Random variables & noise

Processing Prerequisites

Learning objectives

Define a random variable, expectation, variance, and Gaussian distribution
Distinguish white from colored noise by spectrum and autocorrelation
Compute SNR in amplitude and dB and explain when a signal is “invisible” in noise
Connect noise models to denoising strategy: different statistics → different attenuators

Every seismic record is signal plus noise. Every denoising, stacking, deconvolution, and inversion algorithm assumes a model of what the noise looks like. If the model is wrong, the algorithm is wrong. So we spend some time making the assumptions explicit.

1. Random variables in one paragraph

A random variable X is a rule for generating numbers with a probability distribution. Two summary numbers describe it:

\mathbb{E}[X] = \text{mean},\qquad \operatorname{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2].

The square root of variance is the standard deviation σ; for noise, it is the “typical amplitude.”

2. Why Gaussian shows up everywhere

The central limit theorem says: sum many small independent contributions, and the total will look Gaussian no matter how the individuals were distributed. Seismic noise is the sum of many tiny independent effects (thermal noise, ground motion, electronics, nearby activity) — so it is usually very close to Gaussian.

p(x) \;=\; \frac{1}{\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)

In the widget below, switch between white, pink, and blue noise and watch the histogram. All three converge to the Gaussian bell (because all three are built from Gaussian samples — only their temporal correlation differs).

3. White, pink, blue — the color of noise

White noise has a flat magnitude spectrum and a delta autocorrelation: each sample is independent of every other. This is the textbook assumption behind most deconvolution operators.
Pink noise (1/f) has more energy at low frequencies. Produced by integrating white noise, it shows up in tectonic micro-tremor, long-period drift, and some electronic systems.
Blue noise has more energy at high frequencies. Produced by differencing white noise. Appears after over-whitening deconvolution.

4. Signal-to-noise ratio (SNR)

Define

\operatorname{SNR} = \frac{\sum_n s[n]^{2}}{\sum_n n[n]^{2}}\qquad\text{and}\qquad\operatorname{SNR}_{\text{dB}} = 10\log_{10}\operatorname{SNR}.

Rough guide for a seismic trace:

SNR > 20 dB: the signal dominates; you see clear reflections by eye on a single trace.
0 to 10 dB: signal and noise are comparable; visible on a section but needs stacking and attenuation to interpret.
< 0 dB: noise is larger than the signal on a single trace. Only stacking — averaging many independent realizations — recovers the signal. This is the territory of weak reservoir anomalies and deep plays.

Try dropping the SNR slider in the widget from +20 dB down through 0 and into −10 dB. Watch the signal (teal) disappear into the noise (orange), and the fact that stacking is mathematically necessary becomes intuitive.

5. Stacking improves SNR by sqrt(N)

Add N independent realizations of signal + noise. Signal adds coherently (linearly): energy scales as N². Noise adds incoherently (random phase): energy scales as N. Their ratio — the SNR — grows as N. Amplitude SNR grows as √N.

This is why seismic is acquired with lots of fold. Sixty-fold data has ~8× better amplitude SNR than single-fold, for free, with no model assumptions beyond “noise is uncorrelated between traces.”

6. Autocorrelation, or: noise has a fingerprint

The autocorrelation of a signal is its convolution with its own time-reverse:

R_{xx}[\tau] = \sum_n x[n]\, x[n+\tau]

White noise has $R_{xx}[\tau] = \sigma^{2}\delta[\tau]$ — a spike at zero lag and nothing elsewhere. Colored noise has a broadened peak. Coherent noise (like swell on marine data) has periodic peaks at the dominant period. Autocorrelation is the first thing you compute when characterizing unknown noise.

It is also the heart of predictive deconvolution: if the noise (or the multiple train) has structure in its autocorrelation, you can predict and subtract it.

7. Spatial noise is different from temporal noise

Everything above is about noise within one trace. In 2D and 3D, we also care about noise correlation between traces:

Ground roll moves at a fixed low velocity across traces — coherent spatially and temporally. Attenuation happens in the f–k domain.
Swell noise dominates low frequencies on streamers — coherent in time, uncorrelated between streamers.
Random ambient is what the textbook “noise model” usually refers to — uncorrelated trace-to-trace, stacks out beautifully.

Choosing the right noise attenuator depends on knowing what kind of noise you have.

**The one sentence to remember**

Every denoising algorithm has a noise model in it; stacking improves amplitude SNR by √N when noise is uncorrelated between traces; autocorrelation is the diagnostic that tells you which noise model applies.

Where this goes next

Section §0.9 takes the final mathematical piece: optimization. The normal equations of §0.7 solve a quadratic problem exactly, but FWI and most modern inversions minimize non-linear cost functions where you can only iterate toward the answer. Gradients, step sizes, and regularization all live in the next section.

References

Oppenheim, A. V., Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall.
Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.
Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.