 Separating detector noise from pulse-energy variance

Separating detector noise from pulse-energy variance estimates source variability using a model of the readout's own fluctuations. In energy-calibrated units, write $X=U+n$. The exact identity isOnly when the covariance vanishes can a separately estimated noise variance be subtracted directly. Covariance terms are part of measurement-model uncertainty propagation, rather than an optional correction.^[ref]

For an original independent-noise example, suppose the observed energy standard deviation is $0.50\mathrm{nJ}$ and a matched noise-only measurement gives $0.30\mathrm{nJ}$. The inferred source standard deviation is $\sqrt{0.50^2-0.30^2}=0.40\mathrm{nJ}$, not $0.50-0.30=0.20\mathrm{nJ}$. The source contributes $64\%$ of the observed variance in this stipulated model.

A noise-only record should use comparable gain, integration gate and reset conditions. Background subtraction removes an estimated mean, not the random variance remaining around it. Amplifier clipping, unmatched integrator reset or signal-dependent detector noise can invalidate an additive independent-noise model. A negative variance difference is not a physical negative source variance; it can indicate estimation uncertainty or mismatched assumptions.

Splitting pulses between two independently characterized readouts can provide a two-channel covariance estimate of pulse-energy variance under additional noise assumptions. Shared electronics noise and common drift must still be assessed; independence cannot be established just by using two detectors.