Accurate empirical tropospheric mapping functions (MFs) are essential for high-precision geodetic applications such as GNSS and VLBI. However, traditional MFs assume azimuthal symmetry or explicitly model gradients on regular grids, limiting their ability to represent real atmospheric conditions. To address the limitation, this study designs a novel physics-constrained machine learning-based framework implemented with a multilayer perceptron and develops global empirical asymmetric MFs on five years of ERA5-based ray-tracing data. Compared to the widely used GPT3 model, our models reduce global RMSE by up to 12.0% for hydrostatic and 2.8% for wet components. Relative to GPT3 with gradient corrections, they remain globally competitive, with locally improved performance, especially at low elevations, high altitudes, and over many continental regions. The improvements are particularly evident at low elevation angles, high-altitude sites, and over continental regions. To our knowledge, this work marks the first global machine learning-based empirical MF models that directly capture atmospheric asymmetry, offering a flexible and physically consistent alternative to traditional models in space geodesy. The first global machine-learning-based empirical mapping functions for the asymmetric troposphere are developed. A physics-constrained machine-learning framework enables robust and physically consistent mapping function modeling. The models outperform symmetric GPT3, remain competitive with GPT3 with gradients, and support flexible updating and extension. The first global machine-learning-based empirical mapping functions for the asymmetric troposphere are developed. A physics-constrained machine-learning framework enables robust and physically consistent mapping function modeling. The models outperform symmetric GPT3, remain competitive with GPT3 with gradients, and support flexible updating and extension.
Zhen-Yi Zhang, Benedikt Soja· Journal of Geodesy· 1 citation· ⚡1
As low-Earth orbit (LEO) satellites are increasingly equipped with highly precise Global Navigation Satellite Systems (GNSS) receivers, new opportunities for GNSS-based precise orbit determination (POD) arise. The availability of GNSS observations from satellites that are geometrically well distributed over the entire Earth may enable the formation of an independent GNSS network in LEO, which can be processed solely on the basis of double-difference (DD) observations. This study presents a first attempt to compute a network solution of GNSS DD observations in space that combines data from eight satellites, including GRACE-FO C/D, Sentinel-3A/B, Sentinel-6A and Swarm A/B/C. Using satellite laser ranging (SLR) validation, we show that the absolute position accuracy of an ambiguity-float network solution increases from about 4 cm 1D RMS in a constellation of three satellites to about 2 cm–2.5 cm in a constellation of eight satellites. Similar results are obtained by comparing the baseline lengths from our DD network solution with those derived from external ambiguity-fixed zero-difference (ZD) orbits, with differences at a level of 2 cm to 3 cm RMS for a network of eight satellites. For both the SLR residuals and the baseline consistency, an improvement of up to 0.5 cm can be achieved when resolving 60 % to 70 % of the carrier-phase ambiguities of highly dynamic baselines to integer values. Overall, the geometric distribution of the satellites and high dynamics of the constellation prove to be beneficial for computing a DD network solution in space. This is particularly promising in the view of large LEO constellations currently being developed, with a much higher number of spacecraft and a uniform global coverage, where the advantages of the network approach presented in this study will become even more significant.
Lukas Müller, Markus Rothacher, Benedikt Soja et al.· Journal of Geodesy· 0 citations
Accurate representation of atmospheric moisture is essential for reliable weather forecasting, particularly for small-scale convective systems and extreme events. However, determining high-resolution water vapor (WV) fields remains challenging. Conventional global navigation satellite system (GNSS) troposphere tomography reconstructs 4-D atmospheric wet refractivity fields but is limited by sparse and uneven ray paths, an ill-conditioned coefficient matrix, and an ill-posed inverse problem. Stabilization through constraints and regularization may introduce biases, while the low probability of ray–ray intersections in the lowest tropospheric layers reduces observational influence, causing some regions to depend more on background models than observations. To address these limitations, DeepTomo, to the best of the authors’ knowledge, the first artificial intelligence (AI)-based 4-D GNSS troposphere tomography is introduced as an explainable physics-informed deep learning approach that combines hybrid observational constraints with spatiotemporal learning. Beyond tomographic reconstruction, DeepTomo is conceived as an AI-based assimilation of GNSS observations into ERA5 fields; it integrates a 3-D convolutional neural network (CNN) with residual learning and attention mechanisms and employs a hybrid physics-informed loss function that combines GNSS-derived zenith wet delay (ZWD) with radio occultation (RO) and radiosonde refractivity profiles to correct the ERA5 background toward observational constraints. By learning spatiotemporal relationships between observations and background fields, DeepTomo refines wet refractivity estimates and enables physically consistent reconstruction even in voxels with limited observations. Trained and validated over a dense GNSS network in coastal California using a six-month dataset and evaluated against radiosonde and GNSS-derived ZWD data, DeepTomo performs strongly during the extreme weather event of Hurricane Hilary, a tropical cyclone (TC), in August 2023. Compared with conventional voxel-based tomography, it reduces the root mean square error (RMSE) by up to 64.85% during the TC and 41.62% overall, capturing large moisture variability. Explainable AI (XAI) analysis reveals dynamic spatial attention to regions of enhanced variability. A preliminary sensitivity analysis using GraphCast forecasts shows that the moisture corrections introduced by DeepTomo correspond to short-range forecast errors. This analysis provides an initial indication that DeepTomo, by producing physically consistent GNSS-constrained moisture analyses from ERA5 background fields, has the potential to improve initial conditions and forecast performance in next-generation AI weather forecasting systems, such as GraphCast, bridging GNSS observations with AI-based forecast initialization.
Saeid Haji-Aghajany, Benedikt Soja, Kefei Zhang et al.· IEEE Transactions on Geoscie...· 0 citations
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