Vol.:(0123456789) GPS Solutions (2024) 28:212 https://doi.org/10.1007/s10291-024-01754-z ORIGINAL ARTICLE Improving PPP positioning and troposphere estimates using an azimuth‑dependent weighting scheme Shengping He1 · Thomas Hobiger1 · Doris Becker1 Received: 24 May 2024 / Accepted: 22 September 2024 / Published online: 3 October 2024 © The Author(s) 2024 Abstract Asymmetric troposphere modeling is crucial in Precise Point Positioning (PPP). The functional model of the asymmetric troposphere has been thoroughly studied, while the stochastic model lacks discussion. Currently, there is no suitable stochastic model for asymmetric tropospheric conditions, potentially degrading the positioning accuracy and the reliability of Zenith Total/Wet Delay (ZTD/ZWD) estimates. This paper introduces an Azimuth-Dependent Weighting (ADW) scheme that utilizes information from asymmetric mapping functions to adaptively weight Global Navigation Satellite System (GNSS) observations affected by azimuth-dependent errors. The concept of ADW has been validated using Numerical Weather Prediction data and International GNSS Service data. The results indicate that ADW effectively improves the coordinate repeatability of the PPP solution by approximately 10% in the horizontal and 20% in the vertical direction. Additionally, ADW appears to be capable to improve the ZWD estimates during the PPP convergence period and yields smoother ZWD estimates. Consequently, it is recommended to adopt this new weighting scheme in PPP applications when an asymmetric mapping functions is employed. Keywords Asymmetric mapping function · Weighing functions · Azimuth-dependent weighting · Precise point positioning Introduction The consideration of tropospheric asymmetry is of high sig- nificance within Precise Point Positioning (PPP) error mod- elling. Unmodelled tropospheric asymmetry causes azimuth- dependent errors which range from centimeter to decimeter level at low elevation angles (Chen and Herring 1997). The characterization of tropospheric asymmetry begins with the tropospheric horizontal gradient (Gardner 1976). Gra- dients reflect the north–south tilt of the troposphere and provide information on non-isotropic water vapor distribu- tion (Kačmařík et al. 2019) which could even be linked pre- cipitation events (Biswas et al. 2021). Incorporating gradi- ent parameters into Numerical Weather Prediction (NWP) models can significantly improve the prediction capabili- ties of such models (Thundathil et al. 2024). Therefore, the tropospheric gradient has currently become an indispensa- ble parameter in high-precision positioning applications and GNSS meteorology. The two-axis gradient model and its higher-order ver- sions are the most commonly used models for correcting asymmetric errors of the troposphere. Gradient models introduce horizontal gradient parameters in North and East directions, representing tropospheric asymmetry across the full azimuth range using a linear combination of sinusoi- dal functions (MacMillan 1995). Gradient models have been validated to improve the coordinate repeatability of PPP solutions by more than 10% (Meindl et al. 2004), and perform even better under extreme weather conditions (Lu et al. 2016). The VMF3 GRAD product, developed by TUW (Landskron and Böhm 2018), is a global empirical model based on higher-order (second and third order) gra- dient models, offered in grid formats with a spatial resolu- tion of 1◦ × 1◦ or 5◦ × 5◦ and a temporal resolution of six hours. The GRAD products provide empirical reference values of horizontal gradients, allowing users to skip the estimation of gradient parameters while still leading to an * Shengping He shengping.he@ins.uni-stuttgart.de Thomas Hobiger thomas.hobiger@ins.uni-stuttgart.de Doris Becker doris.becker@ins.uni-stuttgart.de 1 Institute of Navigation, University of Stuttgart, 70174 Stuttgart, Germany http://crossmark.crossref.org/dialog/?doi=10.1007/s10291-024-01754-z&domain=pdf GPS Solutions (2024) 28:212212 Page 2 of 14 average improvement of 5% in baseline repeatability in the case of Very Long Baseline Interferometry (VLBI). Directional models (Masoumi et al. 2017), building on the framework of two-axis gradient models, incorporate a compensation term in the form of first-order linear piece- wise function, which relies on eight evenly spaced nodes across all azimuth directions. The B-spline Mapping Func- tion (BMF) relies on cyclic B-spline functions to represent tropospheric asymmetry, with B-spline knots estimated as unknown parameters (He et al. 2024). A quadratic BMF with four knots can assure a 10% improvement in coordi- nate repeatability compared to gradient models. Further- more, tropospheric mapping functions such as the IMF (Niell 2000), AMF (Gegout et al. 2011), and PMF (Zus et al. 2014) are designed specifically for asymmetric tropo- sphere analysis. In terms of stochastic models, the time variation of Zenith Total/Wet Delay (ZTD/ZWD) is commonly represented as the Random Walk (RW) process or the first- order Gaussian Markov model (Tralli and Lichten 1990). The setting of the Random Walk Process Noise (RWPN) for ZWD time series is notably flexible, varying from 1mm∕ √ h to 20mm∕ √ h depending on the specific PPP mode being used (static/kinematic, post-processing/real- time) (Hadas et al. 2017). As for the weighting function, currently there exists no suitable choice specifically designed for asymmetric models. Weighting functions describe the uncertainty of GNSS observations with respect to certain parameters, e.g. elevation angle, and are critical for PPP positioning accuracy. Common weighting functions consider Elevation-Dependent Weighting (EDW) function or make use of carrier-to-noise ratio (C/N0) weighting, whereas the former is typically used for geodetic stations (Hadas et al. 2020), and the latter in urban canyons or with low-cost receivers (Wang et al. 2021). Common EDW functions include sin �-inverse (Dach and Walser 2015), sin �-type (Takasu 2013), sqrt- sin � form (Herring et al. 2010), exponential (Eueler and Goad 1991), and cos �-type (Hadas et al. 2020). The sin � -inverse is the most universally used form, consistent with the approximation of tropospheric mapping function (Niell 2000). However, these EDW funtions do not utilize information provided by asymmetric mapping functions nor effectively describe the distribution of azimuth- dependent errors. This implies that observations with different error levels cannot be appropriately weighted, therefore affecting the positioning accuracy. Additionally, unconstrained asymmetric models could absorb systematic errors from the environment, such as multipath errors (Meindl et al. 2004) or uncorrected antenna pointing errors (Ahmed et al. 2016), potentially causing millimeter to centimeter influence on ZTD/ZWD estimates (Ejigu et al. 2019). To address these shortcomings, this study introduces an Azimuth Dependent Weighting (ADW) scheme. In Sect. 2 an analysis of different EDW functions is conducted by evaluating their performance within PPP. Subsequently, we introduce the concept of the ADW: First the cyclic Slant Wet Delay (SWD) is calculated using asymmetric parameters derived from asymmetric mapping functions, then the ADW function is modelled. In Sect. 3 the impact of ADW on observations affected by azimuth-dependent errors is validated using Numerical Weather Prediction (NWP) data from European Centre for Medium-Range Weather Forecasts (ECMWF). In Sect. 4, the improvement in positioning accuracy and ZWD reliability is discussed using data from the International GNSS Service (IGS). Additionally, a study about the scaling parameter utilized for the ADW is also included in this Sect. 4. Weighting functions Elevation dependent weighing functions As for PPP processing, it is necessary to set up the stochastic model, which should conform to the actual error distribution within the observation system. In general, the observation noise �obs can be expressed as Here, �a represents the uncertainty introduced by the external products used in PPP processing, such as orbit, clock, ionosphere (if an undifferenced-uncombined combination is adopted), differential code biases, etc. This part of observation noise can be acquired once the epoch and satellite are specified. �b is the observation noise caused by errors related to signal propagation and unmodelled hardware errors under actual conditions. � b � b . The weighting function indicates the level of unmodelled errors for the corresponding observation. Commonly there are two types of weighting functions for PPP, namely elevation-dependent and the C/N0 dependent weighting functions. The EDW reflects the noise distribution related to signal propagation path, typically used for stations with good observing conditions (Li et al. 2016) and is closely related to the station’s height (Luo et al. 2014). The C/N0 model better reflects the realistic noise level, usually applied to urban street scenarios or when using low-cost receivers (Wang et al. 2021). The basic form of the EDW function is (1)�2 obs = �2 a + �2 b . (2)�b = �0 ⋅ F(�), GPS Solutions (2024) 28:212 Page 3 of 14 212 where �0 is a priori uncertainty of the observation, and � is the elevation angle. In this study, we have set �0 = 0.3m for pseudorange and �0 = 3mm for carrier-phase observations. Commonly used F(�) include the sin �-inverse (Dach and Walser 2015), the sin �-sqrt (Takasu 2013), the sin �-type (Herring et al. 2010), the exp-type (Eueler and Goad 1991) and the cos �-type (Hadas et al. 2020), which are listed as follows • sin �-inverse • sin �-sqrt • sin �-type with �1 = 0.64, �1 = 0.36. • exp-type with �2 = 1.0, �2 = 3.5, �0 = 9◦. • cos �-type with �3 = 1.0, �3 = 4.0, n = 8. � , � , �0 and n are empirical parameters. The functional behavior of different EDW functions is depicted in Fig. 1. The variation of EDW represents how the assumed (3)F1(�) = 1 sin � . (4)F2(�) = 1√ sin � . (5)F3(�) = √ �1 + �1 � , (6)F4(�) = �2 + �2e −�∕�0 , (7)F5(�) = √ �3 + �3 cos n �, observation uncertainty changes with the elevation angle. The figure reveals that all EDW functions increase sharply at low elevation angles. These weighting models are therefore designed to down-weight observations at low elevation angles. The sin �-inverse is consistent with the simplified form of the tropospheric mapping function sin−1 � (Niell 2000), thus it can be regarded to only represent the noise brought by tropospheric delays. Some other models incorporate representations for other noises, such as antenna pointing errors and height-related errors (Luo et al. 2014). We evaluate the performance of above EDW functions within PPP. The tropospheric delay is modelled through the symmetric tropospheric model without horizontal gra- dients. The PPP settings are listed in Table 2. Observation data from the IGS station WTZR during the spring of 2022 (DOY 91–97) have been used. The ITRF2014 coordinates, including seasonal variation with respect to Earth’s Center of Mass (CM) (Altamimi et al. 2016), are assumed to be the ground-truth against which the estimates can be compared. The test results are summarized in Table 1. This simple test reveals that for this particular station, the sin �-type function exhibits the best performance. Therefore, it is recommended for users considering this weighting function as the first option when the sites are under good observational conditions. However, it is important to note that results in Table 1 should only serve as a reference. In general, the selection of a suitable weighting function is not straightforward, as its actual performance is related to site environment, hardware conditions, height and weather conditions. Consequently, the realistic performance of the weighting models needs to be tested based on actual conditions. Fig. 1 The variation of five different EDW functions in terms of elevation angles, from 3 ◦ to 90◦ GPS Solutions (2024) 28:212212 Page 4 of 14 Towards an azimuth dependent weighting function The commonly used weighting functions are designed solely for elevation-dependent noise and are not particularly suitable for asymmetric mapping functions. Therefore, it has been studied whether an Azimuth-Dependent Weighting is beneficial for space geodetic data processing. An asymmetric mapping function can provide information about the tropospheric asymmetry. Such asymmetric information should be estimated by proper choice of unknown parameters, such as the horizontal gradients gN and gE in the gradient model or the B-spline coefficient pi in the BMF. Unfortunately, there can be residuals when representing the azimuthal variation of SWD using asymmetric mapping functions, especially at low elevations angles (He et al. 2024). Such residuals introduced by mapping functions are related to both the type of mapping functions and the number of asymmetric parameters involved, and they cannot be completely eliminated. In addition, due to the lack of constraints on asymmetric parameters within the system, they can easily absorb azimuth-dependent unmodeled errors, such as multipath errors (Meindl et al. 2004). Based on these facts, we can make the following two inferences: • Condition 1: Asymmetric parameters contain some information about unmodeled azimuth-dependent errors. • Condition 2: When the absolute value of the asymmetric parameters is large, the residual/unmodeled errors in that direction are also greater. These are two conditions that allows us to adopt the asym- metric parameters as indicators to roughly represent the azimuthal-distribution of the unmodeled errors and residu- als. Assuming that sequential estimators, like the Kalman filter, provide meaningful estimates of those gradients or coefficients, one can construct the ADW function. Based on the asymmetric information from the previous epoch, one can establish an ADW function which is applicable to the current epoch. Since in each epoch gradient param- eters or other asymmetric troposphere delay coefficients are estimated it should be always possible to compute the ADW for the consecutive epoch that needs to be pro- cessed. As for the very first epoch, one can start with uni- form weight over all azimuth directions in order to get the sequential estimation scheme running. The corresponding flowchart to this idea can be found in Fig. 2. Some studies have proposed processing strategies for GNSS observations depending on the azimuth, such as azi- muth-dependent elevation masks (Atilaw et al. 2017; Han et al. 2018). However, these methods require prior knowl- edge of the site environment. The advantage of ADW is that it can dynamically adjust the weighting function based on the estimated gradient at each epoch, without the need for predefined masks, and this can be achieved by utilizing the asymmetric tropospheric functions. We expand the weighting function from the function of elevation angle to both elevation and azimuth angle, then the new weighting function yields the following form with Here C(�, �) is a normalized function that describes the tropospheric asymmetry at any given direction. When the elevation angle � is fixed, it becomes a cyclic function describing the azimuthal asymmetry of the tropospheric delay. C̄(𝜀) is the mean value of C(�, �) over the azimuth [0, 360◦] at given � . n is a predefined, positive, scaling parameter of the asymmetric error, ranging within (0, 1] . A smaller n indicates a greater noise contributions for errors caused by asymmetry. A recommended value for n is 1 2 , while the more in-depth discussion about the choice of n can be found in Sect. 4.4. Condition 2 implies that in the direction where the asymmetric mapping function has larger deviation from the mean over all azimuth angles, the unmodeled errors or residuals are also greater. Consequently, Eq.(9) is employed (8)�b = �0 ⋅ F(�, �), (9)F(𝛼, 𝜀) = F(𝜀) ⋅ ( 1 + ||||| C(𝛼, 𝜀) − C̄(𝜀) C̄(𝜀) ||||| n )1∕n . Table 1 Performance of five different EDW functions in terms of PPP coordinate difference w.r.t ITRF2014 expressed as standard deviation (STD), mean absolute error (MAE) (H for horizontal and V for verti- cal), 3D-RMSE as well as WRMS of post-fit residuals. All results are given in centimeter F(�) [cm] STD H STD V MAE H MAE V 3D-RMSE WRMS res sin �-inverse 1.13 1.40 1.10 1.21 2.06 1.54 sin �-sqrt 1.36 1.34 1.13 1.18 2.09 1.53 sin �-type 0.91 1.08 0.91 0.98 1.71 1.51 exp-type 1.05 1.33 1.03 1.28 2.08 1.56 cos �-type 1.14 1.46 1.11 1.33 2.18 1.58 GPS Solutions (2024) 28:212 Page 5 of 14 212 to assign greater observation noise to the GNSS observations in these directions. When the troposphere is isotropic in azimuth direction, ||C(𝛼, 𝜀) − C̄(𝜀)|| is zero, and the weighting function will be only related to the elevation angle; when the troposphere shows a high degree of asymmetry, the azimuth-dependent part will reflect the observation noise at the corresponding azimuth angle. The higher asymmetry in a certain azimuth, the larger the corresponding observation noise will be. For some common asymmetric mapping functions, the cyclic asymmetry C(�, �) and the corresponding mean C̄(𝜀) can be directly computed as • Two-axis gradient model (GRAD) (Herring 1992) (10)CG(�, �) = 1 + (gN cos � + gE sin �) ⋅ mfg(�). with mfg(�) being the gradient mapping function which consists of the total mapping function mf (�) (11) C̄G(𝜀) = 1 2𝜋 ∫ 2𝜋 0 [ 1 + (gN cos 𝛼 +gE sin 𝛼) ⋅ mfg(𝜀) ] d𝛼 = 1. Fig. 2 Flowchart illustrating the difference between the EDW and ADW scheme GPS Solutions (2024) 28:212212 Page 6 of 14 • Second-order gradient model (GRAD-2) (Landskron and Böhm 2018) • B-spline Mapping Function (BMF) (He et al. 2024) (12)mfg(�) = ⎧ ⎪⎪⎨⎪⎪⎩ mf (�) ⋅ cot �, BS model 1 (sin � tan � + C) , CH model − �mf (�) �� , Tilting, model . (13) CG2(�, �) =1 + (gN cos � + gE sin � + gN2 cos 2� + gE2 sin 2�) ⋅ mfg(�). (14) C̄G2(𝜀) = 1 2𝜋 ∫ 2𝜋 0 [ 1 + (gN cos 𝛼 +gE sin 𝛼 + gN2 cos 2𝛼 +gE2 sin 2𝛼) ⋅ mfg(𝜀) ] d𝛼 = 1. (15)CB(�, �) = mf (�) mf (�0) ⋅ m−1∑ i=0 piBi,n(�). For the classical gradient model, hereafter abbreviated as GRAD, we can get its simplified ADW by combining Eq.(9), Eq.(11) and Eq.(14) and obtain Although ADW does not rely on certain types of mapping functions, the two-axis gradient model is being used as one possible choice for a mapping function in order to analyze the effects of ADW. The other two mapping functions, namely second-order gradient and B-spline mapping function, are used as comparisons. Validation of ADW with NWP data SWD asymmtry Both hydrostatic and wet tropospheric delays exhibit dependencies on the azimuth angle. Hydrostatic asymmetry primarily arises from variations in tropopause height, which (16)C̄B(𝜀) = mf (𝜀) mf (𝜀0) ⋅ m−1∑ i=0 [ pi ∫ 2𝜋 m 0 Bi,n(𝛼) d𝛼 ] . (17)F(�, �) = F(�) ⋅ ( 1 + ||CG(�, �) − 1||n )1∕n . Fig. 3 SWD (left) and SWD asymmetry plots (right). For the SWD model, the envelope of the cross-section indicates the cyclic SWD and the color represents the SWD asymmetry. The SWD asymmetry plots have the elevation ranging from 20◦ to 50◦ (right-top) and 3◦ to 15 ◦ (right-bottom), respectively GPS Solutions (2024) 28:212 Page 7 of 14 212 has a strong correlation with geographical latitude (Seidel and Randel 2006). Wet asymmetry is associated with the anisotropic distribution of water vapor, known for its rapid fluctuations and modeling challenges Böhm and Vedel (2014). In this paper we concentrate on the wet part of the tropospheric asymmetry. We simulate the wet delay using Numerical Weather Pre- diction (NWP) data and have conducted a preliminary valida- tion of the ADW performance. All results are based on TUW’s Online Raytracer: https:// vmf. geo. tuwien. ac. at/ raytr acer. html, which makes use of ECMWF’s operational Numerical Weather Models (Nafisi et al. 2011). We generate tropospheric SWD for the IGS station WTZR (Wettzell, Germany) for 2022 DOY97, as illustrated in the Fig. 3. The simulated SWD values range over azimuth angles from 0 to 360◦ and elevation angles from 3◦ to 85◦ . The sampling intervals for azimuth were 3.6◦ , and for elevation, they varied as follows: 5◦ in the interval [20◦, 85◦] , 2.5◦ in the interval [7.5◦, 15◦] , and 1◦ in the interval [3◦, 5◦]. The left image in the figure displays the model of tropospheric delay, where each cross-section’s envelope represents the wet delay across the azimuth [0, 360◦] at corresponding elevations. The SWD model generally resembles an inverted funnel, indicating that the SWD increases rapidly with the decrease of elevation angles. The function describing this variation is the tropospheric wet Mapping Function (MF), which approximates a function of sin−1 � . The model’s color reflects the wet asymmetry in this direction, determined from the ΔLw(𝛼, 𝜀) − ΔL̄w(𝜀) , i.e., the difference between the SWD at a specific direction and the mean SWD at the corresponding elevation across all azimuths. Subplots on the right show variations of SWD asymmetry at different scales. In the upper subplot, one observes step-like SWD variations, which are caused by the resolution of the original data. Notably, SWD asymmetry can reach ±60 ∼ 80mm at 3◦ elevation, approximately ±10mm at about 10◦ , and diminishes to less than 1mm above 30◦ . Consequently, it can be concluded that observations below 10◦ may contain centimeter-level errors solely concerning wet asymmetry, which should not be overlooked in high-precision PPP applications. Downweight observations affected by large azimuth‑dependent errors using ADW When using an asymmetric mapping function to represent tropospheric delays, certain errors are incurred depending on the type of asymmetric model employed. For example, the use of a gradient model can result in misfit errors (residuals) at low elevation angles (Masoumi et al. 2017; Zus et al. 2014). Under 10◦ elevation angle, the residuals can be several centimeters; while at 3◦ elevation angle, the residuals even reach the decimeter level (He et al. 2024). To assess the ADW’s ability to mitigate such azimuth- related residuals, a simple experiment has been conducted. The data fitted were the wet asymmetry values at various elevations described in Sect. 3.1, using a two-axis gradient model as the fitting function. Depending on different weighting function employed, we set three comparison groups: Fig. 4 Absolute residuals in fitting wet asymmetry with gradient model (top row) and the distribution of normalized weights with respect to residuals (bottom row). The range and direction in polar plots indicate the elevation angle and azimuth, respectively. The range is in logarithm scale. For the top figures the color indicates the level of residuals, while for bottom figures the color indicates the density of scatter points. The arrows in bottom figures are referred to the largest residual point https://vmf.geo.tuwien.ac.at/raytracer.html https://vmf.geo.tuwien.ac.at/raytracer.html GPS Solutions (2024) 28:212212 Page 8 of 14 • Group 1 (without WF): No weighting function was applied, and only the constant component of the observation noise was considered. • Group 2 (EDW): An EDW function sin �-type was used. • Group 3 (EDW+ADW): Both an EDW function ( sin �- type) and an ADW (gradient model) were incorporated. The results are shown in the Fig. 4. The first row dis- plays the distribution of absolute residuals (fitting errors) for each group, demonstrating a successive decrease in residu- als caused by wet asymmetry after employing EDW and ADW. The maximum residual in Group 1 was 5mm , which reduced to 0.4mm with EDW and further to 0.12mm with ADW. Additionally, it can be observed bottom row that both EDW and ADW reasonably distribute weights for observa- tions, assigning higher weights to observations with smaller residuals. For EDW group, the location where the scatter points are most concentrated was around 1.5, while for the ADW group it was approximately at 1.8. Moreover, the spike representing larger residuals were lower in the ADW group compared to the EDW group, indicating that ADW derived from gradient parameters down-weight observations with large residuals. Consequently, these results reveal that ADW assigns weights more effectively than EDW, leading to less residuals in low-elevation observations. Furthermore, one recognizes that the magnitude of the residuals is not only related to the elevation but also to the azimuth direction. However, as shown in Fig. 3, extreme values of wet delay only occur at 20◦ and 200◦ azimuths, while fitting residuals in Fig. 4 appear at azimuth directions of 10◦ , 100◦ , 190◦ , and 280◦ . This difference results from the correction of errors using the gradient model, and it also indicates that adjusting the stochastic model with ADW can only mitigate azimuth-dependent errors but will not completely eliminate them but mitigate those errors significantly. Validation of the ADW with GNSS observations from IGS Description of PPP analysis settings and GNSS dataset To validate the effectiveness of ADW, we conduct PPP analysis in static mode. The experiments utilize observa- tional data from eight IGS stations. Each station’s obser- vation period includes data from DOY 1-7, 2022 (winter) and DOY 182-188, 2022 (summer). The parameter settings and processing strategies of PPP are summarized in Table 2. The software utilized in the test is our self-developed PPP software built on RTKLIB (Takasu 2013). It is specifically designed for high-precision PPP post-processing and is cur- rently not open-source. This software includes new features such as a PPP-AR module, multiple observation combina- tion options, and support for various tropospheric strategies. The azimuth-dependent part in the weighting function can be combined with different EDW functions F(�) , which allows to test various combinations to explore their effects separately. We started with a test, in which the sin�-type is used as the EDW function, and the gradient model is adopted in both the asymmetric mapping function and ADW. The group abbreviations and settings are listed as following • NONE: Symmetric MF and EDW, mf (�) + F(�). • GRAD: Asymmetric MF and EDW, mf (�, �) + F(�). • GRAD-ADW: Asymmetric MF and ADW, mf (�, �) + F(�, �). Positioning results display The results are illustrated in Fig. 5. Across all measurement stations, GRAD-ADW exhibits the best performance in Table 2 Processing strategies in terms of PPP settings, involved prod- ucts and EKF parameters 1Deutschse GeoForschungsZentrum (GFZ) 2Technische Universität Wien (TUW) 3International GNSS Service (IGS) 4Chinese Academy of Sciences (CAS) 5Chalmers University of Technology (Chalmers) Strategies Values PPP settings Ele.Mask 10◦ System GPS+Galileo Freq dual-frequency Solution static mode+fixed ambiguities Products Orbit final (.sp3) 1 Clock final, 30 s interval (.clk_30s) 1 Trop GPT3+VMF3 2 DSB DCB (.bsx) 4    Antenna IGS14 standard (igs14.atx) 3 EOPs weekly average (.erp ) 3 Tidal Model FES2004 (.blq) 5 EKF param Process noise �ZWD = 10mm∕ √ h �gN ,gE = 1.0mm∕ √ h �gN2,gE2 = 0.1mm∕ √ h �pos = 1.0 × 10−5 m∕ √ h �clk = 600m∕ √ h Observation noise �0 = 3.0mm for L �0 = 0.3m for P Initial variance PZWD = 0.36m2 PgN ,gE = 1.0 × 10−4 m2 GPS Solutions (2024) 28:212 Page 9 of 14 212 terms of standard deviation and mean absolute errors, both horizontally and vertically. Compared to GRAD, the annual mean improvement in standard deviation for GRAD-ADW in the horizontal/vertical direction is 12%/20% . Addition- ally, the improvement in mean absolute errors in the hori- zontal/vertical direction is 9%/20% . This suggests that for static PPP observations where the gradient model and sin � -type weighting function is used, ADW is performing much better than EDW in characterizing the spatial distribution of unmodelled errors, thereby improving the coordinate repeatability. A crucial question arises from the initial test: Is ADW universally effective for all EDW functions and all asymmet- ric mapping functions? To answer this question, additional tests were conducted to investigate the performance of ADW when combining this strategy with different EDW and asym- metric mapping functions, respectively. As for the EDW test, the GRAD model was used. The test dataset remains identical to that of the previous test, with the results detailed in Table 3. This table illustrates the mean (both winter and summer) across all IGS stations considered in this study. It can be seen that for all EDW groups, GRAD-ADW leads to improvements over GRAD in terms of STD, MAE and RMSE, with improvements ranging approximately between 10% ∼ 20% . The highest improvement is observed in the vertical MAE, where each F(�) group experiences an improvement of 2 ∼ 3mm , approximately 20% in ratio. However, it can also be noted that the improvement in WRMS of residuals is limited, with an average of about 2% . Additionally, across all F(�) groups, the sin �-type exhibits the best coordinate repeatability, which is consistent with the results presented in Table 1. These findings reveal that for static PPP applications with the gradient model, the use of GRAD-ADW contributes to a consistent improvement in coordinate repeatability, which is characterized by approximate 10% in horizontal and 20% Fig. 5 Horizontal and vertical coordinate STD and MAE of 8 IGS stations in winter (left) and summer (right). The MAE is calculated with refer to ITRF2014 coordinates Table 3 Evaluation of the performance of ADW F(�, �) with different EDW F(�) . The asymmetric mapping function used in this test is the gra- dient model GRAD. The results are displayed in the format "EDW/ADW". All results are in centimeters F(�) [cm] STD H STD V MAE H MAE V 3D-RMSE WRMS res sin � 1.13/1.00 1.40/1.16 1.10/0.95 1.21/0.95 2.06/1.72 1.47/1.44 sin �-sqrt 1.15/1.00 1.46/1.24 1.12/0.96 1.33/1.12 2.18/1.86 1.50/1.45 sin �-type 0.91/0.85 1.08/0.85 0.91/0.87 0.98/0.70 1.71/1.43 1.45/1.44 exp-type 1.05/0.92 1.33/1.14 1.03/0.92 1.28/1.11 2.08/1.82 1.48/1.47 cos �-type 1.14/0.98 1.46/1.25 1.11/0.96 1.32/1.14 2.18/1.88 1.50/1.45 GPS Solutions (2024) 28:212212 Page 10 of 14 in vertical directions. Besides, among all EDW options, the sin �-type appears to be the competitive candidate in the PPP processing when using the data from geodetic stations, which generally have good observation environment, i.e. they are not significantly impacted by multipath signals. Additionally, a test has been conducted to investigate dif- ferent asymmetric mapping functions, namely the gradient model (GRAD), the second-order gradient model (GRAD-2) and the B-Spline mapping function (BMF), whereas the sin � -type weighting function was used together with all models. The results, as summarized in Table 4, indicate that ADW effectively improves the coordinate repeatability among all asymmetric mapping functions, yielding an estimated improvement of 5% horizontally and over 20% vertically. Different asymmetric mapping functions adopt different parameters to represent the tropospheric asymmetry, thus it can lead to distinct level of improvements when applying ADW based on different asymmetric mapping functions. In general, when static PPP can provide asymmetric information of the troposphere, it is recommended to use ADW to modify the weighted model. ADW is capable to combine with common asymmetric mapping functions and elevation-dependent weighting models, leading to an improvement in coordinate repeatability of approximately 10% in the horizontal and about 20% in the vertical direction. On the impact of ADW on ZWD and gradient estimates Despite ADW being validated to effectively improve the positioning accuracy of PPP, its performance for GNSS meteorology still requires careful investigation. Gradient parameters are likely to absorb system errors existing in the environment, such as multipath errors (Meindl et al. 2004). This common drawback of the asymmetric mapping function potentially makes the ZWD/ZTD estimates being harmed. Therefore, we consider it necessary to study the impact of ADW on ZWD and gradient estimates. We process the observational data from WTZR during DOY 91-97 in 2022 and compare the estimated ZWD (Fig. 6) and gradient (Fig. 7) time series of EDW and ADW. Results for more IGS stations can be found in the auxiliary data archive of this study which is accessible on https:// sites. google. com/ view/ adwex ample s/% E9% A6% 96% E9% A1% B5. The ZWD time series obtained from both EDW and ADW exhibit good consistency throughout the entire observation Table 4 Evaluation on the performance of ADW F(�, �) with different asymmetric mapping functions mf (�, �) . The EDW used in this test is the sin �-type. The results are displayed in the format "EDW/ADW", and all values are in centimeters mf (�, �) [cm] STD H STD V MAE H MAE V 3D-RMSE WRMS res NONE 0.93 1.00 1.02 0.99 1.70 1.45 GRAD 0.91/0.85 1.08/0.85 0.91/0.87 0.98/0.70 1.71/1.43 1.45/1.44 GRAD-2 0.90/0.86 1.00/0.78 0.88/0.87 0.93/0.69 1.63/1.41 1.45/1.44 BMF 0.95/0.90 0.94/0.78 0.91/0.88 0.86/0.70 1.57/1.40 1.43/1.42 Fig. 6 Illustration of ZWD estimates from PPP using EDW and ADW, respectively. ΔZWD is the difference between the two solutions ZWD EDW − ZWD ADW . The subplot on the right indicates the histo- gram of ΔZWD . The zoom-in window presents the ZWD time series within first two hours https://sites.google.com/view/adwexamples/%E9%A6%96%E9%A1%B5. https://sites.google.com/view/adwexamples/%E9%A6%96%E9%A1%B5. GPS Solutions (2024) 28:212 Page 11 of 14 212 period. According to the histogram statistics, the difference ZWD between the two solutions has mean of −0.6 cm and a STD of 1.6 cm . The subplot window displays the ZWD estimates during the first two hours, where it can be observed that the ZWD estimates from the EDW setting dropped to from 0.15 cm to −0.15 cm in the first 20 minutes, and rapidly returned to positive values. This is a common error in ZWD estimation during the PPP convergence period, caused by the lack of constraints on ZWD in the system. In contrast, ADW maintains positive values and does not present severe fluctuations during the first 20 minutes. This indicates that ADW can better provide ZWD estimates during the convergence period. This probably attributes to ADW’s better representation of the spatial distribution of azimuth- dependent errors, avoiding ZWD estimates being affected by systematic errors and therefore providing more stable estimates during the convergence period. Furthermore, it also can be observed that ZWD estimates based on ADW group are smoother compared to those from the EDW solution, even if the EDW and the ADW have the identical ZWD process noise. High-frequency variations in ZWD time series implies ZWD may be influenced by unmodeled errors, which can reduce the precision of the ZWD estimates. In general, ADW allows for the down- weighting of poor-quality observations, thus being able to filter out potential outliers in ZWD estimates and making it beneficial for GNSS meteorology. As shown in Fig. 7, the gradient estimates under the two processing methods exhibit millimeter-level deviations. The histogram of gradient differences for the whole observation period has a mean value of 0.2 mm, which indicates that the distribution is basically unbiased. The deviation in the gE gradient is higher during the initialization phase, exceeding 5 mm, and gradually decreases as the estimates converge. This is because EDW has a slightly worse behavior during the convergence period. In contrary, the ADW provides better estimation for gradients (around 5 mm). Moreover, it can be observed that after applying ADW, the gradient does not exhibit the smoothing effect as in ZWD estimates, but rather shows a shift at some time periods. This result aligns with our expectations and suggests that ADW has a compensation effect on gradient estimates. Considering that the implementation of ADW improve coordinate accuracy and smooths ZWD time series, it can be inferred that this compensation effect is positive, potentially mitigating some of the systematic errors absorbed by the gradient parameters. Impact of scaling parameter n in ADW In the formulation of ADW, there is the critical parameter n, which represents the amplification level of azimuth- dependent noises (referred to Sect. 2.2). Apparently, when the troposphere is almost uniform and there is barely no wet tropospheric asymmetry, different values of n will only have minimal influence on the weighting function. Conversely, when the troposphere has a high degree of asymmetry, the value of n plays a more significant role. Given that different Fig. 7 Illustration of tropospheric gradients estimates g N and g E from PPP using EDW and ADW, respectively. Δg E and Δg N are the difference between the two solutions. The subplots on the right column are histograms of Δg E and Δg N GPS Solutions (2024) 28:212212 Page 12 of 14 asymmetric mapping functions can provide different asymmetric information as they have different mathematical expressions, they may also have distinguished optimal values of n. To determine the optimal value of n, we carried out a test which has the same framework as in Sect. 4.2 (see Table 2). Three models GRAD, GRAD-2, BMF were selected for evaluation, with the EDW function being sin �-type. The results are listed in the Table 5. It should be noted that results in Table 5 serve merely as a reference, because they are purely applicable to the dataset studied here. In addition to the influence of the asymmetric mapping function, the value of n is also affected by filter configuration, process noise model, the observation noise model, hardware effect, multipath, environmental conditions or the receiver and antenna types. Consequently, the determination of the optimal value of n is a quite complex problem, and users need to find the optimal value suited for the specific software and site conditions. In light of these considerations, it is recommended to use ADW starting from the scaling n = 1 2 , and to adjust the optimal value of n according to the prevailing situation. Conclusion This paper introduces an Azimuth-Dependent Weighting function as a vital compensation mechanism within the sto- chastic model of PPP. The ADW utilizes the output (asym- metric parameters) from asymmetric mapping function to selectively down-weight GNSS observations with large errors at certain azimuth angles. This approach effectively addresses the limitations of traditional Elevation-Dependent Weighting (EDW) functions by accounting for azimuth- dependent errors. Validation using IGS dataset demonstrated that ADW effectively improve the coordinate repeatability of PPP, by approximately 10% horizontally and 20% vertically. Moreover, when assessed against the ITRF2014, ADW lead to 20% improvements in both 3D-RMSE and 3D-MAE. The results also revealed that the ADW could serve as a complementary term alongside various EDW functions, and consistently augment their effectiveness. Additionally, ADW supports different asymmetric mapping functions, as long as these tropospheric models adequately represent the asymmetric characterization of the troposphere. Furthermore, ADW mitigates issues in GNSS meteorology such as the appearance of outliers in ZWD estimates during the PPP convergence period. The use of ADW yields smoother ZWD estimates, which fit better with the physical nature of the troposphere. These improvement are primarily attributed to ADW’s precise representation of azimuth-dependent errors. Although asymmetric mapping functions inherently absorb system errors, the use of ADW effectively mitigates this effect, thereby allowing ZWD estimates to become more robust and smooth. Nonetheless, uncertainty exists in the selection of the scaling parameter n in ADW. This parameter is crucial as it dictates the extent to which asymmetric information is incorporated into the stochastic model. Determination of the optimal n depends not only on the site environment but also on the software-specific configuration such as the parameter settings in the stochastic model within PPP applications. Although this study suggests the potential existence of an optimal n value, commencing with n = 1 2 is recommended until further research identifies the most Table 5 Evaluation of positioning results in terms of coordinate STD, MAE and 3D-RMSE, when tuning the scaling parameter n. All values are in [cm] MF n STD H STD V MAE H MAE V 3D RMSE GRAD 1 0.72 0.86 0.66 0.78 1.53 1/2 0.72 0.82 0.66 0.73 1.49 1/3 0.75 0.74 0.67 0.64 1.45 1/4 0.81 0.63 0.69 0.53 1.43 1/5 0.94 0.55 0.71 0.41 1.51 GRAD-2 1 0.67 0.86 0.63 0.78 1.49 1/2 0.68 0.82 0.64 0.74 1.46 1/3 0.72 0.74 0.65 0.65 1.42 1/4 0.77 0.63 0.67 0.52 1.40 1/5 0.90 0.56 0.70 0.41 1.49 BMF 1 0.81 0.76 0.70 0.68 1.46 1/2 0.83 0.66 0.71 0.61 1.42 1/3 0.76 0.57 0.68 0.57 1.31 1/4 0.82 0.50 0.67 0.55 1.35 1/5 2.46 0.59 0.96 0.58 3.57 GPS Solutions (2024) 28:212 Page 13 of 14 212 suitable value for specific software and site conditions. Future investigations will focus on numerically optimizing n and potentially developing an refined ADW formulation that more accurately characterizes the distribution of azimuth- dependent errors within PPP. Acknowledgements The first author acknowledges the China Schol- arship Council (CSC) for supporting his research. Additionally, the authors would like to acknowledge the Spirent Academia Program for supporting our GNSS research activities. Author contributions SH has implemented the test and wrote the paper. TH gave suggestions on refining the model and revised the paper. TH and DB both provided valuable ideas to complete the work. Funding Open Access funding enabled and organized by Projekt DEAL. The first author is supported by a grant from the China Schol- arship Council (CSC, 202008080118). Data availibility The data and products used in experiments are available from following servers: ∙ IGS/CDDIS (data& products): https:// cddis. nasa. gov/ archi ve/ gnss/ data/ daily/ ∙ GFZ/CDDIS (products): https:// cddis. nasa. gov/ archi ve/ gnss/ produ cts/ latest/ final/ ∙ TUWien (VMF): https:// vmf. geo. tuwien. ac. at/ trop_ produ cts/ GNSS/ ∙ Chalmers (Ocean loading): http:// holt. oso. chalm ers. se/ loadi ng/ ∙ ITRF (Coordinate reference): https:// itrf. ign. fr/ en/ homep age ∙ Online Ray-tracer: https:// vmf. geo. tuwien. ac. at/ raytr acer. html Declarations Ethics approval and consent to participate Not applicable. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 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Radio Sci 49(3):207–216 Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. https://www.rtklib.com/rtklib.htm https://www.rtklib.com/rtklib.htm Improving PPP positioning and troposphere estimates using an azimuth-dependent weighting scheme Abstract Introduction Weighting functions Elevation dependent weighing functions Towards an azimuth dependent weighting function Validation of ADW with NWP data SWD asymmtry Downweight observations affected by large azimuth-dependent errors using ADW Validation of the ADW with GNSS observations from IGS Description of PPP analysis settings and GNSS dataset Positioning results display On the impact of ADW on ZWD and gradient estimates Impact of scaling parameter n in ADW Conclusion Acknowledgements References