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.

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.

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 global gravitational model can be expressed as a series of spherical harmonic coefficients computed up to a certain degree and order. The main ordering characteristic can depend on the degree, order, or type of coefficient. When determining the spherical harmonic coefficients using the least squares method, it is essential to analyze the ordering pattern of these coefficients and their positions (e.g., indices) in the vector or matrix. A systematic analysis on the type of coefficient arrangement is presented in this paper. Moreover, the index algorithm for each coefficient ordering pattern is provided. Additionally, the structure of the normal equation matrix with different coefficient arrangement patterns is analyzed. Based on the analysis and the algorithm presented in this paper: (1) we can calculate the index of each coefficient in the vector or matrix of the normal equation; (2) we can change the structure of the normal equation matrix of the Earth's gravity field from one type of coefficient arrangement to another. Furthermore, (3) we can directly combine different structures of the normal equation matrix, calculated from different types of gravity satellite missions, to form combined normal equations. This approach is beneficial for the determination of Earth's gravitational model using multi-type observation data.

Downward continuation of gravity data is a critical challenge in the practical applications of airborne gravity, such as global geopotential modeling and geophysical interpretation. This study introduces a machine learning algorithm to address this classical problem in physical geodesy. We first constructed a U-Net-based neural network, termed DWC-Unet, specifically designed for gravity anomaly downward continuation. The model was trained using gravity anomalies simulated by the prior global gravity model. Subsequently, the trained DWC-Unet was used on airborne gravity anomalies to calculate the downward continued gravity anomalies at an arbitrary altitude surface. We applied the proposed method to the airborne gravity data in Colorado, Iowa, and regions in the southeast coast of the United States. The experiment results confirm that the DWC-Unet method shows better performance than the analytic continuation algorithm. The standard deviation of downward continuation errors reached 4.39 mGal in Colorado and approximately 3 mGal in other regions when compared to terrestrial gravity data. This study demonstrates the good performance of applying machine learning to the calculation of gravity data continuation, serving as a reference for the application of artificial intelligence in the field of gravity data processing.

Summary Time-varying gravity fields play a crucial role in understanding and analyzing geodynamic processes, particularly the migration of matter across the Earth's surface. However, the current limitations in spatiotemporal resolution hinder their accurate representation. In this context, the use of a giant constellation of low-orbit satellites holds great potential for accurately recovering time-varying gravity fields with high spatiotemporal resolution. Based on the orbital parameters of 5199 satellites in 123 different orbital planes in the first phase configuration of the Starlink constellation and the orbital parameters of the Bender constellation in the next generation gravity mission, we conducted a closed-loop simulation to analyze the recovery ability of time-varying gravity field in 9 days using the short-arc integral method. The errors of aliasing AOHIS signal (Atmosphere, Ocean, Hydrology, Ice, and Solid Earth), ocean tide models, orbit positions, inter-satellite range rates, and accelerometer observations were considered in the numerical simulation. Compared with the Bender constellation, the Starlink-like constellation can effectively decrease the aliasing errors in the spatial- and frequency-domain when the observation noise is not considered. The Starlink-like constellation can also effectively improve the reliability of low-degree coefficients (below degree 15) of retrieved time-varying gravity field models and present higher time resolution (within 9 days) for the full degree spherical harmonic solutions than the Bender constellation when the observation noise is considered. The aliasing effect on the low-degree part of the Bender constellation can be significantly decreased by combining the Starlink-like and Bender constellations, and the accuracy of the recovered time-varying gravity field within degree 30 can be improved by about 0.5 ∼ 1 order of magnitude. Our results can provide a technical reference for the design of future gravity satellite mission.

The high-precision static satellite gravity field models have important applications in fields such as global ocean circulation research and global/regional digital elevation datum determination. In this paper, we discussed the determination of high-degree static satellite gravity field models with GOCE observation, GRACE observation and the joint of them. We first constructed a 300 degree Satellite Gravity Gradiometry (SGG) normal equation with the high-precision gravitational gradient components V-xx, V-xy, Vz(z) and V-xz throughout the entire mission of GOCE by the Direct least squares method, and a 130-degree Satellite-to-Satellite Tracking (SST) normal equation with the SST observation data by the point-wise acceleration approach. The 300-degree GOCEonly satellite gravity field model GOSGO2S is determined by combining SGG and SST normal equations with variance component estimation. The 180-degree model SWPU-GRACE2021S is then determined based on the dynamic approach with the GRACE data throughout the entire mission cycle of 15 years, and the normal equation is combined with GOCE normal equation to determine WHU-SWPU-GOGR2022S, a joint model of GOCE and GRACE. Finally, the XGM2019 model and GPS/leveling data are used for precision analysis of GOSGO2S, SWPU-GRACE2021S and WHU-SWPU-GOGR2022S in frequency domain and space domain respectively. The results show that the accuracy of GOSGO2S and WHU-SWPU-GOGR2022S is comparable to GO_CONS_GCF_2_ DIR_R6, GO_CONS_GCF_2_TIM_R6, GO_CONS_GCF_2_SPW_R5, G00006s and Tongji-GMMG2021S that use the entire mission data of GOCE satellite and the accuracy differences are in the order of millimeters. The SWPU-GRACE2021S model has the same accuracy below degree/order 160 as the international mainstream of GRACE satellite gravity field models, i. e., ITSG-Grace2018s and Tongji-Grace02s.

Abstract The Moho is the interface between crust and mantle, and accurate location of the Moho is important for both resource exploration and deep earth condition and structural change investigations. The theory of the traditional Parker‐Oldenburg (P‐O) method is quite simple and it is widely applied in the frequency domain of Moho depth inversion. However, Moho fluctuation simulations using the P‐O method are not reliable because of the lack of field geographic data constraints during the inversion process and the excessively smoothing of data details caused by using filters to correct the source data signals. To solve those problems, we propose an improved iteration P‐O method with a variable density contrast model, the iterative process is constrained by geological data in the inversion parameters, and the variable depth of the gravity interface is iterated using an equivalent form of upward continuation in the Fourier domain, which is more stable and convergent than downward continuation term in original P‐O method. Synthetic experiments indicate that improved method has the better consistency among the simulations than original method, and our improved method has the smallest root mean square (RMS) of 0.59 km. In a real case, we employed the improved method to invert for the Moho depth of the South China Sea (SCS), and the RMS between our Moho depth model and the seismological data is the smallest value of 2.27 km. The synthetic experiments and application of the model to the SCS further prove that our method is practical and efficient.

The contribution presents the representative research progress on global static gravity field modeling, regional geoid/quasigeoid determination, vertical datum study, as well as the theory, algorithm and software for gravity field study in China from 2019 to 2023, which are the highlights of the chapter 6"Progress in Earth Gravity Model and Vertical Datum"in the"2019—2023 China National Report on Geodesy"that submitted to the International Association of Geodesy ( IAG ) . In addition, suggestions are proposed to promote the research in the fields of earth gravity field, geoid/quasigeoid and vertical datumin China according to trends of international geodesy and related disciplines.

针对物理大地测量学中经典的向下延拓问题,该文尝试全球球谐分析和泰勒级数展开的思路实现航空重力异常的向下解析延拓.首先使用泰勒公式构建向下延拓的迭代结构,随后使用球谐分析方法分析了重力异常数据,并据此求解了其各阶导数,从而实现向下延拓的迭代计算.模拟实验的结果表明,在延拓高度为8000m,且加入了标准差为2mGal白噪声的情况下,模拟重力异常的向下延拓误差的均方差为5.58mGal,证明了该方法的可行性与有效性;使用该方法将台湾地区5156m高度处实测航空重力异常数据向下延拓至地面重力数据点处,向下延拓误差的均方差为9.37mGal,该结果与加入地形改正的最小二乘配置法延拓结果的精度相当.

Global geophysical networks provide powerful databases to infer globally coherent signals, and array processing techniques are useful for inferring them. In this study, we comprehensively analyze seven spherical harmonic-based array processing techniques: spherical harmonic stacking (SHS) as well as its gridded form (SHS_GT) and grid-interval weighted forms (SHS_GK1, SHS_GK2); matrix SHS (MSHS); multi-station experiment (MSE); and optimal sequence estimation (OSE). We first use more specific synthetic tests to evaluate the pros and cons of these techniques, and estimate bias in solutions caused by the station distributions. These methods are applied to four global observation networks, the Global Geodynamics Project (GGP) Network, the Global Seismographic Network (GSN), the Global Geomagnetic (GGM) Network and the Global GNSS Network. For the first time, we restored a much cleaner sequence for one singlet of the S-0(2) mode based on the GGP network, and restored similar result for one singlet of the S-3(1) mode based on the GSN network. We further isolate different Y-lm-related tidal signals from the GGM network for the first time. Moreover, based on global GNSS observations, we estimate the Love number h(21) = 0.6243(+/- 7e - 4) - 0.01(+/- 6e - 3)i at the Chandler Wobble (CW) frequency with OSE/MSHS (The accuracy of the estimate is an order of magnitude higher than the previous results), and further obtain the corresponding lower-mantle anelasticity (f(r)(omega) = 29.5 +/- 0.9, f(i)(omega) = 12.0 +/- 7.2). Our findings confirm that OSE and MSHS methods can more accurately obtain the complex amplitude of any Y-lm-related signal, which is not possible with other methods (and we do obtain more precise results than previous studies upon using them); besides, we also confirm that OSE and MSHS methods can greatly reduce the interference of other signals to the target signals. Hence, we believe the results obtained from the OSE/MSHS will helpful for obtaining reasonable geophysical explanations.

基于拉格朗日中值定理,改进使用重力异常垂直梯度进行重力异常向下延拓的方法,通过重力异常向上延拓后差分以及线性外推,得到更加准确的应用于向下延拓的垂直梯度近似值.首先介绍利用拉格朗日中值定理计算重力异常垂直梯度并进行向下延拓的基本原理,给出进行航空重力向下延拓的基本步骤;然后基于EGM2008全球重力场模型说明重力异常垂直梯度在0～10 000 m的空间范围内在垂向上与高度呈强线性关系,证明进行垂直线性外推的合理性;最后使用EGM2008模型的模拟重力异常在台湾地区进行实验,向下延拓精度达到2.34 mGal,验证方法的可行性与有效性,同时将台湾地区5 156 m高度面上的实测航空重力数据向下延拓到地面的3 360个地面重力观测点,精度达到10.13 mGal.

Based on a satellite constellation composed of two GRACE-type satellite formations with different inclinations (near polar orbit + low inclination) and the theory of repeat orbit cycle, we discuss the methods for selecting medium-low inclinations for global and local gravity fields. The effects of this constellation configuration on gravity field inversion are comparatively analyzed using a whole-course dynamics simulation. The results show that compared with the single GRACE-type satellite formation, the use of satellite constellations with different inclination configurations improves the gravity solution precision by 34%. The inclusion of multi-directional observations can improve the spatio-temporal resolution of the satellite missions, and yield gravity field solutions with higher isotropic sensitivity. Furthermore, it is necessary to select the optimal low inclination according to the study area, which will have a significant influence on the gravity field solution.

The Global Positioning System (GPS) derived bedrock displacements respond to multiple geophysical effects, ranging from surface elastic loads to tectonic sources or viscoelastic uplifts stemming from Earth’s viscous mantle. In this study, the GPS‐inferred vertical crustal velocities are rigorously estimated in mainland China. We integrate the GPS vertical velocity field with Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow‐On (GFO) data, adopting an empirical Spatial Structure Function (SSF), to image tectonic deformation in mainland China with respect to the International Terrestrial Reference Frame (ITRF) 2014. We present four profiles across China, which indicate that our new robust results are superior to kriging. Furthermore, we use the GRACE/GFO products to account for elastic deformation due to surface mass changes to isolate tectonic deformation signals at GPS sites within mainland China from 2002 to 2019. By integrating GPS and GRACE/GFO measurements, our results reveal the long‐term spatial patterns of vertical tectonic motion in different blocks in mainland China. We conclude that significant steep velocity gradients occur at tectonic block boundaries that are attributable to locking and elastic strain accumulation on active block boundary faults.

Abstract We develop an algorithm for a Moho depth recovery from gravity and gravity gradiometry data and apply this method to estimate the Moho depth beneath the Tibetan Plateau. The basic idea of this algorithm is to describe mathematically the Moho depth undulations in terms of a condensation layer with respect to a mean Moho depth, instead of applying more commonly used isostatic compensation schemes. Expressions that functionally relate gravity field quantities with a (Moho) condensation layer are derived in spectral and spatial domains. The main advantage of this algorithm is that a functional relation between gravity field quantities and surface density anomalies, and consequently Moho depth undulations, has a linear form. The proposed algorithm is tested using satellite gravity and gravity gradiometry data. The Moho depth (taken with respect to the geoid surface) estimates obtained based on applying this algorithm are validated against global and regional seismic Moho results at the study area of Tibet. We also compare the result with the corresponding Moho depth estimates obtained by applying the Parker–Oldenburg and Vening Meinesz–Moritz (VMM) methods. The validation shows that results from all three gravimetric methods are similar, and they also closely agree with a regional seismic Moho model. Nevertheless, the VMM method and our algorithm in this comparison overperform the Parker–Oldenburg's method. The analysis of results also reveals that the newly developed algorithm provides better result (in terms of the RMS fit with a regional seismic Moho model) when applied for a Moho determination from gravity gradiometry instead of gravity data.

Here we provide an alternative approach to determine the Earth's external gravitational potential field based on low-orbit target satellite (TS), geostationary satellites (GS), and microwave signal links between them. By emitting and receiving frequency signals controlled by precise clocks between TS and GS, we can determine the gravitational potential (GP) at the TS orbit. We set the TS with polar orbits, altitude of around 500 km above ground, and three evenly distributed GSs with equatorial orbits, altitudes of around 35000 km from the Earth's center. In this case, at any time the TS can be observed via frequency signal links by at least one GS. In this way we may determine a potential distribution over the TS-defined sphere (TDS), which is a sphere that best fits the TS' orbits. Then, based on the potential distribution over the TDS, an Earth's external gravitational field can be determined. Simulation results show that the accuracy of the potential filed established based on 30-days observations can achieve decimeter level if optical atomic clocks with instability of $1\times 10^{-17}\tau^{-1/2}$ are available. The formulation proposed in this study may enrich the approachs for determining the Earth's external gravity field.

In this paper, a new method for regionally improving global geopotential models (GGMs) with global navigation satellite system (GNSS)/levelling data is proposed. In this method, the GNNS/levelling data are at first converted to disturbing potential data with inverse Bruns’ formula. Then the systematic errors in disturbing potential data are removed with a three-parameter correction surface. Afterwards, the disturbing potential data on the Earth's surface are downward continued to the surface of an inner sphere with inverse Poisson's integral equation. Global disturbing potential data on the whole sphere could be achieved with combination of the downward continued data and the GGM-derived data. At last, the final regionally improved geopotential model (RIGM) could be recovered from the disturbing potential data using least-squares method. Four RIGM models for Qingdao (QD) are determined based on four different sets of GNSS/levelling data points to validate the capability of the method. The standard deviation of height anomaly errors of RIGM-QDs are nearly 25 and 30 per cent on average smaller than Earth Gravity Model 2008 (EGM2008) on checkpoints and data points, respectively. This means that the RIGM-QDs fit better to the GNSS/levelling network in this area than EGM2008. The results show that the proposed method is successful at improving GGMs in regional area with regional GNSS/levelling data.

Global ocean mass (GOM) change, which is mainly influenced by water outflow from land, has been identified as one of the important contributors to present‐day sea level rise. Here, we delineated GOM change by dividing the land into five parts, which are exorheic river basins, endorheic basins, polar ice sheets, and land glaciers and ice caps, by the Gravity Recovery and Climate Experiment mass concentration solutions, 2002–2017. We estimated their total contribution to GOM change to be 568.92 ± 21.43 GT•yr − 1 , which is statistically congruent with the Gravity Recovery and Climate Experiment estimated GOM change (579.46 ± 46.62 GT•yr − 1 ). The melting of Earth's ice reservoirs is the dominant contribution in the whole period. However, an abnormal increase is found in the trend of GOM change with an amplitude of approximately 100% since August 2008. Further finding reveals that melting of Antarctica Ice Sheets are increasing fast and significantly dominates among other geophysical contributions to the abnormal increase at 34.28%, following with the water storage change of exorheic river basins at 28.71% and the melting of Greenland Ice Sheets at 22.76%, which is postulated to be resulting from intensified ice wastages and precipitation during the last decade.

Recovering the gravity field with the satellite’s frequency signal might be an alternative measuring mode in the future when the accuracy of the onboard clock was good enough. On the one hand, we analyze the performance of recovering gravity field model from the gravitational potentials with different accuracies on different satellite altitudes (from 200 km to 350 km) based on semi-analytical (SA) method. On the other hand, we analyze the performance based on the numerical analysis. First, the gravitational potentials along the satellite orbit are computed from the clock observations based on the method of satellite’s frequency signal with the accuracies of 10-16 and 10-18s. Then, based on the derived gravitational potentials, we recovered the gravity field models up to degree and order 200 (corresponding to 100 km spatial resolution). At last, the errors of recovered models are validated by comparing with the reference model.