## 1 Introduction  
With GNSS modernization and development, the traditional GPS-only dual-frequency model is gradually transitioning to multi-GNSS (GPS, BDS, GLONASS, Galileo, and regional system) and multi-frequency tendencies. The GPS Block IIF satellites can transmit L1, L2, and L5 signals [1]. The global BeiDou navigation satellite system (BDS-3) satellites can broadcast the BIC, B2a, B2b, and B2 (B2a + B2b) signals as well as the legacy B1I and B3I signals [2]. Similarly, the new GLONASS satellites have begun to transmit code division multiple access (CDMA) G1a, G2a, and G3 signal and frequency division multiple access (FDMA) G1 and G2 signals [3]. Galileo can transmit E1, E5a, E5b, E5, and E6 signals [4]. The multi-frequency observations can be beneficial in precise point positioning (PPP), cycle slip detection, precise clock offset estimation (PCE), and so on [5, 6, 7]. Normally, the legacy precise satellite clock offsets released by GNSS analysis centers (ACs) are generated by a dual-frequency ionosphere-free (IF) combination model [8]. For example, the recommended basic frequency pair for GPS, Galileo, and BDS are L1/L2, E1/E5a, and B1I/B3I signals [1; 9; 10]. Due to the characteristics of multi-frequency multi-channel GNSS signals, there is an inter-frequency bias between different frequencies, and there is an intra-frequency deviation between different channels of the same frequency. The code and phase time-variant hardware delay will be absorbed into satellite clock offsets, which leads to the inconsistency of the precise satellite clock offsets estimated by different frequencies [11]. This inconsistency is known as the inter-frequency clock bias (IFCB) [1]. The IFCB can be divided into code IFCB (CIFCB) and phase IFCB (PIFCB) [12]. The CIFCB is a linear combination of code hardware delays, and it is also called the differential code bias (DCB) or pseudorange observable-specific signal bias (OSB) [13]. The DCB is usually determined by the carrier-to-code leveling (CCL) method [14; 15]. Most International GNSS Service (IGS) and International GNSS Monitoring and Assessment System (iGMAS) ACs such as Centre for Orbit Determination in Europe (CODE), Deutsches Zentrum fur Luft- und Raumfahrt (DLR), Chinses Academy of Sciences (CAS), WhuHan University (WHU), and Shanghai Astronomical Observatory (SHA) use the CCL method to obtain DCB [16; 17; 18; 19]. The PIFCB is a linear combination of phase time-variant hardware delays, which is usually generated by epoch-differenced (ED) geometry-free (GF) ionosphere-free (IF) linear combination (GFIF) [12]. At present, IGS and iGMAS ACs have not yet officially released IFCB, and the OSB files obtained from GNSS AC mainly release pseudorange and phase OSB [3; 20]. Nevertheless, many references have conducted research on it. [PERSON], et al. [21] used the ED GFIF model to obtain GPS L5 PIFCB, and researched characteristics to model and predict it [22]. Similarly, [PERSON], et al. [23] used the ED GFIF model to generate the PIFCB and analyzed its periodic characteristics to develop the corresponding prediction model. [PERSON], et al. [24] converted the PIFCB obtained by the above method into carrier-phase OSB, which greatly simplified the complexity of the client algorithm. Although this method has high computational efficiency, it has certain theoretical flaws. However, the correct method assumes that the PIFCB of the same satellite at different receivers has good consistency and the receiver PIFCB can be combined into satellite PIFCB. Actually, receiver hardware delays are greatly affected by the ground environment variation [25]. If the receiver hardware delays change, it will have a negative impact on the satellite PIFCB estimation. The IFCB can also be calculated by estimating the satellite clock offset for all frequencies. [PERSON] and [PERSON] [26] provided the GPS L1/L2 and L5-only types of satellite clock products for the triple-frequency PPP. [PERSON], et al.