# On the Integration of VLBI Observations to GENESIS into Global VGOS Operations
## 3 Materials and Methods
### Orbit Errors  
We assume that the satellite techniques are utilized for a precise orbit determination of the GENESIS satellite with orbit solutions of 1-day arcs. We also assume that the determined GENESIS orbit is erroneous. In the orbit error simulation module, this is implemented by a modification of the orbital elements. For each of the 1000 simulation runs, the orbital elements are changed randomly by reasonable values to generate a set of erroneous orbital elements. The unchanged orbit parameters and positions are further referred to as true. At the time of each scan, the true and erroneous orbit positions are determined. Each erroneous orbit position is then transformed into the corresponding local coordinate frame of the true orbit position as offset in along-track-, cross-track-, and radial-direction. For scan \(i\), the offset vector is given by  
\[\Delta_{i}= \begin{bmatrix}\text{along}_{i}\\ \text{cross}_{i}\\ \text{radial}_{i}\end{bmatrix} . \tag{2}\]  
To describe the accuracy of the erroneous orbit in each component the root-mean-square (rms) values of the along-track-, cross-track-, and radial-direction are calculated from all scans. The rms value for the along-track component is given by  
\[\text{along}_{\text{RMS}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\text{along}_{i}^{2}}, \tag{3}\]  
where \(N\) is the total number of scans. The values cross\({}_{\text{RMS}}\) and radial\({}_{\text{RMS}}\) are calculated accordingly. In VLBI observations of satellites, the measured delay is significantly less sensitive to changes in the satellite position in the radial direction as compared to the cross-track- and along-track-direction [31]. This is because the VLBI technique is inherently more sensitive to variations that cause changes in the angle of arrival of the signal rather than just the distance. Meanwhile, in precise orbit determination of medium Earth orbit satellites the radial component is typically determined with higher accuracy than the along-track and  
Figure 5: Ground track of the GENESIS orbit over a 24 h period. The small dots represent the position of the satellite over the Earth in a 2.5 min interval. The big dots show the position every hour from 0 h to 24 h.  
cross-track components. To account for this, the along-track and cross-track components of all offset positions are scaled to exceed the offsets in the radial direction. A scale factor of three is selected to ensure that a considerable portion of the orbit error is reflected in the simulated delay. The scaled along-track offsets are given by  
\[\text{along}^{\prime}_{i}=\text{along}_{i}\cdot\frac{3\cdot\text{radial}_{ \text{RMS}}}{\text{along}_{\text{RMS}}}. \tag{4}\]  
The scaled cross-track offsets \(\text{cross}^{\prime}_{i}\) are derived similarly. The radial offsets remain unchanged,  
\[\text{radial}^{\prime}_{i}=\text{radial}_{i}. \tag{5}\]  
The modulated offset vector \(\Delta^{\prime}_{i}\) at scan \(i\) contains the scaled components \(\text{along}^{\prime}_{i}\), \(\text{cross}^{\prime}_{i}\) and \(\text{radial}^{\prime}_{i}\). The 3D-accuracy of the orbit offsets over all scans is given by the rms value  
\[\Delta^{\prime}_{\text{RMS}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\|\Delta^{\prime} _{i}\|^{2}}, \tag{6}\]  
where \(\|\cdot\|\) is the Euclidean vector norm. The orbit errors are assumed to have a certain target orbit accuracy with the value \(\text{target}_{oa}\). The offset vectors \(\Delta^{\prime}_{i}\) are scaled to meet the orbit accuracy  
\[\Delta^{\prime\prime}_{i}=\Delta^{\prime}_{i}\cdot\frac{\text{target}_{oa}}{ \Delta^{\prime}_{\text{RMS}}}. \tag{7}\]  
For the updated offsets of the erroneous orbit positions, the radial component is three times smaller than the along-track and cross-track components. Furthermore, the sizes of the offsets of the erroneous orbit positions meet the target orbit accuracy,  
\[\Delta^{\prime\prime}_{\text{RMS}}=\text{target}_{oa}. \tag{8}\]  
In this work, we consider three different orbit accuracies \(\text{target}_{oa}=\{1\,\text{cm},3\,\text{cm},5\,\text{cm}\}\). [PERSON] et al.