= Receiver-latency uncertainty in the inferred input-time horizon
{title2=$u^2(H)=u^2(T)+u^2(d)-2\operatorname{Cov}(T,d)$}
\b[Receiver-latency uncertainty in the inferred input-time horizon] propagates uncertainty in timing calibration into the latest input time that could have influenced a stored output record. For one channel, the horizon in <input-time horizon of a delayed optical receiver record> is $H=T-d$, with last output time $T$ and effective input-to-output timing offset $d$. Hence
$$
u^2(H)=u^2(T)+u^2(d)-2\operatorname{Cov}(T,d).
$$
This linear <variance> identity retains shared timing influences.<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}
A common calibration offset can cancel when it enters both quantities. In an original model, let errors be $\delta T=c+e_T$ and $\delta d=c+e_d$, with independent components of <standard deviations> $0.8$, $0.2$ and $0.6\,\mathrm{ns}$. Then $\delta H=e_T-e_d$ and $u(H)=\sqrt{0.2^2+0.6^2}\approx0.6325\,\mathrm{ns}$. Discarding their common-offset <covariance> would instead give $\sqrt{0.8^2+0.2^2+0.8^2+0.6^2}\approx1.296\,\mathrm{ns}$.
The shared component must come from the actual calibration model. Here $d$ is the effective timing offset relating the input and output coordinates, so an output-clock alignment error can enter both $T$ and $d$. A physical transport-delay error is not automatically correlated with the last-sample timestamp.
For hard timing limits, propagate validated bounds rather than reinterpret standard uncertainty as a guarantee. If $T=150\,\mathrm{ns}$ is known and $8\leq d\leq12\,\mathrm{ns}$, then $138\leq H\leq142\,\mathrm{ns}$. Claiming the optimistic endpoint as though it held for every allowed delay would overstate the input-time reach.
With several channels, $H=\max_j(T_j-d_j)$. <Channel switching under uncertainty in a combined receiver horizon> makes this a nonsmooth function when the maximizing channel is uncertain. Propagate the justified joint timing set or distribution; a derivative of only one nominally best channel can miss such switching.
NIST's primary pulse calibration measured the detector, cable and <oscilloscope> together using a short known optical input.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37, Figure C1]{ref} The numerical timing errors above are chosen examples, not values from that apparatus. A timing horizon remains a necessary information limit; it does not guarantee invertibility of the receiver or bound future optical illumination.
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