Single-Epoch Decimeter-Level Precise Point Positioning with a Galileo Five-Frequency Ionosphere-Reduced Combination

Abstract

Currently, there are two main methods for single-epoch decimeter-level precise point positioning (PPP); one is a model based on ambiguity-fixed ionosphere-free (AFIF) observations, and the other is based on uncombined (UC) PPP. The implementation of these two models requires both extra-wide-lane (EWL) and wide-lane (WL) ambiguity fixing. Different from the existing methods, this paper proposes a multi-frequency ionosphere-reduced (IR) PPP model suitable for single-epoch decimeter-level positioning. Based on Galileo five-frequency data, the optimal selection strategy of IR combinations is first studied with ionosphere, noise level and wavelength factors considered. Then, based on the selected IR combination, two IR PPP models, namely IR(EST) and IR(IGN), are established according to whether ionosphere parameters are estimated or ignored. Finally, the proposed models are verified with real tracked data from globally distributed stations, and further compared with the existing AFIF/UC models in terms of positioning performance and time consumption. The relationship between the ionosphere equivalent ranging error and satellite elevation in the IR models is analyzed. The lower the elevation is, the larger the residual ranging error is, and its impact on positioning is weakened by downweighting its observations and adjusting the cut-off elevation during the partial ambiguity fixing (PAF) process. The results show that the performance of the two IR models is basically the same, and both can achieve horizontal and vertical accuracies better than 20 cm and 40 cm, respectively. Compared with the existing AFIF/UC models, the proposed IR models can achieve similar decimeter-level accuracy with only one step of EWL ambiguity fixing, and at the same time, the IR models have varying degrees of improvement in time consumption: 38% shorter than the AFIF model and 97% shorter than the UC model.

1. Introduction

The global navigation satellite system (GNSS) precise point positioning (PPP) technology, known for its ability to provide positioning, navigation and timing (PNT) services with high accuracy on a global scale, has been widely used in both military and civilian fields [1,2]. However, traditional float PPP without ambiguity fixing often requires a rather long convergence time to achieve centimeter-level accuracy [3,4], and this also limits the development of PPP technology, which cannot be widely used like real-time kinematic (RTK). By performing ambiguity resolution, the convergence time can be slightly shortened, and the positioning accuracy can also be improved to a certain extent [5,6]. Especially with the development of multi-GNSS, the initialization time is significantly shortened, but even so, the time to first fix (TTFF) is still about 10 min. Improving its real-time performance has always been the pursuit of scholars in the field. Currently, there are two commonly used strategies to improve PPP real-time performance. One is to adopt the similar idea of network real-time kinematic (NRTK) to extract and model the regional undifferenced atmosphere and then achieve fast (or even single-epoch) centimeter-level positioning by means of atmosphere enhancement. This technology is also called PPP-RTK by some scholars [7,8]. Another strategy is to achieve high real-time performance at the cost of degrading positioning accuracy to a certain extent (e.g., from centimeter level to decimeter level) without external atmosphere constraints, and the typical representative is the single-epoch wide-lane ambiguity resolution (WAR) positioning with decimeter-level accuracy [9,10], which takes advantage of the fact that the extra-wide-lane (EWL) and wide-lane (WL) ambiguity could be fixed reliably in single-epoch mode. In general, these two modes have their own advantages and applicable scenarios, and the content of this paper mainly focuses on the latter.

The modernization of GNSS can greatly improve the efficiency and reliability of ambiguity fixing, and especially in the case of good satellite geometry, an instantaneous accuracy of about 20 cm can be achieved [11]. With the fixed EWL and WL ambiguity, the so-called ambiguity-fixed ionosphere-free (AFIF) observation is reconstructed, since its noise and multipath effect outperform the raw pseudorange; this observation can be used to perform instantaneously decimeter-level positioning or to assist narrow-lane (NL) ambiguity fixing [12]. Based on Galileo/BDS multi-frequency data, Li et al. [13,14] studied the performance of various AFIF combinations composed of triple-/quad-/five-frequency, and the results show that decimeter-level accuracy can be obtained within 0.5 min with the instantaneous fixing of EWL/WL ambiguity, and the positioning accuracy of the five-frequency combination is better than that of the triple-/quad-frequency combinations. With Galileo multi-frequency data, Guo and Xin [15] found that compared with other combinations, the E1/E5a/E6 combination has a lower noise level, and the resulting AFIF observation is theoretically expected to improve the single-epoch accuracy to about 10 cm. In addition to the above-mentioned AFIF-based model, some scholars have also achieved similar positioning accuracy based on uncombined (UC) PPP (UCPPP) with EWL/WL/NL ambiguities sequentially fixed [16]. The UCPPP model can make full use of pseudoranges and carrier observations on all frequencies and is an ideal model for processing multi-frequency data. A major advantage of using the UCPPP model is that EWL/WL ambiguities can be estimated by a geometry-based (GB) model, so the ambiguity fixing is theoretically more reliable in single-epoch mode compared with a conventional geometry-free (GF) model, such as the Melbourne–Wübbena (MW) combination [17,18]. The results of Gu et al.’s research [19] based on BDS-2 triple-frequency data show that the EWL and WL ambiguity can be fixed reliably within 2 min, and a horizontal and vertical accuracy of about 0.5 m can be obtained. Gao et al. [20] also analyzed the performance of single-epoch ambiguity-fixed PPP with regional atmosphere enhancement, and the WL ambiguity-fixed solution can achieve an accuracy of several centimeters. Qu et al. [21] also obtained similar positioning results and found that compared with triple-frequency WAR, the performance improvement of quad-/five-frequency cases is not obvious. Generally speaking, UCPPP-based single-epoch positioning can achieve an accuracy of about 20 cm in the horizontal direction on a global scale [9,10].

Compared with a dual-frequency case, multi-frequency can provide more combined observations with excellent characteristics, such as those demonstrated by a ionosphere-reduced (IR) combination [22]. Usually in the ionosphere-free (IF) combination case, in order to eliminate the influence of ionosphere, the combined ambiguity lose its integer nature; in contrast, the IR combination based on integer coefficients can still guarantee the integer solvability of ambiguity under the condition of almost eliminating ionosphere, which provides new opportunities for GNSS data processing. Both Feng [22] and Guo et al. [23] systematically studied the selection criteria of IR combinations. Based on BDS-3 multi-frequency signals, Li et al. [24,25] studied an IR combination suitable for medium-/long-baseline RTK ambiguity resolution, whose ambiguity maintains integer characteristics, and the noise level is comparable to the NL combination. At present, IR-based research mainly focus on differential positioning, such as RTK, and there is little targeted research on PPP.

Based on the above background, this paper mainly studies the theory of a multi-frequency IR combination in single-epoch PPP positioning and further compares it with the existing AFIF and UC models. The article is structured as follows. Section 2 mainly introduces and compares different positioning models, namely the AFIF model, UC model and IR model. Section 3 presents the experiment’s results and analysis. A summary of our results is given in Section 4 and Section 5.

2. Materials and Methods

The zero-differenced observations of pseudorange $P_{r,j}^s$ and carrier phase $L_{r,j}^s$ between the receiver $r$ and satellite $s$ on frequency $j$ can be expressed as:

$$P_{r,j}^s={\rho }_r^s+c·t_r-c·t^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\gamma }_j·I_{r,1}^s+d_{r,j}-d_j^s+e_{r,j}^s$$

(1)

$$L_{r,j}^s={\rho }_r^s+c·t_r-c·t^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}-{\gamma }_j·I_{r,1}^s+{\lambda }_j·\left(N_{r,j}^s+b_{r,j}-b_j^s\right)+{\epsilon }_{r,j}^s$$

(2)

where ${\rho }_r^s$ denotes the geometric distance between the satellite and receiver; $c$ is the speed of light; $t_r$ and $t^s$ denote the receiver and satellite clock offsets; $T_{r,\mathrm{w}}$ is the residual zenith wet-tropospheric delay (ZWD) with mapping function $M_{r,\mathrm{w}}^s$; $I_{r,1}^s$ is the slant ionosphere delay at the first frequency $f_1$, with an amplification factor ${\gamma }_j=f_1^2/f_j^2$ depending on the frequency $f_j$; $d_{r,j}$ and $d_j^s$ denote the receiver and satellite code biases; $b_{r,j}$ and $b_j^s$ denote the receiver and satellite phase biases; ${\lambda }_j$ and $N_{r,j}^s$ are the wavelength and integer ambiguity of frequency $f_j$, respectively; and $e_{r,j}^s$ and ${\epsilon }_{r,j}^s$ denote observation noises. It should be noted that some error items that could be accurately corrected by the existing models are not listed, such as the phase center offset/variation (PCO/PCV), earth tides loading, phase wind-up, etc. [26,27].

Since the zero-differenced PPP data processing cannot eliminate the receiver and satellite hardware bias through differential means used in relative positioning, it is not possible to estimate each parameter accurately, but only the combination of multiple parameters, i.e., re-parameterization. This strategy is widely adopted for each analysis center or client end. In order to simplify the description of subsequent sections, some re-parameterization strategies commonly used in current PPP data processing are explained first.

At present, dual-frequency IF observations are usually adopted for satellite precise orbit determination (POD) and precise clock estimation (PCE); some frequency-related hardware biases are correspondingly absorbed by satellite clock offsets.

$$c·{\tilde{t}}^s=c·t^s+\alpha ·d_1^s+\beta ·d_2^s$$

(3)

where ${\tilde{t}}^s$ denotes the satellite clock offset provided by each analysis center and $\alpha$ and $\beta$ denote the coefficients of a dual-frequency IF combination, with specific forms as:

$$\{\begin{array}{l}\alpha =f_1^2/\left(f_1^2-f_2^2\right)\\ \beta =-f_2^2/\left(f_1^2-f_2^2\right)\end{array}$$

(4)

Similarly, the receiver code bias in the form of the IF combination is also lumped with the re-parameterized receiver clock offset ${\tilde{t}}_r$.

$$c·{\tilde{t}}_r=c·t_r+\alpha ·d_{r,1}+\beta ·d_{r,2}$$

(5)

For an uncombined PPP model with estimated ionosphere parameters, the frequency dependent code biases are merged with ionosphere parameters.

$${\tilde{I}}_{r,1}^s=I_{r,1}^s-\beta ·\left(d_{r,2}-d_{r,1}-d_2^s+d_1^s\right)$$

(6)

The re-parameterization process described in the above Equations (3), (5) and (6) has been widely used in current dual-/multi-frequency PPP data processing, and the subsequent function models are also based on these re-parameterizations. Based on Galileo five-frequency data, as shown in

Table 1. Specific frequency of Galileo E1/E5a/E6/E5b/E5.

Frequency Notation	Frequency ID	Frequency/MHz
$f_1$	E1	1575.42
$f_2$	E5a	1176.45
$f_3$	E6	1278.75
$f_4$	E5b	1207.14
$f_5$	E5	1191.795

, the three positioning models will be described below, namely the AFIF model, UC model and IR model. For the convenience of description, the following sections use $f_i(i=1,2,3,4,5)$ to represent the E1/E5a/E6/E5b/E5 frequencies of Galileo, respectively.

2.1. AFIF Model

The AFIF observation is composed of an IF combination of specific EWL and WL observations. Taking E6/E5a and E1/E5a as the selected EWL and WL combinations, the corresponding observation equations are,

$$\begin{array}{ll}{L}_{r,\mathrm{E}23}^s& =\frac{f_3}{f_3-f_2}·L_{r,3}^s-\frac{f_2}{f_3-f_2}·L_{r,2}^s\\ & ={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}-\frac{f_1^2}{f_2·f_3}·{\tilde{I}}_{r,1}^s+{\lambda }_{\mathrm{E}23}·\left(N_{r,\mathrm{E}23}^s+B_{r,\mathrm{E}23}-B_{\mathrm{E}23}^s\right)+{\epsilon }_{r,\mathrm{E}23}^s\end{array}$$

(7)

$$\begin{array}{ll}{L}_{r,\mathrm{W}12}^s& =\frac{f_1}{f_1-f_2}·L_{r,1}^s-\frac{f_2}{f_1-f_2}·L_{r,2}^s\\ & ={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}-\frac{f_1}{f_2}·{\tilde{I}}_{r,1}^s+{\lambda }_{\mathrm{W}12}·\left(N_{r,\mathrm{W}12}^s+B_{r,\mathrm{W}12}-B_{\mathrm{W}12}^s\right)+{\epsilon }_{r,\mathrm{W}12}^s\end{array}$$

(8)

where $L_{r,\mathrm{E}23}^s$ denotes the EWL observation composed of frequency $f_2$ and $f_3$; $L_{r,\mathrm{W}12}^s$ denotes the WL observation composed of frequency $f_1$ and $f_2$; ${\lambda }_{\mathrm{E}23}$ and ${\lambda }_{\mathrm{W}12}$ are EWL and WL wavelengths, with ${\lambda }_{\mathrm{E}23}=2.93 \mathrm{m}$ and ${\lambda }_{\mathrm{W}12}=0.75 \mathrm{m}$; $N_{r,\mathrm{E}23}^s$ and $N_{r,\mathrm{W}12}^s$ are EWL and WL integer ambiguities; $B_{r,\mathrm{E}23}$ and $B_{\mathrm{E}23}^s$ are receiver and satellite EWL fractional cycle biases (FCBs); $B_{r,\mathrm{W}12}$ and $B_{\mathrm{W}12}^s$ are receiver and satellite WL FCBs; and ${\epsilon }_{r,\mathrm{E}23}^s$ and ${\epsilon }_{r,\mathrm{W}12}^s$ are observation noises of EWL/WL observations.

Due to the long wavelengths of EWL combinations, typically several meters, the EWL ambiguity can be reliably fixed in single-epoch mode. Specifically, the float EWL ambiguity ${\overline{N}}_{r,\mathrm{E}23}^s$ is usually generated by using the well-known MW combination with both geometry and ionosphere terms eliminated. After receiver and satellite FCBs corrections are applied, it can be fixed by integer rounding with a typical threshold of 0.2~0.3 cycles. It should be noted that in order to be consistent with the PCE estimation strategy and avoid the introduction of additional code bias by multi-frequency pseudoranges, when constructing MW combinations, only E1/E5a dual-frequency pseudoranges consistent with the PCE reference are used for various EWL combinations.

$${\overline{N}}_{r,\mathrm{E}23}^s=\frac{L_{r,\mathrm{E}23}^s-\left[\eta ·P_{r,1}^s+\left(1-\eta \right)·P_{r,2}^s\right]}{{\lambda }_{\mathrm{E}23}}=N_{r,\mathrm{E}23}^s+B_{r,\mathrm{E}23}-B_{\mathrm{E}23}^s$$

(9)

with $\eta =\frac{f_1^2·\left(f_3-f_2\right)}{f_3·\left(f_1^2-f_2^2\right)}$.

Once the EWL ambiguity is fixed, the unambiguous EWL observation ${\overline{L}}_{r,\mathrm{E}23}^s$ can be obtained.

$${\overline{L}}_{r,\mathrm{E}23}^s=L_{r,\mathrm{E}23}^s-\left({\stackrel{⌣}{N}}_{r,\mathrm{E}23}^s+B_{r,\mathrm{E}23}-B_{\mathrm{E}23}^s\right)$$

(10)

where ${\stackrel{⌣}{N}}_{r,\mathrm{E}23}^s$ denotes fixed integer EWL ambiguity.

Then, this unambiguous EWL observation is combined with the WL observation in Equation (8) to form the AFIF observation.

$$\begin{array}{ll}{L}_{r,\mathrm{AFIF}123}^s& =\frac{f_1}{f_1-f_3}·L_{r,\mathrm{W}12}^s-\frac{f_3}{f_1-f_3}·{\overline{L}}_{r,\mathrm{E}23}^s\\ & ={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+\frac{f_1}{f_1-f_3}·{\lambda }_{\mathrm{W}12}·\left(N_{r,\mathrm{W}12}^s+B_{r,\mathrm{W}12}-B_{\mathrm{W}12}^s\right)+{\epsilon }_{r,\mathrm{AFIF}123}^s\end{array}$$

(11)

where $L_{r,\mathrm{AFIF}123}^s$ denotes the AFIF observation and ${\epsilon }_{r,\mathrm{AFIF}123}^s$ denotes corresponding observation noise.

Assume that the accuracy of carrier observations at different frequencies is equal, according to the law of error propagation, the prior accuracy of AFIF observations $L_{r,\mathrm{AFIF}123}^s$ can be obtained as:

$${\sigma }_{\mathrm{AFIF}123}^2={\kappa }_{\mathrm{AFIF}123}^2·{\sigma }_L^2=\left({\left(\frac{f_1}{f_1-f_3}\right)}^2·\frac{f_1^2+f_2^2}{{\left(f_1-f_2\right)}^2}+{\left(\frac{f_3}{f_1-f_3}\right)}^2·\frac{f_3^2+f_2^2}{{\left(f_3-f_2\right)}^2}\right)·{\sigma }_L^2$$

(12)

where ${\kappa }_{\mathrm{AFIF}123}$ denotes the noise amplification factor and ${\sigma }_L$ and ${\sigma }_{\mathrm{AFIF}123}$ denote the accuracy of raw carrier phase and AFIF observations, respectively. The ${\sigma }_L$ is usually set to 3 mm.

Based on Equation (11), if the WL ambiguity is successfully fixed, the coordinate parameters can be constrained to about decimeter-level accuracy [12,14]. For WL ambiguity resolution, there are generally two approaches. One is to use the MW combination similarly to EWL ambiguity fixing, that is, the GF method; the other one is to calculate the float WL ambiguity and its variance–covariance based on the GB model and then use the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) algorithm to fix it [28].

If the GF method is adopted for WL ambiguity resolution, the AFIF observation becomes unambiguous, which means that it can be treated as a virtual pseudorange.

$${\overline{L}}_{r,\mathrm{AFIF}123}^s=L_{r,\mathrm{AFIF}123}^s-\frac{f_1}{f_1-f_3}·{\lambda }_{\mathrm{W}12}·\left({\stackrel{⌣}{N}}_{r,\mathrm{W}12}^s+B_{r,\mathrm{W}12}-B_{\mathrm{W}12}^s\right)$$

(13)

where ${\overline{L}}_{r,\mathrm{AFIF}123}^s$ denotes the unambiguous AFIF observation, and ${\stackrel{⌣}{N}}_{r,\mathrm{W}12}^s$ denotes fixed integer WL ambiguity.

In case of sufficient visible satellites, theoretically, absolute positioning can be achieved by only using these unambiguous AFIF observations without the assistance of pseudoranges. However, in order to ensure the stability of the observation model, it can still be combined with pseudorange observations for parameter estimation. On this basis, further combining various AFIF observations composed of other frequencies, such as E1/E5a/E5b and E1/E5a/E5, the following observation model can be obtained:

$$\{\begin{array}{l}{P}_{r,\mathrm{IF}}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+e_{r,\mathrm{IF}}^s\\ {\overline{L}}_{r,\mathrm{AFIF}123}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\epsilon }_{r,\mathrm{AFIF}123}^s\\ {\overline{L}}_{r,\mathrm{AFIF}124}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\epsilon }_{r,\mathrm{AFIF}124}^s\\ {\overline{L}}_{r,\mathrm{AFIF}125}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\epsilon }_{r,\mathrm{AFIF}125}^s\end{array}$$

(14)

where $P_{r,\mathrm{IF}}^s=\alpha ·P_{r,1}^s+\beta ·P_{r,2}^s$ denotes dual-frequency IF pseudoranges. The estimated parameter vector in Equation (14) includes:

$$\widehat{X}=\left[\begin{array}{ccc}x,y,z& {\tilde{t}}_r& T_{r,\mathrm{w}}\end{array}\right]$$

(15)

where $(x,y,z)$ denotes receiver coordinates.

If the GB model is adopted, due to the existence of ambiguity parameters, pseudoranges must be introduced to ensure that the model is full rank:

$$\{\begin{array}{l}{P}_{r,\mathrm{IF}}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+e_{r,\mathrm{IF}}^s\\ L_{r,\mathrm{AFIF}123}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+\frac{f_1}{f_1-f_3}·{\lambda }_{\mathrm{W}12}·{\overline{N}}_{r,\mathrm{W}12}^s+{\epsilon }_{r,\mathrm{AFIF}123}^s\\ L_{r,\mathrm{AFIF}124}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+\frac{f_1}{f_1-f_4}·{\lambda }_{\mathrm{W}12}·{\overline{N}}_{r,\mathrm{W}12}^s+{\epsilon }_{r,\mathrm{AFIF}124}^s\\ L_{r,\mathrm{AFIF}125}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+\frac{f_1}{f_1-f_5}·{\lambda }_{\mathrm{W}12}·{\overline{N}}_{r,\mathrm{W}12}^s+{\epsilon }_{r,\mathrm{AFIF}125}^s\end{array}$$

(16)

where ${\overline{N}}_{r,\mathrm{W}12}^s$ denotes float WL ambiguity and can be expressed as:

$${\overline{N}}_{r,\mathrm{W}12}^s=N_{r,\mathrm{W}12}^s+B_{r,\mathrm{W}12}-B_{\mathrm{W}12}^s$$

(17)

In case of $n$ visible satellites, the estimated vector corresponding to Equation (16) includes:

$$\widehat{X}=\left[\begin{array}{cccc}x,y,z& {\tilde{t}}_r& T_{r,\mathrm{w}}& \underset{n}{\underbrace{{\overline{N}}_{r,\mathrm{W}12}^1,{\overline{N}}_{r,\mathrm{W}12}^2,\cdots ,{\overline{N}}_{r,\mathrm{W}12}^n}}\end{array}\right]$$

(18)

It is not difficult to find from Equation (16) that although three different AFIF combinations are used, only one common WL ambiguity parameter needs to be estimated. For different AFIF combinations, the corresponding WL ambiguity wavelengths are amplified to varying degrees, with magnifications of $\frac{f_1}{f_1-f_3}$, $\frac{f_1}{f_1-f_4}$ and $\frac{f_1}{f_1-f_5}$, respectively [12]. As shown in

Table 2. Noise amplification factors (AF) and equivalent wavelengths for different AFIF combinations.

AFIF Observation	EWL/WL Combination	Noise AF	Prior Accuracy	Wavelength AF	Equivalent Wavelength
$L_{r,\mathrm{AFIF}123}^s$	E6-E5a/E1-E5a	77.7	0.233 m	5.3	3.990 m
$L_{r,\mathrm{AFIF}124}^s$	E5b-E5a/E1-E5a	181.3	0.544 m	4.3	3.214 m
$L_{r,\mathrm{AFIF}125}^s$	E5-E5a/E1-E5a	339.6	1.019 m	4.1	3.086 m

, the equivalent wavelength can reach about 3~4 m, which is obviously larger than the original 0.751 m. The larger equivalent wavelength is also more conducive to efficient WL ambiguity resolution.

For convenience, AFIF(GF) and AFIF(GB) are used in subsequent sections to represent the models in Equation (14) and Equation (16), respectively.

2.2. Uncombined Model

The uncombined PPP model has been demonstrated to be the most ideal model for multi-frequency data processing. To avoid additional code bias correction for multi-frequency pseudoranges, only E1/E5a dual-frequency pseudoranges are used for parameter estimation, with the observation model as:

$$\{\begin{array}{lll}{P}_{r,1}^s\hfill & =\hfill & {\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\tilde{I}}_{r,1}^s+e_{r,1}^s\hfill \\ P_{r,2}^s\hfill & =\hfill & {\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\gamma }_2·{\tilde{I}}_{r,1}^s+e_{r,2}^s\hfill \\ L_{r,1}^s\hfill & =\hfill & {\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}-{\tilde{I}}_{r,1}^s+{\lambda }_1·{\overline{N}}_{r,1}^s+{\epsilon }_{r,1}^s\hfill \\ L_{r,2}^s\hfill & =\hfill & {\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}-{\gamma }_2·{\tilde{I}}_{r,1}^s+{\lambda }_2·{\overline{N}}_{r,2}^s+{\epsilon }_{r,2}^s\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill \\ L_{r,j}^s\hfill & =\hfill & {\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}-{\gamma }_j·{\tilde{I}}_{r,1}^s+{\lambda }_j·{\overline{N}}_{r,j}^s+{\epsilon }_{r,j}^s\hfill \end{array}$$

(19)

where ${\overline{N}}_{r,j}^s$ denotes float ambiguity at frequency $j$, which can be expressed as:

$${\overline{N}}_{r,j}^s=N_{r,j}^s+B_{r,j}-B_j^s$$

(20)

where $B_{r,j}$ and $B_j^s$ denote receiver and satellite FCBs at frequency $j$.

In the case of $n$ visible satellites with $j$ frequencies, the estimated vector contains the receiver position, receiver clock offset, ZWD, slant ionosphere delay and multi-frequency ambiguities for each satellite.

$$\widehat{X}=\left[\begin{array}{lllll}x,y,z\hfill & {\tilde{t}}_r\hfill & T_{r,\mathrm{w}}\hfill & \underset{n}{\underbrace{{\tilde{I}}_{r,1}^1,{\tilde{I}}_{r,1}^2,\cdots ,{\tilde{I}}_{r,1}^n}}\hfill & \underset{n\times j}{\underbrace{{\overline{N}}_{r,\ast }^1,{\overline{N}}_{r,\ast }^2,\cdots ,{\overline{N}}_{r,\ast }^n}}\hfill \end{array}\right]$$

(21)

with ${\overline{N}}_{r,\ast }^s=\left[\underset{j}{\underbrace{{\overline{N}}_{r,1}^s,{\overline{N}}_{r,2}^s,\cdots ,{\overline{N}}_{r,j}^s}}\right]$.

After the float solution $\widehat{X}$ and its variance–covariance $Q_{\widehat{X}\widehat{X}}$ are obtained, the raw ambiguity of each frequency can be mapped to EWL/WL ambiguity with a properly selected transformation matrix. In this paper, the same EWL/WL combinations as shown in Table 2 are adopted, that is, E5–E5a, E5b–E5a, E6–E5a and E1–E5a, with corresponding wavelengths of 19.54 m, 9.77 m, 2.93 m and 0.75 m. According to the error propagation law, the transformed vector ${\widehat{X}}_{\mathrm{LC}}$ and its variance–covariance $Q_{{\widehat{X}}_{\mathrm{LC}}{\widehat{X}}_{\mathrm{LC}}}$ can be derived as:

$${\widehat{X}}_{\mathrm{LC}}=\left[\begin{array}{ll}I\hfill & \hfill \\ \hfill & D_{\mathrm{W}}\hfill \\ \hfill & D_{\mathrm{E}1}\hfill \\ \hfill & D_{\mathrm{E}2}\hfill \\ \hfill & D_{\mathrm{E}3}\hfill \end{array}\right]·\widehat{X}, Q_{{\widehat{X}}_{\mathrm{LC}}{\widehat{X}}_{\mathrm{LC}}}=\left[\begin{array}{ll}I\hfill & \hfill \\ \hfill & D_{\mathrm{W}}\hfill \\ \hfill & D_{\mathrm{E}1}\hfill \\ \hfill & D_{\mathrm{E}2}\hfill \\ \hfill & D_{\mathrm{E}3}\hfill \end{array}\right]·Q_{\widehat{X}\widehat{X}}·{\left[\begin{array}{ll}I\hfill & \hfill \\ \hfill & D_{\mathrm{W}}\hfill \\ \hfill & D_{\mathrm{E}1}\hfill \\ \hfill & D_{\mathrm{E}2}\hfill \\ \hfill & D_{\mathrm{E}3}\hfill \end{array}\right]}^{\mathrm{T}}$$

(22)

where $I$ denotes the identity matrix of dimension $(m+5)$; $D_{\mathrm{W}}$ denotes the E1–E5a WL ambiguity mapping matrix; and $D_{\mathrm{E}1}$, $D_{\mathrm{E}2}$ and $D_{\mathrm{E}3}$ denote the E6–E5a, E5b–E5a and E5–E5a EWL ambiguity mapping matrix, respectively.

To simplify illustration, the first satellite is chosen as the reference satellite to eliminate the effect of receiver FCBs, and the transformed vector ${\widehat{X}}_{\mathrm{LC}}$ contains:

$${\widehat{X}}_{\mathrm{LC}}=\left[\begin{array}{lllll}x,y,z\hfill & {\tilde{t}}_r\hfill & T_{r,\mathrm{w}}\hfill & \underset{n}{\underbrace{{\tilde{I}}_{r,1}^1,{\tilde{I}}_{r,1}^2,\cdots ,{\tilde{I}}_{r,1}^n}}\hfill & \underset{(n-1)\times 4}{\underbrace{{\overline{N}}_{r,\mathrm{W}}^{1\ast },{\overline{N}}_{r,\mathrm{E}1}^{1\ast },{\overline{N}}_{r,\mathrm{E}2}^{1\ast },{\overline{N}}_{r,\mathrm{E}3}^{1\ast }}}\hfill \end{array}\right]$$

(23)

where ${\overline{N}}_{r,\mathrm{W}}^{1\ast },{\overline{N}}_{r,\mathrm{E}1}^{1\ast },{\overline{N}}_{r,\mathrm{E}2}^{1\ast },{\overline{N}}_{r,\mathrm{E}3}^{1\ast }$ denote the between-satellite single-differenced (BSSD) E1–E5a, E6–E5a, E5b–E5a and E5–E5a ambiguity of each satellite, and their specific expressions are as follows:

$$\{\begin{array}{l}{\overline{N}}_{r,\mathrm{W}}^{1\ast }=\left[{\overline{N}}_{r,\mathrm{W}}^{12},{\overline{N}}_{r,\mathrm{W}}^{13}\cdots ,{\overline{N}}_{r,\mathrm{W}}^{1n}\right]\\ {\overline{N}}_{r,\mathrm{E}1}^{1\ast }=\left[{\overline{N}}_{r,\mathrm{E}1}^{12},{\overline{N}}_{r,\mathrm{E}1}^{13},\cdots ,{\overline{N}}_{r,\mathrm{E}1}^{1n}\right]\\ {\overline{N}}_{r,\mathrm{E}2}^{1\ast }=\left[{\overline{N}}_{r,\mathrm{E}2}^{12},{\overline{N}}_{r,\mathrm{E}2}^{13},\cdots ,{\overline{N}}_{r,\mathrm{E}2}^{1n}\right]\\ {\overline{N}}_{r,\mathrm{E}3}^{1\ast }=\left[{\overline{N}}_{r,\mathrm{E}3}^{12},{\overline{N}}_{r,\mathrm{E}3}^{13},\cdots ,{\overline{N}}_{r,\mathrm{E}3}^{1n}\right]\end{array}$$

(24)

The transformation matrix $D_{\ast }$ in Equation (22) can be generically expressed as:

$$D_{\ast }=\left[\begin{array}{ccccc}{C}_{\ast }& -C_{\ast }& & & \\ C_{\ast }& & -C_{\ast }& & \\ ⋮& & & \ddots & \\ C_{\ast }& & & & -C_{\ast }\end{array}\right]$$

(25)

where the symbol $\ast$ represents W, E1, E2 and E3; $C_{\ast }$ denotes a specific combination coefficient; and the expression is as follows:

$$\{\begin{array}{l}{C}_{\mathrm{W}}=\left[\begin{array}{ccccc}1& -1& 0& 0& 0\end{array}\right]\\ C_{\mathrm{E}1}=\left[\begin{array}{ccccc}0& -1& 1& 0& 0\end{array}\right]\\ C_{\mathrm{E}2}=\left[\begin{array}{ccccc}0& -1& 0& 1& 0\end{array}\right]\\ C_{\mathrm{E}3}=\left[\begin{array}{ccccc}0& -1& 0& 0& 1\end{array}\right]\end{array}$$

(26)

Based on the transformed vector ${\widehat{X}}_{\mathrm{LC}}$ and its variance–covariance $Q_{{\widehat{X}}_{\mathrm{LC}}{\widehat{X}}_{\mathrm{LC}}}$, after satellite FCBs are corrected, a stepwise ambiguity resolution process is carried out with EWL/WL ambiguities fixed sequentially. It should be noted that the three selected EWL ambiguities are all fixed in one step, instead of being divided into three steps [16]. Regardless of whether the EWL or WL ambiguity is fixed, the precision of the position parameter can be improved theoretically. When both EWL and WL ambiguities are successfully fixed, decimeter-level positioning accuracy can generally be achieved. Subsequently, the term UC is used to represent the model in Equation (19).

2.3. Ionosphere-Reduced Model

2.3.1. Multi-Frequency Observation Combination

For the E1/E5a/E6/E5b/E5 five-frequency case, the linear combination of observation based on integer coefficients can be expressed as [22,29]:

$$L_{r,(i_1,i_2,i_3,i_4,i_5)}^s=\frac{{\displaystyle \sum _{k=1}^5\left(i_k·f_k·L_{r,k}^s\right)}}{{\displaystyle \sum _{k=1}^5\left(i_k·f_k\right)}}$$

(27)

with the specific observation equation as

$$\begin{array}{ll}{L}_{r,(i_1,i_2,i_3,i_4,i_5)}^s& ={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}\\ & -{\gamma }_{(i_1,i_2,i_3,i_4,i_5)}·{\tilde{I}}_{r,1}^s+{\lambda }_{(i_1,i_2,i_3,i_4,i_5)}·{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^s+{\mu }_{(i_1,i_2,i_3,i_4,i_5)}·{\epsilon }_{r,1}^s\end{array}$$

(28)

where ${\gamma }_{(i_1,i_2,i_3,i_4,i_5)}$, ${\lambda }_{(i_1,i_2,i_3,i_4,i_5)}$, ${\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^s$ and ${\mu }_{(i_1,i_2,i_3,i_4,i_5)}$ denote the ionosphere scalar factor, wavelength, float ambiguity and noise amplification factor of the combined signal, respectively. Their specific forms are as follows:

$$\{\begin{array}{l}{\gamma }_{(i_1,i_2,i_3,i_4,i_5)}=\frac{f_1^2·(i_1/f_1+i_2/f_2+\cdots +i_5/f_5)}{f_{(i_1,i_2,i_3,i_4,i_5)}}\\ {\lambda }_{(i_1,i_2,i_3,i_4,i_5)}=\frac{c}{f_{(i_1,i_2,i_3,i_4,i_5)}}\\ {\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^s=N_{r,(i_1,i_2,i_3,i_4,i_5)}^s+B_{r,(i_1,i_2,i_3,i_4,i_5)}-B_{(i_1,i_2,i_3,i_4,i_5)}^s\\ {\mu }_{(i_1,i_2,i_3,i_4,i_5)}=\frac{\sqrt{{(i_1·f_1)}^2+{(i_2·f_2)}^2+\cdots +{(i_5·f_5)}^2}}{f_{(i_1,i_2,i_3,i_4,i_5)}}\end{array}$$

(29)

where $B_{r,(i_1,i_2,i_3,i_4,i_5)}$ and $B_{(i_1,i_2,i_3,i_4,i_5)}^s$ denote receiver and satellites FCBs of the combined ambiguity, and $f_{(i_1,i_2,i_3,i_4,i_5)}$ denotes the frequency of the combined signal and can be expressed as:

$$f_{(i_1,i_2,i_3,i_4,i_5)}=i_1·f_1+i_2·f_2+\cdots +i_5·f_5$$

(30)

One thing to note is that ${\gamma }_{(i_1,i_2,i_3,i_4,i_5)}$ only reflects the influence of the ionosphere on the ranging accuracy. To further analyze its impacts on ambiguity resolution, it should be converted to ${\beta }_{(i_1,i_2,i_3,i_4,i_5)}$ in the unit of $\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}·{\mathrm{m}}^{-1}$, as shown below:

$${\beta }_{(i_1,i_2,i_3,i_4,i_5)}=\frac{f_1^2·(i_1/f_1+i_2/f_2+\cdots +i_5/f_5)}{c}$$

(31)

2.3.2. Ionosphere-Reduced Combination Selection

For Equation (28), the number of linear combinations is theoretically infinite. In order to find a combination with a sufficiently small ionosphere effect (e.g., IR combination), and at the same time, guarantee the performance of ambiguity resolution and positioning (e.g., decimeter-level), the following criteria are identified:

• The impact of the ionosphere is small. Taking the 100 m ionosphere delay as an example, its impact on ambiguity resolution and ranging accuracy is less than 0.1 cycle and 0.15 m. The first criterion is ${\beta }_{(i_1,i_2,i_3,i_4,i_5)}<0.001$ and ${\gamma }_{(i_1,i_2,i_3,i_4,i_5)}<0.0015$.
• The amplified noise level supports decimeter-level positioning. Considering that the accuracy ratio of carrier phase and pseudorange observations is usually 100:1, and the noise amplification factor of AFIF observations ranges from tens to hundreds, as shown in Table 2, this paper takes ${\mu }_{(i_1,i_2,i_3,i_4,i_5)}<100$ as the second criterion.
• The combination should have a long wavelength to ensure efficient ambiguity resolution and to be able to resist the influence of residual geometry errors. This paper takes the equivalent wavelength in Table 2 as a reference and takes ${\lambda }_{(i_1,i_2,i_3,i_4,i_5)}>3 \mathrm{m}$ as the third criterion.

For coefficients $i_k(k=1,2,3,4,5)$ within range [–10, 10], the IR combination list in

Table 3. Galileo five-frequency ionosphere-reduced (IR) combination.

E1	E5a	E6	E5b	E5ab	${\mathit{\lambda }}_{({\mathit{i}}_1,{\mathit{i}}_2,{\mathit{i}}_3,{\mathit{i}}_4,{\mathit{i}}_5)}$	${\mathit{\gamma }}_{({\mathit{i}}_1,{\mathit{i}}_2,{\mathit{i}}_3,{\mathit{i}}_4,{\mathit{i}}_5)}$	${\mathit{\beta }}_{({\mathit{i}}_1,{\mathit{i}}_2,{\mathit{i}}_3,{\mathit{i}}_4,{\mathit{i}}_5)}$	${\mathit{\mu }}_{({\mathit{i}}_1,{\mathit{i}}_2,{\mathit{i}}_3,{\mathit{i}}_4,{\mathit{i}}_5)}$
${\mathit{i}}_1$	${\mathit{i}}_2$	${\mathit{i}}_3$	${\mathit{i}}_4$	${\mathit{i}}_5$
1	4	−3	1	−3	3.907 m	−0.0012	−0.0003	95.407

is finally selected after traversing each combination. Although the noise amplification factor of the selected IR combination is close to 100, it is derived from raw carrier phase observations, which means that in addition to noise level, the multipath effects are theoretically weaker than the pseudorange.

2.3.3. Ionosphere-Reduced Observation Model

On the basis of the above selected IR combination, pseudorange observations are also needed to build a full-rank model. Like the aforementioned AFIF model and uncombined model, in order to avoid the additional code bias caused by the inconsistency between multi-frequency pseudoranges and precise clock references, only the E1/E5 dual-frequency pseudoranges are used.

For the selected IR observation Equation (28), if the influence of the residual ionosphere is small, it can be ignored directly, thereby reducing the dimension of the estimated parameters and improving the model strength to a certain extent. Depending on whether ionosphere parameters are considered, two models can be established.

If the ionosphere parameters are estimated, raw dual-frequency observations are integrated with Equation (28), and the corresponding observation model is as follows:

$$\{\begin{array}{l}{P}_{r,1}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\tilde{I}}_{r,1}^s+e_{r,1}^s\\ P_{r,2}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\gamma }_2·{\tilde{I}}_{r,1}^s+e_{r,2}^s\\ L_{r,(i_1,i_2,i_3,i_4,i_5)}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -{\gamma }_{(i_1,i_2,i_3,i_4,i_5)}·{\tilde{I}}_{r,1}^s+{\lambda }_{(i_1,i_2,i_3,i_4,i_5)}·{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^s+{\mu }_{(i_1,i_2,i_3,i_4,i_5)}·{\epsilon }_{r,1}^s\end{array}$$

(32)

In case of $n$ visible satellites, the estimated vector in Equation (32) includes:

$$\widehat{X}=\left[\begin{array}{lllll}x,y,z\hfill & {\tilde{t}}_r\hfill & T_{r,\mathrm{w}}\hfill & \underset{n}{\underbrace{{\tilde{I}}_{r,1}^1,{\tilde{I}}_{r,1}^2,\cdots ,{\tilde{I}}_{r,1}^n}}\hfill & \underset{n}{\underbrace{{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^1,{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^2,\cdots ,{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^n}}\hfill \end{array}\right]$$

(33)

If the ionosphere parameters in Equation (28) are directly ignored, a dual-frequency pseudorange in the form of an IF combination is used for parameter estimation. The observation model with the ionosphere ignored can be expressed as:

$$\{\begin{array}{l}{P}_{r,\mathrm{IF}}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+e_{r,\mathrm{IF}}^s\\ L_{r,(i_1,i_2,i_3,i_4,i_5)}^s={\rho }_r^s+c·{\tilde{t}}_r-c·{\tilde{t}}^s+M_{r,\mathrm{w}}^s·T_{r,\mathrm{w}}+{\lambda }_{(i_1,i_2,i_3,i_4,i_5)}·{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^s+{\mu }_{(i_1,i_2,i_3,i_4,i_5)}·{\epsilon }_{r,1}^s\end{array}$$

(34)

The corresponding estimated vector in Equation (34) contains:

$$\widehat{X}=\left[\begin{array}{cccc}x,y,z& {\tilde{t}}_r& T_{r,\mathrm{w}}& \underset{n}{\underbrace{{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^1,{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^2,\cdots ,{\overline{N}}_{r,(i_1,i_2,i_3,i_4,i_5)}^n}}\end{array}\right]$$

(35)

To simplify description, the terms IR(EST) and IR(IGN) are used to represent the models described in Equation (32) and Equation (34), respectively.

2.4. Comparison of Different Models

Table 4. Comparison of AFIF, Uncombined (UC) and Ionosphere-Reduced (IR) models.

Item		AFIF		Uncombined	Ionosphere-Reduced
		AFIF(GF)	AFIF(GB)	UC	IR(EST)	IR(IGN)
Observation configuration		All models use E1/E5a dual-frequency pseudoranges and E1/E5a/E6/E5b/E5 five-frequency carrier phase observations.
Observation model equation		(14)	(16)	(19)	(32)	(34)
Estimated vector equation		(15)	(18)	(21)	(33)	(35)
Number of different types of estimated parameters	● Coordinates	3	3	3	3	3
	● Receiver clock	1	1	1	1	1
	● Ionosphere	0	0	n	n	0
	● Ambiguity	0	n	5n	n	n
Ionosphere delay		Eliminate	Eliminate	Estimate	Estimate	Ignore
Number of observation equation 1		(3 + 1)n	(3 + 1)n	(5 + 2)n	(1 + 2)n	(1 + 1)n
Number of fixed ambiguities 2		(3 + 1)n	(3 + 1)n	(3 + 1)n	(1 + 0)n	(1 + 0)n
EWL ambiguity fixing		GF+ROUND	GF+ROUND	GB+LAMBDA	GB+LAMBDA	GB+LAMBDA
WL ambiguity fixing		GF+ROUND	GB+LAMBDA	GB+LAMBDA	-	-

¹ The first and second numbers in brackets indicate the number of carrier phase and pseudorange observation equations for each satellite, respectively. ² The first and second numbers in brackets indicate the number of fixed EWL and WL ambiguities for each satellite, respectively.

briefly compares the above models in the case of n visible satellites per epoch. Since it is difficult to accurately estimate troposphere delay using only a single epoch of data, and also considering that the accuracy of the prior model can basically meet the decimeter-level positioning requirements, especially since zenith hydrostatic delay (ZHD) can be corrected with high accuracy, therefore, referring to the existing related research [10], the troposphere delay is directly corrected by a prior model instead of being estimated as an unknown, so the ZWD parameter does not appear in the table. The comparisons between various models adopted the same observation configurations, namely E1/E5a dual-frequency pseudoranges and E1/E5a/E6/E5b/E5 multi-frequency carrier phase observations.

In terms of ionosphere processing strategies, except for the UC and IR(EST) models that need to estimate n-dimensional ionosphere parameters, the other three models (i.e., AFIF(GF), AFIF(GB) and IR(IGN)) do not need to estimate the ionosphere. Among them, the two AFIF models use an IF combination to eliminate its influence, while the IR(IGN) model adopts the strategy of directly ignoring it.

In terms of the number of observation equations, the UC model requires the most equations for parameter estimation, up to 7n; the two AFIF models have the same number of equations, both at 4n. In contrast, the IR model requires fewer equations, and the number of IR(EST) and IR(IGN) models are 3n and 2n, respectively.

In terms of the ambiguity number involved in each model, the AFIF(GF) model resolves EWL/WL ambiguities in advance, and there is no need to estimate them in the observation model, while the other four models all need to estimate ambiguities of different dimensions. In particular, the ambiguity dimension of the UC model can reach up to 5n dimensions. In contrast, the AFIF(GB), IR(EST) and IR(IGN) models only need to estimate a small number of ambiguity parameters in n dimensions.

In terms of the number of fixed ambiguities, the UC and two AFIF models all need to fix 3n-dimensional EWL ambiguities and n-dimensional WL ambiguities, while the two IR models only need to fix n-dimensional EWL ambiguities without fixing WL ambiguities.

In terms of ambiguity fixing strategy, the UC and two IR models all adopt the strategy of GB+LAMBDA to fix all ambiguities [28]. For the two AFIF models, the strategies of EWL ambiguity fixing are the same, with both using GF+ROUND, while for WL ambiguity fixing, AFIF(GF) and AFIF(GB) adopt the different fixing strategies of GF+ROUND and GB+LAMBDA, respectively.

Compared with the UC model, the IR model has a smaller dimension of both observation equations and estimated parameters, so the calculation efficiency is higher; compared with the AFIF model, the IR model also has fewer observation equations and uses a more rigorous GB+LAMBDA strategy to resolve all ambiguities, so the performance of ambiguity resolution is better in theory. In summary, the IR model takes into account both performance and efficiency factors.

3. Experiments and Results

About 230 stations from the International GNSS Service (IGS) Multi-GNSS Experiment (MGEX) on 2023 DOY061 with sample rate of 30 s were selected for experiments, and their distribution is shown in Figure 1. All stations could receive Galileo E1/E5a/E6/E5b/E5ab five-frequency data, of which the stations marked by blue solid circles were used for FCBs estimation, and the stations marked with red stars were used for PPP verification. The selected 15 user stations are distributed in different latitudes and equipped with various types of receivers, as shown in

Table 5. Location, receiver and visible satellites of user stations (10 deg cutoff elevation).

Site	Receiver	Latitude/deg	Longitude/deg	Average Number of Satellites per Epoch
AMC4	SEPT POLARX5TR	38.8	−104.5	6.4
AREG	SEPT POLARX5	−16.5	−71.5	6.9
BRUX	SEPT POLARX5TR	50.8	4.4	6.8
GAMG	SEPT POLARX5TR	35.6	127.9	7.3
HARB	SEPT POLARX5TR	−25.9	27.7	7.2
KAT1	SEPT POLARX5	−14.4	132.2	8.0
KIR8	TRIMBLE ALLOY	67.9	21.1	7.2
KIRI	SEPT POLARX5	1.4	172.9	8.3
KITG	SEPT POLARX5	39.1	66.9	7.2
KOUG	SEPT POLARX5TR	5.1	−52.6	7.3
MCHL	TRIMBLE ALLOY	−26.4	148.1	7.5
MET3	JAVAD TRE_3 DELTA	60.2	24.4	7.5
RGDG	TRIMBLE ALLOY	−53.8	−67.8	6.5
SEYG	SEPT POLARX5	−4.7	55.5	8.6
YEL2	SEPT POLARX5TR	62.5	−114.5	7.3

. In data processing, the elevation cut-off angle was set to 10°, and the elevation dependent stochastic model was adopted with the priori precision of 0.003 m and 0.3 m for raw carrier and pseudorange, respectively. The precise orbit and clock products from the Center for Orbit Determination in Europe (CODE) were used. The troposphere delay was corrected by the Saastamoinen model with the pressure, temperature and water vapor information provided by the Global Pressure and Temperature 2 wet (GPT2w) [30,31], and the Vienna mapping function (VMF) was adopted [32]. For ambiguity fixing, a threshold of 0.25 cycles was set for integer rounding, and a ratio threshold of 2.0 was set for the LAMBDA algorithm. In addition, a partial ambiguity fixing (PAF) strategy was used simultaneously to improve the fixing success rate, and the ambiguity subset was determined according to successively increased elevation, until the number of satellites was less than five or the number of ambiguities was less than four [33]. Since there are not as many Galileo satellites as GPS, in order to ensure the redundancy of the model, only epochs with no less than five satellites were calculated. In order to avoid the large interpolation errors caused by orbit day boundary bias at the beginning and end of each day, only observation data from 1:00 to 23:00 were processed [34]. Figure 2 shows the sequence of Kp indexes, and it can be seen that the ionosphere was relatively quiet, and there was no obvious ionosphere scintillation phenomenon during the experiment period. The experimental data were processed by self-designed software, and the operating platform was a desktop computer equipped with an Intel i7-11700 CPU.

3.1. Performance Analysis of Ionosphere-Reduced Model

For the ionosphere-reduced model, although the IR(IGN) model with ionosphere estimated can be used, if the equivalent ranging errors are small and will not affect the ambiguity fixing, it can be ignored, and the IR(IGN) model is used instead. In this way, the dimensions of both estimated parameters and observation equations can be reduced, thereby improving model redundancy and data processing efficiency. Taking the BRUX station as an example, Figure 3 shows the slant ionosphere delay and the corresponding ranging errors of each satellite during the whole period, and the elevation information is also given in the figure. It can be seen from the figure that the variation range of the slant ionosphere was about −5~25 m, and its magnitude was relatively large at around 12:00. The corresponding equivalent ranging error of each satellite did not exceed 3 cm. Considering that the wavelength of the selected ionosphere-reduced combination is about 3.9 m, even if these residual errors are ignored, reliable EWL ambiguity resolution can still be achieved theoretically. Figure 4 further gives the positioning results of the two IR models. It can be seen that ignoring the ionosphere does not have a significant impact on the results, and there are only a few millimeters in difference in the RMS values of the two models. The positioning accuracy of the IR(IGN) and IR(EST) models in the north (N), east (E) and up (U) directions are (8.3, 5.7, 18.9) cm and (8.3, 5.7, 18.5) cm, respectively.

It can be preliminarily found from Figure 3 that the ionosphere delay and its corresponding ranging errors are related to satellite elevation. Therefore, Figure 5 shows the distribution relationship between the ionosphere ranging error and satellite elevation of all 15 user stations. It can be clearly found that low-elevation satellites often have larger residual ranging errors, and the ranging errors gradually decrease as the satellite elevation increases. At intervals of five degrees, the 95th, 90th and 80th percentile values within the range of 10–90 degrees are counted, as given in Figure 5 and

Table 6. The 95th, 90th and 80th percentile ranging errors in different elevation intervals.

Elevation Interval [deg]	Equivalent Ranging Error [cm]
	95th Percentile Value	90th Percentile Value	80th Percentile Value
10–15	4.0	3.4	2.6
15–20	3.9	3.3	2.5
20–25	3.4	2.9	2.2
25–30	3.1	2.8	2.1
30–35	2.9	2.5	1.9
35–40	2.6	2.0	1.6
40–45	2.3	1.9	1.5
45–50	2.2	1.8	1.4
50–55	2.1	1.7	1.3
55–60	2.0	1.6	1.2
60–65	2.0	1.6	1.2
65–70	1.8	1.5	1.1
70–75	1.5	1.4	1.1
75–80	1.7	1.4	1.1
80–85	1.5	1.2	0.9
85–90	1.5	1.4	1.0

. Even for low-elevation satellites (below 20 degrees), the 95th percentile value does not exceed 4 cm, which theoretically still supports EWL ambiguity resolution with a wavelength of several meters and positioning with decimeter-level accuracy. In practical applications, the impact of low-elevation satellites can be avoided or weakened by increasing the cut-off elevation (for example, in the case of multi-GNSS with sufficient satellites) or downweighting their observations.

Figure 6 shows the horizontal and vertical positioning accuracy of each station. The accuracies of the two IR models are almost the same. The horizontal and vertical accuracies of IR(IGN) and IR(EST) are (16.3, 35.8) cm and (16.2, 35.4) cm, respectively, and both can achieve decimeter-level positioning horizontally. Although the large residual ranging errors of low-elevation satellites will theoretically affect the positioning performance, the two models do not show significant differences, mainly for the following two reasons. First, the elevation-dependent stochastic model is adopted, and the weight of low-elevation satellites is small; secondly, in the iterative process of PAF, the low-elevation satellites are excluded, and the selected satellites are all high-elevation with small ionosphere effects.

3.2. Comparison between Ionosphere-Reduced and AFIF/Uncombined Models

In GNSS positioning, in addition to accuracy, real-time performance is also an important evaluation index. For this reason, this section will compare the IR model and the AFIF/UC model from the two aspects of positioning performance and time consumption.

3.2.1. Positioning Performance Comparison

Since the positioning performance of the two IR models is basically the same, the IR(IGN) model is selected in this section as a representative. Figure 7 shows the positioning results of the four models for the KITG station. For the two AFIF models, the main difference between the AFIF(GF) and AFIF(GB) models lies in the ambiguity fixing strategy used. The AFIF(GF) model adopts an integer rounding strategy based on the MW combination, while the AFIF(GB) model adopts an integer least square (ILS) method, namely the LAMBDA algorithm. Although the MW combination could eliminate the geometry and ionosphere terms, due to the enlarged noise of pseudorange observations, multiple epoch smoothing is often required to ensure reliable fixing. However, in single-epoch mode, direct rounding is likely to cause incorrect fixing, especially for combinations with short wavelengths. Figure 8 shows the fractional part distribution of different EWL/WL ambiguities in the AFIF(GF) model, where the FCBs are corrected in advance. It can be clearly seen that compared with the WL combination, the distribution of the three EWL combinations (i.e., E6–E5a/E5b–E5a/E5–E5a) is more concentrated, and the percentages within 0.25 cycles are close to 100%, reaching 99.94%, 100% and 100%, respectively. In contrast, the distribution of WL ambiguity is scattered, and the proportion within 0.25 cycles is only 90.80%. This means that in single-epoch mode, direct rounding is likely to cause incorrect ambiguity fixing, which will lead to larger positioning errors. As shown in Figure 7, the results of the AFIF(GF) model jump at multiple epochs, and the accuracy in the N, E and U directions is only (16.5, 14.0, 47.3) cm, respectively. Except for the AFIF(GF) model, the positioning results of the other three models (i.e., AFIF(GB), UC and IR(IGN)) are basically the same, and the accuracy in the N, E and U directions are (10.6, 10.1, 24.3) cm, (11.1, 9.2, 22.0) cm and (9.4, 8.2, 19.8) cm, respectively.

While the AFIF(GF) model uses an integer rounding strategy, the AFIF(GB), UC and IR(IGN) models all use the LAMBDA method with a ratio test to fix ambiguities. Figure 9 shows the ratio values of the three models. The ratio distributions of the different models are similar, and all three models are able to successfully fix ambiguities with large ratio values.

Figure 10 shows the epoch fixing rate of different models for each station. Due to geographical differences, the fixing rates vary among different stations. For the same station, the fixing rates of different models are very close. Among them, the AFIF(GF) model had the highest fixing rate of 98.9%, but it may include multiple wrongly fixed epochs. The fixing rates of the other three models (i.e., AFIF(GB), UC and IR(IGN)) were 98.6%, 98.7% and 98.7%, respectively.

Figure 11 shows the horizontal and vertical accuracy of different models for all stations. Since the results of the AFIF(GF) model have larger errors caused by incorrectly fixed ambiguities, its accuracy is the worst, with an average horizontal and vertical accuracy of (34.7, 71.5) cm. Except for the AFIF(GF) model, the accuracy of the other three models at different stations is basically the same, and the horizontal and vertical accuracy can basically be maintained within 20 cm and 50 cm, respectively. The horizontal and vertical average accuracies of the AFIF(GB), UC and IR(IGN) models are (17.8, 35.2) cm, (17.2, 33.8) cm and (16.3, 35.8) cm, respectively.

3.2.2. Time Consumption Comparison

Figure 12 shows the time consumption comparison per epoch of different models for each station. For the UC model, due to the high dimensions of both estimated parameters and observation equations, the processing efficiency is the lowest, and the results are not completely shown in Figure 12. The detailed data can be found in

Table 7. Time consumption per epoch of different models for each station.

Site	Time Consumption per Epoch [ms]
	AFIF(GF)	AFIF(GB)	UC	IR(EST)	IR(IGN)
AMC4	0.43	0.63	13.76	0.51	0.38
AREG	0.46	0.64	14.25	0.53	0.42
BRUX	0.52	0.67	14.21	0.52	0.40
GAMG	0.49	0.70	14.72	0.58	0.45
HARB	0.51	0.69	14.58	0.57	0.44
KAT1	0.52	0.71	15.38	0.60	0.43
KIR8	0.46	0.67	14.57	0.53	0.43
KIRI	0.51	0.71	15.28	0.58	0.43
KITG	0.52	0.68	14.38	0.57	0.42
KOUG	0.51	0.68	14.74	0.55	0.42
MCHL	0.49	0.70	14.83	0.56	0.42
MET3	0.48	0.71	14.73	0.56	0.42
RGDG	0.46	0.65	13.62	0.52	0.41
SEYG	0.58	0.73	15.72	0.61	0.45
YEL2	0.51	0.73	14.78	0.56	0.43
Average	0.50	0.69	14.64	0.56	0.42

. Except for the UC model, the time consumption of the other four models is basically at the same level, maintaining around 0.5 ms. At the same time, it can be seen intuitively from the figure that the IR(IGN) model of each station has the shortest time consumption.

It can be seen from Table 7 that the UC model takes an average of 14.64ms to process one epoch data, which is significantly longer than other models. In contrast, the average time consumption of the AFIF(GF), AFIF(GB), IR(EST) and IR(IGN) models is only 0.50 ms, 0.69 ms, 0.56 ms and 0.42 ms, respectively. For the two AFIF models, previous analysis has shown that the positioning performance of the AFIF(GF) model is poor. Therefore, on the premise of ensuring positioning performance, the results of the AFIF(GB) model are selected for subsequent comparison. It can be found that the time consumption of both IR models is shorter than that of the AFIF(GB) model. For the two IR models, since the IR(EST) model estimates the ionosphere parameters and has more observation equations than the IR(IGN) model, its time consumption is also slightly longer.

Overall, the IR(IGN) model takes the least time consumption, being 38% shorter than the AFIF(GB) model and 97% shorter than the UC model. Compared with the AFIF/UC models, the IR model shows varying degrees of advantages in terms of process efficiency under the premise of ensuring positioning performance.

4. Discussion

With the development of multi-frequency GNSS, considering the accuracy, reliability, real-time calculation and other indicators comprehensively, instantaneous decimeter-level positioning with PPP has become a promising positioning mode. Based on existing research, two positioning models, namely AFIF and UC, were derived using different ionosphere processing strategies.

In the AFIF model, the AFIF observations, constructed by the IF combination of EWL and WL observations, are used for positioning. Since the coefficients of the IF combination are not integers, the combined ambiguity does not have integer characteristics, and the EWL and WL ambiguities need to be fixed separately during the implementation process. In the UC model, raw frequency ambiguity is mapped to EWL and WL ambiguity, and then decimeter-level positioning is achieved through the cascading ambiguity resolution. This model can theoretically process observation of any number of frequencies, but as it needs to deal with high-dimensional matrix operations, the computational load is relatively large.

In contrast, the IR model based on integer coefficients greatly eliminates the influence of ionosphere delay under the condition of ensuring the integer characteristics of the ambiguity. By selecting an appropriate wavelength and controlling noise level, in addition to similar decimeter-level accuracy comparable to the AFIF/UC models, the calculation efficiency is also improved to varying degrees due to the avoidance of high-dimensional matrix operations. The proposed IR model takes into account both positioning performance and calculation efficiency. Especially for the IR(IGN) model, at the expense of losing the unbiasedness of the observation model, the redundancy is improved, and the calculation efficiency is also greatly improved. After all, for the practical application of PPP, in addition to positioning accuracy, real-time performance is also an important indicator that users are concerned about.

Of course, the IR model also has certain limitations. First of all, the model is suitable for multi-frequency signal situations. In practical applications, when multi-frequency observations cannot be fully received, the model will not be applicable. Nevertheless, since the model is based on single-epoch mode, that is, there is no parameter transfer between previous and subsequent epochs, it can be flexibly switched to the AFIF or UC models that use fewer frequency signals between different epochs, when a certain frequency is missing. Secondly, the proposed IR models are based on the assumption that the residual ionosphere effect is small, which means that under extreme ionosphere conditions, the applicability of the IR models, especially the IR(IGN) model that ignores ionosphere effect, still needs further study. In a disturbed ionosphere case, as the ionosphere delay increases sharply, the corresponding positioning performance will be greatly reduced [35]. In such case, it is recommended to use the IR(EST) model in Equation (34) instead of blindly using the IR(IGN) model. Since the IR(EST) model properly estimates the ionosphere parameters, the model is theoretically still applicable.

This paper mainly evaluates the positioning accuracy and time-consuming performance of Galileo-only five-frequency IR PPP. On this basis, the next research work will be carried out from the following two aspects: first, to further study the performance and applicability of the proposed IR model in active ionosphere conditions; secondly, to further explore the use of IR combinations from other multi-frequency GNSSs, such as BDS-3, and to evaluate the performance of multi-GNSS IR PPP.

5. Conclusions

This paper investigates the theory and feasibility of a multi-frequency ionosphere-reduced (IR) combination in the application of single-epoch PPP with decimeter-level accuracy. Firstly, the existing AFIF and UC models are introduced. Then, based on multi-frequency observation combination theory, the selection method of an ionosphere-reduced (IR) combination is studied by taking Galileo five-frequency signals as an example. Based on the selected IR combination, two IR positioning models, namely IR(EST) and IR(IGN), are proposed according to whether ionosphere parameters are estimated or ignored, and the proposed models are further compared with the existing AFIF/UC models in various aspects.

Real tracked observations from globally distributed stations were used for model validation. The magnitude of the ionosphere equivalent ranging errors in the IR models and its relationship with satellite elevation were analyzed. The higher the satellite elevation, the smaller the residual ranging errors. The 95th percentile value of low-elevation satellites generally does not exceed 4 cm, which theoretically still supports EWL ambiguity resolution with a wavelength of several meters. In practical applications, the impact of low-elevation satellites can be avoided or weakened by increasing the cut-off elevation or downweighting their observations. The positioning results of the two IR models are almost the same, and the horizontal and vertical accuracies of IR(IGN) and IR(EST) are (16.3, 35.8) cm and (16.2, 35.4) cm, respectively.

The proposed IR models are further compared with the existing AFIF/UC models in terms of positioning performance and time consumption. In terms of positioning performance, the AFIF(GF) model suffers from incorrect ambiguity fixing and has the worst accuracy. In contrast, the IR and AFIF(GB)/UC models all use the LAMBDA method to determine the optimal candidate and can successfully fix ambiguities with large ratio values. The corresponding positioning performance of different models is basically the same, and all can achieve horizontal and vertical accuracy better than 20 cm and 40 cm, respectively. In terms of time consumption, the UC model suffers from high dimensions of both estimated parameters and observation equations, with the longest average processing time of one epoch data, reaching 14.64 ms; meanwhile, the other models take a relatively shorter time, maintaining around 0.5 ms, and the IR(IGN) model takes the shortest time of 0.42 ms.

Overall, the proposed IR models can achieve single-epoch decimeter-level positioning, and this performance is comparable to that of the existing AFIF/UC models. On the basis of guaranteeing positioning performance, the IR model shows varying degrees of advantages in terms of time consumption, compared with the AFIF/UC models. In particular, the IR(IGN) model, with no ionosphere parameters estimated, has the shortest time consumption, being 38% faster than the AFIF(GB) model and 97% faster than the UC model.