The Surface Water and Ocean Topography (SWOT) mission has significantly advanced marine gravity recovery, yet its wide-swath measurements reveal the effects of strong sea level variability, emphasizing the need for effective correction methods. This study reveals that traditional correction strategies, such as stacking method, are insufficient for SWOT gravity recovery in high-variability environments, and quantifies the conditions under which more targeted correction approaches are necessary. Experimental results in the Kuroshio indicate that strong sea level variability directly affects SWOT observations, leading to gravity anomaly disturbances of approximately 2 mGal. To mitigate these disturbances, two strategies are implemented: stacking and sea level anomaly (SLA) model correction. Using SLA model correction, deflection of the vertical (DOV) accuracy improves by similar to 40 % in regions of strong sea level variations, while gravity anomaly accuracy is enhanced by 0.25 mGal (similar to 9%). To determine the conditions under which correction becomes necessary, this study evaluates gravity anomaly accuracy across varying sea level variability levels, quantified by the standard deviation of sea level anomaly (STD-SLA). Results show that when STD-SLA is below 15 cm, gravity recovery remains largely unaffected. However, above this threshold, SLA-induced disturbances become non-negligible, and applying correction improves gravity anomaly accuracy by more than 0.1 mGal on average. Global analysis reveals these regions exceeding this threshold, where correction is essential, are predominantly located along major Western Boundary Currents and the Antarctic Circumpolar Current. These findings underscore the need to reassess standard correction approaches in SWOT-era gravity recovery, and provide quantitative guidance on where SLA-based correction should be applied.

The degree-2 spherical harmonic coefficients of Earth's time-variable gravity field are highly sensitive to large-scale mass redistribution within the hydrosphere and cryosphere. Under contemporary global warming, climate-driven mass changes in these reservoirs are a dominant source, yet their individual contributions remain incompletely quantified. Traditional estimates based on hydrospheric models, filtered GRACE spherical harmonic solutions or GRACE Mascon products are limited by incomplete cryospheric representation, spatial leakage and regularization biases. Here, we apply the Fingerprint Approach that solves the sea-level equation on an elastic Earth to generate geoid fingerprints for four barystatic processes: terrestrial water storage, the Greenland Ice Sheet, the Antarctic Ice Sheet and mountain glaciers. Using these fingerprints and unfiltered GRACE/GRACE-FO Stokes coefficients for 2003-2024, we reconstruct the individual degree-2 coefficients C20, C21 and S21, along with the associated time-series of Earth's dynamic oblateness (J2) and the mass terms of polar motion excitation ($\chi _{\rm{1}}<^>{{\rm{mass}}}$, $\chi _{\rm{2}}<^>{{\rm{mass}}}$). We evaluate the reconstructed contributions against residual geodetic observations from satellite laser ranging and Earth orientation parameters, after removing atmospheric, oceanic and glacial isostatic adjustment effects. The combined hydrospheric and cryospheric reconstructions reproduce both the secular trends and annual cycles of the residual observed J2, $\chi _{\rm{1}}<^>{{\rm{mass}}}$ and $\chi _{\rm{2}}<^>{{\rm{mass}}}$ series. Terrestrial water storage dominates the seasonal variability of J2, $\chi _{\rm{1}}<^>{{\rm{mass}}}$ and $\chi _{\rm{2}}<^>{{\rm{mass}}}$, whereas accelerated ice-mass loss from Greenland and Antarctica controls the secular trend, with mountain-glacier mass loss also contributing to the long-term trend in J2. The resulting polar motion excitation drift closely matches the residual geodetic estimate in both magnitude and direction, indicating that contemporary climate-driven mass redistribution can largely account for recent changes in residual geodetic observations, and demonstrating the value of fingerprint-based reconstructions for monitoring climate impacts on the Earth system.

This paper presents a novel hybrid method named HW-EZ for short-to medium-term forecasting of ΔLOD and UT1-UTC up to 90 days ahead. This method relies on the LOD time series with zonal tidal signals removed, the adaptive Holt-Winters (HW) additive algorithm, and the effective angular momentum (EAM) functions derived from global atmospheric, oceanic, and land water models. The dominant ΔLOD variations are estimated using EAM and tides, while the remaining ones are modeled by HW-EZ with an 8-year sliding window. Numerical validation shows that HW-EZ reduces UT1-UTC mean absolute error (MAE) by 13.77%–67.55% relative to IERS Bulletin A over 1–30-day forecasts. Using 6-day EAM forecasts maintains nearly equivalent accuracy. Comparisons with the Second Earth Orientation Parameters Prediction Comparison Campaign (2nd EOP PCC) confirm that HW-EZ outperforms the average of more than 50 international methods and is competitive with state-of-the-art schemes. The HW-EZ method achieves robust ultra-short-term performance with UT1-UTC MAE ≤0.158 ms for 1–6-day forecasts, making it suitable for real-time geodetic and navigation applications.

Traditional Very Long Baseline Interferometry (VLBI) faces limitations in providing comprehensive geodetic products due to its reliance solely on ground-based observations of extragalactic radio sources. This study investigates the feasibility and benefits of a novel three-level integrated VLBI observation mode that combines quasar, satellite, and space VLBI observations to enhance geodetic parameter estimation. We conducted Monte Carlo simulation using GALILEO, BDS, GPS satellites and ETALON-1/-2 satellites as the Medium Earth Orbit space segment platform, integrating space and satellites VLBI observations with existing continuous geodetic VLBI campaign schedules (CONT17). Our analysis focused on assessing the impact of space VLBI observation quality and quantity on derived geodetic parameters, particularly station positions and Earth Rotation Parameters (ERPs). Results demonstrate that space VLBI observation could perform comparably to satellite VLBI for geodetic references due to providing superior geometric configuration. The integration of space VLBI into combined quasar-satellite schedules, yielded a 10% improvement in station position quality. Assuming 300-picosecond precision for space VLBI observations and GALILEO satellite scans replacing every fourth quasar scan, the derived ERP precision achieved 23.2 las (xp), 20.4 las (yp), 1.5 ls (dUT1). Space VLBI observations may offer the potential to improve the positioning of satellite orbital planes within the celestial reference frame by providing additional spatiotemporal information and broader sky coverage. These observations can also help compensate for the limited arc length of satellite VLBI tracking, particularly when the visible segment of the satellite orbit is short, thereby contributing to amore robust reconstruction of the orbital geometry. This integrated approach demonstrates substantial potential for advancing space geodetic reference frame precision and establishing more consistent space ties between geodetic techniques. (c) 2026 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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

Constraining the physical properties of Earth's deep interior, particularly the viscosity of the solid inner core and the density structure of large low-velocity provinces (LLVPs), remains a major challenge in geophysics. Here we develop a unified dynamical framework that combines mantle-inner core gravitational coupling (MICG) with torsional oscillations in the fluid outer core and show that their interaction can produce a distinct and testable geodetic signature. Guided by this prediction, we analyze satellite gravity observations together with independent corrections for surface mass variability. We identify a robust approximately 6-year signal in the Stokes coefficient Delta S22, while no corresponding stationary signal is detected in Delta C22. A signal with the same periodicity is independently detected in length-of-day variations (Delta LOD), and the two signals exhibit a near anti-phase relationship. Interpreting this coupled signature within the proposed framework allows us to constrain the inner core viscosity to approximately 4.6 (+/- 1.8) x 10^16 Pa s and the equatorial relief of the inner core boundary to a semi-axis difference of about 200 +/- 70 m. The inversion further indicates mean density anomalies of +5.5 (+/- 0.6) per mil at the base of LLVPs. These results indicate that satellite gravimetry provides a direct observational window into deep-Earth dynamics and the physical properties of Earth's deep interior.

We propose a novel initialization and online spatial-temporal calibration method for visual-inertial odometry (VIO), which decouples rotation and translation estimation to achieve higher accuracy and better robustness. Existing initialization methods suffer from limited accuracy or robustness (e.g., in scenarios with small translational motion) and rarely integrate simultaneous spatial-temporal calibration during initialization, despite its considerable practical value. Our proposed method leverages rotation-translation decoupling constraints to enable simultaneous estimation of gyroscope bias, extrinsic rotation, and camera-inertial measurement unit (IMU) time offset-even under pure rotational motion. Moreover, we are the first to conduct observability analysis on rotational constraints in rotation-translation decoupling methods, experimentally identifying the unobservable state-space directions under three degenerate motions within our approach. We also perform extensive experiments to delineate practical parameter solution boundaries for our method, with both efforts substantially enhancing the overall practical applicability of decoupling-based methods. Extensive experiments on simulated and real-world datasets demonstrate that our method outperforms state-of-the-art approaches in accuracy and robustness while maintaining computational efficiency. Furthermore, experiments verify that it significantly improves convergence in VIO systems.

Satellite altimetry-derived and shipborne gravity anomalies exhibit complementary spectral characteristics and spatial resolutions. This study develops an iterative radial basis function (RBF) framework to refine regional marine gravity fields through synergistic fusion of these datasets. Using the Poisson kernel RBF with the spatial and spectral localization properties, we first construct an initial regional gravity model. This model then detects/removes outliers and corrects systematic errors in shipborne data. The updated shipborne measurements are reincorporated into an improved RBF model, with the process iterating until the model residuals converge. Evaluated in the Ogasawara Islands and central-eastern Philippine Sea regions, our iterative RBF approach demonstrates significant accuracy gains over least-squares collocation (LSC) and noniterative RBF methods. The accuracy improvements are 53.14% (versus noniterative RBF), 28.59% (versus DTU17), and 16.68% (versus SS 32.1) in the Ogasawara region, and 16.79% (versus noniterative RBF), 40.23% (versus DTU17), and 40.56% (versus SS 32.1) in the Philippine Sea region. Power spectrum and coherence analyses confirm the RBF model enhances signal content within the 817 km wavelength range, reducing the effective resolution from similar to 17 to <10 km. This demonstrates the framework's capability to resolve finer-scale ocean gravity structures by combining the satellite altimetry-derived and shipborne gravity anomalies critical for geophysical and oceanographic applications.

Coastal zones are closely associated with human activities and socio-economic development. Satellite altimetry is a key technology for coastal sea-level monitoring owing to its global coverage and all-weather capability. Synthetic Aperture Radar (SAR) altimetry has become an important tool for coastal sea-level monitoring due to its enhanced along-track resolution (~300 m). However, the large cross-track footprint (~15 km) allows land contamination and complex nearshore scattering to distort radar echoes, leading to significant deviations from standard ocean waveforms. Waveform retracking is an effective approach for mitigating these effects. Nevertheless, existing retrackers typically rely on independent single-waveform or 20 Hz along-track processing, which has limited effectiveness under complex coastal conditions. Furthermore, commonly used retrackers often require leading-edge fitting to obtain initial significant wave height (SWH) estimates, which is sensitive to leading-edge distortions. As a result, sea surface height (SSH) measurements within 5 km of the coastline, especially in the nearshore zone within 3 km, suffer from low data availability and reduced precision. To address these limitations, we propose a spatiotemporal continuity-constrained multi-parameter sub-waveform retracker (MulPOS-C), based on the multi-parameter optimized sub-waveform (MulPOS) determination framework. MulPOS-C organizes repeated-cycle altimetric observations into a two-dimensional (2D) grid (along-track bin × cycle) and identifies optimal sub-waveforms independently along the spatial and temporal dimensions. A spatiotemporal continuity function is then introduced to reconcile directional discrepancies, yielding a consistent set of optimal sub-waveforms across the 2D domain. Sea surface height (SSH) is subsequently retrieved using a fixed power threshold to define the start and end gates of the water-related sub-waveform. This strategy ensures physical consistency and controllable errors relative to full-waveform physical retrackers, while avoiding the instability associated with initial significant wave height (SWH) estimation. Global validation against four retrackers demonstrates that MulPOS-C substantially improves both data availability and precision. Within 1 km of the coastline, 81.8% of stations achieve the lowest RMSE among all methods. At 1 km offshore, the RMSE and SSH noise are reduced to 9.4 cm and 10.6 cm, respectively, representing reductions of at least 4.3 cm in RMSE and 3.5 cm in SSH noise compared with the four retrackers. At 2 km offshore, the RMSE decreases to 7.7 cm and the proportion of gross errors is reduced to 1.8%, significantly outperforming the other retrackers, with all metrics approaching open-sea levels. At 3 km offshore, all metrics reach open-sea levels. In addition, MulPOS-C is methodologically generic and can be readily extended to conventional Low-Resolution Mode (LRM) altimeter data.

The Gravity Recovery and Climate Experiment and its Follow-On mission have revolutionized the monitoring of terrestrial water storage dynamics since 2002, offering monthly global mass transport data at similar to 300 kilometers resolution. Nevertheless, their temporal resolution remains insufficient to resolve rapid mass fluctuations triggered by extreme hydrometeorological events. Existing methods cannot achieve reliable high spatiotemporal resolution in the absence of external data. Here, we propose a spatiotemporally regularized framework to recover global daily mascon solutions from line-of-sight gravity differences, independent of a priori information, at an effective spatial resolution of similar to 334 kilometers. Our daily results accurately capture extreme flood signals in South Asia (July 2019), Australian (March 2021) and flood storage process at the Three Gorges Dam (July 2020), highlighting its role in attenuating peak flows. This advancement establishes a paradigm for model-independent, high-cadence mass change monitoring and analysis, with applications in extreme event attribution and model validation.

Satellite altimetry constellations offer immense potential for high-resolution marine gravity field recovery. This study investigates the performance of constellations in gravity recovery and addresses key challenges, employing a simulation framework that models unique inter-satellite systematic noise while addressing short-wavelength deficiencies in current sea surface models. Our findings reveal a distinct nonlinear accuracy gain that eventually saturates with increasing constellation size, primarily due to enhanced valid spatial coverage. Expanding the constellation across 2, 5, 10, and 15 satellites increases coverage from 70.65% to 96.19%, yielding a precision improvement of approximately 0.3 mGal. However, the impact of ocean variability escalates with increasing constellation size, constraining further accuracy improvements. To mitigate this, a mask-based adaptive regional filter approach is employed to suppress time-varying signals, which effectively eliminates the strip-like and patch-like residuals induced by ocean variability. Furthermore, orbit convergence at high latitudes reduces cross-track spacing, amplifying cross-track observation noise and degrading gravity recovery accuracy. This is addressed by an adaptive gradient calculation strategy, improving the east component of vertical deflections by ~2 μrad. Validation of short-wavelength features demonstrates that the optimized constellation gravity recovery achieves a resolution comparable to the SWOT mission while providing superior spatial coverage uniformity. This research offers insights into gravity field recovery using altimetry constellations and has certain implications for the design of future constellation missions.

Abstract. River discharge provides one of the clearest measures of how Arctic freshwater systems respond to climate change, yet long-term and continuous monitoring remains challenging. Here we present a gauge-based monthly natural discharge reconstruction dataset for 94 gauging stations in the Eurasian Arctic from 1952 to 2025. The dataset was generated using a basin-scale framework driven by precipitation and air temperature and aided by snow information. Terrestrial water storage (TWS) serves as the key link in the framework. Historical TWS was first reconstructed using an improved state-update empirical model constrained by TWS observations from the Gravity Recovery and Climate Experiment (GRACE), and monthly natural discharge was then estimated from reconstructed TWS using a refined runoff–TWS relationship calibrated against gauge discharge observations. The reconstructed monthly natural discharge agreed well with observations at minimally regulated gauges and during pre-regulation periods at regulated gauges, with median monthly Kling–Gupta efficiency (KGE) values exceeding 0.88. When monthly values were aggregated to annual discharge, the reconstruction still captured observed variability at minimally regulated gauges, with a median annual KGE of 0.73. The dataset generally agreed better with gauge observations than discharge estimates derived from existing GRACE-like TWS products and an existing gridded runoff reconstruction product. Example applications illustrate spatially heterogeneous trends in annual discharge and seasonal allocation over the past seven decades and show how the dataset can be used to assess the effects of human regulation on river discharge. This dataset provides a long-term natural-discharge baseline for characterizing historical discharge variability and assessing the roles of climate variability and human regulation in poorly monitored Arctic basins. The reconstructed monthly natural discharge dataset is available at https://doi.org/10.5281/zenodo.21157816 (Liu et al., 2026).

While numerous studies have been dedicated to the determination of Chandler wobble parameters, namely its period TCW and quality factor QCW, most of their results lack self-consistency. We proposed a rigorous approach to achieve self-consistent estimates of TCW and QCW using only geodetic excitations (χobs) derived from observed polar motion (PM) data in 2023 (denoted as CCRLL). However, a study published in 2025 (denoted as YF) criticized CCRLL's approach and argued that either TCW or QCW cannot be optimized without access to geophysical excitations (χgeo), relying on their misleading numerical simulations using real-valued PM transfer functions. Our present study has established theoretical relations between (TCW, QCW), degree-2 Love numbers and complex-valued transfer functions while proving that traditional Earth's rotation theories, employing real-valued transfer functions, are incapable of providing self-consistent estimates of TCW and QCW. This study also demonstrates that either χobs or χgeo can be used to invert for TCW and QCW, but χgeo must be recalculated using the complex transfer functions derived by this study and χobs-based estimates are more reliable. Thus, CCRLL's approach and results are validated while YF's arguments are refuted. Further, some updates and corrections to CCRLL's Love numbers and transfer functions are also presented.

Potential theory provides rigorous solutions to geodetic boundary value problems involving Laplace's equation, often expressed through global integral transformations. However, the practical application of these solutions is hindered by the incomplete availability of highresolution gravity data. This limitation necessitates the separate evaluation of far-zone contributions-commonly referred to as truncation errors. These errors are typically characterized using truncation coefficients, whose computation via recurrence formulas becomes numerically unstable at high altitudes and degrees when using double precision. This study presents a robust and comprehensive method for accurately computing truncation coefficients for the Poisson, Hotine, and Stokes integrals across all degrees and altitudes. The proposed approach integrates three specialized strategies: (1) an asymptotic expansion for high-degree terms to ensure both accuracy and efficiency; (2) a two-regime altitude model for low-degree terms, with binomial expansions applied at low altitudes and stabilized recurrence series used at high altitudes; and (3) double-precision numerical experiments that validate the method, showing relative errors below 10-5 for the Poisson integral and below 10- 8 for the Hotine and Stokes integrals. These results demonstrate that the method is both accurate and reliable for geodetic applications at all altitudes-from airborne surveys to satellite missions-and across the full range of spherical harmonic degrees.

Geocentre motion, defined as the displacement of Earth's centre of mass relative to its centre of figure, is crucial for maintaining the International Terrestrial Reference Frame origin and quantifying large-scale mass redistribution. However, whether observing geocentre motion by tracking satellite orbits or inferring it using geophysical models, accurately acquiring such subtle motions imposes stringent requirements on the consistency and precision of both tracking data and geophysical models. This study improves geocentre motion estimates derived from the combination of GRACE/GRACE-FO time-variable gravity (TVG) and Ocean Bottom Pressure (OBP) models (the GRACE-OBP method) in two ways. First, we apply a forward modelling technique to mitigate land-ocean leakage in GRACE/GRACE-FO TVG fields, which demonstrably outperforms empirical coastline buffer-zone corrections in controlled simulation experiments. Secondly, we introduce the Bayesian Three-Cornered Hat (BTCH) method to optimally combine geocentre series derived from multiple GRACE solutions and two independent OBP models (Estimating the Circulation and Climate of the Ocean, Phase II and Max Planck Institute Ocean Model), producing an improved geocentre product without requiring a ground-truth reference. Uncertainty analysis shows that the noise level is governed primarily by the GRACE solution, and that BTCH provides a clearer advantage over equal-weighted averaging when the number of input series is limited, reducing the noise level by about 30 per cent. After restoring atmospheric and oceanic contributions, our improved geocentre series shows good agreement with the CSR Satellite Laser Ranging-derived geocentre product. Although uncertainty levels vary among individual solutions, the estimated annual and secular trend signals are broadly consistent and show limited sensitivity to the choice of GRACE TVG solution and OBP model. Using the improved geocentre series, we revisit the annual geocentre oscillation and its drivers; the results indicate that cryospheric mass variability and land-ocean mass exchange (i.e. sea-level fingerprints) provide non-negligible contributions to the annual geocentre cycle and improve consistency with observations. Finally, the improved geocentre series yields the lowest uncertainty in degree-1 mass variations, with a global RMS of 0.55 mm. Incorporating these degree-1 terms into mass budget assessments yields secular trends of 38.8 Gt yr-1 for the Antarctic Ice Sheet and 0.57 mm yr-1 for global mean ocean mass, highlighting the need for accurate geocentre corrections to support reliable long-term climate monitoring.

The deflection of the vertical (DOV) carries abundant gravitational information and serves as a critical input for gravity anomaly derivation, rendering it a core topic in satellite altimetry research. The surface water and ocean topography (SWOT) mission, a new-generation satellite altimetry program, delivers 2-D wide-swath sea surface height (SSH) data via its unique interferometric measurements. While DOV retrieval from along-track and cross-track observations is well established, incorporating additional directional observations into the inversion model remains optional. To optimize multidirectional DOV estimation, we propose a novel multiple directions least squares adjustment (MD-LSA) method, and conduct case studies in a part of the Philippine Sea. Unlike conventional LSA, which employs a fixed set of observation equations, MD-LSA selects an optimal equation subset for each grid point based on predefined criteria, thereby enhancing DOV retrieval accuracy across heterogeneous regions. Simulation results demonstrate that MD-LSA outperforms traditional LSA across varying prior information accuracy levels. Specifically, with random-error-only observations and high-quality prior data, it achieves 64.5% and 54.0% accuracy gains for the north and east DOV components, respectively, relative to the two-direction solution. Real-data experiments verify that MD-LSA achieves the global optimum across different survey lines, with accuracy improved by 27.9% over two-direction LSA and 55.9% over least squares collocation (LSC). Furthermore, in the nearshore zone, MD-LSA residuals exhibit a near-zero concentrated distribution, reflecting its superior capability to retrieve nearshore gravitational signals. These findings confirm that MD-LSA well leverages the 2-D observational capability of SWOT SSH data, mitigates random errors, and accurately recovers coastal marine gravity field signals.

For decades, constructing a three-dimensional density model of the Earth has remained a major challenge, and whether the large low-velocity provinces (LLVPs) at the base of the lower mantle possess positive or negative relative density is still debated. In contrast to conventional seismological approaches, this study proposes a method for recovering large-scale lower-mantle density anomalies using satellite gravity disturbing potentials. In the regularized inversion framework, global seismic tomography models are incorporated as prior structural constraints, together with additional regularization terms that minimize the norms of model parameters, their spatial gradients, and a depth-weighting function. A simplified synthetic model containing both deep and shallow density anomalies was first constructed based on the SMEAN tomography model, and forward-inverse numerical experiments were performed. The results show that, compared with gravity anomalies, disturbing potentials provide more stable inversions for deep mantle structures, validating the effectiveness of the proposed method. Subsequently, the first 20-degree spherical harmonic coefficients of the satellite gravity model GOSG02S were used to compute the disturbing potential at 250 km altitude. After removing the effects of topography, crustal density variations, and Moho undulations, the residual disturbing potential was obtained. An averaged tomography model derived from multiple seismic models was then used to delineate the regions of density anomalies, which served as spatial constraints in the inversion. Using the regularized inversion scheme, a global three-dimensional mantle density anomaly model was estimated with a horizontal resolution of 1 degrees x 1 degrees and a vertical resolution of 100 km. The results indicate that the LLVPs exhibit overall positive density anomalies, particularly near their bases. The anomalies decrease upward, ranging from 10.2 kg.m(-3) to 2.8 kg.m(-3) beneath Africa and from 7.4 kg.m(-3) to 3.35 kg.m(-3) beneath the Pacific. In the lower two-thirds of their vertical extent, the mean density anomaly is approximately 7 kg.m(-3), consistent in magnitude with findings from tidal tomography and other geodynamic studies. In addition, between 1000 similar to 2900 km depth, the ratio of density to seismic-velocity anomalies ranges from-0.4 to 0, supporting the hypothesis that the LLVPs are influenced by both thermal anomalies and compositional heterogeneity.

This paper proposes a method to recover time-varying gravity signals by using relative orbit determination technology between satellites in giant constellations. This study focuses on two relative orbit determination observation modes and conducts a closed-loop simulation experiment with three constellation configurations: A (2 orbital planes, 400 satellites), B (20 orbital planes, 400 satellites) and C (20 orbital planes, 4000 satellites), for recovering the time-varying gravity signal for the HIS (Hydrology, Ice, Solid Earth). The performance was analyzed for seismic and land water monitoring, and compared with the results from the GRACE/Bender observation modes. The closed-loop simulation accounts for error sources like observation noise, tidal model errors and de-aliasing model errors, with relative orbit determination accuracy for giant constellations set to 1 mm. The recovery accuracy of HIS signals across four 7-day periods (before and after the 2004 Sumatra earthquake) was compared for different configurations. The results show the following: (i) Relative orbit determination technique significantly improves the recovery accuracy of middle-to-high-degree (high-frequency) time-varying gravity signals compared to absolute orbit determination. The C constellation demonstrates the largest improvement, with accuracy enhancements of up to 5 similar to 7 times. (ii) The accuracy of the recovered time-varying gravity signal using relative orbit determination in the C constellation is superior to that of GRACE in the degree range of 10 to 50, and about 2 times better than GRACE at degree 20. It also compares favorably with Bender at low degrees. (iii) Spatial domain analysis shows that relative orbit determination in constellation C effectively recovers HIS signals up to degree 30. It can also monitor time-varying gravity signals caused by large earthquakes (such as the Sumatra earthquake), and monitor hydrological and glacial signals in multiple regions, with precision superior to GRACE but slightly inferior to Bender. The research demonstrates that a giant satellite constellation has significant potential for monitoring changes in Earth's gravity field using relative orbit determination, and can be a valuable technical approach for future geoscientific applications in giant satellite constellation missions.

The Newtonian volume integral is the fundamental mathematical framework for modeling the gravitational effects of mass bodies with arbitrary geometry. Tesseroids provide a natural discretization of spherical shells and are therefore well suited for global-scale applications. However, the curved geometry of tesseroids makes it difficult to derive analytical solutions for the Newtonian integral. Numerical approaches, such as Taylor expansions and Gauss–Legendre quadrature, suffer from two major limitations: near-zone errors and polar-region effects, the latter being closely related to the choice of integration elements. In this study, we derive linearly analytical solutions (LASs) for the gravitational effect of tesseroids by applying linear approximations to both the integrands and variables, based on multiple integration elements with and without the cosφ^' term. The main results are as follows: (1) With respect to the integration elements, the cosφ^' term preserves the meridian convergence effect, and both numerical and analytical methods constructed with integration elements containing this term are inherently free from polar-region errors. (2) For Newtonian integration with arbitrary-order polynomial density, numerical integration does not introduce additional errors; in particular, the error does not increase with the polynomial degree. (3) We characterize, for the first time, the radial extent of a tesseroid—defined by its latitudinal and longitudinal coordinates—as a composite function, and derive LAS for the Newtonian integral over a non-rectangular domain. This formulation enables accurate modeling of tesseroids with inclined top or bottom surfaces. Building upon previous developments of LAS in spherical polar coordinates, this study establishes a solid mathematical foundation for modeling tesseroid gravitational effects using linearly analytical expressions. It also advances the understanding of Newtonian integration under different integration elements and represents the first attempt to formulate quadrature rules for numerical integration when the integration elements include the cosφ^' term.