Adaptive Algebraic Reconstruction Algorithms for GNSS Water Vapor Tomography

One of the main challenges of Global Navigation Satellite System (GNSS) tomography is in solving ill-conditioned system equations. Vertical constraint models are typically used in the solution procedure and play an important role in the quality of the GNSS tomography, in addition to helping resolve ill-posed problems in system equations. In this study, based on a water vapor (WV) parameter, namely IRPWV, a new vertical constraint model with six sets of coefficients for six different WV states was developed and tested throughout 2019 in the Hong Kong region with four tomographic schemes, which were carried out with the model and the traditional vertical constraint model using three different types of water vapor scale height parameters. Experimental results were numerically compared against their corresponding radiosonde-derived WV values. Compared with the tests that used the traditional model, our results showed that, first, for the daily relative error of WV density (WVD) less than 30%, the new model can lead to at least 10% and 49% improvement on average at the lower layers (below 3 km, except for the ground surface) and the upper layers (about 5–10 km), respectively. Second, the skill score of the monthly root-mean-square error (RMSE) of layered WVD above 10 accounted for about 83%, 87%, and 64%. Third, for the annual biases of layered WVD, the new model significantly decreased by 1.1–1.5 g/m3 at layers 2–3 (about 1 km), where all schemes showed the maximal bias value. Finally, for the annual RMSE of layered WVD, the new model at the lower (about 0.6–3 km) and upper layers improved by 13–42% and 5–47%, respectively. Overall, the new model performed better on GNSS tomography and significantly improved the accuracy of GNSS tomographic results, compared to the traditional model.

Global Navigation Satellite Systems (GNSS) tomography is a modern technique that leverages GNSS satellites and ground-based network stations for meteorological applications, with a particular focus on studying the dynamics of water vapor. Over the past two decades, various approaches have been proposed, but assessing the accuracy of these methods remains challenging. In this study, we introduce an empirical stochastic model for GNSS tomography. This model is formulated as a fully populated covariance-variance matrix that incorporates the variance of the GNSS-Integrated Water Vapor (IWV) solution, the spherical distance between station-satellite signal paths, and an empirical covariance-variance function. Our findings indicate that combining the functional model with the stochastic model significantly improves the estimation of unknown parameters, suggesting a method for enhancing the accuracy and reliability of GNSS tomographic solutions. We compared the standard tomography solution, both with and without stochastic information, with data from radiosondes, ERA5 reanalysis, and the WRF model. The results demonstrate that incorporating the empirical stochastic model can improve the solution accuracy by 10%. More importantly, it provides a valuable tool for evaluating the trustworthiness of tomographic solutions.

GNSS tropospheric tomography is one of the applications of the Global Navigation Satellite Systems (GNSS) signals which attracts more and more interest in the field of meteorology. This method can reconstruct the water vapour of the atmosphere, which has a considerable effect on weather forecasting and early warning systems of severe weather. In GNSS tomography, traditionally, a regular spaced 3D grid stretches from the GNSS network to the effective height of the troposphere in the area of interest. Therefore, this method can offer a permanent monitoring service for water vapour and wet refractivity fields at a low cost and a reasonable spatial resolution compared to conventional observations, like radiosonde and radio occultation profiles. Nevertheless, the quality of the reconstructed field is still one of the challenges in the GNSS tomography. In this research, we propose the concept of spread as a mathematical tool to provide a quality measure without using the reference field and calculating statistical measures like RMSE and Bias. Thereby, two synthetic and one real datasets (part of Germany and Czechia) covering overlapping periods between 29 May and 14 Jun of the year 2013 (DoY 149–165; DoY 160–165; DoY 160–165, 2013) have been tested to investigate the proposed method. According to the obtained results, the proposed tool shows a strong correlation (up to 0.81 for synthetic and 0.72 for real observations) with the standard deviation of the reconstructed wet refractivity with respect to the radiosonde profile reference. The correlation between spread and the Bias of the retrieved wet refractivity field is also significant. However, there is no clear picture depending on the applied spread computation models. Therefore, the spread of the resolution matrix can be used as a proxy for the accuracy of the tomography reconstruction field based on the quality of the observations, the initial field, and the design matrix.

The global navigation satellite system (GNSS) water vapor tomography technique can retrieve high-quality water vapor profiles and holds significant potential for improving the performance of the initial three-dimensional water vapor field in numerical weather prediction (NWP). However, the empirical voxel-division method of GNSS water vapor tomography has been predominantly used before, and existing tomographic results are rarely incorporated into the NWP models due to the lack of a direct assimilation interface, which becomes the focus of this study. An adaptive voxel-division method for GNSS water vapor tomography is first proposed. The optimal horizontal and vertical steps of water vapor tomography are determined by introducing the principles of maximum grid coverage and the equal amount of layered water vapor. In addition, a two-step variational assimilation method has been developed to address the limitation of existing NWP models without a direct interface for assimilating GNSS water vapor tomographic results. The proposed water vapor tomography method and its application in the weather research and forecasting (WRF) model are comprehensively analysed and evaluated. Numerical results in Hong Kong show the superior performance of the proposed adaptive voxel-division method for GNSS water vapor tomography compared with empirical methods. The average improvement rate of the root mean square error (RMSE) in the water vapor profile ranges from 10.6% to 48.7%, while that in the integrated water vapor is 19.0%–42.9%. Furthermore, the assimilated and forecasted results of assimilating GNSS tomographic results under different weather conditions are validated and proved the significantly positive contributions of water vapor profiles to the WRF model. Specifically, the RMSE reduction in precipitation is 43.5%, while that for relative humidity, temperature, and pressure are 50.7%/58.2%, 41.1%/48.3%, and 24.2%/33.7%, respectively, under no-rain and rain conditions compared with the traditional assimilation method. These results show the good performance of the proposed adaptive voxel-division method for water vapor tomography and the two-step method for assimilating the tomographic water vapor profile, which highlight the significant potential of the GNSS tomographic result for data assimilation.

Global Navigation Satellite System (GNSS) troposphere tomography has become one of the most cost-effective means to obtain three-dimensional (3-d) image of the tropospheric water vapour field. Traditional methods divide the tomography area into a number of 3-d voxels and assume that the water vapour density at any voxel is a constant during the given period. However, such behaviour breaks the spatial continuity of water vapour density in a horizontal direction and the number of unknown parameters needing to be estimated is very large. This is the focus of the paper, which tries to reconstruct the water vapor field using the tomographic technique without imposing empirical horizontal and vertical constraints. The proposed approach introduces the layered functional model in each layer vertically and only an a priori constraint is imposed for the water vapor information at the location of the radiosonde station. The elevation angle mask of 30° is determined according to the distribution of intersections between the satellite rays and different layers, which avoids the impact of ray bending and the error in slant water vapor (SWV) at low elevation angles on the tomographic result. Additionally, an optimal weighting strategy is applied to the established tomographic model to obtain a reasonable result. The tomographic experiment is performed using Global Positioning System (GPS) data of 12 receivers derived from the Satellite Positioning Reference Station Network (SatRef) in Hong Kong. The quality of the established tomographic model is validated under different weather conditions and compared with the conventional tomography method using 31-day data, respectively. The numerical result shows that the proposed method is applicable and superior to the traditional one. Comparisons of integrated water vapour (IWV) of the proposed method with that derived from radiosonde and European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim data show that the root mean square (RMS)/Bias of their differences are 3.2/−0.8 mm and 3.3/−1.7 mm, respectively, while the values of traditional method are 5.1/−3.9 mm and 6.3/−5.9 mm, respectively. Furthermore, the water vapour density profiles are also compared with radiosonde and ECMWF data, and the values of RMS/Bias error for the proposed method are 0.88/0.06 g/m3 and 0.92/−0.08 g/m3, respectively, while the values of the traditional method are 1.33/0.38 g/m3 and 1.59/0.40 g/m3, respectively.

Global Navigation Satellite System (GNSS) tomography has developed into an efficient tool for sensing the high spatiotemporal variability of atmospheric water vapor. The integration of GNSS top signals and side rays for tropospheric tomography systems using a novel height factor model (HFM) is proposed and discussed in this paper. Within the HFM, the sectional slant wet delay (SWD) of inside signals (the part of the side signal inside the tomography area), which is considered a key factor for modeling side rays, is separated into isotropic and anisotropic components. Correspondingly, two height factors are defined to calculate the isotropic and anisotropic part of tropospheric delays in the HFM. In addition, the dynamic tomography top boundary is first analyzed and determined based on 30-year radiosonde data to reasonably divide signals into top and side rays. Four special experimental schemes based on different tomography regions of Hong Kong are performed to assess the proposed HFM method, the results of which show increases of 33.42% in the mean utilization of rays, as well as decreases of 0.46 g/m3 in the average root mean square error (RMSE), compared to the traditional approach, revealing the improvement of tomography solutions when the side signals are included in the modeling. Furthermore, compared with the existing correction model for modeling side rays, the water vapor profiles retrieved from the proposed improved model are closer to the radiosonde data, which highlights the advantages of the proposed HFM for optimizing the GNSS tomography model.

Ground-based GNSS ZTD/IWV estimation system for numerical weather prediction in challenging weather conditions

Acousto-optic devices using lead molybdate and related compounds: Bell Telephone Labs Inc US patent 3617931 (1971) filed 5 May 1969

Water vapor is an important driving factor in the related weather processes in the troposphere, and its temporal-spatial distribution and change are crucial to the formation of cloud and rainfall. Global Navigation Satellite System (GNSS) water vapor tomography, which can reconstruct the water vapor distribution using GNSS observation data, plays an increasingly important role in GNSS meteorology. In this paper, a method to improve the distribution of observations in GNSS water vapor tomography is proposed to overcome the problem of the relatively concentrated distribution of observations, enable satellite signal rays to penetrate more tomographic voxels, and improve the issue of overabundance of zero elements in a tomographic matrix. Numerical results indicate that the accuracy of the water vapor tomography is improved by the proposed method when the slant water vapor calculated by GAMIT is used as a reference. Comparative results of precipitable water vapor (PWV) and water vapor density (WVD) profiles from radiosonde station data indicate that the proposed method is superior to the conventional method in terms of the mean absolute error (MAE), standard deviations (STD), and root-mean-square error (RMS). Further discussion shows that the ill-condition of tomographic equation and the richness of data in the tomographic model need to be discussed separately.

Accurate water vapor information is crucial for improving the quality of numerical weather forecasting. Previous studies have incorporated tropospheric water vapor data obtained from a global navigation satellite system (GNSS) into numerical weather models to enhance the accuracy and reliability of rainfall forecasts. However, research on evaluating forecast accuracy for different rainfall levels and the development of corresponding forecasting platforms is lacking. This study develops and establishes a rainfall forecasting platform supported by the GNSS-assisted weather research and forecasting (WRF) model, quantitatively assessing the effect of GNSS precipitable water vapor (PWV) on the accuracy of WRF model forecasts for light rain (LR), moderate rain (MR), heavy rain (HR), and torrential rain (TR). Three schemes are designed and tested using data from seven ground meteorological stations in Xi’an City, China, in 2021. The results show that assimilating GNSS PWV significantly improves the forecast accuracy of the WRF model for different rainfall levels, with the root mean square error (RMSE) improvement rates of 8%, 15%, 19%, and 25% for LR, MR, HR, and TR, respectively. Additionally, the RMSE of rainfall forecasts demonstrates a decreasing trend with increasing magnitudes of assimilated PWV, particularly effective in the range of [50, 55) mm where the lowest RMSE is 3.58 mm. Moreover, GNSS-assisted numerical weather model shows improvements in statistical forecasting indexes such as probability of detection (POD), false alarm rate (FAR), threat score (TS), and equitable threat score (ETS) across all rainfall intensities, with notable improvements in the forecasts of HR and TR. These results confirm the high precision, visualization capabilities, and robustness of the developed rainfall forecasting platform.

Global Navigation Satellite System (GNSS) tomography has proved its potential for sensing the three-dimensional (3D) distribution of atmospheric water vapor. Although the standard tomography model has improved, the geometry defect of GNSS acquisition remains a key problem and thus becomes the focus of this research. Thanks to the availability of high-resolution Moderate Resolution Imaging Spectroradiometer (MODIS) precipitable water vapor (PWV) maps, the integration of GNSS and MODIS measurements for tropospheric tomography system is introduced and discussed to address this problem. A methodology based on the node tomography model is developed to exploit the MODIS signals, which overcomes the disadvantage of the standard voxel model in combining the MODIS observations. Three experimental schemes based on MODIS data and simultaneous GNSS observations over the Xuzhou area are implemented to validate the proposed approach. The results show that the mean number of newly crossed voxels (i.e., punctured only by the MODIS signals) increases from 2 in the top layer to 24 in the first layer. Consequently, the average number of total crossed voxels is increased from 272 to 447, which highlights the contribution of the MODIS observations to pass through the empty voxels. Besides, the mean number of tomographic observation equations is enhanced by 35.71% when the MODIS signals are combined. Two-dimensional (2D) profiles of water vapor from radiosonde and 3D distributions of it from ERA5 data are considered to validate the proposed method. It is noted that when using the proposed approach, the mean root-mean-square error (RMSE) of the 2D tomographic profiles is decreased by a value of about 1.06 g/m3 and the overall accuracy of the 3D water vapor distribution is improved by 30.40%. Both the 2D and 3D validations demonstrate the satisfactory performance of the new integrated method to optimize the tomographic water vapor field.

Water vapour is a highly variable constituent of the troposphere; thus, its high-resolution measurements are of great importance to weather prediction systems. The Global Navigation Satellite Systems (GNSS) are operationally used in the estimation of the tropospheric state and assimilation of the results into the weather models. One of the GNSS techniques of troposphere sensing is tomography which provides 3-D fields of wet refractivity. The tomographic results have been successfully assimilated into the numerical weather models, showing the great potential of this technique. The GNSS tomography can be based on two different approaches to the parameterisation of the model’s domain, i.e. block (voxel-based) or grid (node-based) approach. Regardless of the parameterisation approach, the tomographic domain should be discretised, which is usually performed in a regular manner, with a grid resolution depending on the mean distance between the GNSS receivers. In this work, we propose a new parameterisation approach based on the optimisation of the tomographic nodes’ location, taking into account the non-uniform distribution of the GNSS information in the troposphere. The experiment was performed using a dense network of 16 low-cost multi-GNSS receivers located in Wrocław and its suburbs, with a mean distance of 3 km. Cross-validation of four different parameterisation approaches is presented. The validation is performed based on the Weather Research and Forecasting model as well as radiosonde observations. The new approach improves the results of wet refractivity estimation by 0.5–2 ppm in terms of RMSE, especially for altitudes of 0.5–2.0 km.

An improved ridge estimation (IRE) method for troposphere water vapor tomography

Abstract. Tropospheric water vapor is one of the most important trace gases of the Earth's climate system, and its temporal and spatial distribution is critical for the genesis of clouds and precipitation. Due to the pronounced dynamics of the atmosphere and the nonlinear relation of air temperature and saturated vapor pressure, it is highly variable, which hampers the development of high-resolution and three-dimensional maps of regional extent. With their complementary high temporal and spatial resolutions, Global Navigation Satellite Systems (GNSS) meteorology and Interferometric Synthetic Aperture Radar (InSAR) satellite remote sensing represent a significant alternative to generally sparsely distributed radio sounding observations. In addition, data fusion with collocation and tomographical methods enables the construction of detailed maps in either two or three dimensions. Finally, by assimilation of these observation-derived datasets with dynamical regional atmospheric models, tropospheric water vapor fields can be determined with high spatial and continuous temporal resolution. In the following, a collection of basic and processed datasets, obtained with the above-listed methods, is presented that describes the state and course of atmospheric water vapor for the extent of the GNSS Upper Rhine Graben Network (GURN) region. The dataset contains hourly 2D fields of integrated water vapor (IWV) and 3D fields of water vapor density (WVD) for four multi-week, variable season periods between April 2016 and October 2018 at a spatial resolution of (2.1 km)2. Zenith total delay (ZTD) from GNSS and collocation and refractivities are provided as intermediate products. InSAR (Sentinel-1A/B)-derived double differential slant total delay phases (ddSTDPs) and GNSS-based ZTDs are available for March 2015 to July 2019. The validation of data assimilation with five independent GNSS stations for IWV shows improving Kling–Gupta efficiency (KGE) scores for all seasons, most notably for summer, with collocation data assimilation (KGE = 0.92) versus the open-cycle simulation (KGE = 0.69). The full dataset can be obtained from https://doi.org/10.1594/PANGAEA.936447 (Fersch et al., 2021).

Water vapour is a crucial and highly variable greenhouse gas in the Earth's atmosphere that plays a major role in the radiative balance, energy transport and photochemical processes. It can also affect the radiative budget indirectly through cloud formation and by altering the size, shape, and chemical composition of aerosol particles. Moreover, monitoring water vapour remains challenging due to its high temporal and spatial variability. Consequently, systematic and accurate observations of water vapour are essential to improve our understanding of its role at both local and global scales and for enhancing climate projections.Advances in remote sensing techniques have enabled continuous acquisition of precipitable water vapour (PWV) measurements using sun/star photometry, microwave radiometry and the Global Navigation Satellite System (GNSS). Nevertheless, none of these instruments provides information on the vertical distribution of water vapour, a critical information considering that water vapour concentrations typically vary by up to three orders of magnitude between the surface and the upper troposphere. In this context, Raman lidar has demonstrated its ability to capture the spatial and temporal evolution of water vapour in the troposphere. Accurate retrievals of the water vapour mixing ratio from Raman lidar measurements rely on robust and well-characterised calibration procedures as well as on an accurate estimation of the differential atmospheric transmission term, which accounts for extinction differences between the molecular reference (nitrogen and oxygen) and water vapour wavelengths.In this study, the lidar calibration constant was determined using a hybrid calibration method, which combines correlative PWV measurements for lidar calibration with Numerical Weather Prediction (NWP) data to reconstruct the water vapour profile within the incomplete overlap region of the lidar system. The differential transmission was estimated using an automated method to account for the aerosol contribution, based on sun photometer Aerosol Optical Depth (AOD) measurements and an exponential decay function with attitude to model aerosol extinction (Díaz-Zurita et al., 2025). Subsequently, a long-term database of water vapour profiles over the period 2009-2022 was generated, providing high vertical and temporal resolution measurements of water vapour over the city of Granada, in Southern Spain. A comprehensive statistical analysis was conducted to characterise the vertical distribution of water vapour over a 14-year period, representing the first long-term vertical characterisation of water vapour in this region. Mean interannual and seasonal water vapour profiles were derived for the entire study period, and trend analyses were performed to assess long-term variations in water vapour content in the lower troposphere. Additionally, lidar-derived PWV values were compared with those obtained from microwave radiometer and GNSS observations.This research was funded by Grant PID2021-128008OB-I00 funded by MICIU/AEI/ 10.13039/501100011033 by ERDF/EU European Union, and by the Spanish national projects CNS2023-145435, PID2023-151817OA-I00 and Marie Skłodowska-Curie Staff Exchange Actions with the project GRASP-SYNERGY (grant agreement no. 10113163). Diaz-Zurita et al. (2025). Remote Sens. 2025, 17(20), 3444; https://doi.org/10.3390/rs17203444