Time & space sampling; aliasing
Learning objectives
- State the Nyquist frequency for both time (f_N = 1/(2Δt)) and space (k_N = 1/(2Δx))
- Predict the aliased frequency f' when a signal of frequency f is sampled below Nyquist
- Recognise why spatial aliasing in a receiver line ruins dip imaging
- Choose Δt and Δx that capture the signal bandwidth of interest
Every digital seismic signal is sampled — in time at Δt and in space at Δx. If you sample too coarsely, energy above the Nyquist frequency folds back into the captured band and masquerades as a ghost low-frequency signal. This is aliasing.
Temporal aliasing
Sample a true sine of frequency f at rate fs = 1/Δt. If f > fs/2 (the Nyquist), the reconstructed signal lives at f′ = |f − k·fs| for some integer k. You cannot tell f and f′ apart from samples alone; the spectrum above Nyquist is indistinguishable from a spectrum below it.
Spatial aliasing
Same thing in space. A plane wave crossing a receiver line with wavenumber k_x, sampled at 1/Δx, aliases when k_x > 1/(2Δx). For dipping events this shows up as a fake dip direction — a steeply-dipping reflector apparently going the wrong way in the CMP stack.
Why this controls bin size
When you pick a CMP bin size Δx, you are picking a maximum dip you can faithfully image. At a target dip angle α and dominant wavelength λ, the spatial Nyquist constraint is Δx < λ/(4 sinα). Push too far and dipping events alias into the stack, ruining migration. The entire “why 25 m bins?” question in Part 3 is answered right here.
References
- Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.
- Vermeer, G. J. O. (1990). Seismic Wavefield Sampling. SEG Geophysical Monograph 4.
- Vermeer, G. J. O. (2002). 3-D Seismic Survey Design. SEG Geophysical References 12.
- Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data (2 vols.). SEG Investigations in Geophysics 10.