## 1 Introduction  
With the Global Positioning System (GPS), the Global Navigation Satellite System (GLONASS) in Russia, the BeilDou Navigation Satellite System (BDS), and the Galileo now providing global services, the Global Navigation Satellite System (GNSS) development has entered a new era. However, due to the inherent characteristics of GNSS, it still faces the challenge of being unable to address all the service requirements using a specific navigation technology. As a result, the navigation systems of countries are targeting higher service accuracy, more diverse functionalities, and more reliable services, initiating the planning and construction of the next generation of navigation systems [1]. In recent years, due to the unique advantages of orbital characteristics and signal strength, augmentation with Low Earth Orbit (LEO) satellites has gained attention and favor in the global satellite navigation field, promising to be an incremental development in the next generation of satellite navigation systems. Worldwide, there is the active implementation and deployment of LEO satellite constellation plans, such as [PERSON]'s Starlink [2], the OneWeb [3], the Iridium [4], the Xona's Pulsar of the United States [5], the Kepler system of Germany [6], and the Centispace [7], Hongyan and Hongyun [8] of China, involving tens of thousands of LEO satellites. Some of these constellations have the capability to actively broadcast navigation signals, such as Xona's Pulsar and the Centispace [7], aiming to augment the performance of GNSS, providing users with high-precision, fast convergence, low-cost, and highly reliable satellite Positioning, Navigation and Timing (PNT) services [9]. The LEO-augmented positioning results using real LEO satellite navigation signals have, e.g., been achieved with the CentiSpace satellites [7]. For cm-level high-accuracy LEO-augmented PNT services, high-precision real-time LEO satellite clock products are indispensable. Therefore, in addition to the LEO satellite real-time orbital products that have been intensively studied over the past decade [10; 11; 12; 13], high-precision LEO satellite clock products are a prerequisite for integrating the LEO navigation augmentation system with GNSS to provide high-performance PNT services. Their contributions to the LEO satellite Signal-In-Space-Ranging-Error (SISRE) to ground users are unignorable. In contrast to GNSS satellites that are predominantly positioned in Medium Earth Orbits (MEOs), LEO satellites have an altitude ranging from a few hundred kilometers to approximately 1500 km. This results in significantly smaller footprints compared to that of GNSS satellites [14]. When employing the same clock determination strategy as that for the GNSS satellite clocks using a network of ground stations, frequent reinitializations and discontinuous estimation cannot be avoided due to the difficulties in building dense network stations in remote areas and oceans [11; 14]. Luckily, LEO satellites serve as users of GNSS satellite signals and can perform continuous clock determination, treating the clock bias as a time-independent white noise estimation parameter [15]. When processing LEO satellite clocks using batch least-squares adjustment in reduced-dynamic mode, a high precision of about 0.15 ns can be achieved in near-real-time when using high-quality GNSS real-time products [16]. This, however, implies a processing time of about 5 min for 24 h processing with a sampling interval of 30 s, even with a powerful processing unit [12]. Considering a time interval between subsequent processing rounds of, e.g., another 5 min, a clock prediction of 10 min is needed to catch up with the real-time applications. The mid-term LEO satellite clock prediction turns out to be a more challenging task for LEO satellite clocks compared to that for the GNSS satellite clocks.