Monitoring time variable gravity fields requires a detailed modelling of mass changes. A stochastic technique is applied to evaluate the gravity signal implied by ice mass variations. The accuracy of the obtained results is analysed with respect to the volume of the modelled masses in order to establish a threshold up to which the stochastic procedure can be implemented. Five ice layers of a regularly monitored glacier in the Austrian Alps are modelled using gridded information provided by Digital Terrain Models of 51 m resolution. Specifically, a layer that describes the melted glacier volume between epochs 2009 and 2016 and four synthetic layers that simulate a uniform elevation change of 10 cm, 1 m, 10 m and 100 m are considered. In order to quantify the convergence behaviour of the involved series, the stochastic modelling technique is implemented by two strategies. Firstly by considering each ice layer as a single polyhedron and secondly by defining a sum of multiple elementary polyhedral masses, coupled to the definition of the gridded DTM values. Three additional algorithms are used for the comparisons of the computed gravity changes, namely the line integral analytical method of a general polyhedron, the analytical solution of a right rectangular prism and the corresponding spherical harmonic expansion due to a polyhedral source, with the line integral analytical method chosen as the reference solution. Up to a volume of 0.01 km3, the stochastic approach using multiple polyhedra provides more accurate results than all other algorithms, compared to the line integral analytical method, up to a 20 km altitude. For larger mass changes, both stochastic modelling techniques derived mean relative differences 1
We aim to evaluate the effect of geopotential coefficients of high degree and order on the Earth tide-generating potential (TGP), not considered before. First, general expressions are derived for calculating the TGP caused by a joint action of the external bodies’ gravity and an arbitrary spherical harmonic in the geopotential. The obtained formulae explicitly separate the TGP coefficients, that depend on masses and positions of external bodies (the Moon, the Sun, major planets), from the coordinates of an observation point at which the TGP is evaluated. At the next step, the TGP coefficients related to the external bodies are tabulated over 30,000+ years with a small sampling step and then developed to precision harmonic series. The modern long-term ephemerides of the Moon and major planets DE441 are taken as the source. The minimum threshold for amplitudes of the harmonic series’ terms is by an order of magnitude less than that used by the previous TGP catalogues HW95 and KSM03. For the first time, the terms caused by the geopotential coefficients of the third degree in the Earth TGP development are found. The number of terms originated by the second degree harmonics in the geopotential is more than doubled. Due to using the modern planetary/lunar ephemerides, the existing terms are updated. The obtained data are released in the form of a tidal potential catalogue in two formats: KSM25 (a modified KSM03) and HW95 (a generalized version). A new type of permanent tides on the Earth’s surface is revealed. It is originated by geopotential coefficients of order 1 (mainly, c31 and s31) and depends on not only the latitude of the observation point, like all previously known permanent tides, but on its longitude as well.
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.
As low-Earth orbit (LEO) satellites are increasingly equipped with highly precise Global Navigation Satellite Systems (GNSS) receivers, new opportunities for GNSS-based precise orbit determination (POD) arise. The availability of GNSS observations from satellites that are geometrically well distributed over the entire Earth may enable the formation of an independent GNSS network in LEO, which can be processed solely on the basis of double-difference (DD) observations. This study presents a first attempt to compute a network solution of GNSS DD observations in space that combines data from eight satellites, including GRACE-FO C/D, Sentinel-3A/B, Sentinel-6A and Swarm A/B/C. Using satellite laser ranging (SLR) validation, we show that the absolute position accuracy of an ambiguity-float network solution increases from about 4 cm 1D RMS in a constellation of three satellites to about 2 cm–2.5 cm in a constellation of eight satellites. Similar results are obtained by comparing the baseline lengths from our DD network solution with those derived from external ambiguity-fixed zero-difference (ZD) orbits, with differences at a level of 2 cm to 3 cm RMS for a network of eight satellites. For both the SLR residuals and the baseline consistency, an improvement of up to 0.5 cm can be achieved when resolving 60
The detection and analysis of the Earth’s free oscillations in the form of normal modes contribute to our understanding of the planet’s structure, dynamics, and seismicity. However, false and dismissed recognitions of the focused normal mode signals prevent the detailed exploration of the Earth’s interior due to the complex patterns of the normal mode spectrum. Here, we develop a deep-learning-based neural network, named ModeNet, which is capable of precisely and efficiently selecting the frequency windows to cover the target normal modes on noisy spectra. ModeNet achieves a remarkable precision rate in the discrimination between normal modes and noises of 98.1
Grazing Global Navigation Satellite System Reflectometry (GNSS-R) altimetry, while promising for precise surface height retrieval, faces challenges from tropospheric refraction. Beyond path delay errors, tropospheric refraction can cause an asymmetry in the incident and reflected signal paths due to tropospheric gradients and the asymmetric observation geometry. The impact of this refraction asymmetry on grazing GNSS-R altimetry remains unclear. This study systematically quantifies this impact and characterizes the resulting errors in grazing observations. We propose a new surface height retrieval model that accounts for refraction asymmetry and applies to both spaceborne and airborne scenarios. Through a 30-day global GNSS-R simulation, in conjunction with European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis data and a ray tracing technique, our results show that decimeter-level systematic bistatic delay biases can be observed in airborne scenarios (3.5 km), increasing exponentially with decreasing elevation angles. In contrast, these errors are millimeter-level in spaceborne scenarios (530 km). A comparison with the traditional height retrieval model reveals that uncorrected refraction asymmetry in airborne observations can introduce decimeter- to meter-level altimetric errors at extremely low elevation angles (2–7 degrees). This finding highlights the critical importance of accounting for the effects of refraction asymmetry on airborne grazing GNSS-R altimetry, especially for continuous coherent observations covering a wide range of low elevation angles.
This study presents a new astro-geodetic deflection of the vertical (DoV) dataset for the Netherlands, combining previously unpublished measurements from the digital zenith camera system TZK2-D with historical visual observations spanning more than a century. The TZK2-D observations have estimated accuracies ranging from 0.03″ at the ‘Sonnenborgh’ observatory in Utrecht to less than 0.1″ at the remaining stations and are in good agreement with independent historical visual observations. No statistically significant differences were detected (α = 0.05), confirming the high quality of both datasets. The combined dataset enables a quantitative comparison with the high-resolution gravimetric quasi-geoid model for the Netherlands. Using a tailored Least Squares 2D Bi-Cubic Spline Approximation (LS-BICSA), gravimetric DoVs were computed from the gridded quasi-geoid model. The observed and computed values (ξ, η) agree at the 0.2″ level (overall weighted RMS), which is an excellent result considering observational, model, and interpolation errors. Based on the distribution of standardized residuals, a mean standard error of approximately 0.08″ was derived for DoVs computed from the gravimetric quasi-geoid model, consistent with independent variance–covariance analysis. These findings underscore the enduring scientific value of both historical and modern astro-geodetic observations for the evaluation of quasi-geoid models.
Multipath and diffraction errors significantly impact the accuracy of precise point positioning (PPP), particularly in challenging environments. Traditional mitigation techniques mainly focus on the multipath, whereas the diffraction is often ignored. This paper proposes a Moran-index-driven classification and correction approach that integrates improved multipath hemispherical map (MHM) and sidereal filtering (SF) models to address both error sources. Specifically, the proposed method leverages a relative Moran index consisting of global and local Moran indices to classify multipath-dominated background grids and diffraction-like clustered residual grids according to their spatial clustering characteristics within the MHM framework. The designed edge- and aperture diffraction experiments show that diffraction-related residuals exhibit stronger local spatial autocorrelation than ordinary multipath in the MHM representation, although strongly clustered anomalies are not assumed to arise exclusively from diffraction. A refined MHM is then applied to grids that are suitable for stable spatial averaging, while a denoised window-matching SF model based on cross-correlation methods is used for grids exhibiting stronger local clustering, local irregularity, and poorer suitability for direct MHM averaging. To validate the effectiveness of the proposed method, two 48-h field experiments were conducted in edge and aperture diffraction environments, complemented by an 8-day real monitoring test. The results from both static and kinematic PPP experiments demonstrate significant enhancements in positioning accuracy, convergence efficiency, and overall reliability. Specifically, compared with the uncorrected and traditional methods, the proposed method achieves average 3-dimensional (3D) root mean square error (RMSE) reductions of 48.46
The development of space navigation and positioning technologies has led to increasing demands for the timeliness and accuracy of Earth Rotation Parameters (ERPs). The enhancement of real-time Global Navigation Satellite System (GNSS) data streaming services has made it possible to estimate ERPs in real-time. Therefore, a filtering method for real-time ERPs estimation is proposed to explore the performance of ERPs estimation by GNSS technology. An external Universal Time (UT1) constraint must be incorporated to ensure its determination. The experiments demonstrate that utilizing the UT1 forecast from the International GNSS Service (IGS) Ultra-rapid (IGU) products and implementing constraints at each processing epoch results in a stable and continuous UT1 estimation. Using the International Earth Rotation and Reference Systems Service (IERS) 20C04 post-processed products as a reference, the filtered ERPs solutions over a 30-day period exhibit Root Mean Square (RMS) values of 70 μ as , 73 μ as , 30 μ s , 215 μ as/day , 187 μ as/day, and 28 μ s for x_p , y_p , UT1, ẋ_p , ẏ_p, and dLOD , respectively. Compared to the real-time accessible IERS predicted products and IGU-ERPs products, the proposed method demonstrates a smoother time series and better agreement with the IERS 20C04 product. Moreover, the filter estimation of ERPs yields more continuous Global Positioning System orbit solutions than directly fixing ERPs from IGU-ERPs products, with RMS values of 2.67 cm, 1.62 cm, 1.68 cm, and 3.55 cm in the along, cross, radial, and 3D directions, respectively, when referenced to the IGS final orbit products. This performance shows a slight improvement in all directions compared to the results obtained from directly fixing ERPs from IERS 20C04 products.
Theoretically, it is possible to reconstruct the 3D distribution of water vapor by means of GNSS tomography using troposphere estimates from a network of GNSS stations, i.e., zenith delays mapped back into satellite directions. However, this technique is still limited by restricted satellite-to-ground observation geometry, a simplified parameterization of troposphere delays in the observation model and mandatory usage of constraints to stabilize the equation system. We propose an alternative approach, called STEPPP, in which a network of ground-based GNSS receivers is used to process GNSS observations, based on the Precise Point Positioning (PPP) technique and, instead of estimating the zenith wet delay and horizontal gradients, the estimation of the whole wet refractivity field in a grid space is performed simultaneously. Contrary to GNSS tomography, STEPPP operates on raw observation data instead of products. Values of wet refractivity at grid nodes are estimated from all stations simultaneously, i.e., they appear as common parameters in PPP. We present the functional and stochastic model of STEPPP, as well as first results of the model performance. We use GPS and Galileo dual-frequency observations generated by the Spirent simulator for 20 evenly distributed stations. The simulated observations are intentionally free of troposphere delays. However, we use a numerical weather model to retrieve reference profiles of wet refractivity and calculate slant wet delays, which are added to the simulated observations. We define a voxel space above the network of stations up to 12 km height and we recover the wet refractivity profiles by means of the STEPPP model. Several numerical experiments are performed using homogenous, inhomogeneous, constant and dynamic wet refractivity profiles. After a convergence time of a few hours, the STEPPP model accurately recovers all model states, including the 3D wet refractivity field and station coordinates.
The Gauss–Listing geopotential value W0 defines the fundamental equipotential surface used as the global vertical reference datum in modern height systems. In this study, the methodology of Dayoub et al. (J Geod 86:681–694, 2012) is revisited using contemporary global datasets and models. The analysis employs the latest mean sea surface (MSS) models DTU25 and CLS22, the mean dynamic topography (MDT) from the ECCO4 ocean state estimate, and the high-resolution global gravity field model XGM2019e_2159, all consistently reduced to the mean-tide system. Compared to earlier realisations, these datasets provide improved spatial coverage and accuracy, particularly in high-latitude and coastal regions, enabling a more robust global determination of W0. This update is timely given the availability of improved global MSS models and ocean state estimates, which enable a more consistent and comprehensive reassessment of W0. A global least‐squares adjustment of geopotential values computed from these datasets yields W0 = 62636853.8 m2s−2 at epoch 2003.0. An alternative computation using the normal gravity field formulation provides the same value of W0 and produces a new best-fitting reference ellipsoid with semi-major axis a0 = 6,378,136.886 m and semi-minor axis b0 = 6,356,751.887 m. Sensitivity analyses indicate that the estimated W0 is robust with respect to model selection and resolution, and that the use of the ECCO4 MDT yields results that agree with the MDT-independent formulation at the centimetre level. Random errors are negligible due to global, multi-year averaging and the large number of data points, while a realistic systematic uncertainty of ± 0.3 m2 s−2 (≈ 3 cm in height) is attributed mainly to altimeter calibration and the input models’ biases. The resulting ellipsoid, satisfying U0 = W0 provides a self-consistent surface for gravity field modelling and geoid computation, eliminating the need for a degree-zero correction. The updated W0 value lies between that of Dayoub et al. (J Geod 86:681–694, 2012) and the IAG conventional value. The result represents a physically consistent modern realisation of W0 and a refined reference ellipsoid suitable for contemporary geodetic applications.
Global Navigation Satellite System (GNSS) satellites are capable of ranging and communicating with one another through Inter-Satellite Links (ISLs), thereby enhancing system autonomy. ISL assignment can be formulated as a complex Constrained Multi-objective Optimization Problem (CMOP). It is critical to GNSS performance but is heavily constrained by compact satellite platforms and point-to-point link types. Prior research has predominantly focused on meeting communication and ranging requirements, often neglecting the load-balancing requirement within the system. To address these challenges, we propose the Constrained ISL-dependent NSGA-II (C-ISL-NSGA-II) method. This algorithm incorporates a novel Constrained Single-Link Rebuilding (CSLR) mutation operator and simultaneously optimizes ranging, communication, and load-balancing performance. Using the BeiDou constellation as a case study, we demonstrate that the CSLR mutation operator outperforms traditional Exchange Mutation (EM) and Displacement Mutation (DM) operators. It achieves faster convergence and superior solution quality across all three objectives. Additionally, we analyze the effects of crossover and mutation rates, finding that a configuration with a high mutation rate and a low crossover rate yields the best Pareto-set approximation for the BeiDou-3 constellation. An empirical parametric study indicates that employing 900 generations with a population of 100 individuals achieves a favorable trade-off among convergence speed, diversity, and computational cost. Compared with randomly generated initial solutions, the mean values of WAPDOP, SDDTD, and WAVWNL in the obtained Pareto-optimal solutions are reduced by 63
The development of techniques for the comparison of remote clocks plays an important role in metrology and fundamental science. Recent progress in the domain of ground-based and space-based atomic clocks requires important improvements in existing time and frequency transfer links. Here, we propose a satellite-based improved laser time transfer (ILTT) scheme based on existing satellite laser ranging (SLR) technique, which has been deployed on the Mengtian Laboratory Module of China’s Space Station (CSS). The ILTT payload comprises a single-photon avalanche diode (SPAD), a multi-level optimized optical system, an event timer, and a laser retroreflector array (LRA). A repetition rate of 10 kHz is adopted to improve normal-point precision. Laboratory experiments demonstrate that the time deviation (TDEV) of the entire ILTT system was better than 1 ps at 1 s and 0.6 ps at 100,000 s. In addition, we introduce a relativistic laser time transfer model and evaluate the impact of relativistic frequency shift terms on time transfer stability, meeting the requirements of the high-precision time–frequency rack (HTFR) mission aboard the CSS. In the on-orbit experiments, we have achieved state-of-the-art satellite-to-ground time synchronization with a precision of 2–5 ps at 1 s and a TDEV of 0.7 ps at 50 s. This approach is also sufficient for future time transfer links to geosynchronous and cislunar orbits.
With the continuous improvement of global satellite navigation system (GNSS), the increase in the number of available satellites and the limited computational capability of mass-market GNSS chips or smartphones require higher time-efficiency in parameter estimation. In this contribution, we leverage the inherent sparse coupling characteristics between parameters and observations in GNSS models to propose a variable-dimension (VD) update strategy. This approach enables high computational efficiency in GNSS data processing by effectively reducing redundant numerical operations. Furthermore, the proposed VD update strategy can be integrated into various sequential Kalman filter (KF) variants, such as the standard sequential KF (SKF), the square-root covariance filter (SRCF), and the U-D factorization filter (UDF). In this work, the VD update strategy is specifically integrated into the UDF to create a variable-dimension UDF (VD-UDF) implementation. To evaluate its performance, the VD-UDF is compared against the standard UDF, as well as the KF in information form (KF-I) and the square-root information filter (SRIF), both of which are widely adopted in GNSS applications. Numerical results and time-efficiency analyses demonstrate that the VD-UDF achieves superior computational efficiency relative to standard UDF, KF-I, and SRIF nearly without compromising numerical accuracy or stability. In summary, the optimized VD-UDF offers a highly time-efficient and robust solution for real-time high-precision GNSS positioning on platforms with limited computational resources.
We derive spherical harmonic coefficients for the external gravitational potential generated by a tesseroid with radially variable density. The Newtonian volume integrals required to compute the coefficients are decomposed into 2D surface integrals and 1D radial integrals. Analytical expressions are derived for the radial integrals corresponding to polynomial, exponential, and parabolic density functions, while an adaptive Gauss–Legendre quadrature is used to numerically evaluate the radial integrals for general smooth density distributions. Simplified expressions for the surface integrals are further obtained of special geometries, including spherical zonal bands, spherical sectors, and spherical shells. The derived formulas extend the tesseroid-based spectral technique for gravimetric forward modeling beyond the common assumption of constant density within each tesseroid. Benchmark tests indicate that our approach is reliable and offers a favorable trade-off of accuracy and efficiency. No pronounced polar-singularity and polar-region problems are observed, whereas a near-zone problem appears as evaluation points approach the source. This behavior is primarily attributed to truncation errors and can be mitigated by increasing the truncated harmonic degree, leading to good agreement with a spatial-domain double exponential quadrature method. Finally, we compute the gravitational field of global marine sediments with 3D variable density and provide the resulting potential coefficients model to support related geodetic and geophysical applications.
Real-time kinematic precise orbit determination (KPOD) is inherently a precise point positioning (PPP)-based method for estimating the precise position of low Earth orbit (LEO) satellites, which relies solely on the onboard global navigation satellite system (GNSS) observations without considering the dynamic model. The high precision and continuous tracking capability of GNSS enable excellent KPOD of LEO satellites. However, the performance of KPOD is severely degraded during ionospheric disturbances, as the geometry-free (GF) cycle-slip detection becomes less reliable and the standard observation stochastic model proves inadequate for characterizing scintillation signals. To address these challenges, two new metrics, the acceleration of total electron content (AOT) and the AOT index (AOTI) are defined in this study. AOT represents the second-order time derivative of the total electron content (TEC), while AOTI is the standard deviation of AOT. AOT and AOTI are proposed to address the unreliable cycle-slip detection and the inadequate stochastic model, respectively, during ionospheric disturbances. By eliminating the first-order variation of TEC, AOT-based GF cycle-slip detection method can prevent the over-detection of spurious cycle slips without increasing the detection threshold. When AOT remains constant over a short interval (e.g., 10 s), AOTI serves as a proportional proxy for phase observation noise, permitting its real-time, on-the-fly estimation. The implementation of AOT and AOTI yields a significant improvement in real-time KPOD of the Swarm-A satellite. During ionospheric disturbances over the last five years, the average improvements in the root mean square (RMS) values computed from orbit errors below the 95th percentile are 21.4
Accurate empirical tropospheric mapping functions (MFs) are essential for high-precision geodetic applications such as GNSS and VLBI. However, traditional MFs assume azimuthal symmetry or explicitly model gradients on regular grids, limiting their ability to represent real atmospheric conditions. To address the limitation, this study designs a novel physics-constrained machine learning-based framework implemented with a multilayer perceptron and develops global empirical asymmetric MFs on five years of ERA5-based ray-tracing data. Compared to the widely used GPT3 model, our models reduce global RMSE by up to 12.0
The time-variable gravity field solutions from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission are generally contaminated by the correlation errors, specifically parameter correlation (strong parameter coupling) and observation noise correlation (colored instead of white noise). In this context, we propose a decorrelation approach to pursue an improved time-variable gravity solution following a step-wise processing. The step-wise decorrelation approach comprises three steps, standard, parameter decorrelation, and observation noise decorrelation processes. First, the standard process serves to establish a reliable signal reference for the following. Then, the parameter decorrelation is implemented through the separate estimation of orbit and gravity field parameters. Finally, to achieve the goal of observation noise decorrelation, the post-fit residuals obtained from the result of parameter decorrelation are used to estimate a colored noise model, which is considered to determine the final gravity field model, specifically termed the step-wise decorrelation solution. The basic idea is that under the regularization constraints of separate estimation for dynamic parameters, the reduced dynamic parameter space allows certain low-frequency perturbative errors to emerge in post-fit residuals, enabling comprehensive characterization of observation noise correlation. Using this step-wise decorrelation approach, we process monthly GRACE-FO gravity field time series and evaluate the performance of them from the aspects of signal and noise. Spectral and spatial domain analyses of noise levels confirm significant noise suppression in the final solution. For instance, it achieves 66
In current conventional precise point positioning (PPP) processing strategies, the tropospheric zenith wet delay (ZWD) is usually dynamically estimated as a stochastic parameter. During the convergence period, ZWD estimates can appear to be negative or unrealistically large due to the low estimation precision, which adversely affects the estimation of other state parameters, especially the Up component of coordinates. To address this issue, we propose a method that incorporates physical constraints on ZWD in PPP processing. This method employs the inequality constrained least squares (ICLS), utilizing Karush–Kuhn–Tucker (KKT) conditions to add boundary conditions on ZWD. The boundary conditions of ZWD are calculated based on the relation between ZWD and relative humidity (RH). The use of physical constraints does not rely on external products or space state representation (SSR) corrections for ZWD during PPP processing and can improve the short-term accuracy of ZWD and coordinate Up component. The efficiency of this approach has been validated using GNSS data and products from GFZ operational networks. For real-time PPP solutions, there is a 30% improvement in short-term accuracy of Up component; for post-processing solutions, the short-term RMSE improvement is about 20% . After convergence, the ZWD upper bound is no longer applied as an ICLS constraint, but is instead used as a diagnostic indicator to identify ZWD anomalies. This indicator demonstrates high sensitivity and reliability under extreme weather conditions, highlighting its potential for application in meteorological hazard early-warning systems.
