BeiDou Global Navigation Satellite System (BDS-3) PPP-B2b service greatly promotes the real-time precise point positioning applications. However, previous research works have revealed that a nonnegligible satellite-specific clock bias (SCB) exists in the PPP-B2b clock offset, causing the degradation of PPP-B2b positioning performance. In this paper, a single-station-augmented PPP-B2b considering the

The prolate spheroidal harmonic series is a well-suited tool for gravity modeling of elongated bodies. In this work, the prolate spheroidal harmonic algorithms for forward modeling of the gravitational field of homogeneous polyhedral bodies are presented. The line integral forms of the prolate spheroidal harmonic coefficients are given using the Gauss divergence and the Stokes theorems, and the discontinuities

Most present-day bathymetric prediction solely addresses the linear mapping relationship between gravity signals and bathymetric data, disregarding nonlinearity’s effects, despite a probable nonlinear mapping between gravity signals and the seafloor topography. This paper investigates the consequences of excluding nonlinear terms in predicting bathymetry and reaches focused conclusions for different

Various vertical reference levels are used in the coastal zone, for various purposes. Being able to transform accurately and efficiently between them is of increasing interest since the need for seamless data over sea and land is growing due to sea-level rise, coastal engineering, and more frequent storm surges. We present a method for simultaneous calculation of models linking the ellipsoid, the geoid

Space missions to small bodies like asteroids, comets, and moons rely on physics-based simulations to test guidance and control systems. However, accurately modeling their gravitational fields is challenging due to their highly irregular shapes and limited knowledge of their internal structures, complicating orbit planning and landing maneuvers. This study presents a new approach to model realistic

Ionospheric sporadic E (Es) layers are thin layers with enhanced ionospheric electron densities (IEDs) which occur frequently in ionospheric E region. Previous detecting method based on ground-based global navigation satellite system (GNSS) observations can only obtain the horizontal maps rather than the vertical distributions and structures of Es layers. This study proposes a computerized ionospheric

GNSS integer ambiguity precision point positioning (IPPP) with satellite integer clock products is currently one of the most precise techniques for time and frequency transfer. However, a challenging issue that hampered the long-term performance of IPPP is the day-boundary discontinuity (DBD) that manifests at UTC (Coordinated Universal Time) midnights during the processing of multi-day GNSS observations

In recent years, the fundamental quantity of the gravitational field has been extended from gravitational potential, gravitational vector, and gravitational gradient tensor to gravitational curvature with its first measurement along the vertical direction in laboratory conditions. Previous studies numerically identified the near-zone and polar-region problems for gravitational curvature of a tesseroid

In urban canyons or natural valleys, diffraction effect occurs when the signal of Global Navigation Satellite Systems (GNSS) transmits to the edge of obstructions, such as buildings, trees, and slopes, resulting in large diffraction error, which is one of the important error sources in carrier-phase-based precise positioning. However, the theoretical formula derivation and numerical modeling of the

The paper presents 3D numerical modelling of the altimetry-derived marine gravity data with the high horizontal resolution 1 × 1 arc min. The finite volume method (FVM) as a numerical method is used to solve the altimetry–gravimetry boundary-value problem. Large-scale parallel computations result in the disturbing potential in every finite volume of the discretized 3D computational domain between an

Spectral gravity forward modelling delivers gravitational fields of mass distributions by evaluating Newton’s integral in the spectral domain. We generalize its spherical harmonic variant to 3D variable densities and to any integration radius. The former is achieved by expressing the density function as an infinite-degree polynomial in the radial direction with polynomial coefficients varying laterally

The Surface Water and Ocean Topography (SWOT) altimeter mission provides a significant opportunity to improve the accuracy of geoid gradients (GGs) and marine gravity fields. This paper presents a new method, namely LSA3, to determine the north and east (\(\xi \) and \(\eta \)) components of GGs from simulated and real SWOT data in the northern South China Sea. To fully use SWOT’s fine-scale sea surface

In this contribution, we introduce some new theory for the classical GNSS ambiguity function (AF) method. We provide the probability model by means of which the AF-estimator becomes a maximum likelihood estimator, and we provide a globally convergent algorithm for computing the AF-estimate. The algorithm is constructed from combining the branch-and-bound principle, with a special convex relaxation

This paper presents a study on depth modernization, paralleling height modernization for land elevations. Depth modernization integrates mean sea surface (MSS) models, ocean tide models, and precise ship positioning to achieve accurate seafloor depth measurements. Conventional methods rely on tidal corrections and chart datum from temporary tide gauges, which can be challenging in regions with complex

The gravitational field of a planetary body is most often modeled by an exterior spherical harmonic series, which is uniformly convergent outside the smallest mass-enclosing sphere centered at the origin of the coordinate system, known as the Brillouin sphere. The model can become unstable inside the spherical boundary. Rarely deliberated or emphasized is an obvious fact that the radius of the Brillouin

For a rapid retrieval of zenith wet delay (ZWD) and multi-global navigation satellite system (GNSS) precise point positioning (PPP) enhancement, a lightweight ZWD retrieval model was constructed by combining ground-based GNSS observations and precipitable water vapor (PWV) data provided by the European Center for Medium-Range Weather Forecasts Reanalysis (ERA5). The proposed model can rapidly produce

Global Navigation Satellite System (GNSS) products are an integral part of a wide range of scientific and commercial applications. The creation of such products requires processing software capable of solving a combined station position and GNSS satellite orbit estimation by least squares adjustment, also known as global GNSS processing. Such processing is routinely performed by the International GNSS

Errors in ocean tide and non-tidal atmospheric and oceanic models are among the largest error sources in gravity field recovery from space. We co-estimate corrections to these background models subject to uncertainty constraints during the adjustment procedure of gravity field spherical harmonic coefficients. Simulations are performed for the Mass-Change and Geoscience International Constellation to

The stochastic modelling of a finite mass distribution can provide a new perspective on the dynamic evaluation of time variable gravity fields. The algorithm for estimating variations of spherical harmonic coefficients implied by corresponding shape changes is implemented for the first-order derivatives of the gravitational potential. The described algorithm uses the spherical harmonic synthesis formula

The gravimetry measurements from the Gravity Recovery and Climate Experiment (GRACE) and its follow-on (GRACE-FO) mission provide an essential way to monitor changes in ocean bottom pressure (\(p_b\)), which is a critical variable in understanding ocean circulation. However, the coarse spatial resolution of the GRACE(-FO) fields blurs important spatial details, such as \(p_b\) gradients. In this study

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission, launched by the European Space Agency, provided high-quality gravitational gradient data with near-global coverage, excluding polar regions. These data have been instrumental in regional gravity field modelling through various methods. One approach involves a mathematical model based on Fredholm’s integral equation of the

While integer ambiguity resolution (IAR) enables GNSS to achieve real-time sub-centimeter-level positioning in open-sky environments, it can be easily hindered if the involved receivers are situated in areas with limited satellite visibility, such as in dense city environments. In such GNSS-challenged cases, commercial Low Earth Orbit (LEO) communication satellites can potentially augment GNSS by providing

Effective angular momentum (EAM) forecasts are widely used as an important input for predicting both polar motion and dUT1. So far, model predictions for atmosphere, ocean, and terrestrial hydrosphere utilized in Earth rotation research reach only 6-days into the future. GFZ’s oceanic and land-surface model forecasts are forced with operational 6-day high-resolution deterministic numerical weather

The stochastic representation of an uncertain shape model allows the dynamic evaluation of its induced gravity signal. This can be also applied for representing a time variable gravity field to model mass changes. The algorithm for estimating variations in gravitational potential is extended for the case of second-order derivatives. Two different harmonic synthesis formulas are used to derive the sought

The geodetic community commonly challenges the composite hypotheses in the statistical testing of mathematical models. Since the composite hypotheses are not specified as opposed to their simple counterparts, they require a prior estimation of the model parameters. However, if the mathematical models are ill-conditioned, the regularized estimation is often applied for the parameters of interest. Due

As a critical parameter in meteorological monitoring, precipitable water vapor (PWV) is widely used in short-term extreme weather forecasting and long-term climate change research. However, as PWV exhibits significant vertical attenuation, especially within 2 km, achieving accurate vertical interpolation is essential for comparisons and fusion across different measurement techniques, such as sampling

In February 2023, the International Laser Ranging Service started the tracking of additional medium Earth orbit satellites from the global BeiDou navigation satellite system (BDS) constellation, increasing the total number of tracked BDS satellites to 27. As an optical space geodesy technique, the Satellite Laser Ranging (SLR) provides another important measurement for BDS other than the microwave

Trends in the deep-ocean M\(_2\) barotropic tide, deduced from nearly three decades of satellite altimetry and recently presented by Opel et al. (Commun Earth Environ 5:261, https://doi.org/10.1038/s43247-024-01432-5, 2024), are here updated with a slightly longer time series and with a focus on potential systematic errors. Tidal changes are very small, of order 0.2 mm/year or less, with a tendency

Latest harmonic developments of the Earth tide-generating potential (TGP), e.g., HW95 (Hartmann and Wenzel in Geoph Res Lett 22:3553, 1995), RATGP95 (Roosbeek in Geophys J Int 126:197, 1996), KSM03 (Kudryavtsev in J Geodesy 77:829, 2004), include a number of terms caused by the joint effect of the Earth’s polar flattening (that can be numerically described by the \({J}_{2}\) geopotential coefficient)

Partial ambiguity resolution (PAR) has been widely adopted in real-time kinematic (RTK) and precise point positioning with augmentation from continuously operating reference station (PPP-RTK). However, current PAR methods, either in the position domain or the ambiguity domain, suffer from high false alarm and miss detection, particularly in challenging environments with poor satellite geometry and

The development of optical clocks has experienced significant acceleration in recent years, positioning them as one of the most promising quantum optical sensors for next-generation gravimetric missions (NGGMs). This study investigates the feasibility of retrieving the temporal gravity field via improved optical clocks through a closed-loop simulation. It evaluates optical clock capabilities in temporal

Lateral inhomogeneity in the Earth’s mantle affects the tidal response. The current study reformulates the expressions for estimating the lateral inhomogeneity effects on tidal gravity with respect to the unperturbed Earth and supplements some critical derivation process to enhance the methodology. The effects of lateral inhomogeneity are calculated using several real Earth models. By considering the

The regional sea level budget and interannual sea level changes around Taiwan and Philippines are studied using altimetry, GRACE, and in-situ hydrographic data during 1993‒2021. Results show that the average sea level trend around Taiwan and Philippines during 1993–2021 derived from the altimetric data is 3.6 ± 0.2 mm/yr. Over 2002–2021, the study shows closure of sea level budget in the eastern ocean

The Kalman filter stands as one of the most widely used methods for recursive parameter estimation. However, its standard formulation typically assumes that all state parameters avail initial values and dynamic models, an assumption that may not always hold true in certain applications, particularly in global navigation satellite system (GNSS) data processing. To address this issue, Teunissen et al

Numerical methods, like the finite element method (FEM) or finite volume method (FVM), are widely used to provide solutions in many boundary value problems. In previous studies, these numerical methods have also been applied in geodesy but demanded extensive computations because the upper boundary condition was usually set up at the satellite orbit level, hundreds of kilometers above the Earth. The

Vertical land motion and the redistribution of masses within and on the surface of the Earth affect the Earth’s gravity field. Hence, studying the ratio between temporal changes of the surface gravity \(\left( {\dot{g}} \right)\) and height (\(\dot{h}\)) is important in geoscience, e.g., for reduction of gravity observations, assessing satellite gravimetry missions, and tuning vertical land motion

LARES-2 is a new geodetic satellite designed for high-accuracy satellite laser ranging. The orbit altitude of LARES-2 is similar to that of LAGEOS-1, whereas the inclination angle of 70° complements the LAGEOS-1 inclination of 110°; hence, both satellites form the butterfly configuration for the verification of the Lense–Thirring effect. Although the major objective of LARES-2 is testing general relativity

For precise orbit determination (POD) and precise applications with POD products, one of the critical issues is the modeling of non-conservative forces acting on satellites. Since the official publication of Galileo satellite metadata in 2017, analytical models including the box-wing model and thermal thrust models have been established to absorb a substantial amount of solar radiation pressure (SRP)

Carrier phase integer ambiguities must be determined for BDS-3/inertial navigation system (INS) tightly coupled (TC) integration to achieve centimetre-level positioning accuracy. However, cycle slip breaks the consistency of the integer ambiguities. Conventional multi-frequency cycle slip methods use the pseudorange; thus, requiring improvement when applied to kinematic situations. Furthermore, a concise

This paper focuses on the retrieval of refractivity fields from GNSS measurements by means of least-squares collocation. Collocation adjustment estimates parameters that relate delays and refractivity without relying on a grid. It contains functional and stochastic models that define the characteristics of the retrieved refractivity fields. This work aims at emphasizing the capabilities and limitations

Deformation phenomena on Earth are inherently three dimensional. With SAR interferometry (InSAR), in many practical situations the maximum number of observations is two (ascending and descending), resulting in an infinite number of possible displacement estimates. Here we propose a practical solution to this underdeterminancy problem in the form of the strapdown approach. With the strapdown approach

Accurate information on satellite antenna phase center offsets (PCOs) and phase variations (PVs) is indispensable for high-precision geodetic applications. In the absence of consistent pre-flight calibrations, satellite antenna PCOs and PVs of global navigation satellite systems are commonly estimated based on observations from a global network, constraining the scale to a given reference frame. As

An attractive feature of PPP-RTK is the possibility of reducing the amount of data that needs to be transferred to users. By leveraging the state-space Representation (SSR) of the corrections, the correction provider (i.e., a GNSS network) can consider distinct transfer rates for each of the individual corrections according to their temporal characteristics. Reducing the transfer rates comes at the

We propose a benchmark for comparing gap-filling techniques used on global time-variable gravity field time-series. The Gravity Recovery and Climate Experiment (GRACE) and the GRACE Follow-On missions provide products to study the Earth’s time-variable gravity field. However, the presence of missing months in the measurements poses challenges for understanding specific Earth processes through the gravity

Inverting fault geometry and slip distribution simultaneously with geodetic observations based on Bayesian theory is becoming increasingly prevalent. A widely used approach, proposed by (Fukuda and Johnson, Geophys J Int 181:1441–1458, 2010) (F-J method), employs the least-squares method to solve the linear parameters of slip distribution after sampling the nonlinear parameters, including fault geometry

The 3D ionosphere structure is of interest in many fields such as radio frequency communication and global navigation satellite system (GNSS) applications. However, the limited temporal and spatial coverage of measurements poses a challenge for 3D electron density modeling. To overcome this challenge, we explore the use of kriging interpolation technique. The kriging interpolation is performed to obtain

The spherical radial basis function (SRBF) approach, widely used in gravity modeling, is theoretically surveyed from a viewpoint of random field theory. Let the gravity potential be a random field which is represented as an integral functional of another random field, namely an isotropic Gaussian random field (IGRF) on a sphere inside the Bjerhammar sphere with the SRBF as the integral kernel. When

In recent years, coseismic velocity from high-rate global navigation satellite systems (GNSS) carrier phase data has been widely utilized to estimate instrumental seismic intensity, thereby guiding earthquake early warning and emergency response. However, using carrier phase data only yields displacement, displacement increment, and average velocity but not instantaneous velocity at the epoch level

The current GNSS meteorology literature focuses on ground-based and space-based GNSS observations separately, without exploring potential synergies. In this study, we propose combining the two data sources using GNSS tomography to overcome current limitations in (1) horizontal resolution of GNSS space-based, (2) low vertical resolution of GNSS ground-based tropospheric retrievals when the number of

Global Navigation Satellite Systems (GNSS) applications require computation of the geometric range between the satellite vehicle at the time-of-signal transmission and the receiver antenna location at the time-of-signal reception. This computation requires attention to the frames of reference due to the rotation of the Earth-Centered Earth-Fixed (ECEF) frame during the time-of-signal propagation. Three