Harmonic Correction for Residual Terrain Modelling (RTM) Technique in Physical Geodesy Applications

<p>       In physical geodesy, the harmonic correction (HC), as one of the main problems when using residual terrain modelling (RTM), has become a research focus of high-frequency gravity field modelling. Over past decades, though various methods have been proposed to handle the HC issues for RTM technique, most of them focused on the HC for RTM gravity anomaly rather than other gravity functionals, such as RTM geoid height and gravity gradient. In practice, the HC for RTM geoid height was generally assumed to be negligible, but a quantification is yet studied. In this study, besides the highlighted HC for gravity anomaly in previous studies, the expressions of HC terms for RTM geoid height are provided in the framework of the classical condensation method under infinite Bouguer plate approximation. The errors involved by various assumption of the classical condensation method, e.g., mass inconsistency between infinite masses in the HC and limited masses in the RTM, and planar assumption of the Earth’s surface, are further studied. Based on the derived formulas, the quantification of HC for RTM geoid height when reference surface is expanded to degree and order of 2,159 is given. Our results showed the significance of HC for RTM geoid height, with values up to ~10 cm, in cm-level and mm-level geoid determination. With integration masses extending up to a sufficient distance, such as 1° from calculation point for the determination of RTM geoid height, the errors due to an infinite Bouguer plate approximation are neglectable small. The validation through comparison with terrestrial measurements proved that the HC terms provided in this study can improve the accuracy of RTM derived geoid height and are expected to be useful for applications of RTM technique in regional and global gravity field modelling.</p>

Stable finite element method for solving the oblique derivative boundary value problems in geodesy

<p><span>We presents local gravity field modelling in a spatial domain using the finite element method (FEM). FEM as a numerical method is applied for solving the geodetic boundary value problem with oblique derivative boundary conditions (BC). We derive a novel FEM numerical scheme which is the second order accurate and more stable than the previous one published in [1]. A main difference is in applying the oblique derivative BC. While in the previous FEM approach it is considered as an average value on the bottom side of finite elements, the novel FEM approach is based on the oblique derivative BC considered in relevant computational nodes. Such an approach should reduce a loss of accuracy due to averaging. Numerical experiments present </span><span>(i) </span><span>a reconstruction of EGM2008 as a harmonic function over the extremely complicated Earth’s topography in the Himalayas and Tibetan Plateau, and (ii) local gravity field modelling in Slovakia with the high-resolution 100 x 100 m while using terrestrial gravimetric data.</span></p><p><span>[1] </span>Macák, Z. Minarechová, R. Čunderlík, K. Mikula, The finite element method as a tool to solve the oblique derivative boundary value problem in geodesy. Tatra Mountains Mathematical Publications. Vol. 75, no. 1, 63-80, (2020)</p>

The Finite Element Method as a Tool to Solve the Oblique Derivative Boundary Value Problem in Geodesy

In this paper, we propose a novel approach to approximate the solution of the Laplace equation with an oblique derivative boundary condition by the finite element method. We present and analyse diverse testing experiments to study its behaviour and convergence. Finally, the usefulness of this approach is demonstrated by using it to gravity field modelling, namely, to approximate the solution of a geodetic boundary value problem in Himalayas.

FVM approach for solving the oblique derivative BVP on unstructured meshes above the real Earth’s topography

<p>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].</p><p>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.</p><p> </p><p>[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</p>

High-resolution combined global gravity field modelling – The d/o 5,400 XGM2020 model

<p>Within this contribution we present the new experimental combined global gravity field model XGM2020. Key feature of this model is the rigorous combination of the latest GOCO06s satellite-only model with global terrestrial gravity anomalies on normal equation level, up to d/o 2159, using individual observation weights. To provide a maximum resolution, the model is further extended to d/o 5400 by applying block diagonal techniques.</p><p>To attain the high resolution, the incorporated terrestrial dataset is composed of three different data sources: Over land 15´ gravity anomalies (by courtesy of NGA) are augmented with topographic information, and over the oceans gravity anomalies derived from altimetry are used.  Corresponding normal equations are computed from these data sets either as full or as block diagonal systems.</p><p>Special emphasis is given to the novel processing techniques needed for very high-resolution gravity field modelling. As such the spheroidal harmonics play a central role, as well as the stable calculation of associated Legendre polynomials up to very high d/o. Also, a new technique for the optimal low-pass filtering of terrestrial gravity datasets is presented.</p><p>On the computational side, solving dense normal equation systems up to d/o 2159 means dealing with matrices of the size of about 158TB. Handling with matrices of such a size is very demanding, even for today’s largest supercomputers. Thus, sophisticated parallelized algorithms with focus on load balancing are crucial for a successful and efficient calculation.</p>