 Simpson baseline integration for pulse-energy correction

Simpson baseline integration for pulse-energy correction subtracts a three-point estimate of integrated background from a measured gate integral. For a baseline $b(t)$ on $[0,T]$, Simpson's rule givesIt is exact for a quadratic or cubic baseline, provided the three values represent background rather than pulse-contaminated samples. For a four-times differentiable baseline, the remainder is ${\int }_0^{T}bdt-B_S=-T^5{b}^{(4)}(\xi )/2880$ for some $0<\xi <T$.^[ref]

For an original chosen quadratic background, take $T=6\mu \mathrm{s}$ with beginning, midpoint and end values $2$, $3$ and $2\mathrm{mW}$. Its integrated baseline is $16\mathrm{nJ}$, whereas endpoint averaging would give $12\mathrm{nJ}$. A raw gate integral of $116\mathrm{nJ}$ therefore gives $100\mathrm{nJ}$ after the three-point correction. Overestimating only the midpoint by $0.50\mathrm{mW}$ would subtract an extra $2\mathrm{nJ}$.

The uncertainty of a three-point baseline integral is separate from its quadrature error. With baseline-sample covariance matrix $C_b$ and $\mathbf{w}=(T/6)(1,4,1{)}^T$, its variance is ${\mathbf{w}}^T{C}_b\mathbf{w}$. Independent samples of common standard uncertainty ${\sigma }_b$ give $u(B_S)=T{\sigma }_b/\sqrt{2}$; a perfectly shared additive baseline error of the same uncertainty gives $T{\sigma }_b$ instead. Covariance terms cannot be discarded.^[ref]

A matched blank acquisition or independently measured background can supply the midpoint. Historical NIST pulse-energy apparatus used before/after baselines for background and thermal-drift compensation.^[ref] The three-point example is an original extension, not a claim about that instrument's processing. Higher polynomial exactness does not compensate for an incorrectly identified baseline.