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The Earth's gravity field is a fundamental physical field for research and analysis in Earth sciences. However, the limited degree of expansion in the gravity field model introduces truncation errors, which hinder the accurate representation of high-frequency information in Earth's gravity field model. To address this issue, this study refined the gravity field model in the spatial domain by constructing a residual terrain model. This study refined the XGM2019e-2159 gravity field model for the study area in Colorado, USA (108°W–104°W, 37°N–41°N). First, the residual terrain model (RTM) was constructed using the high-resolution terrain model SRTMV4.1 and the reference topography model Earth2014. Subsequently, the residual terrain model was discretized into regular grid prisms. Based on Newton's law of universal gravitation, the disturbance potential of each prism within a specified range at the computation point is calculated using the rectangular prism method in the spatial domain. Next, the disturbance potential is used to compute the RTM gravity anomalies and RTM vertical deflections. The results were verified using ground measured gravity anomaly data NGS99 and vertical deflection data GSVS17. The results show that, after RTM correction, the root mean square (RMS) of the difference between modeled and measured gravity anomalies decreased from 19.71 mGal to 13.80 mGal, and the effect of residual terrain correction improves as terrain undulation increases. The RMS of the North–South and East–West component differences between modeled and measured vertical deflections was 1.44″ and 1.82″ before correction, and decreased to 0.89″ and 0.93″ after RTM correction. Finally, a power spectral density analysis of the XGM2019e-2159 gravity anomaly and vertical deflection models before and after RTM correction showed a significant increase in short-wavelength energy after correction. These results indicate that RTM correction effectively compensated for truncation errors in the XGM2019e-2159 gravity anomaly and vertical deflection models, significantly improving data quality. Graphical Abstract

The traditional altimetry satellite, which is based on pulse-limited radar altimeter, only measures ocean surface heights along tracks; hence, leads to poorer accuracy in the east component of the vertical deflections compared to the north component, which in turn limits the final accuracy of the marine gravity field inversion. Wide-swath altimetry using radar interferometry can measure ocean surface heights in two dimensions and, thus, can be used to compute vertical deflections in an arbitrary direction with the same accuracy. This paper aims to investigate the impact of Interferometric Radar Altimeter (InRA) errors on gravity field inversion. The error propagation between gravity anomalies and InRA measurements is analyzed, and formulas of their relationship are given. By giving a group of possible InRA parameters, numerical simulations are conducted to analyze the accuracy of gravity anomaly inversion. The results show that the accuracy of the gravity anomalies is mainly influenced by the phase errors of InRA; and the errors of gravity anomalies have a linear approximation relationship with the phase errors. The results also show that the east component of the vertical deflections has almost the same accuracy as the north component.

Submarine rift systems represent critical tectonic features whose accurate bathymetric characterization remains challenging yet essential for understanding plate boundary dynamics. However, traditional bathymetric inversion methods based on altimetric gravity data exhibit poor performance in resolving rift and steep-slope terrains. To address this limitation and enhance accuracy in complex topographic regions, we propose a multi-gravity-component fusion framework based on an improved DenseNet architecture. By integrating shipborne bathymetry, gravity anomaly (GA), vertical gravity gradient (VGG), vertical deflection components (meridian component ξ and prime vertical component η), and GEBCO_2024, we construct a 16 × 16 × 9 input tensor. The model incorporates adaptive transition layers to preserve fine-scale tectonic features and curvature-based stratification to balance learning across diverse terrains. Validation using 43,035 independent points yields an RMSE of 84.75 m, representing a 47.6% reduction relative to GEBCO_2024. Crucially, in the identified rift targets, errors decreased by 69.3–87.1%. Ablation studies reveal that vertical deflection components (ξ, η) dominate the physical constraints, with their removal increasing the RMSE by 91.08 m (a 107.5% increase relative to the baseline error). Architectural innovations and stratification reduce steep-slope RMSE by 6.1%. These results validate the efficacy of directional gravity derivatives for tectonic feature inversion and demonstrate significant potential for application to mid-ocean ridge systems.

Recently, four global geopotential models (GGMs) were computed and released based on the first 2 months of data collected by the Gravity field and steady-state Ocean Circulation Explorer (GOCE) dedicated satellite gravity field mission. Given that GOCE is a technologically complex mission and different processing strategies were applied to real space-collected GOCE data for the first time, evaluation of the new models is an important aspect. As a first assessment strategy, we use terrestrial gravity data over Switzerland and Australia and astrogeodetic vertical deflections over Europe and Australia as ground-truth data sets for GOCE model evaluation. We apply a spectral enhancement method (SEM) to the truncated GOCE GGMs to make their spectral content more comparable with the terrestrial data. The SEM utilises the high-degree bands of EGM2008 and residual terrain model data as a data source to widely bridge the spectral gap between the satellite and terrestrial data. Analysis of root mean square (RMS) errors is carried out as a function of (i) the GOCE GGM expansion degree and (ii) the four different GOCE GGMs. The RMS curves are also compared against those from EGM2008 and GRACE-based GGMs. As a second assessment strategy, we compare global grids of GOCE GGM and EGM2008 quasigeoid heights. In connection with EGM2008 error estimates, this allows location of regions where GOCE is likely to deliver improved knowledge on the Earth’s gravity field. Our ground truth data sets, together with the EGM2008 quasigeoid comparisons, signal clear improvements in the spectral band ~160–165 to ~180–185 in terms of spherical harmonic degrees for the GOCE-based GGMs, fairly independently of the individual GOCE model used. The results from both assessments together provide strong evidence that the first 2 months of GOCE observations improve the knowledge of the Earth’s static gravity field at spatial scales between ~125 and ~110 km, particularly over parts of Asia, Africa, South America and Antarctica, in comparison with the pre-GOCE-era.

We present results from a new vertical deflection (VD) traverse observed in Perth, Western Australia, which is the first of its kind in the Southern Hemisphere. A digital astrogeodetic QDaedalus instrument was deployed to measure VDs with \\[{\\sim }\\]0.2\\[''\\] precision at 39 benchmarks with a \\[{{\\sim }}1~\\hbox {km}\\] spacing. For the conversion of VDs to quasigeoid height differences, the method of astronomical–topographical levelling was applied, based on topographical information from the Shuttle Radar Topography Mission. The astronomical quasigeoid heights are in 20–30 mm (RMS) agreement with three independent gravimetric quasigeoid models, and the astrogeodetic VDs agree to 0.2–0.3\\[''\\] (north–south) and 0.6–0.9\\[''\\] (east–west) RMS. Tilt-like biases of \\[{\\sim }1\\,\\,\\hbox {mm}\\] over \\[{\\sim }1\\,\\,\\hbox {km}\\] are present for all quasigeoid models within \\[{\\sim }20\\,\\,\\hbox {km}\\] of the coastline, suggesting inconsistencies in the coastal zone gravity data. The VD campaign in Perth was designed as a low-cost effort, possibly allowing replication in other Southern Hemisphere countries (e.g., Asia, Africa, South America and Antarctica), where VD data are particularly scarce.

The first Chinese altimetry satellite, Haiyang-2A (HY-2A), which was launched in 2011, has provided a large amount of sea surface heights which can be used to derive marine gravity field. This paper derived the vertical deflections and gravity disturbances using HY-2A observations for the major area of the whole Earth’s ocean from 60°S and 60°N. The results showed that the standard deviations (STD) of vertical deflections differences were 1.1 s and 3.5 s for the north component and the east component between HY-2A’s observations and those from EGM2008 and EIGEN-6C4, respectively. This indicates the accuracy of the east component was poorer than that of the north component. In order to clearly demonstrate contribution of HY-2A’s observations to gravity disturbances, reference models and the commonly used remove-restore method were not adopted in this study. Therefore, the results can be seen as ‘pure’ signals from HY-2A. Assuming the values from EGM2008 were the true values, the accuracy of the gravity disturbances was about −1.1 mGal in terms of mean value of the errors and 8.0 mGal in terms of the STD. This shows systematic errors if only HY-2A observations were used. An index of STD showed that the accuracy of HY-2A was close to the theoretical accuracy according to the vertical deflection products. To verify whether the systematic errors of gravity field were from the long wavelengths, the long-wavelength parts of HY-2A’s gravity disturbance with wavelengths larger than 500 km were replaced by those from EGM2008. By comparing with ‘pure’ HY-2A version of gravity disturbance, the accuracy of the new version products was improved largely. The systematic errors no longer existed and the error STD was reduced to 6.1 mGal.

Haiyang-2 (HY-2) missions have accumulated sea surface height (SSH) observations on a global scale for more than 10 years. Four satellites, HY-2A, HY-2B, HY-2C and HY-2D, provide even but differently distributed data, which play a complementary role in marine gravity studies with other missions. Therefore, this paper evaluates the performances of HY-2 altimetric data in marine gravity modeling from the following four perspectives: SSH accuracy, geoid signal resolution ability, vertical deflections and gravity anomaly. First, the centimeter-magnitude accuracy level of HY-2 data is proved by analyzing SSH discrepancies at crossover points within a certain time limit. Second, the spectral analysis of repetitive along-track data sequences in a time domain shows a geoid resolution range from 18 to 24 km. Taking HY-2 exact repeat missions (ERM), for example, the resolution could be remarkably enhanced by stacking repetitive cycles. Third, validation with an XGM2019 model showed that vertical deflections were reliably computed for all HY-2 missions, but HY-2A performed slightly worse than the other HY-2 missions. Meanwhile, HY-2C and HY-2D with a ~66° orbital inclination obviously had an improved ability to capture east–west signals compared to HY-2A and HY-2B. Finally, we constructed global marine gravity results based on three input datasets, HY-2 dataset only, multi-satellite dataset without HY-2 and multi-satellite dataset with HY-2. Validations were performed using published models and shipborne gravimetric data. The results showed that the HY-2 dataset is capable of improving marine gravity anomaly recoveries and that the accuracy of NSOAS22 with incorporated HY-2 data is comparable to DTU21 and SS V31.1. Furthermore, HY-2 observations should not be the only input dataset to construct a 1’ × 1’ resolution marine gravity model.

Management of the vertical long-term deflection of a high-speed railway bridge is a crucial factor to guarantee traffic safety and passenger comfort. Therefore, there have been efforts to predict the vertical deflection of a railway bridge based on physics-based models representing various influential factors to vertical deflection such as concrete creep and shrinkage. However, it is not an easy task because the vertical deflection of a railway bridge generally involves several sources of uncertainty. This paper proposes a probabilistic method that employs a Gaussian process to construct a model to predict the vertical deflection of a railway bridge based on actual vision-based measurement and temperature. To deal with the sources of uncertainty which may cause prediction errors, a Gaussian process is modeled with multiple kernels and hyperparameters. Once the hyperparameters are identified through the Gaussian process regression using training data, the proposed method provides a 95% prediction interval as well as a predictive mean about the vertical deflection of the bridge. The proposed method is applied to an arch bridge under operation for high-speed trains in South Korea. The analysis results obtained from the proposed method show good agreement with the actual measurement data on the vertical deflection of the example bridge, and the prediction results can be utilized for decision-making on railway bridge maintenance.

HaiYang-2A (HY-2A, where ‘Haiyang’ means ‘Ocean’ in Chinese) has provided reliable sea surface height observations for gravity with uniform ocean data coverage on a global scale for more than 8 years, particularly with denser across track sampling during the geodetic mission since March 2016. This paper aims at modeling and evaluating the regional marine gravity field at 1′×1′ resolution by incorporating HY-2A altimeter waveform data from 7 complete 168-day cycles in the geodetic mission phase. Initial evaluation indicates that, firstly, the measurements in the geodetic mission stay at a consistent accuracy level with observations at the start-of-life stage according to statistics of discrepancies at crossover points cycle by cycle. Secondly, range precision improvement can be achieved using a two-pass weighted least-squares retracker. Thirdly, a downsampling procedure combined with a low-pass filter is designed for HY-2A 20 Hz data to obtain 5 Hz measurements with enhanced precision. We calculate the 1′×1′ marine gravity field model over the South China Sea area by using the EGM2008 model as a reference field with the remove/restore method. The verifications with published models and shipborne gravimetric data show that HY-2A GM data is capable of improving marine gravity field modeling. Results show slightly higher accuracy than other models with similar input datasets but not including HY-2A. The accuracy is also compared with the latest DTU17 and SIO V27.1 model.

Prestressing methods were used to realize long-span bridges in the last few decades. For their predictive maintenance, devices and dynamic nondestructive procedures for identifying prestress losses were mainly developed since serviceability and safety of Prestressed Concrete (PC) girders depend on the effective state of prestressing. In fact, substantial long term prestress losses can induce excessive deflections and cracking in large span PC bridge girders. However, old unsolved problematics as well as new challenges exist since a variation in prestress force does not significantly affect the vibration responses of such PC girders. As a result, this makes uncertain the use of natural frequencies as appropriate parameters for prestress loss determinations. Thus, amongst emerging techniques, static identification based on vertical deflections has preliminary proved to be a reliable method with the goal to become a dominant approach in the near future. In fact, measured vertical deflections take accurately and instantaneously into account the changes of structural geometry of PC girders due to prestressing losses on the equilibrium conditions, in turn caused by the combined effects of tendon relaxation, concrete creep and shrinkage, and parameters of real environment as, e.g., temperature and relative humidity. Given the current state of quantitative and principled methodologies, this paper represents a state-of-the-art review of some important research works on determining prestress losses conducted worldwide. The attention is principally focused on a static nondestructive method, and a comparison with dynamic ones is elaborated. Comments and recommendations are made at proper places, while concluding remarks including future studies and field developments are mentioned at the end of the paper.