## 1 Introduction  
Position, velocity, and time (PVT) accuracy, integrity, continuity, and availability are the four major performance indicators for satellite navigation systems [1]; innovations and upgrades in navigation systems are constantly being undertaken to technically enhance these properties. The accuracy of the global navigation satellite system (GNSS) has currently been improved from several tens of meters to a meter-level. Typical technical measures adopted include the differential global positioning system (DGPS), represented by a wide area augmentation system (WAAS), for reducing or eliminating GPS measurement errors using the high correlation of the GPS error sources in time and space, thereby improving the user's positioning accuracy to a meter or even a sub-meter level [2; 3]; the integration of pseudolites with the GNSS improves the positioning geometric factors and coverage rate, enhancing the navigation and positioning accuracy [4; 5]; usage of a three carrier ambiguity resolution (TCAR) technology and a binary offset carrier (BOC) signal system to improve the accuracy of the positioning solution [6; 7; 8; 9], etc. Among the various techniques used to improve the GNSS performance, the inter-satellite link (ISL) technology has become a research hotspot in recent years. A National Aeronautics and Space Administration (NASA) senior engineer, [PERSON], who was responsible for the global position system (GPS) III ISL demonstration, has stated that the introduction of ISLs can significantly improve the accuracy of the ephemeris prediction, realize satellite autonomous integrity monitoring, increase the autonomous orbit determination, and improve the navigation system flexibility and expansibility [10]. The European Space Agency (ESA) has initiated several exploratory projects on the ISLs adopted in the Galileo system since 2007 [11]. Recently, with the rapid development of the BeiDou-2 program, ISLs have been used in the Beidou navigation system to achieve autonomous navigation [12]. The ISL has become a significant symbol of the new generation satellite navigation systems and its construction, an important consensus for several current GNSSs [13]. The accuracy of inter-satellite ranging is the key for determining the quality of the ISL construction and ranging consistency is a prerequisite for ranging accuracy. [PERSON] et al. set the precision of inter-satellite ranging to 0.5 m for simulating autonomous orbit determination [14]. When the range variance is 0.45 m and the international GNSS service (IGS) rapid ephemeris is used as the orbit initial value, the accuracy of the autonomous orbit determination in 180 days can be up to 6 m [15]. According to the present conclusion of the autonomous orbit determination and satellite-ground joint orbit determination, the accuracy of inter-satellite ranging should be at a decimeter-level. Owing to the frequency aging drift of the onboard clock and the relativistic effect of the navigation satellite [16], it is necessary to regularly adjust the phase and frequency of the onboard clock. Most of the signal generation and the time counts in the payload of a navigation satellite are directly or indirectly derived from a single 10.23 MHz clock; hence, the 10.23 MHz is adjusted. However, the work processing clock of the ISL payload (here after referred to as \(f_{c}\)) remains at the pre-adjusted state, causing a phase deviation between the 10.23 MHz and the \(f_{c}\), and inducing an offset in the measurement pseudo-range. Moreover, the uncertainty of the initial phase between the 1 PPS (1 pulse per second) and the 10.23 MHz each time the onboard clock device boots also results in a pseudo-range offset, where 1 PPS is obtained by a count of the 10.23 MHz. Currently, most research focusus on ISL signal system design, ISL signal dynamic characteristic analysis, ISL constellation design, and ISL extension applications. Reference [17] proposes a method that uses the navigation data to achieve a rapid acquisition of the ISL signal in highly dynamic conditions.