SNR & detectability

Physics prerequisites for acquisition

Learning objectives

Define the peak signal-to-noise ratio of a single trace in dB, $\mathrm{SNR} = 20\log_{10}(A/\sigma)$
Derive the square-root rule: stacking $N$ aligned traces keeps the reflection and divides the noise rms by $\sqrt{N}$ , a gain of $10\log_{10}N$ dB
Convert a required post-stack SNR into a required fold for a known single-trace SNR
Recognise the two assumptions the rule needs: noise independent from trace to trace, and reflections that line up

Every trace records the reflection you want and noise you do not. How visible the reflection is depends on one ratio. Take $A$ , the peak amplitude of the reflection, and $\\sigma$ , the rms amplitude of the noise around it. The peak signal-to-noise ratio in decibels is

\\mathrm{SNR} = 20\\log\_{10}\\frac{A}{\\sigma}\\ \\mathrm{dB}.

At 0 dB the peak equals the noise rms; at $-6$ dB it is half of it. Random noise on a trace of a second or so reaches about 3 times its own rms somewhere, so a reflection needs roughly 10 dB before its peak stands clear of every noise peak on the trace.

The square-root rule

Take $N$ traces from the same reflection point after NMO correction, so that the reflection lines up across them, and suppose the noise on each trace is independent of the noise on the others. Averaging (stacking) them leaves the reflection unchanged, because every trace contributes the same $A$ . The noise variances add, so the average of $N$ independent noise records has rms $\\sigma/\\sqrt{N}$ :

A\_N = A,\\qquad \\sigma\_N = \\frac{\\sigma}{\\sqrt{N}},\\qquad \\mathrm{SNR}\_N = \\mathrm{SNR}\_1 + 10\\log\_{10}N\\ \\mathrm{dB}.

The amplitude ratio grows as $\\sqrt{N}$ and the power ratio as $N$ , so each doubling of fold buys 3 dB: 16 traces buy 12 dB and 100 traces buy 20 dB.

In the figure, set how deeply one trace buries the reflection and the SNR the stack must reach, then raise the fold and compare what the stack measures with what the rule predicts.

From a target to a fold

Turned round, the rule is a design equation. The figure opens with a reflection at $-6$ dB on each trace and a target of 10 dB after the stack. The stack must gain 16 dB, so

N = 10^{(\\mathrm{SNR}\_{\\mathrm{req}} - \\mathrm{SNR}\_1)/10} = 10^{1.6} \\approx 40,

and the stack of 40 traces in the figure measures 10.3 dB. The cost climbs fast: the next 6 dB needs four times the fold, 160 traces, and traces are what a crew is paid to record. A reflection at $-20$ dB needs a fold of 1000 to reach the same 10 dB. This is why weak, deep targets drive the fold of a survey, and why 1 dB more single-trace SNR, from better coupling or a quieter receiver, saves about a fifth of the fold (21 %).

When the rule fails

The rule rests on two assumptions, and the field breaks both.

The noise must be independent from trace to trace. If a fraction $\\rho$ of the noise power is common to all $N$ traces, that part adds up the way the reflection does. The stacked noise variance is $\\sigma^2\[1 + (N-1)\\rho\]/N$ , so the gain grows as $10\\log\_{10}\\bigl(N/\[1 + (N-1)\\rho\]\\bigr)$ and levels off at $10\\log\_{10}(1/\\rho)$ however many traces you stack. With $\\rho = 0.05$ that ceiling is 13 dB, and in the figure 256 traces gain 12.5 dB where the rule promised 24.1 dB. Coherent noise is not all shared, though. Ground roll and most multiples keep residual moveout after NMO with primary velocities, so the stack does attenuate them, by an amount set by how far their moveout departs from the primary’s (Mayne (1962) used the CMP stack against multiples this way), but not at the $\\sqrt{N}$ rate. What NMO leaves aligned, such as near-offset ground roll, multiples with near-primary moveout and noise common to every channel, stacks like signal. It has to be removed before the stack, by f-k filtering, demultiple or receiver design; recording more of it does not help.

The reflections must line up. Residual statics (uncorrected near-surface delays) shift the reflection by a few milliseconds from trace to trace, so the stack adds it partly out of step. For Gaussian shifts of rms $\\sigma\_t$ the expected stack of a Ricker wavelet of peak frequency $f\_p$ is again a Ricker wavelet, with a lower peak frequency and a smaller peak:

f\_e = \\frac{f\_p}{\\sqrt{1 + 2\\pi^2\\sigma\_t^2 f\_p^2}},\\qquad \\frac{A\_N}{A} \\to \\left(\\frac{f\_e}{f\_p}\\right)^{3}.

At $\\sigma\_t$ = 4 ms that keeps 69 % of a 30 Hz peak but only 32 % of a 60 Hz one: misalignment acts on the stack as a low-pass filter, and no amount of fold buys the lost peak back. Residual statics are corrected before the final stack for this reason, so that the stack sees aligned signal and independent noise, the only case in which the square-root rule holds.

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.