Optimized formulas for the gravitational field of a tesseroid

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton\u27s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid\u27s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45% compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach

External gravitational field of a homogeneous ellipsoidal shell: a reference for testing gravity modelling software

There are numerous applications in geodesy and other geo-sciences in which the gravitational potential effect or other functions of the potential are computed by forward modelling from a given mass distribution. Different volume discretisations, e.g. prisms, tesseroids or mass layers are used. In order to control the numerical realisation of the forward calculation in the practical application, e.g. in reduction tasks, these evaluation programs should be verified against rigorous analytical solutions. In this contribution, a closed analytical solution for the potential of an ellipsoidal shell as a test body is presented. Furthermore, we derive the respective closed formulae for the gravity vector and the gravity gradient tensor. Program implementations of the tesseroid approach are compared on the basis of this ellipsoidal mass arrangement. For the practical usage, fast-converging expansions in spherical harmonics are provided in addition. The derivation of the formulae is based on a closed solution of the potential of a homogeneous ellipsoid for computation points situated on the rotation axis, which then is extended to the external space

Geodätische Woche 2009 : 22. - 24. September 2009, Messe Karlsruhe, Rheinstetten im Rahmen der INTERGEO ­­- Kongress und Fachmesse für Geodäsie, Geoinformation und Landmanagement. Abstracts

Die Geodätische Woche bietet ein Forum für die Diskussion von aktuellen wissenschaftlichen Entwicklungen und Anwendungen in der Geodäsie. Die Geodätische Woche wird im Rahmen der Intergeo®-Kongress und Fachmesse für Geodäsie, Geoinformation und Landmanagement durchgeführt. In dieser Monographie sind die Zusammenfassungen (Abstracts) der für die Geodätische Woche 2009 (Ort: Karlsruhe) eingereichten Beiträge zusammengestellt

A Precise Geoid Model for Africa: AFRgeo2019

In the framework of the IAG African Geoid Project, an attempt towards a precise geoid model for Africa is presented in this investigation. The available gravity data set suffers from significantly large data gaps. These data gaps are filled using the EIGEN-6C4 model on a 15′× 15′ grid prior to the gravity reduction scheme. The window remove-restore technique (Abd-Elmotaal and Kühtreiber, Phys Chem Earth Pt A 24(1):53–59, 1999; J Geod 77(1–2):77–85, 2003) has been used to generate reduced anomalies having a minimum variance to minimize the interpolation errors, especially at the large data gaps. The EIGEN-6C4 global model, complete to degree and order 2190, has served as the reference model. The reduced anomalies are gridded on a 5′× 5′ grid employing an un-equal weight least-squares prediction technique. The reduced gravity anomalies are then used to compute their contribution to the geoid undulation employing Stokes’ integral with Meissl (Preparation for the numerical evaluation of second order Molodensky-type formulas. Ohio State University, Department of Geodetic Science and Surveying, Rep 163, 1971) modified kernel for better combination of the different wavelengths of the earth’s gravity field. Finally the restore step within the window remove-restore technique took place generating the full gravimetric geoid. In the last step, the computed geoid is fitted to the DIR_R5 GOCE satellite-only model by applying an offset and two tilt parameters. The DIR_R5 model is used because it turned out that it represents the best available global geopotential model approximating the African gravity field. A comparison between the geoid computed within the current investigation and the existing former geoid model AGP2003 (Merry et al., A window on the future of geodesy. International Association of Geodesy Symposia, vol 128, pp 374–379, 2005) for Africa has been carried out

Evaluation of the Recent African Gravity Databases V2.x

In the framework of the activities of the IAG Sub-Commission on the gravity and geoid in Africa, a recent set of gravity databases has been established. They are namely: AFRGDB_V2.0 and AFRGDB_V2.2. The AFRGDB_V2.0 has been created using the window remove-restore technique employing EGM2008 as geopotential Earth model complete to degree and order 1800. The AFRGDB_V2.2 has been established using the Residual Terrain Model (RTM) reduction technique employing GOCE DIR_R5 complete to degree and order 280, using the best RTM reference surface. The available gravity data set for Africa, used to establish the above mentioned two independently derived databases, consists of shipborne, altimetry derived gravity anomalies and of land point gravity data. In particular, the data set of point gravity values shows clear deficits with regard to a homogeneous data coverage over the completely African continent. The establishment of the gravity databases has been carried-out using the weighted least-squares prediction technique, in which the point gravity data on land has got the highest precision, while the shipborne and altimetry gravity data got a moderate precision. In this paper a new gravity data set on land and on sea, which became recently available for the IAG Sub-Commission on the gravity and geoid in Africa, located partly in the gap areas of the data set used for generating the gravity databases, has been employed to evaluate the accuracy of the previously created gravity databases. The results show reasonable accuracy of the established gravity databases considering the large data gaps in Africa