Aliasing

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	4
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

The time t is a continuous real variable; that is, time t can take on any real value. However, in digital processing, the signals are sampled at equally spaced time instants. The time increment between adjacent sampling points is called the sampling interval. If we denote continuous time by t, the sampling interval by $\Delta t$, and the discrete time index by the integer n, then we have the fundamental relation $t{\ =}n\Delta t$.

Let us now apply the sampling concept to simple harmonic motion. We will see that the most important consequence of sampling a function at equally spaced time points is the phenomenon called aliasing. For example, in seismic recording, the sampling interval of 4 ms often is used. This causes frequencies greater than 1/(2)(0.004), namely 125 Hz (called the Nyquist frequency), to be aliased (see below). Figure 3a shows the amplitude spectrum of a wavelet with a 4-ms sampling interval. If the wavelet is decimated by resampling it at 8 ms (so that one-half of the values are thrown away), then the new Nyquist frequency becomes 1/(2)(0.008), which is 62.5 Hz. In Figure 3b, the portion of the spectrum beyond 62.5 Hz is folded back, as we see in Figure 3a. For this reason, the Nyquist frequency also is known as the folding frequency. The amplitude spectrum of the decimated wavelet is the sum of the parts shown in Figure 3b. This result is shown in Figure 3c (Robinson and Clark, 1991).

Figure 3.  (a) Amplitude spectrum of a wavelet with 4-ms sampling. (b) The folded amplitude spectrum. (c) The sum of the parts shown in (b).

Aliasing is commonplace, but we are so accustomed to it that we are only vaguely aware of its existence. Suppose we see an object at point A and then at point B, but our eye does not see the actual motion between A and B. Our mind interpolates this motion and makes the apparent path the shortest distance between A and B. Many of the magician’s sleight-of-hand tricks, such as the pea under the walnut shell, are based on this phenomenon.

Many of the old western movies had scenes with wagons and stagecoaches with wheels containing spokes. The spokes let us see the motion of the wheel. But what did we see in our mind? As the stagecoach started up and increased in speed, we saw the wheels turn faster and faster in the forward direction. Then suddenly they would reverse direction and slow to a stop, and the process would be repeated. The wheels would turn faster, reverse direction, slow to a stop, and so on. Our mind saw the wheels go backward as the coach went forward, but our eyes saw only a sequence of still pictures making up the moving-picture film. In fact, a moving-picture film represents a sampling of the actual physical motion being photographed. Any actual high rate of rotation of the wheel appears, as a result of the sampling process, to be aliased into a lower rotational frequency.

Consider the case of a single spoke, which we represent with the vector $e^{i\omega t}$. We sample this exponential function at times $t{=}n\Delta t$ so our sampled function is $e^{i\omega n\Delta t}$. At low speeds, $\omega$ is small; for example, suppose $\omega \Delta t{=}\pi {/6}$ radians, which is $\omega \Delta t{=}{3}0^{o}$. Then our sample function is the sequence of vectors $e^{i\left(\pi {\rm {/6}}\right)n}$. We show these vectors for n = 0, 1, 2 in Figure 4a.

The mind interprets this motion as the slowest motion that can account for the actual observations. In the above case, the apparent rotation is the same as the actual rotation. At a higher speed, $\omega$ is larger. For example, suppose $\omega \Delta t{\ =}{\ 11}\pi {\ /6,}$, which is $\omega \Delta t{\ =33}0^{\ o}$. Then our sample function is the sequence of vectors $e^{i\left({\rm {1l}}\pi {\rm {/6}}\right)n}$. We show these vectors for n = 0, 1, 2 in Figure 4b.

Figure 4.  (a) Three discrete positions of a slowly rotating vector. The apparent motion and the actual motion are the same — counterclockwise. (b) Three positions of a rapidly rotating vector. The actual motion is counterclockwise, but the apparent motion is clockwise.

Again, the mind sees the smallest rotation velocity that accounts for the observations. We thus see an apparent $\omega \Delta t{\rm {\ of}}-\pi {\rm {/6\ =}}-{\rm {3}}0^{\rm {o}}$ instead of the actual $\omega \Delta t$ of ${\rm {11}}\pi /{\rm {6=33}}0^{\rm {o}}$. That is, we see an apparent angular frequency of

${\begin{aligned}&\omega {\ =\ }-{\frac {\pi }{{\rm {6}}\Delta t}}\end{aligned}}$

(10)

instead of the actual angular frequency of

${\begin{aligned}\omega {\ =\ }{\frac {{\rm {ll}}\pi }{{\rm {6}}\Delta t}}.\end{aligned}}$

(11)

We see that the actual angular frequency has been aliased to the apparent angular frequency. In this case, we have to subtract the Nyquist range $2\ \pi {\rm {/}}\Delta t$ from the actual angular frequency to obtain the apparent frequency; that is,

${\begin{aligned}{\frac {{\rm {11}}\pi }{{\ 6}\Delta t}}-{\frac {{\rm {l2}}\pi }{{\ 6}\Delta t}}{\rm {=}}-{\frac {\pi }{{\rm {6}}\Delta t}}.\end{aligned}}$

(12)

The critical points occur when either $\omega {\ =}-\pi {\rm {/}}\Delta t$ or $\omega {\ =}\pi {\rm {/}}\Delta t$. When $\omega {\ =}\pi {\rm {/}}\Delta t,$, for example, the sample function is the sequence of vectors

${\begin{aligned}e^{i\omega {\rm {t}}}{\rm {=}}e^{i{\frac {\pi }{\Delta t}}\Delta tn}{\rm {=}}e^{i\pi n}{\rm {=}}{\left(e^{j\pi }\right)}^{n}{\rm {=}}{\left(-{\rm {l}}\right)}^{n}.\end{aligned}}$

(13)

We show these vectors in Figure 5. The vector just flops back and forth between +1 and –1, so the apparent motion could be either way.

Figure 5.  Positions of a sample vector flipping back and forth. The actual rotation is clockwise.

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Table of Contents (book)

Also in this chapter

• Time series
• The wavelet
• Digitization
• Frequency
• Sinusoidal motion
• The Nyquist frequency
• Sampling geophysical data
• Appendix D: Exercises

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