Remarkable advances over recent years give an evidence that geodesy today develops under a broad spectrum of interactions, including theory, science, engineering, technology, observation, and practice-oriented services. Geodetic science accumulates significant results in studies towards classical geodetic problems and also problems that only emerged or gained new interest, in many cases as a consequence of synergistic activities in geodesy and tremendous advances in the instrumentations and computational facilities. In-depth studies progressed in parallel with investigations that mean a broadening of the traditional core of geodesy. The scope of the session is conceived with a certain degree of freedom, though the session is primarily intended to provide a forum for all investigations and results of theoretical and methodological nature.

Within this concept we seek contributions concerning problems of reference frames, gravity field studies, dynamics and rotation of the Earth, positioning, but also presentations, which surpass frontiers of these topics. We invite presentations illustrating the use of mathematical and numerical methods in solving geodetic problems, showing advances in mathematical modeling, estimating parameters, simulating relations and systems, using high-performance computations, and discussing also methods that enable to exploit data essentially associated with new and existing satellite missions. Presentations showing mathematical and physical research directly motivated by geodetic need, practice and ties to other disciplines are welcome. In parallel to theory oriented results also examples illustrating the use of new methods on real data in various branches of geodetic science and practice are very much solicited in this session.

Convener: Petr Holota | Co-conveners: Robert Cunderlik, Otakar Nesvadba, Nico Sneeuw
| Attendance Mon, 04 May, 08:30–10:15 (CEST)

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Chat time: Monday, 4 May 2020, 08:30–10:15

Chairperson: P. Holota, R. Čunderlík, O. Nesvadba
D1791 |
Saeed Farzaneh and Ehsan Forootan

The Global Ionosphere Maps (GIMs) are generated on a daily basis at the Center for Orbit Determination in Europe (CODE) using the observations from about 200 Global Positioning System (GPS)/GLONASS sites of the International GNSS Service (IGS) and other institutions. These maps contain Vertical Total Electron Content (VTEC) values, which are estimated in a solar-geomagnetic reference frame using a spherical harmonics expansion up to degree and order 15. Although these maps have wide applications, their relatively low spatial resolution limits the accuracy of many geodetic applications such as those related to Precise Point Positioning (PPP) and navigation. In this study, a novel Bayesian approach is proposed to improve the spatial resolution of VTEC estimations in regional and global scales. The proposed technique utilises GIMs as a prior information and updates the VTEC estimates using a new set of base-functions (with better resolution than that of spherical harmonics) and the GNSS measurements that are not included in the network of GIMs. To achieve the highest accuracy possible, our implementation is based on a transformation of spherical harmonics to the Slepian base-functions, where the latter is a set of bandlimited functions that reflect the majority of signal energy inside an arbitrarily defined region, yet they remain orthogonal within this region. The new GNSS measurements are considered in a Bayesian update estimation to modify those of GIMs. Numerical application of this study is demonstrated using the ground-based GPS data over South America. The results are also validated against the VTEC estimations derived from independent GPS stations.

Key words: Spherical Slepian Base-Functions, Spherical Harmonics, Ionospheric modelling, Vertical Total Electron Content (VTEC)

How to cite: Farzaneh, S. and Forootan, E.: A Bayesian Approach to Improve Regional and Global Ionospheric Maps using GNSS Observations , EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-282, https://doi.org/10.5194/egusphere-egu2020-282, 2020.

D1792 |
| solicited
Peter Teunissen

G1.1 Session: Recent Developments in Geodetic Theory


Estimability in Rank-Defect Mixed-Integer Models: Theory and Applications


PJG Teunissen1,2

1GNSS Research Centre, Curtin University, Perth, Australia

2Geoscience and Remote Sensing, Delft University of Technology, The Netherlands

Email: p.teunissen@curtin.edu.au; p.j.g.teunissen@tudelft.nl


Although estimability is one of the foundational concepts of todays’ estimation theory, we show that the current concept of estimability is not adequately equipped to cover the estimation requirements of mixed-integer models, for instance like those of interferometric models, cellular base transceiver network models or the carrier-phase based models of Global Navigation Satellite Systems (GNSSs). We therefore need to generalize the estimability concept to that of integer-estimability. Next to being integer and estimable in the classical sense, functions of integer parameters then also need to guarantee that their integerness corresponds with integer values of the parameters the function is taken of. This is particularly crucial in the context of integer ambiguity resolution. Would this condition not be met, then the integer fixing of integer functions that are not integer-estimable implies that one can fix the undifferenced integer ambiguities to non-integer values and thus force the model to inconsistent and wrong constraints.

In this paper we present a generalized concept of estimability and one that now also is applicable to mixed-integer models. We thereby provide the operationally verifiable necessary and sufficient conditions that a function of integer parameters needs to satisfy in order to be integer-estimable. As one of the conditions we have that estimable functions become integer-estimable if they can be unimodulair transformed to canonical form. Next to the conditions, we also show how to create integer-estimable functions and how a given design matrix can be expressed in them. We then show how these results are to be applied to interferometric models, cellular base transceiver network models and FDMA GNSS models.


Keywords: Estimability, S-system theory, Mixed-integer Models, Integer-Estimability, Admissible Ambiguity Transformation, Interferometry, Global Navigation Satellite Systems (GNSSs)


How to cite: Teunissen, P.: Estimability in Rank-Defect Mixed-Integer Models: Theory and Applications, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-1797, https://doi.org/10.5194/egusphere-egu2020-1797, 2020.

D1793 |
wudong li and weiping jiang

Removal of the Common Mode Error (CME) is very important for the investigation of Global Navigation Satellite Systems (GNSS) technique error and the estimation of accurate GNSS velocity field for geodynamic applications. The commonly used spatiotemporal filtering methods cannot accommodate missing data, or they have high computational complexity when dealing with incomplete data. This research presents the Expectation-Maximization Principal Component Analysis (EMPCA) to estimate and extract CME from the incomplete GNSS position time series. The EMPCA method utilizes an Expectation-Maximization iterative algorithm to search each principal subspace, which allows extracting a few eigenvectors and eigenvalues without covariance matrix and eigenvalue decomposition computation. Moreover, it could straightforwardly handle the missing data by Maximum Likelihood Estimation (MLE) at each iteration. To evaluate the performance of the EMPCA algorithm for extracting CME, 44 continuous GNSS stations located in Southern California have been selected here. Compared to previous approaches, EMPCA could achieve better performance using less computational time and exhibit slightly lower CME relative errors when more missing data exists. Since the first Principal Component (PC) extracted by EMPCA is remarkably larger than the other components, and its corresponding spatial response presents nearly uniform distribution, we only use the first PC and its eigenvector to reconstruct the CME for each station. After filtering out CME, the interstation correlation coefficients are significantly reduced from 0.46, 0.49, 0.42 to 0.18, 0.17, 0.13 for the North, East, and Up (NEU) components, respectively. The Root Mean Square (RMS) values of the residual time series and the colored noise amplitudes for the NEU components are also greatly suppressed, with an average reduction of 25.9%, 27.4%, 23.3% for the former, and 49.7%, 53.9%, and 48.9% for the latter. Moreover, the velocity estimates are more reliable and precise after removing CME, with an average uncertainty reduction of 52.3%, 57.5%, and 50.8% for the NEU components, respectively. All these results indicate that the EMPCA method is an alternative and more efficient way to extract CME from regional GNSS position time series in the presence of missing data. Further work is still required to consider the effect of formal errors on the CME extraction during the EMPCA implementation.

How to cite: li, W. and jiang, W.: Extracting Common Mode Errors of Regional GNSS Position Time Series in the Presence of Missing Data by Expectation-Maximization Principal Component Analysis, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-12635, https://doi.org/10.5194/egusphere-egu2020-12635, 2020.

D1794 |
Xiaopeng Li, Jianliang Huang, Cornelis Slobbe, Roland Klees, Martin Willberg, and Roland Pail

The topic of downward continuation (DWC) has been studied for many decades without very conclusive answers on how different methods compare with each other. On the other hand, there are vast amounts of airborne gravity data collected by the GRAV-D project at NGS NOAA of the United States and by many other groups around the world. These airborne gravity data are collected on flight lines where the height of the aircraft actually varies significantly, and this causes challenges for users of the data. A downward continued gravity grid either on the topography or on the geoid is still needed for many applications such as improving the resolution of a local geoid model. Four downward continuation methods, i.e., Residual Least Squares Collocation (RLSC), the Inverse Poisson Integral, Truncated Spherical Harmonic Analysis, and Radial Basis Functions (RBF), are tested on both simulated data sets and real GRAV-D airborne gravity data in a previous joint study between NGS NOAA and CGS NRCan. The study group is further expanded by adding the TU Delft group on RBF and the TUM group on RLSC to incorporate more updated knowledge in the theoretical background and more in-depth discussion on the numerical results. A formal study group will be established inside IAG for providing the best answers for downward continuing airborne gravity data for local gravity field improvement. In this presentation, we review and compare the four methods theoretically and numerically. Simulated and real airborne and terrestrial data are used for the numerical comparison over block MS05 of the GRAV-D project in Colorado, USA, where the 1cm geoid experiment was performed by 15 international teams. The conclusion drawn from this study will advance the use of GRAV-D data for the new North American-Pacific Geopotential Datum of 2022 (NAPGD2022).

How to cite: Li, X., Huang, J., Slobbe, C., Klees, R., Willberg, M., and Pail, R.: On Downward Continuing Airborne Gravity Data for Local Geoid Modeling, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-3042, https://doi.org/10.5194/egusphere-egu2020-3042, 2020.

D1795 |
Martin Pitoňák, Michal Šprlák, Pavel Novák, and Robert Tenzer

Geodetic boundary-value problems (BVPs) and their solutions are important tools for describing and modelling the Earth’s gravitational field. Many geodetic BVPs have been formulated based on gravitational observables measured by different sensors on the ground or moving platforms (i.e. aeroplanes, satellites). Solutions to spherical geodetic BVPs lead to spherical harmonic series or surface integrals with Green’s kernel functions. When solving this problem for higher-order derivatives of the gravitational potential as boundary conditions, more than one solution is obtained. Solutions to gravimetric, gradiometric and gravitational curvature BVPs (Martinec 2003; Šprlák and Novák 2016), respectively, lead to two, three and four formulas. From a theoretical point of view, all formulas should provide the same solution, but practically, when discrete noisy observations are exploited, they do not.

In this contribution we present combinations of solutions to the above mentioned geodetic BVPs in terms of surface integrals with Green’s kernel functions by a spectral combination method. We investigate an optimal combination of different orders and directional derivatives of potential. The spectral combination method is used to combine terrestrial data with global geopotential models in order to calculate geoid/quasigeoid surface. We consider that the first-, second- and third-order directional derivatives are measured at the satellite altitude and we continue them downward to the Earth’s surface and convert them to the disturbing gravitational potential, gravity disturbances and gravity anomalies. The spectral combination method thus serves in our numerical procedures as the downward continuation technique. This requires to derive the corresponding spectral weights for the n-component estimator (n = 1, 2, … 9) and to provide a generalized formula for evaluation of spectral weights for an arbitrary N-component estimator. Properties of the corresponding combinations are investigated in both, spatial and spectral domains.


How to cite: Pitoňák, M., Šprlák, M., Novák, P., and Tenzer, R.: Overview on the spectral combination of integral transformations , EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-3403, https://doi.org/10.5194/egusphere-egu2020-3403, 2020.

D1796 |
| Highlight
Michal Šprlák, Khosro Ghobadi-Far, Shin-Chan Han, and Pavel Novák

The problem of estimating mass redistribution from temporal variations of the Earth’s gravity field, such as those observed by GRACE, is non-unique. By approximating the Earth’s surface by a sphere, surface mass change can be uniquely determined from time-variable gravity data. Conventionally, the spherical approach of Wahr et al. (1998) is employed for computing the surface mass change caused, for example, by terrestrial water and glaciers. The accuracy of the GRACE Level 2 time-variable gravity data has improved due to updated background geophysical models or enhanced data processing. Moreover, time series analysis of ∼15 years of GRACE observations allows for determining inter-annual and seasonal changes with a significantly higher accuracy than individual monthly fields. Thus, the improved time-variable gravity data might not tolerate the spherical approximation introduced by Wahr et al. (1998).

A spheroid (an ellipsoid of revolution) represents a closer approximation of the Earth than a sphere, particularly in polar regions. Motivated by this fact, we develop a rigorous method for determining surface mass change on a spheroid. Our mathematical treatment is fully ellipsoidal as we concisely use Jacobi ellipsoidal coordinates and exploit the corresponding series expansions of the gravitational potential and of the surface mass. We provide a unique one-to-one relationship between the ellipsoidal spectrum of the surface mass and the ellipsoidal spectrum of the gravitational potential. This ellipsoidal spectral formula is more general and embeds the spherical approach by Wahr et al. (1998) as a special case. We also quantify the differences between the spherical and ellipsoidal approximations numerically by calculating the surface mass change rate in Antarctica and Greenland.



Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: Hydrological and oceanic effects and their possible detection using GRACE. Journal of Geophysical Research: Solid Earth, 103(B12), 30205-30229.

How to cite: Šprlák, M., Ghobadi-Far, K., Han, S.-C., and Novák, P.: Spherical and ellipsoidal surface mass change from GRACE time-variable gravity data, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-13258, https://doi.org/10.5194/egusphere-egu2020-13258, 2020.

D1797 |
Hadi Amin, Lars E. Sjöberg, and Mohammad Bagherbandi

According to the classical Gauss–Listing definition, the geoid is the equipotential surface of the Earth’s gravity field that in a least-squares sense best fits the undisturbed mean sea level. This equipotential surface, except for its zero-degree harmonic, can be characterized using the Earth’s Global Gravity Models (GGM). Although nowadays, the satellite altimetry technique provides the absolute geoid height over oceans that can be used to calibrate the unknown zero-degree harmonic of the gravimetric geoid models, this technique cannot be utilized to estimate the geometric parameters of the Mean Earth Ellipsoid (MEE). In this study, we perform joint estimation of W0, which defines the zero datum of vertical coordinates, and the MEE parameters relying on a new approach and on the newest gravity field, mean sea surface, and mean dynamic topography models. As our approach utilizes both satellite altimetry observations and a GGM model, we consider different aspects of the input data to evaluate the sensitivity of our estimations to the input data. Unlike previous studies, our results show that it is not sufficient to use only the satellite-component of a quasi-stationary GGM to estimate W0. In addition, our results confirm a high sensitivity of the applied approach to the altimetry-based geoid heights, i.e. mean sea surface and mean dynamic topography models. Moreover, as W0 should be considered a quasi-stationary parameter, we quantify the effect of time-dependent Earth’s gravity field changes as well as the time-dependent sea-level changes on the estimation of W0. Our computations resulted in the geoid potential W0 = 62636848.102 ± 0.004 m2s-2 and the semi-major and –minor axes of the MEE, a = 6378137.678 ± 0.0003 m and b = 6356752.964 ± 0.0005 m, which are 0.678 and 0.650 m larger than those axes of the GRS80 reference ellipsoid, respectively. Moreover, a new estimation for the geocentric gravitational constant was obtained as GM = (398600460.55 ± 0.03) × 106 m3s-2.

How to cite: Amin, H., Sjöberg, L. E., and Bagherbandi, M.: A global vertical datum defined by the conventional geoid potential and the Earth ellipsoid parameters, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-15381, https://doi.org/10.5194/egusphere-egu2020-15381, 2020.

D1798 |
Petr Holota and Otakar Nesvadba

When treating geodetic boundary value problems in gravity field studies, the geometry of the physical surface of the Earth may be seen in relation to the structure of the Laplace operator. Similarly as in other branches of engineering and mathematical physics a transformation of coordinates is used that offers a possibility to solve an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. The Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth substantially differs from a sphere or an oblate ellipsoid of revolution, even if these are optimally fitted. The situation may be more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces. The structure of the Laplace operator, however, is more complicated in this case and in a sense it represents the topography of the physical surface of the Earth. The Green’s function method together with the method of successive approximations is used for the solution of geodetic boundary value problems expressed in terms of new coordinates. The structure of iteration steps is analyzed and if useful, it is modified by means of the integration by parts. Subsequently, the individual iteration steps are discussed and interpreted.

How to cite: Holota, P. and Nesvadba, O.: Laplacian structure, solution domain geometry and successive approximations in gravity field studies, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-12839, https://doi.org/10.5194/egusphere-egu2020-12839, 2020.

D1799 |
Marek Macák, Zuzana Minarechová, Róbert Čunderlík, and Karol Mikula

We present a novel approach to the solution of the geodetic boundary value problem with an oblique derivative boundary condition by the finite element method. Namely, we propose and analyse a finite element approximation of a Laplace equation holding on a domain with an oblique derivative boundary condition given on a part of its boundary. The oblique vector in the boundary condition is split into one normal and two tangential components and derivatives in tangential directions are approximated as in the finite difference method. Then we apply the proposed numerical scheme to local gravity field modelling. For our two-dimensional testing numerical experiments, we use four nodes bilinear quadrilateral elements and for a three-dimensional problem, we use hexahedral elements with eight nodes. Practical numerical experiments are located in area of Slovakia that is given by grid points located on the Earth's surface with uniform spacing in horizontal directions. Heights of grid points are interpolated from the SRTM30PLUS topography model. An upper boundary is in the height of 240 km above a reference ellipsoid WGS84 corresponding to an average altitude of the GOCE satellite orbits. Obtained solutions are compared to DVRM05.

How to cite: Macák, M., Minarechová, Z., Čunderlík, R., and Mikula, K.: The finite element method for solving the oblique derivative boundary value problems in geodesy, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-13411, https://doi.org/10.5194/egusphere-egu2020-13411, 2020.

D1800 |
Matej Medľa, Karol Mikula, and Róbert Čunderlík

We present local gravity field modelling based on a numerical solution of the oblique derivative bondary value problem (BVP). We have developed a finite volume method (FVM) for the Laplace equation with the Dirichlet and oblique derivative boundary condition, which is considered on a 3D unstructured mesh about the real Earth’s topography. The oblique derivative boundary condition prescribed on the Earth’s surface as a bottom boundary is split into its normal and tangential components. The normal component directly appears in the flux balance on control volumes touching the domain boundary, and tangential components are managed as an advection term on the boundary. The advection term is stabilised using a vanishing boundary diffusion term. The convergence rate, analysis and theoretical rates of the method are presented in [1].

Using proposed method we present local gravity field modelling in the area of Slovakia using terrestrial gravimetric measurements. On the upper boundary, the FVM solution is fixed to the disturbing potential generated from the GO_CONS_GCF_2_DIR_R5 model while exploiting information from the GRACE and GOCE satellite missions. Precision of the obtained local quasigeoid model is tested by the GNSS/levelling test.


[1] Droniou J, Medľa M, Mikula K, Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling. Journal of Computational Physics, s. 2019

How to cite: Medľa, M., Mikula, K., and Čunderlík, R.: FVM approach for solving the oblique derivative BVP on unstructured meshes above the real Earth’s topography, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-13527, https://doi.org/10.5194/egusphere-egu2020-13527, 2020.

D1801 |
| Highlight
Petr Holota and Otakar Nesvadba

The notion of an equipotential surface of the Earth’s gravity potential is of key importance for vertical datum definition. The aim of this contribution is to focus on differential geometry properties of equipotential surfaces and their relation to parameters of Earth’s gravity field models. The discussion mainly rests on the use of Weingarten’s theorem that has an important role in the theory of surfaces and in parallel an essential tie to Brun’s equation (for gravity gradient) well known in physical geodesy. Also Christoffel’s theorem and its use will be mentioned. These considerations are of constructive nature and their content will be demonstrated for high degree and order gravity field models. The results will be interpreted globally and also in merging segments expressing regional and local features of the gravity field of the Earth. They may contribute to the knowledge important for the realization of the World Height System.

How to cite: Holota, P. and Nesvadba, O.: Differential geometry and curvatures of equipotential surfaces in the realization of the World Height System, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-13418, https://doi.org/10.5194/egusphere-egu2020-13418, 2020.

D1802 |
Gerassimos Manoussakis and Romylos Korakitis

We present a method for the estimation of the components ξ and η of the deflection of the vertical using several parameters of the gravitational potential. Specifically, we assume that we know the geodetic coordinates (φ, λ, h), the magnitude of gravity g, the components ξ, η and the second partial derivatives of the gravitational potential (elements of the Eötvös matrix) at a point P. Knowing only the geodetic coordinates of a neighboring point A (at a distance up to several kilometers from P), we estimate the components ξ and η at A. The proposed method is evaluated with simulated data at several points in Greece. The results show that it may be used for the densification of a given astrogeodetic net.

How to cite: Manoussakis, G. and Korakitis, R.: Local determination of the components of the deflection of the vertical using gravitational potential parameters at a neighboring point, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-20090, https://doi.org/10.5194/egusphere-egu2020-20090, 2020.

D1803 |
Mohamed Eleiche and Ahmed Mansi

The equation of the ellipse in a meridian plan is well known to be which can be defined in its most generic form as where the variable (s) is the indicator value of the ellipsoidal height. For (h=0), the value of  (s) equals (1) , for negative  (h), the value  is lower than (1), and vice versa, for positive (h) , the corresponding  value of (s) is greater than (1). Hence (s) and (h) are highly-correlated. The main goal of this work is to exploit the  (s-h) correlation and to represent it both statistically and mathematically. Moreover, the essential role of (s) in the transformation of the Cartesian coordinates (xp, yp, zp) of any generic point (P) into its geodetic coordinates (φ, λ, h) in reference to the geodetic ellipsoid is thoroughly studied.

How to cite: Eleiche, M. and Mansi, A.: Heuristic Algorithm to Compute Geodetic Height (h) from Ellipse Equation, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-10857, https://doi.org/10.5194/egusphere-egu2020-10857, 2020.

D1804 |
Xie Jian and Long Sichun

The errors-in-variables (EIV) model is applied to surveying and mapping fields such as empirical coordinate transformation, line/plane fitting and rigorous modelling of point clouds and so on as it takes the errors both in coefficient matrix and observation vector into account. In many cases, not all of the elements in coefficient matrix are random or some of the elements are functionally dependent. The partial EIV (PEIV) model is more suitable in dealing with such structured coefficient matrix. Furthermore, when some reliable prior information expressed by inequality constraints is considered, the adjustment result of inequality constrained PEIV (ICPEIV) model is expected to be improved. There are two kinds of algorithms to solve the ICPEIV model under the weighted total least squares (WTLS) criterion currently. On the one hand, one can linearize the PEIV model and transform it into a sequence of quadratic programming (QP) sub-problems. On the other hand, one can directly solve the nonlinear target function by common used programming algorithms.All the QP algorithms and nonlinear programming methods are complicated and not familiar to the geodesists, so the ICPEIV model is not widely used in geodesy.   

In this contribution, an algorithm based on standard least squares is proposed. First, the estimation of model parameters and random variables in coefficient matrix are separated according to the Karush-Kuhn-Tucker (KKT) conditions of the minimization problem. The model parameters are obtained by solving the QP sub-problems while the variables are determined by the functional relationship between them. Then the QP problem is transformed to a system of linear equations with nonnegative Lagrange multipliers which is solved by an improved Jacobi iterative algorithm. It is similar to the equality-constrained least squares problem. The algorithm is simple because the linearization process is not required and it has the same form of classical least squares adjustment. Finally, two empirical examples are presented. The linear approximation algorithm, the sequential quadratic programming algorithm and the standard least squares algorithm are used. The examples show that the new method is efficient in computation and easy to implement, so it is a beneficial extension of classical least squares theory.

How to cite: Jian, X. and Sichun, L.: Weighted total least squares problems with inequality constraints solved by standard least squares theory, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-6362, https://doi.org/10.5194/egusphere-egu2020-6362, 2020.

D1805 |
Tianhe Xu, Song Li, and Nan Jiang

Abstract: With the rapid development of artificial intelligence, machine learning has become an high-efficient tool applied in the fields of GNSS data analysis and processing, such as troposphere, ionosphere or satellite clock modeling and prediction. In this paper, zenith troposphere delay (ZTD) prediction algorithms based on BP neural network (BPNN) and least squares support vector machine (LSSVM) are proposed in the time and space domain. The main trend terms in ZTD time series are deducted by polynomial fitting, and the remaining residuals are reconstructed and modeled by BPNN and LSSVM algorithm respectively. The test results show that the performance of LSSVM is better than that of BPNN in term of prediction stability and accuracy by using ZTD products of International GNSS Service (IGS) of 20 stations in time domain. In order to further improve LSSVM prediction accuracy, a new strategy of training samples selection based on correlation analysis is proposed. The results show that using the proposed strategy, about 80% to 90% of the 1-hour prediction deviation of LSSVM can reach millimeter level depending on the season, and the percentage of the prediction deviation value less than 5 mm is about 60% to 70%, which is 5% to 20% higher than that of the classical random selection in different month. The mean values of RMSE in all 20 stations using the new strategy are 1-3mm smaller than those of the classical one. Then different prediction span from 1 to 12 hours is conducted to show the performance of the proposed method. Finally, the ZTD predictions based on BPNN and LSSVM in space domain are also verified and compared using GNSS CORS network data of Hong Kong, China.

Keywords: ZTD, BP Neural Network, Support Vector Machine, Least Squares, GNSS

Acknowledgments: This work was supported by Natural Science Foundation of China (41874032) and the National Key Research and Development Program (2016YFB0501701)


How to cite: Xu, T., Li, S., and Jiang, N.: Zenith Troposphere Delay Prediction based on BP Neural Network and Least Squares Support Vector Machine, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-5471, https://doi.org/10.5194/egusphere-egu2020-5471, 2020.

D1806 |
xinlin zhang

This study presents a new method to calculate displacement and potential changes caused by an earthquake in a three-dimensional viscoelastic earth model. It is the first time to compute co- and post-seismic deformation in a spherical earth with lateral heterogeneities. Such a method is useful to investigate the 3-dimensional viscoelastic structure of the earth by interpreting precise satellite gravity and GPS data. Firstly, we concern with Maxwell’s constitutive equation, the linearized equation of momentum conservation and Poisson’s equation, and obtain the solution in the Laplace domain in a spherical symmetric viscoelastic earth model. Furthermore, we employ the perturbed method to deal with the effect of lateral heterogeneities and obtain the relation between the solutions of the spherical symmetric earth model, the three-dimension earth model with lateral inhomogeneity and the auxiliary solutions. Then, using the given surface boundary conditions to determine the auxiliary solutions, we obtain the perturbed solutions of lateral increment in the Laplace domain. Finally, taking the inverse Laplace transforms of solutions in a spherical symmetric viscoelastic earth model and perturbed solutions with respect to lateral hetergeneities, we obtain the solutions of deformation in a three-dimensional viscoelastic earth model. 

How to cite: zhang, X.: Dislocation Theory in a 3-dimensional Viscoelastic Earth Model, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-5394, https://doi.org/10.5194/egusphere-egu2020-5394, 2020.

D1807 |
Yosra Afrasteh, Cornelis Slobbe, Martin Verlaan, Martina Sacher, and Roland Klees

Model-based hydrodynamic leveling is an efficient and flexible alternative method to connect islands and offshore tide gauges with the height system on land. The method uses a regional, high-resolution hydrodynamic model that provides total water levels. From the model, we obtain the differences in mean water level (MWL) between tide gauges at the mainland and at the islands or offshore platforms, respectively. Adding them to the MWL relative to the national height system at the mainland’s tide gauges realizes a connection of the island and offshore platforms with the height system on the mainland. Usually, the geodetic leveling networks are based on spirit leveling. So, as we can not make the direct connections between coastal countries, due to the inability of the spirit leveling method to cross the water bodies, they are weak in these regions. In this study, we assessed the impact of using model-based hydrodynamic leveling connections among the North Sea countries on the quality at which the European Vertical Reference System can be realized. In doing so, we combined the model-based hydrodynamic leveling data with synthetic geopotential differences among the height markers of the Unified European Leveling Network (UELN) used to realize the European Vertical Reference Frame 2019. The uncertainties of the latter data set were provided by the BKG. The impact is assessed in terms of both precision and reliability. We will show that adding model-based hydrodynamic leveling connections lowers the standard deviations of the estimated heights in the North Sea countries significantly. In terms of reliability, no significant improvements are observed.

How to cite: Afrasteh, Y., Slobbe, C., Verlaan, M., Sacher, M., and Klees, R.: Model-based hydrodynamic leveling; a power full tool to enhance the quality of the geodetic networks, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-19292, https://doi.org/10.5194/egusphere-egu2020-19292, 2020.

D1808 |
Hussein Abd-Elmotaal and Norbert Kühtreiber

The coverage of the gravity data plays an important role in the geoid determination. This paper tries to answer whether different geoid determination techniques would be affected similarly by such gravity data coverage. The paper presents the determination of the gravimetric geoid in two different countries where the gravity coverage is quite different. Egypt has sparse gravity data coverage over relatively large area, while Austria has quite dense gravity coverage in a significantly smaller area. Two different geoid determination techniques are tested. They are Stokes’ integral with modified Stokes kernel, for better combination of the gravity field wavelengths, and the least-squares collocation technique. The geoid determination has been performed within the framework of the non-ambiguous window remove-restore technique (Abd-Elmotaal and Kühtreiber, 2003). For Stokes’ geoid determination technique, the Meissl (1971) modified kernel has been used with numerical tests to obtain the best cap size for both geoids in Egypt and Austria. For the least-squares collocation technique, a modelled covariance function is needed. The Tscherning-Rapp (Tscherning and Rapp, 1974) covariance function model has been used after being fitted to the empirically determined covariance function. The paper gives a smart method for such covariance function fitting. All geoid are fitted to GNSS/levelling geoids for both countries. For each country, the computed two geoids are compared and the correlation between their differences versus the gravity coverage is comprehensively discussed.

How to cite: Abd-Elmotaal, H. and Kühtreiber, N.: Effect of gravity data coverage on geoid determination: comparison between Stokes and Collocation techniques in Egypt and Austria, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-12900, https://doi.org/10.5194/egusphere-egu2020-12900, 2020.

D1809 |
Yan Ming Wang

The effect of the topographic potential difference and the gravity correction on the geoid-quasigeoid separation are usually ignored in numerical computations. Those effects are computed in a mountainous Colorado region by using the digital elevation model SRTM v4.1 and terrestrial gravity data. The effects are computed at 1′X1′ grid size in the region. The largest effect is the topographic potential difference. It reaches a maximum of 19.0 cm with a standard deviation of 1.8 cm over the whole region. The gravity correction is smaller, but it still reaches a maximum of 3.0 cm with a standard deviation 0.3 cm for the whole region. The combined (ignored) effect ranges from -12.2 to 20.0 cm, with a standard deviation of 1.8 cm for the region. This numerical computation shows that the ignored terms must be taken into account for cm-geoid computation in mountainous regions.

How to cite: Wang, Y. M.: Effect of topographic potential difference and gravity correction on the geoid-quasigeoid separation, EGU General Assembly 2020, Online, 4–8 May 2020, EGU2020-20380, https://doi.org/10.5194/egusphere-egu2020-20380, 2020.

D1810 |
Development and evaluation of the xGEOID20 Digital Elevation Model at NGS
Jordan Krcmaric