Crowdsourcing User-Enhanced PPP-RTK with Weighted Ionospheric Modeling

Abstract

In the conventional PPP-RTK mode, the platform and users act only as the generator and the utilizer of ionospheric corrections, respectively. In sparse reference station networks or regions with an active ionosphere, high-precision modeling still faces challenges. This study utilizes the concept of crowdsourcing and treats users as dynamic reference stations. By continuously feeding back ionospheric information to the platform, high-spatial-resolution modeling is achieved. Additionally, weight factors related to user positions are incorporated into conventional polynomial models to transform the regional ionosphere model from a common model into customized models, thereby providing more personalized services for different users. Validation was conducted with a sparse reference network with an average inter-station distance of approximately 391 km. While increasing the number of crowdsourcing users generally improves modeling performance, the enhancement also depends on their spatial distribution; that is, crowdsourcing users primarily provide localized improvements in their vicinity. Therefore, crowdsourcing users should ideally be uniformly distributed across the whole network. Compared with the conventional common model, the proposed customized model can more effectively characterize the irregular physical characteristics of the ionosphere, and the modeling accuracy is improved by about 12% to 41% in different scenarios. Furthermore, the performance of single-frequency PPP-RTK was verified on the terminal. In general, both crowdsourcing enhancement and the customized model can accelerate the convergence speed of the float solutions and improve positioning accuracy to varying degrees, and the epoch fix rate of the fixed solutions is also significantly improved.

1. Introduction

Global navigation satellite system (GNSS) precision positioning techniques, such as representative precise point positioning (PPP), real-time kinematic (RTK), or network RTK (NRTK), are inseparable from our daily lives [1,2,3]. As an absolute positioning technology, conventional PPP has a poor real-time performance compared with differential positioning technologies, such as RTK or NRTK, due to the correlation between the estimated parameters and the requirement of a certain convergence or initialization time [4,5]. In recent years, with the development of PPP-RTK technology, PPP has also been able to achieve instantaneous centimeter-level positioning comparable to that of NRTK, and this performance improvement is mainly due to the augmentation of high-precision atmosphere modeling [6,7,8,9]. Therefore, there is no doubt that regional atmosphere modeling plays an important role in PPP-RTK.

At present, the commonly used regional atmosphere modeling methods mainly include interpolation and fitting models. The interpolation models include the linear combination model (LCM) [10], linear interpolation method (LIM) [11], lower-order surface model (LSM) [12], distance-based linear interpolation method (DIM) [13], Kriging interpolation (KI) [14], etc. These models usually require two-way communication between the platform and the user during implementation; that is, the user sends their approximate position to the platform, and the platform feeds back corrections. Li et al. adopted the modified LCM (MLCM) to achieve an ionospheric modeling accuracy of about 2 cm in a network with an inter-station distance of about 50 km [7]. Based on different network scales, Gao et al. evaluated the performance of MLCM and found that when the inter-station distance reaches about 369 km, the ionosphere accuracy can be degraded to the decimeter level [15]. In such cases, Zhang et al. pointed out that it is necessary to determine the optimal prior variance of the ionospheric corrections in order to significantly shorten the convergence time for PPP–RTK [16]. Both Geng et al. [17] and Shao et al. [18] used DIM to generate regional atmosphere corrections with centimeter-level accuracy for multi-frequency PPP-RTK vehicle positioning. In order to reduce the burden of two-way communication, in addition to the corrections generated by DIM, Gao et al. synchronously incorporated atmospheric information from Global Forecast System (GFS) products to provide prior constraints for both the server and user ends of PPP-RTK, improving the average convergence time and vertical accuracy [19,20]. Some scholars have conducted theoretical and experimental verifications based on the best linear unbiased predictor (BLUP) framework [21,22]. Psychas and Verhagen used the BLUP model to predict ionospheric corrections epoch by epoch based on reference station networks of varying scales, achieving rapid convergence and ambiguity resolution on the user side [23]. Zhang et al. systematically conducted a theoretical analysis of the uncertainties of the above-mentioned interpolation methods based on the BLUP framework and pointed out that the Kriging method with trend is more suitable for large-scale networks of over 500 km [24].

As for fitting models, the main examples include the polynomial model, the trigonometric series model [25], and the spherical harmonic model [26], among others. In particular, polynomial models are widely used in regional ionospheric modeling due to their easy implementation and high efficiency. Only the model coefficients need to be broadcast to users, so the communication burden is smaller than that of interpolation models. Of course, the above-mentioned interpolation and fitting models can also be used in combination. For large-area PPP-RTK applications, Cui et al. proposed a hierarchical service strategy that combines a fitting model and interpolation model to improve positioning performance with less communication burden [27]. Similarly, Xu et al. also adopted a two-step ionospheric modeling method, combining polynomial and KI models to compensate for the residual modeling errors, which shortened the convergence time by about 10% [28].

In both interpolation and fitting models, performance is closely related to factors such as the quantity and distribution of reference station infrastructure. Considering that the irregularity of the ionosphere is more complex than that of the troposphere, reliable and high-precision modeling is still challenging in sparse networks or in areas with an active ionosphere. The rise of crowdsourcing enhancement in recent years has provided new opportunities to solve this issue. Lehtola et al. conducted a joint estimation of PPP and water vapor based on thousands of sets of simulated GNSS crowdsourcing data [29]. Xu et al. proposed a new crowdsourcing NRTK positioning framework which uses crowdsourcing information to continuously update the regional atmosphere map, thereby improving the spatial resolution and positioning accuracy [30]. Smith et al. used millions of Android phones as crowdsourcing data to fill in large spatiotemporal gaps caused by insufficient reference station coverage, thereby monitoring ionospheric variations on a global scale [31]. Similarly, Pan et al. also used crowdsourcing GNSS data from smartphones to extract and model regional tropospheric delay [32].

Focusing on the development trend of crowdsourcing-enhanced GNSS, this paper, mainly aimed at PPP-RTK within large-scale sparse networks, analyzes some key issues of crowdsourcing enhancement, such as the impact of the number and distribution of crowdsourcing users, and further proposes a weighted ionospheric model that is customized for each user instead of building a common model for all users. The subsequent chapters are as follows: Section 2 explains the relevant concepts and theoretical methods; Section 3 presents a detailed analysis of the experiments; Section 4 discusses potential future research directions; finally, the conclusions are summarized in Section 5.

2. Materials and Methods

2.1. The Concept of Crowdsourcing-Enhanced PPP-RTK

In traditional GNSS-augmented positioning, the interactive information between the platform and user is relatively limited. Usually, the user provides the GPGGA message containing their approximate position to the platform. The platform generates the ionospheric corrections at this approximate position based on high-precision ionospheric delays extracted from the reference stations and then broadcasts them to the user in a one-way manner. After receiving this enhanced information, the weighted ionospheric (WI) model is usually adopted to achieve rapid and precise positioning.

Crowdsourcing-enhanced GNSS is an innovative application mode that combines the crowdsourcing concept with GNSS technology. In this application mode, interactive information between the platform and users is enriched. In addition to conventional GPGGA messages, a large number of users could serve as dynamic reference stations (i.e., crowdsourcing users), and further feed resolved ionospheric delays and even raw observations back to the platform. Then, the ionospheric delays extracted by both reference stations and crowdsourcing users are combined together for regional ionospheric modeling. Meanwhile, as the number of crowdsourcing users increases, the choice of regional ionospheric model becomes more diversified due to the associated increase in redundancy, such as transitioning from linear models to nonlinear models. Through the continuous two-way interaction between the platform and a large number of crowdsourcing users, a regional ionospheric map with a higher spatial resolution is achieved. Thus, the irregularity of the spatial distribution of ionosphere can be better characterized, especially for sparse reference station networks (Figure 1).

2.2. High-Precision Slant Ionospheric Delay Extraction with UCPPP

The uncombined PPP (UCPPP) model can flexibly process multi-frequency observation and retain the ionospheric parameters. The function model can be constructed as follows [15]:

$$\left\{\begin{array}{lll}{P}_{r,1}^s\hfill & =\hfill & {\rho }_r^s+c\cdot t_r-c\cdot t^s+M_{r,\mathrm{w}}^s\cdot T_{r,\mathrm{w}}+{\tilde{I}}_{r,1}^s+e_{r,1}^s\hfill \\ P_{r,2}^s\hfill & =\hfill & {\rho }_r^s+c\cdot t_r-c\cdot t^s+M_{r,\mathrm{w}}^s\cdot T_{r,\mathrm{w}}+{\gamma }_2\cdot {\tilde{I}}_{r,1}^s+e_{r,2}^s\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill \\ P_{r,j}^s\hfill & =\hfill & {\rho }_r^s+c\cdot t_r-c\cdot t^s+M_{r,\mathrm{w}}^s\cdot T_{r,\mathrm{w}}+{\gamma }_j\cdot {\tilde{I}}_{r,1}^s+B_{r,j}-B_j^s+e_{r,j}^s\hfill \\ L_{r,1}^s\hfill & =\hfill & {\rho }_r^s+c\cdot t_r-c\cdot t^s+M_{r,\mathrm{w}}^s\cdot T_{r,\mathrm{w}}-{\tilde{I}}_{r,1}^s+{\lambda }_1\cdot N_{r,1}^s+{\epsilon }_{r,1}^s\hfill \\ L_{r,2}^s\hfill & =\hfill & {\rho }_r^s+c\cdot t_r-c\cdot t^s+M_{r,\mathrm{w}}^s\cdot T_{r,\mathrm{w}}-{\gamma }_2\cdot {\tilde{I}}_{r,1}^s+{\lambda }_2\cdot N_{r,2}^s+{\epsilon }_{r,2}^s\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill \\ L_{r,j}^s\hfill & =\hfill & {\rho }_r^s+c\cdot t_r-c\cdot t^s+M_{r,\mathrm{w}}^s\cdot T_{r,\mathrm{w}}-{\gamma }_j\cdot {\tilde{I}}_{r,1}^s+{\lambda }_j\cdot N_{r,j}^s+{\epsilon }_{r,j}^s\hfill \end{array}\right.$$

(1)

where the superscript $s$ represent different satellites; the subscript $r$ and $j$ represent different receivers and frequencies, respectively; $P_{r,j}^s$ and $L_{r,j}^s$ represent the pseudorange and carrier phase observations; ${\rho }_r^s$ is the geometric distance between the receiver and the satellite; $c$ is the speed of light; $t_r$ and $t^s$ are the receiver and the satellite clock bias, respectively; $T_{r,\mathrm{w}}$ is the zenith wet tropospheric delay (ZWD) with the corresponding mapping function $M_{r,\mathrm{w}}^s$; ${\tilde{I}}_{r,1}^s$ is the slant ionospheric delay of each satellite at the first frequency band with the frequency factor ${\gamma }_j$; $N_{r,j}^s$ represents float ambiguity; ${\lambda }_j$ represents the wavelength of each frequency; $e_{r,j}^s$ and ${\epsilon }_{r,j}^s$ represent the observation noise of the pseudorange and phase observations; and $B_{r,j}$ and $B_j^s$ are the inter-frequency bias (IFB) at the receiver and satellite ends, respectively, introduced by multi-frequency data.

Based on the above fully ranked function model, the Kalman filter is typically employed to estimate the float solutions of unknown parameters epoch by epoch. The estimated parameters can generally be categorized into two types: ambiguity parameters and non-ambiguity parameters (e.g., coordinates, clock bias, and atmospheric delays). After eliminating and correcting the receiver- and satellite-end fractional cycle bias (FCB) in the float ambiguity, the integer nature of ambiguity is restored. Then, the well-known least square ambiguity decorrelation adjustment (LAMBDA) algorithm is applied to achieve ambiguity fixing [33]. Meanwhile, the cascading ambiguity resolution (CAR) and partial ambiguity resolution (PAR) strategies can also be adopted simultaneously to improve the ambiguity fixing performance [34,35]. The constraints of the fixed integer ambiguity further enhance the accuracy of non-ambiguity parameters such as atmospheric delays. Generally, once the ambiguity is successfully fixed, the high-precision slant ionospheric delay for each satellite can be extracted, typically at the centimeter level.

It should be noted that the slant ionospheric parameter ${\tilde{I}}_{r,1}^s$ in Equation (1) is re-parameterized, which also includes the differential code bias (DCB) at the receiver and satellite ends. The specific formula is as follows:

$${\tilde{I}}_{r,1}^s=I_{r,1}^s+{\displaystyle \frac{f_2^2}{f_1^2-f_2^2}}\cdot \left(D_r-D^s\right)$$

(2)

where $I_{r,1}^s$ is the pure slant ionospheric delay, and $D_r$ and $D^s$ represent the receiver and satellite DCBs, respectively.

For a reference station network, the impact of satellite DCBs on all stations is consistent and will not affect the overall performance of subsequent ionospheric modeling; by contrast, the receiver DCBs at different reference stations are usually different, and this effect needs to be deducted before implementing ionospheric modeling. A common strategy is to select a certain satellite as a reference and eliminate its influence through the between-satellite single-difference (BSSD) process [15]. In order to ensure the visibility of the reference satellite, the satellite with the highest elevation angle is usually selected. It should be noted that the receiver DCBs of different GNSS systems are also different, and an independent reference satellite needs to be selected for each GNSS.

$$\nabla {\tilde{I}}_{r,1}^{s,ref}=\nabla I_{r,1}^{s,ref}+{\displaystyle \frac{f_2^2}{f_1^2-f_2^2}}\cdot \nabla D^{s,ref}$$

(3)

where $ref$ represents the reference satellite, and $\nabla$ represents the BSSD operator.

Considering that there is no ideal ionospheric projection function compared to tropospheric delay, if the slant ionospheric delay is mapped in the vertical direction before modeling, decimeter-level errors may be introduced. Therefore, the regional ionospheric model is subsequently established directly for the slant ionospheric delay of each satellite pair.

2.3. Conventional Ionospheric Model Common to All Users

Based on the extracted regional slant ionospheric delay, a certain plane or curved surface model is usually used to describe the spatial distribution of the ionosphere. Compared to spherical harmonic functions and physical models, the polynomial model offers lower complexity and allows for flexible adjustment of the polynomial orders and weights of different reference stations so as to better match the characteristics of regional ionospheric variations, potentially demonstrating superior regional adaptability. Furthermore, by selecting an appropriate polynomial order, the polynomial model can achieve a performance comparable to that of classical interpolation models such as LCM, LIM, and LSM. For these reasons, this study adopts the polynomial model for regional ionospheric modeling. According to the redundancy of prior information, polynomial models of different orders can be selected. Taking the first-, second-, and third-order models as examples, their expressions are as follows:

$$\left\{\begin{array}{l}\nabla {\tilde{I}}_{r,1}^{s,ref}=a_0+a_{1}x+a_{2}y\\ \nabla {\tilde{I}}_{r,1}^{s,ref}=a_0+a_{1}x+a_{2}y+a_3{x}^2+a_4{y}^2+a_{5}xy\\ \nabla {\tilde{I}}_{r,1}^{s,ref}=a_0+a_{1}x+a_{2}y+a_3{x}^2+a_4{y}^2+a_{5}xy+a_6{x}^3+a_7{x}^{2}y+a_{8}x{y}^2+a_9{y}^3\end{array}\right.$$

(4)

where $x$ and $y$ represent the longitude and latitude of the satellite ionospheric pierce point (IPP) or the Gaussian plane coordinates, and $a_i$ represents the model coefficients.

If there are $n$ stations involved in the modeling process, a least square solution model can be constructed:

$$\left\{\begin{array}{l}V=BX-L\\ X={(B^{\mathrm{T}}B)}^{-1}{B}^{\mathrm{T}}L\end{array}\right.$$

(5)

where $L$ and $V$ represent the slant ionospheric and modeling residuals of each station, respectively; $B$ represents the design matrix; and $X$ represents the model coefficients.

The expressions of $L$ and $V$ are as follows:

$$\left\{\begin{array}{l}L={\left[\nabla {\tilde{I}}_{1,1}^{s,ref},\nabla {\tilde{I}}_{2,1}^{s,ref},\cdots ,\nabla {\tilde{I}}_{n-1,1}^{s,ref},\nabla {\tilde{I}}_{n,1}^{s,ref}\right]}^{\mathrm{T}}\\ V={\left[v_1,v_2,\cdots ,v_{n-1},v_n\right]}^{\mathrm{T}}\end{array}\right.$$

(6)

Corresponding to the first-, second-, and third-order polynomial models, the expressions of $B$ and $X$ are as follows:

$$\left\{\begin{array}{l}B=\left[\begin{array}{ccc}1& x_1& y_1\\ 1& x_2& y_2\\ ⋮& ⋮& ⋮\\ 1& x_n& y_n\end{array}\right]\\ X={\left[a_0,a_1,a_2\right]}^{\mathrm{T}}\end{array}\right.$$

(7)

$$\left\{\begin{array}{l}B=\left[\begin{array}{cccccc}1& x_1& y_1& x_1^2& y_1^2& x_1{y}_1\\ 1& x_2& y_2& x_2^2& y_2^2& x_2{y}_2\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& x_n& y_n& x_n^2& y_n^2& x_n{y}_n\end{array}\right]\\ X={\left[a_0,a_1,a_2,a_3,a_4,a_5\right]}^{\mathrm{T}}\end{array}\right.$$

(8)

$$\left\{\begin{array}{l}B=\left[\begin{array}{cccccccccc}1& x_1& y_1& x_1^2& y_1^2& x_1{y}_1& x_1^3& x_1^2{y}_1& x_1{y}_1^2& y_1^3\\ 1& x_2& y_2& x_2^2& y_2^2& x_2{y}_2& x_2^3& x_2^2{y}_2& x_2{y}_2^2& y_2^3\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& x_n& y_n& x_n^2& y_n^2& x_n{y}_n& x_n^3& x_n^2{y}_n& x_n{y}_n^2& y_n^3\end{array}\right]\\ X={\left[a_0,a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9\right]}^{\mathrm{T}}\end{array}\right.$$

(9)

As can be seen, the establishment of the above models is independent of each user; that is, whether there are users or where the users are distributed will not affect the coefficients of the constructed model. In other words, this is a regional ionospheric model common to all users, as the model coefficients are the same for different users. This is obviously unreasonable and difficult to adapt to the diverse characteristics of different users.

In addition, since it is difficult for a specific polynomial model to objectively conform to the spatial distribution of the ionosphere, even for the stations included in ionospheric modeling, their modeling residuals still exist, and this means that the self-consistency of the model at the modeling stations also needs to be improved.

2.4. Weighted Ionospheric Model Customized for Each User

In order to improve the self-consistency of the model and establish diversified models for different users, the weight matrix $P$ related to the user’s position is introduced. Through adjusting the weight of the ionospheric information of each station, a weighted ionospheric model customized for each user can be established.

$$P=diag\left[p_1,p_2,\cdots ,p_n\right]$$

(10)

$$X={(B^{\mathrm{T}}PB)}^{-1}{B}^{\mathrm{T}}PL$$

(11)

where $p_i$ is the weight of the ionosphere for each station.

Considering the geometric distribution of the users and reference stations in the network, distance-related weight factors are used. Taking the spatial correlation of the ionosphere into account, the inverse distance weighting (IDW) method is adopted:

$$p_i={\displaystyle \frac{1/d_i^{\alpha }}{{\displaystyle \sum _{i=1}^n\left(1/d_i^{\alpha }\right)}}}$$

(12)

where $d_i$ is the distance between the user and each reference station, and $\alpha$ is the order, which is usually set to 1 or 2.

In the above weighting strategy, the closer the reference station is to the user, the greater the weight assigned. When the user is infinitely close to a reference station, the model value is naturally infinitely close to the ionospheric delay of the reference station itself, which can ensure that the model residual at the modeling station is 0. In addition, since a different weight matrix is implemented for different users, customized modeling for different users is also achieved. In a sense, the proposed customized model combines the advantages of both the polynomial model and the IDW method. A flowchart of the model is shown in Figure 2.

2.5. Ionosphere-Weighted Single-Frequency PPP-RTK Model for Terminal Positioning

The regional ionospheric model is mainly used for high-precision positioning of users. The terminal positioning performance can also reflect the quality of the model. The model is rank-deficient without the constraint of external ionosphere, especially for single-frequency users, so the impact of the ionosphere on positioning performance is more significant compared with dual-frequency/multi-frequency cases. The ionosphere-weighted (IW) single-frequency PPP-RTK model can be expressed as follows:

$$\left\{\begin{array}{lll}{P}_{u,1}^{ref}\hfill & =\hfill & {\rho }_u^{ref}+c\cdot {\overline{t}}_u-c\cdot t^{ref}+M_{u,\mathrm{w}}^{ref}\cdot T_{u,\mathrm{w}}+{\tilde{I}}_{u,1}^{ref}+e_{u,1}^{ref}\hfill \\ P_{u,1}^1\hfill & =\hfill & {\rho }_u^1+c\cdot {\overline{t}}_u-c\cdot t^1+M_{u,\mathrm{w}}^1\cdot T_{u,\mathrm{w}}+{\tilde{I}}_{u,1}^1+e_{u,1}^1\hfill \\ P_{u,1}^2\hfill & =\hfill & {\rho }_u^2+c\cdot {\overline{t}}_u-c\cdot t^2+M_{u,\mathrm{w}}^2\cdot T_{u,\mathrm{w}}+{\tilde{I}}_{u,1}^2+e_{u,1}^2\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill \\ P_{u,1}^s\hfill & =\hfill & {\rho }_u^s+c\cdot {\overline{t}}_u-c\cdot t^s+M_{u,\mathrm{w}}^s\cdot T_{u,\mathrm{w}}+{\tilde{I}}_{u,1}^s+e_{u,1}^s\hfill \\ L_{u,1}^{ref}\hfill & =\hfill & {\rho }_u^{ref}+c\cdot {\overline{t}}_u-c\cdot t^{ref}+M_{u,\mathrm{w}}^{ref}\cdot T_{u,\mathrm{w}}-{\tilde{I}}_{u,1}^{ref}+{\lambda }_1\cdot {\overline{N}}_{u,1}^{ref}+{\epsilon }_{u,1}^{ref}\hfill \\ L_{u,1}^1\hfill & =\hfill & {\rho }_u^1+c\cdot {\overline{t}}_u-c\cdot t^1+M_{u,\mathrm{w}}^1\cdot T_{u,\mathrm{w}}-{\tilde{I}}_{u,1}^1+{\lambda }_1\cdot {\overline{N}}_{u,1}^1+{\epsilon }_{u,1}^1\hfill \\ L_{u,1}^2\hfill & =\hfill & {\rho }_u^2+c\cdot {\overline{t}}_u-c\cdot t^2+M_{u,\mathrm{w}}^2\cdot T_{u,\mathrm{w}}-{\tilde{I}}_{u,1}^2+{\lambda }_1\cdot {\overline{N}}_{u,1}^2+{\epsilon }_{u,1}^2\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill \\ L_{u,1}^s\hfill & =\hfill & {\rho }_u^s+c\cdot {\overline{t}}_u-c\cdot t^s+M_{u,\mathrm{w}}^s\cdot T_{u,\mathrm{w}}-{\tilde{I}}_{u,1}^s+{\lambda }_1\cdot {\overline{N}}_{u,1}^s+{\epsilon }_{u,1}^s\hfill \\ 0\hfill & =\hfill & {\tilde{I}}_{u,1}^{ref}\hfill \\ \nabla {\overline{I}}_{u,1}^{1,ref}\hfill & =\hfill & {\tilde{I}}_{u,1}^1\hfill \\ \nabla {\overline{I}}_{u,1}^{2,ref}\hfill & =\hfill & {\tilde{I}}_{u,1}^2\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \nabla {\overline{I}}_{u,1}^{m,ref}\hfill & =\hfill & {\tilde{I}}_{u,1}^m\hfill \end{array}\right.$$

(13)

where the subscript $u$ represents the user; $m$ represents the number of satellites with ionospheric constraints; $s$ represents the number of observed satellites, and generally $m\le s$; $\nabla {\overline{I}}_{u,1}^{m,ref}$ represents the pseudo-observation of the ionosphere, and the constraint strength is usually related to the accuracy of ionospheric modeling; and ${\overline{t}}_u$ and ${\overline{N}}_{u,1}^s$ are the re-parameterized receiver clock bias and ambiguity, respectively.

It should be noted that since the ionospheric constraint adopts the form of BSSD and the value of the reference satellite is set to 0, a benchmark related to the ionosphere of the reference satellite is introduced, and the estimated parameters ${\overline{t}}_u$ and ${\overline{N}}_{u,1}^s$ are also affected by this benchmark.

$${\overline{t}}_u=t_u+{\tilde{I}}_{u,1}^{ref}$$

(14)

$${\overline{N}}_{u,1}^s=N_{u,1}^s-2{\displaystyle \frac{{\tilde{I}}_{u,1}^{ref}}{{\lambda }_1}}$$

(15)

In multi-epoch data processing, when the benchmark of ionospheric enhancement changes, a strategy similar to reference satellite handover in differential GNSS positioning needs to be adopted, that is, to synchronously adjust parameters such as receiver clock bias, ionosphere, and ambiguity to ensure the continuity of parameters between adjacent epochs.

3. Results

In order to verify the performance of crowdsourcing PPP-RTK and the weighted ionospheric model, a sparse reference station network from the Australian Regional GNSS Network (ARGN) on 2024 DOY167, as shown in Figure 3, was used for experiments. The stations are distributed across four network cells, with an average inter-station distance of about 391 km. The numbers of base stations, crowdsourcing stations, and user stations are 6, 12, and 31, respectively. All station supports received GPS/BDS/Galileo multi-frequency data with a sample rate of 30 s. For data processing, the precise products provided by the Center for Orbit Determination in Europe (CODE) are adopted, and the frequency-related bias required for multi-frequency PPP, such as inter-frequency clock bias (IFCB) and FCB, are estimated through global monitoring stations. For slant ionospheric delay extraction, both multi-frequency stepwise ambiguity fixing and partial ambiguity fixing are used with a threshold of 2. During the ionospheric modeling, the satellite with the highest elevation of each system is selected as the reference to eliminate the receiver DCB. Based on the first-, second-, and third-order polynomial models, the model performance is evaluated for each of the three strategies—namely no-weighted (NW), first-order IDW (IDW1), and second-order IDW (IDW2)—and a total of 9 schemes are carried out, as shown in

Table 1. Description of different experimental schemes.

#	Scheme	Remark
1	POLY11-NW	1st order polynomial model with No-Weighted strategy
2	POLY22-NW	2nd order polynomial model with No-Weighted strategy
3	POLY33-NW	3rd order polynomial model with No-Weighted strategy
4	POLY11-IDW1	1st order polynomial model with first-order IDW strategy
5	POLY22-IDW1	2nd order polynomial model with first-order IDW strategy
6	POLY33-IDW1	3rd order polynomial model with first-order IDW strategy
7	POLY11-IDW2	1st order polynomial model with second-order IDW strategy
8	POLY22-IDW2	2nd order polynomial model with second- IDW strategy
9	POLY33-IDW2	3rd order polynomial model with second-IDW strategy

. Based on these schemes, the subsequent analysis was carried out from three perspectives. First, the impact of crowdsourcing users on ionospheric modeling was analyzed; then, the performance of the customized weighted ionospheric model was evaluated; finally, the single-frequency PPP-RTK positioning verification was carried out on the terminal.

3.1. Impact of Crowdsourcing Users on Ionospheric Modeling

3.1.1. Impact of Crowdsourcing User Distribution

The impact of crowdsourcing users on the reference station network structure is analyzed as shown in Figure 4 and Figure 5. One and three crowdsourcing users are introduced into four network cells, respectively. By comparing the network structure before and after, it can be seen that the crowdsourcing users cause changes in the topological structure of the surrounding stations. When crowdsourcing users are located at the edge of the whole reference network, they mainly affect the structure of the network cell where they are located and have little impact on other network cells, such as when the crowdsourcing user is located in network cell 3. By contrast, when the user is located near the middle of the reference network, the structures of both the user’s own network cell and other adjacent network cells are affected, such as when the crowdsourcing user is located in network cell 4.

Based on the crowdsourcing user distribution in Figure 4 and Figure 5, taking the model POLY22-IDW2 as an example, Figure 6 presents the corresponding ionospheric modeling results. When a crowdsourcing user is introduced into a certain network cell, the station density of this network cell increases, and the ionospheric modeling accuracy within its coverage area is improved most significantly, while for the remaining network cells, the performance improvement is limited and mainly related to the degree of change in their topological structures. In addition, it can be preliminarily seen from the figure that for certain stations, the performance improvement with three crowdsourcing users is better than that with only one crowdsourcing user.

For network cell 3, regardless of whether it contains one or three crowdsourcing users, only its own topological structure changes, while those of the other three network cells do not change. Therefore, it can also be seen from Figure 6 that only the accuracy within the coverage area of network cell 3 is significantly improved, while no significant improvement is observed in the other three network cells.

For network cell 4, since it is located in the middle of the entire reference station network, except for network cell 3, the topological structures of the other three network cells are all affected with the addition of crowdsourcing users. Correspondingly, the ionospheric modeling accuracy within the coverage of these three network cells shows different degrees of improvement, while that of network cell 3 remains almost unchanged, as shown in Figure 6.

Therefore, from the perspective of the modeling performance of the entire reference station network, crowdsourcing sites should be distributed as evenly as possible across the network rather than concentrated in a small area.

3.1.2. Impact of Crowdsourcing User Number

From the perspective of the entire network, four and twelve crowdsourcing users are evenly selected across the entire network, and the impact of the number of crowdsourcing users on the ionospheric modeling is analyzed. As shown in Figure 7, evenly distributed crowdsourcing users have an impact on the topological structure of the entire network. For the convenience of comparison, Figure 7 also shows the original network structure without crowdsourcing users. Taking three polynomial models as examples (i.e., POLY11-IDW2, POLY22-IDW2, and POLY33-IDW2), the corresponding ionospheric modeling results are shown in Figure 8. It should be noted that since the construction of a third-order polynomial model requires at least ten modeling stations, the redundancy of the original network, which includes only six reference stations and lacks crowdsourcing users, is insufficient; thus, there is no blue curve in the corresponding sub-figure of POLY33-IDW2.

It is particularly important to note that, since the POLY33-IDW2 model requires solving 10 model coefficients, its redundancy drops to zero in scenarios with only six reference stations and four crowdsourced users. Under such conditions, while the model coefficients can still be solved, the IDW strategy effectively becomes inoperative. This may lead to model distortion and poor alignment with actual conditions. Consequently, compared to POLY11-IDW2 and POLY22-IDW2 models, which exclude crowdsourced users, the accuracy at stations such as CNDO and MTBU deteriorated.

The results in the figure show that, with the enhancement of crowdsourcing users, the ionospheric modeling accuracy of users across the entire network has been improved to varying degrees, and that the more crowdsourcing users there are, the higher the modeling accuracy. With the enhancement of four and twelve crowdsourced users, the accuracy of the POLY11-IDW2 model increased from 4.9 cm to 3.4 cm and 2.7 cm, respectively; for the POLY22-IDW2 model, it increased from 4.9 cm to 3.2 cm and 2.4 cm, respectively. For the POLY33-IDW2 model, as the number of crowdsourcing users increased from four to twelve, the accuracy also improved from 4.2 cm to 2.5 cm.

Furthermore, Figure 9 shows the average accuracy of all nine schemes listed in Table 1. With the enhancement of four and twelve crowdsourcing users, the average accuracy increased from 5.1 cm to 3.9 cm and 3.0 cm, with an improvement of 23.2% and 41.2%, respectively.

3.2. Performance of Customized Weighted Ionospheric Model

In the scenario of 12 crowdsourcing users, taking the MENO station as an example, Figure 10 shows the ionospheric modeling errors of all satellites observed. It can be seen intuitively that the modeling error with the NW strategy is the largest, reaching almost 30 cm during certain periods when the ionosphere is active. By contrast, whether the IDW1 or the IDW2 strategy is adopted, the ionospheric modeling error is greatly reduced, and the fluctuation amplitude can be reduced to several centimeters. The main reason is that in conventional NW models, only the spatial distribution of the ionosphere at the reference stations is considered, while user-related factors are neglected. By adjusting the weights of the reference stations through user-related IDW1 and IDW2 strategies, the model achieves better alignment at the user end and demonstrates improved performance. In addition, for the POLY11 and POLY22 models, the error of the IDW2 strategy is generally smaller than that of IDW1, while this difference between the two strategies for the POLY33 model is less pronounced, likely due to its higher polynomial order.

Specifically, E03, G07, and C39 of different GNSSs are selected for analysis, and the results are shown in Figure 11. A similar conclusion can be drawn from the figure; that is, the ionospheric error of the NW strategy fluctuates greatly, and the error can be greatly reduced through the weighted strategies (i.e., IDW1 and IDW2).

In order to more intuitively compare the modeling performance, the spatial distribution of C41 under different strategies is shown in Figure 12. Objectively speaking, the spatial distribution of the ionosphere is irregular. The conventional models with an NW strategy only consider the geometric spatial relationship of the modeling stations, without taking into account this irregular physical characteristic. By contrast, the proposed model with the IDW1 and IDW2 strategies can better characterize this irregular physical characteristic. Although the shortcomings of conventional NW strategy can be compensated by adopting a high-order model (for example, from first-order to third-order), the effect is still not as good as the weighted strategies.

Taking the POLY11 model as an example, the ionosphere of POLY11-NW changes linearly in space, which is obviously inconsistent with the actual situation. By contrast, the results of PLOY11-IDW1 and POLY11-IDW2 are more realistic, and the inhomogeneity of the spatial distribution of the ionosphere is better described. As for the POLY22 and POLY33 models, there is a similar phenomenon; that is, the spatial irregularity of the ionosphere can be better restored by customized weighted models.

The modeling performance of all users across the entire network is evaluated, as shown in Figure 13. It should be noted that for the two scenarios of the POLY22 model without crowdsourcing users and the POLY33 model with four crowdsourcing users, since there is no redundant information during the model’s construction, the results of the three strategies of NW, IDW1, and IDW2 overlap. As can be seen from the figure, regardless of whether there are crowdsourcing users or not, and regardless of which polynomial model is used, the accuracy of the weighted strategy is always better than that of the NW strategy overall. The specific accuracy statistics are listed in

Table 2. Ionospheric modeling accuracy with different numbers of crowdsourcing users and different weighting strategies.

Crowdsourcing User Number	Model	RMS [cm]			Improve [%]
		NW	IDW1	IDW2	IDW1-NW	IDW2-NW
0	POLY11	5.9	5.1	4.9	11.9	15.6
	POLY22	4.9	4.9	4.9	-	-
	POLY33	-	-	-	-	-
4	POLY11	5.0	3.9	3.4	23.5	32.0
	POLY22	3.9	3.4	3.2	14.8	18.3
	POLY33	4.2	4.2	4.2	-	-
12	POLY11	4.5	3.3	2.7	28.0	41.2
	POLY22	3.5	2.7	2.4	22.3	30.2
	POLY33	3.0	2.6	2.5	15.2	18.3

. In scenarios with different numbers of crowdsourcing users corresponding to polynomial models of different orders, the accuracy of IDW1 and IDW2 is improved by approximately 12~28% and 15~41%, respectively, compared with the NW strategy.

3.3. Single-Frequency PPP-RTK Performance Verification

The ionospheric delay is the main error source affecting the performance of single-frequency GNSS users. The simulated single-frequency PPP-RTK is further carried out to verify the above-mentioned regional ionospheric model, and then the results of both the float and fixed solutions are analyzed. For the float solution, its convergence speed and positioning accuracy are analyzed, and for the fixed solution, the epoch fix rate is analyzed. The convergence time refers to the time required for the positioning error to reach a specific threshold and remain below that threshold in subsequent epochs. In this paper, the thresholds for horizontal and vertical directions are set to 10 cm and 15 cm, respectively. The epoch fix rate is defined as the ratio of the number of fixed epochs to the total number of epochs. During data processing, the process noise of the ionosphere is 0.04² m²/s, the data from 2:00 to 22:00 were divided into 2 h intervals, and a total of 310 arc segments of data from all 31 users were processed. The RMS values given in Table 2 are used as a reference to determine the constraint strength of the ionosphere.

3.3.1. Float Solution Performance Verification

Taking the POLY22-IDW2 model as an example, Figure 14 shows the float solution results in scenarios with different numbers of crowdsourcing users. It can be seen that the participation of crowdsourcing users can accelerate the convergence speed of float solution in both horizontal and vertical directions. With the enhancement of four and twelve crowdsourcing users, the time required for the horizontal accuracy to converge to 10 cm was shortened from 43.5 min to 25.5 min and 16.0 min, with a reduction of 41.4% and 63.2% respectively; the time required for the vertical accuracy to converge to 15 cm was shortened from 43.5 min to 25.5 min and 18.5 min, with a reduction of 41.4% and 57.5% respectively.

The customized weighted model is also evaluated, and the comparison with the conventional model using NW strategy is given in Figure 15. The convergence speed of the float solution was significantly accelerated after adopting the IDW2 strategy. The time required for horizontal and vertical accuracy to converge to 10 cm and 15 cm was shortened from 26.0 min and 30.5 min to 16.0 min and 18.5 min, respectively, with a reduction of 38.5% and 39.3%, respectively.

Table 3. The float solution accuracy of different models enhanced by different numbers of crowdsourcing users.

Model	Crowdsourcing User Number	Accuracy [cm]
		Horizontal	Vertical
POLY22-IDW2	0	4.5	7.9
POLY22-IDW2	4	2.9	4.8
POLY22-IDW2	12	2.5	3.7
POLY22-NW	12	3.5	5.7

shows the positioning accuracy statistics of the different schemes. After introducing four and twelve crowdsourcing users, the horizontal and vertical accuracy are improved from (4.5 cm, 7.9 cm) to (2.9 cm, 4.8 cm) and (2.5 cm, 3.7 cm), and the horizontal and vertical accuracy are improved by about 35~45% and 39~53%, respectively. In addition, compared with the (3.5 cm, 5.7 cm) accuracy of the conventional model using the NW strategy, the customized model with the IDW2 strategy has higher accuracy.

In general, both crowdsourcing user enhancement and the customized weighting strategy can accelerate the convergence of the float solution and improve positioning accuracy to varying degrees.

3.3.2. Fixed Solution Performance Verification

To ensure the reliability of ambiguity fixing, only satellites with more than 20 consecutive tracking epochs participate in ambiguity resolution. Figure 16 and Figure 17 show the impact of crowdsourcing users and weighted strategies on the fixed solution, respectively. It can be seen that with the introduction of crowdsourcing users and customized weighting strategies, the overall convergence speed and accuracy over the entire period are significantly improved, especially once the ambiguity is successfully fixed, the accuracy has been significantly improved, and the horizontal and vertical accuracy can instantly reach about 5 cm and 10 cm. After introducing four and twelve crowdsourcing users, the average epoch fix rate increased from 65.1% to 86.4% and 90.1%, respectively. Similarly, after adopting the user-customized strategies, such as IDW2, the epoch fix rate also increased from 81.6% to 90.1%.

4. Discussion

In conventional GNSS augmentation mode, the platform and the user only act as the generator and utilizer of the augmentation information, respectively. By contrast, the crowdsourcing mode makes full use of the information collected by large numbers of users to update and calibrate the regional atmospheric map over time, without increasing the cost of infrastructure construction, thereby achieving high-spatial-resolution atmospheric modeling.

However, in practical applications, taking into account the individual differences of various terminals, there are still some issues to be solved. Crowdsourcing users have a wide variety of devices and are easily affected by factors such as the environment and network communication. For example, signals in urban canyons may be affected by severe multipath and NLOS, resulting in low data quality that cannot be comparable to that of reference stations. In this case, the fusion of data of different qualities is worth further study. In addition, the effect of crowdsourcing enhancement is usually related to the number of crowdsourcing users. The more users there are, the better the improvement will be, and correspondingly, the heavier the computing load of the platform will be. In scenario with large numbers of users, how to select a sample of representative users to achieve a balance between performance improvement and computational load also needs to be studied.

As mentioned earlier, conventional regional ionospheric modeling typically constructs models based solely on reference station information without accounting for spatial distribution differences among various users, resulting in a common model for different users. By contrast, our contribution lies in introducing user position-dependent weighting factors into conventional polynomial models under the framework of crowdsourcing enhancement. This innovation enables the transition of regional ionospheric models from a generic model to customized models so as to better provide personalized services for different users. This research contributes to future PPP-RTK systems in delivering more refined and personalized services to diverse users.

Another important point to note is that, compared with generating corrections, accurately determining their uncertainty presents an even greater challenge and remains a critical topic in PPP-RTK. Although our experiments adopted a post-evaluation method to assess the correction accuracy, practical applications necessitate exploring novel and effective approaches. For instance, the potential advantages of crowdsourcing data in monitoring both uncertainty and integrity deserve further investigation.

5. Conclusions

This paper introduces the concept of crowdsourcing enhancement into PPP-RTK. While utilizing the ionospheric corrections provided by the platform, users also feed their own ionospheric information back to the platform, thereby adjusting and refining the regional ionospheric model with a high spatial resolution. In addition, based on the conventional polynomial model, weight factors related to the users’ positions are introduced, and a weighted ionospheric model customized for each user is proposed to better characterize the spatial irregularity of the ionosphere. Crowdsourcing enhancement and the customized weighting model mentioned above were validated based on a sparse reference station network with an average inter-station distance of approximately 391 km.

First, the impact of crowdsourcing enhancement on ionospheric modeling performance is analyzed. The introduction of crowdsourcing users increases regional station density, triggers changes in the topological structure of surrounding stations, and thus improves the modeling performance in nearby areas. However, for other areas where the topological structure is hardly affected, the impact of crowdsourcing users is negligible. Therefore, crowdsourcing users should be distributed as evenly as possible across the network.

Then, the performance of the proposed weighted ionospheric model is evaluated. Compared with the conventional no-weighted (NW) polynomial model, the proposed weighted model can more effectively describe the irregular physical characteristics of the ionosphere. In the scenarios with four and twelve crowdsourcing users involved, for polynomial models of different orders, the adoption of first-order and second-order IDW strategies can achieve improvements of approximately 12~28% and 15~41%, respectively.

Finally, the proposed model was verified with a single-frequency PPP-RTK. The convergence speed and positioning accuracy of the float solution were significantly improved after the introduction of crowdsourcing users and customized weighted strategies. Similar phenomena can also be seen in the fixed solution. Especially once the ambiguity is successfully fixed; the accuracy is greatly improved, and the horizontal and vertical accuracy can immediately reach about 5 cm and 10 cm. In addition, the epoch fix rate also increased by varying degrees.