 Directional derivative of a clipped logarithmic acquisition time

The 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 $A,\tau ,\epsilon$, writeThe logarithmic tail-duration change at the zero-extension boundary article treats a fixed decay time and logarithmic input coordinate. Here consider a boundary point $A\tau =\epsilon$ and an additive parameter direction $\mathbf{h}=(a,t,e)$.

The one-sided directional derivative uses positive step sizes: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 $\epsilon =1.0\mathrm{nJ}$. 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.^[ref] A derivative can describe parameter sensitivity accurately while the envelope itself still needs experimental justification.