Performance Evaluation of Blind Tropospheric Delay correction Models over Africa

Tropospheric delay is a major error source in positioning by Global Navigation Satellite Systems (GNSS). Many techniques are available for tropospheric delay mitigation consisting of surface meteorological models and global empirical models. Surface meteorological models need surface meteorological data to give high accuracy mitigation while the global empirical models need not. However, most GNSS stations in the African region are not equipped with a meteorological sensor for the collection of surface meteorological data during the measurement. Zenith Tropospheric Delay (ZTD) is often calculated by the various high precision GNSS software packages by utilising standard atmosphere values. Lately, researchers in the University of New Brunswick and Vienna University of Technology have both developed global models (University of New Brunswick (UNB3M) and Global Pressure and Temperature 2 wet (GPT2w) models) for tropospheric delay correction, respectively. This report represents an appraisal of the performance of the GPT2w and UNB3M models with accurate International GNSS Service (IGS)-tropospheric estimations for fifteen IGS stations over a period of 1 year on the Africa continent. Both models perform significantly better at low latitudes than higher latitudes. There was better agreement between the GPT2w model and the IGS estimate than the UNB3m at all stations. Thus, the GPT2w model is recommended as a correction model of the tropospheric error for the GNSS positioning and navigation on the African Continent.


Introduction and background
Tropospheric delay is one of the main error sources in the analysis of space geodetic techniques operating at microwave frequencies, such as Global Navigation Satellite Systems (GNSS), Very Long Baseline Interferometry (VLBI), or Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS).
The tropospheric delay is usually separated into a hydrostatic delay that is modelled a priori, and a wet delay that is estimated from the space geodetic microwave observations.
Modelled hydrostatic delays and the estimated wet delays are usually referred to the zenith direction; corresponding mapping functions are required to convert the slant delays in observation direction to the zenith.In addition, troposphere gradients can be estimated to account for asymmetries of the troposphere.
In GNSS positioning, the tropospheric delay typically ranges between 2.0 m to 2.6 m.
The Zenith Hydrostatic Delay (ZHD) constitutes 90% of the Zenith Tropospheric Delay (ZTD), and Zenith Wet Delay (ZWD) is usually less than 10%.The ZHD can be estimated to an accuracy of better than 90% using empirical models that utilizes meteorological data, such as pressure and temperature as well as the position of the user.Some ZHD models include those of Saastamoinen (1972), Hopfield (1969), Berman (1976), Davis et al (1985), Ifadis(1986), Askne and Nordius (1987) etc.A comprehensive review and validations of some of these model can be found in Tuka and El-Mowafy(2013).The Saastamoinen model is the most used model in geodetic applications and its accuracy has been widely reported (Dodo and Idowu, 2010).
In practice, a user often employs a certain troposphere model based on the popularity of the model without giving enough justification as to why it should be used.Limited comparisons between some of the models have been carried out in the past for local or regional applications.However, in this contribution, this issue is addressed more comprehensively considering the peculiarities of the African GNSS network.Most GNSS stations on the African continent are characterised by the lack of collocated meteorological sensors, as it is required for such to be collocated with the GNSS antenna if the GNSS data are to be processed for integrated water vapour content determination (Isioye et al., 2015).
Thus, the inversion of ground meteorological data into the variable vapour content in the atmosphere is very difficult.Even the Saastamoinen model has difficulties in meeting the needs for high accuracy GNSS positioning and meteorological applications, since most GNSS geodetic software uses the Saastamoinen model with standard atmosphere models for a-poiri estimates.
In view of these shortcomings, it is of practical importance to construct a global model of average tropospheric delay correction with a certain accuracy to be used particularly in the GNSS navigation and positioning in Africa, in which the zenith delay depends only on the latitude, elevation of observing station, and the date of observation.Recently, several of these blind models have been developed such as the University of New Brunswick model; UNB3 (Collins and Langley, 1999), RTCA-Minimum Operational Performance Standards; MOPS (RCTA, 2001); European Geostationary Navigation Overlay Service; EGNOS (Dodson et al., 1999;Penna et al., 2001); UNB3m (where m stands for "modified") (Leandro et al., 2006); European Space Agency; ESA model (ESA Galileo Programme, 2012); Global Pressure Temperature 2; GPT2 (Lagler et al., 2013); and Global Pressure Temperature 2 wet; GPT2w (Boehm et al., 2014).Table 1 provides an overview of the different blind models.1984) Function (Neill, 1996) Function (Neill,

1996)
Mapping Function (Boehm et al., 2006b) function (Boehm et al., 2006a) It is evident from Table1 that the models can be classified into two groups, one based on a set of tabulated climatological data and the other from Numerical Weather Prediction (NWP) models.In the first category, the UNB3m is a refined version of UNB3 model (Leandro et al., 2006) and thus superior to RTCA MOPS, which is the same as the UNB3 model except for the replacement of the Neill mapping function with the Black and Eisner model (Leandro et al., 2006).Considering the other set of models, which are dependent on NWP data, theGPT2w model looks quite outstanding going by the spatial resolution of the model and also for the fact that the Vienna Mapping function is known to model tropospheric delay better that the Neil mapping function adopted by the ESA model (see, Won et al., 2010;Zus et al., 2015).
This paper presents an assessment of the UNB3m and GPT2w tropospheric models.
The Zenith tropospheric estimations were compared from both models with the International GNSS Service (IGS) estimates.The study utilized the new IGS ZTD product (available at ftp://cddis.gsfc.nasa.gov/gps/products/trop_new)for the interval January 2013 to December 2013 and for 15 sites distributed on the African continent as indicated by the squares in Figure 1.The new IGS ZTD product is based on the precise point positioning (PPP) technique.It has a higher sampling rate and lower formal errors than the legacy IGS ZTD product and can be obtained with typical formal errors of 1.5-5 mm from the IGS (Byun and Bar-Sever, 2009).Gaps are common in the data, but at least 3 month of ZTD estimates are available for each site.The IGS data are down sampled from 5 minute to daily intervals.
Detailed method of analysis and inferences are presented in the following sections of this paper.(1972) applied the gas laws to refractivity by considering the atmosphere as a mixture of dry air and water vapour.The model considers the temperature in the troposphere as decreasing with increasing height at a uniform rate, which varies slightly with latitude and season.However, in the polar region, there is a permanent inversion in the lower troposphere where the actual temperature increases with height.Saastamoinen assumed the neutral atmosphere to consist of two layers: the polytropic troposphere, which extends from the earth's surface to an altitude of approximately 11-12 km and the stratosphere, which is an isothermal layer, extending to approximately 50 km.The atmospheric water vapour is confined in the region of the troposphere only.

Saastamoinen
The Saastamoinen model for ZHD, in metres, is expressed as:     6 0.002277 1 0.00266cos 2 0.28 10 In Equation (1), P is the surface pressure in mbar,  is latitude in radians and, h is the height of the surface above the ellipsoid (in metres).
In the zenith wet delay model, Saastamoinen (1972) assumed that there is a linear decrease of temperature with height, and that the water vapour pressure decreases with height.The variation of the water vapour pressure In Equation ( 2), RH is the relative humidity to be determined from local observations, and the surface temperature in Kelvin is s T .
Saastamoinen (1972) gave the expression for the zenith wet delay model using the refractivity constant of Essen and Froome (1951)

2.2
UNB3m Hydrostatic Delay Model Leandro et al. (2006) presented a hybrid neutral atmosphere model designed for radiometric space users.This model, called UNB3m, has its algorithm based on the prediction of meteorological parameter values, which are then used to compute hydrostatic and non-hydrostatic zenith delays using the Saastamoinen model.
In order to account for the seasonal variation of the neutral atmosphere behaviour, a look-up table of meteorological parameters is used.The parameters are barometric pressure, temperature, water vapour pressure (WVP), temperature lapse rate    and water vapour pressure height factor    .This look-up table was derived from the U. S. Standard   Atmosphere Supplements, 1966 (COESA, 1966; Orliac, 2002).Table (2) lists the look-up table values for UNB3m.The data are divided into two groups, to account for the annual average (mean) and amplitude of a cosine function for each parameter.Both amplitudes and averages vary with respect to latitude, for all parameters.In the development of the UNB3m model, water vapour pressure in an earlier version of UNB3 was replaced with relative humidity values in Table (2).This addressed the problem of overestimation of humidity in the UNB3 model.In UNB3m, all computations for the point of interest are done initially using relative humidity, which is subsequently converted to water vapour pressure for use in the zenith delay computation.The conversion is done in line with the conventions of the International Earth Rotation and Reference Frame Services (IERS) (McCarthy &Petit, 2004;Leandro et al., 2006).Further details about the earlier model's (UNB3) development and performance are contained in Collins andLangley (1997, 1998).The first step in the UNB3m algorithm is to obtain the meteorological parameter values for a particular latitude and day of year using the look-up table.By definition, the origin of the yearly variation is day of year (doy) 28.This procedure is similar to the one used in the computation of the Niell mapping functions.The interpolation between latitudes is done with a linear function.The annual average of a given parameter can be computed as: In Equation (4) stands for the latitude of interest in degrees, Avg  is the computed average, i is the index of the nearest lower tabled latitude and Lat is their latitude (from Table 2).The annual amplitude can be computed in a similar manner: , 75 , 15 75 15 In Equation ( 5) Amp  is the computed amplitude.After average and amplitude are computed for given latitude, the parameter values can be estimated for the desired day of year doy .This procedure is followed for each one of the three needed parameters.Once all parameters are determined for given latitude and day of year, the zenith hydrostatic delay can be computed according to: where, T , P , and  are meteorological parameters computed according to (4), ( 5), and ( 6); H is the orthometric height in metres;  Leandro et al. (2006) presented the wet tropospheric refractivity for the station on the Earth's surface as a function of predicted meteorological parameter values.The model is analogous to the hydrostatic component and is expressed as (Farah, 2011): In equation ( 9) T , e  , P , and  are meteorological parameters computed according to equations (4-6); ; 1    (unitless); m T is the mean temperature of water vapour in Kelvin and can be computed from:

Global Pressure Temperature wet (GPT2w) Model
GPT2w is an extension of GPT and GPT2 (Boehm et al., 2007;Lagler et al., 2013) with improved capability to determine zenith delays in blind mode.The tropospheric model GPT2 itself is an enhancement of the Global Pressure and Temperature model (GPT; Boehm et al. 2007) and the Global Mapping Function (GMF; Boehm et al., 2006b).The development and validation of GPT2 as well as the comparison with GPT/GMF have been described in detail by Lagler et al. (2013) The parameters of Equation ( 11) are estimated at the four grid points surrounding the target location before extrapolating the parameters vertically to the desired height and interpolating the data from those base points to the observational site in the horizontal direction.The extrapolation of the hydrostatic mapping function follows Niell (1996), whereas the wet mapping function is assumed to be constant in the vicinity of the Earth surface.The extrapolation of the pressure relies on an exponential trend coefficient related to the inverse of the virtual temperature, and the linear extrapolation of the temperature utilizes the GPT2 inherent temperature lapse rate.Surface grids for specific humidity within the GPT2 model have been derived from linear interpolation between pressure levels in the vicinity of Earth's surface.These parameters are used to determine values of zenith wet delays, by using the expressions of Saastamoinen (1972), although this approach is not optimal, it represents the starting point for the improved version of it.Thus, the GPT2w as an extension to GPT2 comes with an improved capability to determine zenith wet delays in blind mode (Boehm et al., 2014;Moller et al., 2013;Schingelegger et al., 2014).The Saastamoinen formula was replaced with Askne and Nordius (1987) in the GPT2w model as reflected in Equation ( 12).
    In Equation ( 12), 2 k and 3 k are refractivity constants, d R is the specific gas constant for the dry component, m g is the gravity acceleration at the centre of mass of the vertical atmospheric column and s e is the water vapour pressure at the site.
Additionally, the GPT2w blind troposphere delay model provides the mean values plus annual and semi -annual amplitudes of pressure, temperature and its lapse rate, water vapour pressure and its decrease factor λ, weighted mean temperature, as well as hydrostatic and wet mapping function coefficients of the VMF1 (Vienna Mapping Function1).It also benefits from an improved spatial resolution of .
All climatological parameters have been derived consistently from monthly mean pressure level data of ERA-Interim fields with a horizontal resolution of one degree, and the model is suitable to calculate slant hydrostatic and wet delays down to three degrees elevation at sites in the vicinity of the Earth surface using the date and approximate station coordinates as input.

Assessment of the accuracies of the UNB3m and GPT2w Models
The accuracies of the UNB3m and GPT2w models were evaluated using the new IGS ZTD product for the interval January 2013 to December 2013 and for 15 sites distributed on 1 the African continent.A summary of the individual station information is presented in Table 3.The following performance indicators were adopted for the evaluation: Normalised Mean Absolute Error (NMAE) (Shcherbakov et al., 2013), Root Mean Square Error (RMSE), Model Efficiency (MEF) (Murphy, 1988), Reliability Index (RI) (Leggett and Williams, 1981), and Correlation coefficient .They performance indicators are represented as follows;

 
In Equations ( 13) -( 17), N is the number of observations, i O and i P are the "" th i observed and model estimated values, O and P are the mean observed (IGS estimates) and model (UNB3m and GPT2w) estimated values, respectively, and Bias P O .A summary of the results of the different performance evaluator is presented in Table 4.The NMAE measures the absolute deviation of the simulated values (UNB3m and GPT2w) from the observations (IGS estimates), normalised to the mean; a value of zero indicates perfect agreement and greater than zero an average fraction of the discrepancy normalised to the mean, the NMAE value from all the stations are indicative of the good performance of the GPT2w model.Similarly, RMSE measures the average square error with values near zero indicating a close match, the GPT2w model has a minimum RMSE of 22.4626 mm at ADIS and a maximum of RMSE of 44.2097 mm at VACS while for the UNB3m model, a maximum RMSE occur at ZAMB with a value of 70.1788 mm and minimum RMSE of 34.8988 mm at MOIU, thus, again the GPT2w performs better at all the stations.The MEF, which is a measure of the square of the deviation of the model's values (UNB3m and GPT2w) from the observations (IGS), normalised to the standard deviation of the observed data (IGS values).MEF values range from [0, 1] as agreement between predicted values and observations change from no agreement (MEF = 0) to perfect agreement (MEF = 1).From Table 4 it is evident that the GPT2w model performs better at the stations with a range of 0.9013 to 0.3916, except at NKLG where a value of 0.3916 was obtained, the UNB3m model had a range of 0.6419 to 0.3361 which is indicative of a lower variability in the MEF compared to the GPT2w.The RI quantifies the average factor by which the model estimates differ from the IGS solutions.For example, an RI of 2 indicates that a model predicts the observations within a multiplicative factor of two, on average.Ideally, the RI should be close to one.When the RMSE is calculated for log transformed values of the predictions and observations, the RI is the exponentiated RMSE.The RI value for the two models under consideration is indicative of the strength of both models to predict ZTD within an acceptable average factor.
The time series plot of the UNB3m model, the GPT2w model, and the reference model (IGS) is shown in Figure 2. The ZTD estimated from GNSS as provided by the IGS show excellent diurnal characteristics, as the daily variations are very noticeable.However, the UNB3m and GPT2w models do not give good account for the daily variation in the ZTD estimates, but does provide a good estimate of the average daily variation across all the stations.The presence of the semi-annual amplitudes in the ZTDs is also evident in the plot of the GPT2w model across all the stations.Very prominent in the IGS, UNB3m and GPT2w time series is the annual cycle of the ZTD.Furthermore, the time series and Absolute Mean Difference (Error) (MAE) of the difference between each model and the IGS solution is presented in Figures 3(a 4. The correlation coefficient was employed to ascertain the linear inter-relationship among the IGS product, UNB3m, GPT2w, and station elevation.The resultant correlation matrix is presented in Table 5.From Table 5 it is clear that the ZTD estimates from the models under investigation exhibit a very strong negative correlation.Thus, an increase in station elevation results in corresponding decrease in the amount of ZTD over the station.This is further confirmed from  In Figure 6, it is indicative that both the UNB3m and GPT2w models perform better at low latitude ranges, i.e., from .Again, the GPT2w performs better at all latitudes.1 10 

Figure 6: Plot of RMSE versus station absolute latitude
As seen in Figure 7, the MEF value for the GPT2w appears to be small at low latitudes range of , at the same latitude range the UNB3m model is seen to agree with the GPT2w model.Again, the GPT2w have better MEF values for all of the station latitude ranges, except at the stations situated almost at the equator (MBAR, NKLG, and NURK).ftp://cddis.gsfc.nasa.gov/pub/gps/data/daily/.This is a highly accurate meteorological measurement system for GNSS meteorology and environmental monitoring; it measures pressure with an accuracy of +/-0.05hpa from 500 to 1100hpa, temperature +/-0.2deg Celsius, and humidity +/-2% to 100% at standard temperature.
In Figure 8, ZTD was computed with the Saastamoinen formula using measured pressure and temperature at the site and was compared with the IGS product, UNB3m and

Figure 1 :
Figure 1: Map depicting the location of IGS stations in Africa parameter value for latitude  and day of year  

Figure 2 :
Figure 2: Time series plot of the UnB3m, GPT2w, and IGS estimation of ZTD for 2013

Figure 4 ,Figure 4 :
Figure4, that the best line of fit for the IGS, UNB3m and GPT2w when plotted against the corresponding station elevation has a negative gradient, indicating the inverse proportional relationship by all three models under investigation.

Figure 5 :
Figure 5: Plot of RMSE versus station elevation

Figure 7 :
Figure 7: Plot of MEF versus station absolute latitude

02 
Figure 8 it is indicative that the ZTD trend from the Saastamoinen model agrees very well with the IGS solution, with the GPT2w showing very little variation from the IGS solution, and the UNB3m appearing almost constant throughout.The ZHD from IGS product was retrieved from the measured pressure values at the station with the Saastamoinen formula.It can be seen that there is strong agreement among the IGS, Saastamoinen and GPT2w models, this can be interpreted as an indication of the effectiveness of the GPT2w models, and the UNB3m model could still not account for the variation in daily ZHD at the station.Looking at the ZWD estimates, there is again very strong agreement between the Saastamoinen and IGS product.The UNB3m and GPT2w models show weakness in accounting for the daily variation in ZWD estimation, though a careful scrutiny of the data reveal insignificant variations in the ZWD values for the GPT2w model.

Figure 8 :
Figure 8: Estimated ZTD, ZHD, and ZWD from the Saastamoinen formula, IGS product, GPT2w model, and UNB3m model at HRAO for doy of Year 1-31, 2013.The Saastamoinen formula using meteorological parameters measured with a MET 4A unit for ZHD and ZWD estimation, the ZHD from the IGS product was also retrieved utilizing the measured parameter from the Met 4A unit

Table 1 :
Overview of Blind Tropospheric Correction Models

Table 2 :
Look-up table of meteorological parameters for the UNB3m model, the parameters Leandro et al., 2006)ro et al., 2006)

Table 3 :
Station Information for selected IGS stations in Africa

Table 4 :
Performance of the UNB3m and GPT2w for ZTD estimation against the IGS

Table 5 :
Correlation matrix of the IGS product, UNB3m, GPT2w, and station elevation