Yang Ling, School of Surveying, Mapping and Geoinformatics, Tongji University: Refinement of Stochastic Model for Autonomous Integrity Monitoring of GNSS Advanced Receiver | Journal of Surveying and Mapping, Vol. 53, No. 2, 2024

The content of this article comes from the Journal of Surveying and Mapping, Issue 2, 2024 (drawing review number: GS Jing (2024) No. 0297)

The stochastic model of GNSS advanced receiver autonomous integrity monitoring was refined by Yang Ling

, ZHU Jincheng, SUN Nan, YU Yangkang, SHEN Yunzhong, LI Bofeng School of Surveying, Mapping and Geoinformatics, Tongji University, Shanghai 200092 Foundation of China: National Natural Science Foundation of China (42274030) Abstract: In many application fields involving life safety, the user device of GNSS must have the ability to monitor the integrity. Advanced Receiver Autonomous Integrity Monitoring (ARAIM) is the latest advancement in integrity monitoring technology in civil aviation, and is expected to expand to multiple application areas. However, the receiver noise term in the ARAIM algorithm usually uses the constant coefficient altitude angle model recommended by the Aeronautical Radio Technical Committee (RTCA), which can only reflect the noise characteristics of GNSS receivers that meet the specific standards of civil aviation in the civil aviation flight environment. In many application scenarios, the receiver software, hardware and application environment usually do not meet the RAW standard specifications, and the differences are significant, and it is difficult to ensure the effectiveness of the ANALOG algorithm if the solidified constant coefficient random model is still used. To this end, this paper uses Least Squares Variance Component Estimation (LS-VCE) to adaptively construct the ANALOG receiver noise term stochastic model to expand the scope of application of the algorithm, and takes the spaceborne GNSS observation data of GRACE-FO (GRACE Follow-On) as an example to verify the effectiveness. The experimental results show that the refined stochastic model can improve the effectiveness of decoupling fault detection and elimination on the one hand, and effectively reduce the horizontal and vertical protection levels on the other hand, and improve the availability of the integrity monitoring system. Keywords: ARAIM stochastic model, least squares variance component estimation, integrity monitoring, protection level

Citation format: Yang Ling, Zhu Jincheng, Sun Nan, et al. Refinement of Stochastic Model for Autonomous Integrity Monitoring of GNSS Advanced Receiver[J]. Journal of Surveying and Mapping,2024,53(2):286-295. DOI: 10.11947/j.AGCS.2024.20210669

YANG Ling, ZHU Jincheng, SUN Nan, et al. Stochastic model refinement of GNSS advanced receiver autonomous integrity monitoring[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(2): 286-295. DOI: 10.11947/j.AGCS.2024.20210669 阅读全文：http://xb.chinasmp.com/article/2024/1001-1595/20240208.htm

Introduction With the expansion of GNSS in applications such as civil aviation, autonomous driving, and smart cities, its integrity monitoring has gradually attracted the attention of scholars at home and abroad. GNSS integrity monitoring assesses the credibility of the information provided by GNSS navigation and positioning systems, including the ability to warn users when the system is unusable for navigation [1-2]. GNSS integrity monitoring technology originated in the field of civil aviation, and at present, there are three standardized enhancement systems that comply with the International Civil Aviation Organization (ICAO) standards to provide GNSS system integrity services. The three systems are: ground-based augmentation system [3] (GBAS), satellite-based augmentation system (SBAS) [4] and airborne based augmentation system (ABAS) [5]. Receiver autonomous integrity monitoring (RAIM) in ABAS is a classic technology. RAIM relies solely on GNSS systems for positioning, utilizing redundant GNSS observations for integrity monitoring. Its range of services includes the en-route and terminal area flight phases for navigation, as well as the approach phases for horizontal navigation. With the continuous improvement of GNSS infrastructure and the continuous development of user needs, the Federal Aviation Administration (FAA) of the United States has proposed advanced receiver autonomous integrity monitoring (ARAIM) as a technical upgrade of traditional RAIM to meet the needs of vertical navigation performance LPV-200. This technology makes full use of the multi-frequency and multi-constellation characteristics of current GNSS systems, and combines the integrity support message (ISM) provided by ground monitoring stations to provide services [6-8]. GNSS integrity monitoring relies on accurate functional and stochastic models. The accurate construction of stochastic models is more difficult than that of functional models, because it is difficult to accurately determine the variance-covariance matrix of observations to reasonably describe the stochastic characteristics of observations [9-10]. ARAIM's stochastic model usually consists of three parts, including the satellite side (satellite ephemeris and clock error), the signal propagation side (ionospheric and tropospheric delay), and the receiver side (code noise and multipath) noise terms. Among them, the stochastic model of the receiver-end noise term usually uses the constant coefficient altitude angle model recommended by the RTCA in the minimum operational performance standards (MOPS), which can only reflect the noise characteristics of GNSS receivers that meet the specific standards of civil aviation in the civil aviation flight environment [11]. However, in different application scenarios, the receiver software, hardware and application environment usually do not conform to the ANALOG standard specifications, and there are significant diversity, and it is difficult to ensure the effectiveness of the ANALOG algorithm if the solidified constant coefficient stochastic model is still used. Therefore, if the ARAIM algorithm is to be extended to other applications, it is necessary to study the adaptive refinement method of the stochastic model. When there is enough prior observation information, variance component estimation (VCE) can be used to construct a stochastic model that is more consistent with the noise characteristics of the observations. There are many methods for estimating variance components, including minimum norm quadratic unbiased estimator (MINQUE) and least-squares variance component estimation (LS-VCE) [10, 12-15]. Among them, LS-VCE unifies the function model and the stochastic model through the least squares estimation criterion, which is simple, flexible, and practical [12, 16]. Based on LS-VCE, this paper proposes a method for refining the stochastic model of the noise term of the ANALOG receiver. The reliability test of the stochastic model was used to determine the accuracy of the dataset and results required for the refinement of the stochastic model. The effects of ANALOG before and after stochastic model refinement are compared, and the necessity of stochastic model refinement to improve the integrity monitoring ability of ANALOG under different applications is further verified. The on-board GNSS observations from GRACE-FO C from December 2, 2019 to December 17, 2019 were used for verification. The GRACE (Gravity Recovery and Climate Experiment) series of satellites was jointly launched by the National Aeronautics and Space Administration (NASA) and the German Aerospace Center (DLR) to study the Earth's water resources by accurately measuring the Earth's gravitational field. One of the low-orbit satellites of geology and climate [17]. The test data were divided into post-processing part (first 12 days) and forecast part (last 4 days). The general process of the experiment is as follows: (1) Using the data of the post-processing part, the receiver term random model is refined based on LS-VCE, and the altitude angle model of the GRACE-FO spaceborne GNSS receiver is obtained, and the appropriate fitting dataset is selected through the reliability test. (2) The ANALOG algorithm was verified by using the stochastic model before and after refinement to verify the fitting dataset and the forecast dataset, and the effect of ANALOG before and after the stochastic model refinement was evaluated. 1 Principle 1.1 The basic principle of least squares variance component estimation is as follows: the linear variance-covariance component model is as follows

(1) Wherein

(2) where, E(·) and D(·) x is the parameter to be estimated, Qyy is the variance-covariance matrix of the observed values, Q0 is the known part of the variance-covariance component matrix, σk(k=1, 2, ..., p) is the matrix element in Qyy, Qk is the positioning matrix of the variance-covariance component, and the function is to determine the position of the variance to be calculated in the matrix. The parameter estimates are obtained by the weighted least squares criterion, and the corrections for the observations are

(3) From the law of error propagation, the mathematical expectation and variance-covariance matrix of the parameter estimation and correction numbers are

(4) In the formula,

，因为E(v)=0，D(v)=E(vvT)=RTQyyR，将式(1)代入式(4)可得

(5) In the equation, the matrix in the expectation operator on the left is a symmetric square matrix, and the matrix equation is composed of r2 scalar observation equations, r is the number of redundant observations, and the semi-vectorized operator vh(·) is taken on both sides of the equation , the linear model of LS-VCE can be obtained as

(6)式中，yvh=vh(vvT-RTQ0R)； Avh=[vh(RTQ1R), vh(RTQ2R), …, vh(RTQpR)]；σ=[σ1, σ2, …, σp]T。 式(6)的加权最小二乘估值为

(7) where N=AvhTWvhAvh;l=AvhTWvhyvh. In the LS-VCE theory, there are different ways to take the weight matrix Wvh, and this paper will use one of the special weight matrices [18]

(8)式中，D为复制矩阵，作用是将半向量算子转化为向量算子；Wy为观测值的权矩阵Qyy-1；⊗为克罗内克积。 由向量、半向量化算子，克罗内克积和矩阵的迹的性质，vh(S)=D+ vec(S)，tr(UVST)=vec(T)T(ST⊗U)vec(VT)可知，此时N、l矩阵中的元素为

(9)

(10), where k and l are the number of rows and columns where the matrix elements are located, respectively, can be obtained to obtain the variance-covariance component. 1.2 The accuracy of the stochastic model is the key to the refinement and reliability testing of the stochastic model, which affects the availability and effectiveness of integrity monitoring. The core of fault detection and elimination in integrity monitoring is to construct test statistics that conform to the actual situation, and these test statistics are very sensitive to random models, so the construction of accurate random models is the prerequisite for effective fault detection and elimination [9, 19]. In addition, in order to evaluate the effectiveness and rationality of stochastic model refinement, it is necessary to test the reliability of the results of stochastic model refinement [20]. In general, the stochastic model of ARAIM is represented as follows [11, 21]

(11) where σURA2 is the variance of the satellite ephemeris and clock error error, σtropo2 is the variance of the residual tropospheric delay error, and σuser2 is the variance of the receiver noise. Since the ionospheric first-order term is eliminated by using a non-ionosphere combination, the ionospheric term is ignored in the stochastic model. The satellite-side error σURA2, including the standard deviation σorbit and the standard deviation σclk of the satellite orbit, is expressed as

(12) Since civil aviation ANALOG uses a broadcast ephemeris [22], it is necessary to redefine its stochastic model when applying precision ephemeris, Table 1 shows the accuracy of various orbit and clock products officially announced by IGS [23], if a certain type of product is used, the standard deviation of the satellite side can be obtained from Table 1.

表 1 IGS精密产品服务的数据质量Tab. 1 Product quality of IGS products

Table options

For the tropospheric delay term σtropo2, the standard deviation of the model-corrected residual tropospheric delay error is modeled as in the ARAIM user benchmark algorithm

(13) where σZPD is the standard deviation of the residual tropospheric error in the zenith direction, El is the satellite altitude angle, and the fractional part is the projection function [24]. Since the orbital altitude of the GRACE-FO satellite is about 500 km [17], which is above the troposphere, the tropospheric influence can be ignored. For the receiver noise term σuser2, for civil aviation flight applications, the constant coefficient altitude angle model is usually used in ARAIM

(14) where fL1 and fL2 are the frequencies of the L1 and L2 bands, respectively, σMP is the multipath error, and σnoise is the receiver noise error, which is in the form of

(15) However, in localization applications in different environments, the height angle model must be redefined to improve the ability to monitor anomalous observations. In this paper, the variance obtained by LS-VCE is used to fit the coefficients a and b in the height angle model of Eq. (16) [25] to obtain a refined receiver-end stochastic model

(16) The reliability of the stochastic model can be assessed by testing the degree to which the sample set matches the preset probability density function [20]. In this paper, the statistic T based on the positional parameter solution vector and its estimated variance-covariance matrix is constructed

(17) In the formula,

Satellite coordinates from the GNV file officially published by the GRACE-FO satellite are used. The statistic T theoretically obeys a chi-square distribution with 3 degrees of freedom (χ2(3)). Therefore, a histogram of the probability distribution of the statistic T can be plotted to evaluate its compatibility with the probability density function of χ2(3). In order to quantitatively analyze the degree of conformity, the index IND is calculated by equation (18).

(18) where hbinCount is the number of groups in the probability distribution histogram, ycentor, i represents the probability of each group in the probability distribution histogram, and ychi2, i represents the probability density function value of χ2(3) corresponding to the midpoint of each group. The smaller the ind indicator, the more consistent the metric T is.

1.3 Integrity monitoring

1.3.1 Fault detection and troubleshootingFault detection and troubleshooting is one of the important tasks of integrity monitoring [26], and in the ARAIM user benchmark algorithm, the method of fault detection and troubleshooting adopts the method of decoupling. The core idea of solution separation is to perform fault tolerance subset positioning and solution by eliminating faulty satellites, and then perform a continuity risk test on the difference between the fault-tolerant subset solution and the fault-free full set solution. The specific algorithm steps are as follows: (1) Fault-free full set and fault-tolerant subset positioning. The parameter estimate of the localization solution obtained by the least squares criterion is

(19) where m is the index of the fault subset, when m=0, the solution result is the trouble-free full set positioning solution, the design matrix A is a Jacobian matrix, the matrix size is Nsat×(3+Nconst), Nsat is the number of satellites participating in the positioning solution, Nconst is the number of constellations participating in the positioning solution, y is the residual of the pseudorange, and the weight matrix W is defined as follows

(20) That is, the weight of the faulty satellite in the mth fault subset is set to zero, and D is the variance covariance matrix of the stochastic model. (2) Dissociation test. Use the location solution for each fault-tolerant subset

Locate the solution with a trouble-free full set

After doing the difference, the decoupling test quantity is obtained

(21) and calculate its variance

(22) where q∈{1, 2, 3} represent the components of the three directions, and eq is a one-dimensional vector, where the q element is 1 and the rest are 0. By dissociating the variance of the test quantity, the test threshold can be calculated

(23) where Kfa, q is derived from the continuity risk probability, and each fault subset is assigned a continuity risk probability, and the continuity risk probability of these subsets is assigned to three directions, which is defined as

(24) In Eq. (24), PFA_HOR and PFA_VERT are the horizontal and vertical continuity risk probabilities defined in the ANALOG user benchmark algorithm, with magnitudes of 9×10-8 and 3.9×10-6, respectively [11]; Nfaultmodes is the number of failure modes;Q(·) is a standard normal distribution probability density function. The following describes how to test whether there is an anomaly in a fault-tolerant subset

(25) The distance test quantity composed of the positioning solution of each fault-tolerant subset and the positioning solution of the fault-free full set needs to be tested in three directions and the tests in all three directions are passed, before the fault subset can be considered to be fault-free, otherwise the fault subset has failed, and the corresponding faulty satellite needs to be eliminated.

1.3.2 Protection level calculationIn practical applications, the true positioning error is not known, so the protection level needs to be calculated to measure the integrity risk of exceeding the alarm limit. The alarm limit refers to the upper bound of the true positioning error allowed when the system does not send an alarm, and the integrity risk refers to the probability that the true positioning error exceeds the alarm limit [5]. Protection levels are usually decomposed into horizontal and vertical protection levels in the location domain and calculated separately. where the horizontal protection level represents the boundary of the horizontal position error for a given integrity risk, and similarly, the vertical protection level represents the boundary of the vertical position error, which is calculated as follows

(26)

In Eq. (27), PHMI_HOR (VERT) and ADJ are a priori horizontal (vertical) integrity risk, the value of which is given by the specification of the specific application field. In the ARAIM user benchmark algorithm, the horizontal integrity risk is typically 10-8 and the vertical integrity risk is typically 9×10-8 [11]. q=1,2 is the horizontal component subscript identification, q=3 represents the vertical component subscript identification, PLq is the protection level component, bq(m) is the projection of the satellite nominal deviation in the positioning domain, σq(m) is the standard deviation of the positioning solution, Pfault, m is the probability of m-order failure, PHMI is the total integrity risk, PHMIHOR(VERT) is the integrity risk assigned in the horizontal (vertical) direction, Psat not monitored and Pconst not monitored are the probability of unmonitored risk for satellites and constellations, respectively. The vertical and horizontal protection levels are denoted separately as

(28)

(29) The unknown parameter in Eq. (26) is the protection component PLq to be solved, which can be solved by numerical calculation dichotomy. The integrity monitoring system will be declared unavailable when the horizontal protection level of the current epoch is greater than the horizontal alarm limit or the vertical protection level is greater than the vertical alarm limit, misleading information (MI) events occur when the system declares it available and the positioning error exceeds the protection level but not the alarm limit, and hazardously occurs when the system declares it available and the positioning error exceeds the alarm limit misleading information (HMI) events. 2 ExperimentIn this paper, the measured data of GRACE-FO C from December 2, 2019 to December 17, 2019 (including the observation data of the spaceborne GPS receiver and the data of the precision orbit of the GRACE-FO satellite) were selected, and the precise orbit and clock error of the GNSS satellite were calculated by using the post-event precision ephemeris and clock error products given by the IGS analysis center, and the dual-frequency ionospheric non-ionospheric combination was used to eliminate the influence of the primary term of ionospheric delay and ignore the influence of tropospheric delay. The test data were divided into post-processing part (first 12 days) and forecast part (last 4 days). Since the effect of tropospheric delay is ignored, σtropo2=0 in Eq. (11). Q0 in Eq. (2) is the error of the satellite side, that is, σURA2 is taken as the known part of the LS-VCE, and the variance of the error term at the receiver side is estimated. Since the variance component estimation uses redundant observational information to estimate the variance, the data of the precision orbit of the GRACE-FO satellite in the post-processing part can be substituted into the equation as a reference value to reduce the unknown parameters and improve the accuracy of the variance component estimation (the above practice does not affect the real-time monitoring of the integrity of the forecast part of the data). The variance less than or equal to 0 generated by the misestimation was eliminated, and then the standard deviation and the height angle were fitted by the height angle model, and the epochs involved in the fitting met the criterion that the positioning errors in the three directions were less than 3 standard deviations, and finally the coefficient value of the height angle model was obtained. Figure 1 shows the fitting of the height angle model coefficients according to Eq. (16), with the number of days on the horizontal axis representing the results obtained from the data processed after the first i days, where i∈{1, 2, ..., 12}. It can be found that the coefficients a and b show roughly the same change law, and the change from the first 1 day to the first 7 days fluctuates greatly, and tends to be stable after the first 8 days. This is due to the fact that the fitting effect is unstable due to the small amount of data, and the fitting coefficient tends to be stable when the amount of data is large. Table 2 lists the specific values of coefficients a and b. In order to determine the appropriate sample size of the fitting data, the stochastic model reliability test was carried out on the above stochastic model refinement results, and the statistical results are shown in Table 2. As can be seen from Table 2, the ind index of the first 10 days is the smallest, indicating that the statistic T is more in line with the chi-square distribution with 3 degrees of freedom. Therefore, the post-processing data of the first 10 days were selected as the fitting dataset and the corresponding height angle model coefficients were used.

图 1 高度角模型拟合系数Fig. 1 LS-VCE fitting curve

Diagram options

表 2 高度角模型系数值及ind值Tab. 2 Coefficients of elevation-dependent model and ind

Table options

Figure 2 shows the results of the stochastic model reliability test for the fitted dataset (Fig. 2(a), Fig. 2(c)) vs. the forecast dataset (Fig. 2(b), Fig. 2(d)). where the column is the probability distribution histogram of the statistic T, and the red curve is the probability density function curve of the chi-square distribution with a degree of freedom of 3. Figures 2(a)—2(b) show that the distribution of the bars and the preset probability density function curve are significantly biased before the stochastic model is refined, with the ind indicators of the statistic T being 8.25 (fitted dataset) and 7.77 (predicted dataset). In this case, the probability density function of the statistic T and χ2(3) is less matched. Fig. 2(c)—Fig. 2(d), the probability density function matching degree with χ2(3) is significantly improved after the LS-VCE refined stochastic model, and its ind indexes are 1.43 (fitting dataset) and 1.66 (forecasting dataset), respectively. From Eq. (17), it can be seen that the statistic T is affected by the positioning error and its matrix of variance. Before the refinement of the stochastic model, the statistic T is distributed on the left side, indicating that the definition of the variance covariance matrix before refinement is too conservative, and the refined stochastic model more accurately reflects the observation noise level, and the fitting dataset and the prediction dataset have such a trend, which verifies the effectiveness of the stochastic model refinement.

图 2 随机模型精化前后统计量T的概率分布Fig. 2 Probability distribution of T before and after stochastic model refinement

Diagram options

Figure 3 shows the Stamford plot of the horizontal protection level and the horizontal positioning error before and after the refinement of the stochastic model. Stanford diagrams are used to interpret and illustrate most integrity events and their relationships, as well as to evaluate positioning system performance. In a Stanford diagram, the positioning error and protection level of each epoch are plotted as a point in the diagram, the horizontal axis is the positioning error, the vertical axis is the protection level, and the horizontal and vertical components are plotted separately. The Stamford diagram provides a quick view of whether an integrity event has occurred, the normal positioning area (white) and the MI event area (pink) are separated diagonally, the red area indicates the occurrence of an HMI event, and the Stanford diagram can also evaluate the availability of the system, indicated by an area whose ordinate is greater than the alarm limit (yellow, orange). The color bars in the graph represent the concentration of the scatters. In order to evaluate the impact of stochastic model refinement on the availability of integrity and the probability of various integrity events, the horizontal and vertical alarm thresholds are set to 40 m and 35 m, respectively [11]. Table 3 lists the metrics before and after the refinement of the stochastic model, including the average level of protection, system availability, the number of epochs of MI events, and the number of epochs of HMI events. By comparing the yellow area scatters in Fig. 3(a), Fig. 3(b), Fig. 3(c), and Fig. 3(d), it can be seen that the epoch scatters that are not available in the integrity monitoring system after the refinement of the stochastic model are significantly reduced, regardless of whether it is a fitting dataset or a forecast dataset. The distribution of the overall scatter showed a downward concentration trend after the refinement of the stochastic model, which indicated that the protection level of the stochastic model was generally reduced after refining, and the system availability was significantly increased. Table 3 further shows that the average protection level of the fitted dataset and the forecast dataset after the refinement of the stochastic model is reduced by 7.78 m and 7.68 m, respectively, while the average vertical protection level is reduced by 6.39 m and 6.24 m, respectively, and the availability of the integrity monitoring system is increased by 4.6% and 5.3%, respectively. The above results show that stochastic model refinement can effectively reduce the horizontal and vertical protection levels and improve the availability of the integrity monitoring system.

图 3 随机模型精化前后的水平斯坦福图Fig. 3 Horizontal Stanford diagram before and after stochastic model refinement

Diagram options

表 3 随机模型精化前后的斯坦福图Tab. 3 Stanford diagram before and after stochastic model refinement

Table options

Figure 4 shows the probability plot of the vertical positioning error distribution before and after the refinement of the stochastic model, and the blue and green scatters in the figure represent the stochastic model before and after refining, respectively. The dashed line is a normal distribution probability chart, and the closer the scatter is to the dashed line distribution, the more consistent the data is. Table 4 lists the root mean square error (RMSE) after the detection and elimination of the stochastic model before and after refinement, and the corresponding positioning errors under the probabilities of 0.990 0, 0.999 0, and 0.999 9 in Figure 4. As can be seen from Figure 4, both the fitted and forecasted datasets show the same pattern: the blue scatters are distributed closer to the dashed line than the green scatters, and there are significantly fewer outliers. Table 4 shows that after the stochastic model is refined, the positioning error of the fitting dataset is reduced by 0.05, 2.63 and 19.84 m at the probabilities of 0.990 0, 0.999 0 and 0.999 9, respectively, and the positioning error of the forecast dataset is reduced by 1.93, 27.22 and 22.70 m at the probabilities of 0.990 0, 0.999 0 and 0.999 9, respectively. The above analysis shows that the refinement of the stochastic model has a limited improvement on the overall data solution accuracy, but it can significantly reduce the extreme positioning error under the condition of small probability. This indicates that the stochastic model refinement effectively improves the ability of ANALOG decoupling fault detection and elimination.

图 4 随机模型精化前后垂直定位误差分布的概率图Fig. 4 Vertical position error probability plot before and after stochastic model refinement

Diagram options

表 4 随机模型精化前后故障探测与排除后的定位误差Tab. 4 Position error after fault exclusion before and after stochastic model refinement m

Table options

From the results of Figure 3, Figure 4 and Table 4, it can be seen that the stochastic model refinement can improve the integrity monitoring ability of ARAIM, so that the ANALOG algorithm can be applied to more fields. On the one hand, stochastic model refinement can effectively reduce the protection level and improve the availability of system integrity monitoring, and on the other hand, stochastic model refinement can improve the ability of ARAIM's decoupling fault detection and elimination, and reduce the extreme positioning error under small probability conditions. 3 ConclusionThe two main tasks of ARAIM are fault detection and troubleshooting, and the calculation of protection level to determine system availability. In order to apply the ARAIM algorithm, which only meets the special specifications of civil aviation, to more fields, it is necessary to adaptively refine the stochastic model of the receiver noise term, so that it can accurately describe the noise characteristics of various types of GNSS receivers. In this paper, an ARAIM stochastic model refinement method is proposed based on LS-VCE, and the measured data of GRACE-FO C satellite from December 2, 2019 to December 17, 2019 are selected for verification, and the results show that: (1) Stochastic model refinement can effectively reduce the protection level of ANALOG and improve the availability of ARAIM. After the stochastic model refinement, the average protection level of the fitting dataset and the forecast dataset decreased by 7.78 m and 7.68 m, the average vertical protection level decreased by 6.39 m and 6.24 m, respectively, and the availability of ARAIM increased by 4.6% and 5.3%, respectively. (2) Stochastic model refinement can improve the effectiveness of ANALOG decoupling fault detection and elimination. After the refinement of the stochastic model, although the RMSE of the fitting dataset and the prediction dataset did not change significantly, the positioning error decreased by 19.84 m and 22.70 m under the 99.99% probability, respectively. In conclusion, stochastic model refinement is a necessary step to extend ANALOG to other application areas. On the one hand, stochastic model refinement can improve the fault detection and elimination ability of ARAIM, and reduce the extreme positioning error under the condition of small probability, and on the other hand, stochastic model refinement can effectively reduce the protection level and improve the availability of the system. About author:YANG Ling (1986-), female, Ph.D., associate professor, research interests in GNSS anomaly detection, robustness estimation and integrity monitoring. E-mail: [email protected]

First trial: Zhang Yanling review: Song Qifan

Final Judge: Jin Jun

information