Journal of Geodesy

Multipath remains one of the major challenges in high-precision GNSS positioning. The multipath hemispherical map (MHM) based on satellites’ location repeatability in space is a popular method to mitigate GNSS multipath effects, but its performance depends on the availability of sufficient satellite orbital tracks in the skyplot. For instance, for BDS-3 medium Earth orbiters and Galileo satellites with 7-day and 10-day orbital repeat times, respectively, the skyplot of their orbital tracks will be too sparse to cover the shifting orbital tracks on the succeeding days, if only a few days of observations are used to construct MHMs. In this study, we establish an interoperable MHM using the overlap frequency signals of GPS, Galileo and BDS-3 (i.e., GPS L1/L5, Galileo E1/E5a and BDS-3 B1C/B2a). We compared the performance of GPS/Galileo/BDS-3 MHM (i.e., MP_GEC) and single-constellation MHMs (i.e., MP_G, MP_E and MP_C). The mean reduction rates of the L1/E1/B1C and L5/E5a/B2a carrier-phase residuals for the MP_GEC applied to GPS, Galileo and BDS-3 are 36% and 48%, respectively, which are 10–30% points larger compared to the MP_G, MP_E and MP_C. The MP_GEC constructed using 4 days of observations reduced the Galileo RMS positioning errors by 26%, 31% and 29% for the east, north, and up components, respectively, showing improvements of about 16, 18 and 17% points compared to the MP_E, and even approaching the RMS errors of the MP_E constructed using 10 days of observations. The results show that the interoperable GPS/Galileo/BDS-3 MHM is able to improve the spatial resolution, modeling efficiency and correction performance in mitigating multipath effects for high-precision GNSS positioning.

The ionosphere crucially impacts on Global Navigation Satellite System (GNSS) positioning accuracy and integrity. Recently some network-based methods have shown the potential to construct a regional/global vertical total electron content (VTEC) or grid ionospheric vertical delay (GIVD) map for accuracy augmentation purposes. However, how to use these advanced methods for integrity augmentation has not been adequately investigated. The authors have investigated a regional ionospheric integrity monitoring strategy based on the radial basis function neural network (RBF-NN), using GNSS TEC observations. Similar to the SBAS approach, the GIVD map is constructed so as to enhance positioning accuracy, and the corresponding grid ionospheric vertical error (GIVE) map is constructed for protection level calculation to enhance positioning integrity. To reduce the GIVD residuals and the GIVE values, the local ionospheric spatial activity index (LISAI) is proposed as an indicator of local ionospheric spatial activity level. The RBF-NN structure parameters are able to be adaptively determined via hierarchical clustering. Modeling results in the China region have verified that the proposed GIVD modeling method is slightly better than the classical WAAS-Kriging method. The proposed GIVE modeling method significantly outperforms WAAS-Kriging, achieving an improvement of around 46% and 25% during the ionospheric calm and active periods, respectively.

Reliable integer ambiguity resolution (IAR) is essential for carrier phase-based centimeter-level accurate positioning using global navigation satellite systems (GNSSs). In all IAR methods, the best integer equivariant (BIE) estimator is optimal in the sense of minimizing the mean-squared errors. However, the BIE estimator comprises an enumeration in the integer space of ambiguities, and its complexity grows exponentially with the number of ambiguities. Moreover, in a complex urban environment, the positioning performance of the BIE estimator is also reduced due to larger observation errors and even outliers. To address this problem, an efficient and reliable IAR method is proposed in this paper, which consists of two major steps. First, we apply the vectorial integer bootstrapping (VIB) (Teunissen et al. in J Geod 95(9):1–14, 2021) by implementing BIE in each sequential block-by-block integer estimation to improve computation efficiency, which is denoted as VIB-BIE. Second, a measure, named the acceptable probability (ACP), is defined to control the reliability of VIB-BIE estimation. Both simulated and real multi-GNSS data are employed to evaluate the performance of the proposed method and conventional BIE. The results show that the flexibility and efficiency of IAR are both improved by VIB-BIE. In a complex urban environment, the ACP-based VIB-BIE outperforms the BIE in terms of IAR reliability and positioning accuracy. Compared to the BIE, the positioning accuracies are improved by 42.4%, 34.2%, and 31.8% in the east, north, and upward directions, respectively.

One of the most challenging environments for accurate geoid models is in high, rugged mountain areas. Orthometric heights derived from GNSS and a geoid model can easily have errors at the decimeter level. To investigate the effect of geoid model variability on the elevations of peaks in high, rugged mountain areas, this paper is focused on the “Fourteeners” of Colorado, USA (a group of about 60 peaks that are above 14,000 feet = 4267.2 m). Airborne LiDAR data are used to determine geometric (ellipsoidal) heights, which first requires removing a hybrid geoid model, as the LiDAR data is originally provided as orthometric heights. We quantify a significant improvement when using these derived ellipsoidal heights compared with the original orthometric heights: from ± 0.074 to ± 0.054 m (RMSE), an improvement of 28%. Next, a mean geoid model is determined with a relative accuracy of ± 0.06 to 0.08 m and used as a “stand in” realization of the future, official geopotential datum of the USA, NAPGD2022. Using the LiDAR ellipsoidal heights and geoid model, elevations (and uncertainties) for each of the Fourteener summits are determined and found to be, on average, 1.6 m lower than currently published values. This is a much larger change than the 0.5 m decrease expected from the new datum shift alone. The bulk of the difference is due to the original treatments of the vertical angle, triangulation data. A reanalysis of 32 of the 60 peaks shows that the historic data were indeed too high by about 1.0 m or more. Ultimately, no peak falls below the 14,000-foot level nor are any peaks elevated above this level.

We investigate using the GIRAFE cold-atom gravimeter during an airborne gravity survey for improving gravity field and quasigeoid modelling. The study is conducted over the Bay of Biscay, France. Geoid/quasigeoid determination is usually a major challenge over such coastal areas due to scarce and inconsistent gravity data. In a first step, the GIRAFE dataset is analysed and compared with available surface gravity data as well as with global altimetry models from UCSD and DTU. The comparisons indicate that the DTU model is better than the UCSD model within around 10 km from the coastline. Furthermore, recent satellite altimeter missions significantly improve the altimetry models in coastal areas. A significant bias (− 4.00 mGal) in shipborne data is also found from this comparison. In a second step, eight quasigeoid solutions are calculated to evaluate the contribution of GIRAFE data. This contribution reaches 3 cm in terms of height anomaly for DTU21 while being much larger for UCSDv31 and shipborne data. Finally, the quasigeoid solutions are validated using GNSS-levelling data. The results indicate that using GIRAFE data improves by approximately 50% the quality of quasigeoid models over land near the coast. The highest accuracy, around 1 cm, is achieved when GIRAFE data are merged with refined gravity data. Importantly, the standard deviation is just 1.2 cm when compared with GNSS-levelling points if using only GIRAFE data over marine areas, which is very close to the 1 cm goal of geoid/quasigeoid model determination in modern geodesy. This study thus confirms the benefits of performing airborne gravity survey using quantum sensors.

The Atmosphere and Ocean non-tidal De-aliasing Level-1B (AOD1B) product is widely used in precise orbit determination and satellite gravimetry to correct for transient effects of atmosphere–ocean mass variability that would otherwise alias into monthly mean global gravity fields. The most recent release is based on the global ERA5 reanalysis and ECMWF operational data together with simulations from the general ocean circulation model MPIOM consistently forced with fields from the corresponding atmospheric dataset. As background models are inevitably imperfect, residual errors will consequently propagate into the resulting geodetic products. Accounting for uncertainties of the background model data in a statistical sense, however, has been shown before to be a useful approach to mitigate the impact of residual errors leading to temporal aliasing artefacts. In light of the changes made in the new release RL07 of AOD1B, previous uncertainty assessments are deemed too pessimistic and thus need to be revisited. We here present an analysis of the residual errors in AOD1B RL07 based on ensemble statistics derived from different atmospheric reanalyses, including ERA5, MERRA2 and JRA55. For the oceans, we investigate the impact of both the forced and intrinsic variability through differences in MPIOM simulation experiments. The atmospheric and oceanic information is then combined to produce a new time-series of true errors, called AOe07, which is applicable in combination with AOD1B RL07. AOe07 is further complemented by a new spatial error variance–covariance matrix. Results from gravity field recovery simulation experiments for the planned Mass-Change and Geosciences International Constellation (MAGIC) based on GFZ’s EPOS software demonstrate improvements that can be expected from rigorously implementing the newly available stochastic information from AOD1B RL07 into the gravity field estimation process.

Diurnal tidal oscillations in the coupled atmosphere–ocean system generate important contributions to the Earth’s free core nutation (FCN) and annual and sub-annual components of forced nutation in the celestial pole offsets. The determination of FCN parameters cannot avoid the influence of geophysical fluid excitation neither with the direct analysis of FCN signal (direct approaches) nor with the resonance analysis of forced nutation (resonance approaches). There is a significant difference in the FCN parameters obtained with resonance and direct approaches from celestial pole offsets observed through very long baseline interferometry (VLBI). The source of the difference between the two lacks quantitative analysis, which causes difficulties in interpreting the validity of the derived FCN parameters. Using both approaches, we conducted a simulation of celestial pole offsets to quantitatively demonstrate how geophysical fluid excitation affects the determination of FCN parameters from VLBI observations. Using the same excitation source, the FCN period obtained by the direct approach deviated from the set value (430.21 d) by more than 10 d, while the FCN period obtained by the resonance approach showed no deviation from the set value by more than 1 d. The results indicate that the resonance approach more accurately reflects the intrinsic period of the FCN. The impact of atmospheric and oceanic contributions on the determination of the FCN period with the resonance approach was within 2 d. Numerical simulation shows that discrepancies in FCN parameters caused by geophysical excitation were nonnegligible in constructing accurate FCN models.

The thermal expansion effects of GNSS stations are influenced by not only temperature variations, but also bedrock depths and types. Unfortunately, the current studies treat the subsurface GNSS monument and their nearby bedrock as a whole, without taking into account the inconsistencies among bedrock depths and types, while the existing full-spectrum finite element method (FEM) cannot be easily extended to consider the bedrock information. To solve this problem, we propose a refined full-spectrum temperature-induced subsurface thermal expansion model (FSHBDT) that considers both seasonal and non-seasonal temperature variations as well as bedrock information based on the half-space harmonic model. Results show that the full-spectrum half-space harmonic model (FSH), which considers only seasonal and non-seasonal temperature variations, can obtain comparable results to the FEM and even outperform the FEM for inland stations. In addition, the depth and type of bedrock have significant effects on the annual amplitude and phase of thermal expansion-induced vertical displacement. In particular, we find that the station displacement increases by more than 1 mm and the annual phase delays by up to 10° for high-latitude and deeper bedrock stations when bedrock depths are taken into account. The FSHBDT improves the correlation coefficient between GNSS height and mass load displacements by up to 42.3% compared to the FEM and explains up to 8.2% of the nonlinear variation in the GNSS height time series. Our work confirms the advantage of rigorous subsurface thermal expansion modeling to correct the nonlinear variations of global GNSS stations, which might provide a potential opportunity to improve the terrestrial reference frame toward the goal of 1 mm accuracy.

The demand for high-precision positioning has risen substantially in modern urban settings. In that regard, Global Navigation Satellite Systems (GNSS) offer several advantages such as global coverage, real-time capability, high accuracy, ease of use, and cost-effectiveness. The accuracy of GNSS-based positioning, however, suffers in urban environments due to signal blockage, reflection, and diffraction, which makes it difficult to fix ambiguities correctly within a real-time kinematic (RTK). To address this issue, this paper applies random sample consensus (RANSAC) to develop a novel single-epoch triple-frequency RTK positioning method. In our proposed method, the ambiguities of the extra-wide-lane, wide-lane, and original frequencies are resolved sequentially. RANSAC then detects and excludes incorrectly fixed ambiguities. To validate the effectiveness of the proposed method, two static experiments (cases 1 and 2) and one dynamic experiment (case 3) were conducted in representative urban areas. The findings demonstrate that the proposed method outperforms all comparative methods in positional availability, with comparable positional accuracy in terms of root-mean-square errors (RMSEs). In cases 1, 2, and 3, the proposed method achieves 3D RMSEs of 2.74, 4.29, and 20.35 cm, and the positional availabilities of 100%, 75.0%, and 73.1%, using a 10-degree mask angle (and a carrier-to-noise ratio (C/N0) threshold 35 dB-Hz). The corresponding RMSEs (positional availabilities) of comparative methods are from 1.51 to 4.04 cm (75.7 to 96.3%) in case 1, 4.19 to 7.78 cm (34.5 to 49.9%) in case 2, and 23.52 to 37.54 cm (15.4 to 33.9%) in case 3, respectively. Compared to these methods, the proposed method shows improvements of positional availabilities between 3.7 and 24.3 percentage points in case 1, between 25.1 and 40.5 percentage points in case 2, and between 39.2 and 57.7 percentage points in case 3.

Radio signals transmitted by Global Navigation Satellite System (GNSS) satellites experience tropospheric delays. While the hydrostatic part, referred to as zenith hydrostatic delay (ZHD) when mapped to the zenith direction, can be analytically modelled with sufficient accuracy, the wet part, referred to as zenith wet delay (ZWD), is much more difficult to determine and needs to be estimated. Thus, there exist several ZWD models which are used for various applications such as positioning and climate research. In this study, we present a data-driven, global model of the spatial ZWD field, based on the Extreme Gradient Boosting (XGBoost). The model takes the geographical location, the time, and a number of meteorological variables (in particular, specific humidity at several pressure levels) as input, and can predict ZWD anywhere on Earth as long as the input features are available. It was trained on ZWDs at 10718 GNSS stations and tested on ZWDs at 2684 GNSS stations for the year 2019. Across all test stations and all observations, the trained model achieved a mean absolute error of 6.1 mm, respectively, a root mean squared error of 8.1 mm. Comparisons of the XGBoost-based ZWD predictions with independently computed ZWDs and baseline models underline the good performance of the proposed model. Moreover, we analysed regional and monthly models, as well as the seasonal behaviour of the ZWD predictions in different climate zones, and found that the global model exhibits a high predictive skill in all regions and across all months of the year.

Predicting Earth Orientation Parameters (EOP) is crucial for precise positioning and navigation both on the Earth’s surface and in space. In recent years, many approaches have been developed to forecast EOP, incorporating observed EOP as well as information on the effective angular momentum (EAM) derived from numerical models of the atmosphere, oceans, and land-surface dynamics. The Second Earth Orientation Parameters Prediction Comparison Campaign (2nd EOP PCC) aimed to comprehensively evaluate EOP forecasts from many international participants and identify the most promising prediction methodologies. This paper presents the validation results of predictions for universal time and length-of-day variations submitted during the 2nd EOP PCC, providing an assessment of their accuracy and reliability. We conduct a detailed evaluation of all valid forecasts using the IERS 14 C04 solution provided by the International Earth Rotation and Reference Systems Service (IERS) as a reference and mean absolute error as the quality measure. Our analysis demonstrates that approaches based on machine learning or the combination of least squares and autoregression, with the use of EAM information as an additional input, provide the highest prediction accuracy for both investigated parameters. Utilizing precise EAM data and forecasts emerges as a pivotal factor in enhancing forecasting accuracy. Although several methods show some potential to outperform the IERS forecasts, the current standard predictions disseminated by IERS are highly reliable and can be fully recommended for operational purposes.

The PPP-RTK system, which is capable of providing a centimeter-level real-time positioning service for an unlimited number of users, is becoming a promising tool in mass-market applications such as smartphones, the Internet of Things (IoT), and the automotive industry. The extended Kalman filter (EKF) is the conventional method for parameter estimation in the existing PPP-RTK system. Recently, an alternative method known as factor graph optimization (FGO), which fully leverages the time correlation among current and historical measurements, has the potential to further improve the accuracy and robustness of PPP-RTK solutions. In this contribution, a factor graph optimization-based PPP-RTK framework is developed, where raw pseudorange, phase measurements, precise atmospheric corrections, and time-differenced carrier-phase (TDCP) measurements serve as factors in FGO estimators. The continuously tracked phase ambiguities are estimated as the time-invariant state node and propagated by marginalization while ambiguity resolution is conducted independently between epochs. A second optimization process with the utilization of ambiguity-resolved solutions and time-differenced carrier-phase (TDCP) measurements is conducted to further improve the reliability of positioning results. The effectiveness of the proposed method is evaluated by vehicular tests in urban environments. Results indicate that the FGO method could improve the performance of ambiguity resolution by reducing the ambiguity search space and increasing the ratio values, leading to a significant accuracy improvement of 55% in an open-sky environment compared to the traditional EKF-based method. Furthermore, in GNSS signal partly block scenes, the FGO-based PPP-RTK is capable of obtaining more robust and accurate positioning solutions with fewer outliers compared to the EKF method.

The lunar orbiter laser altimeter (LOLA) onboard the lunar reconnaissance orbiter has performed high-precision, full-coverage, and high-density laser ranging observations for the entire lunar surface since its launch. Statistics have shown that LOLA has collected 6.94 billion effective altimeter data up to June 2022. Most of the typical orbits in the LOLA dataset have a high quality and exhibit horizontal offsets of almost 7 m and radial offsets of almost 0.5 m. However, there is still a category of orbits in the dataset that will cause apparent noise in the constructed DEM, which is attributed to the orbits with large or anomalous errors. We call such orbits as flawed orbits in this paper. The flawed orbits can be identified and screened by the elevation discrepancy at the crossovers of the orbits. The results show that the flawed orbits are caused by significant along-track errors, which also result in the radial error of up to several kilometers. Moreover, most of the flawed orbits are concentrated in several consecutive time intervals. A correction method is then proposed to correct the flawed orbits in the local region. The position of the flawed orbits is reconstructed using the feature points matching of the DEMs before and after they are removed. Some experimental analyzes show that the apparent terrain artifacts have been eliminated and more identifiable terrain details are reappeared. Identifying and correcting these flawed orbits with significant along-track offsets paves the way for improving the quality of the LOLA data and reconstructing the topography of the Moon.

By introducing the auxiliary variable with respect to the reduced latitude, a new closed-form method for transforming Cartesian to geodetic coordinates has been proposed based on the solution of a special constructed unary quartic equation. The algorithm comes with rigorous and concise procedure of root-finding. Moreover, through theoretical analysis, different approaches with respective pros and cons to determine the geodetic latitude and height have been explored. Besides fast computation, numerical experiments covering the magnitude of the geodetic height from \(- 6.33 \times 10^{6} {\text{m}}\) to \(10^{10} {\text{m}}\) have also shown that the new method can be operational with high precision at almost any point including the region near or at the pole, the equator and the center of the reference ellipsoid. Considering the accuracy, efficiency and adaptability simultaneously, it is prospective to be applied into computation and inspection on critical occasions in comparison to existing methods.

Geodetic time series are usually modeled with a deterministic approach that includes trend, annual, and semiannual periodic components having constant amplitude and phase-lag. Although simple, this approach neglects the time-variability or stochasticity of trend and seasonal components, and can potentially lead to inadequate interpretations, such as an overestimation of global navigation satellite system (GNSS) station velocity uncertainties, up to masking important geophysical phenomena. In this contribution, we generalize previous methods for determining trends and seasonal components and address the challenge of their time-variability by proposing a novel linear additive model, according to which (i) the trend is allowed to evolve over time, (ii) the seasonality is represented by a fractional sinusoidal waveform process (fSWp), accounting for possible non-stationary cyclical long-memory, and (iii) an additional serially correlated noise captures the short term variability. The model has a state space representation, opening the way for the evaluation of the likelihood and signal extraction with the support of the Kalman filter (KF) and the associated smoothing algorithm. Suitable enhancements of the basic methodology enable handling data gaps, outliers, and offsets. We demonstrate the advantage of our method with respect to the benchmark deterministic approach using both observed and simulated time series and provide a fair comparison with the Hector software. To that end, various geodetic time series are considered which illustrate the ability to capture the time-varying stochastic seasonal signals with the fSWp.

The Earth’s crust is exposed by tectonic processes and is not static over time. Modelling of the Earth’s surface velocities is of utmost importance for research in geodesy, geophysics, structural geology, and other geosciences. It may support positioning, navigation, seismic risk, and volcano notification services, for example. Space geodetic techniques can be used to provide high-quality velocities in a network of geodetic sites. Velocity field modelling should, however, expand the velocities from a discrete set of points to any location in-between. This paper presents four new methods for the Earth’s surface velocity interpolation. Contrary to the widely used approach dividing the velocity field to the horizontal and vertical components, a full 3D interpolation approach is proposed based on the Delaunay triangulation and the n-simplex interpolation. The use of a combination of all three components is advantageous for geophysical interpretation. The proposed interpolation approach is entirely local but enables global modelling, which does not suffer from map projection distortions and singularities at the poles. Various global and regional position/velocity datasets are used to evaluate the performance of the proposed velocity interpolation methods. The latter provide practically the same results when applied to regional velocity field modelling. However, the so-called continuous piecewise quasi-radial 3D velocity field interpolation method is recommended for its favourable properties. It introduces an ellipsoidal Earth model, appropriately considers vertical/up and horizontal velocity components, tends to radial symmetry, and provides continuity for the interpolated velocity components as well as for the estimated uncertainties.

PPP-RTK corrections, aiding GNSS users to achieve single-receiver integer ambiguity-resolved parameter solutions, are often estimated in a recursive manner by a provider. Such recursive, multi-epoch, estimation of the corrections relies on a set of \(\mathcal {S}\) -basis parameters that are chosen by the provider so as to make the underlying measurement setup solvable. As a consequence, the provider can only estimate estimable forms of the corrections rather than the original corrections themselves. It is the goal of the present contribution to address the consequence of the corrections’ dependency on the provider’s \(\mathcal {S}\) -basis, showcasing the characteristics of their multi-epoch solutions, thereby identifying potential pitfalls which the PPP-RTK user should avoid when evaluating such solutions. To this end, we develop a simulation platform that allows one to have full control over the properties of PPP-RTK corrections and demonstrate various misleading temporal behaviors that exist when interpreting the multi-epoch solutions of their estimable forms. The roles of the correction latency and time correlation in the multi-epoch user positioning performance are quantified, while the deviation of the user-reported positioning precision description from its user-actual counterpart is measured under a misspecified user stochastic model.