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The availability of many high-degree Global Geopotential Models (GGMs), namely EGM2008, EIGEN-6C4, GECO, SGG-UGM-1, SGG-UGM-2, XGM2019e_2159, and GGMPlus, challenges users regarding which model is best for Vietnam. This study, therefore, evaluates their performance by comparing them with GNSS/leveling data over Vietnam. Results show that their absolute and relative performances are largely independent of topographic conditions and geographical location and can be ranked into three classes: (1) XGM2019e_2159 has the highest accuracy, (2) the models EIGEN-6C4, GECO, SGG-UGM-1, SGG-UGM-2, and GGMPlus, have a very similar level of medium accuracy, while (3) EGM2008 is found to be the least accurate. In an absolute sense, the differences between GNSS/leveling and EGM2008-based height anomalies have a standard deviation (STD) of 0.290 ± 0.010 m, whereas, for XGM2019e_2159, this is 0.156 ± 0.006 m. All other models have STDs of (0.18–0.19) ± 0.007 m. Regarding relative performance without fitting, all GGMs have comparable accuracies for baseline length of 5–20 km, while for baselines longer than 20 km, the STD of XGM2019e_2159 is 1.5 ppm–0.5 ppm (approximately 19%–40%) lower compared with EGM2008, and 0.5 ppm–0.25 ppm (approximately 7%–36%) lower compared with EIGEN6C4, GECO, SGG-UGM-1, SGG-UGM-2, and GGMPlus. In addition, the STDs decrease significantly from 20 to 12 ppm in the range of 5–10 km, slightly from 12 to 6 ppm for 10–35 km, very slightly from 6 to 2.5 ppm for 35–200 km, and then remain almost unchanged for longer baselines. After fitting, the relative accuracies of all GGMs are at the same level with negligible STD/RMSE values. Furthermore, only EGM2008 experiences significant regional differences, while other GGMs show more homogeneous spatial variation of absolute accuracy over Vietnam. These findings can contribute to the development of local quasigeoid models in Vietnam and may be helpful with the improvement of GGMs in the future.

Climate extremes such as droughts and intense rainfall events are expected to strongly influence global/regional water resources in addition to the growing demands for freshwater. This study examines the impacts of precipitation extremes and human water usage on total water storage (TWS) over the Ganges-Brahmaputra-Meghna (GBM) River Basin in South Asia. Monthly TWS changes derived from the Gravity Recovery And Climate Experiment (GRACE) (2002–2014) and soil moisture from three reanalyses (1979–2014) are used to estimate new extreme indices. These indices are applied in conjunction with standardized precipitation indices (SPI) to explore the impacts of precipitation extremes on TWS in the region. The results indicate that although long-term precipitation do not indicate any significant trends over the two subbasins (Ganges and Brahmaputra-Meghna), there is significant decline in rainfall (9.0?±?4.0 mm/decade) over the Brahmaputra-Meghna River Basin from 1998 to 2014. Both river basins exhibit a rapid decline of TWS from 2002 to 2014 (Ganges: 12.2?±?3.4 km3/yr and Brahmaputra-Meghna: 9.1?±?2.7 km3/yr). While the Ganges River Basin has been regaining TWS (5.4?±?2.2 km3/yr) from 2010 onward, the Brahmaputra-Meghna River Basin exhibits a further decline (13.0?±?3.2 km3/yr) in TWS from 2011 onward. The impact of human water consumption on TWS appears to be considerably higher in Ganges compared to Brahmaputra-Meghna, where it is mainly concentrated over Bangladesh. The interannual water storage dynamics are found to be strongly associated with meteorological forcing data such as precipitation. In particular, extreme drought conditions, such as those of 2006 and 2009, had profound negative impacts on the TWS, where groundwater resources are already being unsustainably exploited. Extreme droughts have profound negative impacts on TWS in the GBM River BasinDeclining TWS in the Brahmaputra-Meghna River Basin likely due to declining rainfallTWS variations over Ganges and Bangladesh are strongly affected by excessive groundwater withdrawal

Radiosonde data are important for understanding and monitoring the upper troposphere and lower stratosphere (UTLS) region. Over much of Africa, however, such data are lacking; consequently, the African UTLS is understudied, and potential proxies such as climate models and reanalysis products fail to fully capture the behavior of the UTLS. This study pioneers the use of Global Navigation Satellite System‐Radio Occultation (GNSS‐RO) data from 2001 to 2020 to address the radiosonde data gap over Africa and contributes to a better understanding of the tropopause (TP) characteristics under the influence of global and regional climate drivers over the continent. As a first step to using GNSS‐RO for infilling the radiosonde data gap over Africa, we analyzed the performance of GNSS‐RO (2001–2020) and reanalysis products (European Centre for Medium‐Range Weather Forecasts Reanalysis 5 (ERA5) and Modern‐Era Retrospective analysis for Research and Applications version 2 (MERRA‐2)) against radiosonde observations applying the Kling‐Gupta Efficiency metric. The analyses show that GNSS‐RO data from Challenging Mini‐satellite Payload, Gravity Recovery and Climate Experiment, Meteorological Operational, Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC), and COSMIC‐2 are in good agreement with radiosonde measurements with differences being smaller than 1 K in the UTLS, thereby enabling infilling of missing radiosonde data in Africa during 2001–2020. By contrast, the smoothed vertical temperature profiles of reanalysis products lead to a warm bias of +0.8 K in ERA5 and +1.2 K in MERRA‐2 and these biases alter some vertical and temporal structure details, with possible implications on climate change detection and attribution. Furthermore, the analysis of GNSS‐RO data over Africa revealed: (a) the teleconnections of El Niño‐Southern Oscillation (ENSO), Quasi‐Biennial Oscillation (QBO), Indian Ocean Dipole (IOD), Madden‐Julian Oscillation (MJO), North Atlantic Oscillation (NAO) and Southern Annular Mode (SAM) at the tropopause boundary; (b) multiple coupled global climate drivers such as ENSO‐IOD, ENSO‐MJO, ENSO‐NAO, QBO‐IOD, and ENSO‐NAO‐MJO; (c) coupled global and regional climate drivers that influence the TP variability, for example, ENSO‐Inter Tropical Convergence Zone; and (d), the deep convection associated with the Asian Summer Monsoon and Tropical/African Easterly Jet also locally influence TP height. In conclusion, this study demonstrates the capability of GNSS‐RO to fill the vast radiosonde data gap over Africa. This opens the opportunity for further detailed studies toward a better understanding of the tropopause characteristics including localization, quantification of trends, and influences of global, regional, and coupled climate drivers. Global Navigation Satellite System‐Radio Occultation (GNSS‐RO) is capable of infilling the vast radiosonde data gap over AfricaGNSS‐RO captures global and regional climate drivers of tropopause behaviorGNSS‐RO reveals multiple coupled global and regional climate drivers Global Navigation Satellite System‐Radio Occultation (GNSS‐RO) is capable of infilling the vast radiosonde data gap over Africa GNSS‐RO captures global and regional climate drivers of tropopause behavior GNSS‐RO reveals multiple coupled global and regional climate drivers

The nonlinear system of equationsfor solving GPS atmospheric sounding’s bending angles are normally solved using Newton’s method. Because of the nonlinear nature of the equations, Newton’s method applies linearization and iterations. The method assumes the refraction angle to be small enough such that the dependency of the doppler shift on these angles are linear. The bending angles are then solved iteratively. Since the approach assumes the dependency of doppler shift on bending angles to be linear, which in actual sense is not, some small nonlinearity erroris incurred. The Newton’s iterative method is often used owing to the bottleneck of solving in exact form the nonlinear system of equations for bending angles. By converting this system of trigonometricnonlinear equations into algebraic, the present contribution proposes an analytic (algebraic) algorithm for solving the bending angles and presents the geometry of the solution space. The algorithm is tested by computing bending angles of three CHAMP occultation data and the results compared to those of iterative Newton’s approach. Occultation 133 of 3rd May 2002 is selected as it occurred during diurnal solar radiation maximum past afternoon. During this time, the effect of ionospheric noise is high. Occultations number 3 of 14th May 2001 and number 6 of 2nd February 2002 were selected since they occurred past mid-night, a time of low solar activity and thus less effect of ionospheric noise. The results for occultation 133 of 3rd May 2002 indicate that the nonlinearity errorsin bending angles increase with decrease in height to a maximum absolute value of 0.00069° (0.1%) for the region 5–40 km during period of high solar activity. Such nonlinearity errorsare shown to impact significantly on the computed impact parameters to which the bending angles are referred. During low solar activity period, the nonlinearity errorwas relatively small for occultation number 3 of 14th May 2001 with maximum absolute value of 0.00001°. The analytical algorithm thus provide an independent methodfor controlling classical iterative procedures and could be used where very accurate results are desired.

In the recently developed Spatial Reference Systemthat is designed to check and control the accuracy of the three-dimensional coordinate measuring machines and tooling equipment (Metronom US., Inc., Ann Arbor: http://www.metronomus.com), the coordinates of the edges of the instrument are computed from distances of the bars. The use of distances in industrial application is fast gaining momentum just as in Geodesyand in Geophysical applicationsand thus necessitating efficient algorithms to solve the nonlinear distance equations.Whereas the ranging problem with minimum known stations was considered in our previous contribution in the same Journal, the present contribution extends to the case where one is faced with many distance observationsthan unknowns(overdetermined case) as is usually the case in practise. Using the Gauss-Jacobi Combinatorial approach, we demonstrate how one can proceed to position without reverting to iterativeand linearizingprocedures such as Newton’sor Least Squaresapproach.

In GPS atmospheric sounding, geodetic positioning, roboticsand photogrammetric(perspective center and intersection) problems, distances (ranges) as observables play a key role in determining the unknown parameters. The measured distances (ranges) are however normally related to the desired parameters via nonlinear equationsor nonlinear system of equationsthat require explicitor exactsolutions. Procedures for solving such equations are either normally iterative, and thus require linearization or the existing analytical procedures require laborious forward and backward substitutions. We present in the present contribution direct proceduresfor solving distance nonlinear system of equationswithout linearization, iteration, forward and backward substitution. In particular, we exploit the advantage of faster computerswith large storage capacities and the computer algebraic softwares of Mathematica, Maple and Matlab to test polynomial basedapproaches. These polynomial (algebraic based) approaches turn out to be the key to solving distance nonlinear system of equations. The algebraic techniques discussed here does not however solve all general typesof nonlinear equations but only those nonlinear system of equationsthat can be converted into algebraic (polynomial)form.

The nonlinear adjustment of GPS observations of type pseudo-ranges is performed in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in ℝ3is generated via the Error Propagation law/variance-covariance propagation. The Fast Nonlinear Adjustment Algorithm (FNon Ad Al) has been already proposed by Gauss whose work was published posthumously and Jacobi (1841). The algorithm, here referred to as the Gauss-Jacobi Combinatorial algorithm, solves the over-determined GPS pseudo-ranging problem without reverting to iterative or linearization procedure except for the second moment (Variance-Covariance propagation). The results compared well with the solutions obtained using the linearized least squares approach giving legitimacy to the Gauss-Jacobi combinatorial procedure. © 2002 Wiley Periodicals, Inc.

Several procedures for solving, in a closed form the GPS pseudo-ranging four-point problem P4P in matrix form already exist. We present here alternative algebraic procedures using Multipolynomial resultant and Groebner basis to solve the same problem. The advantage is that these algebraic algorithms have already been implemented in algebraic software such as “Mathematica” and “Maple”. The procedures are straightforward and simple to apply. We illustrate here how the algebraic techniques of Multipolynomial resultant and Groebner basis explicitly solve the nonlinear GPS pseudo-ranging four-point equations once they have been converted into algebraic (polynomial) form and reduced to linear equations. In particular, the algebraic tools of Multipolynomial resultant and Groebner basis provide symbolic solutions to the GPS four-point pseudo-ranging problem. The various forward and backward substitution steps inherent in the clasical closed form solutions of the problem are avoided. Similar to the Gauss elimination techniques in linear systems of equations, the Multipolynomial resultant and Groebner basis approaches eliminate several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of univariate polynomial equations (in this case quadratic equations for the range bias expressed algebraically using the given quantities) whose roots can be determined by existing programs (e. g., the roots command in MATLAB). © 2002 Wiley Periodicals, Inc.