a Mac and losing MATLAB, has cast me out into the cold, and my Python skillz aren't up to snuff yet.

October 25, 2012 | Evan

Great post Evan! Almost exactly as I think of Nyquist/aliasing!

I think with the case of the spinning wagon wheel, i believe you will first see the wheel appear to slow down (I think we could think about this as the freqs above nyquist folding about the nyquist to lower freqs, as you mention), until the point where the wheel accelerates so much that we then see the wheel appear to reverse direction (I think we can think of this as the point where the freqs are above twice nyquist so fold past 0hz to negative frequencies). I remember having a discussion about this in a processing course I was in once, but can't remember exactly how it goes so i could be wrong..I think we ended up in a debate about whether we observe it in real life driving down the freeway (which I think we do, not due to discrete sampling of our eyes but due to the amount of info our brain can process in a certain period of time)..i might have to do some reading on this now haha!

See you in Vegas? cheers for the post, Matt

October 25, 2012 | MattS

Nice post! With the geophones that my research group deployed in a glacier this summer we went for 1KHz 12-bit sensors oversampled to 16-bit and down to 500KHz. Excess data is a massive issue though as we send it back each day and only have so much battery life to do that. To deal with that we run an algorithm that only prompts recording of data above a certain threshold which hopefully gets any events. I've no idea if that's used throughout seismology though, I'm very much a novice in the subject.

October 25, 2012 | Alex Clayton

Excellent post but t does not answer a thought that has been with me for some time, in the frequency domain does a higher sample rate, that is more statistical evidence help with PSTM and PSDM or are we still bound by the frequencies that we have recorded. Intuitively having more samples should help with resolution but resolution is a function of bandwidth.

October 25, 2012 | Ea Craigie

@MattS,
I think you are right about the wagon wheel effect, and that the perceived slowing down and reversing of the motion has to do with the changing of the folding frequencies. However I don't know what the relationship is between reversing and slowing relative to Nyquist. I'm sure it's measurable if not derivable.
Sadly, I won't be at SEG, (divide and conquer right?), but Matt will. Be sure to track him down.

@Alex,
Your problem of long recordings reminds me of this behind the scenes look of how they did the amazing slow motion footage of the Great White Shark attacks offshore South Africa. Watch the video.

@Ea,
You are right. Temporal resolution is a function of bandwidth. As long as you are above Nyquist, over-sampling does not help or does not change resolution. However if you are under-sampling, you are (by definition) destroying resolution because of folding. There may be some subtleties that I am not aware of but, I think it is important to remember that the sampling frequency has nothing to do with the bandwidth of the signal itself. But, the sampling frequency should be set to record the entire signal properly. So I guess the two ideas are connected. I am not sure if I totally understand your notion of "statistical evidence". I would say that if you are over-sampling, you are not improving the "statistics" of the signal beyond a certain point. You cannot add any more information by sampling more data points, you can only take away information by not sampling enough. Does that make sense? It might be a good exercise to prove this to yourself, if you find yourself working this out on a real problem. There might be some issues around dealing with noise that I am not aware of, but that's my understanding of it. If you need better "statistical evidence" of the signal, then maybe what you are really trying to do is characterize the noise in the signal. If that is the case, then your acquisition needs to record the entire bandwidth for the noise (as well). It makes me think, "what is noise, and what is signal"? and "is noise white"?

October 25, 2012 | Evan

Most marine seismic I see is recorded at a 2ms sample interval also, with 1ms or even 0.5ms being used for high resolution work such as site surveys. Much more rapid sample intervals are used for very high resolution work, with boomer or sparker sources.

Originally, the source tow depth for marine data recorded at 2ms sample interval is typically designed so that the ghost "notch" cuts in around 125Hz - this acting in part as a "field" anti-alias filter so that the data could be resampled to 4ms (for processing) more easily

With "broadband" seismic the latest buzzword in the industry, of course, this is now an unwanted effect.

October 25, 2012 | Guy

@Guy,

Thanks for the comment. I need to learn (actually re-learn) what ghosting is. It is one of those things that I am sure I had an understanding of at one time, but I have forgotten after not being in the thick of it. Is ghosting caused by some kind of wave guide (or is it like a multiple) between the tow depth of the the hydrophones and the free water surface? Is it a simple function of depth, water velocity, and therefore wavelength?

I feel that the word broadband, like all buzzwords, actually does more harm than good. Everybody seems to have their own definition of what broadband means, so if there is no consensus, it ends up confusing people as opposed to actually quantifying what is being described. Is broadband being used to describe "above ghosting" , or "above Nyquist" or something?
I don't like when technical terms become part of marketing strategy.

I am also wondering if it would be helpful to more geoscientists if they were shown a to-scale cartoon of the streamer tow depth, and geometry. They might get an appreciation for vertical resolvability, no? Maybe it's something we should put in the headers and plot on sections.

Thanks for connecting!

October 25, 2012 | Evan Bianco