Mitigating Integrity Risk in SBAS Positioning Using Enhanced IGG III Robust Estimation

Highlights

What are the main findings?

• The improved IGG III robust estimation algorithm significantly enhances the user-end positioning performance of SBAS. In various scenarios, it greatly improves the positioning accuracy in both the horizontal and vertical directions, reduces the integrity risk, and enhances the availability.
• This method effectively suppresses the influence of outliers in the observed data, avoids overly conservative protection level (PL) estimation, and obtains more reasonable and reliable protection level values without affecting real-time performance.

What is the implication of the main finding?

• This research offers a powerful and computationally efficient solution that can enhance the service reliability of SBAS in challenging environments such as urban dynamics and UAV maneuvering flight, making it highly suitable for safety-critical applications in aviation and intelligent transportation.
• This algorithm has strong adaptability and stability in multiple practical scenarios, providing an important reference for the performance optimization and integrity guarantee of future SBAS user-end under poor data quality conditions.

Abstract

To address the limitations in positioning accuracy and the risk of integrity degradation in Satellite-Based Augmentation Systems (SBAS) user-end after applying augmentation information, this study proposes a positioning algorithm integrating an improved IGG III robust estimation method. By using integrity information from SBAS, this method improves protection level calculations and better adjusts observed weights by adding new factors to the weight function model. This improvement allows for better discrimination between reliable and anomalous measurements, thereby enhancing positioning accuracy, reducing integrity risks, and improving availability. Experimental results show that, compared to conventional SBAS user positioning, the proposed method achieves notable performance improvements across various scenarios. In static environments, it reduces horizontal integrity risk by up to 6.7%, increases availability by up to 6.6%, and improves positioning accuracy by up to 71.3%. In urban vehicular environments, horizontal integrity risk is reduced by 0.5%, availability is increased by 0.5%, and accuracy improves by up to 58.7%. In Unmanned Aerial Vehicle flight scenarios, horizontal integrity risk is reduced by 2.8%, availability increases by 2.8%, and accuracy improves by up to 50.38%. In all scenarios, vertical integrity risk is completely eliminated and availability improves slightly. Additionally, compared to the conventional IGG III estimator, the improved method offers more effective control over weight adjustment during solution estimation, thereby avoiding excessive down-weighting and mitigating overbounding of protection levels. These results demonstrate the potential of the proposed method to improve the performance and reliability of SBAS user-end under both static and dynamic conditions.

1. Introduction

Satellite-Based Augmentation Systems (SBAS) have been widely implemented to enhance the accuracy, integrity, and availability of Global Navigation Satellite Systems (GNSS) positioning, particularly for safety-critical applications such as aviation and intelligent transportation [1]. Despite the provision of correction and integrity information, SBAS user-end may still experience degraded positioning performance in challenging environments [2], where observation quality is uneven and outliers are more frequent. In such cases, ensuring robust positioning and reliable Protection Level (PL) estimation becomes critical to maintain user trust and service availability. This motivates the development of improved positioning algorithms that can effectively handle observation anomalies while preserving integrity assurance.

At present, with the continuous development of SBAS technology, its user-end positioning method has become more mature and can show better positioning results and integrity performance when the data quality is better. However, in further expanding the application field of SBAS, the complexity of the environment has become more prominent, and the unstable quality of observation data has become more frequent, which leads to poor positioning results and integrity risks despite the use of enhancement information in messages to improve positioning accuracy. Therefore, the SBAS user-end positioning method still needs to be further improved in scenarios with more complex environments and poor data quality.

In recent years, there have been studies from the aspect of multi-system fusion and weighted model optimisation in SBAS user-end positioning. To improve the accuracy of Unmanned Aerial Vehicle (UAV) positioning, Krasuski et al. integrated the augmented data from multiple European Geostationary Navigation Overlay Service (EGNOS) satellites through an average weighted model to improve the accuracy of positioning [3]. Meanwhile, a proposed weighting algorithm based on dual GNSS receivers optimises the weighting model by taking the number of satellite tracking as a weighting factor, which effectively improves the positioning accuracy of System for Differential Correction and Monitoring (SDCM) and EGNOS in air navigation [4]. Thereafter, in order to better apply SBAS to UAV, a positioning accuracy improvement algorithm based on the joint solution of multiple augmentation systems is proposed, which integrates the ionospheric correction data provided by multiple SBAS through the average weighting model to enhance the positioning accuracy of UAV [5]. SBAS in urban environments where Geostationary Earth Orbit (GEO) visibility is poor and SBAS signals are obscured; Yoon et al. proposed online transmission of SBAS correction information through the communication link of the ground control station, which achieves the improvement of positioning accuracy in complex environments [6]. Liang Li et al. used SBAS enhancement parameters for real-time single-frequency Precise Point Positioning (PPP), which significantly improved the positioning accuracy compared to the traditional SBAS method [7]. Park et al. proposed an efficient multi-GNSS and multi-SBAS positioning method by independently calculating the positioning results of each GNSS and each SBAS combination, and then determining the final position by weighted summing, which ultimately improves the positioning accuracy [8]. To address the problem that single-frequency SBAS is affected by ionospheric storms and scintillation, Li et al. proposed a new dual-frequency user positioning algorithm, which improves the positioning accuracy and integrity [9]. Based on the concept of flexible Positioning Navigation Timing (PNT) [10], Zhang et al. proposed an integration framework without ground network, which achieves multi-system error correction by flexibly integrating B1C, B2a and broadcast ephemeris messages, and improves the positioning stability of BeiDou Satellite-Based Augmentation System (BDSBAS) in complex environments [11]. To address the problem that the overlapping areas between different SBAS systems cannot be effectively fused, Wang et al. proposed a fusion positioning algorithm for BDSBAS and Multi-functional Satellite Augmentation System (MSAS) based on fully connected neural networks, which can provide more stable and high-precision augmented navigation services [12]. Wang et al. proposed a novel service weighting model for road transport SBAS Dual-Frequency Multi-Constellation (DFMC), which integrates the Carrier-to-Noise Ratio (CNR) and the smoothing time factor to dynamically adjust the weights of the observed data, which effectively reduces the protection level and improves the availability [13]. Zheng et al. proposed two improved integrity monitoring approaches: one based on covariance analysis and another based on pseudorange residuals combined with satellite-station geometry [14,15]. These methods significantly enhanced system integrity and usability. Wang et al. employed a Kalman filter to amplify the covariance matrix of observation noise, thereby reducing the observation weight when the number of stations decreased. This adjustment improved both the accuracy and continuity of the system [16]. Shao et al. introduced a Dual-Frequency Range Error (DFRE) estimation technique based on the projection method, which establishes an envelope for the maximum correction error with a specified probability [17]. In addition, a dual-frequency SBAS integrity optimization algorithm and an improved over-limit pairing method have been developed [18,19]. These approaches enhance system availability by optimizing weight assignments and modeling error characteristics more precisely. More recently, with the rapid increase in Low Earth Orbit (LEO) satellites and their global coverage, new opportunities have emerged. Xin et al. improved monitoring performance and service coverage by optimizing the observation link, effectively mitigating the impact of satellite entry and exit, and substantially enhancing integrity performance [20]. In summary, there are fewer current improved SBAS user-end positioning methods, and most of them still use traditional user-end positioning methods. In addition, the existing weighted model optimisation has unspecified positioning performance in more complex environments with poorer data quality, or the weighted model is complex and dependent on smoothing time. And although multi-system fusion can improve the positioning performance of SBAS users, it cannot reduce the effect of coarseness in the observations when the data quality is poor, and it is not applicable to single-system users, which has certain limitations.

To address these limitations, this study introduces an improved robust estimation algorithm into the SBAS user positioning process, based on the IGG III framework. The proposed method is designed with consideration of both the weighting behavior of the user algorithm and the PL computation. Through extensive experiments in static, urban vehicular, and UAV flight scenarios, the method is shown to significantly reduce integrity risk, improve availability, and enhance positioning accuracy compared to traditional approaches. The results also demonstrate that the improved estimator avoids overbounding by maintaining controlled weight variation compared to the traditional IGG III robust estimation, thereby providing a more balanced and effective solution for SBAS user-end positioning.

The remainder of this paper is organized as follows. Method presents the improved IGG III-based robust estimation algorithm, including the design of the weight function and its integration with PL computation. The results describe the experimental setup and test scenarios, covering static, urban, and UAV environments and further discusses the experimental results, focusing on positioning accuracy, integrity risk, and availability performance. Finally, we conclude the study and outline potential directions for future work.

2. Method

2.1. IGG III Variance Expansion Function

The IGG III variance expansion function divides the observed data into three categories according to quality standards: effective information, available information and harmful information. Based on the principle of robust estimation, differentiates information of different quality, that is, effective information retains the original weight value, the available information is estimated with a down-weighting, and the harmful information is given a zero-weighting [21].

The IGG III weight function can be expressed as follows [22]:

$${\overline{P}}_i=\left\{\begin{array}{c}{P}_i \left|{\tilde{v}}_i\right|\le k_0\\ P_i{\displaystyle \frac{k_0}{\left|{\tilde{v}}_i\right|}}{\left({\displaystyle \frac{k_1-\left|{\tilde{v}}_i\right|}{k_1-k_0}}\right)}^2 k_0<\left|{\tilde{v}}_i\right|\le k_1 \\ 0 \left|{\tilde{v}}_i\right|>k_1\end{array}\right.$$

(1)

where $P_i$ is the weight of the i-th satellite, $k_0$ and $k_1$ are harmonic coefficients that partition the standardized residual $\left|{\tilde{v}}_i\right|$ into three intervals, corresponding, respectively, to effective information, available information, and harmful information. In practical applications, $k_0$ is typically set in the range of 1.0 to 2.0, $k_1$ is chosen between 2.5 and 3.0 and ${\tilde{v}}_i$ is the standardised residuals.

It can be seen from the weight function that the selection of harmonic coefficients directly influences whether observation weights are appropriately reassigned, thereby enabling the effective utilization of reliable observations, the cautious handling of suspicious data, and the exclusion of outliers. Assuming that the observation residuals $v_i$ follow a Gaussian distribution, the corresponding test statistic can be expressed as follows:

$${\tilde{v}}_i={\displaystyle \frac{v_i-\mu }{{\sigma }_0\sqrt{q_{v_i}}}}~N\left(\mathrm{0,1}\right)$$

(2)

where $q_{v_i}$ denotes the weight of the i-th observation, ${\sigma }_0$ is the standard deviation of the residuals, and $\mu$ is the mean of the residuals from all observations. Based on the statistical principle of three-sigma $\sigma$ outlier detection, the IGG III weight function is applied with harmonic coefficients set as $k_1$ = 3$\sigma$ and $k_0$ = 2$\sigma$, where $\sigma$ is the standard deviation of the residuals. Under this configuration, the standardized residuals are divided into three intervals: outliers ($-\infty$, −3$\sigma$)∪(3$\sigma$, $+\infty$), accounting for approximately 0.3% of the data; suspicious values (−3$\sigma$, −2$\sigma$)∪(2$\sigma$, 3$\sigma$), accounting for about 4.3%; and valid values (−2$\sigma$, 2$\sigma$), covering roughly 95.4% of the total. This classification effectively distinguishes between outliers, questionable data, and reliable observations, enhancing the robustness of the estimation process.

2.2. SBAS User-End Positioning and Protection Level Calculation

The SBAS user-end consists primarily of two components: the augmented positioning solution and the PL. The augmented positioning solution utilizes GNSS observation data, broadcast ephemerides, and SBAS correction messages to estimate the user position and receiver clock offset using a least squares estimation approach. The PL computation relies on the integrity information provided by SBAS. If PL exceeds a predefined Alarm Limit (AL), an alert is triggered to prevent potential hazards during navigation. The relationship among PE, PL, and AL, along with the corresponding system service status, is summarized in

Table 1. Relationship among PE, PL, and AL.

Relationship	Service State	Affect
PE ≤ PL ≤ AL	Available	None
PL < PE ≤ AL	Misleading Information (MI)	Integrity risk
AL < PL < $\mathrm{P}E$	MI	Integrity risk
PL $<A$L < $\mathrm{P}E$	Hazardous MI (HMI)	Integrity risk (missing detections)
PE $<A$L < $\mathrm{P}\mathrm{L}$	Unavailable	Alarm

. The integrity risk rate refers to the probability that the location information error provided by the navigation system exceeds the specified tolerance range within a given service time, but the system fails to issue an alarm to the user. Availability rate is the ability of a navigation system to provide the required functionality and performance within the expected service time, usually expressed in percentage.

(1)

Enhanced positioning solution

SBAS user can improve satellite orbit and satellite clock accuracy using satellite-based augmentation corrections. The pseudorange single-point positioning observation equation (dual-frequency ionosphere-free combination) using the satellite-based enhancement corrections can be expressed as follows [23]:

$$P_j^i\left(IF\right)={\rho }_j^i+c\ast \left(∆t_j-∆t^i\right)+T_j^i+{\epsilon }_j^i+{cor}^i$$

(3)

where ${\rho }_j^i$ is the geometric distance from the station to the satellite, $c$ is speed of light, $∆t_j$ and $∆t^i$ are the receiver and satellite clock offsets, respectively, $T_j^i$ is tropospheric slant delay, ${\epsilon }_j^i$ is the pseudorange noise, and ${cor}^i$ is the correction using SBAS corrections.

The specific process is as follows: first, the unit direction vector between the receiver and each satellite is computed; then, the satellite orbit corrections are projected onto the line-of-sight vector from the receiver to the satellite; finally, these are combined with the satellite clock corrections to form the total range correction. Unlike conventional single-point positioning, SBAS user positioning incorporates integrity information to determine the variance of the spatial signal error, which is used in the weighting of observations.

$$P=\left[\begin{array}{ccc}{\displaystyle \frac{1}{{\sigma }_1^2}}& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & {\displaystyle \frac{1}{{\sigma }_n^2}}\end{array}\right]$$

(4)

where ${\sigma }_i^2$ = ${\sigma }_{i,DFC}^2+{\sigma }_{i,trop}^2+{\sigma }_{i,air\_DF}^2+{\sigma }_{i,iono}^2$, where ${\sigma }_{i,DFC}^2$ is the variance of the spatial signal error (specifically, the variance of the orbital clock corrections mapped in the line-of-sight direction), ${\sigma }_{i,trop}^2$ is the variance of tropospheric residual error, ${\sigma }_{i,air\_DF}^2$ is the variance of the dual-frequency measurement noise, multipath, and antenna group delay errors, and ${\sigma }_{i,iono}^2$ is the variance of the residuals in the ionosphere from the dual-frequency ionosphere combination, and they can be expressed as

$$\left\{\begin{array}{c}{\sigma }_{i,DFC}^2={({\sigma }_{DFRE} \ast {\delta }_{DFRE})}^2+{\epsilon }_{corr}^2+{\epsilon }_{er}^2\\ {\sigma }_{i,trop}^2=0.12 \ast {\displaystyle \frac{1.001}{\sqrt{0.002001+\sin {\left(Ele\right)}_{i,deg}^2}}}\\ {\sigma }_{i,air\_DF}^2=0.36+0.4 \ast {\mathrm{e}}^{-{\displaystyle \frac{{Ele}_i}{{14}^{°}}}}\\ {\sigma }_{i,iono}^2={\displaystyle \frac{40.0}{261.0+{\left({ele}_{i,deg}\right)}^2}}+0.018\end{array}\right.$$

(5)

where ${\sigma }_{DFRE}$ is the sigma value of Dual Frequency Range Error (DFRE), which can be retrieved from the correspondence table of DFRE and Dual Frequency Range Error Indicator (DFREI) [24], ${\epsilon }_{corr}^2$ and ${\epsilon }_{er}^2$ are both degradation parameters, and ${\delta }_{DFRE}$ can be computed from the user positional information and the covariance array information of the satellite.

(2)

Protection level solution

The calculation of the PL must balance both integrity and availability requirements. As such, the PL should neither be overly conservative (loose) nor overly optimistic (tight). The PL computation formula adopted in this study is given as follows [25]:

$$\left\{\begin{array}{c}HPL=K_H \ast d_{major}\\ VPL=K_V \ast d_U\end{array}\right.$$

(6)

where $K_H$ and $K_V$ are constants. For non-precision approach, $K_H$ = 6.18 and no PL is solved in vertical direction. For precision approach, $K_H$ = 6.0 and $K_V$ = 5.33 [26]; $d_{major}$ and $d_U$ are obtained from the elements in the covariance matrix $invG$, and they can be expressed as

$$\left\{\begin{array}{c}{d}_{major}=\sqrt{{\displaystyle \frac{d_{east}^2+d_{north}^2}{2}}+\sqrt{{\left({\displaystyle \frac{d_{east}^2-d_{north}^2}{2}}\right)}^2+d_{EN}^2}}\\ d_{east}^2=invG\left(\mathrm{1,1}\right)\\ d_{north}^2=invG\left(\mathrm{2,2}\right)\\ d_U=invG\left(\mathrm{3,3}\right)\end{array}\right.$$

(7)

The covariance matrix can be expressed as follows:

$$invG={\left(G^{\mathrm{T}}WG\right)}^{-1}$$

(8)

where $G$ is the spatial geometric matrix and $W$ is the weight matrix for position solving as follows [27]:

$$G=\left[\begin{array}{cc}-\cos \left({Ele}^1\right) \ast \sin \left({Azl}^1\right)& \begin{array}{ccc}-\cos \left({Ele}^1\right) \ast \cos \left({Azl}^1\right)& -\sin \left({Ele}^1\right)& 1\end{array}\\ \begin{array}{c}⋮\\ -\cos \left({Ele}^n\right) \ast \sin \left({Azl}^n\right)\end{array}& \begin{array}{ccc}\begin{array}{c}⋮\\ -\cos \left({Ele}^n\right) \ast \cos \left({Azl}^n\right)\end{array}& \begin{array}{c}⋮\\ -\sin \left({Ele}^n\right)\end{array}& \begin{array}{c}⋮\\ 1\end{array}\end{array}\end{array}\right]$$

(9)

$$W=\left[\begin{array}{ccc}{\displaystyle \frac{1}{{\sigma }_{1,DFC}^2+{\sigma }_{1,trop}^2+{\sigma }_{1,air\_DF}^2+{\sigma }_{1,iono}^2}}& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & {\displaystyle \frac{1}{{\sigma }_{n,DFC}^2+{\sigma }_{n,trop}^2+{\sigma }_{n,air\_DF}^2+{\sigma }_{n,iono}^2}}\end{array}\right]$$

(10)

where $Ele$ and $Azl$ are the elevation angle and azimuth angle of the satellite relative to the user, n is the number of satellites.

2.3. Improved SBAS User Positioning Algorithm Combined with IGG III Robust Estimation

In SBAS user positioning, the observation weight is generally composed of three parts: spatial signal influence, signal propagation process and receiver-related effects. In addition to position estimation, the PL must also be calculated using the same observation weight applied during positioning. When the standardized residual exceeds the threshold, although the traditional IGG III method can reduce the weight of unreliable data and improve positioning accuracy by increasing the observation variance, it adjusts the above three parts at the same time, which has the risk of excessive weight changes, which may lead to the calculated PL overbounding issue. Therefore, directly applying the traditional IGG III robust estimation method is not suitable for SBAS user positioning. To solve this problem, this study uses the spatial signal error variance accurately calculated from integrity information to divide the total variance into two parts: (1) the spatial signal influence, expressed as ${\sigma }_{i,DFC}^2$; (2) signal propagation, and the comprehensive influence of the receiver-related factors is expressed as ${\sigma }_{i,trop}^2+{\sigma }_{i,air\_DF}^2+{\sigma }_{i,iono}^2$. The former has higher accuracy, while the latter is mainly related to the height angle. In robust estimation, the two components are adjusted according to the variance of spatial signal error.

To facilitate the selection of appropriate harmonic coefficient values for robust estimation, it is necessary to further analyze the distribution characteristics of the spatial signal error variance, ${\sigma }_{i,DFC}^2$. In this study, statistical analysis was performed on the values of ${\sigma }_{i,DFC}^2$ for all GPS satellites on Day of Year (DOY) 159 in 2023. The Cumulative Distribution Function (CDF) of these values is presented in Figure 1. It shows the probability that a random variable takes a value less than or equal to a certain value. The horizontal axis represents the value range of the random variable, and the vertical axis represents the cumulative probability, that is, the probability that the value of the random variable is less than or equal to the corresponding value on the horizontal axis. It can be observed that in 90% of the cases, ${\sigma }_{i,DFC}^2$ remains below 0.25, with only a small proportion of values exceeding this threshold. The rapid rise in the curve indicates that ${\sigma }_{i,DFC}^2$ is highly concentrated near a specific value, which further highlights the characteristics of ${\sigma }_{i,DFC}^2$ distribution: most of the values are closely clustered in a concentrated area, while only a few values are scattered far away from the concentrated area.

Meanwhile, ${\sigma }_{i,DFC}^2$ reflects the correction accuracy of the satellite orbit and clock information. When the correction accuracy is low, the value of ${\sigma }_{i,DFC}^2$ becomes larger, indicating that the residual ephemeris errors have a more significant impact on user positioning accuracy. Therefore, when the pseudorange residual satisfies the robust estimation condition, the magnitude of ${\sigma }_{i,DFC}^2$ can be used to selectively adjust the observation variance, thereby controlling the extent of deweighting. This approach helps to avoid excessive suppression of valid measurements. Based on this idea, the improved weight function for SBAS user positioning is formulated as follows:

$${\overline{P}}_i=\left\{\begin{array}{c}{P}_i \left|{\tilde{v}}_i\right|\le k_0\\ {\displaystyle \frac{1}{\raisebox{1ex}{${\sigma }_{i,DFC}^2$}\!\left/ \!\raisebox{-1ex}{$\frac{k_0}{\left|{\tilde{v}}_i\right|}{\left(\frac{k_1-\left|{\tilde{v}}_i\right|}{k_1-k_0}\right)}^2$}\right.+{\sigma }_{i,trop}^2+{\sigma }_{i,air\_DF}^2+{\sigma }_{i,iono}^2}}{k}_0<\left|{\tilde{v}}_i\right|\le k_1,{\tilde{\sigma }}_{i,DFC}^2\ge k_2 \\ {\displaystyle \frac{1}{{\sigma }_{i,DFC}^2+\raisebox{1ex}{${(\sigma }_{i,trop}^2+{\sigma }_{i,air\_DF}^2+{\sigma }_{i,iono}^2)$}\!\left/ \!\raisebox{-1ex}{$\frac{k_0}{\left|{\tilde{v}}_i\right|}{\left(\frac{k_1-\left|{\tilde{v}}_i\right|}{k_1-k_0}\right)}^2$}\right.}}{k}_0<\left|{\tilde{v}}_i\right|\le k_1,{\tilde{\sigma }}_{i,DFC}^2<k_2 \\ \\ 0 k_1<\left|{\tilde{v}}_i\right|\end{array}\right.$$

(11)

For ease of representation, the spatial signal error variance ${\sigma }_{i,DFC}^2$ is standardized in this paper and denoted as ${\tilde{\sigma }}_{i,DFC}^2$. The threshold for normalization is defined as twice the median value of the error. The harmonic coefficients $k_0$ and $k_1$ are retained as in the traditional IGG III formulation, where $k_0$ is typically chosen between 1.0 and 2.0, and $k_1$ is set between 2.5 and 3.0. Here, ${\tilde{v}}_i$ denotes the standardized residual, $P_i$ is the weight of the i-th satellite. ${\sigma }_{i,trop}^2$, ${\sigma }_{i,air\_DF}^2$ and ${\sigma }_{i,iono}^2$ have explained in Equation (4). In addition, it can be seen from the calculation formula of the protection level in Equations (6)–(10) that the covariance matrix of the positioning robust estimation output will be used to calculate the protection level. Therefore, the improved IGG III robust estimation also has corresponding optimization effects on the protection level calculation while finely adjusting the weight.

The flowchart of the improved robust algorithm for SBAS user positioning is shown in Figure 2. First, an initial positioning solution is obtained, and convergence is checked. If the solution has not yet converged, the pseudorange residuals are computed and standardized, while the spatial signal error variances are also statistically analyzed and standardized. Next, a decision is made on whether to apply down-weighting based on whether the standardized residual exceeds a predefined threshold. If down-weighting is required, the observation variance is selectively adjusted according to the magnitude of the standardized spatial signal error variance. The updated weights are then used in the augmented positioning solution to improve positioning accuracy. If no weight adjustment is needed, the original weights are retained. After convergence is reached through iterative updates, the PL is computed accordingly.

3. Experiment and Result Analysis

3.1. Processing Strategy and Data Quality Analysis

This study conducts experimental analysis through three scenarios: static, urban environment, and UAV maneuvering flight to verify the effectiveness of the algorithm in terms of accuracy, availability, integrity, and PL. Among them, the static experimental data selects the DOY 159–165 in 2023 actual measured data of three monitoring stations in China (green) CSH2, FUZ2, and KAS2. The sampling rate is 1 s. There are 21 enhanced stations (red). They are used to calculate the correction information of SBAS and serve the user-end positioning. The station distribution is shown in Figure 3.

To verify that the corrective values employed in this study are neither anomalous nor in violation of SBAS end-user requirements, data from the enhanced stations illustrated in Figure 3 were used to compute the corrections. The methodology and procedural flow are outlined in Figure 4, and the key steps are summarized as follows: First, dual-frequency ionosphere-free combinations of pseudorange observations were generated to eliminate the majority of ionospheric delays. The combined observations were subsequently smoothed, after which the effects of satellite clock errors and tropospheric delays were removed, yielding pseudorange residuals that primarily contained errors from satellite orbit, satellite clock, and receiver clock. Following clock synchronization, the receiver clock error was eliminated. Finally, the post-synchronization residuals were utilized to estimate the satellite orbit and clock corrections. It should be emphasized that the corrections derived in this study are directly integrated into the user positioning algorithm rather than being broadcast. In most practical applications, such corrections are typically transmitted to users through data messages; however, this delivery mechanism does not compromise the accuracy of the estimated parameters.

Figure 5 counts the pseudorange residuals before and after enhancement of all GPS satellites over a total of 159–165 days, and uses the pseudorange residuals before and after enhancement as the evaluation criteria. The specific calculation is as follows: first calculate the average value of the pseudorange residuals before enhancement from all enhancement stations for a given satellite, and then calculate the average value of the pseudorange residuals after enhancement using the correction number, repeating the process until all satellites are evaluated. Compared with the values before enhancement, the pseudorange residuals after enhancement for all satellites are reduced to varying degrees, and the values of the pseudorange residuals after enhancement fall between 0.1 m and 0.3 m, indicating that the corrections used in this study are not abnormal and the accuracy is in line with the current level.

To demonstrate that the improved IGG III robust estimation method proposed in this study does not introduce substantial computational overhead for SBAS users, Figure 6 presents a comparison of the computation delays of SBAS-enhanced positioning with and without the improved algorithm. The results indicate that the average delays are 28.3 ms and 32.69 ms, respectively. The overall difference is minimal, corresponding to an average increase of 4.4 ms. Such an increment is negligible relative to current hardware capabilities and therefore does not compromise real-time applicability. Moreover, when deployed on high-performance servers, the delay difference is expected to be further reduced.

The vehicle dataset in the urban environment is collected using a passenger car driving through city streets. The data acquisition lasts for approximately one hour, with a sampling interval of 1 s. The collection environment and trajectory are illustrated in Figure 7. The UAV maneuvering flight dataset is obtained from a flight conducted in a relatively open environment, lasting around 20 min, also with a sampling rate of 1 s. It is worth noting that only GPS data are used in this experiment, with no additional onboard sensors. Therefore, the data collection environment is relatively unobstructed and does not include areas with severe signal occlusion. The experimental equipment in the two scenarios is shown in Figure 8.

A comparison of the positioning strategies and solutions for dual-frequency SBAS user is summarized in

Table 2. Solution and enhanced positioning strategy.

Item	Solution 1	Solution 2
Signal frequency	GPS L1&L2	GPS L1&L2
Orbital correction	Broadcast ephemeris + DFMC SBAS	Broadcast ephemeris + DFMC SBAS
Clock correction	Broadcast ephemeris + DFMC SBAS	Broadcast ephemeris + DFMC SBAS
Ionospheric correction	Dual-frequency combination	Dual-frequency combination
Tropospheric correction	UNB3m	UNB3m
Elevation cut-off	10°	10°
Robust	No	Yes

. Solution 1 represents enhanced positioning without the application of robust estimation, while Solution 2 incorporates the improved robust algorithm into the positioning process. In both cases, the PL is calculated using the observation weights derived from their respective enhanced positioning strategies.

In order to show that the experimental data quality in the three scenarios meets the premise of poor data quality, the GPS (L1 and L2) observation data in the three scenarios are analyzed in terms of Signal-to-Noise Ratio (SNR), multipath error, and Position Dilution of Precision (PDOP) value. This study uses a box plot to count different values. The upper and lower boundaries of the box represent the first quartile (Q1) and the third quartile (Q3), respectively. A line in the box represents the median (Q2) of the data. The values beyond the whiskers are considered outliers, which are values other than 2.7 times the median value of the error. The data quality results are illustrated in Figure 9. Experimental data are affected to varying degrees by environmental factors, which cause outliers to appear in different indicators. In the static scenario, CSH2, KAS2, and FUZ2 stations show large outliers in PDOP values and multipath errors. Although there are few outliers in the SNR (L2), the dispersion is relatively high. In the on-board scenario of urban environments, there are large outliers in the SNR (L1), PDOP value, and multipath error, among which the outliers in the SNR are the most significant. In the drone maneuvering flight scenario, outliers appear in the SNR, PDOP value, and multipath error, and the range is very large in the SNR (L1). In addition, this study also counts the average values of each indicator under different scenarios in

Table 3. Average of data quality analysis indicators in different scenarios.

Station ID	SNR (L1)/dB-Hz	SNR (L2)/dB-Hz	Multipath Error (L1)/m	Multipath Error (L2)/m	PDOP/m
CSH2	42.16	37.25	0.17	0.14	1.98
KAS2	43.29	36.34	0.35	0.25	2.44
FUZ2	42.42	36.47	0.16	0.14	2.2
CAR	44.15	41.00	0.13	0.17	2.51
FLY	41.59	38.83	0.17	0.31	4.13

. It can be seen that except for some outliers, the overall quality of observation data in each scenario can meet the positioning requirements, and the overall quality of observation data in the drone maneuvering flight scenario is the worst. In summary, the data quality in the three scenarios is poor, and many outliers appear, which eventually have a great impact on the positioning results and integrity. This can be used as experimental data in this paper to effectively verify the effectiveness of the algorithm.

3.2. Static Scenario

The three stations used two methods of solution 1 and solution 2 for positioning and protection level calculation, and compared with the APV-I standard (HAL is 40 m, VAL is 50 m), the percentage of availability improvement and the percentage of integrity risk reduction in solution 2 is DOY 159-165 compared with solution 1, as shown in Figure 10. The CSH2, KAS2 and FUZ2 stations used the solution 2 method to reduce the percentage of horizontal integrity risk by 6.5, 11.15 and 0.66, respectively, and the percentage of horizontal availability by 6.5, 11.12 and 0.66, respectively, and the percentage of vertical integrity risk by 5.18, 2.45 and 0.28, respectively, and the percentage of vertical availability by 5.08, 2.09 and 0.18, respectively. It shows that in static scenarios, adding an improved robust algorithm can effectively reduce the risk of integrity and improve availability.

Taking the annual accumulation date of 159 as an example, we will analyze the performance of solution 2 in three aspects: integrity, positioning accuracy and protection level. In this study, Stanford diagrams were used to calculate its integrity performance according to the APV-I standard, as shown in Figure 11, Figure 12 and Figure 13. The horizontal axis is the positioning error, the vertical axis is the protection level, and the red line is the alarm limit. The figure is divided into 5 sub-graphs according to the size relationship between them, representing different navigation states (Table 1). The area in the upper left corner of the figure indicates that the system is unavailable when the position error exceeds the protection level. The area in the lower left corner indicates that the positioning information provided by the system is available within the protection level. The orange area in the lower right corner indicates that the position error is below the protection level, but the positioning information provided by the system may mislead the user. The red area in the lower right corner indicates that the position error exceeds the protection level. The positioning information provided by the system will not only mislead the user, but may also pose danger.

Solution 1: Although the three stations use SBAS correction information, the positioning error (PE) still large, and there are still occurring MI and HMI. Solution 2: The PE of the three stations is significantly improved, reducing the percentage of MI and HMI, and the availability rate is also improved. Details are outlined in

Table 4. Statistics of integrity monitoring performance of three stations.

ID	Solution	Horizontal				Vertical
		Availability	HMI	MI	Availability	HMI	MI
CSH2	Solution 1	99.07%	0.001%	0.93%	98.38%	0.007%	0.27%
	Solution 2	100%	0%	0%	98.55%	0%	0%
FUZ2	Solution 1	98.82%	0.003%	0.86%	98.13%	0.02%	0.38%
	Solution 2	99.66%	0%	0.002%	98.27%	0%	0%
KAS2	Solution 1	89.49%	0.001%	10.44%	95.91%	0.002%	0.48%
	Solution 2	96.12%	0%	3.72%	96.06%	0%	0%

. Specifically, in the horizontal direction, MI rates dropped from 0.931%, 0.863%, and 10.441% to 0%, 0.002%, and 3.72%—decreasing by approximately 0.93%, 0.85%, and 6.72%, respectively—while HMI was eliminated entirely. Availability correspondingly rose from 99.07%, 98.82%, and 89.49% to 100%, 99.66%, and 96.12%, an improvement of 0.93%, 0.84%, and 6.62%. Vertically, MI and HMI were completely eliminated, and availability increased modestly from 98.38% to 98.55%, from 98.13% to 98.27%, and from 95.91% to 96.06%, gains of 0.17%, 0.14%, and 0.15%. The results show that the improved robust algorithm can further reduce the missing detections rate and integrity risk rate while weakening the impact of the coarse errors, so as to achieve effective alarms, and to improve the availability rate.

Through the presence of outliers in the data quality indicators, it can be observed that under the influence of the surrounding environment, the quality of certain observations is significantly degraded, leading to substantial errors. These inaccuracies are likely attributable to factors such as receiver quality, short-term adverse atmospheric conditions, signal reflections from buildings in urban settings, and temporary unfavorable satellite geometric configurations. In order to more clearly analyze the improvement effect of the improved robust algorithm on the positioning results, we also compared and analyzed with the M-estimators method, and the PE sequence diagrams of the horizontal and vertical directions of the three stations are drawn and their internal and external consistency accuracy are counted, as shown in Figure 14, Figure 15 and Figure 16 and

Table 5. Statistics of positioning errors of three stations.

Station ID	Solution 1	Horizontal		Vertical
		STD/m	RMSE/m	STD/m	RMSE/m
KAS2	Solution 1	1.22	2.73	1.87	1.99
	Solution 2	0.35	2.4	0.62	0.82
	M-estimators	0.85	2.64	1.43	1.61
FUZ2	Solution 1	0.52	1.13	1.28	1.48
	Solution 2	0.2	1.01	0.62	0.82
	M-estimators	0.29	1.04	0.89	1.07
CSH2	Solution 1	0.42	1.08	0.95	1.04
	Solution 2	0.14	0.99	0.3	0.43
	M-estimators	0.14	0.99	0.3	0.46

. The positioning results of Solution 2 are smoother than those of Solution 1, which effectively weakens the influence of coarse errors. As shown in Table 5, the Standard Deviation (STD) and Root Mean Square Error (RMSE) of horizontal and vertical directions of the three stations in Solution 2 are reduced compared with that of Solution 1. Among them, the horizontal and vertical internal consistency accuracy of the KAS2 station increased from 1.22 to 0.35 and from 1.87 to 0.62, respectively, which increased by 71.31% and 66.84%, respectively; the external consistency accuracy increased from 2.73 to 2.4, and from 1.99 to 0.82, respectively, which increased by 12.09% and 58.79%, respectively. The horizontal and vertical internal consistency accuracy of the FUZ2 station increased from 0.52 to 0.2 and from 1.28 to 0.62, respectively, which increased by 61.54% and 51.56%, respectively; the external consistency accuracy increased from 1.13 to 1.01 and from 1.48 to 0.82, which increased by 10.62% and 44.59%, respectively. The horizontal and vertical internal consistency accuracy of the CSH2 stations increased from 0.42 to 0.14 and from 0.95 to 0.3, respectively, which increased by 66.67% and 68.42%, respectively; the external consistency accuracy increased from 1.08 to 0.99 and from 1.04 to 0.43, which increased by 8.33% and 58.65%, respectively. The results show that when the data quality is poor and the PE is still large even with the addition of SBAS correction information, adding the improved robust algorithm to the SBAS user positioning can effectively weaken the effect of coarse errors and improve the positioning accuracy. The M-estimators method can also effectively reduce the impact of coarse errors, but compared with the improved IGG III robust estimation algorithm, the improvement effect is weaker. After the robust estimation, a very small number of elements still have large errors, and this study considers that there are two reasons: firstly, the number of satellites participating in the positioning of this element is relatively small, and no robust treatment; secondly, this element has too many satellites involved in robust estimation, which makes the robust estimation ineffective.

To verify that the proposed improved robust algorithm offers better control over weight adjustment in SBAS user positioning compared to the traditional IGG III method, and to prevent excessive conservatism in protection level (PL) calculation caused by robustness strategies, a comparative analysis was also conducted with the PL results derived from the M-estimators method. The protection levels calculated using the traditional IGG III method, the improved robust algorithm, and the M-estimators approach are presented in Figure 17, Figure 18 and Figure 19. The three stations use the Horizontal Protection Level (HPL) and Vertical Protection Level (VPL) of the improved robust algorithm in part of the epoch is smaller than the traditional IGG III robust algorithm, which indicates that the improved robust algorithm will not be too ‘loose’ on the PL solving under the premise of meeting the integrity and positioning accuracy. The PL calculation under the M-estimators method is the most conservative, further explaining the superiority of the improved algorithm. In other epochs, the PL difference between the traditional IGG III robust algorithm and the improved IGG III robust algorithm is not obvious, because when the enhancement information is better, the variance of the corrections is smaller compared with the variance of other errors, and at this time, the variance of the corrections has less influence in the deweighting, which makes the PL calculated by the two algorithms comparable.

3.3. Urban Environment Vehicle Scenario

The data in the urban environment are used to perform positioning and protection-level solutions using two methods: Solution 1 and Solution 2, respectively. The RTK result is used as the true value, and the APV-I standard is selected to count its integrity performance. The results are shown in Figure 20. In Solution 1, in the urban environment, which is susceptible to environmental interference, using SBAS correction information, the PE still exists in a large situation, and the MI and HMI occurs; in Solution 2, the horizontal and vertical PE is improved, the percentage of MI and HMI are reduced, and the availability rate is improved. The percentage of MI in the horizontal direction decreased from 0.6% to 0.08%, a decrease of 0.52%, and the availability increased from 99.39% to 99.91%, an increase of 0.52%. The vertical percentages of MI and HMI both decreased to 0%, and the availability increased from 98.73% to 98.79%, an increase of 0.06%. The results show that the improved robust algorithm can reduce the missing detections rate and integrity risk by weakening the effect of the coarse errors in the SBAS-enhanced urban environment vehicle scenario, achieve effective alarms, and improve the availability rate.

In order to more clearly analyze the improvement effect of the improved robust algorithm on the positioning results, the PE sequence and three-dimensional motion trajectory in the horizontal and vertical directions, which are illustrated in Figure 21. Compared with Solution 1, the positioning results of Solution 2 are more stable, eliminating larger error jumps. In addition, the STD and RMSE in Solution 2 are both lower than Solution 1. Significant improvements were observed across all accuracy metrics. The horizontal internal consistency accuracy improved by 23.28%, increasing from 0.73 to 0.56. The horizontal external consistency accuracy showed a 5.4% enhancement, rising from 1.48 to 1.4. In the vertical direction, the internal consistency accuracy increased by 58.77%, from 2.28 to 0.94, while the external consistency accuracy improved by 49.8%, from 2.63 to 1.32. The M-estimators method effectively improves positioning accuracy, but the improvement effect is not as good as Solution 2. Figure 22 further demonstrates the effect of the improved robust algorithm in the motion trajectory, with a smoother elevation direction. The results show that the use of improved robust algorithms in the positioning of the SBAS-enhanced urban environment vehicle scenario can effectively weaken the influence of coarse errors and improve positioning accuracy.

The PL results calculated by the positioning weights after robust estimation using the traditional IGG III robust algorithm, the improved robust algorithm and the M-estimators method are shown in Figure 23. In the SBAS-enhanced Urban environment vehicle scenario, compared with traditional IGG III robust algorithm, HPL and VPL using improved robust algorithms have been reduced in most epochs. The result show that the improved robust algorithm is better in the calculation of PL under the premise of integrity and positioning accuracy. The M-estimators method can envelop positioning errors, but the results are too conservative and will reduce usability.

3.4. UAV Maneuvering Flight Scenario

Using the data in the UAV maneuvering flight scenario to perform positioning and PL solutions using the two methods of Solution 1 and Solution 2, respectively, the RTK results are taken as the true values and the APV-I standard are chosen to count the integrity, and the result are shown in Figure 24. In Solution 1, using SBAS correction information, the PE is still large and there are MI and HMI. In Solution 2, the horizontal and vertical PE are improved, the percentage of MI and HMI are reduced, and the availability rate is improved. The horizontal percentage of MI is reduced by 2.76%, and the availability rate is improved by 2.76%; the vertical percentages of MI are reduced by 1.71%, and the availability rate is improved by 1.65%. The results show that the improved robust algorithm can reduce the missing detections rate and integrity risk by weakening the effect of the coarse errors in the SBAS-enhanced UAV maneuvering flight scenario, achieve effective alarms, and improve the availability rate.

The positioning error sequences in the horizontal and vertical directions are illustrated in Figure 25, respectively, showing in detail the improved effect of the improved robust algorithm and M-estimators methods on the positioning results. Compared with Solution 1, the positioning results of Solution 2 are smoother and the larger error jump points are eliminated; the positioning accuracy of the M-estimators method has been improved, but the improvement effect is not as good as that of Solution 2. In addition, the STD and RMSE of Solution 2 in the horizontal and vertical directions are reduced compared with that of Solution 1, of which the internal and external consistency accuracies in the horizontal direction can be improved by 47.1% and 16.45%, the internal and external consistency accuracies in the vertical direction can be improved by 50.38% and 38.15%, respectively. The results show that in the SBAS-enhanced UAV maneuvering flight scenario, the use of the improved robust algorithm in the user-end positioning can effectively weaken the influence of the coarse errors and improve the positioning accuracy.

The results of the PL calculations using the traditional IGG III robust algorithm, the improved robust algorithm and M-estimators method in the SBAS-enhanced UAV maneuvering flight scenario are shown in Figure 26. In SBAS-enhanced UAV maneuvering flight scenario, compared with traditional IGG III robust algorithm, HPL and VPL using improved robust algorithms are reduced most of epochs. The result show that, while ensuring integrity and positioning accuracy, the improved robust algorithm is better than the traditional IGG III robust algorithm when calculating the PL. In addition, the M-estimators method is too conservative in PL calculations, which will affect the availability of positioning users, further explaining the advantages of improved robust algorithms in PL calculations.

3.5. Statistical Test

This study conducted a statistical analysis (t-test) on the experimental results of the above three scenarios. The differences between the positioning errors and protection level results of Solution 1 and Solution 2 are shown in

Table 6. t-test of the experimental results in three scenarios.

Item	T	p	D
CAR-vertical-errors	10.509	<0.001	2.151
CAR-vertical-PL	18.525	<0.001	5.916
CAR-horizontal-errors	0.477	0.633	0.542
CAR-horizontal-PL	21.942	<0.001	0.551
FLY-vertical-errors	6.705	<0.001	2.466
FLY-vertical-PL	26.699	<0.001	0.931
FLY-horizontal-errors	2.084	<0.037	1.244
FLY-horizontal-PL	16.346	<0.001	2.083
FUZ2-vertical-errors	51.247	<0.001	1.145
FUZ2-vertical-PL	22.317	<0.001	1.920
FUZ2-horizontal-errors	5.282	<0.001	0.498
FUZ2-horizontal-PL	−1.860	0.063	0.689
KAS2-vertical-errors	22.743	<0.001	1.811
KAS2-vertical-PL	5.642	<0.001	1.552
KAS2-horizontal-errors	16.181	<0.001	1.18
KAS2-horizontal-PL	11.656	<0.001	1.087
CSH2-vertical-errors	35.629	<0.001	0.933
CSH2-vertical-PL	17.022	<0.001	1.56
CSH2-horizontal-errors	11.257	<0.001	0.408
CSH2-horizontal-PL	4.91	<0.001	0.454

(p > 0.05: Not Significant; p ≤ 0.05: Significant; p ≤ 0.01: Highly Significant; p ≤ 0.001: Extremely Significant).

. The t-test results show that, except for the horizontal positioning error results in the urban scenario, there are significant differences in the experimental results of all other scenarios, and the degree of effect is above moderate. The differences in the results of horizontal positioning errors in urban scenarios did not reach statistical significance, but a moderate effect was observed, indicating that there might be differences of practical significance.

4. Discussion

This study addresses the issues of degraded positioning performance, increased integrity risk, and higher rates of missed detection events under poor data quality conditions in SBAS user-end positioning by proposing an improved IGG III robust estimation positioning algorithm. Through validation using real-world data from three scenarios—static, urban vehicular, and UAV maneuvering flight—several noteworthy observations were made.

In terms of system performance, the improved robust algorithm significantly enhances the availability of SBAS and effectively reduces integrity risks and missed detection rates in both horizontal and vertical directions. This improvement markedly strengthens system stability in complex environments. The influence of coarse errors in observation data on positioning accuracy is also effectively suppressed by the improved robust estimation, leading to substantial improvements in both horizontal and vertical positioning accuracy. Particularly noteworthy is the outstanding performance of the improved algorithm in protection level computation, avoiding the common issue of overbounding seen in traditional methods and resulting in more reasonable and reliable protection level estimates.

Comparative analysis with the M-estimators method further demonstrates the unique advantages of the proposed improved IGG III robust estimation for SBAS user-end applications. It not only effectively addresses the limitations of traditional methods in suppressing coarse errors but also avoids potential overconservatism in protection level computation.

From the perspective of current research trends, previous improvements in SBAS performance have predominantly focused on the system end, such as data preprocessing, time synchronization, and model optimization. While these efforts have enhanced overall system performance, there has been relatively little attention paid to user-end performance under challenging conditions, especially when performance remains unsatisfactory even after enhancements. Although various methods exist to suppress coarse errors and improve positioning accuracy, few are well-suited for SBAS user-end applications. This study focuses on this practical issue and offers a new approach to addressing positioning problems at the user end in degraded environments.

It should be noted that this study has certain limitations regarding datasets. Due to constraints in data acquisition, the current research is validated only based on data from China. This geographical limitation may affect the assessment of the algorithm’s applicability in other regions, and future work will expand the study area to further verify the method’s universality.

5. Conclusions

This study demonstrates that the enhanced IGG III robust estimation-based SBAS user-end positioning algorithm effectively improves positioning performance under challenging environmental conditions. The proposed method not only preserves strong robustness but also leads to notable enhancements in system availability, positioning accuracy, and the reliability of protection level calculations. It offers a practical and efficient solution for mitigating positioning challenges in degraded operational environments. Future work will focus on extending the validation to broader geographical regions and more diverse scenarios to further assess the algorithm’s adaptability and robustness.