# ILRS Reference Point Determination Using Close Range Photogrammetry
## 3 Satellite Observing System Wettzell
### Analysis and Results
#### 3.2.4 Impact of the Stochastic Model
\begin{array}{cccc}\sigma_{x_{p}}^{2}&\sigma_{x_{ p}y_{p}}&\sigma_{x_{p}z_{p}}&\dots&\sigma_{x_{p}z_{q}}\\ \sigma_{y_{p}x_{p}}&\sigma_{y_{p}}^{2}&\sigma_{y_{p}z_{p}}&\dots&\sigma_{y_{p}z _{q}}\\ \sigma_{z_{p}x_{p}}&\sigma_{z_{p}y_{p}}&\sigma_{z_{p}}^{2}&\dots&\sigma_{z_{p}z _{q}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \sigma_{z_{q}x_{p}}&\sigma_{z_{q}y_{p}}&\sigma_{z_{q}z_{p}}&\dots&\sigma_{z_{ q}}^{2}\end{array} \right)_{l}\] is the symmetric block matrix of the \(l\)th mounted marker \(\mathbf{p}_{l}\). The number of block matrices is equal to the number of mounted markers \(m\). The resulting block-diagonal matrix neglects correlations between the positions obtained by different markers. Figure 8 depicts the resulting standard deviations derived the by five different stochastic models and the three realized measurement configurations. The measurement configuration has only a minor effect on the estimated standard deviations, and all configurations under consideration yield comparable results. In contrast to the configuration, the chosen stochastic model affects the resulting standard deviations clearly. The results are similar for the simplified stochastic models \(\mathbf{\Sigma_{I}}\), \(\mathbf{\Sigma_{D}}\), and \(\mathbf{\Sigma_{P}}\), and only differ by about 10 %. These three models neglect most of the correlations in \(\mathbf{\Sigma}\). In particular, the benefit of \(\mathbf{\Sigma_{P}}\), which considers intrapoint correlations of \(\mathbf{P}_{i}\), is quite small in comparison to the most simplified model \(\mathbf{\Sigma_{I}}\). The block-diagonal matrix \(\mathbf{\Sigma_{M}}\) is more complex and considers intramarker correlations. The obtained standard deviations increase by about 40 % and 10 % for the reference point coordinates \(\mathbf{P}_{\text{IRP}}\) and the axis offset \(e_{\text{AO}}\), respectively. However, introducing the fully populated dispersion matrix \(\mathbf{\Sigma}\), which is by far the most complex stochastic model, yields the largest standard deviations. In comparison to the most simplified model \(\mathbf{\Sigma_{I}}\), the resulting standard deviations derived by the fully populated dispersion matrix \(\mathbf{\Sigma}\) are three times and two times larger for the reference point coordinates \(\mathbf{P}_{\text{IRP}}\) and the axis offset \(e_{\text{AO}}\), respectively. Neglecting the correlations between the observed positions overestimates the standard deviations, and overly optimistic results are obtained. The reason of the large differences depicted in Figure 8 is a consequence of the oversimplified stochastic models, which neglect most of the correlations. The frequency of the correlation coefficients \(\rho\) derived by the correlation matrix  
\[\mathbf{K}=\mathbf{\Sigma_{D}^{-\frac{1}{2}}}\mathbf{\Sigma\Sigma_{D}^{-\frac{ 1}{2}}} \tag{42}\]  
is depicted in Figure 9. Obviously, most of the parameters are positively correlated, cf. Figure 9a. About 88 % of the correlation coefficients are in a range of about \(\pm 0.3\). A second large accumulation is visible between 0.3 and 0.6, which contains about 10 % of the coefficients. About 2 % of the coefficients exceed 0.6 and indicate highly correlated parameters. The reason of this frequency is the structure of the correlation matrix \(\mathbf{K}\). Figure 9b depicts the component-wise intracorrelation coefficients \(\rho_{x_{i}x_{j}}\), \(\rho_{y_{i}y_{j}}\), and \(\rho_{z_{i}z_{j}}\), with \(i\
eq j\), of an arbitrary marker position at the telescope, obtained from the \(i\)th row in \(\mathbf{K}\). For the first 11 coefficients, the correlations exceed 0.7. Here, the corresponding coordinate components refer to positions of a dedicated telescope orientation defined by \(\omega_{k}\), \(\kappa_{k}\). These coordinate components are highly correlated because they are observed (nearly) simultaneously by identical images. If the telescope orientation is changed, new coordinates are obtained for the mounted markers and the correlations strongly decrease.