= Two-channel covariance estimate of pulse-energy variance
{title2=$\operatorname{Var}(U)=\operatorname{Cov}(X,Y)/(ab)$}
A \b[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,
$$
\operatorname{Cov}(X,Y)=ab\operatorname{Var}(U).
$$
This follows by expanding the covariance. Correlation assumptions are essential measurement-model inputs; the uncertainty guide retains covariance terms explicitly.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[AppendixA, equationA-3]{ref}
For an original example, choose $a=2\,\mathrm{V/nJ}$, $b=4\,\mathrm{V/nJ}$ and measured covariance $0.08\,\mathrm{V^2}$. The inferred source variance is $0.01\,\mathrm{nJ^2}$, 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.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.4, calibrated beam splitting]{ref} Matching pulse indices, bandwidth and integration windows is part of the proposed model.
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