 Covariance of differential spectral-calibration gains

The covariance of differential spectral-calibration gains describes the joint errors of gain differences used in several spectral ratios. Let the calibration-error vector be $\mathbf{g}$ with covariance ${\Sigma }_g$, and form differences $\mathbf{d}=B\mathbf{g}$. ThenThis covariance identity is exact for the linear differences; no normal-distribution assumption is required. The first-order relation between gain differences and isotope stiffness ratios is given in separate spectral-calibration gains in isotope stiffness inference.^[ref]

For an original shared-reference model, take three gain errors $g_i=c+e_i$, where $c$ is a common scale error and the $e_i$ are independent residual errors with variances $s_i^2$. Two sample spectra are compared with the same reference spectrum:The common scale cancels, but the residual reference error remains in both differences:Cancellation of one common influence therefore does not establish independent comparisons.

Choose all three residual standard uncertainties as $0.1\%$. Then ${\Sigma }_d=10^{-6}\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$, with correlation $1/2$ between the differences. Even a much larger common scale uncertainty cancels in this stipulated model. Each inferred log stiffness ratio has error approximately twice its gain difference. Averaging the two log-ratio estimates gives variance $6\times 10^{-6}$ and standard uncertainty about $0.245\%$. Incorrectly treating them as independent would give only $0.200\%$.

This shared-reference floor in averaged isotope stiffness ratios matters when many sample spectra reuse one calibrated reference. Independent extra sample readings reduce their own errors, while a reference component shared by all comparisons remains. Assess covariance from the actual reference reuse, calibration drift and repeated acquisition protocol. The matrix calculation concerns multiplicative calibration errors; uncertain peak assignment, isotope composition and additive spectral offsets need their own contributions.