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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.

The uncertainty of a three-point baseline integral depends on both the sample uncertainties and their correlations. With a known fixed gate duration $T$, Simpson baseline integration for pulse-energy correction uses the linear combinationIf $C_b$ is the covariance matrix of the baseline estimates, thenFor fixed weights this variance identity is exact; it does not require Gaussian errors. Its use as a complete measurement uncertainty still requires an adequate baseline model. The general covariance propagation rule retains these cross terms.^[ref]

An original equal-precision model assigns each sample standard uncertainty ${\sigma }_b$ and common pairwise correlation coefficient $\rho$, with $-1/2\le \rho \le 1$. Substitution givesFor $T=10\mu \mathrm{s}$ and ${\sigma }_b=0.10\mathrm{mW}$, independent samples give $u(B)=0.707\mathrm{nJ}$. At $\rho =0.60$ this rises to $0.894\mathrm{nJ}$; a perfectly common additive error gives $1.00\mathrm{nJ}$. Three readings under one uncertain offset need not behave like independent measurements.

The final corrected pulse integral has another covariance. If $Y$ is the raw gate integral and $\hat{U}=Y-B$, thenThe covariance between a raw pulse integral and its baseline correction matters when both share a readout offset. If that offset is $c$, it contributes $Tc$ to both $Y$ and $B$ and cancels exactly in their difference. Treating these appearances as independent would invent uncertainty.

Use repeated matched blank acquisitions to estimate the relevant covariance structure, keeping the gate and readout sequence comparable. A contaminated midpoint, an inadequate baseline shape or uncertain timing adds effects outside the fixed-weight sample-noise calculation. Numerical quadrature exactness alone does not establish a sound pulse-energy uncertainty budget.