Compressed sensing fundamentals
Learning objectives
- State the compressed-sensing recovery bound: M ≳ K·log(N/K)
- Distinguish sparse recovery by L1 from Shannon sampling
- Identify the sparsifying transform for typical seismic data (curvelet / Radon)
- Quote realistic survey-cost savings (50–75% at equal image quality)
Shannon–Nyquist sampling says: to faithfully record a signal bandlimited to f_max, sample at 2·f_max. Translated to acquisition geometry, that is half-wavelength station spacing over the whole survey. Compressed sensing offers an alternative: if the signal is SPARSE in some basis (Fourier, wavelet, curvelet), as few as M ≳ K·log(N/K) random measurements suffice, where K is the number of non-zero sparsifying-domain coefficients.
Why sparsity matters
A shot gather dominated by a small number of dips is sparse in the curvelet or τ–p domain — most coefficients in those bases are zero. CS theory says such a signal can be recovered exactly from a random subset of M samples provided M clears the K·log(N/K) threshold. The recovery is NOT a simple inverse Fourier transform; it is the L1 minimisation min |x|₁ subject to A_M x = y, where A_M is the sampling operator.
Why not linear reconstruction?
If you zero-fill the missing samples and inverse-DFT, the missing-sample spectrum spreads energy across every frequency — the reconstructed signal is not sparse, and the true non-zero coefficients are buried under noise. L1 reconstruction explicitly searches for the sparsest solution consistent with the measurements; there is no cheaper way to recover the true signal.
Practical acquisition savings
In seismic, CS typically allows 50–75% shot-count reduction at equal final-image quality. Mobil–Delaware compressed sensing tests (2014) reduced land-survey shot count by 60% with equivalent imaging; Shell marine NOS surveys achieved similar savings. Processing cost rises by a factor of 3–10 (L1/curvelet solvers are slow) but that is compute, not crew day-rate — a much cheaper commodity.
References
- Berkhout, A. J. (2008). Changing the mindset in seismic data acquisition. The Leading Edge, 27(7), 924–938.
- 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.
- Vermeer, G. J. O. (2002). 3-D Seismic Survey Design. SEG Geophysical References 12.
- Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.