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’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 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.     

The Rock–Water–Ice Topographic Gravity Field Model RWI_TOPO_2015 and Its Comparison to a Conventional Rock-Equivalent Version

RWI_TOPO_2015 is a new high-resolution spherical harmonic representation of the Earth’s topographic gravitational potential that is based on a refined Rock–Water–Ice (RWI) approach. This method is characterized by a three-layer decomposition of the Earth’s topography with respect to its rock, water, and ice masses. To allow a rigorous separate modeling of these masses with variable density values, gravity forward modeling is performed in the space domain using tesseroid mass bodies arranged on an ellipsoidal reference surface. While the predecessor model RWI_TOPO_2012 was based on the \(5'\times 5'\) global topographic database DTM2006.0 (Digital Topographic Model 2006.0), the new RWI model uses updated height information of the \(1'\times 1'\) Earth2014 topography suite. Moreover, in the case of RWI_TOPO_2015, the representation in spherical harmonics is extended to degree and order 2190 (formerly 1800). Beside a presentation of the used formalism, the processing for RWI_TOPO_2015 is described in detail, and the characteristics of the resulting spherical harmonic coefficients are analyzed in the space and frequency domain. Furthermore, this paper focuses on a comparison of the RWI approach to the conventionally used rock-equivalent method. For this purpose, a consistent rock-equivalent version REQ_TOPO_2015 is generated, in which the heights of water and ice masses are condensed to the constant rock density. When evaluated on the surface of the GRS80 ellipsoid (Geodetic Reference System 1980), the differences of RWI_TOPO_2015 and REQ_TOPO_2015 reach maximum amplitudes of about 1 m, 50 mGal, and 20 mE in terms of height anomaly, gravity disturbance, and the radial–radial gravity gradient, respectively. Although these differences are attenuated with increasing height above the ellipsoid, significant magnitudes can even be detected in the case of the satellite altitudes of current gravity field missions. In order to assess their performance, RWI_TOPO_2015, REQ_TOPO_2015, and RWI_TOPO_2012 are validated against independent gravity information of current global geopotential models, clearly demonstrating the attained improvements in the case of the new RWI model.     

A Wavelet-Based Assessment of Topographic-Isostatic Reductions for GOCE Gravity Gradients

Gravity gradient measurements from ESA’s satellite mission Gravity field and steady-state Ocean Circulation Explorer (GOCE) contain significant high- and mid-frequency signal components, which are primarily caused by the attraction of the Earth’s topographic and isostatic masses. In order to mitigate the resulting numerical instability of a harmonic downward continuation, the observed gradients can be smoothed with respect to topographic-isostatic effects using a remove–compute–restore technique. For this reason, topographic-isostatic reductions are calculated by forward modeling that employs the advanced Rock–Water–Ice methodology. The basis of this approach is a three-layer decomposition of the topography with variable density values and a modified Airy–Heiskanen isostatic concept incorporating a depth model of the Mohorovi?i? discontinuity. Moreover, tesseroid bodies are utilized for mass discretization and arranged on an ellipsoidal reference surface. To evaluate the degree of smoothing via topographic-isostatic reduction of GOCE gravity gradients, a wavelet-based assessment is presented in this paper and compared with statistical inferences in the space domain. Using the Morlet wavelet, continuous wavelet transforms are applied to measured GOCE gravity gradients before and after reducing topographic-isostatic signals. By analyzing a representative data set in the Himalayan region, an employment of the reductions leads to significantly smoothed gradients. In addition, smoothing effects that are invisible in the space domain can be detected in wavelet scalograms, making a wavelet-based spectral analysis a powerful tool.     

On High-Frequency Topography-Implied Gravity Signals for a Height System Unification Using GOCE-Based Global Geopotential Models

National height reference systems have conventionally been linked to the local mean sea level, observed at individual tide gauges. Due to variations in the sea surface topography, the reference levels of these systems are inconsistent, causing height datum offsets of up to ±1–2 m. For the unification of height systems, a satellite-based method is presented that utilizes global geopotential models (GGMs) derived from ESA’s satellite mission Gravity field and steady-state Ocean Circulation Explorer (GOCE). In this context, height datum offsets are estimated within a least squares adjustment by comparing the GGM information with measured GNSS/leveling data. While the GNSS/leveling data comprises the full spectral information, GOCE GGMs are restricted to long wavelengths according to the maximum degree of their spherical harmonic representation. To provide accurate height datum offsets, it is indispensable to account for the remaining signal above this maximum degree, known as the omission error of the GGM. Therefore, a combination of the GOCE information with the high-resolution Earth Gravitational Model 2008 (EGM2008) is performed. The main contribution of this paper is to analyze the benefit, when high-frequency topography-implied gravity signals are additionally used to reduce the remaining omission error of EGM2008. In terms of a spectral extension, a new method is proposed that does not rely on an assumed spectral consistency of topographic heights and implied gravity as is the case for the residual terrain modeling (RTM) technique. In the first step of this new approach, gravity forward modeling based on tesseroid mass bodies is performed according to the Rock–Water–Ice (RWI) approach. In a second step, the resulting full spectral RWI-based topographic potential values are reduced by the effect of the topographic gravity field model RWI_TOPO_2015, thus, removing the long to medium wavelengths. By using the latest GOCE GGMs, the impact of topography-implied gravity signals on the estimation of height datum offsets is analyzed in detail for representative GNSS/leveling data sets in Germany, Austria, and Brazil. Besides considerable changes in the estimated offset of up to 3 cm, the conducted analyses show that significant improvements of 30–40% can be achieved in terms of a reduced standard deviation and range of the least squares adjusted residuals.     

Comparison of two different approaches for computing the gravitational effect of a tesseroid

Studia Geophysica et Geodaetica - Forward modelling in the space domain is a very important task in geodesy and other geosciences. From topographical or isostatic information in the form of digital...