 Quadratic baseline curvature in gated energy integration

Quadratic baseline curvature in gated energy integration describes a baseline whose second time derivative is nonzero over the measurement gate. Averaging only its two endpoint values is exact for a linear baseline but generally not a quadratic one. The baseline-drift bias in pulse-energy integration article derives the resulting endpoint-subtraction error.

A focused correction follows from Simpson's rule. For a quadratic baseline on $[0,T]$,For an original example, take $T=12\mu \mathrm{s}$ and baseline readings $10,11,14\mathrm{mW}$ at the beginning, midpoint and end. The integrated baseline is $0.136\mu \mathrm{J}$. Endpoint averaging would instead subtract $0.144\mu \mathrm{J}$, biasing the inferred signal by $-8\mathrm{nJ}$.

This Simpson baseline integration for pulse-energy correction needs a genuine baseline at the midpoint. A sample taken during the pulse contains signal as well as background. A separate matched blank run or independently characterized reference can supply additional information; treating a contaminated pulse sample as baseline does not improve the correction. If the midpoint estimate is biased by $\Delta b$, the Simpson integral is biased by $2T\Delta b/3$.

Historical NIST measurements used before/after baselines and averaged them for background and thermal-drift compensation.^[ref] The three-point quadratic example is an independent mathematical extension, not a description of that apparatus's processing.