= Covariance of differential spectral-calibration gains
{title2=$\Sigma_d=B\Sigma_gB^T$}
The \b[<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 $\mathbf d=B\mathbf g$. Then
$$
\Sigma_d=B\Sigma_gB^T.
$$
This <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>.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3, sensitivity and covariance propagation]{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:
$$
\mathbf d=\begin{pmatrix}g_2-g_1\\g_3-g_1\end{pmatrix},\qquad
B=\begin{pmatrix}-1&1&0\\-1&0&1\end{pmatrix}.
$$
The common scale cancels, but the residual reference error remains in both differences:
$$
\Sigma_d=\begin{pmatrix}s_1^2+s_2^2&s_1^2\\s_1^2&s_1^2+s_3^2\end{pmatrix}.
$$
Cancellation of one common influence therefore does not establish independent comparisons.
Choose all three residual standard uncertainties as $0.1\%$. Then $\Sigma_d=10^{-6}\begin{pmatrix}2&1\\1&2\end{pmatrix}$, 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> $6\times10^{-6}$ 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.
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