Exploring the role of PPP–RTK network configuration: a balance of server budget and user performance

With atmospheric corrections generated from the server, precise point positioning real-time kinematic (PPP–RTK) can achieve high-precision solutions in a fast convergence. PPP–RTK users are concerned about how to use the corrections and the level of performance that can be achieved; thus, our research has focused on correction methods, a priori stochastic modeling, and positioning performance evaluation. Conversely, it is crucial for the server to improve the precision of corrections provided and to consider the balance between cost, computation burden and user performance, especially for commercial applications. We use different scales of the national GPS network of France to generate ionospheric and tropospheric corrections, and corresponding uncertainty information is generated by establishing error functions with respect to an inter-station distance. The quality of corrections and corresponding user performance are analyzed with inter-station distances varying from 22 to 251 km. The results show that the precision of atmospheric corrections can be improved with an increasing number of stations in the network, but the improvement is not significant when the inter-station distances are smaller than 50 km. Regarding user performance, compared to conventional PPP solutions with ambiguity resolution, the convergence time can be reduced by a maximum of 93% and 85% in the horizontal and vertical components, respectively, when the inter-station distance is about 23 km. However, a station spacing within 100 km can still support a 3-min convergence; thus, a balance of server budget and user performance should be considered instead of a dense network.


Introduction
Precise point positioning (PPP) usually suffers from a long initial convergence time to reach centimeter-level positioning accuracy.Introducing satellite phase biases and atmosphere corrections to conventional PPP processing enables PPP real-time kinematic (PPP-RTK) technology to reduce convergence time to a few minutes, benefiting various application fields, especially for real-time cases.Wübbena et al. (2005) first proposed the PPP-RTK concept, which uses RTK networks to perform an ambiguity resolution within a regional network and generate state space representation (SSR) corrections.By applying these corrections, users can reach an RTK-like centimeter accuracy in a few seconds.Since then, research on both the server and user sides has been carried out.Investigations on both sides vary in different modes and different network scales (Li et al. 2011(Li et al. , 2021)).In some research, satellite biases are estimated, and ionospheric corrections are optional, which are called the ionosphere-float and ionosphere-weighted solutions, respectively.The generated ionospheric corrections can be in slant or vertical format, while for other approaches, the tropospheric corrections are also modeled together with the satellite biases and ionospheric corrections.The corresponding impact of different network configurations on user performances is assessed in these studies.Psychas et al. (2018) found that when the precision of ionospheric corrections at the user is better than 5 cm, a fast PPP-RTK solution can be achieved.Zhang et al. (2018) revisited three variants of the un-differenced and un-combined (UDUC) PPP-RTK network model and investigated different linear combinations of the generated satellite phase biases to be transmitted to the users.Using the high-precision atmosphere corrections generated from continuous operating reference stations (CORS) networks, Li et al. (2020) obtained centimeter-level accuracy with GPS in an average of 1.5 epochs and BDS-2+3 in 1.6 epochs, respectively.
By generating ionospheric corrections from an ultradense regional network (e.g., 10 km inter-station distance), the corrections are theoretically expected to be closer to the physical error sources and can thus better represent the error characteristics.In other words, the denser the network used for generating the corrections, the shorter the convergence time for users.Nadarajah et al. (2018) estimated satellite corrections, phase biases, and optional ionospheric corrections with Australianwide and small-scale networks of 30 km, respectively, and the user position can converge within 2 min with the small-scale network.No tropospheric corrections are applied in this contribution, and the ionospheric corrections (if applicable) are determined using inverse-distance weighting (IDW).Zhao et al. (2021) simultaneously estimated the vertical ionospheric delays and the receiver hardware biases with two different scales of networks, while ionospheric corrections at the user were calculated using the IDW with an empirical variance of 0.15 m.The results indicate that the ambiguity resolution can be achieved within three and several minutes for smaller and larger networks, respectively.Psychas et al. (2020) mainly investigated ionosphere-weighted PPP-RTK, which estimates the slant ionospheric delays and satellite biases in the network solution using the best linear unbiased predictor.The results demonstrated that the user's convergence times bear a linear relationship with the network density.In practice, the station density in the network around the service area is usually limited due to the cost.A denser network implicitly requires more loads for the solution computing and the atmosphere modeling.
Realistic estimates of the precision of atmospheric corrections are another factor possibly influencing user performance.This knowledge is usually considered in the user stochastic model, and its realistic values can speed up the extended Kalman filter (EKF) convergence.In addition to the fixed variances in the stochastic model of ionospheric corrections, the distance linear-dependent variance (Nadarajah et al. 2018) and distance exponential-dependent stochastic model (Zha et al. 2021) are commonly used.Research by Zhang et al. (2022) indicates that when applying different a priori precisions to the ionospheric delay constraints, a smaller variance leads to a long convergence time, while a larger variance leads to the same convergence time compared to conventional PPP-AR.Recently, Li et al. (2022) proposed to generate uncertainty information for ionospheric corrections by (1) calculating differences between the ionospheric model value and the estimated value at individual stations and (2) establishing a distance-related model to determine the interpolation error with different scales of networks.This method has been proven to be useful when generating an ionospheric uncertainty map compared to a fixed or empirical precision that most researchers use.But whether it is applicable and what is the resulting performance for users with various network scales is still worth investigating.
Aside from the ionosphere correction, the troposphere zenith hydrostatic delay (ZHD) calculated from the existing model is precise enough to be applied.But the zenith wet delay (ZWD) correction, which can reach the centimeter level, is still worth investigating for whether, and to what extent, it can further speed up the convergence when applied to PPP-RTK.Oliveira Jr. et al. (2017) modeled the ZWD with dense and sparse networks.The results show no significant change in positioning accuracy, but the convergence time for ZWD with the dense network is faster than the sparse one.
Previous research focuses mainly on the impact on the user side, and there was not much effort on realistic assessment of the atmospheric model and most researchers usually exploit a stochastic model based on empirical values.For the current work, the precision of ionospheric and tropospheric corrections is estimated with different scales of networks at the server, and the performance is assessed using these corrections at the user side.In this way, the research aims to provide an overview and assessment of (1) how the inter-station distance in the server network influences both generation of atmospheric corrections and user positioning performance, (2) which density of the network can be enough to satisfy users with different requirements in terms of positioning accuracy and convergence time, (3) whether an ultra-dense network will still significantly improve user solution when the network is dense enough.
The methodology for PPP-RTK is first introduced, together with different scales of networks and detailed processing strategies for both server and user.The assessment of the ionosphere uncertainty maps using different networks followed by the user performance in terms of convergence time, time to first fix, fixing rate, and positioning precision is then analyzed.Finally, conclusions and future work are given.

Methodology
Figure 1 illustrates the simplified flowchart at the server and user ends in this study.A two-dimension processing strategy is applied to generate atmosphere corrections at the server side and to conduct PPP-RTK at the user side.The UDUC model is highly suitable for the PPP-RTK server, as the raw GNSS measurements can be used to estimate directly both slant ionospheric and zenith tropospheric wet delays.To ensure the precision of the estimated atmospheric corrections, the integer ambiguities are resolved at the server side with the method described in Laurichesse et al. (2009).The international GNSS service (IGS) integer clock products are used, in which the widelane (WL) fractional cycle bias (FCB) is given in the file header and the narrow-lane (NL) FCBs are absorbed into the satellite clock corrections.In this way, after applying the necessary corrections such as satellite phase center offsets/variations (Schmid et al. 2005), phase wind-up (Wu et al. 1993), differential code biases, ocean tide loading, and relativistic effects according to the existing models (Kouba 2009), the ambiguity-fixed solution for the server can be obtained site-by-site with the least-squares ambiguity decorrelation adjustment (LAMBDA) method (Teunissen 1995).Therefore, the corresponding high-precision ionospheric and tropospheric delays can then be derived.Finally, regional tropospheric corrections are modeled using the modified optimal fitting coefficients (MOFC) model (Cui et al. 2022), with model coefficients and the modeling precision factor broadcasted to the user.The corresponding uncertainty maps can also have a significant influence on user performance.In this contribution, the ionosphere uncertainty map described in Li et al. (2022) is also generated, which was demonstrated to outperform empirical values.
For PPP-RTK users, the derived slant ionospheric corrections of all reference stations and the tropospheric delay model coefficients are transmitted.Users can interpolate corrections from nearby reference stations and apply them to their locations.The raw UDUC mathematic model of the PPP-RTK user can be simply given as: where P and L are code and phase measurement prefit resid- uals with error models and corrections applied, s and j are satellite and frequency indices, respectively.e s x , e s y , and e s z are line-of-sight unit direction vectors from satellite to receiver.mf s represents the wet mapping function of the tropospheric delay, and j is the wavelength of the phase measurement at j th frequency.u j = f 2 1 ∕f 2 j is a constant value, where f is the frequency.For the estimates, x, y, and z are coordinates of the user station (the coordinate system is consistent with the used precise products), dtr denotes the receiver clock offset, T is the zenith wet tropospheric delay, I s 1 is the slant ionospheric effect on the first frequency and the satellite S, and N is the ambiguity term in phase measurements.With the corrections broadcasted by the server, the tropospheric constraints ( T ) and between-satellite differenced STEC with respect to reference satellite r ( Ĩ1 − Ĩr , ⋯ , Ĩs − Ĩr ) can be applied as virtual observations when appended to the raw functional model, as with representing the residuals.The corresponding stochastic model can therefore be defined as Q y , which is a diagonal matrix reflecting the precision of phase noise ( L ), code noise ( P ), a priori tropospheric constraint (  T ), and ionospheric constraint ( ̃I ).Hence, the covariance matrix for all the estimates after the filter can be computed as (Zhang and Li 2016;Banville and Langley 2013) where A is the design matrix, which is a combined form of the raw and virtual design matrices, as In this way, an analysis of the impacts of correction precision (  T and ̃I ) on position precision and WL AR can be easily performed by evaluating the covariance of the coordinate and ambiguity parameters of Q yy , i.e., the upper left 3 × 3 submatrix is the position precision, and the lower right s × s submatrix is the ambiguity variance and covariance information, which can be further utilized to assess WL AR.This knowledge is used to perform the following simulation analysis.
Single-epoch decimeter-level positioning accuracy is expected for most users in daily applications.To have a theoretical sense of which level of quality can satisfy users' needs, Fig. 2 illustrates the relationship between position precision and ionospheric/tropospheric delay correction precision and the benefit of fixing WL ambiguities.Typically, the success rate (SR) of bootstrapping (Teunissen and Verhagen 2008) is an important indicator to assess the possibility of fixing ambiguities.Therefore, (2) it is applied in this simulation process to investigate the effect of atmospheric correction precision on WL ambiguity resolution.In this simulated assessment, the first epoch on Day Of Year (DOY) 224 in 2022 for station AJAC is selected, and a least-square filter is used together with the atmospheric corrections applied as virtual observations.With the observation from this epoch, the design matrix A can be parameterized with user position and actual satellite coordinates.At this station, nine GPS satellites are tracked with dual-frequency measurements, the code and phase measurement precisions in zenith direction are set to 0.6 m and 0.003 m, respectively, and an elevation-dependent weighting is applied; therefore, Q y is then formed.When evaluating the impact of iono- spheric delay precision, the tropospheric delay precision is fixed and set to 0.03 m.Similarly, ̃I is set to 0.1 m when investigating the relationship between tropospheric precision and position precision.The variance and covariance information derived from Eq. ( 3) is used to assess the horizontal and vertical positioning precision and calculating the SR.The covariance information for estimated coordinates expressed in the earth-centered, earth-fixed (ECEF) is converted into east, north, and up components.
A threshold of 0.99 is set for bootstrapping, which indicates when SR is larger than 0.99, the WL ambiguity can be fixed and tightly constrained as virtual observation and hereby benefit the positioning performance.NL ambiguity resolution is not presented here, because it is difficult to fix NL ambiguities with a single epoch in the simulation result, and, therefore, the solution would be the same as the float solution.
In Fig. 2, for float PPP-RTK solution, with the improvement of correction precision, the corresponding simulated position precision can be improved.The precision improvement in the vertical direction is more significant than the horizontal component.Compared to the smaller improvement in the horizontal direction, the tropospheric correction precision contributes mainly to the vertical component, which is as expected according to the real data processing experiences.A higher correction precision can enable WL ambiguity resolution and significantly improve positioning accuracy with WL ambiguity fixing for even a single epoch.With the precision change for tropospheric corrections, the SR stays as a constant value, because the default ̃I is set to 0.1 m, and is optimal according to the results in the upper right figure.Also, WL ambiguities are less correlated with tropospheric refraction.From the simulation results in Fig. 2, it can be concluded that for centimeter-level single-epoch WL AR solutions, the precision of atmospheric corrections is expected to be within several centimeters.

Data and experiment description
The National GNSS Network of French research laboratories (https:// www.resif.fr/ en/ actio ns/ perma nent-gnssnetwo rk/) aims to provide GNSS observations for scientific communities and is dedicated to scientific research and Earth observation in internal and external geophysics and geodesy.It is an ideal network to be used in this contribution because of the relatively average and dense station distribution.At the server, all stations are grouped into five networks with different average inter-station distances consisting of a different number of stations varying from 13 to more than 200, namely PPPRTK-13, PPPRTK-27, PPPRTK-50, PPPRTK-160 and PPPRTK-ALL.The postfix numbers of each scheme name are the number of used sites.Figure 3 illustrates the site distributions for different station-spacing networks, from which it can be noticed that all the selected networks cover France, and each green dot represents a server station, while red stars represent individual user stations exploited in the performance assessment.The average inter-station distance for these networks is also provided in the bottom right for easier comparison.The average inter-station distance for all the experimental networks varies from about 22 to 251 km, which could represent different PPP-RTK applications with different scales of networks.
To evaluate how the corrections generated from different networks will influence the user performance, 14 user stations not included in the server side are chosen, as shown in Table 1.These stations are equipped with different receiver and antenna types, which can better indicate the average performance at the user.One month of observations with a sample rate of 30 s from DOY 244 to DOY 273 (September) in 2022 is processed, and the user station observation files are split into one-hour data and processed in a simulated kinematic mode, which is to estimate the receiver coordinates epoch by epoch to assume the user is in a kinematic mode.Therefore, there are 10,080 data batches processed in total.Only dual-frequency observations of GPS satellites are used because most stations can observe GPS constellation only, and the integer satellite clock and precise orbit products from GFZ, as well as the P1C1 differential code bias (DCB) products from CODE, are used in the data processing.More details of the processing strategy can be found in Table 2.

Results
In this section we assess atmospheric corrections generated at the server side and the PPP-RTK positioning performance at the user side.In terms of the atmospheric corrections, the ionosphere error map is first displayed to illustrate the precision of ionospheric corrections at different areas of the networks, followed by the ZWD residual map.Then, the user performance is evaluated by analyzing the time-to-first-fix (TTFF), fix rate, positioning errors, and convergence time for the users under different stationspacing networks.TTFF is defined as the time needed for the first integer ambiguity resolution, and the fix rate is defined as the ratio of the number of fixed epochs to the total number of epochs.Positioning errors of a kinematic process are regarded as the root-mean-square error of the differences between the solution and the reference coordinates.The convergence time is the time needed to converge to 1 dm.The reference coordinates are generated using the weekly least-square solution.

Evaluation of ionosphere error map
As shown in Fig. 4, the ionosphere error maps for different station-spacing networks at two certain epochs (UTC 0:00 and 14:00) are illustrated with a grid resolution of  0.5° × 0.25°.The maps are generated using the method proposed by Li et al. (2022), which consists in (1) calculating differences between ionospheric interpolated and PPP-derived values at the server station-by-station and (2) establishing a distance-related function to determine the interpolation errors.The errors are distributed between 0 and 0.2 m, while the transparent areas contain ionosphere errors larger than 0.2 m, which is regarded as less contribution to users with such a large error.It is obvious from Figure 4 that the area with correction accuracy better than 0.2 m becomes larger with an increasing number of sites at the server side, but when more than 160 sites are used, the trend is not that significant.By comparing the error maps at different epochs, it is concluded that the correction errors at 14 o'clock are larger than those at 0 o'clock, especially at the serve domain margins attributed to the ionosphere's active level during the afternoon compared to midnight.With a large number of sites used in the server, e.g., more than 160, the ionosphere errors are within 0.05 m for main service area.This precision is already expected to contribute significantly to the quick convergence and accurate positioning performance according to the simulation analysis in the section methodology.
Noticing that the ionospheric uncertainty in the serve domain margins of PPPRTK-all is worse than PPPRTK-160, it may be the reason that the corrections from the three closest server stations of PPPRTK-all used in the interpolation are accidentally worse than PPPRTK-160, resulting in a worse uncertainty in PPPRTK-all.However, it does not matter much because in the PPP-RTK user case, the user position should be located in the coverage area of the server stations.

Evaluation of ZWD residual map
As demonstrated in the previous simulation, a high-precision tropospheric correction is expected to improve the PPP-RTK vertical positioning performance.The ZWD residuals map is generated to provide a clear view of the quality of the tropospheric corrections.The tropospheric corrections are generated every five minutes, but only a specific epoch (14:00, September 15, 2022) is selected for analysis.Figure 5 gives an overview of the station-by-station residuals for five network configurations, which is the absolute value of differences between the model and "true" values derived from the server end.For different network configurations, most of the ZWD residuals are within 3.0 cm, and the built ZWD model can fit perfectly for the server sites.This indicates that the precision of tropospheric correction for users around the service areas can also be at a high level.According to the simulation results, when the inter-station distance is 50 km, the maximum residual is 6.7 cm, and the mean residual is 1.9 cm, they can benefit the user performance, especially in the vertical component.

TTFF comparison
The TTFF is an important indicator for evaluating atmosphere correction performance implemented in PPP-RTK.The shorter the TTFF, the faster the expected convergence.
Once the ambiguities are correctly fixed, position accuracy can be improved significantly. Figure 6 summarizes the TTFFs for different schemes and enables assessing the impact on the TTFF.The standard PPP-AR is included as a reference too.Corresponding improvements compared to the PPP-AR are then illustrated with the red line.It is worth mentioning that the results are highly related to the interval of data update.It can be noticed that the average TTFF for GPS PPP-AR is 38 min.Even with atmospheric corrections generated with 13 stations, the TTFF can be reduced to 23 min, representing an improvement of 40%.With the increase of the number of sites used at the server, the TTFF can be, at most, reduced to 2 min.The most significant improvement occurred when the used sites increased from 50 to 160, with the TTFF shortened from 10 to 3 min, representing an improvement of 72%.The results are similar to those in Nadarajah et al. (2018) with an average inter-station distance of 30 km.With more than 160 sites, the TTFF can be further shortened by around 30 s.This slight improvement is consistent with the ionosphere error map assessed in the above section, indicating the corrections remain similar even over 160 stations.The results in the analysis of TTFF indicate that the TTFF can benefit from a greater number of sites used at the server.But for some use cases, a balance between cost and user performance should be taken into consideration.

Fix rate
The fix rate for different schemes is evaluated as an indicator that reflects the ambiguity resolution performance.Figure 7 depicts the fix rate after TTFF, which is defined as the ratio of the number of epochs with ambiguity resolution and the total number of epochs after TTFF. Figure 8 then summarizes the fix rate of all the processed epochs.Noting that the definitions for the two fix rates are different, the fix rate after TTFF is expected to be higher than the overall fix rate.
Figure 7 shows that once the ambiguities are fixed, the fix rate remains at a high level, with more than 97% of epochs being fixed.When comparing the PPPRTK-13 and PPP-AR solutions, there is a slight decrease in the fix rate with a value of 0.64%, which may be the reason that the correction generated with 13 server stations is not accurate enough for all the user stations, as described in the analysis of the ionosphere map errors.Therefore, the constraint applied is not reasonable enough.With an increasing number of sites at the server, the fix rate after TTFF also increases slightly, 184 Page 8 of 12 reaching a maximum of 99%.Compared to the results in Fig. 7, there is a more obvious increasing trend when assessing the fix rate of all the processed epochs.The overall fix rate in Fig. 8 is correlated with the interval of data batches due to the change of denominator.And a larger overall fix rate usually indicates a faster convergence and a reliable positioning performance.Even with 13 sites used at the server to generate the atmospheric corrections, the overall fix rate can be improved by 23% compared to PPP-AR with data arcs of one hour.However, with increasing the number of sites over 160, the PPP-RTK overall fix rate does not improve significantly.The analysis of fix rate infers that the more stations used at the server, the higher the fix rate, and therefore, the more reliable the user performance.The overall fix rate is slightly smaller than reported in Zhang et al. (2022); the main reason is the different data length.

Positioning performance analysis
The 90 th percentile of positioning errors for horizontal and vertical components is first evaluated.As shown in Fig. 9, different colors represent positioning error time series for different schemes.For both horizontal and vertical components, the PPPRTK-160 and PPPRTK-all perform similarly.For 90% of the test datasets, it takes less than 5 min to converge to 0.1 m and solutions can reach a positioning accuracy of 0.03 m and 0.05 m within 20 min for the horizontal and vertical components, respectively.The PPP-AR performance without any external atmosphere correction has the largest positioning errors and the longest convergence time.Table 3 summarizes the detailed positioning performance for both the average and the 90 th percentile convergence time, and the corresponding median server-user distances.The average convergence time of PPP-AR is 25 and 21 min for horizontal and vertical components, respectively, and it can be shortened to 2 and 3 min, correspondingly, when using all the sites for generating corrections, i.e., representing improvements of 93% and 85%.
There is a common decreasing trend in the convergence time when the number of sites used at the server is increasing, and even with only 13 sites at the server, the convergence time can be shortened largely.However, when more than 160 sites are used, the improvements are limited to 10% and 9% for horizontal and vertical directions, respectively.The convergence time in this study is comparable with the results in Psychas et al. (2020), in which it is reported with partial AR, and the 90th percentile of convergence time for 68 km network density is 5 min, which infer that a reliable a priori precision can improve the convergence time.Compared with the state-of-the-art Network RTK (NRTK) performance, which can achieve centimeter-level positioning accuracy in around 5 s with the network configuration of 70-100 km (European GNSS Agency 2019), PPP-RTK requires a longer time to converge with a comparable network configuration.The reason may be the greater number of estimates in PPP-RTK compared to NRTK.But at the same time, the broadcasted information for NRTK will increase with the extension of satellite frequencies, while it does not change for PPP-RTK.

Conclusions and remarks
With different-scaled networks, the PPP-RTK corrections at the server and positioning performance at the user have been assessed.The results can be a reference on deciding the balance of the server budget and user performance.
The simulation results of PPP-RTK show a strong positive correlation between the accuracy of the atmospheric corrections and achieved positioning precision.The precision improvement in the vertical direction is more significant than in the horizontal component.It is mainly attributed to tropospheric corrections contributing mainly to the vertical component.In addition, the results indicate a priori precision of atmospheric correction is of vital importance and highly related to positioning precision.Second, results from the server PPP-RTK corrections show that when inter-station distances are shortened from 251 to 50 km, the improvement of the ionospheric uncertainty map becomes significant; however, any further shortening of inter-station distances improves the maps only slightly.The ionospheric precision becomes worse with an increase in altitude at the server margin areas because of the coverage of server stations.And the corrections are more precise at 0 o'clock compared with 14 o'clock, which is due to a more active ionosphere in the afternoon compared to midnight.
Finally, user positioning results show that ionosphere refractions impact the convergence time, TTFF, and fixing rate, while the troposphere effects impact mainly the improvements of vertical positioning.With corrections generated by all server stations, the TTFF can reach 2 min, which is an obvious improvement compared to PPP-AR.Even using corrections from 13 stations with an interstation distance of 251 km, the TTFF can be shortened by 40%.With more stations included at the server, the interstation distances are smaller, the atmospheric corrections are more reliable, and therefore the position is achieved in shorter TTFF.Aside from TTFF, the fix rate is also evaluated, and results show that the more dense network at the server, the higher the fix rate, and hence, the more reliable the user performance.However, when the interstation distances are within 50 km, the improvement is not significant.This result is consistent with the assessment of the ionospheric uncertainty maps as well as the evaluation of the user performance with almost identical positioning  errors in PPPRTK-160 and PPRTK-ALL simulations.The convergence can be speed up when inter-station distances become shorter, and even with only 13 stations used, the average convergence time can be shortened by 93% and 85% in horizontal and vertical components, respectively, compared to PPP-AR.In this study, only GPS constellation is used to generate the atmosphere correction since the ionospheric delay, which contributes to the majority of the atmosphere errors, is implemented satellite-wise.In future work, PPP-RTK correction generation using multi-constellations and multifrequency will be investigated, and there is a potential that with multi-constellations, a sparse network may contribute to the same performance as described in this contribution and will satisfy user applications.An accurate and reliable a priori precision is the key factor to speed up the convergence and TTFF; therefore, it is also worth investigating a best fit precision model for atmospheric corrections to further improve the correction quality and user performance.

Fig. 1
Fig. 1 Simplified PPP-RTK flowchart at the server and the user.The server side (upper) contains generation of correction products, and the user side (bottom) utilizes these corrections for PPP-RTK

Fig. 2
Fig. 2 Relationship between horizontal/vertical position precision and ionospheric/tropospheric correction precision, as well as the benefit of fixing WL ambiguity

Fig. 3
Fig. 3 Site distributions at the server (green) and user (red) with different inter-station distances

Fig. 6 Fig. 7
Fig. 6 Time-to-first-fix of users using corrections from different scales of networks

Fig. 8
Fig.8Fix rate of all processed data batches of users using corrections from different scales of networks

Table 1
Receiver and antenna information for user stations

Table 3
Statistics of convergence time for different schemes (unit: minute)