= Acquisition-time sensitivity to tail-model parameters
{title2=$L=\tau\log(A\tau/\varepsilon)$}
\b[Acquisition-time sensitivity to tail-model parameters] measures how an inferred recording extension changes when the assumed optical tail changes. In the positive-duration branch of <acquisition duration from an exponential tail-energy tolerance>, write
$$
L=\tau\ell,\qquad \ell=\log\frac{A\tau}{\varepsilon}>0.
$$
Here $A$ bounds the initial tail power, $\tau$ is its decay time and $\varepsilon$ the tolerated missed energy. Their first-order relative effects are
$$
\frac{\delta L}{L}\simeq
\frac1\ell\frac{\delta A}{A}
+\left(1+\frac1\ell\right)\frac{\delta\tau}{\tau}
-\frac1\ell\frac{\delta\varepsilon}{\varepsilon}.
$$
These <logarithmic condition numbers of a tail-acquisition requirement> distinguish a fractional amplitude error from a fractional decay-time error. For a stringent tolerance with large $\ell$, decay-time sensitivity stays near unity while amplitude sensitivity decreases.
For an original example, take $A=0.20\,\mathrm W$, $\tau=20\,\mathrm{ns}$ and $\varepsilon=0.040\,\mathrm{nJ}$. Then $A\tau/\varepsilon=100$ and $L\approx92.10\,\mathrm{ns}$. Let the independent standard uncertainties of $A$ and $\tau$ be $10\%$ and $5\%$, with the tolerance fixed. The amplitude contribution to $u(L)$ is $2.00\,\mathrm{ns}$; the decay-time contribution is $(1+\log100)\times1\,\mathrm{ns}\approx5.605\,\mathrm{ns}$. Adding their <variances> gives $u(L)\approx5.95\,\mathrm{ns}$, about $6.46\%$.
Correlated fitted parameters require <covariance> terms in this propagation.<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 standard uncertainty is not a guaranteed recording margin: a worst-case requirement needs a justified allowed parameter set, as in the parent article. The zero-duration boundary is also outside this smooth positive-branch calculation.
Physically, these parameters must describe the unrecorded optical power, not merely an electrical detector tail. NIST separately measured the detector/cable/oscilloscope response before correcting pulse measurements.<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]{ref} A precise fit to the wrong response component does not justify the optical envelope or its acquisition-time bound.
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