G1.5

We present a numerical approach for solving the oblique derivative boundary value problem (BVP) based on the finite element method (FEM) with mapped infinite elements. To that goal, we formulate the BVP consisting of the Laplace equation in 3D semi-infinite domain outside the Earth which is bounded by the approximation of the Earth's surface where the oblique derivative boundary condition is given. At infinity, regularity of the disturbing potential is prescribed. As the numerical method, we have implemented the FEM with mapped infinite elements, where the computational domain is divided into
two centrical parts, one meshed with finite elements and one with infinite ones. In numerical experiments, we firstly test a convergence of the proposed numerical scheme and then we deal with global gravity field modelling using EGM2008 data. To perform such numerical experiments, we create a special discretization of the Earth's surface to fulfil the conditions that arise from correct geometrical properties of finite elements. Then a reconstruction of EGM2008 aims to indicate efficiency of the presented numerical approach.

How to cite: Macák, M., Minarechová, Z., Čunderlík, R., Mikula, K., and Tomek, L.: Global gravity field modelling by the finite element method involving mapped infinite elements., EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-2429, https://doi.org/10.5194/egusphere-egu22-2429, 2022.

Spherical harmonic transforms aiming at degrees as high as a few tens of thousands are vital in geodesy to improve our knowledge of the Earth's gravity field.  A prominent example is spectral gravity forward modelling of topographic masses, which is able to approximate fine gravity field structures up to the sub-km-level and beyond (degree ~20,000 and higher).  Driven by these applications, we have developed CHarm, a C library to perform spherical harmonic transforms.  CHarm is centered around (but not limited to) high-degree expansions, say, well beyond degree 2700.  Its goal is to be numerically stable on the one hand, while achieving reasonable computational efficiency with minimized memory requirements on the other hand.  Supported are surface spherical harmonic analysis and solid (3D) synthesis, both with point and area-mean data values.  Standard quadratures due to Gauss--Legendre and Driscoll--Healy are implemented for exact harmonic analysis of point data values.  The library can be compiled in double precision or, in case higher numerical accuracy is sought, in quadruple precision.  For efficient FFT transforms along the latitude parallels, the state-of-the-art FFTW library is employed to boost the performance.  Unique to CHarm is a routine integrating solid spherical harmonic expansions on band-limited undulated surfaces.  It can deliver, for instance, area-mean potential values on planetary surfaces.  Available are also routines to compute Fourier coefficients of Legendre functions and integrals of a product of two spherical harmonics or of two Legendre functions over a restricted domain.  To utilize the power of multicore processors, CHarm can be compiled with enabled parallelization on shared-memory architectures (OpenMP).  A significant effort is put into the documentation of the library (HTML, PDF) to allow its easy use.

In this contribution, we discuss the motivation behind the development of CHarm, explain its main functionalities and demonstrate some usage case studies.  Within a high-degree closed-loop synthetic environment, we assess the numerical accuracy, the computational speed and the memory management of the library.  A discussion on the future work closes the contribution.  CHarm is available at https://edisk.cvt.stuba.sk/~xbuchab/charm/doc/index.html.

How to cite: Bucha, B.: CHarm: C library to work with spherical harmonics up to almost arbitrarily high degrees, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-11206, https://doi.org/10.5194/egusphere-egu22-11206, 2022.

For the representation of the Earth’s global gravity field, Spherical Harmonics (SH) are widely used in geodetic community. On the other hand, for the representation of a local or regional gravity field, Spherical Cap Harmonics (SCH) and Rectangular Harmonics (RH) are alternative techniques with important advantages over SH. Although SCH are extensively presented in literature, RH are found in very few applications, especially of the gravity field. This work derives different functional forms of the disturbing potential, outside of the Earth’s masses, using RH. Also, the necessary transformation from geocentric into local rectangular coordinates is presented. The Rectangular Harmonic Coefficients (RHC) of the different mathematical models of the disturbing potential can be estimated through a least squares’ adjustment process. In order to select the best mathematical model, numerical experiments, based on data generated from a geopotential model, are conducted and the results are validated. Then, for the best model of the disturbing potential, its functionals (gravity anomaly and disturbance, height anomaly, geoid undulation and deflection of vertical) are given in terms of RHC. We conclude that RH representations are both suitable and convenient for the modelling of the local or regional gravity field.

How to cite: Panou, G. and Korakitis, R.: Modelling the local gravity field by rectangular harmonics with numerical validations, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-1880, https://doi.org/10.5194/egusphere-egu22-1880, 2022.

Gravity forward modelling is one of the fundamental topics in geodesy and geophysics. A spherical shell is a commonly used reference model among the mass bodies for the spatial domain of gravity forward modelling. The reason is that it has simple analytical expressions for gravitational effects (e.g. gravitational potential (GP), gravity vector (GV), gravity gradient tensor (GGT), and gravitational or gravity curvatures (GC)). The finer grid size will need more computation time when adopting the numerical strategy of a spherical shell discretized using tesseroids. This contribution presents the simpler analytical expressions for the GV and GGT of a homogeneous zonal band. The new analytical formula of the GC of a homogeneous zonal band is derived. The computation time and relative errors of the GP, GV, GGT, and GC between a spherical zonal band and a spherical shell discretized using tesseroids are quantitatively investigated with different grid sizes. Numerical results reveal that the computation time of a spherical zonal band discretized using tesseroids is about 180/n (i.e. n is the grid size) times less than that of a spherical shell discretized using tesseroids in double and quadruple precision. The relative errors' mean values of the GP, GV, GGT, and GC for a spherical zonal band discretized using tesseroids are smaller than those for a spherical shell discretized using tesseroids. In short, the benefit of a spherical zonal band in comparison with a spherical shell discretized using tesseroids regarding both the computation time and errors is confirmed numerically. The numerical approach of a spherical zonal band discretized using tesseroids can be applied instead of the classical numerical strategy in numerical evaluation of a tesseroid or other spherical mass bodies in gravity field modelling. This study is supported by the project funded by China Postdoctoral Science Foundation (Grant No. 2021M691402).

How to cite: Deng, X.-L.: A comparison of gravitational effects between a spherical zonal band and a spherical shell discretized using tesseroids, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-875, https://doi.org/10.5194/egusphere-egu22-875, 2022.

Geodetic boundary-value problems (BVPs) and their solutions represent an important tool for describing and modelling potential fields such as the Earth’s gravitational field. Solutions to spherical geodetic BVPs lead to spherical harmonic series or surface convolution integrals with Green’s kernel functions. New BVPs have recently been formulated reflecting development of sensors. BVPs have been also developed for observables measured by kinematic sensors on moving platforms, i.e., airplanes and satellites. Solutions to BVPs for higher-order derivatives of the gravitational potential as boundary conditions are represented by multiple integral transforms. For example, solutions to gravimetric, gradiometric and gravitational curvature BVPs are represented by two, three and four integral transforms, respectively. Theoretically, each of the nine transforms provides an identical value of the gravitational potential, but practically, when discrete noisy observations are exploited, they provide different estimates. Combination of solutions to the above mentioned geodetic BVPs in terms of surface integrals with Green’s kernel functions by a spectral method is investigated in this contribution. It is assumed that the first-, second- and third-order directional derivatives of the Earth’s gravitational potential can be measured at the satellite altitude. They are downward continued to the Earth’s surface and converted into height anomalies. Thus, the spectral combination method serves in our numerical procedure also as the downward continuation technique. The spectral combination method requires deriving corresponding spectral weights for all nine estimators. A generalized formula for evaluation of spectral weights for the estimators is formulated. Properties of spectral combinations are investigated in both spatial and spectral domains.

How to cite: Pitoňák, M., Šprlák, M., and Novák, P.: Combination of integral transforms by spectral weighting – an overview , EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-9610, https://doi.org/10.5194/egusphere-egu22-9610, 2022.

Theoretically, solving the geodetic boundary-value problems, we don't need any height systems to include them into integral equations. E.g. so called telluroid and the normal gravity on it are determined not by the normal height itself, but by the curvilinear coordinates of the points with the normal geopotential difference equal to the real geopotential difference. The length of the normal forceline, determining normal height value, is secondary. Similarly, gravity anomalies include normal gravity, determined also by the same geopotential difference in normal field.

Practically, using of the measured geopotential differences in geodesy is uncomfortable, since the corresponding levelling staff would have the variable step of the measuring scale, depending on the position of the point on the earth's surface and in space. Comparison and standardization of that staff is impossible. Then all the height systems we introduce to convert the geopotential values into the linear measure are non-optimal.

To determine the geoid at the same time with the orthometric height, the three only practically ways are possible (first fig.).

First way is the vertical spirit levelling, when the gravimeter is lowered into a vertical well and readings are taken from it at equal distances (a). The point with the geopotential number equal to zero will show us the point “on” the geoid, the rope length is the orthometric height. The second way is similar to the first with the spirit levelling along the paths on the walls of the quarry (b). The third way is a mechanical construction of a tunnel, the floor of which starts from the sea level and is built at a constant zero elevation (c).

Even if we know the upper crust mass distribution (with accuracy we need we must consider it completely unknown), the difficult volume integrals must be calculated for any benchmark.

The normal height is determined when M. S. Molodensky (1945) formulate his integro-differential equation (p. 55 of the English translation): “we compute the [curvilinear] coordinate q corresponding to the known potential of the real Earth..., neglecting the disturbing potential and the deflection of the vertical – an obvious first approximation”. In other words we may reformulate this, that the normal height is the ortometric height in the normal field. Moreover, the role of the geoid in normal field plays the level ellipsoid, not the quasigeoid (second fig.)!

In general, we don't need in “quasigeoid” in any physical or geometrical meaning, e.g. for the height measuring, as a “brother” of the geoid or in the BVP solving. So, strictly speaking, the quasigeoid is not a “vertical reference surface”, and the normal heights they are counted/measured not from ellipsoid nor from quasigeoid. The height mark is calculated and assigned as a “passport value” to each point of the earth’s surface.

How to cite: Popadyev, V. and Rakhmonov, S.: Some remarks about orthometric and normal height systems, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-10246, https://doi.org/10.5194/egusphere-egu22-10246, 2022.

A study by Afrasteh et al. (2021) has shown that combining model-based hydrodynamic leveling data with data of the Unified European Leveling Network (UELN) has great potential to improve the quality of the European Vertical Reference Frame (EVRF). In the current study, we made our first attempt to actually include the model-based hydrodynamic leveling data as new observations and compute a new realization for the European Vertical Reference System (EVRS). Please note, at this stage our results are provisional and should not be considered as an official realization for EVRS. For the spirit leveling data, we have used the potential differences from UELN, including the third leveling epoch in Great Britain. To generate the model-based hydrodynamic leveling data, 3D DCSM-FM hydrodynamic model that covers the North-east Atlantic Ocean including the North Sea is used to simulate the mean water level for January 1997 to January 2019. The tide gauges records covering the same period have been collected for the North Sea countries to compute the observed water level time series. The difference between observation- and the model-derived mean water level is used to generate the noise model for the hydrodynamic leveling data. We observe an improvement in the precision of the estimated heights in all coastal countries surrounding the 3D DCSM-FM domain. Moreover, our results show that adding model-based hydrodynamic leveling connections significantly reduces the south-north tilt in Great Britain, comparing the EVRF heights with the EGG2015 geoid model. Such a tilt in the British vertical datum, which is caused by a systematic error in the British leveling observations, has been reported in several studies. Our results show that using the model-based hydrodynamic leveling data could solve this problem in the British spirit leveling-based network and provide a stronger tie between Great Britain and other North Sea countries.

Y. Afrasteh, D. C. Slobbe, M. Verlaan, M. Sacher, R. Klees, H. Guarneri, L. Keyzer, J. Pietrzak, M. Snellen, and F. Zijl. The potential impact of hydrodynamic leveling on the quality of the European vertical reference frame. Journal of Geodesy, 95(8), 2021. doi: 10.1007/s00190-021-01543-3.

How to cite: Afrasteh, Y., Slobbe, C., Verlaan, M., Sacher, M., Klees, R., Guarneri, H., Keyzer, L., Pietrzak, J., Snellen, M., and Zijl, F.: New realization for European vertical reference system; a first attempt to include the hydrodynamic leveling data, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-6136, https://doi.org/10.5194/egusphere-egu22-6136, 2022.

The quality of a geodetic network is classically determined with the precision criteria computed from the cofactor matrix of the unknown parameters. One of the precision criteria having more information compared to the Helmert position error and the mean error of the unknown parameters is the Helmert mean error ellipsoid. In two-dimensional networks – and the celestial reference frame realized by extragalactic radio sources can be considered as such - the Helmert mean error ellipse consists of three parameters which are the semi-major and semi-minor axis of the error ellipse and the direction of the semi-major axes. In a well-designed geodetic network, the error ellipses should have homogenous structures. In other words, the semi-axes of the error ellipses for all radio sources should be similar. In this study, daily IVS sessions of the CONT17 were evaluated with The Potsdam Open Source Radio Interferometry Tool (PORT) and the parameters of the Helmert mean error ellipses were computed for the radio sources in each session. Also, the results are compared with the number of observations and the angular position of the radio sources. As a result of this study, it can be seen how the precision criteria are affected depending on the angular position of the radio sources.

How to cite: Küreç Nehbit, P., Glaser, S., Lunz, S., Heinkelmann, R., Schuh, H., and Konak, H.: Applying Precision Criteria to the Radio Sources in the Daily IVS Sessions, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-5626, https://doi.org/10.5194/egusphere-egu22-5626, 2022.

Since 2014, total station-based QDaedalus astrogeodetic measurement systems have been used to observe astronomical latitudes and longitudes to determine the astrogeodetic deflection of the vertical (DoV). In this study, the Leica Nova MS60 MultiStation-based QDaedalus system’s precision was determined at the HEIG-VD test station, located on the university campus in Yverdon-les-Bains, Switzerland. The data were collected over 13 nights (in an observation period of 44 days from February-April 2021), and comprise 115 series of observations performed over 26 sessions. The term “series” here describes the DoV data obtained during a specified period; QDaedalus observations were executed at ~15 minutes per series, and up to seven series of observations were conducted per session. The standard deviations (SDs) were calculated as 0.11″ and 0.09″ for the N-S and E-W components of the DoV, respectively. The SDs of the results from the HEIG-VD test station show that the N-S DoV components are not as precise as the E-W DoV components. There is a systematic trend in the observed N-S DoV data; the spread of the data in the N-S direction (0.92″) is larger than in the E-W direction (0.71″). The large trend in the N-S direction may be explained by the 0.008″/day trend (0.38″ over the 44-day observation period) in the N-S DoV components; however, this will require further investigation.

This study is the most extensive thus far for determining the precision of the QDaedalus astrogeodetic measurement system. We can conclude that the precisions of the two components lie on the same order of magnitude of 0.1″. These results and the applied method show that the MS60-based QDaedalus system is at least as reliable as the previously-reported total station-based QDaedalus systems. As a result, the MS60-based QDaedalus system can be used effectively in astrogeodetic applications that require high precision. Also, this study demonstrates that astrogeodetic test observations can be conducted at the HEIG-VD test station to determine the precision of newly installed QDaedalus systems.

How to cite: Albayrak, M., Willi, D., and Guillaume, S.: A Performance Analysis of the Leica MS60 MultiStation-Based QDaedalus Astrogeodetic Measurement System at the HEIG-VD Test Station, Switzerland, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-2610, https://doi.org/10.5194/egusphere-egu22-2610, 2022.

We present results for geodetic SAR which is a technique in the field of geodesy and remote sensing that enables the localization of specifically designed radar targets. It might help to connect the GNSS network to tide gauge stations and then to link the sea level records of tide gauge stations to the geometric network. In further perspective it will also enable the determination of vertical movements of the Earth's crust at these stations, allowing the estimation of the absolute value of sea level changes in various regions of the world, which are important in the study of climate change. In order to investigate the feasibility of using active SAR transponders ESA Project Baltic+ Theme No. 5. ‘Geodetic SAR for Baltic Height System Unification and Baltic Sea Level Research (SAR-HSU)’ was completed in 2019 – 2021 by international consortium. During the project SAR novel active transponders were located around the Baltic Sea. Among the locations, two transponders were placed in Wladyslawowo and Leba, Poland under the care of the Space Research Centre of the Polish Academy of Sciences (SRC PAS).

The installation of permanent radar targets allows for long-term position monitoring. The technique is a particularly interesting for displacement and height changes observations. The research illustrates the results acquired from the electronic corner reflectors operating in Poland. For purpose of this research SAR images captured by the Sentinel-1 are used as ESA offers unrestricted access to all the data acquired at study region. Level 1 SLC products together with geodetic data are the main input for the study. With a repeat cycle of 6 days, the number of Sentinel-1 SAR observations per test site amounts to about 180 measurements for one year.

We present the outcomes of ECR positioning from July 2021 to January 2022 when further tests of active transponders were conducted beyond the end of the project. The research is carried out with the software developed in SRC PAS and designed for purposes of geodetic SAR. Software consists of several modules e.g. for data preparation (including SAR data, EOP, precise orbits, ionosphere and troposphere models) or for processing data related to geodynamic effects and corrections to radar measurements. Here we present results for Absolute Location Error in the azimuth and range. We show our experience in processing data for active transponders and our comments on their maintenance and exploitation.

How to cite: Kur, T., Zdunek, R., Nastula, J., Gisinger, C., and Śliwińska, J.: Geodetic SAR – the use of electronic corner reflectors in Wladyslawowo and Leba, Poland, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-9425, https://doi.org/10.5194/egusphere-egu22-9425, 2022.