SNR & detectability

Physics prerequisites for acquisition

Learning objectives

Define signal-to-noise ratio (SNR) in dB for a single trace
Derive the √N stacking rule: coherent signal grows as N, random noise grows as √N
Convert a required post-stack SNR into a required fold for known per-trace SNR
Recognise the limits of stacking (coherent noise doesn’t attenuate)

A reflection event you cannot see on a single trace may become obvious after stacking tens of traces. That is the stacking miracle: it is the single most important noise-fighting tool in seismic.

The √N rule

Suppose every trace has the same reflection (signal) and uncorrelated Gaussian noise. Stacking N traces (averaging them) gives:

\mathrm{signal\,power}_{\mathrm{stack}} = \mathrm{signal\,power}\quad\text{(coherent; unchanged after averaging)}

\mathrm{noise\,power}_{\mathrm{stack}} = \frac{\mathrm{noise\,power}}{N}

So the SNR improves by a factor of N in power, which is √N in amplitude, which is 10·log₁₀(N) dB. Stack 16 traces, gain 12 dB. Stack 100, gain 20 dB.

When stacking stops working

The √N gain assumes noise is uncorrelated trace-to-trace. Ground roll is coherent, it stacks just like signal does. Multiples are coherent, same problem. Residual statics (uncorrected time shifts) misalign the signal before summing, destroying coherence and eating into signal. This is why f-k filters, demultiple, and statics corrections exist: they clean up the traces so that stacking sees only uncorrelated residual noise, making the √N rule hold again.

References

Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data (2 vols.). SEG Investigations in Geophysics 10.
Mayne, W. H. (1962). Common reflection point horizontal data stacking techniques. Geophysics, 27(6), 927-938.
Krey, T. (1987). Attenuation of random noise by 2-D and 3-D CDP stacking and Kirchhoff migration. Geophysical Prospecting, 35(2), 135-147.