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Local troposphere augmentation for realtime precise point positioning
Earth, Planets and Space volume 66, Article number: 30 (2014)
Abstract
The IGS realtime service (RTS) enables realtime precise point positioning (PPP) at a global scale. A long convergence time however is still a challenging factor. In order to reduce the convergence time, external troposphere corrections could be introduced to remove the troposphere effects on the coordinate solution. This paper proposes the use of a local troposphere model to augment realtime PPP. First, undifferenced observations from a network of multiple stations are processed to estimate the stationbased troposphere zenith wet delay (ZWD). A set of local troposphere fitting coefficients are then derived using a proposed optimal fitting model. Finally, the determined troposphere fitting coefficients are broadcast to users to reduce the convergence time in the user solution. A continuous operating reference station (CORS) network is utilized to assess the performance of the proposed approach under quiet and active troposphere conditions. The numerical results show that the overall fitting precisions of the local troposphere model can reach 1.42 and 1.05 cm under the two troposphere conditions. The convergence time of the positioning solutions, especially the height solution, can be greatly reduced using the local troposphere model. The horizontal accuracy of 9.2 cm and the vertical accuracy of 10.1 cm are obtainable under the quiet troposphere condition after 20 min of initialization time, compared to the 14.7 cm horizontal and 21.5 cm vertical accuracies in the conventional troposphere estimation approach. Moreover, the horizontal accuracies of 13.0 cm and the vertical accuracies of 12.4 cm have also been obtained after 20 min under the active troposphere condition.
Background
Investigations on realtime satellite orbit and clock products for realtime precise point positioning (PPP) began about one decade ago. Gao and Chen (2004) carried out the performance analysis of PPP using Jet Propulsion Laboratory (JPL)’s realtime orbit and clock corrections for static and kinematic applications. The obtainable positioning accuracies are comparable to those using International Global Navigation Satellite System (GNSS) Service (IGS) final orbit and clock products that however are available with at least 17h latency (Kouba 2009). Tao (2008) investigated the performance of PPPinferred troposphere estimates using JPL’s realtime products and concluded that the obtained realtime PPPinferred zenith wet delay had an accuracy of approximately 13 mm. In 2007, IGS initiated the realtime pilot project (RTPP) with the infrastructure of realtime GNSS data streams on a global basis. Based on the realtime GNSS observations, several realtime orbit and clock products are generated by participating IGS RTPP agencies. Using BKG’s realtime corrections, Altiner et al. (2010) demonstrated a horizontal accuracy of 10 cm and a vertical accuracy of 20 cm for realtime PPP with 17h observations at station CONZ. As to the convergence time, 10 min were found sufficient to achieve 10cm horizontal accuracy at station FFMJ. A bias of 40 cm, however, was identified in the vertical solution. Sturze et al. (2012) demonstrated a horizontal accuracy of 4 to approximately 5 cm after convergence based on daily solutions from six participating IGS RTPP agencies. Wang et al. (2013) assessed the near realtime PPPinferred troposphere parameter, using Centre National d’Etudes Spatiales (CNES)’s realtime corrections, and found a mean bias of approximately 6.5 mm and a root mean square (RMS) error of approximately 13 mm for the zenith wet delay compared to those using postmission products. Using a satellitedifference and epochdifference method, Li et al. (2010) and Chen et al. (2013) demonstrated a horizontal accuracy of 5 cm using hourly observations in the static mode and 3D precision of 10 cm after 20min convergence time in the kinematic mode. After the 6year experimental tests, IGS officially announced the realtime service (RTS) on 1 April 2013, which provides Global Positioning System (GPS) realtime orbit and clock corrections and experimental GLONASS corrections to support realtime PPP at a global scale (Caissy et al. 2012).
Although IGS RTS has solved the latency issue of precise satellite orbit and clock products, the long convergence time still remains a challenging factor for realtime PPP. In fact, the PPP convergence highly depends on the estimation of the troposphere delays. Therefore, regional troposphere models have been studied to reduce the convergence time for (near) realtime PPP. Ibrahim and EIRabbany (2011) evaluated the numerical weather prediction based on the National Oceanic and Atmospheric Administration (NOAA) tropospheric signal delay model within the North America. The results demonstrate that the PPP convergence using the NOAA troposphere model can be improved by 1%, 10%, and 15% for latitude, longitude, and height components, respectively, compared to that using the Hopfield model. Hadas et al. (2013) assessed the benefits of near realtime regional troposphere model for PPP. The IGS’s ultrarapid orbit and clock products are used to generate the near realtime regional troposphere model which greatly improves the height accuracy in simulated realtime PPP scenarios. Li et al. (2011) investigated the regional atmosphere augmentation for realtime PPP with instantaneous ambiguity resolution. Li’s method however requires the user to send approximate coordinates to the server in order to receive an interpolated troposphere delay from the server. Such twoway communication mode would increase the user’s communication cost and limit the server’s maximum number of allowed connections.
A realtime troposphere augmentation method is proposed in this paper to eliminate the limitation of the maximum number of allowed users and reduce the convergence time of realtime PPP. We first utilize the IGS realtime orbit and clock corrections to estimate the stationbased troposphere zenith wet delays based on a continuous operating reference station (CORS) network. A novel method is then proposed to determine the fitting coefficients of the optimal local troposphere model which does not require the user’s approximate coordinates and can be broadcast to unlimited users. Since this approach is based on the oneway communication mode, it eliminates the need for users to communicate with the server and also has no limit on the number of allowed users.
The paper will be organized as follows. The Section ‘Methods’ describes the flowchart and the mathematics of the proposed local troposphere augmentation for realtime PPP. Section ‘Results and discussion’ analyzes two GPS daily observations under quiet and active troposphere conditions in terms of the accuracy of IGS realtime orbit and clock products, the precisions of the local troposphere model, and the benefits of local troposphere augmentation to realtime PPP. Finally, some conclusion remarks and future works are given in Section ‘Conclusions.’
Methods
Figure 1 shows the flowchart that describes the proposed realtime local troposphere model determination at the server end and the realtime PPP with local troposphere augmentation at the user end. For the determination of the realtime local troposphere model, the IGS realtime orbit and clock corrections are needed at each reference station to perform realtime PPP which estimates four types of parameters: the coordinates, the receiver clock error, the troposphere delay, and the ambiguity parameters. As the troposphere zenith hydrostatic delay (ZHD) can be corrected using the global pressure and temperature (GPT) model and the Saastamoinen model (Boehm et al. 2007; Gérard and Luzum 2010), the estimated troposphere parameter is the troposphere zenith wet delay (ZWD). The stationbased ZWD values of the reference stations are utilized to determine a set of optimal troposphere ZWD fitting coefficients which are then broadcast to the user. The user needs to calculate the ZWD value based on its geographic location and the received troposphere ZWD fitting coefficients. In order to maximize the performance, the ZHD at the user end must also be corrected using the GPT and Saastamoinen models as those used at the server end. The calculated ZWD is then applied to remove the residual troposphere delay at the user end. As a result, realtime PPP at the user end only needs to estimate three types of parameters: the coordinates, the receiver clock error, and the ambiguities.
The methods for determining a local troposphere model can be classified into two categories. The first category requires the user to send its approximate coordinates to the server and receive the interpolated troposphere delay from the server in a twoway communication manner. One implementation is the virtual reference station (VRS) technology (Alves and Monico 2011). Dai et al. (2004) summarized the equivalence of the different troposphere interpolation methods requiring the twoway communication connection. On the other hand, if a set of local troposphere fitting coefficients could be broadcast from the server to the user, then there is no need for the user to communicate with the server and the maximum number of allowed users becomes unlimited. Xiong et al. (2006) and Zhang et al. (2013) investigated the local troposphere model determination using several empirical fitting functions. Although the user communication cost has been eliminated and the maximum number of allowed users becomes unlimited, whether the utilized empirical fitting functions were optimal has not been investigated.
This paper proposes a method to determine the optimal fitting coefficients for local troposphere modeling. The general form of the secondorder fitting model consists of the observation equations
and the constraint equations
where n denotes the number of reference stations, the subscript i denotes the index of the reference stations, ZWD_{ i } is the troposphere zenith wet delay at the i th station, (x_{ i }, y_{ i }) are the Gaussian projection horizontal coordinates; h_{ i } is the ellipsoid height; (a_{0}, a_{1}, …, a_{9}) are ten fitting coefficients; the subscript j denotes the index of the constraint coefficients; φ_{ j } ∈ {0, 1} is the constraint coefficient of a_{ j }.
The key to determining the optimal local troposphere model is to compute the optimal fitting coefficients. Unlike the previous work using empirical fitting functions, the optimal local troposphere model is always expressed as (1) with one set of ten fitting coefficients (a_{0}, a_{1}, …, a_{9}) satisfying the userdefined optimization criterion. The determination of the optimal fitting coefficients is described as follows. In step I, no constraint equation is constructed. The number of the constraint equation candidates is {\mathit{C}}_{10}^{0}=\frac{10!}{0!*\left(100\right)!}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}1. Then only n observation Equation 1 is used to estimate the set of fitting coefficients (a_{0}, a_{1}, …, a_{9}) in a leastsquare estimator. In step II, one constraint equation is constructed. The number of the constraint equation candidates is {\mathit{C}}_{10}^{1}=\frac{10!}{1!*\left(101\right)!}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}10. We can form ten combinations of one constraint Equation 2 and n observation Equation 1 which are then used to estimate ten sets of fitting coefficients (a_{0}, a_{1}, …, a_{9}). In step III, two constraint equations are constructed. The number of the constraint equation candidates is {\mathit{C}}_{10}^{2}=\frac{10!}{2!*\left(102\right)!}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}45. We can form 45 combinations of two constraint Equation 2 and n observation Equation 1 which are then used to estimate 45 sets of fitting coefficients (a_{0}, a_{1}, …, a_{9}). In step IV, we increment the size of constraint equations by 1 until the size increases to 10. The total number of the fitting coefficients sets is {\displaystyle \sum _{\mathit{k}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}0}^{10}{\mathit{C}}_{10}^{\mathit{k}}}=1,024. In step V, an optimization criterion is defined such as the minimization of the sum of square (SOS) of the troposphere fitting errors at n reference stations. The set of the fitting coefficients meeting the optimization criterion is determined as the optimal fitting coefficients for local troposphere modeling.
Note that in order to determine the optimal secondorder local troposphere model with ten fitting coefficients, a minimum of ten reference stations are required. If the number of reference stations is less than 10, we can remove several fitting coefficients from Equation 1. For example, if we have nine reference stations and the height distribution of these reference stations is relatively smooth, we can remove the heightdependent coefficients (a_{3}, a_{5}, a_{6}, a_{9}) and keep only the horizontaldependent coefficients (a_{0}, a_{1}, a_{2}, a_{4}, a_{7}, a_{8}).
Using the general fitting model (Equations 1 and 2) and defining the optimization criterion, the optimal fitting coefficients can be obtained and broadcast to the users. With the user’s approximate coordinates (x_{ i }, y_{ i }, h_{ i }) and the received fitting coefficients (a_{0}, a_{1}, …, a_{9}), the troposphere ZWD at the user end can be computed by Equation 1.
Results and discussion
In this section, the datasets and the processing strategies used to evaluate the proposed method are first described. Two case studies concerning quiet and active troposphere conditions are analyzed afterwards to assess the performance of realtime PPP by local troposphere augmentation.
Data description and processing strategy
Two GPS daily observations with quiet and active troposphere conditions are processed based on a CORS network of nine stations shown in Figure 2. The longest distance is 191 km between no. 3 and no. 7, while the shortest distance is 18 km between no. 2 and no. 6. The maximum height difference is 26 m between no. 3 and no. 5. GPS observations are collected at the sampling interval of 30 s.
In this paper, we concern two different situations with quiet and active troposphere conditions on the day of year (DOY) 2013 127 and 185, respectively. The weather condition on DOY 127 is relatively stable compared with those on the preceding and following days. A severe rainfall event, however, occurred on the following days of DOY 185, indicating an accumulating phase of the troposphere ZWD on DOY 185. For each case study, we first evaluate the accuracies of the realtime satellite orbit and clock products. The determination of the optimal ZWD fitting coefficients are then explained, followed by an assessment of the performance improvements of the realtime PPP with local troposphere augmentation over the conventional troposphere estimation approach. PPP software package used in this paper is P^{3} developed by the Positioning and Mobile Information Systems Group of Department of Geomatics Engineering at The University of Calgary (Gao 2005) with some general processing settings summarized in Table 1. It should be noted that although we use realtime satellite corrections, the experiments are implemented in a simulated realtime mode. However, it will not reduce the general applicability and performance of the realtime PPP.
Case study 1: quiet troposphere condition
Realtime satellite orbit and clock corrections
In addition to the IGS combined correction stream, orbit and clock correction streams from several participating IGS RTS agencies are also provided for special considerations. For example, GFZ has been hosting a realtime PPP project with local reference network augmentation (Li et al. 2011); CNES has been developing a realtime zerodifference PPP integer ambiguity resolution demonstrator (Laurichesse 2011). Therefore, realtime PPP users can choose the proper correction stream to match their specific application. In this paper, the realtime correction stream from CNES with IGS mountpoint ID CLK90 is selected because of its potential to support realtime PPP with integer ambiguity resolution. In this subsection, we will evaluate the accuracy of CNES realtime orbit and clock corrections which will be used for the determination of local troposphere model at the server end and the realtime PPP with local troposphere augmentation at the user end.
In order to obtain the precise satellite coordinates, an initial coordinate vector r and the velocity vector \dot{\mathit{r}} should be first calculated based on the broadcast ephemeris. Then, the IGS realtime orbit corrections encoded as the Radio Technical Commission for Maritime Services (RTCM) space state representative (SSR) messages (RTCM Special Committee 104 2011) are applied to the initial coordinate vector by
where r_{SSR} is the corrected coordinate vector; {\mathit{e}}_{\mathrm{radial}}=\frac{\dot{\mathit{r}}}{\left\dot{\mathit{r}}\right},{\mathit{e}}_{\mathrm{along}}=\frac{\mathit{r}\times \dot{\mathit{r}}}{\left\mathit{r}\times \dot{\mathit{r}}\right},{\mathit{e}}_{\mathrm{cross}}={\mathit{e}}_{\mathrm{radial}}\times {\mathit{e}}_{\mathrm{along}};\mathit{\delta O} is the SSR orbit correction vector in radial, along, and crosstrack components.
As to the precise satellite clock error, the application of SSR clock correction is represented by
where dt^{s}_{SSR} is the corrected satellite clock error; δC = C_{0} + C_{1}(t  t_{0}) + C_{2}(t  t_{0})^{2} is the clock correction; t is the broadcast clock time; t_{0} is the reference time obtained from SSR clock correction messages; C_{0}, C_{1}, C_{2} are three clock correction coefficients in the SSR clock correction messages; c is the speed of light in vacuum.
Oneday CNES realtime orbit and clock corrections are collected on DOY 127 to conduct the accuracy evaluation. The IGS 15min final orbit and 30s final clock products are used as reference. Lagrange interpolation is applied for the orbit product to match the GPS observations at 30 s interval. The orbit accuracy for each satellite is expressed as the RMS error of the differences between the calculated coordinates and the reference coordinates. As to the clock’s accuracy, a reference satellite is first selected to make a single difference with the other satellites in order to remove the clock datum inconsistency between the CNES realtime and IGS final clock products. In this paper, GPS PRN no. 1 is chosen as the reference satellite. The satellite clock accuracy is calculated as the RMS error of the differences between the calculated singledifferenced clocks and the reference clocks.
Figures 3 and 4 illustrate the RMS accuracies of broadcast and SSRcorrected orbits and clocks with respect to the IGS final products on DOY 2013 127. Obviously, the orbit accuracies with SSR corrections (0.025/0.025/0.025 m in XYZ directions) are much better than the orbit accuracies without corrections (0.839/0.798/0.798 m in XYZ directions). As to the SSRcorrected clocks, an overall accuracy of 0.441 ns is obtained while that for the broadcast clocks is 3.085 ns.
Determination of optimal local troposphere fitting coefficients
CNES realtime orbit and clock corrections are applied to conduct realtime PPPinferred troposphere estimation at the reference stations. The a priori troposphere zenith hydrostatic delay is calculated based on GPTderived climatological data and Saastamoinen model. The residual troposphere zenith wet delay is estimated in a random walk pattern. Other processing settings are the same as those for the troposphere estimation approach listed in Table 1.
As the number of the reference stations is less than 10, the general secondorder fitting model cannot be conducted. We remove the secondorder terms from Equations 1 and 2 to perform the general firstorder fitting model which only considers four fitting coefficients (a_{0}, a_{1}, a_{2}, a_{3}):
The troposphere ZWD values are chosen at the interval of 30 min between UTC 02:00 and 22:00 so that 42 samples are available for each station. At each sampling epoch, we use eight troposphere ZWD values to determine the troposphere fitting coefficients and interpolate the troposphere zenith wet delay ZWD_{interpolated} for the ninth station. The interpolated troposphere delay is then utilized to evaluate the precision of the local troposphere model along with the estimated troposphere zenith wet delay ZWD_{estimated}. The optimum criterion is defined as
According to the optimum criterion (6), the optimum local troposphere fitting coefficients for the test network are determined with a SOS value of 1.42 cm. The troposphere ZWD fitting precisions for each station are listed in Table 2. The interpolated ZWD series at station no. 5 are taken as an example in Figure 5. For the purpose of comparison, the estimated ZWD values at station no. 5 are also illustrated. The interpolated ZWD values match the estimated ZWD values quite well with a maximum discrepancy of approximately 3 cm and an average discrepancy of 1.55 cm.
Realtime PPP with local troposphere augmentation
To evaluate the performance of realtime PPP by local troposphere augmentation, we split the daily observations into 2h data sets for the test stations. A total number of 90 2h datasets are obtained and have been processed using the CNES realtime corrections described in subsection ‘Realtime satellite orbit and clock corrections.’ The local troposphere model has also been determined in subsection ‘Determination of optimal local troposphere fitting coefficients.’ The troposphere ZHD is corrected by the GPT and Saastamoinen models, whereas the troposphere ZWD is corrected by the interpolated ZWD based on the local troposphere fitting coefficients so that no troposphere estimation is needed at the user end. All other settings can be found as the troposphere augmentation approach in Table 1.
Figure 6 demonstrates the north/east/up coordinate solutions with troposphere estimation and augmentation during UTC 18:00 and 20:00 of station no. 5 as an example. The two horizontal coordinate solutions with troposphere augmentation converge slightly faster than those with troposphere estimation. On the other hand, the convergence of the vertical solution is significantly improved using the local troposphere augmentation. When the troposphere parameter is estimated, the height solution converged after 1 h, but it requires only about 15 min with local troposphere augmentation.
Although the improvements by the local troposphere augmentation as shown in Figure 6 are found significant, the satellite geometry during the observation period is also an important factor which would affect the convergence of realtime PPP. Therefore, we apply the same processing strategy used for station no. 5 to the rest sessions and the rest stations. Given in Figures 7 and 8 are the horizontal and vertical positioning RMS accuracies after three selected initialization periods, namely 20, 60, and 100 min. The horizontal RMS accuracy after 20 min is 13.7 cm with troposphere estimation whereas it reduces to 9.4 cm when the local troposphere augmentation is applied. A longer initialization time of 60 or 100 min, however, does not bring significant improvements as the horizontal RMS accuracy differs slightly (i.e., 7.5 versus 5.6 cm or 6.4 versus 5.3 cm). As for the height solution, the RMS accuracy is improved from 18.3 to 10.1 cm after the 20min initialization time, a much more significant improvement than the horizontal counterpart. It can also be seen from Figure 8 that there is still an approximately 10cm bias for the height solutions with local troposphere augmentation. This bias is mainly caused by the discrepancy between the estimated and interpolated troposphere ZWD listed in Table 2. In summary, the local troposphere model can augment realtime PPP in terms of positioning accuracy and convergence by eliminating the troposphere effects on the coordinate solution. However, the positioning accuracies are still limited by other factors including the satellite geometry and ambiguity fractional biases which cannot be corrected by the local troposphere model. To further improve the positioning performance as shown in Figures 7 and 8, the integer phase ambiguity resolution technique should be considered in the future for realtime PPP with local troposphere augmentation.
Case study 2: active troposphere condition
The second case study describes a troposphere ZWD accumulating phase on DOY 2013 185, 1 day before a severe rainfall event on the following days. All the processing settings and strategies are the same as those used in the first case study. First, the CNES realtime orbit and clock products are analyzed with results shown in Figures 9 and 10. The overall accuracies of approximately 5 cm and 0.523 ns are obtained for satellite orbits and clock errors, respectively. Second, the precisions of local troposphere model for each station are summarized in Table 3 with an overall precision of 1.05 cm. The interpolated and estimated ZWD time series at station no. 5 are again taken as an example in Figure 11. Unlike the relatively stable pattern of the ZWD time series on DOY 127, an ascending pattern has been identified for the daily ZWD time series on DOY 185. Third, Figures 12 and 13 illustrate RMS accuracies of the horizontal and vertical coordinate solutions. Similar to the first case study, more significant improvements can be detected in the vertical component than the horizontal component. The horizontal RMS accuracies of 12.4 cm and the vertical RMS accuracies of 13.0 cm have been obtained after 20min initialization time.
Conclusions
The IGS realtime service enables realtime PPP at a global scale. But realtime PPP is still limited by the long convergence time. In order to reduce the convergence time, this paper proposes a local troposphere augmentation method to help eliminate the troposphere effects and accelerate PPP convergence. An optimal local troposphere fitting model is proposed in this paper. The determined optimal local troposphere model is broadcast to the users, so there is no need for the user to establish communication connection to the server and no limit on the maximum number of allowed users.
The proposed method has been evaluated by a CORS network under quiet and active troposphere conditions on DOY 2013 127 and 185, respectively. First, the realtime orbit and clock corrections are assessed against the IGS final products. The results show that the current IGS CLK90 realtime stream can provide <5 cm orbit and approximately 0.5 ns clock accuracies on both days. Second, using the proposed method the optimal local troposphere model has been determined with the overall precision of 1.42 cm on DOY 127 and 1.05 cm on DOY 185. Third, the determined local troposphere model is applied to augment realtime PPP at the user end. The results indicate that realtime PPP with local troposphere augmentation can provide 9.2cm horizontal and 10.1cm vertical positioning accuracies after 20 min of initialization time under the quiet troposphere condition, which is a significant improvement over the 13.2cm horizontal and 18.3cm vertical accuracies of realtime PPP with troposphere estimation. Moreover, the horizontal accuracies of 13.0 cm and the vertical accuracies of 12.4 cm have been obtained after 20min initialization time under the active troposphere condition, compared to the 16.0cm horizontal and 23.4cm vertical accuracies in the conventional troposphere estimation approach.
In addition to local troposphere augmentation, the ambiguity fractional bias is another limiting factor to realtime PPP. We have already demonstrated the improvement of PPP ambiguityresolved height solution using troposphere corrections in a postmission mode (Shi and Gao 2014). An investigation should be conducted in the future to further improve the positioning performance by considering the impacts of local troposphere augmentation on realtime PPP with integer ambiguity resolution.
So far this paper only concerns the local CORS network within a small coverage. More efforts will also be considered for the ZWD fitting model determination and transition for regional CORS networks with larger coverage in the future.
Authors’ information
SJ is a lecturer of Wuhan University. His current research involves GNSS precise point positioning and GNSS meteorology. XC is a PhD candidate of Wuhan University. His current research is regional troposphere and ionosphere modeling. GJ is a professor of Wuhan University. His current research involves GNSS positioning and engineering surveying. GY is a professor of The University of Calgary. His research expertise includes both theoretical aspects and practical applications of satellite positioning and navigation systems.
Abbreviations
 CORS:

Continuous Operating Reference Station
 DOY:

day of year
 GNSS:

Global Navigation Satellite System
 GPS:

Global Positioning System
 GPT:

global pressure and temperature
 IGS:

International GNSS Service
 PPP:

precise point positioning
 RMS:

root mean square
 RT:

real time
 RTCM:

Radio Technical Commission for Maritime Services
 RTPP:

RealTime Pilot Project
 SOS:

sum of square
 SSR:

space state representative
 VRS:

virtual reference station
 ZHD:

zenith hydrostatic delay
 ZWD:

zenith wet delay.
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Acknowledgements
IGS and CNES are acknowledged for providing the postmission and realtime satellite precise orbit and clock products. We would like to acknowledge the anonymous reviewers and the editor for their valuable comments on this manuscript. This work has been supported by the National Key Developing Program for Basic Sciences of China (Grant No. 2012CB719902), National Natural Science Foundation of China (Grant No. 41371432), Liaoning Talent Program (Grant No. LR2011007), and Key Laboratory of Precise Engineering and Industry Surveying, National Administration of Surveying, Mapping and Geoinformation (Grant No. PF201213).
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SJ initiates and leads this research. XC participates in the data processing, analysis, and makes the figures. GJ organizes the group discussion and involves in drafting the manuscript. GY involves in the analysis and discussion of the results, critically revises the manuscript, and proofreads the submission. All authors read and approved the final manuscript.
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Shi, J., Xu, C., Guo, J. et al. Local troposphere augmentation for realtime precise point positioning. Earth Planet Sp 66, 30 (2014). https://doi.org/10.1186/188059816630
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DOI: https://doi.org/10.1186/188059816630
Keywords
 Realtime precise point positioning
 Local troposphere augmentation
 Optimal fitting model
 Convergence time