= Resolution-limit conditioning in finite-input detector calibration
{title2=$\kappa_q=\sigma_q^2/(\sigma_q^2-\sigma_p^2)$}
\b[Resolution-limit conditioning in finite-input detector calibration] is the sensitivity of a small inferred detector width to uncertainty in two nearly equal measured widths. In <finite-input correction in pulse-detector calibration>, the nonnegative normalized <convolution> model gives
$$
D=\sigma_q^2-\sigma_p^2,\qquad\sigma_h=\sqrt D\quad(D>0).
$$
The signed relative sensitivities are
$$
\frac{\partial\log\sigma_h}{\partial\log\sigma_q}=\frac{\sigma_q^2}{D},\qquad
\frac{\partial\log\sigma_h}{\partial\log\sigma_p}=-\frac{\sigma_p^2}{D}.
$$
They diverge as the measured output width $\sigma_q$ approaches the known input width $\sigma_p$. Subtracting two well-measured squared widths can therefore be a poorly conditioned inference.
For an original example, $\sigma_q=1.00\,\mathrm{ns}$ and $\sigma_p=0.99\,\mathrm{ns}$ give $D=0.0199\,\mathrm{ns^2}$ and $\sigma_h\approx0.1411\,\mathrm{ns}$. The two relative sensitivity coefficients are about $50.25$ and $-49.25$. A small independent fractional width error can produce a much larger fractional response-width error; a common multiplicative timebase error instead partly cancels because the coefficients sum to one. Correlations must therefore be retained.<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}
Near the boundary, report uncertainty in the <variance> difference before transforming it. For a chosen interval $D\in[-0.010,0.050]\,\mathrm{ns^2}$, imposing the physical constraint $D\geq0$ gives a detector-width interval $[0,\sqrt{0.050}]\,\mathrm{ns}$, approximately $[0,0.2236]\,\mathrm{ns}$. A <coverage-preserving detector-width bound from a variance interval> uses this monotone square-root transformation on the physical intersection. It need not resemble a symmetric error bar around a point estimate. An interval wholly below zero contains no admissible width under this model; replacing it silently with a zero-width result hides the inconsistency.
NIST characterized its detector/cable/oscilloscope chain with an optical pulse much shorter than the receiver response.<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; approximately 120 ps optical input and Figure C1]{ref} That design avoids subtracting two nearly equal broad widths. The <variance> identity still requires nonnegative profiles and finite second moments; arbitrary full widths at half maximum and responses with negative ringing need a different model.
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