1. Introduction
The concept and technology of Precise Point Positioning (PPP) were first proposed and implemented for the Global Positioning System (GPS) by the American Jet Propulsion Laboratory (JPL) in the late 1990s [
1]. PPP has attracted significant interest over the intervening years due to its high accuracy without needing a specific reference station, providing correctional information, simple operations, and cost effectiveness due to reductions in labor and equipment costs. Therefore, it has been extensively used in the areas, for instance, determines the precise orbit, surface ice flow speed, as well as positioning, navigation and timing (PNT) applications [
2,
3,
4]. The traditional PPP models are based on Ionosphere-Free (IF) combinations with dual-frequency raw phase and code observations for the removal of the first-order effect of ionospheric refraction [
5]. However, the second- and third-order ionospheric effects still exist and they may cause measurements errors of sub-centimeters in GPS [
6,
7]. Therefore, traditional PPP technology based on IF combinations cannot obtain ionospheric information, and the ionospheric error is not completely eliminated. In the last few years, PPP used raw observations has received an increasing amount of research attention because of its several advantages as compared to traditional IF PPP, particularly the development of new multi-frequency GNSS [
8].
The ionosphere refers to the atmospheric space starting at 60 km above the ground and extending to the magnetosphere. The Global Navigation Satellite System (GNSS) signal will generate ionospheric delay after passing through the ionosphere. A large number of models are currently being used to describe the delay that the ionosphere produces for electromagnetic signals propagating from satellites to receivers. If the delay is not corrected, it can have an important impact on the positioning accuracy of GNSS [
9], especially for single-frequency PPP (SF PPP). Shi et al. proposed an improved method in which the deterministic representation is further refined by a stochastic process for each satellite with an empirical model for its power density [
10]. The results of this method show that the single- and dual-frequency PPP exhibited enhanced convergence time, and the positioning accuracy of SF data is only improved by 25% [
11]. Abd Rabbou et al. [
12] developed an SF PPP model, and the improved model uses between-satellite-single-difference quasi-phase constrained GNSS observations. The GRoup and PHase Ionospheric Correction (GRAPHIC) method uses an SF code and carrier phase data to form an IF combined observation [
13]. The positioning accuracy can reach several centimeters based on the GRAPHIC method, but the method requires two hours of convergence time [
14].
The International GNSS Service (IGS) Ionosphere Working Group (IWG) was created in 1998. Several analysis centers (CODE (Center for Orbit Determination in Europe), ESA (European Space Agency), JPL (Jet Propulsion Laboratory), UPC (Universitat Politecnica de Catalunya), CAS (Chinese Academy of Sciences), and WHU (Wuhan University)) produce and release the post-processed Global Ionospheric Map (GIM) products, where the format is IONosphere EXchange (IONEX) [
15,
16]. The data portion of the GIM products utilized contain a total of 25 maps, with the latitude variation ranging from 87.5° to −87.5° with an interval −2.5° and the longitude ranging from −180° to 180° at 5° intervals. The accuracy of the GIM is improved based on the Spherical Harmonic (SH) expansion model [
17]. The VTEC of each ionospheric grid point is obtained while using different interpolation methods, such as bilinear interpolation [
18], inverse distance weighted function [
19], and Kriging interpolation [
20]. Lanyi and Roth [
21] proposed a polynomial model for single-station TEC derivation, and a single-station receiver bias can be estimated while using this model [
22]. This method was used by Lu et al. [
23] to study the effects of ionospheric shell height on GPS-based TEC derivation by a single station, while Kao et al. [
24] applied this method to multi-stations. Chen et al. [
25] analyzed the applicability of the sophisticated Klobuchar model for VTEC in China. Liu et al. [
26] had based on GPS to observe and analyze the fluctuation characteristics of TEC over China.
In GNSS data processing, the slant tropospheric delay on the signal propagation path between the satellite and the receiver is usually mapped to the zenith direction via a mapping function. The Zenith Tropospheric Delay (ZTD) consists of the Zenith Hydrostatic Delay (ZHD) and the Zenith Wet Delay (ZWD, i.e., the zenith non-hydrostatic delay). The models of tropospheric delay estimation usually adopt the Hopfield model and Saastamoinen model. There are roughly two types of mapping functions. The first type are empirical models, which merely require the epoch time and the approximate coordinates of the receiver station, such as the Niell Mapping Function (NMF) [
27] and the Global Mapping Function (GMF) [
28]. The second type of mapping function is based on a large number of weather model analyses at the epoch of the observations, such as Isobaric Mapping Functions (IMF) [
29] and Vienna Mapping Functions (VMF1) [
28]. Huang et al. [
30] proposes a new Asian single site tropospheric correction model, called the Single Site Improved European Geostationary Navigation Overlay Service model (SSIEGNOS).
SF PPP cannot be combined or differenced to be eliminated or attenuated by a part of the error, like dual-frequency PPP. If there are some information related to the parameters, and this information is accurate enough, the precision of traditional SF PPP can be largely improved and the convergence time can be shortened. Zhang et al. [
31] studied real-time GIM and its application in SF positioning, Aggrey and Bisnath [
32] studied the effect of atmospheric-constrained on the convergence time of dual- and triple-frequency PPP, and Gao et al. [
33] applied the Inertial Navigation System (INS) to the ionosphere-constrained PPP to overcome the drawbacks that accompany unexpected and unavoidable substandard observation environments.
This study uses the GIM products and the tropospheric zenith path delays from the IGS as the constrained information for SF PPP. The organization of this study is as follows: the next step details the methods for mathematical models of standard SF GPS PPP, troposphere-constrained SF GPS PPP, and ionosphere-constrained SF GPS PPP, and the details of the four interpolation methods of GIM products. Afterwards, in
Section 3 we compare and analysis the influence of interpolation methods and constraint methods on SF GPS PPP convergence time, and finally in
Section 4 we draw conclusions. Later, we will study ionospheric delay and tropospheric delay prediction models to provide virtual atmospheric delay observations for real-time PPP and also provide a priori information for the constraint processing.
4. Conclusions
In this study, we discussed the effects of four methods, including NENE, BILI, BICU, and JUNK, for the percentage of convergence time, the average convergence time and the calculation time of single station of SF TIC2 PPP. The numerical results show that: (1) the percentage of the convergence time of the estimated parameters through the BILI method is the shortest in every condition; and, (2) in the static mode, when the RMS is less than 5 cm, only the percentage of the BILI method calculation exceeds 40%. While in the kinematic mode, when the RMS is less than 15 cm, only the BILI method is more than 60% and 40% in the horizontal and vertical component, respectively. Under the other identical thresholds of the RMS, the percentage of the convergence time of the estimated parameters through BILI method is higher than the other three methods; (3) in the static mode, the average convergence time of the estimated parameters through BILI method is reduced by at least 2.2% when compared to the other three methods and the maximum is shortened by 23.6%; while, in the kinematic mode, they are 1.9% and 16.6%, respectively; and, (4) for the average computation time of a station, the BILI method has 0.121 s more than the shortest calculation time in the static mode and 0.038 s in the kinematic mode. Therefore, we chose the BILI method for calculating the TEC at the required time when using the TIC2 or TIC3 calculation based on the GIM grid products.
In order to verify the different constraint methods, a total of 18,720 tests were performed with 78 stations. The experimental results revealed the following findings: (1) the TIC1 convergence time percentage is slightly higher than the TIC0 in each convergence time threshold, while the TIC2 is larger and the TIC3 is the largest. That is to say, the convergence can be accelerated under the constraint conditions; (2) as compared with TIC0 method, the average convergence time of the TIC1 method is shortened by at least 4.0% and, at most, 28.9% in static mode. The TIC2 method has a larger percentage of shortening, at least up to 16.6%, the maximum up to 88.4%. However, in the vertical component, the TIC1 method is larger than the TIC2 method at the 68% level. Compared with TIC0, the TIC3 method is shortened by 46.5% at least and 88.8% at most. Additionally TIC3 method is better than TIC1 method and TIC2 method. In kinematic mode, the percentage of time that is shortened after the constraint is less than in static mode. The average convergence time of the TIC1 method is at least 2.1% shorter than the TIC0 method, and the maximum is shortened by 11.7%. The TIC2 method has a larger percentage of shortening compared to the TIC1 method, with a minimum of 2.5% and a maximum of 73.9%. The TIC3 method is superior to the TIC1 method and the TIC2 method, with a minimum of 5.4% reduction and a maximum reduction of 74.5% as compared to TIC0; and, (3) through comparative analysis, it is concluded that, the final positioning error of TIC0, TIC1, TIC2 and TIC3 methods are basically the same, no method is better. After the above analysis and conclusions, under the constraints of atmospheric error products, convergence can be accelerated, and there is no impact on the final accuracy improvement.