## 2 Observation Equation of Multi-GNSS CP Time Transfer  
In general, the mathematic model of time transfer based on a CP technique is the same as in the precise point positioning (PPP) model. The ionosphere-free combination for individual GNSSs has typically been utilized for dual-frequency carrier phase and pseudo-range observations to eliminate the first-order effects of the ionosphere. The observation equations can be written as [21; 24]:  
\[ \begin{split} P&=\rho+c\cdot(dt_{r}-dt_{s})+T_{ trop}+\varepsilon_{P}\\ L&=\rho+c\cdot(dt_{r}-dt_{s})+T_{trop}+N_{IF}+ \varepsilon_{L}\end{split}  \tag{1}\]  
where, \(P\) and \(L\) are the ionosphere-free combinations of carrier phase and pseudo-range. The geometric distance is denoted as \(\rho\), \(c\) is the speed of light, \(T_{trop}\) is the tropospheric delay, \(N_{IF}\) is the phase ambiguity of the ionosphere-free (IF) combination in metres, \(\varepsilon_{P}\) and \(\varepsilon_{L}\) are the measurement noise and multipath error for pseudo-range and carrier phase, respectively, \(dt_{s}\) is the satellite clock offset and \(dt_{r}\) is the receiver clock offset, which is the difference between the external time frequency reference and the GNSS time scale. It is important to note that the phase centre offset and variation, tidal loading, ocean tides, phase wind-up and relativistic delay must be corrected according to the existing models [24], although they are not explicitly expressed in Equation (1). In the combined GPS/BDS/Galileo time and frequency transfer based on the CP technique, there are three receiver clock offsets that represent differences between the common external time frequency reference and their corresponding system time scales. When the GPS time scale (GPST) is selected as a unique reference time scale to provide the common receiver clock offset based on three CDMA GNSSs, additional inter-system bias (ISB) parameters in BDS and Galileo are introduced, which also refer to inter-system receiver hardware bias. The formula may be expressed as:  
\[\left\{ \begin{array}{l}ISB_{sys}^{GC}=dt_{r}^{C}-dt_{r}^{G}\\ ISB_{sys}^{GE}=dt_{r}^{E}-dt_{r}^{G}\end{array} \right. \tag{2}\]  
The combined observation equation using GPS/BDS/Galileo measurements can therefore be expressed as [18]:  
\[\left\{ \begin{array}{l}P_{i}^{G}=\rho_{i}^{G}+c\cdot(dt_{r}^{G}-dt_{is}^{G}) +T_{trop}+\varepsilon_{i,P}^{G}\\ L_{i}^{G}=\rho_{i}^{G}+c\cdot(dt_{r}^{G}-dt_{is}^{G})+T_{trop}+N^{G}+ \varepsilon_{i,P}^{G}\\ P_{i}^{C}=\rho_{i}^{G}+c\cdot(dt_{r}^{G}+ISB_{sys}^{GC}-dt_{is}^{C})+T_{trop }+\varepsilon_{i,P}^{G}\\ L_{i}^{C}=\rho_{i}^{G}+c\cdot(dt_{r}^{G}+ISB_{sys}^{GC}-dt_{is}^{C})+T_{trop }+N^{C}+\varepsilon_{i,P}^{C}\\ P_{i}^{E}=\rho_{i}^{E}+c\cdot(dt_{r}^{G}+ISB_{sys}^{GE}-dt_{is}^{E})+T_{trop }+\varepsilon_{i,P}^{E}\\ L_{i}^{E}=\rho_{i}^{E}+c\cdot(dt_{r}^{G}+ISB_{sys}^{GE}-dt_{is}^{E})+T_{trop }+N^{E}+\varepsilon_{i,P}^{E}\end{array} \right. \tag{3}\]  
where the superscripts \(G\), \(C\), and \(E\) represent the GPS, BDS, and Galileo satellite systems, respectively. The subscript \(i\) denotes a single satellite. It is notable that the \(dt_{r}^{G}\) becomes a parameter that is jointly determined by GPS, BDS and Galileo satellite systems measurements, which are the quantities of interest when applying precise time and frequency transfer. When the two remote stations \(m\) and \(n\) equipped with different time and frequency references \(t_{m}\) and \(t_{n}\), are used for time and frequency transfer, the common GPST plays a role in the intermediate quantity to achieve time and frequency transfer progress; the corresponding formula can be expressed as follows:  
\[\Delta T_{m,n}=dt_{r}^{G}(m)-dt_{r}^{G}(n)=(t_{m}-GPST)-(t_{n}-GPST)=t_{m}-t_{n} \tag{4}\]  
Equation (4) indicates that the receiver clock offset is a parameter of interest in time and frequency transfer.