Retrieving Precipitable Water Vapor from Real-Time Precise Point Positioning Using VMF1/VMF3 Forecasting Products

: Real-time precise point positioning (RT-PPP) has become a powerful technique for the determination of the zenith tropospheric delay (ZTD) over a GPS (global positioning system) or GNSS (global navigation satellite systems) station of interest, and the follow-on high-precision retrieval of precipitable water vapor (PWV). The a priori zenith hydrostatic delay (ZHD) and the mapping function used in the PPP approach are the two factors that could affect the accuracy of the PPP-based ZTD signiﬁcantly. If the in situ atmospheric pressure is available, the Saastamoinen model can be used to determine ZHD values, and the model-predicted ZHD results are of high accuracy. However, not all GPS/GNSS are equipped with an in situ meteorological sensor. In this research, the daily forecasting ZHD and mapping function values from VMF1 forecasting (VMF1_FC) and VMF3 forecasting (VMF3_FC) products were used for the determination of the GPS-derived PWV. The a priori ZHDs derived from VMF1_FC and VMF3_FC were ﬁrst evaluated by comparing against the reference ZHDs from globally distributed radiosonde stations. GPS observations from 41 IGS stations that have co-located radiosonde stations during the period of the ﬁrst half of 2020 were used to test the quality of GPS-ZTD and GPS-PWV. Three sets of ZTDs estimated from RT-PPP solutions using the a priori ZHD and mapping function from the following three VMF products were evaluated: (1) VMF1_FC; (2) VMF3_FC (resolution 5 ◦ × 5 ◦ ); (3) VMF3_FC (resolution 1 ◦ × 1 ◦ ). The results showed that, when the ZHDs from 443 globally distributed radiosonde stations from 1 July 2018 to 30 June 2021 were used as the reference, the mean RMSEs of the ZHDs from the three VMF products were 5.9, 5.4, and 4.3 mm, respectively. The ZTDs estimated from RT-PPP at 41 selected IGS stations were compared with those from IGS, and the results showed that the mean RMSEs of the ZTDs of the 41 stations from the three PPP solutions were 8.6, 9.0, and 8.6 mm, respectively, and the mean RMSEs of the PWV converted from their corresponding ZWDs were 1.9, 2.4, and 1.7 mm, respectively, in comparison with the reference PWV from co-located radiosonde stations. The results suggest that the a priori ZHD and mapping function from VMF1_FC and VMF3_FC can be used for the precise determination of real-time GPS/GNSS-PWV in most regions, especially the VMF3_FC (resolution 1 ◦ × 1 ◦ ) product. converted from the ZWDs at the 41 stations were 1.9, 2.4, and 1.7 mm, respectively, compared to the reference PWV from co-located radiosonde data. These results suggest that the a priori ZHD and mapping function from both VMF1_FC and VMF3_FC can be applied to the PPP approach for obtaining real-time GNSS-PWV in most regions.


Introduction
The GNSS (global navigation satellite systems), initially designed for positioning, navigation, and timing, has now been used in the field of atmospheric sensing. GNSS measurements are subject to various types of errors, particularly atmospheric errors, i.e., the tropospheric and ionospheric delay errors. The tropospheric delay needs to be estimated along with all the other unknown parameters in the GNSS data processing. For reducing the number of unknown parameters to be solved for, the slant tropospheric delay (STD) of mid-latitude and high-latitude regions, compared to that at low-latitude regions. Moreover, atmospheric pressure obtained from NWM data (including reanalysis data and forecasting data) can also be used as the input variable of a standard ZHD model to obtain the ZHD. The reanalysis data, such as ERA5 reanalysis, are only available in the post-processing mode; thus, it is not suitable for real-time applications. A priori ZHD and mapping functions obtained from forecasting NWM data can be used for real-time GNSS data processing, but this may increase the computational burden of parameter estimation. Fortunately, the Vienna mapping functions 1 (VMF1) [2] and Vienna mapping functions 3 (VMF3) [37] data, which contain forecast results (named VMF1_FC and VMF3_FC, respectively), are routinely generated by the Vienna University of Technology (TU Wien, TUW) and published on the VMF Data Server [38], which provides access to the ZTD (including the ZHD and ZWD) and mapping function from the NWM. VMF-like products are also provided by the University of New Brunswick (UNB, http://unb-vmf1.gge.unb.ca/, accessed on 26 July 2021) and GeoForschungsZentrum Potsdam (GFZ, ftp://139.17.3.3/, accessed on 26 July 2021). Yao et al. [39] evaluated the accuracy of the ZTD derived from VMF1_FC grids by comparing it against the IGS tropospheric product. Yuan [40] stated that the ZTD and position estimated from RT-PPP were improved by the use of VMF1_FC, compared to the one that utilizes empirical tropospheric models. However, little work has been reported on the accuracy of the a priori ZHD from VMF1_FC and VMF3_FC, as well as the ZTD and PWV resulting from RT-PPP utilizing ZHD and the mapping function from VMF1_FC and VMF3_FC. This research mainly focused on the assessment of the performance of these products. This paper is organized as follows. Section 2 describes the mathematical model of real-time uncombined PPP and the method of converting GNSS-ZWD to PWV. Data and processing strategies employed in this research are introduced in Section 3. In Section 4, the accuracies of various results, including the a priori ZHDs from VMF1_FC and VMF3_FC, the ZTD estimated from RT-PPP, the application of VMF1_FC and VMF3_FC, and the ZWD-converted PWV were evaluated. The summary and conclusions are presented in the last section.

Materials and Methods
In this section, the data sources and processing strategies employed in this research are depicted.

Real-Time Orbits and Clocks
In this research, CLK93 real-time orbit and clock corrections from DOY (day of year) 1 to 180 in 2020 were decoded and stored to the local disk through the BNC software [41]. However, due to an internet connection problem of the computer for recording the CLK93 corrections, only 154 days of corrections were available for the data processing of the RT-PPP. It is noted that the RT-PPP algorithm was carried out in a simulated real-time mode as the observation data were from data files, rather than streams (which are truly real-time data).

GPS Data and Processing Strategies
GPS observation files from DOY 1 to 180 in 2020 downloaded from the FTP server of Wuhan University were used for testing. The 41 IGS stations, as is shown in Figure 1, were selected based on the following criteria: (1) As the reference of the PPP-derived ZTD, the ZTD from IGS products is available. (2) A radiosonde station that has a less than 10 km horizontal distance and a less than 100 m height difference from the IGS station is available. A modified BNC software was developed to support the real-time dual-frequency uncombined PPP algorithm for this study. The strategies utilized in the RT-PPP data processing are shown in Table 1.
where is the a priori sigma of observations; = 1, = 4, and = 8 ar is the elevation angle.  The elevation-dependent stochastic model recommended by Hadas [24] was used to determine the weight of the observation: where σ 0 is the a priori sigma of observations; a = 1, b = 4, and n = 8 are constants; e is the elevation angle. The ZWD estimated from RT-PPP is converted into PWV using: where Π is the conversion factor, which is a function of the weighted mean temperature (T m ) along the vertical direction of the atmosphere over the receiver's site: where ρ w is the density of liquid water; R v = 461.5J/(kg • K) is the specific gas constant for water vapor; k 2 = 22.1 K/hPa and k 3 = 373, 900 K 2 /hPa are atmospheric refractivity constants. The procedure of the RT-PPP-based PWV retrieval utilizing VMF1/VMF3 forecasting products is shown in Figure 2.
where is the density of liquid water; = 461.5 J (kg • K ⁄ ) is the specific gas constant for water vapor; = 22.1 K hPa ⁄ and = 373,900 K hPa ⁄ are atmospheric refractivity constants.
The procedure of the RT-PPP-based PWV retrieval utilizing VMF1/VMF3 forecasting products is shown in Figure 2.

VMF Forecasting Data
The two types of VMF forecasting data provided by the VMF Data Server are gridbased data for any location around the globe and site-based data for specific GNSS, DO-RIS, VLBI stations. Only three grid-based VMF forecasting products, see Table 2, were tested in this research. The horizontal resolution of VMF1_FC is 2.5° × 2.0°, while VMF3_FC has two resolutions: 1.0° × 1.0° and 5.0° × 5.0°. The a priori ZHD and ZWD, as well as the coefficients of their corresponding mapping functions, are given for each of the grid points. All of these grid-based VMF forecasting data in 2020 were downloaded from the VMF Data Server [38] for the evaluation of the accuracy of their corresponding a priori ZHDs and the application for real-time retrieval of GPS-PWV. The vertical reduction model and the bilinear interpolation method recommended by Kouba [30] were utilized to obtain the ZHD for the target GPS station based on the height difference and horizontal distance between the station and its four surrounding grid points. First, the atmospheric pressure at the grid point can be calculated according to the Saastamoinen model [32,42]: where is the atmospheric pressure at the grid point; and H are the latitude and height of the grid point. Then, the atmospheric pressure at the target height over the grod point can be obtained by:

VMF Forecasting Data
The two types of VMF forecasting data provided by the VMF Data Server are gridbased data for any location around the globe and site-based data for specific GNSS, DORIS, VLBI stations. Only three grid-based VMF forecasting products, see Table 2, were tested in this research. The horizontal resolution of VMF1_FC is 2.5 • × 2.0 • , while VMF3_FC has two resolutions: 1.0 • × 1.0 • and 5.0 • × 5.0 • . The a priori ZHD and ZWD, as well as the coefficients of their corresponding mapping functions, are given for each of the grid points. All of these grid-based VMF forecasting data in 2020 were downloaded from the VMF Data Server [38] for the evaluation of the accuracy of their corresponding a priori ZHDs and the application for real-time retrieval of GPS-PWV. The vertical reduction model and the bilinear interpolation method recommended by Kouba [30] were utilized to obtain the ZHD for the target GPS station based on the height difference and horizontal distance between the station and its four surrounding grid points. First, the atmospheric pressure at the grid point can be calculated according to the Saastamoinen model [32,42]: where P 0 is the atmospheric pressure at the grid point; ϕ and H are the latitude and height of the grid point. Then, the atmospheric pressure at the target height over the grod point can be obtained by: where h is the height of the target station. Finally, the ZHD at the target height over the grid point can be calculated using the Saastamoinen model, followed by the horizontal interpolation of ZHD at the target station.

Radiosonde Data for Evaluating VMF1/VMF3 Forecasting ZHD
The radiosonde data were downloaded from the FTP server of the Integrated Global Radiosonde Archive (IGRA). Although radiosonde balloons at most radiosonde stations are released two to four times a day, only the profiles at 00:00 and 12:00 UTC were selected in this research. For the evaluation of the ZHD resulting from VMF1_FC and VMF3_FC, atmospheric pressures over the earth surface of 443 stations from 1 July 2018 to 30 June 2021 were selected as more than 1500 profiles were observed at these stations. The ZHD at the surface level of a selected radiosonde station was calculated using the Saastamoinen model. It should be noted that as radiosonde balloons were not necessarily released exactly at 00:00 and 12:00 UTC, the ZHD obtained from the surface atmospheric pressure was in fact of the release time rather than 00:00 and 12:00 UTC.

Radiosonde Data for Evaluating RT-PPP-Based PWV
For the evaluation of the accuracy of GNSS-PWV resulting from the above-mentioned VMF forecasting products, the reference PWV for each of the 41 selected IGS stations shown in Figure 1 was calculated from the observation profiles at its co-located radiosonde station through the integration defined below: where ρ is the density of liquid water; q is the specific humidity; g is the gravitational acceleration, which was set to 9.80665 m/s 2 in this research; p is the atmospheric pressure. It should be noted that in practice, a discretization is used for an approximation of Equation (7), as radiosonde observations are given at the observed pressure levels. In consideration of the height difference between the selected IGS station and its co-located radiosonde station, the atmospheric pressure for the height of the IGS station is obtained by the following reduction model: where P s and P 0 are the pressures at the height of the IGS station and the reference pressure level, respectively, and h s and h 0 are their heights; g m is the mean gravitational acceleration (9.80665 m/s 2 ). M is the constant for the molar mass of dry air (0.0289644 kg/mol); R is the universal gas constant (8.3143 J/K/mol); T v is the virtual temperature (in K) at the reference height, which can be calculated by: where T is the temperature (in Kelvin). The specific humidity was obtained by: where E is the water vapor pressure. A linear interpolation or extrapolation was performed to obtain the water vapor pressure at the height of the IGS station from its closest two radiosonde pressure levels. The reference PWV can then be obtained from the radiosonde profile starting with the specific humidity and atmospheric pressure at the height of the IGS station.

A Priori ZHD from VMF1_FC and VMF3_FC
The overall accuracy of the a priori ZHD derived from the three VMF products are listed in Table 2, and all testing data were measured by the bias and RMSE of all the n samples: where ZHD V MF and ZHD RS are the ZHD from the VMF product and radiosonde observations, respectively. Figure 3 shows the bias and RMSE of the ZHD predicted by the three VMF forecasting products at each of the 443 stations, and Table 3 shows the statistical results of all the stations. From the left panes (a1, a2, and a3) in Figure 3, we can see that most biases ranged from −0.5 to 0.5 cm, suggesting little biases in the VMF-based ZHDs in most regions. Those stations that had a warm bias were mainly located in low-latitude regions, especially in China and Southeast Asia; and those stations that had a large cold bias were mainly located in the Asian continent, northwest America, and Antarctica, especially from the first two products. The mean biases of the ZHDs predicted by products 1 to 3 were −2.2, 0.5, and 1.7 mm, respectively (see Table 3).
The right panels (b1, b2, and b3) in Figure 3 indicate that the RMSEs at most stations were in the range (0-5) mm, suggesting that the ZHDs from both VMF1_FC and VMF3_FC can be used to retrieve PWV from the RT-PPP-based approach at most stations. The mean RMSEs resulting from the three VMF products at all the 443 stations were 5.9, 5.4, and 4.3 mm, respectively, see Table 3. Those stations that had a large RMSE from the first two products were mainly located in Asia, North America, and Antarctica. According to Equation (3), if Π = 0.15, for a 5 mm error in ZHD, 0.75 mm in its resultant PWV is expected. Thus, these grid-based VMF1/VMF3 forecasting products should be evaluated before being applied to real-time PWV retrieval in some regions. s. 2021, 13, x FOR PEER REVIEW 8 of 16  Table 3. Statistical results of all the 443 stations shown in Figure 2 for the three products. Given that the ZHD above the target GNSS station is intepolated from four nearby grid points and the height difference between the target station and the grid points is adjusted using an empirical model, the accuracy of the VMF-predicted ZHD may be affected by the height differences [43]. In this research, a new parameter, named weighted absolute heigh difference (WAHD) was used for measuring the correlation between the abovementioned statistics and the height difference between the target station and the grid points: where ∆ is the height difference between the target station and the i ℎ nearby grid point; is the weight of the i ℎ height difference:  Table 3. Statistical results of all the 443 stations shown in Figure 2 for the three products. Given that the ZHD above the target GNSS station is intepolated from four nearby grid points and the height difference between the target station and the grid points is adjusted using an empirical model, the accuracy of the VMF-predicted ZHD may be affected by the height differences [43]. In this research, a new parameter, named weighted absolute heigh difference (WAHD) was used for measuring the correlation between the above-mentioned statistics and the height difference between the target station and the grid points: where ∆H i is the height difference between the target station and the ith nearby grid point; w i is the weight of the ith height difference: where d i is the horizontal distance between the target station and the ith nearby grid point. Figure 4 shows the correlation between WAHD and two statistics (the bias and RMSE in Figure 3). The mean WAHD of the three products were 286.9, 260.3, and 147.7 m, respectively, at 442 selected stations (the station that is located at 90 • S, 0 • E was not taken into consideration here). The mean WAHD of product 3 reduced significantly compared to those of products 1 and 2, probably due to its high spatial resolution. As is shown in the figure, stations with a large WAHD tend to have a large bias and RMSE, especially products 1 and 2. These results suggest that the topography should be considered when using these grid-based products. where is the horizontal distance between the target station and the i ℎ nearby grid point. Figure 4 shows the correlation between WAHD and two statistics (the bias and RMSE in Figure 3). The mean WAHD of the three products were 286.9, 260.3, and 147.7 m, respectively, at 442 selected stations (the station that is located at 90°S, 0°E was not taken into consideration here). The mean WAHD of product 3 reduced significantly compared to those of products 1 and 2, probably due to its high spatial resolution. As is shown in the figure, stations with a large WAHD tend to have a large bias and RMSE, especially products 1 and 2. These results suggest that the topography should be considered when using these grid-based products. The time series of the ZHD obtained from the GPT3 model and VMF1/VMF3 forecasting products in July 2018 at four radiosonde stations are shown in Figure 5. As is shown in the figure, the variation in ZHD can be captured by VMF1/VMF3 forecasting products. The GPT3 model performed worst as only the mean, annual, and semi-annual variations in the asmospheric pressure were modeled.

RT-ZTD Estimated from PPP
The RT-ZTD estimated from PPP that applied the three VMF products for the tropospheric delay at each of the 41 selected IGS stations (shown in Figure 1) was compared to the IGS-provided ZTD of the same station for the accuracy evaluation of the VMF product, and the statistical results of all the 154 days in 2020 are shown in Figure 6. It should be mentioned that the estimated ZWDs during the first 2 h of the convergence process of PPP were excluded from the evaluation results to ensure that all the ZTDs used to calculate the RMSE were converged results. As is shown, the RMSE of the ZTD estimates at any station was under 15 mm, and most of them were under 10 mm. In addition, the mean RMSEs of the ZTDs at all the 41 stations resulting from the three VMF products were 8.6, 9.0, and 8.6 mm, meaning that product 2 performed worst (especially at the DAV1 station (−68.58°S, 77.97°E), see the top pane in Figure 6). Besides, differences between the heights of the DAV1 station resulting from products 2 and 3 were also found from the testing. The RMS of the error in the height component at DAV1 station resulting from RT-PPP using products 1 to 3 were 2.92, 5.13, and 2.80 cm, respectively, compared to the IGS-provided

RT-ZTD Estimated from PPP
The RT-ZTD estimated from PPP that applied the three VMF products for the tropospheric delay at each of the 41 selected IGS stations (shown in Figure 1) was compared to the IGS-provided ZTD of the same station for the accuracy evaluation of the VMF product, and the statistical results of all the 154 days in 2020 are shown in Figure 6. It should be mentioned that the estimated ZWDs during the first 2 h of the convergence process of PPP were excluded from the evaluation results to ensure that all the ZTDs used to calculate the RMSE were converged results. As is shown, the RMSE of the ZTD estimates at any station was under 15 mm, and most of them were under 10 mm. In addition, the mean RMSEs of the ZTDs at all the 41 stations resulting from the three VMF products were 8.6, 9.0, and 8.6 mm, meaning that product 2 performed worst (especially at the DAV1 station (−68.58 • S, 77.97 • E), see the top pane in Figure 6). Besides, differences between the heights of the DAV1 station resulting from products 2 and 3 were also found from the testing. The RMS of the error in the height component at DAV1 station resulting from RT-PPP using products 1 to 3 were 2.92, 5.13, and 2.80 cm, respectively, compared to the IGS-provided coordinate. The mean RMS of the height error at the 41 selected stations resulting from the above-mentioned schemes were 3.16, 3.31, and 3.15 cm, respectively. These results suggest that the height component of the station coordinate is also affected by the a priori ZHD and mapping functions.
Sens. 2021, 13, x FOR PEER REVIEW 11 of 16 coordinate. The mean RMS of the height error at the 41 selected stations resulting from the above-mentioned schemes were 3.16, 3.31, and 3.15 cm, respectively. These results suggest that the height component of the station coordinate is also affected by the a priori ZHD and mapping functions.

RT-PWV Estimated from PPP
The PWV over each of the above 41 selected IGS stations was obtained from the conversion of the RT-ZWD using the weighted mean temperature predicted by the GGNTm model [44], followed by the comparison of the PWV against the reference PWV obtained from the co-located radiosonde station. The RMSE of the converted PWVs of the 154 days in 2020 at each station is shown in Figure 7. As we can see, most of the RMSEs of the PWVs resulting from the three VMF products were under 3 mm, except for the GAMB (top panel) and MAJU (bottom panel) stations, at which all the RMSEs from the three VMF products exceeded 3 mm. The mean RMSEs of the PWVs of all the 41 stations resulting from the three schemes were 1.9, 2.4, and 1.7 mm, meaning that product 2 is the worst performer, again, especially at the DAV1 (top pane) and MAW1 (bottom pane) stations. Figure 8 shows the correlation between the PWV converted from PPP-ZWD at nine IGS stations and the reference PWV from their co-located radiosonde stations. It can be seen that product 3 (red) was best, while the PWVs resulting from the other two products at the DAV1 and MCM4 stations were largely underestimated, and both stations are located in Antarctica.

RT-PWV Estimated from PPP
The PWV over each of the above 41 selected IGS stations was obtained from the conversion of the RT-ZWD using the weighted mean temperature predicted by the GGNTm model [44], followed by the comparison of the PWV against the reference PWV obtained from the co-located radiosonde station. The RMSE of the converted PWVs of the 154 days in 2020 at each station is shown in Figure 7. As we can see, most of the RMSEs of the PWVs resulting from the three VMF products were under 3 mm, except for the GAMB (top panel) and MAJU (bottom panel) stations, at which all the RMSEs from the three VMF products exceeded 3 mm. The mean RMSEs of the PWVs of all the 41 stations resulting from the three schemes were 1.9, 2.4, and 1.7 mm, meaning that product 2 is the worst performer, again, especially at the DAV1 (top pane) and MAW1 (bottom pane) stations.    Figure 8 shows the correlation between the PWV converted from PPP-ZWD at nine IGS stations and the reference PWV from their co-located radiosonde stations. It can be seen that product 3 (red) was best, while the PWVs resulting from the other two products at the DAV1 and MCM4 stations were largely underestimated, and both stations are located in Antarctica.

Conclusions
RT-PPP has been proven to be an efficient technique for the retrieval of the ZTD over a GNSS station from all GNSS signals observed at the station [24], and the accuracy of the ZTD estimate is affected by various factors, including the a priori ZHD and the mapping function applied in the observation equations [40]. In this research, the ZHD and mapping function from the VMF forecasting products (VMF1_FC and VMF3_FC) were applied to RT-PPP for testing their resultant GPS-PWV, which is converted from GNSS-ZWD. The accuracy of the a priori ZHD derived from three selected grid-based VMF forecasting products was evaluated by comparing them against the reference ZHD obtained from sounding profiles observed at 443 globally distributed radiosonde stations. GPS observations from 41 IGS stations that have co-located radiosonde stations during the period of the first half year of 2020 were used to test GNSS-ZTD and GNSS-PWV. Three sets of ZTDs estimated from RT-PPP and the application of the a priori ZHD and mapping function obtained from the following three VMF products were evaluated: (1) VMF1_FC; (2) VMF3_FC (resolution 5 • × 5 • ); (3) VMF3_FC (resolution 1 • × 1 • ). It is shown that when the ZHDs from the above 443 radiosonde stations from 1 July 2018 to 30 June 2021 were used as the reference, the mean RMSEs of the ZHDs from the three VMF products were 5.9, 5.4, and 4.3 mm, respectively. The ZTDs estimated at the above 41 selected IGS stations were compared against IGS-provided ZTDs, and results indicated that the second VMF product was the worst performer at some stations, and the mean RMSEs of the ZTD estimates of the 41 IGS stations from the three products were 8.6, 9.0 and 8.6 mm, respectively. The mean RMSEs of the PWV converted from the ZWDs at the 41 stations were 1.9, 2.4, and 1.7 mm, respectively, compared to the reference PWV from co-located radiosonde data. These results suggest that the a priori ZHD and mapping function from both VMF1_FC and VMF3_FC can be applied to the PPP approach for obtaining real-time GNSS-PWV in most regions.  on the frequency f 1 ; N s r,j is the integer ambiguity of the carrier phase; d s r,j and d s j are the uncalibrated code delays (UCD) of the receiver and satellite, respectively; b s r,j and b s j are the uncalibrated phase delays (UPD) of the receiver and satellite, respectively; ε s r,j and ξ s r,j are the observation noise of the code and phase observations, respectively. The IGS GPS satellite clock products are conventionally generated through a linear ionosphere-free (IF) combination of L1 and L2 dual-frequency observations. As a result, the linear combination of the satellite's UCDs are contained in the satellite clock error dt s : where α 12 = where ∆P s r,j and ∆L s r,j are the observed-minus-computed (O-C) code and phase observations, respectively; e s r is the unit vector of the component from r to s; ∆x is the increment of the receiver's (or station's) coordinate with respect to its a priori value; c(d s IF,12 − d s j ) is the satellite code bias item that can be corrected by the differential or observation-specific code bias product. In this research, the code bias is corrected by the observation-specific code bias corrections contained in the CLK93 real-time orbit and clock correction stream.
To solve the rank-deficient problem when estimating the position, the receiver's clock offset, the ZWD, the ionospheric delays, and ambiguities, a re-parameterization process is often performed on the L1/L2 dual-frequency observation equations. After the code bias is corrected, Equation (4) is simplified to: ∆P s r,j = e s r • ∆x + c • dt r + ZWD • m f s w r + γ j • I s r,1 + ε s r,j ∆L s r,j = e s r • ∆x + c • dt r + ZWD • m f s w r − γ j • I s r,1 + N s r,j + ξ s r,j where dt r , I s r,1 , and N s r,j are the re-parameterized receiver clock offset, slant ionospheric delay, and ambiguities, respectively: where d r,IF12 = α 12 • d r,1 + β 12 • d r,2 ; DCB is the differential code bias: DCB r,12 = d s r,1 − d s r,2 . If n GPS satellites are tracked by the receiver at an epoch, the estimated parameters of the GPS dual-frequency uncombined PPP at the epoch can be expressed as: are the vectors of the slant ionospheric delay and the ambiguities on the two frequencies, respectively.