A reduced-condition-number algorithm for single-frequency precise point positioning based on regularized Kalman filter

Single-frequency receivers are low cost and portable, thus being widely applied in engineering; the extended Kalman filter (EKF) is commonly used to perform single-frequency precise point positioning (SF-PPP). However, the positioning performance of SF-PPP is seriously influenced by various errors. Due to the large process noise and initial variance of the estimated parameters, the weight matrix of state parameters will be ill-conditioned, and since the noise of the pseudo-range is much higher than that of the carrier phase, the weight matrix of observations presents as ill-conditioned. Additionally, the condition number of the normal matrix will jump on the conditions of cycle slip, new emerging satellites, and signal outages. To reduce the condition number of the normal matrix, the regularized Kalman filter (RKF) algorithm is proposed, with additional support for the maximum variance matrix and singular value decomposition, thereby improving the accuracy and stability of SF-PPP. Through static and dynamic experiments, it is found that the proposed method can reduce both the ill-conditioning of the weight matrices of the observations and the state parameters. The condition number of the normal matrix is <500 per epoch, and the convergence time is shortened by >40%. Compared with the SF-PPP using EKF, centimeter-level static positioning accuracies of 1.13, 0.73, and 2.92 cm and decimeter-level kinematic positioning accuracies of 12.5, 10.8, and 27.3 cm in the east, north, and vertical components, respectively, using RKF; this yielded 38.3, 29.8, and 45.2% and 39.6, 41.9, and 21.3% improvement in the static and kinematic scenarios, respectively.


Introduction
With global coverage and excellent accuracy, global navigation satellite system (GNSS) has been widely used in daily life [1].The single-frequency (SF) receiver, which is simple to operate and low-cost, is especially popular with scientific and engineering fields such as time transfer [2], atmospheric delay inversion [3], and differential code bias solution [4].With abundant research on SF precise point positioning (PPP) observation model refinement [5], ionospheric delay enhancement [6], and performance evaluation with different types of antennas [7], the models and theories of SF-PPP are becoming more mature.
Three models are usually utilized in SF-PPP: utilizing the original pseudo-range and phase observations, the pseudorange as well as group and phase ionospheric correction (GRAPHIC) observations, and both GRAPHIC and phase observations [8].Since the high-accuracy phase observation plays a major role in the latter two models, they have almost the same positioning performance [9].To solve for the estimated parameters, receiver coordinates, receiver clock, tropospheric delay, ionospheric delay, and ambiguity parameter, the extended Kalman filter (EKF) is usually applied.
Using the parameter elimination scheme, all the estimated parameters are computed through modified square root information filtering and smoothing, and no external ionospheric model is needed [9].In the second scheme, relatively accurate initial values are calculated and assigned to some estimated parameters, coordinates are obtained from single point positioning, and the initial value of the ambiguity parameter is the difference between the phase and pseudo-range observations.Ionospheric delay, the critical factor in SF positioning, must be dealt with carefully [10].The Klobuchar model [11], broadcast by GNSS satellites, and global ionosphere maps (GIM) [12] provided by the international GNSS services (IGS) are the most widely used.GIM products can provide better positioning accuracy but are limited to 2 ∼ 8 total electron content unit.Le and Tiberius [13] suggested an optimal filter to consider the impact of ionospheric errors and confirmed its efficiency.Choy [14] confirmed the impact of a weighting strategy for considering the noise of ionospheric corrections.Based on the relatively accurate initial values and weighting scheme, the last scheme further applies the constraint model on the ionospheric delay, including a temporal loosening constraint [5], temporal and spatial constraint [15], a stepwise relaxed model [16], a low-order polynomial model with a refined stochastic process for each satellite [17], and ionospheric-weighted model [18].
In the parameter elimination scheme, the weight matrix of state parameters will have a very large condition number.While those of the other two schemes are relatively small, elaborate processing is required on the ionospheric delay, and on the condition of inaccurate initial values, the weight matrix of the state parameters will be ill-conditioned [19,20].In addition, in the circumstance of a cycle slip, an emerging satellite, or a signal outage, the weight matrix of the observations is also ill-conditioned.These two ill-conditioned matrices will lead to a sudden increase in the condition number of the normal matrix, and the accuracy and stability of SF-PPP will be affected [21].
To solve the ill-posed problem, the ridge estimation algorithm [22], truncated singular value decomposition (TSVD) [23], and Tikhonov regularization [24] are commonly used.It is difficult to determine the ridge parameters in ridge estimation, and by directly cutting off the smaller singular values and their corresponding singular value vectors, the unreliable part of the model is essentially deleted in TSVD, which may reduce the accuracy of the solution.Therefore, Tikhonov regularization is most effective.In SF GPS rapid positioning with double-difference phase observations, Wang et al [25] utilized Tikhonov regularization to weaken the ill-condition of the normal equation.To solve the problem of approximate linear correlation caused by a short observation time during rapid positioning, Gui and Han [26] proposed the regularized least square method.In view of the ill-condition of the normal matrix which is formed by a few epoch observations in GNSS rapid positioning, an improved Tikhonov regularization method was proposed to construct a regularization matrix [27].The above methods are all based on least square estimation, and the condition number for the weight matrix of the estimated parameters is decreased.However, using EKF, the normal matrix of SF-PPP also contains the weight matrix of state parameters, and the condition number will also be affected by the process noise and initial variance of the estimated parameters.Therefore, the ill condition of the SF-PPP normal matrix must be further analyzed.
To reduce the condition number of the SF-PPP normal matrix and improve its positioning performance, we summarize the causes of the ill condition and propose a regularized Kalman filter (RKF).The remainder of this study is organized as follows.Section 2 analyzes the condition number of the normal matrix, an RKF model is proposed, and the construction of the regularization matrix is described.The positioning performance of the proposed method is analyzed and evaluated in section 3. We discuss our conclusions in section 4.

Methodology
We discuss three SF-PPP models, analyze the numerical stability of SF-PPP using EKF, compare the normal matrix of SF-PPP and the condition numbers of its internal elements based on the above three schemes, and propose a reduced-conditionnumber algorithm based on RKF.

Observation equation and parameter estimation of SF-PPP
The pseudo-range hardware delay is combined into other parameters due to its correlation with the receiver clock error, ionospheric delay, and phase ambiguity.The sorted SF undifferenced observation equations of code C and phase P can be expressed as where ρ s k is the satellite-to-receiver range at epoch k for satellite s; T s k and I s k are the respective tropospheric and ionospheric delay; I s k = ι s k + d s is the combination of satellite code hardware delays d s and real ionospheric delay ι s k ; dt r,k = δt r,k + d r is the combination of receiver code hardware delay d r and real receiver clock errors δt r,k ; and b s = λN s − δ r + δ s − d r + d s is the ambiguity in meter, which, except for the integer carrierphase ambiguity (in cycles) with carrier wavelength λ, absorbs the satellite phase hardware delay δ s , satellite code hardware delay d s , receiver phase hardware delay δ r , and receiver code hardware delay d r ; ε s C and ε s P are the observation noises of the code and phase, respectively.
Three commonly used SF-PPP models can be expressed as follows: (1) As shown in equation ( 1), the CP model consists of the original code C and phase P; (2) The ionosphere-free-half CG model is composed of code C and GRAPHIC, where G s k = (P s k + L s k ) /2 is a GRAPHIC observation combining the ionosphere-free code and phase, where ε s G = (ε s P + ε s C ) /2 is observation noise.The weight between the GRAPHIC and pseudo-range is 4:1 [8]; (3) The GP model of the GRAPHIC G and phase P is The accuracy of the phase observations is much higher than that of GRAPHIC, and the weight between the carrier phase and GRAPHIC is 5000:1 [8].Hence, since GP and CP have basically the same performance, only these models are selected.Compared with the phase observation, the accuracy of GRAPHIC observation is lower, and hence the CP and GP models are more accurate than the CG model.
The estimated parameter of SF-PPP can be obtained using EKF as where Xk and Xk are the predicted value and corrected value of the estimated parameters at epoch k, respectively; Z k constitutes the measurements; Φ k−1 and H k are the state transfer and design matrices, respectively; Γ k−1 is the mapping matrix; and W k−1 and V k are the independent white noises with Gaussian probability of the state parameters and observations, respectively.For ease of presentation, the parameter elimination scheme is named as the first scheme, and the relatively accurate initial values and weighting scheme is referred to simply as the second scheme.Additionally, the second scheme with additional ionospheric constraint is called as the third scheme.

Numerical stability analysis of SF-PPP using EKF
To analyze the numerical stability, we first analyze Xk , which can be expressed as where K k is the gain matrix; and Σk and Σ k are the respective variance matrices of the predicted state parameters and the observations.The ill-condition of H k Σk H T k + Σ k can have two causes.The precision of pseudo-range observation may be poorer than that of phase observation, which will result in a large condition number of Σ k ; or the noise of the estimated parameters and the initial variance of the newly joined parameters may be large, leading to a large value in Σk .The illconditioned H k Σk H T k + Σ k will have large eigenvalues and obtain unstable inversion, causing the abnormal update of EKF [28,29].
Xk can also be expressed as Perturbation analysis is carried out based on equation ( 6), and the perturbation equation can be expressed as where are the respective weight matrices of observations and state parameters.
Xk + δ Xk can be expressed as where is the free item; δN and δb are the perturbation terms of the normal matrix and free item, respectively; and δ Xk is the deviation caused by the perturbation.
The matrix (N + δN) is nonsingular as long as It is clear that the effects of δN and δb on δ Xk are closely related to N −1 .To overcome numerical calculation defects using EKF and improve the stability of the solution, a regularization matrix can be introduced.Once the condition number of N is large, a small change of N and b will lead to large fluctuation of the solution.
Table 1 lists the processing strategies and parameter settings of SF-PPP.It can be found that only the processing for the receiver coordinate is slightly different for dynamic and static  positioning.Therefore, the analysis of the condition number is carried out using CG and CP models in a static scenario.
Taking the static observation data at the IGS SHAO station on 19 March 2019, as an example, the condition number of the normal matrix and its internal elements at the first epoch are shown in table 2. Under the first scheme, the condition number of the normal matrix using the CG model is >10 7 , and that of the CP model is >10 8 .This is mainly because both P k and Pk are ill-conditioned, and the latter is mainly affected by the initial value of the parameter.By assigning relatively accurate initial values for the estimated parameters, the condition numbers of Pk and N drop sharply to 4.71 × 10 4 and 2.15 × 10 4 , respectively, in the second scheme.In the third scheme, the temporal and spatial constraint model is imposed on the ionospheric delay, and due to the limited accuracy of GIM products, the variances of the ionospheric pseudo-observations gradually increase over time to decrease the contributions of the ionospheric pseudo-observations [5]; as a result, their respective condition numbers are reduced to 1.20 × 10 4 and 9.13 × 10 3 .
The singular is introduced to analyze the reversibility and stability of N. Larger singular values and their corresponding singular value vectors indicate more certain and reliable parts of the model parameters, while smaller singularity values and their corresponding vectors indicate unreliable parts.With different visible satellites, the number of estimated parameters and the normal matrix are different, and hence it is necessary to set different visible satellites for analysis.With 7, 8, and 9 visible satellites, the singular value distribution at the first epoch using the CP model is as shown in figure 1, from which it can be seen that the number of satellites has little effect on the distribution of singular values.With 9, 8, or 7 available satellites, there are 14, 13, or 12 singular values, respectively, that are close to 0. The singular normal matrix shows poor reversibility and stability, decreasing the performance of SF-PPP using EKF.
Figure 2 further shows the singular values of the normal matrix at the 100th epoch using the above three schemes.It can be seen that, compared with the first scheme, the difference between the minimum and maximum singular values in the second and third schemes can be narrowed significantly.In addition, comparing the second and third schemes, the constraint on the ionospheric delay has little effect on the singular values.
The normal matrix changes epoch by epoch; its daily condition numbers along with the available satellite number and position dilution of precision (PDOP) under three schemes are shown in figure 3. It can be concluded that (1) when the number of available satellites remains the same, the condition number of the normal matrix decreases gradually to a stable value; (2) with the decrease of available satellites, the condition number of the normal matrix will also decrease; and (3) with the increase of available satellites, the newly added ambiguity parameter and ionospheric parameters cannot be assigned with accurate initial values, and their large initial variances increase the condition number.The greater the number of satellites, the more the condition number changes.It is worth noting that, due to the signal outage at epochs 625 and 1236, the condition numbers of all three schemes change sharply.
When cycle slip occurs, the ambiguity parameter must be reinitialized, and the condition number of the normal matrix will increase.Cycle slip has less influence than satellite change on the condition number.This is because cycle slip will only lead to ambiguity initialization, while a newly available satellite will bring a new ambiguity parameter and ionospheric delay parameters.Although the ill-condition of the normal matrix can be improved through the reformation of the weight matrix Pk , influences including cycle slip, signal outage, and newly emerging satellites cannot be removed.It is the ill-condition of the weight matrix P k that should be mitigated further, for which we propose an improved algorithm.

Reduced condition-number algorithm based on RKF
From the above analysis, we can conclude that SF-PPP using EKF is a typical ill-posed problem.Regularization can effectively solve the above unsteady problem.The basic idea is to add some regularization conditions to the unsteady equations, lose the unbiased nature of EKF, and transform them into steady equations, to obtain an accurate and stable solution.To decrease the condition number of the normal matrix, RKF is utilized to keep it within a certain threshold.Referring to [35] and the time-series of SF-PPP condition number, the threshold of the condition number is set to 500 for SF-PPP.If <500, the normal matrix is considered good, and otherwise as illconditioned.If the threshold is too large, the small singular value will not be corrected.And if the threshold is too small, frequent regularization will bring too much computational burden.
The objective function of RKF is: min where Vk = Xk − Xk is the innovation vector, Xk is a stable regularizing function, R is a symmetric nonnegative definite regularization matrix, and α is the regularization parameter, and can be determined by the most widely used L-curve method [19,36], the X-axis ξ and Y-axis η are lg ∥HX α − Z∥ P and lg ∥X α ∥ K , respectively.The crurve is obtained based on α The optimum value of α is located where the curvature of the curve is the largest: where, ξ , η, ξ , and η are the first-order and second-order derivatives, respectively.Usually, a difference approach can be taken to calculate these derivatives.
The effect of eigenvalues on the variance can be expressed as the effect of singular values on standard deviation (STD): where σ 0 and λ are the median error of unit weight and singular value, n is the total number of parameters to be estimated.The smaller the singular values, the greater the effect on the estimated parameters.There are usually multiple small singular values in the SF-PPP, and to mitigate their effect, the regularization matrix needs to be determined based on these small singular values.Based on the characteristics of the large difference between the larger singular values and smaller singular values of the ill-conditioned matrix, it is set that when the sum of the STD of the small singular values accounts for more than 95% of the total STD, these singular values are the small singular values with serious influence, and they should be regularized to alleviate the influence on the STD, and the judgment criteria is expressed as: where , and λ k is the cutoff for determining the small singular values.
And the regularization matrix is constructed through the following two methods.The first method is to simply construct additional support for maximum variance matrix [25], it equals to directly specify the regularization matrix corresponding to the small singular value as 0 The second method constructs the regularization matrix using the eigenvector corresponding to the small singular value [37]: where G i is the eigenvector corresponding to the small singular value shown in figures 1 and 2.
The solution of RKF is obtained as Compared with the EKF algorithm, the constraint of requiring the stable function Ω( Xk ) to be extremely small is added to the regularizing standard, to solve the ill-posed problem. Figure 4 shows the flowchart of SF-PPP using RKF, which has three steps: (1) Preprocessing: the cycle slip is detected through the combination of Doppler and phase pseudo-range, and the gross errors and clock slips are detected and removed [38] 15).Using the first scheme, the regularization matrix R = R Pk + RP k is adopted, and regularization matrix R = R Pk is utilized using the second and third schemes; (3) In subsequent epochs, the condition number of the normal matrix is calculated epoch by epoch to judge whether the normal matrix is ill-conditioned.Once the condition number is >500, it is necessary to judge whether there exist the three circumstances of a new emerging satellite, cycle slip occurrence, or signal outage, and R Pk must be added to the regularization matrix for all three schemes until the condition number of the normal matrix no longer exceeds 500.Otherwise, EKF is applied.
Notably, EKF means completely using EKF for parameter estimation, while RKF means that RKF is used only when the condition number of the normal matrix is >500, and otherwise EKF is used.

Experimental results and analysis
The 14 d of observation data from 19 March 2019, to 1 April 2019, recorded by 15 globally distributed IGS tracking stations with a sampling interval of 30 s, were selected.Figure 5 shows the distribution of the selected stations.The results were evaluated from the aspects of convergence time and positioning accuracy.Only when the east, north, and vertical deviations of 20 consecutive epochs were within the limit of 10 cm was filtering considered to have converged at the current epoch.

Condition number analysis using RKF
Using RKF, the condition number of the normal matrix at Shao station on 19 March 2019, is as shown in figure 6.Compared with figure 3, where EKF is applied, by improving the illcondition of P k and Pk by adding some regularization conditions to the unsteady equations, basically all the condition numbers of the normal matrix can be maintained at <500 for  all three schemes.This is due to the fact that correcting for the smaller eigenvalues without changing the larger eigenvalues, which ultimately improves the ill condition of the normal equation and reduces the variance and bias of KF.The effect of ill condition on the estimated variance is centered on the amplification of the variance by smaller singular values, with larger singular values not adversely affecting the variance, resulting in an improvement of the stability of regularization method.The statistical results show that the condition numbers of the normal matrix using the first, second, and third schemes are reduced by 200, 50, and 33%, respectively.Although all three schemes are still affected by the variation of satellite number, cycle slip, and signal outage, the condition number after a sharp jump remains <500.
With the largest condition number, the first scheme is not commonly used.And the third scheme can be simplified into the second without additional ionospheric constraints.Therefore, the positioning performance was evaluated using the third scheme with the CG and CP models, respectively.the supplemental correction for the smaller singular values can more efficiently reduce the variance, while retaining the larger portion of the singular values with sufficient information reduces the information corruption and thus the bias of the estimation.Benefiting from transforming the ill-conditioned normal equation into a steady one, the positioning accuracy and convergence speed using RKF show a significant improvement compared to EKF.It takes 620 epochs for EKF to converge, and 280 epochs for RKF, an improvement of 54.8%.In the vertical component, the fluctuation using EKF is greater than that of RKF, and the positioning accuracy of RKF is stable at about 0.04 m after 200 epochs.

Analysis of static positioning accuracy and convergence time
Since the CP model differs from the CG model in the weight matrix of observations, to verify whether RKF is also applicable to the CP model, figure 8 shows its static positioning accuracy at Shao station on 19 March 2019, where it can be seen that the improvement using RKF is similar to that of the CG model, and the positioning accuracy and stability using RKF are better than those of EKF.Additionally, it is worth noting that the positioning deviations of CG and CP models using RKF are smoother and more stable that these of using EKF, which is mainly attributed to the small condition number in figure 6.
Table 3 lists the average positioning accuracy and convergence time of 15 IGS stations over the selected period.The positioning accuracies in the east, north, and vertical components for the CG model are improved from 2.14, 1.42, and 6.22 cm, respectively, to 1.41, 0.91, and 3.51 cm, respectively through RKF; the achieved improvement is 34.1, 35.9, and 43.6%, respectively; the convergence time is reduced by 46.9%.The positioning accuracies in the east, north, and vertical components for the CP model are improved from 1.83, 1.04, and 5.33 cm, respectively, to 1.13, 0.73, and 2.93 cm, respectively, using RKF; the achieved improvement is 38.3, 29.8, and 45.2%, respectively; the convergence time is reduced by 40.2%.The CP model shows better performance than the CG model in the aspects of positioning accuracy and convergence speed, presumably because the accuracy of phase observation is greater than that of graphic observation.

Analysis of dynamic positioning accuracy and convergence time
Since the observation environment changes quickly in the dynamic environment, and the satellite is easily unlocked, the situations of cycle slip, signal outage, and newly emerging satellites can easily occur, which will result in a rapid variation of the weight matrix P k .A dynamic experiment was carried out with onboard observation data collected in the campus playground of Information Engineering University, China from GPST 08:00:00-09:07:30 on 26 October 2018.The sampling rate was 1 s, and the trajectory is shown in figure 9, the campus playground was surrounded by a six-floor building and tall trees.A nearby reference station in an open environment was set up, and a fixed real-time kinematic solution was used as a reference.
Figures 10 and 11 show the dynamic positioning accuracy under the CG and CP models, respectively.It can be seen that the positioning accuracy using the CP model is still better than when using the CG model.The fluctuation of positioning errors using EKF is mainly caused by the variation of condition number, which are originated from signal outage and newly emerging satellites.In contrast, the proposed RKF can   improve positioning stability and accuracy, and the convergence time can also be shortened significantly.In the aspect of positioning accuracy, decimeter-level dynamic positioning accuracy of much better than 0.5 m in the east, north, and vertical components can be realized, and most importantly, owing to the reduced condition number, the time-series of positioning deviations is especially smooth, no fluctuation of positioning deviations is observed after convergence, which is beneficial for location-based services.experiment.In dynamic positioning, filter convergence is defined as positioning deviations in the east, north, and vertical components better than 3 dm.Compared with EKF, the positioning accuracies of the CG model in the east, north, and vertical components are improved from 0.243, 0.198, 0.421 m, respectively to 0.157, 1.124, and 0.318 m, respectively, with improvements of 35.4,37.4, and 24.5%, respectively.These three components are improved by 39.6, 41.9, and 21.3%, using the CP model.RKF can also shorten the convergence time in dynamic positioning and can realize fast positioning.

Conclusions and discussion
To reduce the condition number of the SF-PPP normal matrix using EKF, we analyzed the condition numbers under different schemes and proposed a reduced condition-number algorithm based on RKF to improve positioning accuracy and stability.We concluded the following.
(1) Among the three SF-PPP models, CP and GP were more accurate than CG, and the performance of GP and CP was basically the same.(2) The condition number of the SF-PPP normal matrix was affected by the weight matrices of the observations and the state parameters.Among the three solution schemes, the condition number of the less commonly used first scheme, which estimates all the estimated parameters, was the largest, while that of the third scheme with external constraints was the lowest.The third scheme could be simplified to the second scheme without additional ionospheric constraints.(3) On the conditions of a cycle slip, signal outage, and a newly emerging satellite, the initial large variance of new emerging parameters caused the condition number of the normal matrix to increase; as a result, the positioning results fluctuated sharply.(4) With the adjustment of regularization parameter and regularization matrix, RKF could improve the ill-condition of the weight matrices of the observations and the state parameters and could decrease the fluctuation of the condition number in the circumstances of a cycle slip, signal outage, and a newly emerging satellite.As a result, the convergence speed could be accelerated, the accuracy and stability of SF-PPP could be improved, and centimeter-level static positioning accuracy and decimeter-level kinematic positioning accuracy could be achieved.Additionally, with reduced condition number, the smooth and stable positioning accuracy is beneficial for the location-based service.

Figure 1 .
Figure 1.Distribution of singular values at the first epoch with 7, 8, and 9 visible satellites, using the CP model.

Figure 2 .
Figure 2. Singular values of the normal matrix at the 100th epoch under the aforementioned three schemes.

Figure 3 .
Figure 3. Number of available satellites, PDOP, and condition number under three schemes.Top panel: available satellites and PDOP; bottom panel: variation of condition number with three schemes.

Figure 4 .
Figure 4. Flowchart of reduced condition-number algorithm for SF-PPP based on RKF.

Figure 5 .
Figure 5. Distribution of 15 globally distributed IGS stations used in the experiment.

Figure 6 .
Figure 6.Condition-number variation of the normal matrix using RKF with three schemes.Red, green, and orange lines: first, second, and third schemes, respectively.

Figure 7
Figure 7 presents the static positioning accuracy of the CG model at Shao station on 19 March 2019, using EKF and RKF.By selecting the eigenvectors corresponding to the smaller singular values and constructing the regularization matrix, the smaller singular values can be supplementally corrected, and since the singular value decomposition concentrates the information-sufficient portion on the smaller singular values,

Figure 7 .
Figure 7. Static positioning accuracy of CG model at SHAO station on 19 March 2019.Green and orange lines: positioning accuracy of SF-PPP using EKF and RKF, respectively.

Figure 8 .
Figure 8. Static positioning accuracy of CP model at SHAO station on 19 March 2019.Green and orange lines: positioning accuracy of SF-PPP using EKF and RKF, respectively.

Figure 10 .
Figure 10.Dynamic positioning accuracy in east, north, and vertical components using CG model.Green and orange lines: dynamic positioning accuracy of SF-PPP using EKF and RKF, respectively.

Figure 11 .
Figure 11.Dynamic positioning accuracy in the east, north, and vertical components using CP model.Green and orange lines: dynamic positioning accuracy of SF-PPP using EKF and RKF, respectively.

Table 1 .
Processing strategies and parameter settings of different parameters in SF-PPP.

Table 2 .
Condition numbers of normal matrix and its internal elements at first epoch of SHAO station on 19 March 2019.

Table 3 .
Average RMS and convergence time of static positioning at 15 IGS stations during 14 d.

Table 4
lists the statistical results of the average root mean square (RMS) errors and convergence time in the dynamic

Table 4 .
Average RMS and convergence time of dynamic positioning.