## 6 Uncertainties Associated with Distances Derived from Point Clouds
### Distance Measurement: Uncertainty Contribution from Spatially Correlated Location Effects  
See Section 5.3. Suppose that  
\[\mathbf{x}_{i}=\mathbf{x}_{i}^{*}+\mathbf{e}_{i},\quad\mathbf{x}_{j}=\mathbf{x}_{j}^{*}+\mathbf{e}_{j},\]  
where \(\mathbf{e}_{i}\) and \(\mathbf{e}_{j}\) are correlated effects. Suppose the \(x\)-components of the correlated effects are such that  
\[e_{i,x^{\prime}}e_{j,x}\sim\mathcal{N}(0,\sigma_{x}^{2}),\]  
and the coefficient of correlation for these two effects is \(\rho_{ij,x}\), and the \(y\)- and \(z\)-components are similarly distributed. Assuming that the \(x\)-, \(y\)- and \(z\)-components are mutually independent, then  
\[u^{2}(d_{ij})=\frac{2}{d_{ij}^{2}}\Big{(}\chi_{ij}^{2}\sigma_{x}^{2}(1-\rho_{ ij,x})+y_{ij}^{2}\sigma_{y}^{2}(1-\rho_{ij,y})+z_{ij}^{2}\sigma_{z}^{2}(1-\rho_{ ij,z})\Big{)}.\]  
(Here \(x_{ij}=x_{j}-x_{i}\), etc., as before). If the correlation is described in terms of a correlation kernel as in Equation (21), then \(\rho_{ij,x}=e^{-d_{ij}^{2}/\Lambda_{2 ET}^{2}}\). If the correlation behaviour is the same in each axis with \(\sigma_{x}=\sigma_{y}=\sigma_{z}=\sigma_{ET}\) and \(\rho_{ij,x}=\rho_{ij,y}=\rho_{ij,z}=\rho_{ij}=e^{-d_{ij}^{2}/\Lambda_{2 ET}^{2}}\), then  
\[u^{2}(d_{ij})=2\sigma_{ET}^{2}\Big{(}1-e^{-d_{ij}^{2}/\Lambda_{2 ET}^{2}}\Big{)}.\]  
Let \(r=d_{ij}/\lambda_{ET}\). Then  
\[1-e^{-d_{ij}^{2}/\Lambda_{2 ET}^{2}} =1-\bigg{\{}1-r^{2}+\frac{r^{4}}{2!}-\frac{r^{6}}{3!}+\cdots\bigg{\}},\] \[=r^{2}-\frac{r^{4}}{2!}+\frac{r^{6}}{3!}-\cdots\]  
If \(\lambda_{ET}\) is much greater than \(d_{ij}\), then \(e^{-r^{2}}\) is close to 1 and the uncertainty is the distance close to zero. For this case, the effects \(\mathbf{e}_{i}\approx\mathbf{e}_{j}\) act like a fixed offset associated with the measurements (similar to a probe qualification effect) and do not contribute to the uncertainty associated with the distance. For \(d_{ij}\) somewhat less than \(\lambda_{ET}\), the term on the right above is dominated by the first term \(r^{2}\) so that  
\[u^{2}(d_{ij})\approx 2\sigma_{ET}^{2}\frac{d_{ij}^{2}}{\lambda_{ET}^{2}}.\]  
In this case, the uncertainty associated with \(d_{ij}\) is approximately proportional to \(d_{ij}\), showing that the correlated effects behave somewhat like a scale effect. If \(d_{ij}\) is much greater then \(\lambda_{ET}\), the \(e^{-d_{ij}^{2}/\Lambda_{2 ET}^{2}}\approx 0\) and  
\[u^{2}(d_{ij})\approx 2\sigma_{ET}^{2}.\]  
For this case the correlated effects behave more like independent random effects. Figures 1-3 also give insight into the dependence of uncertainties associated with distances on spatial correlation length. We now evaluate the uncertainty \(u(d_{ij}-d_{rs})\) in the difference in two distances associated with four points associated with an isotropic spatial correlation model defined by statistical parameters \(\sigma_{ET}\) and length scale parameter \(\lambda_{ET}\). Let \(\mathbf{n}_{ij}\) be the unit normal point in the direction \(\mathbf{x}_{j}-\mathbf{x}_{i}\), and let \(\mathbf{n}_{rs}\) be defined similarly. Then,  
\[u^{2}(d_{ij}-d_{rs})=\sigma_{ET}^{2}\mathbf{g}^{\top}V_{N}\mathbf{g},\quad\mathbf{g}=(1,-1,- 1,1)^{\top},\]  
with  
\[V_{N}=\left[ \begin{array}{cccc}1&e^{-d_{ij}^{2}/\lambda_{ET}^{2}}&ce^{-d_{ij }^{2}/\lambda_{ET}^{2}}&ce^{-d_{ii}^{2}/\lambda_{ET}^{2}}\\ e^{-d_{ij}^{2}/\lambda_{ET}^{2}}&1&ce^{-d_{ij}^{2}/\lambda_{ET}^{2}}&ce^{-d_{ij }^{2}/\lambda_{ET}^{2}}\\ ce^{-d_{ij}^{2}/\lambda_{ET}^{2}}&ce^{-d_{ij}^{2}/\lambda_{ET}^{2}}&1&e^{-d_{ ii}^{2}/\lambda_{ET}^{2}}\\ ce^{-d_{ii}^{2}/\lambda_{ET}^{2}}&ce^{-d_{ij}^{2}/\lambda_{ET}^{2}}&e^{-d_{ ii}^{2}/\lambda_{ET}^{2}}&1\end{array} \right],\]  
where \(c=\mathbf{n}_{ij}^{\top}\mathbf{n}_{rs}\), the cosine of the angle between the two normal vectors. Performing the matrix-vector multiplications, we end up with  
\[u^{2}(d_{ij}-d_{rs})=2\sigma_{ET}^{2}\big{(}2-e_{ij}-e_{rs}+c(e_{ip}+e_{is}-e_{ ir}-e_{js})\big{)},\]  
where \(e_{ij}=e^{-d_{ij}^{2}/\lambda_{ET}^{2}}\), etc.