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.