## 2 Methods  
### Ionosphere-Free PPP Observation Model  
Ionosphere-free (IF) observations can be used to remove first-order ionospheric delays from pseudo-range and carrier-phase measurements. For a receiver r that has observed a satellite s (such as GPS, BDS-3, or Galileo) at a specific epoch, the linearization of the ionosphere-free code and carrier-phase observation equations are perhaps defined as  
\[p_{r,\mathrm{IF}}^{s,\mathrm{M}}=p_{r}^{s,\mathrm{M}}+c(dt_{r,\mathrm{IF}}^{ \mathrm{M}}-dt_{\mathrm{IF}}^{s,\mathrm{M}})+T_{r}^{s,\mathrm{M}}+b_{r,\mathrm{ IF}}^{s,\mathrm{M}}-b_{\mathrm{IF}}^{s,\mathrm{M}}+\delta_{r,\mathrm{IF}}^{s, \mathrm{M}} \tag{1}\]  
\[L_{r,\mathrm{IF}}^{s,\mathrm{M}}=p_{r}^{s,\mathrm{M}}+c(dt_{r,\mathrm{IF}}^{ \mathrm{M}}-dt_{\mathrm{IF}}^{s,\mathrm{M}})+T_{r}^{s,\mathrm{M}}+\tilde{N}_{r,\mathrm{IF}}^{s,\mathrm{M}}+\varepsilon_{r,\mathrm{IF}}^{s,\mathrm{M}} \tag{2}\]  
where  
\(p_{r,\mathrm{IF}}^{s,\mathrm{M}}\) and \(L_{r,\mathrm{IF}}^{s,\mathrm{M}}\) represent the observed minus calculated values of ionosphere-free code and carrier-phase observables, respectively;  
the superscript M corresponds to G, C, or E, which indicate the satellite systems GPS, BDS-3, or Galileo;  
\(\rho_{r}^{s,\mathrm{M}}\) stands for the geometric distance between the receiver and the satellite; \(c\) is the speed of light; \(dt_{r,\mathrm{IF}}^{\mathrm{M}}\) and \(dt_{\mathrm{IF}}^{s,\mathrm{M}}\) denote the receiver clock offset and satellite clock offset, respectively; \(T_{r}^{s,\mathrm{M}}\) is the tropospheric delay;  
\(b_{r,\mathrm{IF}}^{\mathrm{M}}\) and \(b_{\mathrm{IF}}^{s,\mathrm{M}}\) are the uncalibrated code hardware delays (UCDs) by the receiver and satellite, respectively;  
\(\delta_{r,\mathrm{IF}}^{s,\mathrm{M}}\) and \(\varepsilon_{r,\mathrm{IF}}^{s,\mathrm{M}}\) represent the total of code and carrier-phase observation measurement noise and multi-path errors, respectively;\(\bar{N}_{r,\rm IF}^{S,\rm M}\) is the carrier-phase ambiguity, which consists of the ionosphere-free combined receiver and satellite UCDs (\(b_{r,\rm IF}^{\rm M}\) and \(b_{\rm IF}^{\rm s,\rm M}\)), the receiver and satellite uncalibrated phase hardware delays (UPDs) (\(B_{r,\rm IF}^{\rm M}\) and \(B_{\rm IF}^{\rm s,\rm M}\)), the ionosphere-free wavelength (\(\lambda_{\rm IF}\)), and the ionosphere-free combined integer-phase ambiguity (\(N_{r,\rm IF}^{s,\rm M}\)). \[\bar{N}_{r,\rm IF}^{s,\rm M}=\lambda_{\rm IF}N_{r,\rm IF}^{s,\rm M}+\left(b_{r,\rm IF}^{\rm M}-b_{\rm IF}^{\rm s,\rm M}\right)-\left(B_{r,\rm IF}^{\rm M}-B_ {\rm IF}^{s,\rm M}\right) \tag{3}\]  
The error components not mentioned above are supposed to be properly adjusted using their respective models, such as PCO and PCV, phase wind-up, solid earth tides, polar tides, ocean loading, and earth rotation parameters (ERPs). The tropospheric delay can be divided into the hydrostatic component and the wet component. The zenith hydrostatic delay (ZHD) is adjusted with the Saastamoinen model. The zenith wet delay (ZWD) has to be approximated as an uncertainty. The slant hydrostatic delays (SHDs) and slant wet delays (SWDs) can be corrected using the mapping function, which simulates the connection between the SHD and SWD to the corresponding zenith delays. For a single-system PPP, the clock parameter is often regarded as an epoch-wise parameter. A GNSS clock and an inter-system time difference parameter (referred to as ISB) are estimated for the multi-system combined PPP. There are various methods for modeling the ISB parameter. Epoch-wise ISB should be calculated for robust data analysis. The difference between the common pseudo-range and phase hardware delays is assimilated into the ambiguity parameter in the multi-system combined PPP model, while the common pseudo-range hardware delay is assimilated into the clock parameter. The Ionosphere-free PPP model with multi-constellation can be described as the following by taking the aforementioned concepts into consideration:  
\[\left\{