A Hybrid Machine Learning Approach for Modeling Tropospheric Zenith Wet Delay with Enhanced Generalization Performance.

<p>Tropospheric delays have been a major error source for space geodetic techniques and the performance of their modeling is significantly limited due to the high spatiotemporal variability of the moisture in the lower atmosphere. In this study, tropospheric zenith wet delay (ZWD) modeling was realized based on the machine learning (random forest approach, RF) and using 10 years (2010-2019) of radiosonde measurements at 586 globally distributed stations. Subsequently, the ZWD modeling accuracy was validated based on the sounding profiles across the globe for the year 2020. We find that ZWD modeling accuracy is significantly improved by taking account meteorological parameters in the functional formulation, especially for surface water vapor pressure. When surface meteorological data are available, the RF-based ZWD models with meteorological parameterization can achieve an overall accuracy of 2.9 cm and the bias close to zero across the globe, which clearly outperforms current empirical models, such as the GPT3, or other models based on surface meteorological measurements. From the analyses of spatial characteristics of the ZWD accuracy, it can be concluded that the RF-based ZWD models especially mitigate the systematic biases in the regions with monsoon climate and tropical rainforest climate types. </p>

<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>

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.

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.

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

In recent years, geodesy based on spaceborne microwave remote sensing has gained significant advances. However, whether the observations from the Global Navigation Satellite System (GNSS) or Interferometric Synthetic Aperture Radar (InSAR), the results are inevitably influenced by atmospheric tropospheric delay. Although the tropospheric zenith total delay (ZTD) can be estimated through the gridded meteorological data products and empirical models provided by the ERA5 reanalysis product, its accuracy is still insufficient to meet the needs of modern geodesy. To overcome this challenge, we propose leveraging machine learning techniques to learn local spatio-temporal patterns of tropospheric delay for inferring total zenith delay (ZTD) and zenith wet delay (ZWD) at any location within the learning area.Our findings indicate that artificial neural networks can establish a robust mapping between ZTD estimated by empirical models and GNSS-measured ZTD. Then employing the ensemble learning strategy and the time series dynamics model, the ZTD at any location within the sample area can be inferred. To evaluate our approach, we conducted tests during the active water vapor season in the Tübingen region of Baden-Württemberg, Germany, from June 25 to July 9, 2022. In comparative experiments with the root mean square error (RMSE) of Zenith Total Delay (ZTD) derived from ERA5, our proposed method yielded a significant reduction in RMSE, decreasing it from 16.4292mm to 7.2108mm. This reflects a remarkable accuracy improvement of 56.11%.The proposed approach holds promise for enhancing the precision of GNSS positioning, InSAR earth observation, and generating more dependable water vapor products.

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.

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 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.

Tropospheric wet delay remains a key error source for space geodesy, including GNSS, VLBI, and InSAR. Empirical models such as GPT3 are widely used, yet they rely on simplified parameterizations and fixed coefficient tables that limit modeling capacity. Frequent updates are difficult because the entire archive must be reprocessed. With the rapid progress of machine learning, it is natural to seek ML-based tropospheric models that improve both accuracy and efficiency. To date, most work has focused on zenith wet delay (ZWD), which is essentially one-dimensional, while fully data-driven slant modeling has been largely unexplored. Slant wet delays (SWD) are inherently anisotropic, which makes the task more challenging.We propose a hybrid ML framework that embeds a physical layer inside the network to predict SWD end-to-end and yields consistent ZWD and wet mapping function as internal outputs. Training uses hundreds of millions of ERA5 ray-traced samples from 2018 to 2022 with global coverage. The resulting ML model outperforms GPT3 for SWD, with markedly lower errors over continental regions where most space-geodetic stations operate and with the largest gains at low elevation angles and along coasts. The learned mapping is asymmetric in elevation and azimuth, which removes the need for explicit horizontal gradients. As ancillary products, the framework provides ZWD that surpasses GPT3 and a wet mapping function that exceeds the symmetric GPT3 variant and is comparable to the asymmetric one. We also develop augmented variants that accept surface temperature and water vapor pressure as inputs and obtain further accuracy gains. To our knowledge, this is the first ML-based model that directly predicts SWD. The model is compact and faster than GPT3 when applied to large sample sets. The hybrid design supports efficient fine-tuning with new observations and provides a practical path to maintainable routine processing and continued advances in space-geodetic troposphere modeling.

Tropospheric delay is an important error source in Global Navigation Satellite System (GNSS) positioning and can also be used in water vapor monitoring. Many models have been built to correct tropospheric delays or to convert zenith wet delays to precipitable water vapor. However, these models suffer from limited resolutions (spatial resolution lower than 1° and temporal resolution lower than 6 hr), which affects their performance. The release of European Centre for Medium‐Range Weather Forecasts ReAnalysis 5 (ERA5) provides the opportunity to lift this limit. In this study, we use the ERA5 hourly 0.5° × 0.5° data to build a new model over China, which integrates tropospheric delay correction for GNSS positioning and weighted mean temperature calculation for GNSS meteorology. By modeling the diurnal variations of zenith hydrostatic delay, zenith wet delay, and weighted mean temperature and the seasonal variations in their lapse rates, this model has the state‐of‐the‐art spatial resolution of 0.5° × 0.5° and temporal resolution of 1 hr. We validate this new model by the ERA5 data, the radiosonde data, and the GNSS data in comparison with the canonical GPT2w model. The results show that the new model has better accuracies in terms of root‐mean‐square than the GPT2w model in all parameters. Especially, the new model well captures the diurnal variations in tropospheric delay and weighted mean temperature. This new model provides accurate tropospheric delays and weighted mean temperature simultaneously, which enables GNSS receivers to measure precipitable water vapor directly and also benefits GNSS positioning.

Tropospheric delays from GNSS for application in coastal altimetry

In this paper, we demonstrate the advantage of applying a Kalman filter for the parameter estimation in very-long-baseline interferometry (VLBI) data analysis. We present the implementation of a Kalman filter in the VLBI software VieVS@GFZ. The performance is then investigated by looking at the accuracy obtained for various parameters, like baseline lengths, Earth Orientation Parameters, radio source coordinates, and tropospheric delays. The results are compared to those obtained when the classical least squares method (LSM) is applied for the parameter estimation, where clocks and zenith wet delays are estimated with 30-min intervals and gradients with 120-min intervals. We show that the accuracy generally is better for the Kalman filter solution, for example, the baseline length repeatabilities are on average about 10 % better compared to the LSM solution. We also discuss the possibilities to use the Kalman filter to estimate sub-diurnal station position variations and show that the variations caused by solid Earth tides can be retrieved with an accuracy of about 2 cm.

&amp;lt;p&amp;gt;Water vapor (WV) is the most variable greenhouse gas in the atmosphere, which acts as a key feature in climate change studies and plays a crucial role in global warming. Its spatiotemporal distribution is necessary for understanding the hydrological cycle, and consequently can be used as an input factor in climatological studies at global, regional, and local scales. Integrated water vapor (IWV), which is defined as the amount of vertically integrated water vapor, can also augment atmospheric modeling at local and regional scales because it is frequently used in energy budget and evapotranspiration assessments. Currently, there are numerous existing atmospheric models which are able to estimate IWV amount, nevertheless, they fail to obtain extremely accurate results compared with in-situ measurements such as radiosondes. Here, we present a new methodology for improving Weather Research and Forecast (WRF) model predictions accuracy, by using data assimilation technique, which combines estimated 2D IWV regional maps, derived from GPS tropospheric path delays, along with the WRF numerical model output to generate an optimal approximation of the evolving sate of the system. This is done as opposed to pervious works, which assimilated single point measurements, either from radiosondes or GPS zenith wet delay (ZTD) estimation, demonstrating some extant of improvement in the WRF prediction accuracy compare to the standalone WRF numerical runs. Using the suggested technique, our results shows a decrease of up to 30% in the root mean square difference relative to the radiosonde data for WRF predictions assimilated with 2D GPS-IWV regional maps compare to the standalone WRF numerical runs.&amp;lt;/p&amp;gt;

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.