In this section, two tests were conducted using a set of static data and a set of ship-borne dynamic data to evaluate the performance of the real-time estimation model. In each test, we first analyzed and compared the estimation accuracy of the real-time estimation model and the empirical stochastic model. To this end, the following three weighting schemes for pseudorange observations are designed:

**Scheme 1**: The empirical model is adopted, the parameter ({upsigma }_{0}) in (6) is set to a general value, (sigma_{0} = 0.3). (the general empirical model)**Scheme 2**: The empirical model is adopted, the parameter ({upsigma }_{0}) in (6) is set to the optimal value, which is obtained through post-processing analysis. (the optimal empirical model)**Scheme 3**: The real-time estimation m.del is adopted, and the amplification coefficient is set to (s = 3.0). (the real-time estimation model)

The performance of the real-time estimation model was analyzed from the perspective of the positioning domain by comparing the positioning accuracy and convergence time of kinematic PPP. The positioning accuracy is described by the Root Mean Square (RMS) of positioning errors, which is calculated by

$$s_{{{text{RMS}}}} = sqrt {frac{{sumlimits_{i = 1}^{n} {(hat{x}_{i} – x_{{{text{ref,}}i}} )^{2} } }}{n}}$$

(17)

where (hat{x}) and (x_{{{text{ref}}}}) are the state estimate and reference, and *n* is the number of epochs. The convergence time is measured as the time until the positioning error in the three directions of East (*E*), North (*N*) and Up (*U*) directions are all less than 0.5 m, which continues for 1 h. The detailed processing strategies of PPP are listed in Table 1.

### Tests with static data

The static test data was collected from 00:00:00 to 08:23:16 on Jan 07, 2019 (GPS time) by the C200-AT high-precise GNSS receiver produced by BDStar Navigation company at a sampling frequency of 1 Hz. The receiver antenna was placed on the top of a building, and its surrounding environment is shown in Fig. 3. The coordinates of the station is (− 2 610 957.4 m, 4 232 178.5 m, 3 980 666.8 m) in Earth-Centered Earth-Fixed (ECEF) coordinate system, which is calculated by Canadian Spatial Reference System (CSRS) PPP (Tétreault et al., 2005). The sky plot of visible GPS satellites is shown in Fig. 4, and the number of visible satellites in the test period is 15. It should be noted that due to the long connecting cable (about 50 m) between the receiver and antenna, and the high-rise buildings around the antenna, and the serious multipath errors, the quality of the static observation data is not very good. For this test, the parameter ({upsigma }_{0}) of the empirical model in Scheme 2 is set to the optimal value of 1.5 m.

### Evaluation of variance estimation

Figure 5 shows the RMS of pseudorange measurement residuals and the standard deviation estimates at each epoch. Figure 6 illustrates the pseudorange measurement residuals and standard deviation for each observed satellite. In figures, the blue curve represents the pseudorange measurement residuals. The red, green, and black curves represent the pseudorange Standard Deviations (SD) estimated by the general empirical model in Scheme 1, the optimal empirical model in Scheme 2, and the real-time estimation model in Scheme 3, respectively. The statistical results for each satellite are listed in Table 2.

It can be seen from Figs. 5, 6, and Table 2 that the standard deviation of pseudorange estimated by the general empirical model in Scheme 1 is very conservative, and the average standard deviation is about 1.19 m, which cannot reflect the true error level of the pseudorange observations. The average pseudorange standard deviation calculated by the optimal empirical model in Scheme 2 and the real-time estimation model in Scheme 3 are 6.69 m and 8.01 m, respectively. The standard deviation of pseudorange obtained by these two models are very close to each other, and they can obtain more accurate evaluation results. The statistical results in Table 2 show that the average pseudorange standard deviation estimated by the optimal empirical model and the real-time estimation model is about 2 to 3 times the RMS of pseudorange measurement residuals. In addition, it can be seen from the graph of the pseudorange measurement residuals for each satellite in Fig. 6 that the pseudorange measurement residuals are correlated with satellite elevation angles, and the measurement residuals decrease as the satellite elevation angles increase.

### Evaluation of Positioning performance

Figure 7, 8 and 9 show the positioning errors in E, N, and U directions, respectively. Table3 lists the statistical results of the 2 Double (2D) RMS of positioning errors in East (*E*),North (*N*) and Up (*U*) directions and the convergence time.

As can be seen from Figs. 7, 8, 9, the convergence time of PPP using the optimal empirical model in scheme 2 and the real-time estimation model in scheme 3 are better than that of the general empirical model in scheme 1, and the positioning results of PPP using the optimal empirical model and the real-time estimation model are more accurate and stable. It can be also seen from the statistical results in Table 3 that the positioning accuracy of PPP using the general empirical model in scheme 1 is the worst, and the convergence can not be reached in the whole process. Compared with the general empirical model in scheme 1, the positioning performance of PPP using the optimal empirical model in scheme 2 and the real-time estimation model in scheme 3 are significantly improved.

The statistical results in Table 3 show that compared with the general empirical mode in scheme 1, the RMSs of PPP using the optimal empirical model in scheme 2 are decreased by 0.538, 0.265, and 0.648 m in *E**, **N**, **U* directions respectively, and the convergence time is reduced from 30 195 to 4 087 s; the RMSs of PPP using the real-time estimation model in scheme 3 are decreased by 0.616, 0.303, and 0.68 m in *E**, **N**, **U* directions respectively, and the convergence time is reduced to 1 964 s. By comparing the positioning results of PPP with scheme 2 and scheme 3, it is found that the positioning performance of PPP with scheme 3 is slightly better than that with Scheme 2. Compared with the optimal empirical model, the real-time estimation model reduces the RMSs of PPP by 0.078, 0.038, and 0.032 m in *E*, *N*, *U* directions respectively, and the convergence time by 2123 s.

In summary, both the optimal empirical model and the real-time estimation model can obtain more accurate pseudorange variance estimates, and therefore can achieve better positioning performance than the general empirical model. Compared with optimal empirical model, the real-time estimation model can adaptively adjust the pseudorange standard deviation estimates according to the changes in the measurement error level at each epoch, so the real-time estimation model can better reflect the actual error level of the pseudorange observations at each epoch, and achieve better positioning performance.

### Tests with ship-borne dynamic data

To further assess the performance of the real-time estimation model in dynamic environment, a ship-borne test was carried out on January 1, 2019, which lasted for about 3.5 h. The test data was collected by the NovAtel SPAN PwrPak7D-E1 receiver, and the sampling frequency is 1 Hz. Figure 10 shows the reference trajectory, which is calculated by the NovAtel Inertial Explore 8.80 post-processing software through the carrier phase Differential Global Navigation Satellite System (DGNSS) positioning mode, with centimeter-level positioning accuracy. Figure 11 shows the GPS satellite sky plot, the number of visible satellites during the test period is 15. In this test, the parameter ({upsigma }_{0}) in Scheme 2 is set to the optimal value of 2.5 m.

#### Evaluation of variance estimation

Figure 12 shows the RMS of pseudorange measurement residuals and the standard deviation estimates at each epoch. Figure 13 illustrates the pseudorange measurement residuals and standard deviation for each satellite. The statistical results for each satellite are listed in Table 4.

It can be seen from the statistical results in Table 4 that the RMS of pseudorange measurement residuals is about 3.63 m, while the average pseudorange standard deviation estimated by the general empirical model in Scheme 1 is about 1.26 m, which is obviously, conservative and cannot reflect the actual error level of pseudorange observations. The average pseudorange standard deviations estimated by the optimal empirical model in scheme 2 and the real-time estimation model in Scheme 3 are 8.84 m and 8.96 m, respectively. They are equivalent, about 2–3 times the RMS of pseudorange measurement residuals. It can be seen that the real-time estimation model and the optimal empirical model have the same estimation accuracy and can better reflect the pseudorange error level.

### Evaluation of Positioning performance

The convergence behavior of the positioning for the ship-borne test data in *E*, *N*, *U* directions are illustrated in Figs. 14, 15, 16. The statistical results of positioning errors and convergence time are listed in Table 5.

It can be seen from the position errors graph in Figs. 14, 15, 16 and the statistical results of positioning performance in Table 5 that the positioning results of PPP with Scheme 1 are the worst. This is because the model cannot reflect the actual error level of pseudorange observations. Compared with the model in Scheme 1, the model in Scheme 2 can reduce the positioning RMS of PPP by 0.207, 0.362, and 1.067 m in E, N, U directions respectively, and the convergence time from 12974 to 2225 s; the model in Scheme 3 can reduce the positioning RMS of PPP by 0.269, 0.57, and 1.213 m in *E*, *N*, *U* directions respectively, and the convergence time to 1126 s. The positioning performance of PPP with the optimal empirical model and real-time estimation model has been greatly improved.

In addition, compared with the model in scheme 2, the positioning accuracy of PPP with model in scheme 3 is further improved by 0.062, 0.208 and 0.146 m in *E*, *N*, *U* directions directions respectively, and the convergence time is reduced by 1099 s. The positioning performance of PPP with model in scheme 3 is slightly better than that with the model in scheme 2. This is because the real-time estimation model can dynamically adjust the pseudorange standard deviation according to the degree of the influence of surrounding environment on the measurement error, so it can obtain a more accurate estimation than the optimal empirical model.

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