# Modelling and Assessment of a New Triple-Frequency IF1213 PPP with BDS/GPS
## 1 Introduction  
In the late 1990s, precise point positioning (PPP) technology was proposed by [PERSON] et al. [1] and implemented [2]. With the modernization of GPS and the completion of BDS, an increasing number of navigation satellites provide signals at three or more frequencies, and the research and application of triple-frequency PPP has become increasingly extensive and in depth [3; 4; 5; 6]. The available triple-frequency signals create new prospects for integrity monitoring [7], for facilitating cycle slip detection and repairing [8; 9], for fast ambiguity resolution (AR) [10] and for ionospheric analysis [11]. While triple-frequency signals have many benefits, new bias has also been introduced. [PERSON] et al. [12] first demonstrated the existence of a bias between L1/L2 and L1/L5 in ionosphere-free (IF) combination, including a periodic line bias between the satellite and the signal, based on geometry-free and ionosphere-free (GFIF) phase combination. The inconsistency of L1/L2/L5 was defined as inter-frequency clock bias (IFCB) [13]. Precise clock estimation(PCE) is obtained through dual-frequency IF combination. When using signals of different frequencies for the dual-frequency IF combination PCE, the satellite clock estimation will have a bias, which is the IFCB [14]. The bias consists of the receiver and satellite code hardware bias and phase hardware bias time-varying components, which generate the code-based IFCB (CIFCB) and PIFCB, respectively. The PIFCB consists of a satellite PIFCB and a receiver PIFCB. However, not all satellites of constellations have PIFCB, e.g., BDS-3, Galileo and QZSS constellations could ignore the satellite PIFCB [15; 16]. GPS Block IIF satellite PIFCB varies throughout the day, with peak-to-peak amplitudes of a few to several tens of centimeters [13]. [PERSON] et al. [14] proposed a satellite PIFCB estimation method for triple-frequency PPP for GPS and established an uncombined (UC) and IF123 PPP model with satellite PIFCB estimations followed by corrections. The experimental results showed that after GPS satellite PIFCB corrections, the corresponding positioning accuracy of the UC123 and IF123 PPP models could be improved to 5, 4, and 9 mm and 4, 3, and 10 mm in the E, N and U directions, respectively. For BDS-2, there was a small bias in B1I/B2I/B3I, and its satellite PIFCB varied throughout the day with a peak-to-peak amplitude of approximately 2 cm [17; 18]. [PERSON] et al. [19] proposed a GNSS IFCB estimation and correction generic model, which was implemented and validated using BDS-2 and BDS-3 data. It was experimentally demonstrated that the BDS-2 satellite PIFCB showed a periodic variation and with some of the satellites corrected by PIFCB, the mean root mean square error (RMS) value of the GFIF phase combination was 5 mm, which was an improvement of 50%. [PERSON] et al. [20] and [PERSON] et al. [21] systematically studied the long-term characteristics of the GNSS satellite PIFCB through GFIF phase combination. In addition to the study of the characteristics of the satellite PIFCB, a number of researchers have analyzed different triple-frequency PPP models based on the consideration of the satellite PIFCB [22; 3]. [PERSON] et al. [23] conducted a systematic study of the UC123, IF1213 and IF123 PPP models based on BDS-2 B1I/B2I/B3I triple-frequency observation data. [PERSON] et al. [24] systematically analyzed different triple-frequency PPP models based on GPS triple-frequency observation data. Different studies have shown that triple-frequency PPP positioning performance considering satellite PIFCB could reach the level of dual-frequency PPP. As GPS, GLONASS, BDS and Galileo continue to improve and modernize, an increasing number of researchers are focusing on multi-constellation combinations [25; 26; 27; 28], while multi-constellation multifrequency PPP is also a new trend. [PERSON] et al. [29] investigated the performance of BDS/Galileo for triple-frequency PPP AR, and [PERSON] et al.