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Signal, noise and stacking
Combining many exposures so that the signal adds linearly while random noise adds as the square root, improving the ratio as √N.
Why it matters
It is the entire reason astrophotography is possible from a suburban garden. Ten hours of integration is not a boast; it is the mechanism.
Try it
More frames, or the same time divided?
Explore expected signal-to-noise for one pixel in a simplified detector. Values are illustrative electron rates, not camera presets.
4.00 times the SNR of one exposure of the same length in this model.
The vertical scale adapts to the controls. The point marks your selected stack; this curve shows an expectation, not a noisy measurement.
Compare exact values
| Frames | Seconds each | Total minutes | SNR |
|---|---|---|---|
| 1 | 60.00 | 1.0 | 4.36 |
| 4 | 60.00 | 4.0 | 8.73 |
| 16 | 60.00 | 16.0 | 17.46 |
| 64 | 60.00 | 64.0 | 34.91 |
| 128 | 60.00 | 128.0 | 49.38 |
| 256 | 60.00 | 256.0 | 69.83 |
Where the noise variance comes from
Source photon statistics: 31.7% / Sky photon statistics: 63.5% / Read noise: 4.8%
These are shares of variance, not shares of noise amplitude. Removing the mean sky level does not remove its shot noise.
More frames here also mean more total time. Four times as many equal exposures gives twice the expected SNR when their noise is independent.
Model, assumptions and source
For N frames of t seconds, source rate s, sky rate b and read noise r: SNR = Nst / sqrt(N(s+b)t + Nr²). Rates are detected electrons per second per pixel; r is electrons RMS per pixel per frame.
Assumes independent identical exposures, linear unsaturated response, perfect registration and a known background mean. Omits dark current, calibration uncertainty, correlated noise, gradients, tracking errors, rejection and overheads. It does not recommend exposure lengths; a single long exposure may saturate or trail.
Model adapted to one pixel from the STScI ACS exposure-time equation.
The maths, in layers
1 · Intuition
Four times as many frames gives twice the signal-to-noise. Diminishing returns are built in, which is why the second hour helps far more than the tenth.
2 · The equation
SNR ∝ √N
- N
- number of sub-exposures stacked
Rests on
2 concepts in the full chain back to first principles.
Leads to
Nothing here builds on it yet.