# On the Integration of VLBI Observations to GENESIS into Global VGOS Operations
## 3 Materials and Methods
### Orbit Errors
[6] state that the GENESIS orbit could be determined with accuracies better than 1 cm. For Low-Earth Orbit missions, GNSS data usually provide one of the strongest contributions to precise orbit determination due to the almost continuous tracking of GNSS satellites. However, [PERSON] et al. [32] underline that at an altitude of 6000 km, the GNSS tracking coverage for a spaceborne receiver is significantly reduced. Furthermore, the authors underline that GENESIS will receive GNSS signals outside the Terrestrial Service Volume. A lack of detailed data on antenna transmit patterns in this region currently hinders a precise evaluation of the potential measurement and orbit-determination accuracy. Given that the expected final orbit accuracy is unclear, we consider three different levels of orbit accuracies. Figure 6 shows three exemplary sample realizations of offsets in along-track-, cross-track-, and radial-direction over a period of 24 h. The target orbit accuracy is 3 cm. The offsets oscillate around the true satellite position. Additionally, trends and constant offsets are present. For each simulation, orbit errors over 24 h are simulated 1000 times. In the following, we calculate the delay difference resulting from the offset between the true and erroneous satellite positions as observed by the stations. For each observation, the true satellite position, the offset satellite position, as well as the positions of station 1 and station 2 are determined as absolute coordinates in a common coordinate frame. The geometric configuration is illustrated in Figure 7. The delays are calculated on the pure geometric delay described in the formalism by [PERSON] [33]. We neglect other effects like the retarded baseline effect and relativistic effects. As this is a simulation study and we are considering the differences between the true and observed delay, the simulation results are hardly affected. The geometric delay observed from the stations to the true satellite position is given by  
\[\tau_{\text{t}}=\frac{\|L_{\text{2}}\|-\|L_{\text{t}}\|}{\cwhere \(c\) is the speed of light and \(L_{\text{ti}}\) is the geometric distance between station \(i\) and the true satellite position. Similarly, the geometric delay observed from the stations to the erroneous offset satellite position is given by  
\[\tau_{\text{o}}=\frac{\|L_{\text{o2}}\|-\|L_{\text{o1}}\|}{c}. \tag{10}\]  
The delay difference resulting from an erroneous orbit determination of GENESIS is calculated by  
\[\tau_{orb}=\tau_{\text{t}}-\tau_{\text{o}}. \tag{11}\]  
The value \(\tau_{orb}\) is added to the observed minus computed value as described in Section 3.2. Figure 5:0 Three exemplary sample realizations of a thousand simulated orbit errors in along-track-, cross-track-, and radial-direction over a period of 24 h. The 3D target orbit accuracy is 3 cm. Figure 5:1 The orbit error delay calculation takes into account the geometric configuration of the true satellite position, the offset satellite position, station 1, and station 2. The distances between station 1 and station 2 to the true satellite position are \(L_{\text{t1}}\) and \(L_{\text{t2}}\), respectively. The distances to the erroneous offset satellite position are \(L_{\text{o1}}\) and \(L_{\text{o2}}\).