Hybrid Machine-Learning Framework for Slant Wet Delay Modeling

Developing a new combined model of zenith wet delay by using neural network

The tropospheric wet delay is a significant systematic error of GNSS positioning, nevertheless it carries important information to meteorologists. It is closely related to the integrated water vapour that is the upper limit of precipitable water. The zenith wet delay can be converted to the integrated water vapour using a simple conversion factor. This conversion factor can be determined with the empirical formulae derived from radiosonde observations. In the past decades, numerous models were derived for this purpose, but all of these models rely on radiosonde observations stemming from a limited area of the globe. Although these models are valid for the area, where the underlying radiosonde observations were measured, there are several examples that these empirical formulae are used to validate GNSS based integrated water vapour estimations all over the globe. Our aim is to create a global model for the conversion of the zenith tropospheric delay to the integrated water vapour for realtime and nearrealtime applications using globally available Numerical Weather Models (NWM). Thus our model takes into consideration the fact that the model parameters strongly depend on the geographical location. 10 years of monthly mean ECMWF (European Center for Medium-Range Weather Forecast) dataset were used for the derivation of the model parameters in a grid with the resolution of 1° × 1°. The empirical coefficients of the developed models depend on two input parameters, namely the geographical location and the surface temperature measured at the station. Thus, the new models can be used for both realtime and near-realtime GNSS meteorological applications. The developed models were validated using 6 years of independent global ECMWF monthly mean analysis datasets (2011–2016). The results showed, that the application of the original models outside the area of the underlying radiosonde data sets can result in a relative systematic error of 7–8% in the estimation of the conversion factor as well as the estimated IWV values.

The wet delay is one of the most difficult sources of error to quantify in global navigation satellite systems (GNSS) due to its spatial and temporal variability. However, its estimation is vital in some applications, such as in Precise Point Positioning (PPP), network real-time kinematic and weather forecast. Such estimation is usually carried out for its projected component along the zenith, defined as the zenith wet delay (ZWD). Prediction of the ZWD is important if there is a break in its estimation, and in providing initial values for future processing. In this paper, prediction modeling for the ZWD is investigated. An autocorrelation study was first performed on ZWD data over three different days at the international GNSS service station ONSALA to provide an insight into the temporal correlations among ZWDs. The choice of this station was based on the availability of reliable ZWD data measured by water vapor radiometers, which provide a reference for assessment of the accuracy of predicted values. Results have shown that successive ZWD estimates are significantly correlated for up to 1.7 hours. Different trend and smoothing prediction models were then investigated. Since prediction of ZWD requires continuous data sequence, interpolation methods of possible missing ZWD values are discussed. The use of linear interpolation or a cubic Hermite polynomial was found to be sufficient for interpolation of ZWD. Test results of prediction models show that the single-exponential smoothing model was the best performer where ZWD estimates were forecasted with a root mean square error of less than 1 cm for up to two hours.

Global Navigation Satellite System (GNSS) signals arrive at the Earth in a nonlinear and slightly curved way due to the refraction effect caused by the troposphere. The troposphere delay of the GNSS signal consists of hydrostatic and wet parts. In particular, tropospheric wet delay prediction and interpolation are more difficult than those of the dry component due to the rapid temporal and spatial variation of the water vapor content. Wet delay estimation and interpolation with a sufficient accuracy is an important issue for all parameters obtained by GNSS positioning techniques. In particular, in real-time positioning applications, errors caused by interpolation of the wet troposphere delay are reflected in the height component of about 1 to 2 cm. Furthermore, the amount of water vapor in the troposphere is very important information in weather forecast applications obtained as a function of wet delay. Therefore, real-time monitoring of the troposphere can be achieved with a higher resolution and accuracy by utilizing neural network models for interpolation of the wet tropospheric delay. In addition, in the absence of the GNSS station, wet delays can be interpolated by means of the surrounding stations to the desired location. In this study, a back propagation artificial neural network (BPNN) model based on meteorological parameters obtained from The New Austrian Meteorological Measuring Network (TAWES) was used to interpolate wet troposphere delay. Analysis was carried out at 40 reference stations of the Echtzeit Positionierung Austria (EPOSA) GNSS Network covering the whole of Austria. The interpolation of zenith wet delays based on the artificial neural network was performed by using latitude, longitude, altitude and meteorological parameters (temperature, pressure, weighted mean temperature, and water vapor pressure). These parameters were then subtracted from the artificial neural network model one by one and six different artificial neural networks were designed. In addition, the linear interpolation method (LIN) and inverse distance weighted interpolation method (IDW) were used as conventional interpolation methods. In order to investigate the effect of the network density on interpolation methods, three networks, including 40, 30, and 20 reference stations, were formed and the increased distance effect on interpolation methods was evaluated. In addition, analyses were conducted in winter, spring, and summer to evaluate the seasonal effects on interpolation methods. According to the statistical analysis, the root mean square error (RMSE) values of the IDW, LIN, and BPNN methods were found to be 12.6, 13.4, and 5.9 mm, respectively.

Global Positioning System (GPS) signals experience ranging errors due to propagation through the neutral atmosphere. These range delays consist of a hydrostatic component, dependent on air pressure and temperature, and a wet delay dependent on water vapour pressure and temperature. Range delays arising from the hydrostatic component can be computed with accuracies of a few millimeters using existing models, provided that surface barometric or meteorological data are available. By using a regional network of GPS reference stations, it is possible to recover estimates of the Slant Wet Delay to all satellites in view. Observations of the Slant Wet Delay (SWD) can be used to model the vertical and horizontal structure of water vapour over a local area. These techniques are based on a tomographic approach using the SWD as input observables, where 4-D models of the wet refractivity may be derived. This method allows improved resolution of water vapour estimates for precise positioning applications and assimilation into Numerical Weather Predictions (NWP). In this paper we present strategies for real-time modeling of wet refractivity, with simulations and preliminary results of data processing for a regional GPS network in Southern California.

Using the observations from local and regional GPS networks, the estimation of slant wet delays (SWDs) is possible for each line of sight between satellite and receiver. The observations of SWD are used to model horizontal and vertical variations of the wet refractivity in the atmosphere above the study area. This work is done using the tomography method. In tomography, the horizontal variations of tropospheric wet refractivity are modeled with the polynomial in degree and rank of 2 with latitude and longitude as variables. Also, altitude variations are modeled in the form of discrete layers with constant heights. The main innovation is to estimate the tropospheric parameters for each line of sight by the artificial neural networks (ANNs). The SWD obtained from GPS observations for the different signals at each station is compared with the SWD generated by the ANNs (SWDGPS–SWDANNs). The square of the difference between these two values is introduced as the cost function in the ANNs. To evaluate, we used observations from October 27 to 31, 2011. The availability of GPS and radiosonde data is the main reason for choosing this timeframe. The correlation coefficient, root mean square error (RMSE), and relative error allow for evaluation of the proposed model. The results were also compared with the results of the voxel-based troposphere tomography method. For a more detailed evaluation, four test stations are selected and ANN zenith wet delays (ZWDANN) are compared with the ZWDGPS. Observations of test stations are not used in the modeling step. The correlation coefficient in the testing step for TomoANN and Tomovoxel is 0.9006 and 0.8863, respectively. The mean RMSE at 5 days for TomoANN and Tomovoxel is calculated as 0.63 and 0.71 mm/km, respectively. Also, the average relative error at the four test stations for TomoANN is 15.37% and for Tomovoxel it is 19.69%. The results demonstrate the better capability of the proposed method in the modeling of the tropospheric wet refractivity in the region of Iran.

Global pressure and temperature 2 wet (GPT2w) is an empirical troposphere delay model providing the mean values plus annual and semiannual amplitudes of pressure, temperature and its lapse rate, water vapor pressure and its decrease factor, weighted mean temperature, as well as hydrostatic and wet mapping function coefficients of the Vienna mapping function 1. All climatological parameters have been derived consistently from monthly mean pressure level data of ERA-Interim fields (European Centre for Medium-Range Weather Forecasts Re-Analysis) with a horizontal resolution of 1°, and the model is suitable to calculate slant hydrostatic and wet delays down to 3° elevation at sites in the vicinity of the earth surface using the date and approximate station coordinates as input. The wet delay estimation builds upon gridded values of the water vapor pressure, the weighted mean temperature, and the water vapor decrease factor, with the latter being tuned to ray-traced zenith wet delays. Comparisons with zenith delays at 341 globally distributed global navigation satellite systems stations show that the mean bias over all stations is below 1 mm and the mean standard deviation is about 3.6 cm. The GPT2w model with the gridded input file is provided at http://ggosatm.hg.tuwien.ac.at/DELAY/SOURCE/GPT2w/.

The vertical structure of the atmospheric water vapor induces phase errors in interferometric synthetic aperture radar (SAR) data. This paper presents a simulation study to investigate whether spaceborne submillimeter radiometric observations, which can be realized with fairly high spatial resolution, are able to derive the vertical structure of the atmospheric wet delay. The accuracy of the retrieved zenith wet delay (ZWD) trend as a function of surface height is assessed in order to correct the associated height dependence of the interferometric phase error in a SAR interferogram. Using a simulated benchmark, we evaluate the errors associated with the use of both a linear and an exponential model of the behavior of ZWD as a function of the surface height. This paper shows a fairly accurate reconstruction of the trend parameters estimated from radiometer brightness temperature images, with respect to realistic atmospheric profiles provided by radiosounding observations (RAOBs). The trend parameters that we consider in this paper are the slope K for the linear model and scale height H for the exponential one. An overall better accuracy is found for the exponential model, which is more representative of the actual behavior of ZWD with height, resulting in a residual uncertainty in the path delay due to the atmospheric stratification of approximately 0.2-0.3 cm and nearly zero bias, as compared to RAOBs.

Accurate tropospheric delays are of great importance for both Global Navigation Satellite System (GNSS)-based positioning and precipitable water vapor monitoring. The gridded tropospheric delay products, including zenith hydrostatic delays (ZHD) and zenith wet delays (ZWD), are the most ideal method for accessing accurate tropospheric delays. The vertical adjustment method is critical for implementing the gridded tropospheric products. In this work, we consider the different models used for grid products and assess their performance over Yunnan mountains with complex topography. We summarize the main results as follows: (1) The products can provide accurate ZHD with mean biases of −2.6 mm and mean Standard Deviation (STD) of 1.5 mm while the ZWD results from grid products show a performance with biases of −0.4 mm and STD of 1.3 cm over the Yunnan area. (2) The Tv-based model shows a better performance than the T0-based model and IGPZWD in rugged areas with large height differences. The grid products can provide hourly ZHD with biases of 3 mm and wet delay with mean biases of within 2 cm and mean STD of below 3 cm in the Yunnan mountains, which exhibit a large height difference of around 1.5 km. (3) The radiosondes results confirm that the Tv-based model has an obvious advantage in calculating ZHD height corrections for differences within 2 km while the T0-model suffers from a loss in accuracy in the case of large height differences. If the site is located more than 1 km below the reference height, the IGPZWD model can provide a better ZWD with a mean bias of 1.5 cm and a mean STD of 1.7 cm. With vertical reduction models, the grid products can provide accurate ZHD and ZWD in real time, even if in complex area.

<p>Radio signals transmitted by Global Navigation Satellite System (GNSS) satellites propagate through the atmosphere before being received on Earth. Thereby, the signal is delayed and tropospheric parameters can be estimated. The good global coverage of GNSS receivers, combined with the high temporal resolution and the high accuracy, make GNSS a suitable tool for studies on the atmosphere.</p><p>Atmospheric delays are differentiated into a zenith hydrostatic (ZHD) and a non-hydrostatic, or zenith wet delay (ZWD). The hydrostatic part has a larger contribution (causing a delay of roughly 2.4 meters in the zenith direction) but can be modeled with sufficient accuracy using analytical methods. The ZWD has a smaller contribution (causing a delay between 0 to 40 centimeters) and depends mainly on the water vapour content in the atmosphere. However, due to the variable nature of water vapour, the ZWD is difficult to model and is therefore typically estimated. Its quantification is essential since it drives weather systems and climate change to a great extent. For many applications, such as weather forecasting or positioning using low-cost GNSS receivers such as smartphones, global real-time monitoring or even predictions of ZWD would be required and be beneficial.</p><p>In the last decade, machine learning (ML) algorithms have gained a lot of interest and are successfully utilized in many different fields. Thereby, ML algorithms have proven to be able to efficiently process and combine large amounts of data and solve problems of various kinds.</p><p>This motivated us to investigate the feasibility of ML algorithms for the prediction of tropospheric parameters, in particular ZWD, with the help of meteorological data such as the water vapour content. The work aims to develop a global model capable of predicting ZWD in space and time. Therefore, different ML algorithms are used to train a model based on meteorological features. The performance of the utilized algorithms is evaluated based on commonly used performance metrics, such as Root Mean Squared Error (RMSE) and R².</p><p>Preliminary investigations are carried out utilizing 3000 GNSS stations distributed over Europe. The performance of various ML methods, such as Linear Regression methods, Random Forest, (Extreme) Gradient Boosting, and Multilayer Perceptron is compared. Furthermore, different feature combinations, as well as training sample sizes are investigated. It is revealed that linear methods are not able to properly reflect the observations. Instead, our Random Forest approach provides, so far, the highest model accuracy with an RMSE of 1.7 centimeters and an R² value of 0.88.</p>

We obtain a set of pointed wet delay observations at a total of 81 stations at 9 separate time epochs by ray tracing through the output fields of a Japanese numerical weather model with a grid spacing of 10 km. For each station-epoch, we invert the simulated data set, consisting of 4000 pointed delays, using an isotropic and an anisotropic delay model. The isotropic model has only one parameter-the zenith wet delay (ZWD). The anisotropic delay model of Chen and Herring has two additional lateral gradient parameters. We find that the anisotropic model leads to a much better fit between the delay model and the delay observations. On the other hand, the ZWD parameters estimated using the differing delay models are almost identical. This implies the ZWD estimates are not greatly perturbed by lateral heterogeneity of the atmosphere.

The study of climate change is an important field of research. Monitoring of atmospheric variability especially the tropospheric precipitable water vapor (PWV) is a powerful way to investigate climate change. Global navigation satellite systems (GNSS) provide a good tool for studying atmospheric parameters as GNSS signals along its path from the satellites to the ground based receivers suffer a significant delay due to the refractivity of earth’s atmosphere. GNSS signals are not only delayed but also refracted in the neutral atmosphere. The zenith wet delays (ZWD) caused by the troposphere can be estimated during the geodetic processing of GNSS signals. Since the ZWD is tightly correlated to the PWV, GNSS observations can be used to study the changes of PWV. In this study GNSS data from the Egyptian permanent GNSS network (EPGN), International GNSS Service (IGS), EUREF Permanent Network (EPN) and Scripps Orbit and Permanent Array Center (SOPAC), for the years 2013 and 2014, were used. These GNSS stations provided a good geometrical coverage over the respective region. All GNSS data were processed using Bernese V 5.2 software. The needed product from GNSS data processing is the tropospheric zenith total delay (ZTD) over each GNSS site. The ZTD is divided based on physical parameters into zenith hydrostatic delay (ZHD) and zenith wet delay (ZWD). The ZWD is the basic observable used to calculate PWV. The values of the PWV were calculated and its variation over the study area was investigated. The quality of calculated PWV values from GNSS data evaluated against traditional Radiosonde (RS) measurements. The results of GNSS PWV showed good agreement with RS PWV when the PWV values were low, but this agreement became worse at high PWV values as the differences between the two techniques increased. The correlation coefficient between RS PWV and GNSS PWV varied from 0.31 to 0.84. Standard deviation of the differences between RS PWV and GNSS PWV ranged from 2.369 to 5.973 mm. The PWV estimated from GNSS observations had annual cycle, and its cycle in 2013 was different from that in 2014. The PWV differences between the 2 years clarified that the water vapor content over the east Mediterranean in most of 2014 days was higher than that in 2013. Furthermore, PWV variations were noted on both temporal and spatial scales. The highest temporal variation value was 25.41 mm whereas the maximum value of the spatial variation was 19.67 mm. The present study illustrated the importance of using geodetic networks to provide atmospheric information.

The Global Navigation Satellite System (GNSS) plays an important role in retrieving high temporal–spatial resolution precipitable water vapor (PWV) and its applications. The weighted mean temperature (Tm) is a key parameter for the GNSS PWV estimation, which acts as the conversion factor from the zenith wet delay (ZWD) to the PWV. The Tm is determined by the air pressure and water vapor pressure, while it is not available nearby most GNSS stations. The empirical formular is often applied for the GNSS station surface temperature (Ts) but has a lower accuracy. In this paper, the temporal and spatial distribution characteristics of the coefficients of the linear Tm-Ts model are analyzed, and then a piecewise-linear Tm-Ts relationship is established for each GPS station using radiosonde data collected from 2011 to 2019. The Tm accuracy was increased by more than 10% and 20% for 86 and 52 radiosonde stations, respectively. The PWV time series at 377 GNSS stations from the infrastructure construction of national geodetic datum modernization and Crustal Movement Observation Network of China (CMONC) were further obtained from the GPS observations and meteorological data from 2011 to 2019. The PWV accuracy was improved when compared with the Bevis model. Furthermore, the daily and monthly average values, long-term trend, and its change characteristics of the PWV were analyzed using the high-precision inversion model. The results showed that the averaged PWV was higher in Central-Eastern China and Southern China and lower in Northwest China, Northeast China, and North China. The PWV is increasing in most parts of China, while the some PWVs in North China show a downward trend.

The accurate modeling of tropospheric path delay is significant for data processing of radio space-geodetic techniques such as Global Navigation Satellite System (GNSS) and Very-Long-Baseline Interferometry (VLBI). The Vienna Mapping Function 1 (VMF1) model, based on continuous update of Numerical Weather Prediction (NWP) data from the European Centre for Medium-Range Weather Forecasts (ECMWF), is recommended for this purpose in post-processing. The VMF1 coefficients determined from forecast data of the ECMWF are now readily and freely available. However, little or no implementation of this forecast VMF1 (VMF1-FC) model in real-time GNSS or VLBI applications has occurred. This study investigates the performance of the VMF1-FC model in terms of its three components which are critical for the modeling of tropospheric path delay: the Zenith Hydrostatic Delay (ZHD), the Zenith Wet Delay (ZWD) and mapping functions. All three components are assessed in the context of GNSS Precise Point Positioning (PPP) using 28 global stations over a 70-day period. The Zenith Total Delays (ZTD) derived with the VMF1-FC (implemented in real-time PPP) are shown to agree well with the tropospheric delay product from the Center for Orbit Determination Europe (CODE). Root mean square (RMS) errors associated with these ZTD estimates are < 10 mm at all 28 stations. The results also show that the VMF1-FC model performs better than empirical models such as the widely used Global Pressure and Temperature 2 (GPT2) and GPT2 wet (GPT2w), with smaller RMS errors associated with the ZTD estimates. It is recommended that VMF1-FC be applied for future tropospheric delay modeling in real-time GNSS and VLBI applications.

Precise point positioning (PPP) is one of the well-known applications of Global Navigation Satellite System (GNSS) and provides precise positioning solutions using accurate satellite orbit and clock products. The tropospheric delay due to the neutral atmosphere for microwave signals is one of the main sources of measurement error in PPP. As one component of this delay, the hydrostatic delay is usually compensated by using an empirical correction model. However, how to eliminate the effects of the wet delay during a weather event is a challenge because current troposphere models are not capable of considering the complex atmosphere around the receiver during situations such as typhoons, storms, heavy rainfall, et cetera. Thus, how positioning results can be improved if the residual wet delays are taken into account needs to be investigated . In this contribution, a real-time procedure of recursive detection, identification and adaptation (DIA) is applied to detect the model errors which have the same effects on both phase and code observables; e.g., the model error caused by the tropospheric delay. Once the model errors are identified, additional parameters are added to the functional model to account for the measurement residuals. This approach is evaluated with Global Positioning System (GPS) data during two rainfall events in Darwin, Australia, proving the usefulness of compensated residual slant wet delay for positioning results. Comparisons with the standard approach show that the precision of the up component is improved significantly during the periods of the weather events; for the two case studies, and improvements of root mean squared error (RMS) resulted, and the precision of the horizontal component obtained by the proposed approach is also improved more than compared to the standard approach. The results also show that the identified model errors are concentrated at the beginning of both heavy rainfall processes when the front causes significant spatial and temporal gradients of the integrated water vapor above the receiver.