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's 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's 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 inte-T. Grombein · K. Seitz · B. Heck Geodetic Institute, Karlsruhe Institute of Technology (KIT) Englerstr. 7, Karlsruhe, D-76128 Germany Tel.: +49-721-608-42306 Fax: +49-721-608-46808 E-mail: grombein@kit.edu gral 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.
a b s t r a c tThis paper describes the computation and analysis of the Earth's short-scale gravity field through highresolution gravity forward modelling using the Shuttle Radar Topography Mission (SRTM) global topography model. We use the established residual terrain modelling technique along with advanced computational resources and massive parallelisation to convert the high-pass filtered SRTM topographycomplemented with bathymetric information in coastal zones -to implied short-scale gravity effects. The result is the ERTM2160 model (Earth Residual Terrain Modelled-gravity field with the spatial scales equivalent to spherical-harmonic coefficients up to degree 2160 removed). ERTM2160, used successfully for the construction of the GGMplus gravity maps, approximates the short-scale (i.e., $ 10 km down to $ 250 m) gravity field in terms of gravity disturbances, quasi/geoid heights and vertical deflections at $ 3 billion gridded points within 7601 latitude. ERTM2160 reaches maximum values for the quasi/geoid height of $ 30 cm, gravity disturbance in excess of 100 mGal, and vertical deflections of $ 30″ over the Himalaya mountains. Analysis of the ERTM2160 field as a function of terrain roughness shows in good approximation a linear relationship between terrain roughness and gravity effects, with values of $ 1.7 cm (quasi/geoid heights), $ 11 mGal (gravity disturbances) and 1.5″ (vertical deflections) signal strength per 100 m standard deviation of the terrain. These statistics can be used to assess the magnitude of omitted gravity signals over various types of terrain when using degree-2160 gravity models such as EGM2008. Applications for ERTM2160 are outlined including its use in gravity smoothing procedures, augmentation of EGM2008, fill-in for future ultra-high resolution gravity models in spherical harmonics, or calculation of localised or global power spectra of Earth's short-scale gravity field. ERTM2160 is freely available via http://ddfe.curtin.edu.au/ gravitymodels/ERTM2160.
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