= Directional derivative of a clipped logarithmic acquisition time
{title2=$D^+L=\tau\max(0,a/A+t/\tau-e/\varepsilon)$}
The \b[<directional derivative> of a clipped logarithmic acquisition time] describes the first-order recording extension when several envelope parameters change together at the acquisition threshold. For positive $A,\tau,\varepsilon$, write
$$
L(A,\tau,\varepsilon)=\tau\max\bigl(0,\log(A\tau/\varepsilon)\bigr).
$$
The <logarithmic tail-duration change at the zero-extension boundary> article treats a fixed decay time and logarithmic input coordinate. Here consider a boundary point $A\tau=\varepsilon$ and an additive parameter direction $\mathbf h=(a,t,e)$.
The one-sided <directional derivative> uses positive step sizes:
$$
D^+L(\mathbf h)=\lim_{s\downarrow0}
\frac{L(A+sa,\tau+st,\varepsilon+se)-L(A,\tau,\varepsilon)}s
=\tau\max\left(0,\frac aA+\frac t\tau-\frac e\varepsilon\right).
$$
The logarithmic chain rule gives the expression inside the maximum. Differentiating the prefactor contributes zero at the boundary because the clipped logarithm there is zero.
For an original example, let $A=0.10\,\mathrm W$, $\tau=10\,\mathrm{ns}$ and $\varepsilon=1.0\,\mathrm{nJ}$. Increase amplitude in direction $a=0.002\,\mathrm W$ while decreasing decay time in direction $t=-0.10\,\mathrm{ns}$, with $e=0$. The net fractional direction is $0.02-0.01=0.01$, so $D^+L=0.10\,\mathrm{ns}$ per unit step. Reversing the direction gives derivative zero, not $-0.10\,\mathrm{ns}$.
The derivative is positively homogeneous in the direction but is not linear, so no single gradient represents all threshold-crossing perturbations. Above threshold the usual smooth derivative applies; strictly below threshold all sufficiently small changes leave the extension zero. <Gaussian input uncertainty at a clipped acquisition-time threshold> can consequently produce a mass of outcomes exactly at zero rather than a symmetric distribution around a nominal duration. The directional calculation alone does not assign a confidence level.
This is a conditional optical-envelope calculation. Separating optical power from the receiver's electrical response remains necessary before its parameters describe missed light. NIST independently characterized the detector/cable/oscilloscope response 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 derivative can describe parameter sensitivity accurately while the envelope itself still needs experimental justification.
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