= Logarithmic condition numbers of a tail-acquisition requirement
{title2=$\kappa_x=\partial\log L/\partial\log x$}
\b[Logarithmic condition numbers of a tail-acquisition requirement] quantify the fractional response of a positive recording extension to fractional parameter changes. For $L=\tau\ell$ with $\ell=\log(A\tau/\varepsilon)>0$, define the signed sensitivity $\kappa_x=\partial\log L/\partial\log x$. Then
$$
\kappa_A=\frac1\ell,\qquad\kappa_\tau=1+\frac1\ell,\qquad\kappa_\varepsilon=-\frac1\ell.
$$
Their absolute values are componentwise relative <condition numbers>. <Acquisition-time sensitivity to tail-model parameters> gives the corresponding dimensional derivatives and uncertainty propagation.
These numbers depend on which coordinates are held fixed. If $M=A\tau$ denotes the initial tail-energy allowance, then $L=\tau\log(M/\varepsilon)$. In coordinates $(M,\tau,\varepsilon)$, the decay-time sensitivity at fixed $M$ is exactly one, rather than $1+1/\ell$. This is the same physical formula: holding $M$ fixed while changing $\tau$ also changes $A$. Transforming the parameter <covariance> consistently leaves the propagated output uncertainty unchanged.<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}
An original boundary example takes $A\tau/\varepsilon=1.01$ and $\tau=20\,\mathrm{ns}$. The required extension is only $L\approx0.1990\,\mathrm{ns}$, but $\kappa_A\approx100.50$. Increasing $A$ by $1\%$ multiplies the ratio by $1.01$, giving $1.0201=(1.01)^2$ and doubling the required extension exactly. Its relative change is large because the original extension was very small.
This does not mean the absolute requirement is singular there. In the active branch $\partial L/\partial\log(A\tau/\varepsilon)=\tau$ remains finite. At and below the threshold, the physical extension is zero, so $\log L$ is undefined. A <logarithmic tail-duration change at the zero-extension boundary> must instead use the piecewise requirement or absolute changes. Large relative conditioning alone is not evidence of a large absolute recording-time risk.
Finally, conditioning describes the stated envelope model, not its experimental validity. The optical power tail must be distinguished from detector response, which NIST measured independently in its pulse calibration procedure.<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 well-conditioned calculation based on an invalid optical envelope still gives an unjustified acquisition guarantee.
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