## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
The optimal choice has seven points along the boundary in the \(xy\)-plane and one point in the interior. Any interior point not too far from the origin would also result in a good selection. In terms of the aggregate measure \(\bar{d}_{k}\) defined in (4) associated with the three datasets, we have \(\bar{d}_{1}=6.0\bar{d}_{3}\) and \(\bar{d}_{2}=2.2\bar{d}_{3}\). Table 3 gives the square roots \(v_{i}\) of the diagonal elements of \((\int_{k}^{\top}I_{k})^{-1}\), where \(I_{k}\) is the Jacobian matrix associated with the \(k\)th dataset. If the variance matrix associated with the measurements is \(\sigma^{2}I\), then \(v_{i}=u(a_{j})/\sigma\), where \(u(a_{j})\) is the estimate of the standard uncertainty associated with the \(j\)th parameter [36; 37]. In terms of standard uncertainties, the D-optimal dataset \(X_{3}\) is far better for estimating the surface parameters than the other two datasets. Figure 10 plots the values of \(t_{q}\), \(q=1,\ldots,100\), and its analytical approximation \(\bar{I}(q)\) given by (12), determined by the sequential D-optimal Algorithm 4, with and without repeat measurements. For higher values of \(q\), there is good agreement between \(t_{q}\) and \(\bar{t}_{q}\). Figure 6: The CMI hyperbolic paraboloid reference artefact. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(u(a_{j})/\sigma\) & \(\alpha\) & \(\beta\) & \(b_{1}\) & \(b_{2}\) & \(b_{3}\) & \(b_{4}\) & \(b_{5}\) & \(b_{6}\) \\ \hline \(X_{1}\) & 2.3867 & 2.3867 & 0.0049 & 0.0049 & 0.0139 & 2.6219 & 2.6219 & 10.5328 \\ \(X_{2}\) & 0.7259 & 0.7259 & 0.0042 & 0.0042 & 0.0035 & 0.4999 & 0.4999 & 2.8188 \\ \(X_{3}\) & 0.2049 & 0.2413 & 0.0012 & 0.0014 & 0.0014 & 0.2883 & 0.2520 & 1.4690 \\ \hline \end{tabular}
\end{table} 
Table 3: Standard uncertainty factors \(u(a_{j})/\sigma\) associated with the hyperbolic paraboloid parameters \(a=(\alpha,\beta,b^{\top})^{\top}\) for three eight-point datasets \(X_{k},k=1,2,3\). Figure 7: Hyperbolic paraboloid calibration points: ‘expert’ guess at a good set of calibration points. Figure 8: Hyperbolic paraboloid calibration points: second ‘expert’ guess at a good set of calibration points, a modification of \(X_{1}\), Figure 7.

Figure 9: Hyperbolic paraboloid calibration points: D-optimal points determined by the SSQR-GE algorithm. ### Calibration of a Network of Standards Using a Comparator

In this section, we consider the calibration of a number of standards, starting with one calibrated standard and using a comparator to calibrate the other standards relative to the calibrated standard. An example application is in mass calibration using a mass balance [38; 39; 40]. Given a calibrated standard \(A_{1}\), nominally of 1 kg, and two uncalibrated masses \(A_{2}\) and \(A_{3}\) each of nominal mass 0.5 kg, \(A_{2}\) and \(A_{3}\) can be calibrated using two measurements, the first comparing \(A_{1}\) with the combined mass of \(A_{2}\) and \(A_{3}\), and the second comparing \(A_{2}\) and \(A_{3}\). A second example from length metrology relates to the calibration of gauge blocks [41], where it is possible to align two gauge blocks end to end, a process called 'wringing', to define a combined length. This discussion is relevant to the calibration of any set of artefacts defining an extensive quantity where the artefacts can be grouped to define a quantity that is (nominally) the sum of the individual quantities. We use the term 'network' to reflect the fact that the estimates of all the quantities are statistically correlated and an experiment that updates the information about one artefact will also update information about the other artefacts. Let \(\mathbf{a}=(a_{1},\dots,a_{n})^{\top}\) represent the values of the quantities associated with the artefacts where the nominal values \(\mathbf{a}^{\sharp}=(a_{1}^{\sharp},\dots,a_{n}^{\sharp})^{\top}\) of the quantities are such that the artefacts can be grouped into sets of two groups, with each group being associated with the same nominal value.