= Covariance between a raw pulse integral and its baseline correction
{title2=$\operatorname{Cov}(Y,B)$}
A raw <laser pulse energy> estimate and its baseline correction can contain the same readout error. Let $Y$ be the raw gate integral, $B$ the estimated integrated background and $\widehat U=Y-B$ the corrected result. <Linearity of covariance> gives the exact identity
$$
\boxed{\operatorname{Var}(\widehat U)=\operatorname{Var}(Y)+\operatorname{Var}(B)-2\operatorname{Cov}(Y,B)}.
$$
For this linear measurement model, the <law of propagation of uncertainty> is exact whenever the input variances exist. Its general form includes the sensitivity coefficients and <covariance> terms.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref} Positive <covariance> reduces the <variance> of this difference; it would instead increase the <variance> of a sum.
An original shared-offset model makes the mechanism explicit. For fixed gate duration $T$, write
$$
Y=U+Tc+\varepsilon_Y,\qquad B=Tc+\varepsilon_B,
$$
where $c$ is a common additive power-readout offset with <standard uncertainty> $\sigma_c$, while $\varepsilon_Y$ and $\varepsilon_B$ are independent zero-mean energy errors, independent also of $c$. Then $\operatorname{Cov}(Y,B)=T^2\sigma_c^2$, and <shared-offset cancellation in a baseline-corrected pulse integral> gives $\widehat U=U+\varepsilon_Y-\varepsilon_B$. \b[The shared offset contributes no uncertainty to this difference.] It need not be small or separately known.
Choose $T=10\,\mathrm{\mu s}$, $\sigma_c=0.20\,\mathrm{mW}$, $u(\varepsilon_Y)=0.30\,\mathrm{nJ}$ and $u(\varepsilon_B)=0.40\,\mathrm{nJ}$. The shared energy contribution is $2.0\,\mathrm{nJ}$ in each reading. Thus $u^2(Y)=4.09\,\mathrm{nJ}^2$, $u^2(B)=4.16\,\mathrm{nJ}^2$ and $\operatorname{Cov}(Y,B)=4.00\,\mathrm{nJ}^2$, yielding \b[$u(\widehat U)=0.50\,\mathrm{nJ}$]. Incorrectly discarding the <covariance> gives $\sqrt{8.25}\approx2.87\,\mathrm{nJ}$.
Cancellation weakens if the offset changes between acquisitions. Replace $c$ by two offsets with the same <standard uncertainty> $\sigma_c$ and <correlation coefficient> $\rho$. Under the same independent residual-error assumptions,
$$
u^2(\widehat U)=u^2(\varepsilon_Y)+u^2(\varepsilon_B)+2T^2\sigma_c^2(1-\rho).
$$
At $\rho=0.90$, the chosen example gives $u(\widehat U)=\sqrt{1.05}\approx1.025\,\mathrm{nJ}$. A baseline taken far away in time cannot simply be assigned the covariance of an unchanged offset.
\Image[/pulse-baseline-covariance-cancellation.png]
{title=Shared baseline errors cancel according to their correlation}
{description=Original numerical model with a 2 nJ offset uncertainty in each acquisition and independent residual uncertainties of 0.3 nJ and 0.4 nJ. The comparison curve incorrectly treats the shared offsets as independent.}
A sampled implementation has the same structure. If a vector of readings $\mathbf x$ has <covariance matrix> $C$, and fixed integration weights give $Y=\mathbf q^T\mathbf x$ and $B=\mathbf w^T\mathbf x$, then
$$
\operatorname{Cov}(Y,B)=\mathbf q^TC\mathbf w,\qquad
\operatorname{Var}(Y-B)=(\mathbf q-\mathbf w)^TC(\mathbf q-\mathbf w).
$$
A constant offset cancels when the weights sum to the same gate duration: $\mathbf q^T\mathbf1=\mathbf w^T\mathbf1=T$. This does not justify using pulse-contaminated readings as background. Matched blank sequences, collected with the actual timing, help characterize the needed noise correlations.
The historical NIST TC-24 apparatus used a thermocouple-based absorbing plate and averaged baseline measurements before and after laser injection to compensate background and thermal drift.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.5, printed p. 8; §7.2, Leakage Effect]{ref} The offset model above is an independent illustration, not a reconstruction of that instrument's uncertainty budget.
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