1. Introduction
Baseline processing based on Global Navigation Satellite Systems (GNSS) differenced carrier phases observation is a standard high-accuracy post-positioning technique. However, various high accuracy positioning tasks need real-time operations. Traditional Real Time Kinematic (RTK) was developed in the mid-1990s, which provided cm-level accuracy positioning in real-time based on GNSS measurements [
1]. It involved one reference receiver that transmits raw measurements or observation corrections to a rover receiver via some sort of data communication link (e.g., VHF or UHF radio, and cellular telephone). To provide positioning services to larger region, multiple reference stations have to be set up and maintained, and the development of Network RTK (NRTK) resulted in tremendous reduction of the investment costs, which serves as a prerequisite for starting an RTK positioning service [
1]. Compared with traditional RTK, NRTK has many other advantages: capable of modeling GPS errors over the entire network area, increasing mobility and efficiency, reducing initialization times for rovers, extending surveying range, capability of supporting multiple users and applications, and continuous operation. Continuously Operating Reference Stations System (CORS) is the fundamental infrastructure for NRTK. In the United States, the National Geodetic Survey (NGS) manages a network of CORS of more than 2000 stations [
2]. The Canadian Active Control System (CACS) provides improved GPS positioning capability for the Canadian surveying and geophysical community as well as for other spatial referencing needs. By mid-2015, there were 102 sites in CACS [
3]. In Australia, GNSS data of approximately 100 stations from the Australian Regional GPS Network (ARGN), South Pacific Regional GNSS Network (SPRGN), and the AuScope Network are collected by Geoscience Australia [
4,
5]. A Satellite Positioning Service of the Satellite Positioning Service of the German National Survey (SAPOS) is operated in Germany. It consists of over 270 reference stations with mean station distances of 25–60 km [
6]. The Geospatial Information Authority of Japan (GSI) operates a nationwide GPS observation array, GEONET. Over 1200 GEONET stations are operated with real-time data transmission and a high sampling rate mode. The construction of CORS in China began at the end of 20th century and the development gradually expanded from a city-based system to a nationwide system. In China, different administrations and provinces have built over 4000 reference stations. One of the major national networks in China is the Crustal Movement Observation Network of China, which includes 260 reference stations [
7]. It is a network project for comprehensive observations of Earth Sciences, which employs the GNSS system and other complementary space and precise observation technologies to monitor the real-time dynamic changes of the continental tectonic environment and explore influences of these changes on resources, environment, and disasters.
Besides CORS, the NRTK software/platforms also reflect the development of the NRTK. The most commonly used commercial NRTK software/platforms are Trimble Pivot Platform of Trimble, Spider of Leica, and GNNET-RTK of GEO++. Trimble pivot is the fourth generation of CORS system software from Trimble, which is designed to connect a large number of reference stations to become a network correction number, ideal for high-precision positioning. GNSS Spider is a highly integrated software suite for the central control and operation of a single reference station or reference station network. The Spider v5.2 series software fully supports China’s independent research and development of the Beidou satellite navigation system (BDS). GNNET-RTK of GEO++ is able to achieve accuracies of a few millimeters by using antenna and multipath calibrations. However, GNNET-RTK is only currently suitable for GPS + GLONASS. In addition to the above-mentioned commercial NRTK software/platforms, universities and research institutions have also developed their own NRTK software such as the Multi-Purpose GPS Processing Software (MPGPSTM) of The Ohio State University, the software for Multiple Reference stations real-time kinematic GPS application (MultiRef™) by the University of Calgary, and the PowerNetwork of Wuhan University [
8]. The network correction modes that these software packages support are different from each other. Based on SSR (State Space Representation) technology, developed and promoted by Geo++, all of today’s GNSS network correction modes (FKP + VRS/PRS + MAC + SSR) are effectively supported by Geo++. The NRTK software of Trimble mainly supports VRS (Virtual Reference Station), and Leica uses the MAC (Master Auxiliary Corrections) technology. The PowerNetwork of Wuhan University applies the modified combined bias interpolation method.
Although NRTK has been successfully applied in civilian and military fields, the performance of NRTK in urban areas is affected by many factors. In Hong Kong, for instance, there are too many tall buildings and narrow streets, which will block satellite signals and, therefore, the availability of GPS positioning in urban Hong Kong is extremely low. For example, in the streets of Wanchai, the availability of GPS positioning is only 7% and the extremely large multipath of GPS signals greatly reduces the accuracy of GPS positioning in Hong Kong [
9,
10]. Hong Kong Satellite Positioning Reference Station Network (SatRef) is a local satellite positioning system established by the Survey and Mapping Office of Lands Department of Hong Kong. The network consists of 18 continuously operating reference stations evenly distributed in Hong Kong (
Figure 1). Although SatRef has been successfully applied in surveying applications, there are still some problems that need to be solved. For example, the initialization speed, ambiguity successfully fixed rate, and accuracy of NRTK may sometimes be affected by blocked satellite signals and high ionospheric effects.
Fortunately, GNSS are evolving to a new era. The US GPS system is currently improving through the GPS modernization program. BDS with a global coverage will be completed by 2020 [
11]. Among the new GNSS, GPS introduces the L5 signal at 1176.45 MHz in addition to the current L1 at 1575.42 MHz and L2 at 1227.6 MHz, while the Chinese BDS navigation satellite system operates in three frequency bands: 1561.098 MHz; 1207.14 MHz; and 1268.52 MHz. The multiple-constellation and multiple-frequency GNSS data will bring great benefit to the NRTK in urban areas. Most research [
12,
13,
14,
15,
16,
17] has shown that the centimeter level of positioning accuracy can be achieved in a very short initialization time using triple-frequency observations. The first BDS results outside the Chinese mainland were reported by Montenbruck et al., Steigenberger et al., and Nadarajah et al. [
18,
19,
20,
21]. Teunissen et al. analyzed the instantaneous RTK positioning capabilities of a combined GPS + BDS system for cut-off elevation angles ranging between 10° and 40° in terms of the ADOP, the bootstrapped success rate, and the positioning precision. Test results showed that the GPS/BDS combination had good high-precision positioning performance for up to a 40° cut-off elevation [
20]. The result is important as such a measurement set-up will significantly increase the GNSS applicability in constrained environments such as in urban canyons or when a low-elevation multipath is present. However, only a few RTK tests have been conducted in the urban environment. The GPS/BDS dual/triple frequency NRTK performance assessment in urban areas especially in Hong Kong is limited.
In this study, we developed a new NRTK server platform. This platform integrates multiple-constellation and multiple-frequency GNSS data to support reliable NRTK operation in Hong Kong. It is worth mentioning that the BDS observation collected in our study was from the BDS-2 system [
19]. Based on this platform, the performance of NRTK in urban areas was examined using a series of experiments. In this paper, we first introduce the development of CORS, NRTK software, and the multiple-constellation and multiple-frequency GNSS, and discuss the issues that affect the performance of NRTK in urban areas. Second, the functions and key technology of the Hong Kong GNSS Network RTK Service Platform are discussed. Third, the ambiguity resolution (AR) technique for triple-frequency signals used in this study is presented. Initialization time test, positioning accuracy test, ambiguity successfully fixed rate test and the triple-frequency NRTK test were carried out to assess the NRTK performance in Hong Kong. Finally, we present our conclusions.
3. Ambiguity Resolution for Triple-Frequency Signals
For the AR of the triple-frequency signals, the definitions of the virtual signals and related parameters should first be given. The general form of the GNSS linear carrier phase and pseudo-range observation combination equations of three fundamental signals can be generally formulated as [
16]
where
and
are the pseudo-range and carrier phase combination in meters, respectively, and
is the phase combination in cycles.
i, j, and
k are the combination coefficients, which are integers.
and
are the pseudo-range and phase measurements in meters, and
is the frequency of the carrier phase.
is the phase measurement in cycles. The corresponding virtual frequency, wavelength and the ambiguity of the observation combination are
where
denotes the light speed.
is the ambiguity of the triple-frequency carrier phase measurement.
The virtual double difference pseudo-range and phase signals can be described as
where
is the double difference geometric distance between the satellite and receiver,
is the double difference tropospheric delay,
is the ionospheric scale factor defined with respect to the first-order ionopheric delay on the L1 carrier (
), and
and
are the pseudo-range and phase observation noise of the triple-frequency combination, respectively.
Theoretically, there are infinite choices of linear integer combinations, and three of them are linearly independent. For ambiguity resolution purposes, two optimal combinations of extra-wide lane (λ ≥ 2.93 m, EWL) and wide lane (0.75 m ≤ λ < 2.93 m, WL) should first be selected. Generally, the optimal combinations proposed by different researchers are often based on the smallest ionospheric scale factor or largest wavelength-to-noise ratio. In this study, the
, which has been the straightforward choice, was selected as the first EWL signal [
16]. The observation combination with the minimal or near minimal first-order ionospheric scale factor (
) was chosen as the second best EWL/WL observation in this study, which was
for GPS and
for BDS [
16].
Three/Multiple Carrier Ambiguity Resolution (TCAR/MCAR) and Cascading Integer Resolution (CIR) are typical three/multiple-carrier ambiguity resolution method proposed by Forssell et al. [
26], Vollath et al. [
27], De Jonge et al. [
28] and Hatch et al. [
29]. The integer estimation principles of TCAR and CIR are both examples of integer bootstrapping. The integer ambiguity of the observation can generally be determined by rounding the float ones to its nearest integer values. However, these methods are biased by the residual ionospheric delay [
30,
31]. Following these studies, a large amount of work has been carried out on the TCAR/CIR or modified TCAR/CIR methods. Feng et al. [
16] resolved the ambiguities of the optimized virtual signals in a three-step procedure. Feng and Li [
32] used both geometry-based and geometry-free TCAR model to process the ambiguity resolution. A geometry-free and ionosphere-free for distance-independent reliable TCAR method was imposed in 2010 by Li et al. [
33], which was free from both ionospheric effects and geometric terms. Ji et al. [
14] presented an improved CAR method which included the advantages of both integer least-squares (ILS) and CAR. Tang et al. [
34] proposed a modified stepwise AR method based on the TCAR. Teunissen et al. [
30,
31]compared the performance of LAMBDA and TCAR/CIR when they were applied to the multiple-frequency geometry-free case and the multiple-frequency geometry-based cases. For the Geometry-free case, TCAR and CIR ambiguity resolution perform as well as The LAMBDA method. For Geometry-based case, although the reliability of LAMBDA is generally better than that of TCAR and CID, the LAMBDA is computationally more intensive. As a result, we combined the TCAR and LAMBDA method to fix the ambiguity for triple-frequency signals. Research has shown that the ionospheric variability in low-latitude regions is much greater than that in mid-latitude areas [
35,
36,
37]. During ionosphere storms, the ionospheric delay for a 10 km baseline can reach tens of meters. As a result, the double difference ionospheric delay residual cannot be ignored or well modeled even for the baseline of 15–30 km in Hong Kong. In this study, we tried to fix the ambiguity of the triple-frequency signals for baselines between reference stations in three steps without the effect of the ionospheric delay. The whole process was:
Step 1. We fixed the ambiguity of
using the geometry-free method as suggested by most authors (Equation (10)), which is free of ionospheric effects and nearly minimally affected by code noise [
16].
Step 2. We formed the double difference ionosphere-free (IF) code observation (
) first, and then computed the
using the code observation
() with Equation (11). Since the first-order ionospheric scale factor of the second best EWL/WL was especially small (
, for instance), the estimation error due to the code observation in Equation (11) only showed limited impact on the ambiguity resolution [
33]. Equation (12) was applied to compute the float ambiguity of the second best EWL/WL (
, for instance).
where
is the vector of unknown parameters of the baseline components and
is the baseline design matrix.
Step 3. The unknown parameter of ionospheric delay was involved. The float ambiguity of was computed using Equation (13). The LAMBDA method was used to fix the ambiguity in Steps 2 and 3.
The R-ratio test has been developed for decades with several improved versions [
38,
39]. In this research, we used the R-ratio test with the definition shown in Equation (14) for ambiguity validation. Let the float ambiguity vector and its variance matrix be given as
and
, respectively. Furthermore, let
be the integer least square solution, i.e., the integer minimizer of
, and let
be the integer vector that returns the second smallest value of the quadratic form
, and the equation of the ratio-test can be written as:
In this research, the threshold value was set to 2.5 [
40].