Nonuniform optimal sampling (NOS)

Compressed sensing (§8.1) only delivers its cost savings if the subsampling is INCOHERENT. A regular sub-Nyquist sampling pattern has sharp spectral peaks at 1/dx in wavenumber, aliased energy deposits onto coherent lines that an L1 solver cannot tell apart from true signal. A randomly jittered subsample has a broad white spectrum; aliased energy scatters as incoherent noise, which L1 treats as an outlier to be removed.

Sampling-operator spectrum

For a 1D station pattern x₁, x₂, …, x_M, the wavenumber spectrum |S(k)|² is the squared magnitude of ∑ e^(-ikx_j). A periodic set gives sharp delta-like peaks at k = 2πn/dx; a random set gives a flat pedestal with peaks only at k = 0. Jittering each station by ≈ 40-60% of the nominal spacing converts the first to the second.

Jitter amplitude sweet spot

Too little jitter (< 20% of spacing) and the spectrum still has coherent peaks, L1 recovery fails at the peak wavenumbers. Too much jitter (> 80%) and neighbouring samples overlap and add little new information. Industry rule of thumb: ≈50% jitter. The widget’s spectrum panel shows the transition visually.

Real-world NOS deployments

Mobil-Delaware NOS land survey (Mosher et al., 2014), 60% shot reduction with equal image quality. Shell marine nodal NOS (Gulf of Mexico, North Sea, multiple programmes 2015-2022), up to 75% shot reduction. BG Egypt offshore programmes also use NOS geometries. The common theme: crew-day savings dominate the extra processing cost by 5-10×.

References

• Berkhout, A. J. (2008). Changing the mindset in seismic data acquisition. The Leading Edge, 27(7), 924-938.
• Vermeer, G. J. O. (2012). 3D Seismic Survey Design (2nd ed.). SEG.
• Mahdad, A., Doulgeris, P., Blacquière, G. (2011). Separation of blended data by iterative estimation and subtraction of blending interference noise. Geophysics, 76(3), Q9-Q17.