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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.

Reference-versus-sample allocation in repeated isotope-ratio measurements distributes a finite measurement budget between sample readings and the reference used to calibrate them. Suppose independent sample and reference readings have first-order gain-error variances $s^2$ and $r^2$. Averaging $n_s$ sample readings and $n_r$ reference readings gives log stiffness-ratio variancewhere $s_c^2$ is the variance of a differential calibration error shared by all ratio estimates. The shared-reference floor in averaged isotope stiffness ratios explains why that last component does not average down. Covariance propagation must reflect which observations and calibration influences are actually shared.^[ref]

Let one sample reading cost $c_s$ budget units and one reference reading cost $c_r$, with $c_s{n}_s+c_r{n}_r=T$. Treating counts continuously for the initial optimization, a constraint multiplier givesThe counts are proportional to standard uncertainty divided by the square root of cost, rather than simply to each variance. A budget-independent common component changes the attainable floor but not this optimum ratio under the stated model.

For an original example, take equal costs, $s=0.3\%$, $r=0.1\%$ and 40 total readings. The optimum is 30 sample and ten reference readings. Without a common component, the predicted log-ratio standard uncertainty is about $0.1265\%$, compared with $0.1414\%$ for twenty of each. A common standard uncertainty $s_c=0.05\%$ raises those values to about $0.1612\%$ and $0.1732\%$, respectively.

An integer allocation near a continuous isotope-reference optimum must compare feasible neighboring count choices and any acquisition-order constraints. More reference readings help only to the extent that their errors are independent; repeatedly measuring an unchanged unknown calibration bias does not create independent references. If reference drift depends on elapsed time, a single constant cost and common-variance term may be inadequate.

In practice, record which reference observations calibrate which sample spectra and account for any correlated peak fitting, isotope-composition uncertainty and spectral offsets. The variance and cost model above is a chosen allocation problem, not an experimentally established precision specification for an isotope measurement system.