Sampling & aliasing

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 $t = n\\Delta t$, where $\\Delta t$ is the sampling interval and $f\_{\\mathrm s} = 1/\\Delta t$ is the sampling rate. A trace sampled every 2 ms has $f\_{\\mathrm s}$ = 500 Hz; one sampled every 4 ms, 250 Hz.

2. The Nyquist theorem

The threshold $f\_{\\mathrm N} = f\_{\\mathrm s}/2 = 1/(2\\Delta t)$ is called the Nyquist frequency: 250 Hz for 2 ms data, 125 Hz for 4 ms data. Frequencies below it are recoverable. Frequencies above it are not: they fold back down and impersonate lower frequencies, and that folding is aliasing. A sinusoid exactly at $f\_{\\mathrm N}$ is not safe either, because what the samples keep of it depends on where they fall on its cycle. The inequality is strict.

3. See it, then believe it

Below, you sample a sinusoid every 4 ms and compare it with what the samples say. Drag its frequency $f$ up past the 125 Hz Nyquist frequency and watch the dashed curve, which is rebuilt from the samples alone: below Nyquist it lies on the input, above it the same dots describe a different, slower signal. Then switch on the anti-alias filter, and try a broadband Ricker wavelet in place of the sinusoid.

The figure opens on the case that matters: a 200 Hz signal sampled every 4 ms. Its samples are exactly the samples of a 50 Hz sinusoid running backwards, because $200 - 250 = -50$ Hz, and the reconstruction in (a) passes through every dot while looking nothing like the input. From the samples alone there is no way to tell 200 Hz from 50 Hz. Information has been destroyed, and plate (c) shows where every other frequency lands.

4. The folding formula

For a sinusoid at frequency $f$, the apparent (aliased) frequency in \[0, f\_{\\mathrm N}\] is

Practical recipe: take $r = f \\bmod f\_{\\mathrm s}$. If $r > f\_{\\mathrm N}$, replace it with $f\_{\\mathrm s} - r$. The result is $f\_{\\mathrm{app}}$. The sign carries information too: when $f - k f\_{\\mathrm s}$ is negative, the samples trace a sinusoid whose phase runs backwards. The check under the figure asks for 170 Hz at 4 ms: $170 - 250 = -80$ Hz, so 80 Hz, running backwards.

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 removes content above $f\_{\\mathrm N}$. Switch it on in the figure at 200 Hz and the record goes silent instead of showing a false 50 Hz. Then drag $f$ down to 110 Hz: the figure’s filter ramps from $0.8 f\_{\\mathrm N}$ to zero at $f\_{\\mathrm N}$, so it passes only 0.60 of that signal’s amplitude. Protection costs the top of the band.

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 $f\_{\\mathrm s}$, 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 $\\Delta x$ (bin size or group interval). The spatial Nyquist wavenumber is $k\_{\\mathrm N} = 1/(2\\Delta x)$ cycles per metre, or $\\pi/\\Delta x$ radians per metre. An event crossing the spread with apparent velocity $V\_{\\mathrm{app}}$ has wavenumber $f/V\_{\\mathrm{app}}$ at frequency $f$, so it aliases above $f = V\_{\\mathrm{app}}/(2\\Delta x)$. A plane wave arriving at angle $\\theta$ from the vertical in rock of velocity $V$ has $V\_{\\mathrm{app}} = V/\\sin\\theta$, so steep events alias first. 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. Keeping every fourth sample of 2 ms data is sampling at 8 ms: in the figure, 8.3 % of a 40 Hz Ricker wavelet’s energy lies above the new 62.5 Hz Nyquist frequency, and the reconstruction misses the wavelet by 35 %. Always low-pass before you downsample. The rule is symmetric: always anti-alias before reducing the sample rate, in time or in space.

7. Stroboscopes and helicopter blades

Aliasing is what makes a helicopter rotor or a wagon wheel appear to spin backwards on film: the camera’s frames cannot follow the rotation, so the wheel aliases to a low, sometimes negative, apparent rate. Try the last exercise under the figure: a 240 Hz signal sampled every 4 ms lands at $240 - 250 = -10$ Hz, a slow 10 Hz sinusoid running backwards, and 260 Hz lands at +10 Hz, the same wheel turning forwards.

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.