Sampling and Aliasing

Part 1, Part 1: Wavelets, the Source

Learning objectives

Define the Nyquist frequency as half the sample rate
See that frequencies above Nyquist fold back to lower aliases
Recognise that an alias cannot be removed once it is in the samples
Choose a sample interval that keeps a wavelet below Nyquist

Continuous Physics on a Discrete Grid

Every wavelet in this course lives in continuous time, but the moment you compute a synthetic you put it on a sample grid at some interval $\Delta t$ . That grid has a hard ceiling. It can faithfully carry frequencies only up to the Nyquist frequency,

$f_N = \dfrac{1}{2\,\Delta t}$ N=dfrac12,Deltat

half the sample rate. Frequencies below $f_N$ N are recorded honestly. Frequencies above it are not lost, which would be merciful; they fold back and reappear as lower frequencies. That impostor is an alias.

An Alias Cannot Be Undone

Raise the signal frequency past the dashed Nyquist line and watch the reconstruction, the smooth curve the samples actually support, peel away from the true signal and settle onto a lower frequency. The dots are identical to those a genuine low-frequency signal would produce, so no later processing can tell the two apart. Aliasing is not noise you can filter; it is ambiguity baked into the samples.

The folding chart on the right says it in one line: feed in any true frequency and read off the apparent frequency, folding at every multiple of Nyquist. For synthetic modelling the rule is simple. Pick $\Delta t$ small enough that the highest frequency in your wavelet stays comfortably below $f_N$ N. A 2 ms sample interval gives a 250 Hz Nyquist, ample for a 60 Hz seismic wavelet, which is why 1 to 4 ms is the usual modelling grid.