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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, writeHere $A$ bounds the initial tail power, $\tau$ is its decay time and $\epsilon$ the tolerated missed energy. Their first-order relative effects areThese 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 $\epsilon =0.040\mathrm{nJ}$. Then $A\tau /\epsilon =100$ and $L\approx 92.10\mathrm{ns}$. 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 $(1+\log 100)\times 1\mathrm{ns}\approx 5.605\mathrm{ns}$. Adding their variances gives $u(L)\approx 5.95\mathrm{ns}$, about $6.46\%$.

Correlated fitted parameters require covariance terms in this propagation.^[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.^[ref] A precise fit to the wrong response component does not justify the optical envelope or its acquisition-time bound.

Logarithmic tail-duration change at the zero-extension boundary describes a recording requirement whose input ratio crosses the point where no further acquisition is needed. For fixed decay time $\tau >0$, put $R=A\tau /\epsilon$ and $x=\log R$. The physical extension isHere the logarithm applies to the input ratio, not to a zero output duration. Logarithmic condition numbers of a tail-acquisition requirement applies only where $L>0$.

At $x=0$, the left derivative is zero and the right derivative is $\tau$. The directional derivative of a clipped logarithmic acquisition time along a signed input change $h$ isThus no single ordinary derivative describes threshold-crossing perturbations. Nevertheless, the duration is stable in absolute units:This follows because the positive-part map has Lipschitz continuity with constant one.

For an original interval example, take $\tau =25\mathrm{ns}$ and a justified input-ratio range $0.99\le R\le 1.02$. The corresponding duration range is $0\le L\le 25\log 1.02\approx 0.4951\mathrm{ns}$. Its lower part is clipped at zero. Linearizing only on the zero-duration side would incorrectly predict no possible extension even though the allowed set crosses the threshold. A relative error divided by a nominal $L=0$ is undefined, while this absolute bound remains meaningful.

If $\tau$ is uncertain too, optimize the full expression over the justified joint parameter set rather than treating this fixed-$\tau$ inequality as a complete uncertainty analysis. The tail model must bound optical power throughout the missed interval. NIST's calibration apparatus measured detector/cable/oscilloscope response separately, illustrating why a recorded electrical decay cannot automatically certify such an optical envelope.^[ref] The clipping analysis is an original property of the stated conditional acquisition model.