Two-channel covariance estimate of pulse-energy variance ($\mathrm{Var}(U)=\mathrm{Cov}(X,Y)/(ab)$)

A two-channel covariance estimate of pulse-energy variance uses the component shared by paired readouts to estimate source fluctuations. In a stipulated linear model let $X=aU+n_X$ and $Y=bU+n_Y$, with calibrated nonzero gains. If both noise terms are uncorrelated with $U$ and with each other,This follows by expanding the covariance. Correlation assumptions are essential measurement-model inputs; the uncertainty guide retains covariance terms explicitly.^[ref]

For an original example, choose $a=2V/nJ$, $b=4V/nJ$ and measured covariance $0.08{\mathrm{V}}^2$. The inferred source variance is $0.01\mathrm{n}{\mathrm{J}}^2$, giving a standard deviation of $0.10\mathrm{nJ}$. Independent readout-noise variances $0.36$ and $0.64{\mathrm{V}}^2$ would give total variances $0.40$ and $0.80{\mathrm{V}}^2$ while preserving that cross-covariance.

But a shared noise covariance of $0.04{\mathrm{V}}^2$ raises the observed cross-covariance to $0.12{\mathrm{V}}^2$. Ignoring it would infer a source standard deviation of $0.1225\mathrm{nJ}$, about $22.5\%$ too high.

A split-beam experiment records both channels for the same pulses, checks gains and measures dark or independent-input controls for correlated electronics. NIST's calibrated splitting apparatus supplies an example of the optical geometry, not a claim that it used this covariance estimator.^[ref] Matching pulse indices, bandwidth and integration windows is part of the proposed model.