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# Assessment of GNSS zenith tropospheric delay responses to atmospheric variables derived from ERA5 data over Nigeria

*Satellite Navigation*
**volume 4**, Article number: 15 (2023)

## Abstract

Tropospheric delay is a major error caused by atmospheric refraction in Global Navigation Satellite System (GNSS) positioning. The study evaluates the potential of the European Centre for Medium-range Weather Forecast (ECMWF) Reanalysis 5 (ERA5) atmospheric variables in estimating the Zenith Tropospheric Delay (ZTD). Linear regression models (LRM) are applied to estimate ZTD with the ERA5 atmospheric variables. The ZTD are also estimated using standard ZTD models based on ERA5 and Global Pressure and Temperature 3 (GPT3) atmospheric variables. These ZTD estimates are evaluated using the data collected from the permanent GNSS continuously operating reference stations in the Nigerian region. The results reveal that the Zenith Hydrostatic Delay (ZHD) from the LRM and the Saastamoinien model using ERA5 surface pressure are of identical accuracy, having a Root Mean Square (RMS) error of 2.3 mm while the GPT3-ZHD has an RMS of 3.4 mm. For the Zenith Wet Delay (ZWD) component, the best estimates are derived using ERA5 Precipitable Water Vapour (PWV). These include the ZWD derived by the LRM having an average RMS of 20.9 mm and Bevis equation having RMS of 21.1 mm and 21.0 mm for global and local weighted mean temperatures, respectively. The evaluation of GPT3-ZWD estimates gives RMS of 45.8 mm. This study has provided a valuable insight into the application of ERA5 data for ZTD estimation. In line with the findings of the study, the ERA5 atmospheric variables are recommended for improving the accuracy in ZTD estimation, required for GNSS positioning.

## Introduction

Over the years, Global Navigation Satellite Systems (GNSS) have been applied for positioning and navigation. The system is weather-proof and in continuous operation providing real-time navigation information (Orliac, 2009). While most of the errors associated with this system are easily mitigated by adopting suitable differencing or precise point positioning techniques, the errors due to atmospheric refraction remain major problems in GNSS positioning. GNSS observations made on or near the earth’s surface are usually degraded by the effect of tropospheric error, resulting from slowing and refraction of the GNSS signals propagating from the satellite to the receiver and consequently leading to significant positioning error, thereby lowering the positioning accuracy. Tropospheric delay considered vertically above the receiver's horizon is termed Zenith Tropospheric Delay (ZTD), mapped to a slant delay using a mapping function. Mapping functions project the original slant path of the GNSS signals to the zenith, which is dependent on the satellite’s azimuthal position and elevation angle. The ZTD in GNSS signals consists of Zenith Hydrostatic Delay (ZHD) and Zenith Wet Delay (ZWD).

The ZHD which contributes approximately 90% to the ZTD has been described in Tregoning and Herring (2006) and Wang et al. (2017). In the ZHD estimation, the errors in the a-priori ZHD can degrade the accuracy of GNSS solutions (Kouba, 2009; Tregoning & Watson, 2009; Wang et al., 2008). The a-priori ZHD errors have been identified to project into Global Positioning System (GPS) height estimates with a typical sensitivity of up to ~ 0.2 mm/hPa with a satellite elevation cut-off angle of 7° (Tregoning & Herring, 2006). Therefore, to obtain a height solution with an accuracy better than 1 mm, the pressure errors for calculating the a priori ZHD should be below 5 hPa equivalent to the a-priori ZHD error of ~ 10 mm (Zhang et al., 2021). Also, the study of Zhang et al. (2021) suggests that the sensitivity of PPP-ZTD/height solutions to the a-priori ZHD errors decreases by adding GLObal NAvigation Satellite System (GLONASS) data at high latitudes but increases at low latitudes. Therefore, an accurate ZHD is essential for high-accuracy GNSS positioning and tropospheric delay modelling (Douša et al., 2018). The ZWD, which contributes approximately 10% to ZTD depends on the Atmospheric Water Vapour Component (AWVC) which exhibits high spatiotemporal variations (Namaoui et al., 2017; Shrestha, 2003). The Saastamoinen model is the most used in geodetic applications, and its accuracy is widely reported. However, the Saastamoinen model cannot meet the needs for high-accuracy GNSS positioning and meteorological applications, since most GNSS geodetic software uses the Saastamoinen model with standard atmospheric models for the a-priori estimates (Isioye et al., 2015). Evaluations of empirical models (Hopfield, 1969; Neill, 1996; Saastamoinen, 1972) over Africa and Nigeria's GNSS networks have shown the standard deviation ranging from 45.0 to 53.0 mm (Dodo & Idowu, 2010; Dodo et al., 2015).

Global Pressure and Temperature 3 (GPT3) is a blind empirical tropospheric delay model, optionally on a 5° × 5° or 1° × 1° grid (Landskron & Böhm, 2018). Ding and Chen (2020) evaluated the GPT3 model based on Nevada Geodetic Laboratory (NGL) global tropospheric data. Based on this study, the global average Root Mean Square (RMS) of the ZTD estimated by GPT3 is 44.1 mm. Currently, there are 4 versions: GPT, GPT2, Global Pressure and Temperature 2 wet (GPT2w), and GPT3. The accuracy of GPT2 evaluated using the GNSS data in China is 46.5 mm (Wang et al., 2016). GPT2w adds two parameters of temperature lapse rate and mean temperature of the water vapour based on GPT2, considered to be the most accurate tropospheric model for quite a long time. GPT3 is an upgraded version of GPT2w, developed along with VMF3 (Vienna Mapping Functions 3), which is based on ERA-Interim (ERA-I) pressure-level data. It adds two parameters of gradients in the north and east direction to GPT2w (Landskron & Böhm, 2018). A study on the blind tropospheric delay models: University of New Brunswick (UNB3) (Leandro et al., 2006) and GPT2w (Böhm et al., 2015) using the African International GNSS Service (IGS) stations has been carried out (Isioye et al., 2015), which records an RMS of ~ 50.0 mm.

GNSS, an efficient meteorological tool, has been applied in estimating atmospheric variables such as Precipitable Water Vapour (Alshawaf et al., 2015). The GNSS-PWV data have several applications, such as calibrating or validating AWVC observations generated from other sources such as radiosonde observations satellite water vapour observations with high accuracy, and monitoring climate change (Bock et al., 2010; Buehler et al., 2012). GNSS-PWV data is also applied in assimilation into numerical weather prediction models for improving accuracy (Ssenyunzi et al., 2020). A comparison between GNSS and ERA-I PWV showed a difference of 4.0 mm in the Algerian highland region (Namaoui et al., 2017). The development of empirical models for estimating PWV based on the measured surface meteorological data (ambient temperature, relative humidity, vapour pressure and dew-point temperature), collected from the Nigerian Meteorological Agency (NIMET) was carried out (Abimbola & Falaiye, 2016). These models were validated against the National Center for Environmental Prediction (NCEP) Reanalysis II and the Department of Energy (DOE) data. The PWV estimated from the dew-point temperature performed better than the estimates from relative humidity. In Nigeria, a comparison of PWV derived from 3 GNSS stations and the NCEP II for 2012, gave RMS ranging from 2.8 to 3.4 mm (Abimbola et al., 2017). Again, evaluations of the ray-traced Numerical Weather Model (NWM) ZTD over the Nigerian GNSS Network (NIGNET) resulted in standard deviations of ~ 33.1–44.9 mm based on IGS-ZPD estimates (Mayaki et al., 2018). Evaluations of site-wise versions of the Vienna Mapping Function 3 ZTD (VMF3-ZTD) at West Africa IGS stations, based on IGS-ZTD yielded RMS of ~ 11.0 mm (Osah et al., 2021). This is better compared to the gridded version evaluated over the NIGNET which yielded RMS ranging from ~ 41.0–95.0 mm, locally refined over Nigeria, leading to an improved accuracy with RMS of ~ 30.0–37.0 mm (Nzelibe & Idowu, 2023).

In July 2017 the European Centre for Medium-Range Weather Forecasts (ECMWF) released the fifth-generation Reanalysis dataset (ERA5) (Yang et al., 2020). The dataset uses updated observation data and improved physical processes compared to previous reanalysis data, including ERA-I. The dataset offers global atmospheric variables with a spatial resolution of 0.25° × 0.25° regular latitude/longitude grid and a temporal resolution of 1 h. In ERA5 the atmosphere is divided into 137 model levels from the earth's surface up to a height of 80 km which are interpolated to 37 pressure levels (Albergel et al., 2018). Several studies have been conducted around the globe on the assessment of the ERA5 atmospheric variables in GNSS meteorology. Assessment of ERA5-ZTD data over China based on GNSS-ZTD has shown an average RMS of ~ 39.3 mm and 11.5 mm for surface-level and pressure-level ERA5-ZTD, respectively (Jiang et al., 2020). In Tropical East Africa, the validation of ERA5-PWV based on GNSS-PWV resulted in an RMS range of 1.4–2.3 mm (Ssenyunzi et al., 2020). Based on global radiosonde data, the evaluations of the spatiotemporal variations of PWV from 1994 to 2020 for over 14,000 NGL stations worldwide show the average RMS of ~ 1.1–2.6 mm (Ding et al., 2022). In Nigeria, the seasonality of ERA5-PWV was assessed based on the NGL-PWV between 2012 and 2013**,** resulting in an average RMS of 3.7 mm, 3.4 mm and 7.0 mm for sub-daily, diurnal, and seasonal scales, respectively (Bawa et al., 2022). The ERA5 data can improve the accuracy of estimated ZTD compared to the previous reanalysis data, which is indicated in the higher accuracies of ZTD derived from ERA5 pressure-level and surface-level data by approximately 13.8% and 10.9%, respectively than those from ERA-I in China (Jiang et al., 2020). However, not much work has been reported on the applications of the ERA5 dataset in Nigeria and the sub-Saharan African region. This calls for more studies to validate the ERA5 atmospheric datasets based on observed datasets over the region. Again, considering the data gaps in the NIGNET observations, attributed to poor network maintenance (Mayaki et al., 2018), an improved alternative solution is required for estimating ZTD for improving geodetic positioning within the study area.

This study focuses on assessing the potentials of the ERA5 atmospheric variables in estimating ZTD and developing local empirical models required for estimating ZTD based on the surface-level atmospheric data over Nigeria. ERA5 atmospheric variables considered are surface pressure, dew point temperature, PWV, surface temperature, relative humidity, and specific humidity. The estimates from a simple linear regression and standard models with the ERA5 data are used in fitting with the GNSS-ZTD data for the hydrostatic and wet components.

## Materials and methodology

### Study area

The coverage of the study is bordered by latitude 4° N and 14° N and longitude 2.4° E and 15° E with Nigeria being the focus of the study. The Federal Government of Nigeria through the Office of the Surveyor General of the Federation (OSGoF) and the Presidential Task Force on Land Reform (PTFLR) has set up surveying infrastructure throughout the country known as the Nigerian GNSS network (NIGNET). This consists of a GNSS Continuously Operating Reference Stations (CORS) network set up at different locations in Nigeria for satellite positioning-related applications. The details on the stability of the NIGNET CORS can be found in Moses et al. (2021). For this study, six GNSS CORS are used, while some GNSS-CORS found within the study area are unused, attributed to large data gaps resulting from either non-streaming of data at the CORS or the inability of the data to be processed within the period of study. However, the stations used have a good spread, covering the study area as shown in Fig. 1. This figure represents the map of the study area indicating the locations of the GNSS stations. The coordinates of the GNSS stations used are presented in Table 1.

### GNSS data and processing with GAPS

Daily Receiver INdependent EXchange format (RINEX) files at 6 GNSS Stations within the study area, for a period of 3 years (2014–2016), were downloaded from the NIGNET and IGS sites and processed using the GNSS Analysis and Processing Software (GAPS). GAPS is considered preferable in processing GNSS observations, since it employs the Precise Point Positioning (PPP) technique for processing GNSS observations from a single receiver, compared to those based on differential positioning that depends on baselines. This is so since very long baselines between the GNSS stations in the study area are observed and it has been shown that an increase in the length of baseline produces a larger tropospheric effect (Dodo & Idowu, 2010). The PPP applies the un-differenced observations to determine the absolute position of a receiver, which makes it prone to errors in ZTD or mapping functions. The PPP takes advantage of the precise carrier phase measurements, the a priori orbit**,** and clock products to obtain geodetic level accuracy**,** assuming the parameters have sufficient time to converge. The precise clock and orbit products derived from the IGS information from Script Orbit and Permanent Array Center (SOPAC) used are to ensure high-quality ZTD estimation. To eliminate the ionosphere's effect, PPP takes advantage of carrier phase ionosphere-free linear combinations of the GNSS undifferenced L1 and L2 carrier-phase and pseudo-range measurements. GNSS observations are processed over 24 h based on forward-only, sequential, and weighted least squares to estimate the unknown parameters. The forward-only, sequential, and weighted least squares filter used is in the Eq. (1).

where \({\varvec{\delta}}_{i}\) is the update vector, \({\varvec{A}}_{i}\) is the design matrix, \({\varvec{P}}_{i}\) is the observation weight matrix, and \({\varvec{w}}_{i}\) is the misclosure vector for epoch \(i\). The covariance matrix of the estimated parameters, \({}^{{{\varvec{\delta}}_{i} }}{\varvec{C}}\), is updated at each epoch according to Eq. (2).

where \({}^{{{\varvec{\delta}}_{i - 1} }}{\varvec{C}}\) is the covariance matrix from epoch \(i - 1\) and \({\varvec{Q}}_{\sigma }\) is the process noise matrix.

The parameters used in processing the GNSS observations using GAPS are summarized in Table 2. These parameters were adopted to ensure geodetic level accuracy in estimating ZTD. The VMF1 (Boehm et al., 2006) was adopted as the a priori hydrostatic delay model and mapping function, and an elevation angle cut-off of 10°. The satellites and receivers’ antennae for phase centre offsets and phase centre variations were corrected. The International Terrestrial Reference Frame (ITRF) 2008 was adopted for the stations' coordinates (Altamimi et al., 2011), while the ambiguities were estimated as real numbers. The ZTD was estimated at every epoch. The horizontal gradients the delay in the north–south and east–west directions are included in the estimation, and the results have shown the improvement especially under the presence of extreme weather conditions (Balidakis et al., 2018; Nikolaidou et al., 2018).

The quality of the ZTD derived by processing GNSS observations using the PPP technique with GAPS over the study area was assessed based on the IGS Zenith Path Delay (IGS-ZPD). This was carried out for BJCO, an IGS station with the result having an RMS of ~ 11.0 mm (Nzelibe & Idowu, 2023). This satisfies the 20.0 mm accuracy requirement stated in (De-Haan, 2006) for GNSS-Meteorology.

### ERA5 Atmospheric parameters and calculation of the vertical adjustment

The ERA5 data are downloaded from https://cds.climate.copernicus.eu/, with a spatial coverage between latitude 4° N and 14° N and longitude 2.4° E and 15° E covering the study area for 3 years between 2014 and 2016. The ERA5 atmospheric variables are downloaded in netCDF file format derived mainly from two ERA5 data stores. The first is the “ERA5 hourly data on single levels from 1959 to present” from which data downloaded are surface pressure (P) in Pascal, 2 m Dew Point Temperature (DPT) in Kelvin and PWV in mm. The details of this data can be found in Hersbach et al. (2018b). The second is “ERA5 hourly data on pressure levels from 1959 to present” from which data downloaded are: Relative Humidity (RH) in %, Temperature (T) in Kelvin, Specific Humidity (SH) in kg/kg and geopotential height in m^{2}/s^{2}. These datasets are provided at 37 pressure levels. The details on this data are provided by Hersbach et al. (2018a). All atmospheric variables acquired from the ERA5 have a uniform spatial resolution of 0.25° × 0.25° latitude/longitude and temporal resolution of 1 h. This dataset can be classified based on the ZTD applications as hydrostatic and wet. The ZHD is estimated based on the ERA5-surface pressure for the hydrostatic component. For the wet component, the ZWD is estimated based on AWVC. The AWVC is expressed using various measures**:** vapour pressure, specific humidity, mixing ratio, DPT, relative humidity, Integrated Water Vapour (IWV), and PWV. However, ERA5-AWVC data considered in this study are RH (%): the water vapour pressure as a percentage of the value at which the air becomes saturated, SH (kg/kg): the mass of water vapour per kilogram of moist air, 2 m DPT (K): the temperature to which the air at 2 m above the surface of the Earth would have to be cooled for saturation to occur and PWV (mm): the total amount of water vapour in a column extending from the surface of the earth to the top of the atmosphere (Hersbach et al., 2018a, 2018b). The data from the ERA5 were visualised and extracted with Panoply software.

To model the ZTD parameters, the ERA5 atmospheric parameters acquired on the surface and pressure levels at elevations that differs from that of the GNSS site are reduced to the elevations of the GNSS stations by vertical adjustment methods. Thereafter, horizontal interpolation is applied using bilinear interpolation between four surrounding grid points. The vertical temperature reduction is performed based on the temperature lapse rate \(L\) derived from ERA5 successive temperature pressure levels, applied in the formula shown in Eq. (3). The pressure at the GNSS site \(P_{S}\) is obtained by vertically interpolating the surface pressure \(P_{0}\) using the barometric formula expressed in Eq. (4). The vertical reduction method in the AWVC is achieved based on the ERA5 Tropospheric humidity profile (g/kg). This is defined as the ratio of the mass of water vapour in the air to the total mass of the mixture of air and water vapour (Gleisner et al., 2019). The ERA5 tropospheric humidity profile data has a global gridded coverage with a horizontal resolution of 5.0° in latitude, and 0–12 km vertical coverage at a vertical resolution of 0.2 km. The vertical reduction of the PWV is obtained by its definition, expressed by the Eq. (5).

In Eq. (3) \(T_{S}\) is the reduced temperature at the elevations of the GNSS station, \(T_{0}\) is the temperature at the ERA5 model base pressure level, \(h_{S}\) is the elevation of the observation station above Mean Sea Level (MSL) and \(h_{0}\) is the elevation of the ERA5 model base pressure level above MSL, obtained by dividing the geopotential heights (m^{−2} s^{−2}) at the base pressure level by the acceleration due to gravity \(g\) (9.806 65 m s^{−2}). In Eq. (4), \(T\) (288.15 K) is the standard temperature of the atmosphere at sea level, \(R\) (8.314 32 J mol^{−1} K^{−1}) is the universal gas constant for air, \(M_{d}\) (0.028 96 kg mol^{−1}) is the molar mass of dry air and \(g\) is the gravitational acceleration. In Eq. (5) \(\rho_{w}\) is the density of water vapour (1000 kg m^{−3}), \(g\) is the gravitational acceleration at the station ERA5 model level altitude, \(q(p)\) is the mixing ratio of water vapour obtained from the ERA5 Tropospheric humidity profile. This data is provided at pressure levels starting from the ERA5 model base pressure level \(q(p)_{0}\), while \(q(p)_{S}\) at the GNSS station is obtained along the vertical profile by linear interpolation.

### Computations of ZTD from atmospheric parameters

The ERA5 computed ZTD *V*_{ZID} comprises the hydrostatic ZHD *V*_{ZHD} and wet ZWD *V*_{ZWD} components of the delay related by Eq. (6)

The \(V_{{\text{ZHD}}}\) can be accurately estimated with the Saastamoinen model as revised by Davis et al., (1985) with surface pressure. This is functionally expressed in Eq. (7).

where \(P_{S}\) is the total surface pressure at the site of the GNSS station in hPa, \(\phi\) is the latitude of the observation station in radians and \(h_{S}\) is the station’s elevation above MSL in kilometres. However, Eq. (7) is insensitive to the difference between orthometric and ellipsoidal heights. Therefore, ellipsoidal height can be used without loss of accuracy (International Earth Rotation Services, 2010).

The relation between ZWD and the AWVC in the atmosphere is expressed by IWV \(V_{{{\text{IWV}}}}\) and PWV \(V_{{{\text{PWV}}}}\) as provided in Eqs. (8), (9) and (10) (Bevis et al., 1994; Kleijer, 2004).

where \(\pi\) is the water vapour conversion factor, \(\rho_{w}\) is the water density, \(R_{w}\) = (461.525 ± 0.003) J kg^{−1} K^{−1}, is the specific gas constant for water vapour, \(K_{2}^{^{\prime}}\) and \(K_{3}\) are refraction constants with typical values of (22.13 ± 2.20) K/hPa and (373,900 ± 12,000) K^{2}/hPa, respectively and \(T_{m}\) represents the weighted mean temperature in Kelvin (K). A globally adopted linear regression model for estimating \(T_{m}\) based on the relationship between surface temperature (\(T_{S}\)) and \(T_{m}\), based on 8700 radiosonde profiles collected at 13 stations over the United States for 2 years was proposed by Bevis et al., (1992), as given in Eq. (11):

Although Eq. (11) is adopted worldwide, the accuracy varies. For the best results, it was recommended that the constants be fine-tuned to areas of interest for the different seasons (Bevis et al., 1992, 1994; Ross & Rosenfeld, 1997). Consequently, a fine-tuned model recommended for Nigerian users (Isioye et al., 2016) is given in Eq. (12):

Another approach to approximate is the formula by Askne and Nordius (1987) which requires three input parameters: water vapour pressure \(e\), mean temperature weighted with water vapour pressure \(T_{m}\), and water vapour decrease factor \(\lambda\) and is provided in Eq. (13).

where \(R_{d}\) (287.046 4 J kg^{−1} K^{−1}) is the specific gas constant for dry constituents and \(g_{m}\) (9.806 65 m s^{−2}) is the mean gravity.

### Model development and validations

Several curve-fitting models exist, which are applied in fitting independent variables to response variables. Common examples are linear regression, polynomial, exponential, Fourier, and others. This study evaluates several fitting models to identify the best-fit model. The evaluation results indicate that the polynomial, Fourier, and linear regression have similar best goodness of fit in fitting the ERA5 data to GNSS-ZTD. However, since the study is focused on assessing the ERA5 data, the Linear Regression Model (LRM) is adopted for simplicity. A simple LRM in Eq. (14) is applied with the atmospheric variables derived from ERA5 data to fit the GNSS-ZTD (wet and hydrostatic components).

In Eq. (14), \(y_{i}\) represents the GNSS-ZTD, \(x_{i}\) represents the ERA5 atmospheric datasets, \(\varepsilon_{i}\) is the residual at the *i*th sample and the coefficients \(a_{1}\) and \(a_{2}\) represent the gradient and intercepts of the LRM, respectively. The fitting was carried out using 3 years (2014–2016) of GNSS and ERA5 data at 6 selected GNSS stations spatially distributed within the study area. The datasets are prepared by selecting only intersecting pairs of corresponding data samples in the GNSS and ERA5 datasets. This is necessary to resolve the data gaps observed in the GNSS data and the varying temporal intervals of 5 min and 1 h for GNSS and ERA5 data, respectively. For the hydrostatic component, the LRM is implemented using the ERA5 surface pressure (\(P_{S}\)) as the independent variable, obtained from ERA5 data after vertical adjustment in Eq. (4) is applied, while the GNSS-ZHD data is the response variable. Furthermore, the ERA5-ZHD is computed using the Saastamoinen formula in Eq. (7) with ERA5-\(P_{S}\), latitude, and ellipsoidal elevation at GNSS stations. For the wet component, the LRM with ERA5- AWVC data (RH, SH, DPT and PWV) fitting the GNSS-ZWD. The ZWD is also computed using the Bevis formula in Eq. (9) with ERA5-PWV and water vapour conversion factor (\(\pi\)) as input. The \(\Pi\) is computed using Eq. (10) with the weighted mean temperature \(T_{m}\) as an input. Two different models are used to estimate the \(T_{m}\). The first is the globally adopted model proposed by Bevis et al. (1992) as in Eq. (11) while the second is a fine-tuned model recommended for Nigerian users proposed by Isioye et al. (2016) as provided in Eq. (12). The \(T_{m}\) are computed based on both approaches using surface temperature (\(T_{S}\)) obtained from ERA5 after applying the vertical adjustment formula in Eq. (3). ERA5-ZWD computed based on the globally adopted \(T_{m}\) model is referred to ERA5-ZWD1, while that based on the Nigerian-recommended \(T_{m}\) model is referred to ERA5-ZWD2. The GPT3 provided in Landskron and Böhm (2018), as an upgraded version of GPT2w, has long been recognized as the high-precision tropospheric delay model (Ding & Chen, 2020; Sun et al., 2019). The ZTD derived with the GPT3 model (GPT3-ZTD) is evaluated over the study area and compared with ERA5-ZTD. The GPT3 gridded and script files are downloaded from https://vmf.geo.tuwien.ac.at/. The GPT3 script accepts the input parameters: mjd modified Julian date, *ϕ* Geographic latitude (rad), \(\lambda\) Geographic longitude (rad), and *h*ell Ellipsoidal height (m) to derive output parameters: *p* pressure (hPa), *T* Temperature (℃), d*T* temperature lapse rate (K·km^{−1}), *T*_{m} mean temperature weighted with water vapour pressure (K), *e* water vapour pressure (hPa), *λ* water vapour decrease factor, *N* geoid undulation (m), mapping function coefficients, and gradient parameters (Landskron & Böhm, 2018). To compute the GPT3-ZTD, the outputs from the GPT3 script evaluated using 1° × 1° grid data are used as input to the Saastamoinen formula in Eq. (7) to compute GPT3-ZHD and Askne and Nordius formula in Eq. (13) to compute GPT3-ZWD, as recommended in Landskron and Böhm (2018).

The model evaluation metrics adopted to assess the ZTD derived using various estimation techniques adopted in this study are RMS error and coefficients of determination (\(R^{2}\)). These evaluations are carried out based on the GNSS-ZTD data derived by processing GNSS data in PPP mode using GAPS. The RMS of the model predicted (\(f(x_{i} )\)) and observed (\(y_{i}\)) data is the average magnitude of the error in the set of observation (*s*), which can be calculated by Eq. (15). RMS is in the same units of the original measurements as stated in Ojigi and Opaluwa (2019): while the coefficient of determination (\(R^{2}\)) is the square of *the* correlation coefficient and is represented by Eq. (16).

where \(n\) is the number of samples, while \(\hat{x}\) and \(\hat{y}\) are the mean of the designated predicted variables and observed, respectively.

## Results and discussion

### Characteristics of the ERA5 atmospheric parameters used in the study

The study area overlaid with the atmospheric parameters collected from the ERA5 is presented as colour maps on Fig. 2a–f. These colour maps represent the 3-year mean of surface pressure (Fig. 2a), relative humidity (Fig. 2b), specific humidity (Fig. 2c), dew point temperature (Fig. 2d), PWV (Fig. 2e) and surface temperature (Fig. 2f). These plots showcase the mean characteristics of the ERA5 atmospheric parameters applied in estimating the ZTD over the study area.

The plots of the 3-year mean ERA5-surface pressure variations in Fig. 2a indicate an inverse relationship with elevation. This is evident from the relatively low pressure at the highlands around Jos (Plateau State), Mambilla (Taraba State extending to Cameroon), and their surrounding areas, while relatively higher pressure is observed at low-lying areas towards the coast of the Atlantic Ocean. Figure 2b–e show the ERA5 atmospheric variables depicting the 3-yeas mean AWVC in the study area. These plots show the similarities in the variations of the AWVC. A relatively low AWVC can be observed in the arid region in the north with an increase while moving southwards towards the coast of the Atlantic Ocean. These places can be easily identified in the study area map in Fig. 1.

### Evaluations of ZHD responses to ERA5 surface pressure

For the hydrostatic component, the simple LRM with ERA5 surface pressure fitting to GNSS-ZHD is presented in Fig. 3 while the regression coefficients and validation parameters (goodness of fits) are presented in Table 3. Furthermore, the ZHDs computed by the Saastamoinen model using the atmospheric pressure derived from surface ERA5 and GPT3 models as inputs are presented as regression plots in Fig. 4a and b respectively, while the validation parameters evaluated based on GNSS-ZHD are presented in Table 4.

Evaluating the linear regression plot of GNSS-ZHD against ERA5 surface pressure in Fig. 3, the scatter plots are approximated by a linear line with positive slope at all GNSS sites under study. These lines are represented by the LRM given in Eq. (14) having the values of their respective models at each GNSS Site determined by the coefficients *a*1 and *a*2, with the confidence bounds at 95% confidence level and the goodness of fit presented for all 6 GNSS stations in Table 3. From this table, the ZHD LRM average RMS and \(R^{2}\) are 2.3 mm and 0.75, respectively. This accuracy is identical to the ERA5-ZHD computed with the Saastamoinen formula using ERA5 surface pressure, while the GPT3-ZHD has an average RMS and \(R^{2}\) of 3.4 mm and 0.46. This means that the ERA5 model is of better accuracy compared to the GPT3 model in estimating ZHD within the study area when assessed based on the GNSS-ZHD derived by PPP processing with GAPS. This is an indication that the application of ERA5 surface pressure in estimating ZHD can yield a better height solution in GNSS positioning, in line with the assertions by Douša et al. (2018), which state that “an accurate ZHD is not only essential for high-accuracy GNSS-based positioning but also tropospheric delay modelling”. From the plots and evaluation parameters, uniform accuracies are observed in each atmospheric models (ERA5 and GPT3) with only little variations, at all GNSS sites. This indicates consistency in the accuracies of the ERA5 and GPT3 datasets since the GNSS stations used are widely spread over the study area. A weak correlation is observed between the GPT3-ZHD and the GNSS-ZHD in comparison to the ERA5-ZHD, based on the values of their respective \(R^{2}\). This may be attributed to the fact that the ERA5 data is a product of reanalysis data provided at 0.25° × 0.25° grid, while the GPT3 is a product of the empirical model provided at 1° × 1° grid. Therefore, short-term variations in ZHD cannot be captured by the empirical model as stated in Zhang et al. (2021).

### Evaluations of ZWD responses to ERA5 AWVC data

For the wet component, the plots of a simple LRM fit of the ERA5-AWVC data and GNSS-ZWD are shown in Fig. 5a–d for RH, SH, DPT and PWV respectively. The LRM coefficients with the confidence bounds at 95% confidence level and the goodness of fit are presented in Table 5. The regression plots of GNSS-ZWD against ZWD computed with standard ZWD models are presented in Fig. 6a–c. These standard models are: Bevis equation using ERA5-PWV and globally adopted T_{m} model (ERA5-ZWD1) presented in Fig. 6a, Bevis equation using ERA5-PWV and locally fine-tuned T_{m} model for Nigeran users (ERA5-ZWD2) presented in Fig. 6b and Askne and Nordius formula with GPT3 atmospheric model (GPT3-ZWD) presented in Fig. 6c. While the validation parameters of the ERA5-ZWD1, ERA5-ZWD2, and GPT3-ZWD evaluated based on GNSS-ZWD derived by processing GNSS PPP observations with GAPS are presented in Table 6.

For the wet component of the ZTD, the results of the LRM fitting of GNSS-ZWD and ERA5-AWVC shown in Fig. 5a–d and fitting parameters in Table 5 indicate that LRM accuracies improve in the order: relative humidity, dew point temperature, specific humidity, and PWV having averages of RMS and \(R^{2}\) as: [48.2 mm and 0.56], [43.6 mm and 0.60], [40.5 mm and 0.64] and [20.9 m and 0.92], respectively, assessed based on GNSS-ZWD derived by processing in PPP mode with GAPS. Overall comparison of the performances of all ZWD estimation techniques adopted in this study, shows that the LRM with ERA5-PWV is of the best performance, closely followed by the computed ERA5-ZWD2, ERA5-ZWD1, and GPT3-ZWD having averages of RMS and \(R^{2}\) as: [21.0 mm and 0.94], [21.1 mm and 0.92] and [45.8 mm and 0.58], respectively. The performance of the ERA5-PWV can be traced to the definition, being the total amount of AWVC in a column extending from the surface of the earth to the top of the atmosphere (Hersbach et al., 2018b), while the other ERA5-AWVC data is considered at the earth's surface level. The slightly better performance of ERA5-ZWD2 compared to the ERA5-ZWD1 can be attributed to the fact that ERA5-ZWD2 is derived using \({T}_{m}\) computed by a fine-tuned model recommended in Isioye et al. (2016) for Nigerian users, unlike the ERA5-ZWD1 derived using T_{m} computed by the globally adopted model (Bevis et al., 1992). The GPT3-ZTD within the study area performs slightly poorer compared to the global average having RMS of 44.1 mm, based on a global assessment of GPT3-ZTD using the NGL global tropospheric data (Ding & Chen, 2020). Furthermore, a marginal improvement is seen in the GPT3 model compared to the GPT2w assessed within Africa, having an RMS of ~ 50.0 mm (Isioye et al., 2015). Considering spatial variations in the accuracy of the ERA5-AWVC data in the study area, only slight differences are observed between the GNSS stations in each AWVC variable evaluated, as shown in Table 5. This indicates consistency in the accuracy of the ERA5-AWVC data over the study area.

This study has shown improved accuracy in applying the ERA5 datasets in estimating the ZTD compared to empirical and older reanalysis atmospheric models such as UNB3, GPT2w, GPT3 and NCEP II and ERA-I as evaluated in Abimbola et al. (2017) Isioye et al. (2015) and Mayaki et al. (2018) over the study area. Also, the accuracy of the ZTD estimates obtained using surface-level ERA5 data over Nigeria in this study, is better when compared to ZTD derived from ERA5 surface-level data over China assessed by Jiang et al. (2020). This may be due to the difference in the ZTD modelling approach and ERA5 atmospheric variables adopted in both studies.

## Conclusions

The study has assessed the potential of surface-level ERA5 atmospheric variables in estimating ZTD over Nigeria, based on the GNSS-ZTD derived from PPP processing with GAPS. A series of LRM with ERA5 data are developed for estimating hydrostatic and wet components of ZTD. Also, ZTD estimates are derived by the computations with standard ZTD models using the ERA5 and GPT3 atmospheric data as inputs. The accuracies of the results obtained are analysed and compared. Based on the evaluations of the hydrostatic component, it may be inferred that the ZHD derived based on ERA5 surface pressure is of high accuracy, since the ZHD estimates derived by LRM and Saastamoinien equation are both based on the ERA5 surface pressure and have an identical average RMS of 2.3 mm. The evaluations of the GPT3-ZHD give an average RMS of 3.4 mm. For the wet component, the best performance is observed in the ZWD estimates derived based on ERA5-PWV with LRM having an RMS of 20.9 mm. This is closely followed by the ZWD derived based on ERA5-PWV using Bevis equation with fine-tuned T_{m} model for Nigeran users and the globally adopted T_{m} model having average RMS of 21.0 mm and 21.1 mm, respectively. The ERA5-ZWD estimates also perform better than the GPT3-ZWD since the GPT3-ZWD has an average RMS of 45.8 mm. Considering spatial variations in the accuracies of the ZTD derived from ERA5 and GPT3 data, approximately uniform accuracies are observed at all GNSS sites, in each atmospheric variable. Improved accuracies are observed in the ZTD derived from ERA5 compared to several earlier atmospheric models evaluated in the past studies.

The contributions of the study include providing insight into the potential of the ERA5 atmospheric variables in ZTD estimation and development of a model based on ERA5 surface-level data for estimating ZTD within the Nigerian region for improving GNSS positioning. However, some limitations are identified in the study which include limited number of GNSS-CORS are used, which is due to large data gaps at some GNSS stations, making then unsuitable for the assessment; also, the unavailability of radiosonde data for validations within the study period and beyond. These limitations can easily be overcome in the future with adequate maintenance of the infrastructure required for collecting the said datasets.

Further studies are recommended on improving the accuracies of the ZTD estimates by the application of ERA5 pressure level data in modelling ZTD over Nigeria and the African continent. Possibility of improvement may occur in the accuracy of the ZTD estimated from ERA5 data by the application of more complex modelling approach such as the machine learning.

## Availability of data and materials

The ERA5 data is available at the European Centre for Medium-range Weather Forecast (ECMWF) repository, https://cds.climate.copernicus.eu/. For other datasets please contact the corresponding author upon reasonable request.

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## Acknowledgements

The authors wish to extend their sincere gratitude to the Tertiary Education Trust Fund (TETFund) and The Federal University of Technology Akure (FUTA), Nigeria for the Institution Based Research (IBR) Intervention funding, the Office of Surveyor-General of the Federation (OSGOF) of Nigeria for making available the GNSS CORS data, University of New Brunswick (UNB) for processing the GNSS data with their GNSS Analysis and Processing Software (GAPS), European Centre for Medium-range Weather Forecast (ECMWF) for providing the atmospheric data (ERA5) and the International GNSS Service (IGS) data centres for their open-access tropospheric delay products. We are also grateful to Robert B. Schmunk of NASA (U.S.A) for making the Panoply software freely available.

## Funding

This work was sponsored by the Nigeria Tertiary Education Trust Fund (TETFund) for Institution Based Research (IBR) Intervention funding.

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### Contributions

Conceptualization, IUN and TOI; methodology, IUN, HT and TOI; software, IUN; writing original draft, IUN; formal analysis, IUN; writing review and editing, IUN, HT and TOI; and Supervision TOI. All authors read and approved the final manuscript.

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Nzelibe, I.U., Tata, H. & Idowu, T.O. Assessment of GNSS zenith tropospheric delay responses to atmospheric variables derived from ERA5 data over Nigeria.
*Satell Navig* **4**, 15 (2023). https://doi.org/10.1186/s43020-023-00104-7

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DOI: https://doi.org/10.1186/s43020-023-00104-7