# ILRS Reference Point Determination Using Close Range Photogrammetry
## 3 Satellite Observing System Wettzell
### Analysis and Results
#### 3.2.1 Bundle Adjustment  
The experiments of the measurement campaign are analysed individually as free-network adjustment, introducing NNT and NNR conditions to solve the rank deficiency of the normal equation matrix, cf. Equations (11)-(16). The NNS condition is omitted because three calibrated scale bars are observed, which are traced to the SI meter. The remaining \(D=6\) datum conditions are exclusively applied to the nine DN points because these points are permanently marked, are interoperable with other measurement systems, and ensure the connection to the existing station network, cf. Section 3.1. According to [PERSON] [19; 21], distortion parameters of the camera are estimated within the bundle adjustment, which leads to the self-calibration approach described in Section 2.1. Figure 6 depicts the horizontal distribution of the observed markers at the telescope for the three measurement configurations. Ellipses indicate the \(1\sigma\) confidence level of the points obtained by the fully populated dispersion matrix of the bundle adjustment. Further global adjustment results are summarized in Table 2. The mounted markers have unique positions in the telescope fixed frame (object frame) but in the frame of the measurement instrument (reference frame) these markers yield trajectories. The number of observed positions per marker depends on the visibility of the marker and the number of different telescope orientations used during the experiment, cf. Table 1. For that reason, the number of unknowns \(n_{\text{par}}\) as well as the number of observations \(n_{\text{obs}}\) increases, if the number of telescope orientations gets larger, see Figure 6. With one exception, all estimated \(\vartheta_{0}\) are close to the expectation value \(\text{E}\big{\{}\hat{\sigma}_{0}^{2}\big{\}}=1\) and \(\tilde{\sigma}_{3D}\) does not exceed \(50\,\upmu\text{m}\). For DOY 259, \(\tilde{\sigma}_{3D}\) is slightly larger and \(\tilde{\sigma}_{0}\) exceeds its expectation value. The reason is not yet entirely clear and the effect is still under investigation. This configuration contains the smallest number of telescope orientations. However, DOY 260 uses an equivalent configuration but is inconspicuous. Thus, the reduction of telescope positions can be excluded as a possible cause. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**DOY** & \(n_{\text{obs}}\) & \(n_{\text{par}}\) & \(\vartheta_{0}\) & \(\sigma_{X}\) & \(\sigma_{Y}\) & \(\sigma_{Z}\) & \(\tilde{\sigma}_{3D}\) & \(\tilde{\sigma}_{3D}^{*}\) \\ \hline
255 & 64,367 & 10,015 & 0.886 & 0.03 & 0.03 & 0.02 & 0.05 & 0.04 \\
256 & 82,905 & 13,003 & 0.917 & 0.03 & 0.03 & 0.02 & 0.05 & 0.04 \\
257 & 91,437 & 12,139 & 0.949 & 0.02 & 0.02 & 0.02 & 0.04 & 0.03 \\
258 & 92,817 & 12,130 & 0.939 & 0.02 & 0.03 & 0.02 & 0.04 & 0.04 \\
259 & 62,549 & 8383 & 1.671 & 0.04 & 0.04 & 0.03 & 0.07 & 0.06 \\
260 & 75,731 & 9277 & 0.923 & 0.02 & 0.02 & 0.02 & 0.03 & 0.03 \\
264 & 120,005 & 13,663 & 0.844 & 0.02 & 0.02 & 0.01 & 0.03 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experiment-wise benchmark data of the bundle adjustments derived by JAiCov: \(n_{\text{obs}}\) and \(n_{\text{par}}\) are the number of observations and unknowns, respectively. \(\vartheta_{0}^{2}\) is the (unitless) variance of the unit weight. The mean standard deviations of the telescope points are given by \(\tilde{\sigma}_{X}\), \(\tilde{\sigma}_{Y}\), \(\tilde{\sigma}_{Z}\), and \(\sigma_{3D}=\sqrt{\varrho_{X}^{2}+\varrho_{Y}^{2}+\sigma_{Z}^{2}}\). \(\tilde{\sigma}_{3D}^{*}\) denotes the corresponding mean overall standard deviation of the bundle adjustment. Standard deviations are given in mm. The mean standard deviations of the horizontal components \(\bar{\sigma}_{X}\) and \(\bar{\sigma}_{Y}\) are comparable to each other but the vertical component \(\bar{\sigma}_{Z}\) is slightly smaller than the horizontal components. The reason is the distribution of the markers and the measurement configuration.