Tao Jiang

The accurate modeling of the Earth gravity field and geoid is critical for geodesy, yet traditional methods face limitations in handling the growing complexity and heterogeneity of modern geodetic data. To address these challenges, this study proposes a physics-informed neural network (PINN) framework for high-precision geoid modeling. The PINN employs convolutional neural networks (CNNs) to extract multi-scale features from terrestrial and airborne gravity data, which are then processed by a multilayer perceptron (MLP) to establish an accurate mapping between these features and the disturbing potential. Physical constraints, including Laplace’s equation and differential equations governing gravity anomaly and gravity disturbance, are embedded into the loss function to enhance both accuracy and interpretability. The proposed method is applied to the Colorado 1 cm geoid experiment. Compared to GNSS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17), the PINN-derived geoid model achieves a standard deviation (STD) of 2.1 cm. This represents a 12.5%–27.6% improvement over traditional methods and purely data-driven networks (DDNs). The PINN exhibits strong generalization under sparse data conditions, achieving 28.5% higher accuracy than the DDN with only 500 samples. Furthermore, analysis of geoid slopes and physical constraint contributions demonstrates that PINN’s dual physical constraints effectively balance global characteristics and localized fidelity of the geoid. This study establishes the PINN as a robust, physically interpretable machine learning approach for geoid modeling, outperforming classical methods and offering a promising pathway for gravity field estimation in regions with sparse or heterogeneous data. By bridging purely data-driven machine learning with fundamental geodetic principles, this work paves the way for future advancements in physics-informed machine learning-based geodetic modeling.

Greenland, home to the largest ice sheet in the Northern Hemisphere, provides a crucial digital elevation model (DEM) for understanding polar climate evolution and valuable data for global climate change research. Based on ICESat-2 laser altimetry data collected from satellite observations over Greenland between November 2020 and November 2021, the Shandong University of Science and Technology 2021 DEM (SDUST2021DEM) with 500 m grid resolution at the epoch of May 2021 was constructed using a spatiotemporally fitted subgrid least squares method. The precision of the DEM was evaluated by comparison with National Aeronautics and Space Administration IceBridge data and supplemented by GNSS station measurements. The median difference between the DEM and IceBridge data was −0.33 m, the mean deviation −0.58 m, and the median absolute deviation 2.31 m. The accuracy of SDUST2021DEM exhibits a clear spatial pattern: it is higher in the central ice sheet than at the margins, decreases in regions with complex terrain, and remains more reliable in areas characterized by gentle slopes and flat terrain. Overall, the SDUST2021DEM demonstrates stable accuracy and can reliably produce high-precision DEMs for a specific temporal epoch.

Currently, research on seafloor topographic inversion primarily focuses on various gravity data and inversion methods, while little attention has been given to the seafloor sediment effects. Additionally, there is ongoing controversy regarding the extent to which sediment effects influence the seafloor topographic inversion. To address these, we first apply Parker’s method to estimate the sediment gravity anomalies (GAs) using sediment thickness data, thereby resolving the difficulty of directly inputting sediment thickness into the inversion. In previous regression methods, the coefficient factors, when treated as outputs, significantly deviated from theoretical value. Therefore, we then propose the coefficient-constrained multivariate regression (MR-CC) method. By constraining the output range of the coefficient factors, we investigates the contribution of incorporating sediment effects to improving the accuracy of the coefficient factors and enhancing the effectiveness of seafloor topographic inversion. The accuracy of the seafloor topography models was evaluated over the study area located in the Sea of Japan. The results show that, compared to the linear regression (LR) model, the MR-CC series models, which consider the sediment contributions, improves RMS by 19.54% and MAPE by 26.27%, effectively enhancing the regression and inversion accuracy. Furthermore, we evaluated the impact of various sediment thickness ranges on the inversion accuracy and concluded that the sediment thickness data within the range of 0-4000 m offer higher accuracy, making them more suitable for seafloor topographic inversion. Finally, we employ the Convolutional Neural Network (CNN) to fuse sediment thickness and sediment GAs separately, demonstrating the effectiveness of data preprocessing in mitigating feature dependency in artificial intelligence (AI) applications, which further clarifies the previous controversies on sediment effects in seafloor topographic inversion.

Although machine learning has become increasingly important in geodesy related fields such as geophysics, seismology and remote sensing, its applications in geodesy, especially in physical geodesy, are still in its early stages. The main reason for this can be attributed to the black box nature of pure data-driven machine learning, which lacks physical interpretability and credibility, making it difficult for machine learning approaches to be used in physical geodesy that takes reliability and accuracy as its core criteria. Physics-Informed Neural Networks (PINNs) is a class of deep learning algorithms aims to seamlessly integrate data and physical prior knowledge including ordinary or partial differential equations, it can yield more physically interpretable machine learning models that provide robust and accurate predictions. We present the PINN approach for gravimetric geoid modeling from Earth gravity model, terrestrial and airborne gravity datasets. A convolutional neural network (CNN) deep learning architecture is used, gravity measurements and physical laws are integrated by embedding the Laplace’s equation of disturbing potential and the fundamental equation of gravity anomaly into the loss function of the neural network using automatic differentiation. The PINN based geoid computation approach is tested in the area of the Colorado 1-cm geoid experiment. Simulated gravity observations and GNSS leveling derived geoid heights based on EIGEN-6C4 are used to validate the theoretical correctness and validity of the proposed PINN approach, and its performance on precise geoid modeling in this challenging area is evaluated using the actual terrestrial and airborne gravity observations, GNSS leveling measured geoid heights and high resolution DEM provided by NGS/NOAA.

The non-tectonic deformation caused by hydrological loads is an important influencing factor in GNSS vertical displacement. Limited by the temporal and spatial resolution of global models and model errors, the hydrological load results calculated by traditional methods are difficult to meet the high temporal and spatial resolution requirements of small to medium-scale regions. This paper introduces the idea of the remove–restore method, assimilates regional high-resolution hydrological data, and obtains higher temporal and spatial-resolution hydrological load results. Subsequently, utilizing data from 12 CORS observed in the western Yunnan region between January 2018 and December 2020, the quantitative relationship and variation characteristics between GNSS vertical displacement and hydrological load displacement were analyzed in detail. Furthermore, the annual signals of both were extracted using the SSA method for comparative analysis. After removing the effects of atmospheric load and non-tidal ocean load, the average correlation coefficient between GNSS vertical displacement and hydrological load displacement is 0.84, with an average reduction of WRMS (%) reaching 37.17%. The average correlation coefficient of the annual signals between GNSS vertical displacement and hydrological load deformation is 0.94, with an average reduction of WRMS (%) reaching 46.5%, indicating that the contribution of hydrological load to the GNSS non-tectonic vertical displacement annual signal is close to 50%. The research results provide scientific support and important references for studying surface tectonic deformation by removing non-tectonic deformations such as hydrological loads from GNSS vertical displacement. Additionally, it helps to explore the mechanisms of interaction between water storage migration and surface deformation.

The North China Plain in China is one of the areas with the scarcest water resources, and it is an area with more groundwater extraction. The form of groundwater in this area is severe, and it is necessary to monitor the changes in groundwater reserves in this area. Traditional use of groundwater wells to monitor changes in groundwater levels makes it difficult to obtain accurate changes in groundwater reserves. GRACE/GRACE-FO satellite gravity has made great achievements in the changes of land water reserves and groundwater reserves, but due to the low spatial resolution, it is not possible to obtain the changes of groundwater reserves in the North China Plain. This paper integrates GNSS and GRACE/GRACE-FO satellite gravity data, based on the load deformation theory, effectively integrates GNSS and satellite gravity data, separates groundwater balance effect and groundwater load impact, obtains the change of groundwater reserves in the North China Plain from 2018 to the end of 2021, and conducts comparative analysis on the observation data of groundwater wells and precipitation data, finding that the downward trend of groundwater level has slowed down, The groundwater reserves in the North China Plain declined as a whole, and the groundwater in some areas of Beijing rose slightly.

This paper presents a method for dynamically maintaining a regional elevation datum using CORS stations as core nodes. By utilizing CORS station data and surface mass loading data (including land water storage, sea level, and atmospheric pressure), the normal height changes of each station can be determined and dynamically maintained. The validity of this method is verified using multiple leveling survey results from five CORS stations in Beijing’s subsidence area between January 2012 and June 2021. Results show that it is necessary to derive and correct the height anomaly variation of CORS stations caused by surface mass loading using the remove-calculate-restore method and the Green’s function integration method, with the influence of surface mass changes reaching a subcentimeter level. CORS stations exhibiting great observation quality achieve a mean accuracy of 2.7 mm in determining normal height changes. Such accuracy surpasses the requirements of second-class leveling surveys covering route lengths exceeding 1.35 km, as well as conforming/closed loop routes with distances greater than 0.46 km. By strategically selecting CORS stations with long-term continuous observations and high-quality data as core nodes within the elevation control network, dynamic maintenance of the regional elevation datum can be achieved based on CORS station data.

To achieve the goal that China and Nepal jointly announce the new height of Mount Qomolangma (MQ), the campaign of Qomolangma Height Survey (QHS) was carried out from 2019 to 2020. A high precision geoid model realizing the common height datum for both sides is necessary for determining the unique height of MQ. However, high altitude and rugged topography make it extremely difficult to conduct terrestrial gravity measurements in this region, the accuracy of geoid model is restricted by terrestrial gravity data gaps. In the campaign of 2020 QHS, the airborne gravity survey was carried out over MQ and its surrounding areas, the airborne gravity data covering an area of 12,700 km² were successfully collected. For the first time, the high precision observations of terrestrial gravity and BeiDou Navigation Satellite System (BDS) at the peak of MQ were collected. These datasets pave the way for the precise determination of the orthometric height of MQ. According to the definition of the International Height Reference System (IHRS), we developed the IHRS-based gravimetric quasigeoid model by combining the airborne and terrestrial gravity data. Validations against highly accurate GNSS leveling data at 61 benchmarks demonstrate that the accuracy of the quasigeoid model is 3.8 cm, and the addition of airborne gravity data improves the accuracy by 51.3%. Based on the IHRS, the final orthometric height of the snow surface of the peak of MQ is determined to be 8848.86 m.

The redistribution of surface mass (e.g., atmosphere, soil water, oceans, and groundwater) can cause load responses, resulting in vertical deformations of the crust. Indeed, the global navigation satellite system (GNSS)-based continuously operating reference stations (CORS) are able to accurately measure the vertical deformation caused by surface mass loads. In this study, the CORS was used to invert groundwater storage anomalies (GWSA), represented by the equivalent water height (EWH), after removing the effect of the non-groundwater surface mass load (atmospheric, groundwater, and non-tidal oceanic loads) from the vertical deformation monitored by CORS. In addition, the global and regional high-resolution surface mass models were combined to calculate the high-precision load deformation field in in western Yunnan using the remove–restore method, thereby obtaining more accurate surface mass load data and improving the accuracy of the inverted GWSA results. In order to assess the feasibility of the CORS inversion for the GWSA used, 66 CORS stations in western Yunnan Province were considered, presenting weekly GWSA data from 10 January 2018 to 31 December 2020. The results revealed significant seasonal variation in GWSA in the study area, showing an amplitude range of −200–200 mm. This approach is based on the already-established CORS network without requiring additional set-up costs. In addition, the reliability of CORS inverse results was assessed using Gravity Recovery and Climate Experiment (GRACE) inverse results and actual groundwater monitoring data. According to the obtained results, GWSA can be monitored by both CORS and GRACE data; however, CORS provided a more effective spatiotemporal resolution of GWSA. Therefore, the CORS network combined with surface mass load data is able to effectively monitor the spatiotemporal dynamics of GWSA in small-scale areas and provides important references for the study of hydrology.