# ILRS Reference Point Determination Using Close Range Photogrammetry
## 3 Satellite Observing System Wettzell
### Analysis and Results
#### 3.2.4 Impact of the Stochastic Model  
The estimated parameters of a linear model are known to be unbiased, even if the stochastic model of the observation is misspecified ([57] p. 180). However, the dispersion of the estimated parameters is affected by an erroneous stochastic model [58]. To evaluate the impact of the chosen stochastic model onto the dispersion of the reference point and the axis offset, the fully populated dispersion matrix \(\mathbf{\Sigma}\) obtained by the bundle adjustment is replaced by four simplifications. 1. The most simplified stochastic model reads \[\mathbf{\Sigma_{\mathrm{I}}}=\sigma_{0}^{2}\mathbf{\mathrm{I}},\] (38) which disregards all prior results derived by the bundle adjustment, i.e., the variances of the points and the dependencies between the points are neglected. Such a model is usually applied in curve and surface analyses [59; 60]. Here, matrix \(\mathbf{\mathrm{I}}\) is the identity matrix, and \(\sigma_{0}^{2}\) is the variance of the unit weight. 2. Using the variances given in \(\mathbf{\Sigma}\) yields the most common approach [61], because commercial software packages often provide the standard deviations \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{z}\) of the estimated object points. Having \(n\) points, this stochastic model reads \[\mathbf{\Sigma_{\mathrm{D}}}=\mathrm{diag}\left(\sigma_{x_{i}}^{2},\sigma_{y_{i} }^{2},\sigma_{z_{i}}^{2},\ldots,\sigma_{z_{n}}^{2}\right).\] (39) Like before, all correlations \(\rho\) are neglected. 3. Especially in laser scanning applications [61; 62], a block-diagonal matrix \[\mathbf{\Sigma_{\mathrm{P}}}=\mathrm{blkdiag}\left(\mathbf{\Sigma_{i}},\ldots,\mathbf{ \Sigma_{n}}\right)\] (40) is frequently used to define the stochastic model. The number of block matrices is equal to the number of observed points \(n\), and the matrix \(\mathbf{\Sigma}_{i}\) is the \(i\)th symmetric block matrix that corresponds to the \(3\times 3\) sub-matrix of the \(i\)th observed position \(\mathbf{\mathrm{P}}_{i}\) in \(\mathbf{\Sigma}\), i.e., \[\mathbf{\Sigma}_{i}=\left( \begin{array}{ccc}\sigma_{x}^{2}&\sigma_{xy}&\sigma_{ xz}\\ \sigma_{yx}&\sigma_{y}^{2}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{z}^{2}\\ \end{array} \right)_{i}.\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**DOY** & \(\mathbf{\Delta X}\) & \(\mathbf{\Delta Y}\) & \(\mathbf{\Delta Z}\) & \(\mathbf{\Delta e_{\mathrm{AO}}}\) & \(\mathbf{\Delta\hat{\sigma}_{X}}\) & \(\mathbf{\Delta\hat{\sigma}_{Y}}\) & \(\mathbf{\Delta\hat{\sigma}_{Z}}\) & \(\mathbf{\Delta\hat{\sigma}_{\epsilon_{\mathrm{AO}}}}\) \\ \hline
255 & 0.0 & \(-0.1\) & 0.0 & 0.0 & 2.1 & 2.2 & 3.3 & 1.2 \\
258 & 0.0 & \(-0.1\) & 0.0 & 0.0 & 1.2 & 1.3 & 2.0 & 0.7 \\
260 & 0.0 & 0.0 & 0.0 & 0.0 & 1.3 & 1.4 & 2.1 & 0.8 \\ \hline \end{tabular}
\end{table} 
Table 4: Differences between the second-order approximation derived by the SSUT and the linear substitute solution derived by the GMM of the coordinates of the reference point \(\mathbf{\mathrm{P}}_{\mathrm{IRP}}\), the axis offset \(e_{\mathrm{AO}}\), and the related standard deviations, i.e., \(\Delta=\mathrm{SSUT}-\mathrm{GMM}\). All estimates are given in \(\upmu\upmu\). Here, \(\sigma_{xy}\), \(\sigma_{zz}\), and \(\sigma_{yz}\) are the covariances between the coordinate components. Thus, correlations \(\rho\) between the coordinate components of the point \(\mathbf{P}_{i}\) are taken into account, but correlations between the observed positions are still neglected. 4. The stochastic model, which takes the correlations \(\rho\) between the observed positions realized by a single marker \(\mathbf{p}_{l}\) into account, is given by \[\mathbf{\Sigma_{M}}=\text{blkdiag}\left(\mathbf{\Sigma}_{l},\dots,\mathbf{ \Sigma}_{m}\right),\] (41) where \[\mathbf{\Sigma}_{l}=\left(