= Shared-blank covariance in carbon recovery
\b[Shared-blank covariance in carbon recovery] is the statistical dependence between net-carbon results that subtract the same estimated background. Independent specimen measurements can become correlated through that common subtraction. The covariance matters when averaging results, comparing specimens or deciding how much effort to allocate to blank measurements.
Let $Q_i=T_i-\overline B$, with independent specimen totals $T_i$ and a shared blank estimate independent of those totals. Then
$$
\operatorname{cov}(Q_i,Q_j)=u_B^2\quad(i\ne j),\qquad
\Sigma_Q=\operatorname{diag}(u_{T_i}^2)+u_B^2\mathbf1\mathbf1^{\mathsf T}.
$$
Here $u_B$ is the standard uncertainty of the blank estimate. The added matrix term represents one shared random quantity, not a separate new blank error for each specimen. Common-reference covariance is explicitly retained in the GUM.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.2.2–5.2.4]{ref}
For a weighted mean with $\sum_iw_i=1$,
$$
\operatorname{Var}\!\left(\sum_iw_iQ_i\right)=\sum_iw_i^2u_{T_i}^2+u_B^2.
$$
The specimen terms can decrease through averaging, but the shared blank term remains. With independent unequal specimen uncertainties, the minimum-variance weights remain proportional to $1/u_{T_i}^2$ under this particular common-additive model; the common term does not depend on the normalized weights.
A contrast behaves differently: $Q_i-Q_j=T_i-T_j$, so the same blank cancels exactly. This cancellation requires the identical additive correction on the same carbon basis. Different blank batches, collection durations or conversion factors need not cancel. Dividing each result by a different uncertain initial carbon inventory adds another measurement layer.
An original planning example chooses $n_T$ specimen-total measurements and $n_B$ independent blank measurements, with fixed total count $N=n_T+n_B$. If individual standard deviations are $\sigma_T,\sigma_B$, the net mean has variance
$$
\frac{\sigma_T^2}{n_T}+\frac{\sigma_B^2}{n_B}.
$$
A continuous minimum at equal cost satisfies $n_T/n_B=\sigma_T/\sigma_B$. With stipulated $N=12$, $\sigma_T=1$ mg and $\sigma_B=2$ mg, four specimen measurements and eight blanks give variance $0.75\,\mathrm{mg^2}$. Ten specimen measurements and two blanks give $2.1\,\mathrm{mg^2}$. More specimen readings alone can therefore be a poor allocation. <Blank-allocation design for carbon recovery> also needs actual preparation costs and biological independence.
The OECD carbon-dioxide evolution test measures gas production relative to inoculum blanks, but notes that endogenous activity with a test chemical may differ from the control.<Test No. 301: Ready Biodegradability by OECD>[1992 guideline, General principle §4; method 301 B §§1–3]{ref} Replication estimates random variation under the chosen background model. It cannot establish that the treated system's background is exactly additive. Preserve the experimental blank lineage and distinguish sampling uncertainty from a possible change in the background process.
Back to article page
