Journal of Geodesy

The ionospheric mapping function (MF) for Global Navigation Satellite System (GNSS), a mutual projection method for the slant total electron content (STEC) and vertical total electron content, is one of the significant factors affecting the performance of ionospheric models. The commonly used MF assumes isotropic TEC variations and takes into account only the satellite elevation angle, which may result in significant ionospheric projection errors, especially at low elevation angles. Based on the single-layer model, we propose an additional azimuth parameter mapping function (APMF). The APMF was estimated and evaluated by the NeQuick model during the periods of January 2014 and January 2022 from the aspect of simulation and measured STEC during the periods of 2014 and 2022 from the aspect of actual measurements over China, respectively. Compared to the modified single-layer model mapping function (MSLM-MF), the experimental results indicate that (1) The APMF can significantly reduce the ionospheric projection error, and the fluctuation in errors with different azimuth angles is small. (2) According to the evaluation based on the NeQuick simulation during the TEC peak time, when the ionosphere is quite active, the upper and lower quartiles of the absolute projection error boxplot of the APMF relative to the MSLM-MF in January 2014 are reduced by 56.1% and 60.0%, respectively, and in January 2022, they are reduced by 67.7% and 65.2%, respectively. Similarly, the upper whiskers in the boxplot are reduced by 54.7% and 67.5% in January 2014 and January 2022, respectively; the APMF performance in terms of the root mean square error (RMSE) is improved by 47.0% in January 2014 and 58.3% in January 2022. (3) According to the evaluation based on the measured STEC from GNSS raw data during the TEC peak time, the upper and lower quartiles of the absolute mapping error boxplot of the APMF relative to the MSLM-MF in 2014 are reduced by 48.9% and 46.9%, respectively, while in 2022, they are reduced by 48.3% and 41.2%, respectively. The upper whiskers in the boxplot are reduced by 41.8% and 35.2% in 2014 and 2022, respectively; the APMF performance in terms of RMSE is improved by 44.6% in 2014 and 39.2% in 2022.

The short-term variability in receiver code biases (RCBs) has been identified as a prominent source of error leading to the degradation of precise point positioning (PPP) performance and ionospheric total electron content (TEC) estimation accuracy. To minimize the adverse impact of RCB variability, this study extends the modified PPP (MPPP) method from the GPS only dual-frequency (DF) model to multifrequency (MF) and multiconstellation cases. In the MF MPPP method, multi-GNSS (GPS, BDS and Galileo) dual-, triple- or even arbitrary-frequency observations can be jointly processed in a flexible and reliable way by taking the time-varying RCBs of all available signals into account. Benefiting from this, the between-epoch fluctuations experienced by RCBs for all constellations and frequencies can be detected and their adverse impacts on the ionospheric observables and ambiguity parameters are mitigated. Compared to the traditional MF PPP method, the retrieval accuracy of the multi-GNSS-based ionospheric observables using our proposed method can be improved by more than 74% in the presence of significant intraday RCB variations. The variation trends are not always consistent for RCBs in different frequency bands for different satellite systems. The dependence of multi-GNSS and MF RCB variations on the ambient temperature is also verified. The percentages of the stations analyzed with the absolute Pearson correlation coefficient (PCC) values above 0.8 for BDS are higher than those of GPS and Galileo, and the temperature dependence of RCB on the second frequency band is higher than those of the first frequency band for all the three constellations.

We propose a nonparametric minimum entropy method for estimating an optimal velocity from position time series, which may contain unknown noise, data gaps, loading effects, transients, outliers and step discontinuities. Although nonparametric, the proposed method is based on elementary statistical concepts familiar to least-squares and maximum-likelihood users. It seeks a constant velocity with a best possible (realistic) variance rather than a best variable velocity fit to the closest position data. We show, based on information theory, synthetic and real data, that minimum-entropy velocity estimation: (1) accounts for colored noise without assumptions about its distribution or the extent of its temporal correlations; (2) is unaffected by the series deterministic content such as an initial position and the heights of step discontinuities and insensitive to small-amplitude periodic variations and transients; (3) is robust against outliers and, for long time series, against step discontinuities and even slight non-stationarity of the noise; (4) does not involve covariance matrices or eigen/singular value analysis, thus can be implemented by a short and efficient software; (5) under no circumstances results in a velocity variance that decays as \(1/N\) , where \(N\) is the number of observations. The proposed method is verified based on synthetic data and then applied to a few hundred NGL (Nevada Geodetic Lab) position time series of different characteristics, and the results are compared to those of the Median Interannual Difference Adjusted for Skewness (MIDAS) algorithm. The compared time series include continuous and linear ones used to test the agreement between the two methods in the presence of unknown noise, data gaps and loading effects, discontinuous but linear series selected to include the effect of a few (1–4) discontinuities, and nonlinear but continuous time series selected for including the effects of transients. Both the minimum-entropy and MIDAS methods are nonparametric in the sense that they only extract the velocity from a position time series with hardly any explicit assumptions about its noise distribution or correlation structure. Otherwise, the two methods differ in every single possible technical sense. Other than pointing to a close agreement between the derived velocities, the comparisons consistently revealed that minimum-entropy velocity uncertainties suggest a smaller degree of temporal correlations in the NGL time series than the MIDAS does.

The Total Electron Content (TEC) computed from ionospheric models is a widely used parameter for characterizing the morphological structure of the ionosphere. The global TEC maps from empirical models, like the International Reference Ionosphere (IRI) model, have limited accuracy compared to those calculated by dual-frequency measurements from the global navigation satellite systems (GNSS). We have developed a reconstructed IRI TEC model for generating high-precision global TEC maps based on a deep learning method. For this, we have collected 48,204 pairs of global TEC maps from the IRI-2020 model and Global Ionosphere Maps (GIM) model with 2-h time resolution from 2009 to 2019 covering the whole solar cycle 24. The daily solar radio flux (F10.7), sunspot number (SSN), Dst, and Kp indices are also introduced as input features to train the model. We have investigated the optimum combination of the input parameters for the reconstructed TEC model and compared the performance of the model during the years with high and low solar activity levels. Results show that the reconstructed TEC model with F10.7 and Kp features has a better performance compared to that considering all solar and geomagnetic indices. The global TEC maps predicted from our model are much more consistent with the corresponding TEC maps from the GIM model than those from the IRI-2020 model. Especially, the large-scale equatorial ionospheric anomaly (EIA) crests and the pronounced enhancement of TEC are well predicted by the reconstructed TEC model. From statistical metrics, the accuracy of the reconstructed TEC model increased by 40.8% during the high solar activity year 2015 and 43.0% during the low solar activity year 2018 compared with the IRI-2020 model. The prediction performance of the reconstructed TEC model also shows better accuracy during the storm periods.

Satellite Laser Ranging (SLR) is an established technique providing very accurate position measurements of satellites in Earth orbit. However, despite decades of development, it remains a complex and expensive technology, which impedes its further growth to new applications and users. The miniSLR implements a complete SLR system within a small, transportable enclosure. Through this design, costs of ownership can be reduced significantly, and the process of establishing a new SLR site is greatly simplified. A number of novel technical solutions have been implemented to achieve a good laser ranging performance despite the small size and simplified design. Data from the initial six months of test operation have been used to generate a first estimation of the system performance. The data include measurements to many of the important SLR satellites, such as Lageos, Etalon and most of the geodetic and Earth observation missions in LEO. It is shown that the miniSLR achieves sub-centimetre accuracy, comparable with conventional SLR systems. The miniSLR is an engineering station in the International Laser Ranging Service and supplies data to the community. Continuous efforts are undertaken to further improve the system operation and stability.

Two innovations are presented for coordinate time-series computation. First, an improved solution is given to a century-old problem, that is the non-iterative computation of conventional geodetic (CG: latitude, longitude, height) coordinates from geocentric Cartesian (GC: x, y, z) coordinates. The accuracy is 1 nm for heights < 500 km and < 10−15 rad for latitude at any point, terrestrial or outer space. This can be 3 orders of magnitude more accurate than published non-iterative methods. Secondly, CG time series are transformed into a practical system of “graticule distance” (GD: easting, northing, height) curvilinear coordinates that, unlike the commonly used system of topocentric Cartesian (TC: east, north, up) coordinates, is global in nature without arbitrary specification of GC reference coordinates for every geodetic station. Since 2011, Nevada Geodetic Laboratory has publicly produced time series for 20,000 GPS stations in GD form that have been cited by hundreds of studies. The GD system facilitates direct comparison of positions for co-located stations. Users of GD time series are able: (1) to resolve different historical station names that have been assigned to the same physical benchmark and (2) to resolve different physical benchmarks that have been assigned the same name. This benefits historical reconstruction of benchmark occupation and local site tie analysis for reference frame integrity. GD coordinates have archival value, in that inversion back to GC coordinates is practically exact. For geodetic stations, GD time series closely emulate TC time series with rates agreeing to 0.01 mm/yr, and so can be used interchangeably.