1. Introduction
Galileo has been under development by the European Union (EU) since 1999 with the objective of providing a free, open and independent high-precision positioning system for civil users. Two Galileo in-orbit validation element (GIOVE) satellites were launched in 2005 [
1] and 2008, respectively, followed by the launch of four In-Orbit Validation (IOV) satellites in 2011 and 2012 to validate the signals and system [
2]. By the beginning of 2018, three IOV satellites (PRNs E11, E12 and E19) were still operational, while the fourth IOV satellite (PRN E20) is declared “unavailable”. The launch of the Full Operational Capability (FOC) satellites started from 2014 [
3], and to this day, there are 20 Galileo FOC satellites providing positioning service to the global users and two FOC satellites suffered from orbit injection failure [
4,
5]. The full constellation of the Galileo system is expected to consist of 30 active in-orbit satellites by 2020, with the signals transmitted on three frequency bands, i.e., E1 (1575.42 MHz), E5 (1191.795 MHz) and E6 (1278.75 MHz), and the E5 band consists of E5a (1176.45 MHz) and E5b (1207.14 MHz), which signals could be used either separately or in a combined way. The E6 signal is encrypted for commercial use and cannot be tracked by the receivers in open data source [
6].
The Galileo system is a developing and valuable navigation system; the responding research is under development and is not as rich as GPS. Since its early development stage, the Galileo system has attracted great interest and been studied in various applications. Zaminpardaz et al. (2017) analyzed the Galileo signals for IOV and FOC satellites and conducted the E5 real-time kinematic (RTK); the result found that the E5 signal has a better performance than other signals [
7]. Steigenberger et al. (2017) comprehensively evaluated Galileo orbits, clocks and positioning status, and found that the Galileo-only single-point positioning (SPP) accuracy is at meter level and its precise-point positioning (PPP) solutions could have 1–2 cm precision [
8]. Due to the overlapping frequency bands in the Galileo and GPS systems, more studies were conducted on the combination of the two systems. Cai et al. (2014) and Gioia et al. (2015) evaluated Galileo IOV signals and positioning performance, and found that Galileo is able to achieve similar accuracy as GPS; the Galileo and GPS combined positioning can improve the vertical accuracy by 10% compared to the GPS-only solutions [
9,
10]. Odijk et al. (2014) conducted combined Galileo and GPS RTK with dual-frequency signals (E1 + E5a, L1 + L2) and Odolinski et al. (2015) combined four constellations to realize single-frequency RTK positioning; the solutions accelerated the ambiguity-fixed time and improved the accuracy significantly [
11,
12]. Tian et al. (2019) found that the RTK results of the GPS/Galileo/BeiDou Navigation Satellite System (BDS) are not significantly different; the double-differenced carrier phase and code residuals of E5 is the smallest [
13]. Luo et al. (2020) analyzed the benefits of Galileo to high-precision global navigation satellite system (GNSS) positioning under strong multipath conditions; Galileo still could provide more accurate solutions [
14].
NRTK mainly includes three steps: ambiguity resolution between reference stations, generation of error correction, and rover station positioning. Among them, the ambiguity resolution between reference stations is the premise and basis for NRTK to provide reliable navigation and position services. Currently, the ambiguity resolution between reference stations is generally obtained by the dual-frequency observations. Based on the dual-frequency model, the error processing method and the distance between reference stations are limited. The use of three-frequency observations can effectively improve the accuracy and reliability of ambiguity resolution, which could enlarge the distance between the reference stations and the effective service range of NRTK.
Aiming at taking full advantage of multifrequency GNSS signals, Forssell et al. (1997) proposed a three-carrier ambiguity resolution (TCAR) method for geometry-free integer bootstrapping in GNSS-2 (Galileo), but at that time, this method was lacking in the assessment with the real data [
15]. Zhang et al. (2003) and Julien et al. (2004) combined the GPS and Galileo system to propose their cascading TCAR methods; however, their methods have a large number of ambiguities to estimate and cannot be applied to the Galileo-only scenario very well [
16,
17]. Li et al. (2010, 2013) introduced the semi-simulation method to generate the GPS third-frequency signals to realize the ambiguity resolution (AR) of triple-frequency signals; the navigation precision and availability were both improved [
18,
19]. In this work, the ambiguities of medium-lane or narrow-lane observations could be fixed correctly after several minutes without a distance constraint. However, the results need to be further examined by the real triple-frequency data. With the broadcast of GNSS real triple-frequency signals, Li et al. (2015, 2017) conducted RTK with BDS triple-frequency signals and extra-wide-lane (EWL) RTK for all satellites of BDS, Galileo and for some GPS satellites, and centimetre-level positioning precision could be achieved in the baselines over 50 km [
20,
21]. However, the AR for the Galileo original signal were not conducted, and the TCAR methods for the Galileo-only system are yet to be developed.
For GNSS AR, LAMBDA introduced by Teunissen (1995) is the most reliable method for all available satellites as a whole [
22], but for reference stations with the known coordinates, the ambiguities of all satellites are not necessary to be fixed and the atmosphere biases could be eliminated by multifrequency combination. In addition, some methods for the reference station AR could be more efficient than LAMBDA. In this study, we will describe the geometry-free (GF) TCAR method which was firstly introduced by Feng and Li (2008) and Feng and Rizos (2009), and also the ionospheric-free (IF) TCAR method which was introduced by Tang et al. (2018) to resolve the ambiguity of the independent satellites [
23,
24,
25]. Both methods have been applied in GPS and BDS processing, respectively, and in this study they were applied in the Galileo-only processing while the IOV satellites are still in the verification stage and have poor satellite stability compared to FOC satellites. We will focus on the AR of the double-differenced (DD) Galileo measurements, which are collected from reference stations in the Hong Kong area, which is an ionosphere active area, and could benefit from the IF TCAR method. We will evaluate and compare the IF TCAR method with the GF TCAR method to illustrate the triple-frequency AR performance of the current early-stage Galileo service in the Hong Kong area. For a comparison, we will also show the GPS TCAR performance, and in the following, the IF TCAR method is modified to combine GPS and Galileo for an improved performance.
In this paper, the fundamental mathematical models and the procedure of triple-frequency signals processing are described first. Then, two TCAR methods for the Galileo processing are introduced. In the following, the IF TCAR method is modified to combine a multi-system in the original signal AR. Lastly, the real data experiments in Hong Kong are presented to evaluate the performance of these TCAR methods, and the conclusion is drawn in the final section.
4. Discussion
Since the Galileo system is a developing and valuable navigation system and the responding research is under development, we focused on analyzing the triple-frequency ambiguity resolution performance between reference stations in the Galileo system. Meanwhile, the work we analyzed and assessed here was the first step of the NRTK procedure, which was followed by the generation of atmosphere corrections as the second step and the solving of the positioning domain as the third step. The improvement was only on the float ambiguity between reference stations, improving both the precision and convergence speed. Further, the results of this paper are expected to provide useful references for future research of Galileo triple-frequency ambiguity resolution. Meanwhile, the effectiveness and reliability of ambiguity resolution between reference stations is the key for NRTK to provide reliable navigation and position services. For the ambiguity resolution of reference stations in NRTK, the coordinates of reference stations have been precisely known, and we do not need to resolve the ambiguity of all satellites. The TCAR method focuses on resolving the ambiguity based on a single double-differenced satellite pair and will be more efficient and appropriate. Therefore, the research and analysis of the TCAR method is of great significance to GAL NRTK. The number of satellites of the Galileo in the Hong Kong area is limited, and its three-frequency ambiguity has a large difference between the two TCAR algorithms. The ambiguity residuals solved by the GF TCAR method are difficult to converge to the true E1 or L1 value, while the IF TCAR method could still resolve its ambiguity resolution very reliably. The accuracy and the convergence time of the ambiguity resolution could be effectively improved by the GPS and Galileo combined IF TCAR method.
However, the test data are not large enough. So, further experiments need to continue with large data considering the state of the ionosphere, included the geographical latitude, season and time of day, as well as solar activity. With the opening of the global service of BDS3, in the future, we could study and analyze the TCAR method with BDS. The study Galileo and multi-system TCAR is expected to further improve the ambiguity resolution between reference stations, so as to provide GNSS users with more effective NRTK services.
5. Conclusions
In this paper, we discussed the Galileo triple-frequency optimal combinations for TCAR using E1, E5a, and E5b signals. Then, we introduced two TCAR methods for the Galileo system, i.e., the GF TCAR method and the IF TCAR method. The GF TCAR method is free from geometry-based errors, which include the tropospheric delay and the orbital bias, while the IF TCAR method could cancel most of the ionospheric delay. Then, the IF TCAR method is modified to resolve multi-system triple-frequency ambiguity. Finally, three experiments were conducted in the Hong Kong area; the first one was to evaluate GF TCAR and IF TCAR AR performance in the current stage Galileo service, the second test was to compare the Galileo with GPS in terms of TCAR performance, the last experiment was to further analyse the advantage of a combined system in the TCAR performance.
In the Galileo system, all methods choose a HMW combination with E5b and E5a to determine the EWL combination ambiguity correctly and reliably. For the WL AR, the GF and IF TCAR methods could also fix the ambiguity correctly after a smoothing processing, while the IF TCAR method could converge to the true ambiguity in a shorter time and also could be more reliable. Although both methods take different strategies to determine the WL ambiguities, they both could fix the WL ambiguity correctly. In terms of fixing the E1 ambiguity, these two methods have different performances. The GF TCAR method calculates the ionospheric delay and refines the result over 120 s, in which period the ionospheric delay is assumed to be stable. At the last step, the E1 ambiguity is resolved with the help of refined ionospheric delay in GF model after smoothing. The IF TCAR method takes KF to estimate the IF combined ambiguity with a higher precision, then determines the E1 ambiguity. The IF TCAR method improves the WL accuracy by 21.6% and the convergence time by 37.6%, respectively. Meanwhile, the E1 accuracy of the IF TCAR method is improved by 72.7% compared with the GF TCAR method. From the result of E1 signal in two TCAR methods, we can see IF TCAR outperforms GF TCAR. For comparison of GPS TCAR performance, Galileo satellites show similar performance, while the CIF TCAR method could speed up the convergence time of E1 ambiguity obviously with the help of GPS data. Compared to the single GPS or Galileo, the ambiguity accuracy and the convergence time of the combined GPS and Galileo are improved by about 25.7% and 47.1%, respectively.
Suffering from the current early-stage Galileo service in the Hong Kong area, where the number of visible satellites is limited, more advantage of Galileo triple-frequency signals could be reflected with more Galileo satellites. Further, the ambiguity resolution between reference stations will benefit from the combined Galileo, GPS and BDS.