## 2 Method
### LEO Satellite Clock Determination in Near-Real-Time
\(d\tilde{t}_{r}\) represents the estimable receiver clocks with the inclusion of the IF receiver code bias \(d_{r,IF}\) and the relativistic effects \(d_{rel}\). Recall that \(dt_{r}\) denotes the LEO satellite clock bias. It is assumed here that the IF code hardware delay of the GNSS satellite is already included in the GNSS satellite clock products. Depending on the specific type of pseudorange dual-frequency observations used, Differential Code Bias (DCB) corrections may be applied. For multi-GNSS processing, Inter-System Biases (ISBs) need to be additionally considered [13; 17]. The Kalman filter is a commonly used parameter estimation method in the field of satellite navigation. It first requires the establishment of a linearized kinematic observation model and a state-transition model for filtering. Subsequently, by recursion, the parameters to be estimated are determined. For detailed information, see [24]. In this study, ambiguity parameters are constrained as constants if there is no cycle slip. This study aims to determine LEO satellite clocks timely, which involves firstly the near-real-time clock determination as introduced above. Therefore, the choice of real-time GNSS products needs to be carefully considered. Currently, many analysis centers of the International GNSS Service (IGS) and other analysis systems like the International GNSS Monitoring and Assessment System (iGMAS) provide rich real-time GNSS orbit and clocks products [25; 26; 27; 28; 29], such as those offered by the National Centre for Space Studies (CNES) in France [30], The Federal Agency for Cartography and Geodes (BKG) in Germany [31], the GMV Aerospace and Defense (GMV) in Spain [32], the Wuhan University in China [33] and the GeoForschungsZentrum (GFZ) in Germany [34]. In this study, the selection of real-time GNSS products is based on the following steps:  
1. Analyze the availability and continuity of each real-time product over a past period. 2. Analyze the accuracy of GNSS satellite orbits, the precision of the GNSS satellite clocks, and their Signal-In-Space Range Error (ISRE) over a past period through comparison with the final GNSS products of, e.g., the Center for Orbit Determination in Europe (CODE) [35; 36]. 3. Based on the results of the first two steps, identify a set of optional real-time GNSS products and establish an initial ranking. The ranking list is updated on a daily basis. 4. Upon each processing round, conduct an assessment of the availability/continuity of the real-time products of GNSS satellites in the corresponding processing period. For those passing the pre-defined thresholds for availability and continuity, select the real-time GNSS products according to the ranking list, so that real-time GNSS products with both good completeness and precision can be used for the real-time determination of LEO satellite clocks. Figure 2 shows, e.g., the precision of real-time GPS satellite clocks from the BKG, CNE, GMV, GFZ, and WHU on Day of Year (DOY) 358 to 364, 2023 with a sampling interval of  
10 s. It can be seen that the average clock precision from all the analysis centers is better than 0.2 ns, with BKG and WHU exhibiting the best performances during the test period, approximately at 0.12 ns. Using the above-mentioned selection strategy for real-time GNSS products, the estimated LEO satellite clocks determined may face two issues:  
1. Inconsistent time references for LEO satellite clocks are determined in each session. This is caused by the different time references of different real-time GNSS satellite clocks provided by different analysis centers. 2. Poor stability of the real-time LEO satellite clock time reference due to the poor stability of the time reference of the real-time GNSS satellite clocks, as shown in Figure 3 (red). In Figure 3, the real-time time reference is calculated as the epoch mean difference between all the usable real-time GPS satellite clocks from the CNES and the CODE final GPS satellite clocks [18].