# Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations
## 1 Introduction  
Currently, Global Navigation Satellite Systems (GNSSs) are developing towards multi-systems and multi-frequencies. BeiDou Navigation Satellite System (BDS) completed its global satellite network on 31 July 2020. Among them, 30 BDS-3 satellites provide four public service signals--B1I, B1C, B2a, and B3I--in three frequency bands--B1, B2, and B3. The European system Galileo Satellite Navigation System is a multi-frequency global satellite navigation system that was independently developed by Europe and not controlled by the military. As of June 2020, Galileo has launched 26 satellites, of which 24 are in orbit, 2 in orbit verification (IOV) satellites and 22 full operational capability (FOC) satellites, and is close to achieving global deployment [1, 2]. With the combination of BDS-3 and Galileo, the number of visible satellites increases. Therefore, some scholars have also studied the positioning performance of GPS, BDS-2, and BDS-3 or Galileo data fusion. [PERSON] et al. combined BDS-2 and BDS-3 and evaluated their positioning performance [3]. [PERSON] et al. also analyzed the positioning performance of the combination of Galileo and BDS-2 [4]. The combined use of these satellite constellations can improve the reliability, accuracy, and stability of GNSS positioning performance [5]. However, previous studies only used dual-frequency or triple-frequency data. [PERSON] et al. analyzed the triple-frequency, quad-frequency, and penta-frequency linear combinations provided by BDS-3. The number of high-quality signals of quad-frequency is about ten times that of triple-frequency [6]. [PERSON] et al. used precise ephemeris and Galileo geometry-based data after multipath correction to analyze the ambiguity resolution (AR) performance of known baselines. The empirical success rate of combinations after multipath correction increases by 4-5 percentage points in the case of five frequencies, but it needs a multipath calculation at least one week in advance and cannot be applied to real-time calculations [7]. [PERSON] et al. analyzed the short-baseline positioning experiment of the multi-frequency observation data from the Galileo system. Among them, the quad-frequency positioning performance is poor due to the bad observation quality of the E1 frequency, and the positioning precision of quad-frequency is not as good as that of triple-frequency. In addition, the fourth frequency used in the experiment is E5 instead of E6 [1]. The traditional short-baseline RTK can also resolve ambiguities in a single epoch and can obtain centimeter-level positioning accuracy [8]. However, when the operating range increases, the correlation of the spatial error gradually weakens and the positioning performance deteriorates. Therefore, in order to ensure a wide range of high-precision and rapid positioning, a dense reference station network is required. On the contrary, precise point positioning (PPP) can also provide centimeter-level positioning performance and does not require reference stations, but its major limitations are a slow convergence speed and a long initialization time [9; 10; 11]. Even with multiple GNSSs, PPP still takes about 10 min to obtain reliable integer ambiguity resolutions. In addition, if an open sky-view is not constantly available, frequent re-convergences can destroy the practicability of real-time PPP completely [12; 13]. Multi-frequency signals provide opportunities for high-precision and rapid positioning. Generally, multi-frequency signals can increase the number of combinations [14] and provide additional long-wavelength, small ionospheric scalar factor, and easy-to-fix carrier phase combinations. Moreover, it will reduce location-blind areas (such as special environments) [15] and weaken the influence of various error sources on the combined ambiguity [16; 17] to apply the long-baseline RTK [18] and large-scale network RTK [19].