Comparative Evaluation and Analysis of Different Tropospheric Delay Models in Ghana

Tropospheric delay prediction models have become increasingly important in Global Navigation Satellite System (GNSS) as they play a critical role in GNSS positioning applications. Due to the different atmospheric conditions over the earth regions, tropospheric effect on GNSS signals also differs, influencing the performance of these prediction models. Thus, the choice of a particular prediction model can significantly degrade the positioning accuracy especially when the model does not suit the user’s environs. Therefore, a performance assessment of existing prediction models in various regions for a suitable one is very imperative. This paper evaluates and analyses seven commonly used tropospheric delay models in Ghana in terms of performances in Zenith Tropospheric Delay (ZTD) estimation and baseline positional accuracies using data from six selected Continuously Operating Reference Stations (CORS). The 1˚x1˚ gridded Vienna Mapping Function s 3 (VMF3) ZTD product and coordinates solutions from the CSRS-PPP positioning service were respectively used as references. The results show that the Black model performed better in estimating the ZTD, followed by Askne and Nordius model. The Saastamoinen, Marini and Murray, Niell, Goads and Goodman and Hopfield models respectively performed poorly. However, the result of the baseline solutions did not show much variation in the coordinate difference provided by the use of the prediction models, nonetheless, the Black and Askne and Nordius models continue to dominate the other models. Of all the models evaluated, either Black or Askne and Nordius model is recommended for use to mitigate the ZTD in the study area, however, the choice of the Black model will be more desirable.


Introduction
Global Navigation Satellite System (GNSS) is well noted for its 24-hour provision of all-weather precise three-dimensional (3D) Positioning, Navigation, Surveillance, and Timing capabilities on a global scale to all manner of users. Regardless of GNSS's enormous capabilities, the system is still faced with some major limitations and shortcomings that decrease the precision and accuracy of GNSS positioning. One of such limitations is the problem of delays in signal propagation through the Earth's atmospheric layers, primarily, the ionosphere and the troposphere. GNSS signals propagating through the Earth's atmospheric layers, experience refraction due to the variations in refractivity along the signal transmission path, which changes the propagation speed and direction before arriving at the receiver. This results in time delay and bending and consequently causes an increase in the optical path lengths travelled by the signals more than the true geometric distances between the satellites in view and the receiver (Mendes, 1999;El-Rabbany, 2002). Thus, positioning accuracy is degraded by this phenomenon as range errors are introduced in the position solution.
Nonetheless, each atmospheric layer has a different impact on the GNSS signals, of which the ionospheric delay can to a large extent be removed by an ionosphere-free linear combination of dual or multi-frequency GNSS observables owing to its dispersive nature (Wang and Li, 2016). This leaves tropospheric delay as the major error source in GNSS applications (Ma et al., 2019). The impact of tropospheric delay ranges from about 2.0 to 2.6m in the zenith direction to about 20 to 28m near the horizon and at lower elevation angles (Sanlioglu and Zeybek, 2012). The extent of the impact of tropospheric delay on GNSS signals is further aggravated in the low-latitude or equatorial region due to the thicker tropospheric layer (up to 16km) and the hot conditions in this region (Musa et al., 2005).
The thicker tropospheric layer exposes the low-latitude region to huge volumes and imbalance of water vapour content which significantly affects the propagating signals via the troposphere (Amir and Musa, 2011). However, Ghana's proximity to the equator makes Ghana more susceptible to the impact of the tropospheric delay. Tropospheric effects on GNSS signals, therefore, remain a major challenge and a growing concern to GNSS users particularly, in the low-latitude region (Musa et al., 2005). Modelling and/or mitigation is therefore required for precise GNSS positioning and navigational applications.
The most common technique used in mitigating tropospheric delay in GNSS applications is the use of empirical prediction models (Hu and Yao, 2019). A number of these prediction models have been developed and implemented in the various GNSS processing software for use to estimate and mitigate tropospheric delay error. The availability of these models provides users with countless means of selection in their applications to reduce the tropospheric effect on GNSS signals (Chen and Liu, 2016). However, since positioning accuracy is affected by the accuracy of the tropospheric delay model, an improperly selected tropospheric model may result in mismodelling of tropospheric delay which may significantly impair the performance of tropospheric delay mitigation, resulting in degrading the positioning accuracy up to an error of about 10m or more (Penna et al., 2001;Farah, 2020). But then again, the problem that arises is, which model best-suit the prevailing environs or atmosphere since the accuracies and performances of these prediction models differ under different atmospheric or climatic conditions (Opaluwa et al., 2013;Tuka and El-Mowafy, 2013). Consistent with Isioye et al. (2015) users in many instances employ a particular tropospheric model based on its popularity or those which are set as default in their processing software without giving adequate background as to why it should be used. Therefore, it is important to evaluate the impact and performance of these prediction models in different places or regions. This paper aims at evaluating and analysing seven commonly used tropospheric delay models in Ghana in terms of performances in estimating the ZTD and positional accuracies using one-month GNSS data from six selected CORS. The 1˚x1˚gridded VMF3 ZTD product and coordinates solutions from the CSRS-PPP online positioning service were used as references and the optimal tropospheric model for the study area is recommended.

Tropospheric Delay Modelling
The troposphere forms the lower portion of the neutral atmosphere closest to the earth. It extends at a height of about 8 km over the poles and 16 km over the equator (Kos et al., 2009;Misra and Enge, 2011). The effect of the troposphere on GNSS signals is called tropospheric delay which induces an excess propagation path length on GNSS signals and emanates from the sum of the propagation delay and signal bending (Mendes, 1999). From Fermat's principle and following (Bevis et al., 1992;Mendes, 1999;Kleijer, 2004) the excess propagation path ( t r L ∆ ) can be calculated by integration of the refractive index along the signal transmission path, s, from satellite, t to receiver, r at elevation angle (θ) via the troposphere as: where S is the optical or electromagnetic path length [m], c = speed of light in vacuum [ms-1], ν= ds dt , the propagation speed through the troposphere [ms-1], n= c v , the refractive index, G is the geometric range along the straight path between t and r assuming the signal travel through a vacuum.
In the zenith direction (ZD) where satellites are directly overhead the receiver (i.e. θ = 90ᵒ), the signal path is a straight line (i.e. S and G become identical) and the bending effect disappears with the assumption that the troposphere is horizontally stratified (Bevis et al., 1992;Mendes, 1999).
Expressing n in terms of total refractivity N, where path and can be separated into hydrostatic or dry (Nd) and wet (Nw) components caused by dry gases (mainly N2, O2) and water vapour respectively (Sanz et al., 2013).
The refractivity (N) can be related to the meteorological parameters (P, T, e) as as (Smith and Weintraub, 1953): , , k k k are refractivity constants whose values according to Bevis et al (1994)  where, h is the height of station or receiver antenna, and TOA is the top of the atmosphere. It can be seen from equation (8-10) that, ZTD is the sum of the Zenith Hydrostatic Delay (ZHD) and the Zenith Wet Delay (ZWD) expressed as With an appropriate mapping function and using the satellite elevation angle(θ) as input, the ZTD can be resolved into the Slant Tropospheric Delay (STD) as: where MFh and MFw represent the hydrostatic and wet mapping functions respectively, SHD and SWD are the slant hydrostatic and wet tropospheric delays respectively.
The ZHD accounts for about 90% of the ZTD and can easily be modelled or predicted to submillimetre accuracy by empirical prediction models using surface meteorological observations; whereas the ZWD accounts for the remaining 10% of the ZTD. In contrast to the ZHD, the ZWD cannot be precisely modelled or predicted due to its large spatial and temporal variability (Younes, 2016;Zhang et al., 2016). The troposphere is characterized as being a non-dispersive medium for radio frequencies up to 15 GHz, hence its effect is independent of GNSS frequencies and therefore affects carrier phase and code measurements equally (Sanz et al., 2013).

Tropospheric Delay Models
Tropospheric delay models are categorized into two (Sun et al., 2019). The first category, the traditional models, require real-time surface meteorological observations at the user's environs to predict the tropospheric delays at decimeter to centimetre level of accuracy. The second category, the blind models, is not dependent on observed meteorological parameters but require user location and time information for the prediction of tropospheric delay computed from climatological data or forecast of Numeric Weather Prediction (NWP) models. This paper compares the performances of the first category (traditional models). The following sub-sections give a brief description of the various traditional tropospheric models used in this study.

Refined Saastamoinen Model (SAAS)
The general form of the Refined Saastamoinen model with mapping function for the computation of STD is written as: where P is the pressure in hPa, e is the partial pressure of water vapour in hPa, T is the temperature in Kelvin, z is the zenith angle, β and δR are correction terms dependent on the height of the station and the zenith angle given in a tabular form. Details of table can be inferred from (Hofmann-Wellenhof et al., 2008) and (Xu and Xu, 2016).

Hopfield Model (HOP)
By using two-quartic refractivity profiles expressed as a function of height of the 4th degree above the Earth surface, Hopfield (1969)

Goad and Goodman Model (GG)
Goad and Goodman model (Goad and Goodman, 1974), also known as the modified Hopfield model stems from the modification of Hopfield model by the use of lengths of position vectors instead of heights (Hofmann-Wellenhof et al., 2008). The STD by Goad and Goodman can be summarized as (Goad and Goodman, 1974;Hofmann-Wellenhof et al., 2008;Xu and Xu, 2016): the subscript i is used to identify the dry and wet components of the tropospheric delay (i.e. ), N is the refractivity, r is the range to the top of the dry or wet component, i h is the topmost point where the wet or dry component becomes zero, E is the satellite elevation angle.
Nd and Nw can be expressed as (Xu and Xu, 2016):

Black Model
By utilizing Hopfield model, (Black, 1978) proposed a simple analytical form in the integration of the refractivity. The hydrostatic and wet components of the model are as follows: is the mapping function given as (Black, 1978): At higher elevations (i.e. E >30˚), Black (1978) proposed E is the Satellite elevation angle in [degrees], s r is the distance from the centre of the earth to the station in [meters]. For elevation angles of 5˚ and above, a choice of gives an error no greater than 4.5 cm.
Considering the difficulty in modelling the wet component of the tropospheric delay, (Black, 1978) suggested the following regional empirical constants: Kw = 0.28 m for summer in tropic or mid-latitude regions, 0.20 m for spring or fall in mid-latitudes, 0.12 m for winter in maritime mid-latitudes, 0.06 m for winter in continental mid-latitudes, and 0.05 m for polar regions.

Askne and Nordius Model (AN)
Askne and Nordius model (Askne and Nordius, 1987) for the computation of tropospheric path delay is based on a two-parameter closed-form model of atmospheric profile analysis. The ZHD and ZWD are given as (Askne and Nordius, 1987): where Tm is the weighted mean temperature(K) given as: The Askne and Nordius model provides only zenith delays and has no internal mapping function.
We employed the Black and Eisner mapping function model (Black and Eisner, 1984) for the computation of the slant delays which can be expressed as (Black & Eisner, 1984):

Marini and Murray Model (MM)
Marini and Murray Model (Marini and Murray Jr, 1973)  Niell model is the blend of Saastamoinen zenith delays and the Niell mapping function (Niell, 1996) model (Dodo et al., 2019). The Niell Mapping Function (NMF) formulation is a function of station latitude and observation time and it is based on the interpolation of the average and seasonal amplitude values given in a tabular form at 15˚ latitudinal interval. The general form of the model for both hydrostatic and wet mapping functions can be expressed as: However, the addition of height correction term to the hydrostatic term differentiates the two models. Thus, the final hydrostatic mapping function mfh is given as:

Data Acquisition
Data acquired in this study include GNSS data from GNSS CORS in Ghana, Meteorological parameters derived from the GPT3 model, and ZTD data obtained from the gridded VMF3-ZTD products. The following sub-sections summarize the various datasets used in this study.

GNSS Data
One month of GNSS data spanning from September 1 to September 30, 2019, were collected from six GNSS CORS located in four regions of Ghana, namely, Bolgatanga (Upper East region), Kumasi (Ashanti region), Accra (Greater Accra region), Takoradi and Tarkwa (Western region). Details of the stations and their locations in WGS 84 are shown in Table 1 and Figure 1.

Meteorological Data
None of the CORS used in this study is equipped with ground-based meteorological sensors for the provision of in-situ meteorological parameters. However, in the absence of in-situ surface meteorological observations, the empirical blind tropospheric delay models can be used for the provision of accurate surface meteorological parameters (Böhm et al., 2015). This study utilized the recent blind tropospheric delay model, Global Pressure and Temperature 3, GPT3 (Landskron and Böhm, 2018). GPT3 is a refinement of the GPT2w (Böhm et al., 2015)

VMF ZTD Data
The Vienna Mapping Functions (VMF) formally the Global Geodetic Observing System (GGOS) atmosphere provides global gridded VMF1 and VMF3 zenith tropospheric products (http://vmf.geo.tuwien.ac.at/trop_products/GRID). They are generated from the ECMWF reanalysis data and are provided every 6 hours daily at four epochs 00, 06, 12 and 18 UTC. VMF3 is the successor of VMF1 realized on both 1˚ and 5˚ global grids (Landskron and Böhm, 2018). This study utilized the 1˚ gridded VMF3-ZTD products as a reference for evaluating the prediction models ZTD following (Yao et al., 2016;Sun et al., 2017) and also owing to the fact that the CORS employed in this study are not International GNSS Service (IGS). The precision and accuracy of the VMF-ZTD data have been investigated and validated against the IGS-ZTDs by (Yao et al., 2017(Yao et al., , 2018 to be accurate enough for tropospheric delay mitigation, model development and validation.

Data Processing
(i) GNSS data processing -24 hours GNSS data in rinex format collected from the six selected CORS were processed at 60-seconds sampling rate and 10˚ elevation cut-off using goGPS software version 0.5.2 beta 1 (Realini and Reguzzoni, 2013;Herrera et al., 2016). The relative positioning technique on both code and carrier-phase double-difference was utilized with Kalman filtering using the LISAG_SPINTEX station as the base and the remaining stations as rovers. A total of five baselines, shown in table 2, were processed. Two processing strategies were employed: Strategy I: processing without the application of tropospheric delay model.
Strategy II: processing with the application of the tropospheric delay models.
(ii) Extraction of VMF3-ZTD -To obtain the VMF3-ZTD at the various CORS, we needed to interpolate the VMF3-ZTD to the exact location and time of each COR station. So, the stations' ellipsoidal coordinates (latitude, longitude, and height) and modified Julian (mjd) date obtained from the rinex observation data were used. The extraction was done using the MATLAB coded script or m-file 'SearchReadVMFgrids.m', which is a modified version of the file vmf3_grid.m written by Landskron and Böhm (2018). The vmf3_grid.m file can be obtained at http:// vmf.geo.tuwien.ac.at/codes/ (last access 23 September 23, 2020). For each epoch grid file (e.g. 00 UT), a spatial interpolation to the specified locations of the CORS is first performed using the four surrounding grid points that are closest to the specified locations. A bilinear interpolation is then employed to interpolate the zenith delays at the grid points heights. By employing formulae suggested by Kouba (2008), the zenith delays at the grid heights are then transferred to the respective height of the COR stations using the orography file, orography_ell_1x1. Finally, the zenith delays (ZHD and ZWD) at the current epoch are obtained by a linear interpolation using values from two surrounding epochs. The average daily ZTD is obtained by adding the average daily ZHD and ZWD values.
It is worth mentioning that the prediction models used in this study and other ancillary information relating to tropospheric delay modelling were all coded and implemented in the goGPS v0.5.2 beta 1 software. The processed station coordinates in Universal Transverse Mercator (UTM) from the two processing strategies as well as the estimated ZTD from strategy II were extracted for analysis.

Results and Discussion
In order to investigate the performance of the prediction models, SAAS, HOP, GG, Black, Niell, AN and MM in the study area, the estimated ZTD obtained from each of the prediction models was compared with the interpolated gridded VMF3-ZTD dataset. Likewise, the final station UTM coordinate solutions [North(N), East(E), Height(h)] from the two processing strategies were also compared to their corresponding reference coordinates obtained from the CSRS-PPP online positioning service. The statistical quantities, mean Bias and Root Mean Square Error (RMSE) of each model ZTD and coordinate solution were computed for each station using equation (48) and a summary of the results are presented in Table 3 and Figure 2, 3, 5, 6, 7, 8, 9 and 10.
( ) where n is the number of observations,

Models Assessment Based on estimated ZTD
The performances of the prediction models are evaluated by comparing their estimated ZTDs with those derived from the gridded VMF3 ZTD data. From the results in Table 3 and Figure 2, 3 and 5 it can be seen that the mean ZTD bias and RMSE for the black and Askne and Nordius models are very small compared to Saastamoinen, Marini and Murray, Niell, Goads and Goodman, and Hopfield models which gave large bias and RMS errors respectively. The smaller the bias and RMS errors, the better the model's predictions.
The difference in performance is mainly due to the modelling of the wet component of the tropospheric delay in the models particularly the assumption made about the distribution of water vapour within the troposphere (Hofmann-Wellenhof et al., 2008;Li et al., 2008). For instance, the Black model assumes regional empirical constants that vary according to season and climate. Though Black model assumes constant wet delays but seems to perform better in the study area. The Saastamoinen model assumes that the water vapour pressure decreases with height in a similar manner as the total pressure as: Equation (14) and equations (32 and 33 combined) according to Askne and Nordius (1987) can be made numerically identical when 6.2 K km α = and 3 λ = . Besides, α and λ are assigned constant values in the Saastamoinen model while variable in the Askne and Nordius model to meet the prevailing environs. Therefore, models, where α and λ can be varied to suit the prevailing environs, produces better accuracy (Askne and Nordius, 1987;Li et al., 2008). Moreover, the numerical values for the refractivity constants k1, k2, k3 used by the Saastamoinen model differ slightly from those used by the Askne and Nordius model. These slight differences in refractivities also contribute to the varying performances of the models in the study area. The Marini and Murray and Niell models performed comparably to that of the Saastamoinen model since they both stem from the Saastamoinen model. Hopfield model, on the other hand, assumes that the wet refractivity decreases with height above the earth surface in about the same manner as the dry refractivity where 1 4 d g R α − = with g = 980cms -2 and α = 6.828°C/km. This assumption according to Hopfield (1969) corresponds in regions of the earth where this lapse rate value is a realistic value in the troposphere. However, this assumption may differ from reality since g and α obviously vary with latitude and/or season respectively (Li et al., 2008) (Chen et al., 2014) and (Li et al., 2008).
Furthermore, the result from the multiple comparison test as shown in Figure 4 and Table 4 indicates that the mean of group 1 (i.e. AN) in blue is considered to be significantly different from group 3, 4, 5, 6 and 7 (i.e. GG, HOP, MM, NIELL and SAAS) in red and with small p-values (0.000) at 95% confidence level, but not significantly different from group 2 (Black) shown in grey and with p-value > 0. Thus, the performance of Black and AN are significantly different from that of GG, HOP, MM, NIELL and SAAS models at 95% confidence level in the study area.

Models assessment based on the positional accuracies
To further analyse the performance of the prediction models and their impact on positional accuracies, the baseline coordinate solutions from each prediction model were analysed. The results in Figure 6, 7 and 8 indicate that ignoring the application of tropospheric model (strategy I) leads to unreliable baselined results with large bias and RMS errors compared to strategy II (application of tropospheric models). The results also did not show much significant variation in the coordinate difference provided by the application of the prediction models. However, the Black and Askne and Nordius models continue to show superior performance over the other models. It can also be seen from Figure 6, 7 and 9 that positional impact of the tropospheric delay in the horizontal component is more on the North than the East. This is in agreement with the study conducted by Odumosu et al. (2015). It can further be observed that the longer the baseline the more the impact of the tropospheric delay resulting in large biases or residuals. This affirms the assertion that tropospheric delay is a distant-dependent error (Yahya and Kamarudin, 2005;Dodo et al., 2019). The results further revealed in Figure 10 that the error in the height component increases with increasing baseline height difference (∆H).

Conclusion
The accuracy of tropospheric delay prediction models plays a critical role in precise GNSS positioning applications. This paper evaluated and analysed the performances of seven tropospheric delay models for modelling and mitigating tropospheric delays using six selected CORS over four regions of Ghana. Firstly, the models were evaluated based on the estimated ZTD using the 1˚ gridded VMF3 ZTD product as a reference. The analysis showed that the Black model performed better followed by Askne and Nordius model. Saastamoinen, Marini and Murray, Niell, Goad and Goodman, and Hopfield models performed poorly in the order given. The MCT performed on the RMSE of the models ZTD revealed that the Black and Askne and Nordius models produced results that are significantly different from the other models at a 5% significance level in the study area.
To further investigate the impact of the tropospheric delay models on positional accuracies, the baseline coordinates solutions from each prediction model were also analysed using corresponding coordinates solutions from the CSRS-PPP online positioning service as a reference. The results showed that: I. Neglecting the application of a tropospheric delay model in the baseline processing leads to unreliable baseline results.

II.
There was not much variation in the coordinate difference provided by using the prediction models. However, the Black and Askne and Nordius models remained dominated over the other models.
III. Positional impact of the tropospheric delay in the horizontal component is more on the North than the East.
IV. Increase in the baseline length increases the impact of the tropospheric delay leading to large biases or residuals.
V. The residuals in the height component increase with increasing baseline height difference.
Based on the results, either Black or Askne and Nordius model is recommended for estimating and mitigating tropospheric delay in the study area for positioning and navigational applications, however, the choice of Black model will be optimal.