# Integrated Precise Orbit Determination of Multi-Gnss and Large LEO Constellations
## 2 Constellations and Observation Simulation
### Observations Simulation Configuration
\((b_{g,j},b_{leo,j},b_{j}^{s})\) and \((B_{g,j},B_{leo,j},B_{j}^{s})\) represent the code hardware delay and carrier phase hardware delay of the ground receiver, onboard receiver, and satellites respectively. \((\varepsilon_{g,j}^{s},\varepsilon_{leo,j}^{s})\) and \((\omega_{g,j}^{s},\omega_{leo,j}^{s})\) are the combination of multipath and noise for code and carrier phase observations. The main task of the GNSS observation simulations is to accurately compute the components on the right side of Equation (1) using the existing models, and to guarantee that the simulated GNSS observations reflect a real-world environment as much as possible. Before the observation simulation, the orbits of both GNSS and LEO satellites are firstly simulated. Then, we performed a standard multi-GNSS PPP using the real GNSS measurements of each ground station to provide a weekly solution of ground station position, receiver clock offset, the wet components of tropospheric delay, inter-system bias (ISB), and inter-frequency bias (IFB) for the subsequent observation simulation. The satellite-to-receiver distance is calculated based on the position for station at the signal receiving time and mass center position for satellites at the signal transmitting time. The previous PPP processing provides a weekly solution of the ground station position, while the GNSS satellites position can be obtained from the simulated orbits. The phase center offset (PCO) and phase center variation (PCV) values of satellites and ground stations should be considered using the values from \"igs14.atx\" [26], though they are not presented in Equation (1). The multi-GNSS receiver clock offsets are simulated using the PPP-derived receiver clock offsets, as well as both ISB and IFB. The multi-GNSS precise clock products from GeoForschungsZentrum (GFZ) are used to provide the values of satellite clock offsets. Given the ongoing development of BDS and Galileo constellations, the clock offsets of the unoperated satellites are replaced with the existing one, e.g., the clock offset of BDS C27 satellite is considered the same as that of C13 satellite. It's noteworthy that both the satellite precise clock products and PPP-derived receiver clock offsets are generated based on an ionosphere-free (IF) combination, and they usually absorb the IF combination of code hardware delays for the satellite and receiver, respectively,due to the strong coupling between the clock offsets and the code hardware delay. Hence, Equation (1) can be rewritten as  
\[\left\{ \begin{array}{c}P_{g,j}^{s}=\rho_{g,j}^{s}+c(\overline{\delta t}_{g}- \overline{\delta t}^{s})+I_{g,j}^{s}+T_{g}^{s}+(b_{g,j}-b_{g,IF})-(b_{j}^{s}+b_{ IF}^{s})+\varepsilon_{g,j}^{s}\\ P_{lo,j}^{s}=\rho_{lo,j}^{s}+c(\overline{\delta t}_{lo}-\overline{\delta t}^{s}) +I_{lo,j}^{s}+(b_{lo,j}-b_{lo,IF})-(b_{j}^{s}+b_{IF}^{s})+\varepsilon_{lo,j}^{ s}\\ L_{g,j}^{s}=\rho_{g,j}^{s}+c(\overline{\delta t}_{g}-\overline{\delta t}^{s}) -I_{g,j}^{s}+T_{g}^{s}+\lambda_{j}N_{g,j}^{s}+\lambda_{j}(B_{g,j}-B_{j}^{s})-b_ {g,IF}-b_{IF}^{s}+\omega_{g,j}^{s}\\ L_{lo,j}^{s}=\rho_{lo,j}^{s}+c(\overline{\delta t}_{lo}-\overline{\delta t}^{s}) -I_{lo,j}^{s}+\lambda_{j}N_{lo,j}^{s}+\lambda_{j}(B_{lo,j}-B_{j}^{s})-b_{lo, IF}-b_{IF}^{s}+\omega_{lo,j}^{s}\end{array} \right. \tag{2}\]  
with  
\[\left\{ \begin{array}{c}\overline{\delta t}_{g}=\delta t_{g}+b_{g,IF}\\ \overline{\delta t}_{lo}=\delta t_{lo}+b_{lo,IF}\\ \overline{\delta t}^{s}=\delta t^{s}-b_{IF}^{s}\end{array} \right. \tag{3}\]  
The combination of code hardware delays in the code observation equation can be calculated using known differential code biases (DCB) from Center for Orbit Determination in Europe (CODE) [27; 28]. The ionospheric delay along the direction of the signal propagation at a specific frequency can be modeled using the international GNSS service (IGS) global ionosphere maps (GIM).