## 2 Materials and Methods
### PPP Time Transfer Model with a Priori ISB Constraints Considering Satellite Clock Datums  
In view that the ISB estimates are usually very stable, it is deduced that the a priori ISB estimates, such as the estimates of the previous day, can be used as constraints to enhance the PPP time transfer performance. Following Equation (4), the ISB estimates are composed of the differences in time scales, satellite clock datums, and receiver-dependent code hardware delays among different satellite systems. As the time system is aligned to the GPS time for all satellite systems in both multi-system observation files and precise satellite clock files, the effects of time scales are marginal and can be ignored. As for the receiver-dependent code hardware delays, they are considered to be stable within a short period of time, for example, several days [20; 21]. In contrast, the satellite clock datums are redefined day-by-day. Thus, the different satellite clock datums between two adjacent days will result in the distinct ISB estimates for the two days, which is an obstacle for the introduction of a priori ISB constraints in the data processing of multi-system integrated PPP time transfer. If the variations of the satellite clock datums between two adjacent days can be computed in advance, this problem will be solved.  
With the use of the satellite clock corrections at the GPS time 23:58:30, 23:59:00, and 23:59:30 in the precise satellite clock file of the previous day, those at the GPS time 00:00:00 on this day can be predicted based on the second-order polynomial fitting. The factors that represent the stability of satellite clocks in the time-frequency domain include phase, frequency, and frequency drift. Usually, a second-order polynomial model containing the three factors can be selected as the clock offset prediction model [22]. The complicated periodic terms can be ignored with the datasets of only several minutes. Due to the effects of different satellite clock datums, the satellite clock corrections at the GPS time 00:00:00 in the precise satellite clock file of this day differ from the predicted ones. However, the difference between them is roughly consistent for all the satellites (see Equation (2)). It is noted that the satellite-dependent code hardware delays are also stable within a short period of time. Thus, the difference in the satellite clock datums between two adjacent days can be calculated as:  
\[dD_{diff}^{AC,Q}=dD_{t}^{AC,Q}-dD_{p}^{AC,Q}=\left[\sum_{s=1}^{n}\left(cdt_{IF,e}^{ s,Q}-cdt_{IF,p}^{s,Q}\right)\right]/n \tag{9}\]  
where \(dD_{t}^{AC,Q}\) is the satellite clock datum of this day, \(dD_{p}^{AC,Q}\) is the satellite clock datum of the previous day, \(cdt_{IF,e}^{s,Q}\) is the satellite clock corrections at the GPS time 00:00:00 in the precise satellite clock file of this day, \(cdt_{IF,p}^{s,Q}\) denotes the predicted ones, and \(n\) is the number of satellites.  
To constrain the ISB estimates with the consideration of satellite clock datums, the following pseudo-observation equation can be introduced into the observation model of multi-system integrated PPP time transfer:  
\[ISB_{r,p}^{G,Q}+dD_{diff}^{AC,Q}-dD_{diff}^{AC,G}=ISB_{r}^{G,Q} \tag{10}\]  
where \(ISB_{r,p}^{G,Q}\) denotes the fully converged ISB estimates of the previous day. As to the stochastic model for the pseudo-observation, the STD of a priori ISBs including the adjustment of satellite clock datums is empirically set to 1 ns.