## 3 Simulation Configuration
### The Adaption of PANDA Software  
The PANDA [25] software has been developed as a multi-functional data processing software package for various applications, such as precise positioning services and POD of GNSS and LEO satellites. We implemented our LeGNSS PPP on the PANDA platform in this study. The LeGNSS PPP model and its processing strategies are presented as follows.  
Usually, the IF combination is used in PPP to eliminate the first-order ionospheric delays. For the tropospheric delay, its dry component is precisely corrected via an a priori model, while the residual (wet) part is captured in data processing by setting up a zenith wet delay (ZTD). Then the linearization of Equations (1) and (2) read as:  
\[p^{s}_{r,IF,k}=-u^{s}_{r}\mathbf{r}+\Delta t_{r,k}+b_{r,IF}+m_{r}z_{r,k}+\varepsilon _{p^{s}_{r,IF,k}} \tag{3}\]  
\[l^{s}_{r,IF,k}=-u^{s}_{r}\mathbf{r}+\Delta t_{r,k}+\delta_{r,IF}+m_{r}z_{r,k}+ \lambda_{IF}\widehat{N}^{s}_{r,IF}+\varepsilon_{L^{s}_{r,IF,k}} \tag{4}\]  
where \(p^{s}_{r,IF,k}\) and \(l^{s}_{r,IF,k}\) denote the observed-minus-computed code and phase observations from satellite \(s\) to receiver \(r\) at epoch \(k\). \(u^{s}_{r}\) is the unit vector of the direction from receiver to satellite. \(\mathbf{r}\) denotes the vector of receiver position increments relative to an a priori position. \(z_{r,i}\) is the ZTD parameter with the mapping function \(m_{r}\), \(\widehat{N}^{s}_{r,IF}=N^{s}_{r,IF}+\left(b^{s}_{IF}-\delta^{s}_{IF}\right)/ \lambda_{IF}\). The remaining symbols have the same meanings as in Equations (1) and (2). For the LeGNSS case with combined GPS, BDS, and LEO systems, the PPP model reads:  
\[p^{G}_{r,IF,k}=-u^{G}_{r}\mathbf{r}+\Delta\bar{t}_{r,k}+m_{r}z_{r,k}+\varepsilon_ {p^{G}_{r,IF,k}} \tag{5}\]  
\[p^{C}_{r,IF,k}=-u^{C}_{r}\mathbf{r}+\Delta\bar{t}_{r,k}+ ISB_{GC}+m_{r}z_{r,k}+ \varepsilon_{p^{C}_{r,IF,k}} \tag{6}\]  
\[p^{L}_{r,IF,k}=-u^{L}_{r}\mathbf{r}+\Delta\bar{t}_{r,k}+ ISB_{GL}+m_{r}z_{r,k}+ \varepsilon_{p^{L}_{r,IF,k}} \tag{7}\]  
\[J^{G}_{r,IF,k}=-u^{G}_{r}\mathbf{r}+\Delta\bar{t}_{r,k}+m_{r}z_{r,k}+\lambda_{IF,G} \overline{N}^{G}_{r,IF}+\varepsilon_{L^{G}_{r,IF,k}} \tag{8}\]  
\[I^{C}_{r,IF,k}=-u^{C}_{r}\mathbf{r}+\Delta\bar{t}_{r,k}+ ISB_{GC}+m_{r}z_{r,k}+\lambda_{IF,C}\overline{N}^{C}_{r,IF}+ \varepsilon_{L^{G}_{r,IF,k}} \tag{9}\]  
\[l^{L}_{r,IF,k}=-u^{L}_{r}\mathbf{r}+\Delta\bar{t}_{r,k}+ ISB_{GL}+m_{r}z_{r,k}+\lambda_{IF,L}\overline{N}^{L}_{r,IF}+ \varepsilon_{L^{L}_{r,IF,k}} \tag{10}\]  
where the superscripts \(G\), \(C\), and \(L\) refer to GPS, BDS, and LEO satellites. \(\Delta\bar{t}_{r,k}=\Delta t_{r,k}+b_{rG,IF}\) is the lumped (estimable) receiver clock biases. \(ISB_{GC}=b_{rC,IF}-b_{rG,IF}\) and \(ISB_{GL}=b_{rL,IF}-b_{rG,IF}\) are the receiver inter-system biases (ISB) parameters between GPS and BDS, as well as between GPS and LEO systems. \(b_{rG,IF}\), \(b_{rC,IF}\), and \(b_{rL,IF}\) are the IF receiver code biases for GPS, BDS, and LEO satellites, respectively. The reparametrized ambiguity is \(\overline{N}^{G}_{r,IF}=\widehat{N}^{G}_{r,IF}+(\delta_{rG,IF}-b_{rG,IF})/ \lambda_{IF,G}\) for GPS and they are completely analogously for BDS and LEO.  
Then the unknown parameter vector is:  
\[\mathbf{X}=\left[r^{T},\Delta\bar{t}_{r,k}, ISB_{GC}, ISB_{GL},z_{r,k}\left(\overline{N}^{s}_{r,IF}\right)^{T}\right]^{T} \tag{11}\]In the PPP model, the observations from all satellites are processed simultaneously in a single adjustment. The receiver positions are estimated in kinematic mode, while the ZTD is estimated as a random walk process with noise of 1 cm/\(\sqrt{1}\) hour. The receiver clock is estimated epoch-wise as white noise. The ambiguities are estimated as constant for each arc, and the ISB is estimated as constant over time.