Journal of Geodesy

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.

It has been observed that when using sea levels derived from GPS (Global Positioning System) signal-to-noise ratio (SNR) data to perform tidal analysis, the luni-solar semidiurnal (K2) and the luni-solar diurnal (K1) constituents are biased due to geometrical errors in the reflection data, which result from their periods coinciding with the GPS orbital period and revisit period. In this work, we use 18 months of GNSS SNR data from multiple frequencies and multiple constellations at three sites to further investigate the biases and how to mitigate them. We first estimate sea levels using SNR data from the GPS, GLONASS, and Galileo signals, both individually and by combination. Secondly, we conduct tidal harmonic analysis using these sea-level estimates. By comparing the eight major tidal constituents estimated from SNR data with those estimated from the co-located tide-gauge records, we find that the biases in the K1 and K2 amplitudes from GPS S1C, S2X and S5X SNR data can reach 5 cm, and they can be mitigated by supplementing GLONASS- and Galileo-based sea-level estimates. With a proper combination of sea-level estimates from GPS, GLONASS, and Galileo, SNR-based tidal constituents can reach agreement at the millimeter level with those from tide gauges.

The algorithms given in Karney (J. Geodesy 87:43–55, 2013), to compute geodesics on terrestrial ellipsoids, are extended to apply to ellipsoids of revolution with arbitrary eccentricity. For the direct and inverse geodesic problems, this entails implementing the formulation in terms of elliptic integrals given by Legendre and Cayley. The integral for the area of geodesic polygons is computed in terms of the discrete sine transform of the integrand. In all cases, accuracy close to full machine precision is achieved. An open-source implementation of these algorithms is available.

In geodesy, Tikhonov regularization and truncated singular value decomposition (TSVD) are commonly used to derive a well-defined solution for ill-conditioned observation equations. However, as single-parameter regularization methods, they may face some limitations in application due to their lack of flexibility. In this contribution, a kind of multiparameter regularization method is considered, called generalized ridge regression (GRR). Generally, GRR projects observations into several orthogonal spectral domains and then uses different regularization parameters to minimize the mean squared error of the estimated parameters in corresponding spectral domains. To find suitable regularization parameters for GRR, an iterative and shrinking generalized ridge regression (IS-GRR) is proposed. The IS-GRR procedure starts by introducing a predetermined approximation of unknown parameters. Subsequently, in each spectral domain, the signal and noise of the observations are estimated in an iterative and shrinking manner, and the regularization parameters are updated according to the estimated signal-to-noise ratio. Compared to conventional regularization schemes, IS-GRR has the following advantages: Tikhonov regularization usually oversmooths signals in the low-spectral domains and undersuppresses noise in the high-spectral domains, whereas TSVD usually undersuppresses noise in the low-spectral domains and oversmooths signals in the high-spectral domains. However, IS-GRR strikes a balance between retaining signals and suppressing noise in different spectral domains, thereby exhibiting better performance. Two experiments (simulation and mascon modelling examples) verify the effectiveness of IS-GRR for solving ill-conditioned equations in geodesy.