Real-time estimation of multi-GNSS integer recovery clock with undifferenced ambiguity resolution Abstract Precise satellite clock product is an important prerequisite to support the real-time precise positioning service. In this contribution, the multi-GNSS integer recovery clock (MIRC) model is developed to improve both the accuracy and efficiency of real-time clock estimates. In the proposed method, the undifferenced ambiguities of GPS, BDS, Galileo and GLONASS are fixed to integers, and thus the integer properties of the ambiguities are recovered and the accuracy of the clock estimates is also improved. In addition, benefiting from the removal of large quantities of ambiguity parameters, the computation time is greatly reduced which can guarantee the high processing efficiency of real-time clock estimates. Multi-GNSS observations from 151 globally distributed Multi-GNSS Experiment tracking stations are processed with the proposed MIRC model over a one-month period (DOY 240–270, 2018). Compared to the float satellite clocks, the precision (standard deviation, STD) of the real-time MIRC with respect to CODE 30 s final multi-GNSS satellite clock products was improved by 53.0% from 0.046 to 0.022 ns for GPS, 42.7% from 0.096 to 0.055 ns for BDS, 63.7% from 0.097 to 0.035 ns for Galileo and 33.9% from 0.153 to 0.101 ns for GLONASS, respectively. With the proposed method, the average computation time per epoch with multi-GNSS observations for 50-, 100- and 150-station networks was improved by 7.3%, 82.7% and 97.1% compared to that of standard float clock estimation. Multi-GNSS kinematic precise point positioning (PPP) ambiguity resolution was also performed with the derived real-time MIRC products. Compared to the float PPP solutions, the position accuracy of the multi-GNSS MIRC-based fixed solutions was improved by 77.2%, 49.7% and 52.7% from 24.2, 13.3 and 30.7 mm to 5.5, 6.7 and 14.5 mm for the east, north and up components, respectively.

Coastal gravity field refinement by combining airborne and ground-based data Abstract Gravity field modelling in coastal region faces challenges due to the degradation of the quality of altimeter data and poor coverage of gravimetric measurements. Airborne gravimetry can provide seamless measurements both onshore and offshore with uniform accuracies, which may alleviate the coastal zone problem. We study the role of airborne data for gravity field recovery in a coastal region and the possibility to validate coastal gravity field model against recent altimetry data (CryoSat-2, Jason-1, and SARAL/Altika). Moreover, we combine airborne and ground-based gravity data for regional refinement and quantify and validate the contribution introduced by airborne data. Numerical experiments in the Gippsland Basin over the south-eastern coast of Australia show that the effects introduced by airborne gravity data appear as small-scale patterns on the centimetre scale in terms of quasi-geoid heights. Numerical results demonstrate that the combination of airborne data improves the coastal gravity field, and the recent altimetry data can be potentially used to validate the high-frequency signals introduced by airborne data. The validation against recent altimetry data demonstrates that the combination of airborne measurements improves the coastal quasi-geoid, by ~ 5 mm, compared with a model computed from terrestrial and altimetry-derived gravity anomalies alone. These results show that the recently released altimetry data with relatively denser spatial resolutions and higher accuracies than older altimeter data may be beneficial for gravity field model assessment in coastal areas.

GRACE gravity field recovery with background model uncertainties Abstract In this article, we present a computationally efficient method to incorporate background model uncertainties into the gravity field recovery process. While the geophysical models typically used during the processing of GRACE data, such as the atmosphere and ocean dealiasing product, have been greatly improved over the last years, they are still a limiting factor of the overall solution quality. Our idea is to use information about the uncertainty of these models to find a more appropriate stochastic model for the GRACE observations within the least squares adjustment, thus potentially improving the gravity field estimates. We used the ESA Earth System Model to derive uncertainty estimates for the atmosphere and ocean dealiasing product in the form of an autoregressive model. To assess our approach, we computed time series of monthly GRACE solutions from L1B data in the time span of 2005 to 2010 with and without the derived error model. Intercomparisons between these time series show that noise is reduced on all spatial scales, with up to 25% RMS reduction for Gaussian filter radii from 250 to 300 km, while preserving the monthly signal. We further observe a better agreement between formal and empirical errors, which supports our conclusion that used uncertainty information does improve the stochastic description of the GRACE observables.

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New analytical solution and associated software for computing full-tensor gravitational field due to irregularly shaped bodies Abstract We present a new analytical solution to compute the full-tensor gravity gradient due to a body mass of uniform density with arbitrary geometry. The solution is an extension of an existing analytical computation of gravitational anomalies of a polyhedron source, based on a transition of the general expressions from surface to line integrals. These developments enable the computation of the gravity gradient tensor using the same simple procedures as the gravitational field. The method is validated by comparing with a closed analytical solution, including on/in the near field of the body surface. The algorithm is implemented in the freely available MATLAB-based software called Gal Eötvös Earth Calculator. It is tested successfully for various measurement distances and body mass sizes, enabling applications from local geophysical prospecting to global topographic effect for satellite data. Due to its flexibility, the new solution, and the associated software, is particularly well suited for joint analyses of all types of gravity measurements regardless of the extent, altitude and irregularity of their spatial distribution.

GPS satellite inter-frequency clock bias estimation using triple-frequency raw observations Abstract This study proposes a unified uncombined model to estimate GPS satellite inter-frequency clock bias (IFCB) in both triple-frequency code and carrier-phase observations. In the proposed model, the formulae of both phase-based and code-based IFCBs are rigorously derived. Specifically, satellite phase-based IFCB refers to its time-variant part and it is modeled as a periodic function related to the sun–spacecraft–earth angle. A zero-mean condition of all available GPS satellites that support triple-frequency data is introduced to render satellite code-based IFCB estimable. Three months of data from 40 globally distributed stations of the International GNSS Service Multi-GNSS Experiment are used to test our method. The results show that the four-order periodic function is suitable for eliminating the 12-h, 6-h, 4-h, and 3-h periods that exist in the a posteriori phase residuals when no periodic function is used. For comparison, the geometry-free and ionosphere-free (GFIF) phase combination and differential code bias (DCB) products released by DLR (German Aerospace Center) and IGG (Institute of Geodesy and Geophysics, China) are also used to calculate the satellite phase-based and code-based IFCBs, respectively. The results show that (1) the average root mean square (RMS) of the phase-based IFCB difference between the proposed method and the GFIF phase combination is 4.3 mm; (2) the average RMS in the eclipse period increased by 50% compared with the average RMS in the eclipse-free period; (3) the mean monthly STD for code-based IFCB from the proposed method is 0.09 ns; and (4) the average RMS values of code-based IFCB differences between the proposed method and the DCB products released by DLR and IGG are 0.32 and 0.38 ns. This proposed model also provides a general approach for multi-frequency GNSS applications such as precise orbit and clock determination.

Impact of network constraining on the terrestrial reference frame realization based on SLR observations to LAGEOS Abstract The Satellite Laser Ranging (SLR) network struggles with some major limitations including an inhomogeneous global station distribution and uneven performance of SLR sites. The International Laser Ranging Service (ILRS) prepares the time-variable list of the most well-performing stations denoted as ‘core sites’ and recommends using them for the terrestrial reference frame (TRF) datum realization in SLR processing. Here, we check how different approaches of the TRF datum realization using minimum constraint conditions (MCs) and the selection of datum-defining stations affect the estimated SLR station coordinates, the terrestrial scale, Earth rotation parameters (ERPs), and geocenter coordinates (GCC). The analyses are based on the processing of the SLR observations to LAGEOS-1/-2 collected between 2010 and 2018. We show that it is essential to reject outlying stations from the reference frame realization to maintain a high quality of SLR-based products. We test station selection criteria based on the Helmert transformation of the network w.r.t. the a priori SLRF2014 coordinates to reject misbehaving stations from the list of datum-defining stations. The 25 mm threshold is optimal to eliminate the epoch-wise temporal deviations and to provide a proper number of datum-defining stations. According to the station selection algorithm, we found that some of the stations that are not included in the list of ILRS core sites could be taken into account as potential core stations in the TRF datum realization. When using a robust station selection for the datum definition, we can improve the station coordinate repeatability by 8%, 4%, and 6%, for the North, East and Up components, respectively. The global distribution of datum-defining stations is also crucial for the estimation of ERPs and GCC. When excluding just two core stations from the SLR network, the amplitude of the annual signal in the GCC estimates is changed by up to 2.2 mm, and the noise of the estimated pole coordinates is substantially increased.

Preface to the second special issue on Laser Ranging

Time bias service: analysis and monitoring of satellite orbit prediction quality Abstract The performance of satellite laser ranging (SLR) station operations relies to a large extent on the quality of the required satellite orbit predictions. Poor predictions with large along-track offsets, so-called time biases, increase the target acquisition time and thus reduce the performance of stations and the International Laser Ranging (ILRS) network as a whole. There is currently no established process to evaluate or monitor the quality of predictions. This paper presents a method for such a process that uses normal point data uploaded to data centers by ILRS stations worldwide. The first analysis results show systematic trends over time for most targets and prediction providers. These trends were used to predict the development of time bias values. We also present a service that provides these predicted values for the latest satellite orbit predictions of selected targets and providers in real time. Using these values during tracking allows for faster target acquisition and thus better tracking performance at ILRS SLR stations. Through monitoring, the service further enables stations to select the best available predictions during tracking and to notify prediction providers if issues are encountered. This tool benefits not only the stations by improving their tracking performance but also allows for prediction improvement and greater support of missions.

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