Journal of Geodesy最新文献

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 sphere, which is the maximum radius coordinate of the body, changes with the origin. The sphere can thus be adjusted to fit a certain convex portion of irregular body shape via an appropriate coordinate translation, thereby maximizing the region of model stability above the body. We demonstrate that it is, while perhaps counterintuitive, rational to displace the coordinate origin from the center of figure, or even off the body entirely. We review concisely the theory and a method of spherical harmonic translation. We consider some textbook examples that illuminate the physical meaning and the practical advantage of the transformation, the discussion of which, as it turns out, is not so easily encountered. We provide seminormalized as well as fully normalized version of the algorithms, which are compact and easy to work with for low-degree applications. At little cost, the proposed approach enables the spherical harmonics to be comparable with the far more complicated ellipsoidal harmonics in performance in the case of two small objects, Phobos and 433 Eros.

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 ZWD without relying on the meteorological profile parameters. The proposed ZWD retrieval model achieved an RMSE and STD of 1.74 cm, with a correlation coefficient of 0.98. The enhanced performance of PWV-generated ZWD in GNSS PPP was tested in this study. The results showed that the ZWD constraint in GNSS PPP mainly affects the convergence time of the standard PPP solution, with the most significant effect in the U-direction. The PPP convergence time can be shortened by a maximum of 43%, with an average reduction of 24% for the eight sites over the four seasons. In the PPP-ambiguity resolution solution, the time to first fix (TTFF) was shorter for all sites with ZWD enhancement than for those without ZWD enhancement. The TTFF of the eight sites was significantly shortened in all four seasons, with an average improvement of 31%. The ZWD retrieval method based on the ERA5 PWV proposed in this study can quickly generate ZWD with high accuracy and resolution over a large area and significantly enhance GNSS PPP. The methodology proposed in this study is valuable for utilizing multi-source PWV-generated ZWD services for GNSS PPP enhancement.

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 Service (IGS) and its Analysis Centers. For the IGS Reprocessing Campaign 3 (repro3), Graz University of Technology (TUG) participated as an AC using the raw observation approach, which uses all measurements as observed by the receivers. However, a common feature of almost all global multi-GNSS processing strategies is the use of diagonal covariance matrices as stochastic models for simplicity. This implies that any spatial or temporal correlations are ignored. However, numerous studies have shown that GNSS processing is indeed affected by spatial and temporal correlations. For global GNSS processing, research on stochastic modeling and its challenges is rather scarce. In this work, a detailed insight into the problems of stochastic modeling in global GNSS processing using the raw observation approach is given along with a detailed overview of the intended TUG approach. An analysis of the impact of temporal correlation modeling on the resulting GNSS products and GNSS frame estimation is also given.

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 evaluate the effect of such a constrained procedure on monthly gravity field retrievals for the planned ESA-NASA double-pair mission. The influence of co-estimating background model corrections subject to known uncertainty information is evaluated separately for both types of background models and is then combined and used to retrieve monthly gravity fields over one year. Retrieval errors are compared to those obtained with the standard recovery procedure, which neglects these corrections. It is shown that gravity field retrieval errors are reduced by up to 36%. In addition, the one-year simulation is used to estimate residual corrections for eight major tidal constituents in order to improve ocean tide background modelling. Adding these residual corrections to the applied a priori ocean tide model shows that ocean tide errors are decreased by up to 27%.

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 expressed in Cartesian coordinates that includes the derived Legendre functions (DLFs). Here, we expand the estimation process by implementing also the traditional spherical harmonic synthesis formula of normalized associated Legendre functions (ALFs) expressed in spherical coordinates. The variations obtained by applying the two approaches are compared with gravity signal differences induced by the modelled shape changes using the line integral analytical approach. The numerical comparisons refer to three asteroid shape models of Eros, Didymos and Dimorphos. The first-order derivative values provided by the DLF expressions and their variations using ALF are closer to the analytical method’s results. The highest calculated differences refer to ΔV_z with their mean value reaching 37% with respect to the other components obtained by all methods. Finally, the respective harmonic series converge to a fixed numerical value at a maximum expansion degree equal to 15 near Brillouin sphere and 5 as the distance of the computation point increases.

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, we employ a self-supervised deep learning algorithm to downscale global monthly (p_b) anomalies derived from GRACE(-FO) observations to an equal-angle 0.25 ( ^{circ }) grid in the absence of high-resolution ground truth. The optimization process is realized by constraining the outputs to follow the large-scale mass conservation contained in the gravity field estimates while learning the spatial details from two ocean reanalysis products. The downscaled product agrees with GRACE(-FO) solutions over large ocean basins at the millimeter level in terms of equivalent water height and shows signs of outperforming them when evaluating short spatial scale variability. In particular, the downscaled (p_b) product has more realistic signal content near the coast and exhibits better agreement with tide gauge measurements at around 80% of 465 globally distributed stations. Our method presents a novel way of combining the advantages of satellite measurements and ocean models at the product level, with potential downstream applications for studies of the large-scale ocean circulation, coastal sea level variability, and changes in global geodetic parameters.

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 first kind, which relates surface gravity anomalies to satellite gradient data. Solving this equation requires discretising a surface integral and applying further regularisation techniques to stabilise the numerical solution of a resulting system of linear equations. This study examines four methods for modifying the system of linear equations derived by discretising the Fredholm integral equation. The methods include direct inversion, remove-compute-restore, truncation reduction of the integral formula, and inversion of a modified integral for estimating surface gravity anomalies from satellite gradient data over a test area in Central Europe. Since the system of linear equations is ill-conditioned, the Tikhonov regularisation is applied to stabilise its numerical solution. To assess the precision and reliability of the estimated gravity anomalies, the study introduces mathematical models for estimation of biased and de-biased noise variance–covariance matrices of estimated surface gravity anomalies. The results indicate that the signal-to-noise ratio of reduced satellite gradient data in the remove-compute-restore method is smaller compared to other methods in the study, necessitating stronger stabilisation of the model to recover surface gravity anomalies. This, in turn, leads to a more optimistic uncertainty propagation than the other considered methods.

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 additional measurements. However, LEO satellites often lack code measurements, mainly transmitting satellite-specific frequency-varying carrier phase signals. This contribution aims to study the ambiguity-resolved baseline positioning performance of such phase-only signals, addressing the extent to which LEO constellations can realize near real-time positioning in standalone and GNSS-combined modes. Through a simulation platform, we analyze the distinct response of each LEO constellation (Iridium, Globalstar, Starlink, OneWeb, and Orbcomm) to IAR under various circumstances. Although achieving single-receiver high-precision positioning can be challenged by inaccuracies in the LEO satellite orbit products, the relative distance between two receivers can help overcome this limitation. As a result, centimeter-level relative positioning over short baselines can be made possible, even with a satellite elevation cut-off angle of 50 degrees, making it suitable for GNSS-challenged environments. This can be achieved with high-grade receiver clocks over very short baselines ((sim )5 km) and access to decimeter-level orbit products.

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 variations: one expressed in spherical coordinates using the traditional associated Legendre functions (ALF) and their derivatives up to the second order, while the other expressed in Cartesian coordinates by including the derived Legendre functions (DLF). The obtained variations are compared in terms of convergence with gravity signal differences referring to the specific shape changes using the line integral analytical approach for three asteroid shape models. Both approaches provide results that differ from the analytical method at a 1E−1 level, while the differences between them are at the 1E−15 level. The obtained results are highly influenced by the geometry of the examined shape model, with the ALF approach providing variations with closer agreement with the analytical method only for the almost spherical shape. Both harmonic synthesis expressions can be used to derive accurate results, as they differ at a very low level, and one can choose based on the convenience of their algorithmic characteristics.