Sampling & aliasing

Processing Prerequisites

Learning objectives

State and justify the Nyquist sampling theorem: fₛ > 2·fₘₐₓ to avoid aliasing
Compute the apparent (aliased) frequency of a signal that violates Nyquist
Explain why every acquisition system uses an analog anti-alias filter before the A/D
Recognize spatial aliasing in seismic data and its connection to bin size and wavelength

Continuous signals live on a continuum. Digital recorders do not. Somewhere between the geophone and the disk, a sampler says: “not every time, just these moments.” If the signal wiggles faster than the sampler can keep up, the record does not merely lose detail — it actively lies about what happened. That lie has a name: aliasing.

1. What sampling is

A continuous signal x(t) becomes a discrete sequence x[n] by evaluating it at regular instants t = n·Tₛ, where Tₛ is the sampling interval and fₛ = 1/Tₛ is the sampling rate. Every 2-millisecond seismic trace is fₛ = 500 Hz.

2. The Nyquist theorem

\boxed{\,\text{To record a signal of maximum frequency } f_{\max}\;\text{faithfully, you must sample at}\; f_s > 2 f_{\max}.}

The threshold fₛ / 2 is called the Nyquist frequency. Frequencies below Nyquist are recoverable; frequencies above it are not — they fold back down and impersonate lower frequencies. That folding is aliasing.

3. See it, then believe it

The widget below shows a continuous sinusoid in teal and the samples a sampling grid produces in orange. The dashed curve is the reconstructed signal — the lowest-frequency sinusoid that passes through the same sample dots. When the signal is below Nyquist, the reconstruction traces the truth. When the signal exceeds Nyquist, the reconstruction appears at an aliased frequency and the widget colors it red.

Try the preset “Aliased: 15 / 20.” The signal is 15 Hz, Nyquist is 10 Hz, so 15 Hz folds to |15 − 20| = 5 Hz. The samples are identical to what a true 5 Hz sinusoid would produce — there is no way, from the samples alone, to tell 15 Hz from 5 Hz. Information has been destroyed.

4. The folding formula

For a sinusoid at frequency fₛᵢᵍ, the apparent (aliased) frequency in [0, fₛ/2] is

f_{\text{app}} \;=\; \bigl|\,f_{\text{sig}} - k f_s\,\bigr|\qquad\text{for the integer } k \text{ that minimizes this distance.}

Practical recipe: take fₛᵢᵍ mod fₛ. If the result exceeds fₛ / 2, subtract it from fₛ. The remainder is f_app.

5. Anti-alias filtering

Since you cannot un-fold aliased frequencies after sampling, you must prevent them from being sampled in the first place. Every digital recorder has an analog anti-alias filter right before the A/D converter — a steep low-pass that kills all content above fₛ / 2.

This has a cost. The filter must be sharp (close to brick-wall) or high frequencies leak through; but sharper filters have worse time-domain behavior (ringing, phase nonlinearity). Modern seismic recorders use delta-sigma oversampling A/Ds that sample far above the final fₛ, apply a digital brick-wall filter, then decimate — shifting most of the pain to digital logic where it is easier.

6. Aliasing in seismic processing

Time-domain aliasing is rarely an issue in practice because acquisition vendors have solved it. The two places aliasing still bites processors every day are:

Spatial aliasing. Traces are sampled in space at intervals Δx (bin size or group interval). The spatial Nyquist wavenumber is kₙ = π / Δx. A dipping reflector at steep angle has high apparent wavenumber; if the bin size is too coarse, the reflector aliases in the f–k domain. You see this as “stair-stepping” on steep events and as f–k fans that wrap. Every migration algorithm has spatial-aliasing limits built in.
Processing-step downsampling. If you decimate a trace without first anti-alias filtering, you alias the high-frequency content into the new baseband. Always low-pass before you downsample. The rule is symmetric: always anti-alias before reducing sample rate, in time or in space.

7. Stroboscopes and helicopter blades

Aliasing is what makes a helicopter rotor appear to spin backwards on video — the camera’s 30-fps sampling is not fast enough to follow the true rotation frequency, so the rotor aliases to a low, sometimes negative, apparent rate. Try the preset “Stroboscope: 30 / 8”: a 30 Hz signal sampled at 8 Hz. The apparent frequency is 2 Hz — almost motionless.

**The one sentence to remember**

Sample above twice the highest frequency, or prevent higher frequencies from reaching the sampler. Without one of those, the record tells you lies about the signal — and there is no way to recover the truth after the fact.

Where this goes next

Section §0.6 introduces the Z-transform, the discrete-time counterpart of the Fourier transform that gives us a clean way to talk about digital filters — including the anti-alias filters we just said every recorder needs. Poles, zeros, causality, and stability all live in the Z-plane.

References

Oppenheim, A. V., Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall.
Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.