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

Gravity forward modeling with a tesseroid-based Rock-Water-Ice approach – Theory and applications in the context of the GOCE mission and height system unification

Detailed information on the gravitational effect of the Earth’s topographic and isostatic masses can be calculated by gravity forward modeling. Within this book, the tesseroid-based Rock-Water-Ice (RWI) approach is developed, which allows a rigorous separate modeling of the Earth’s rock, water, and ice masses with variable density values. Besides a discussion and evaluation of the RWI approach, applications in the context of the GOCE satellite mission and height system unification are presented

Gravity forward modeling with a tesseroid-based Rock-Water-Ice approach – Theory and applications in the context of the GOCE mission and height system unification

Detailed information on the gravitational effect of the Earth\u27s topographic and isostatic masses is needed for various applications in geodesy and geophysics and can be calculated by gravity forward modeling (GFM). Within this thesis, the tesseroid-based Rock-Water-Ice (RWI) approach is developed, which provides an important contribution to state-of-the-art GFM. The basis of this approach is a rigorous separate modeling of the Earth’s rock, water, and ice masses with variable density values and a modified Airy-Heiskanen isostatic concept. For the numerical evaluation, optimized tesseroid formulas are elaborated that significantly reduce the computational demand. Besides a discussion and evaluation of the newly developed methods, applications in the context of the GOCE satellite mission and height system unification are presented. For the use in other research, several topographic-isostatic gravity field models are generated and made publicly available in terms of spherical harmonic coefficients

Determination and combination of monthly gravity field time series from kinematic orbits of GRACE, GRACE-FO and Swarm

Dedicated gravity field missions like GRACE and GRACE-FO use ultra-precise inter-satellite ranging observations to derive time series of monthly gravity field solutions. In addition, any (non-dedicated) Low Earth Orbiting (LEO) satellite with a dual-frequency GNSS receiver may also serve as a gravity field sensor. To this end, GPS-derived kinematic LEO orbit positions are used as pseudo-observations for gravity field recovery. Although less sensitive, this technique can provide valuable information for the monitoring of largescale time-variable gravity signals, particularly for those months where no inter-satellite ranging measurements are available. Due to a growing number of LEO satellites that collect continuous and mostly uninterrupted GPS data, the value of a combined multi-LEO gravity field time series is likely to increase in the near future. In this paper, we present monthly gravity field time series derived from GPS-based kinematic orbit positions of the GRACE, GRACE-FO and Swarm missions. We analyse their individual contribution as well as the additional benefit of their combination. For this purpose, two combination strategies at solution level are studied that are based on (i) least-squares variance component estimation, and (ii) stochastic properties of the gravity field solutions. By evaluating mass variations in Greenland and the Amazon river basin, the resulting gravity field time series are assessed with respect to superior solutions based on inter-satellite ranging

Reprocessed precise science orbits and gravity field recovery for the entire GOCE mission

ESA’s Gravity field and steady-state Ocean Circulation Explorer (GOCE) orbited the Earth between 2009 and 2013 for the determination of the static part of Earth’s gravity field. The GPS-derived precise science orbits (PSOs) were operationally generated by the Astronomical Institute of the University of Bern (AIUB). Due to a significantly improved understanding of remaining artifacts after the end of the GOCE mission (especially in the GOCE gradiometry data), ESA initiated a reprocessing of the entire GOCE Level 1b data in 2018. In this framework, AIUB was commissioned to recompute the GOCE reduced-dynamic and kinematic PSOs. In this paper, we report on the employed precise orbit determination methods, with a focus on measures undertaken to mitigate ionosphere-induced artifacts in the kinematic orbits and thereof derived gravity field models. With respect to the PSOs computed during the operational phase of GOCE, the reprocessed PSOs show in average a 8–9% better consistency with GPS data, 31% smaller 3-dimensional reduced-dynamic orbit overlaps, an 8% better 3-dimensional consistency between reduced-dynamic and kinematic orbits, and a 3–7% reduction of satellite laser ranging residuals. In the second part of the paper, we present results from GPS-based gravity field determinations that highlight the strong benefit of the GOCE reprocessed kinematic PSOs. Due to the applied data weighting strategy, a substantially improved quality of gravity field coefficients between degree 10 and 40 is achieved, corresponding to a remarkable reduction of ionosphere-induced artifacts along the geomagnetic equator. For a static gravity field solution covering the entire mission period, geoid height differences with respect to a superior inter-satellite ranging solution are markedly reduced (43% in terms of global RMS, compared to previous GOCE GPS-based gravity fields). Furthermore, we demonstrate that the reprocessed GOCE PSOs allow to recover long-wavelength time-variable gravity field signals (up to degree 10), comparable to information derived from GPS data of dedicated satellite missions. To this end, it is essential to take into account the GOCE common-mode accelerometer data in the gravity field recovery

On the combination of gravity field time series derived from kinematic positions of Low Earth Orbiting satellites

The Earth’s time-variable gravity field provides important information for the monitoring of changes in the Earth’s system. Dedicated satellite missions like GRACE and GRACE-FO use ultra-precise inter-satellite ranging observations to derive time series of monthly gravity field solutions. Alternative gravity field information can be obtained from the analysis of GPS-based kinematic orbit positions of Low Earth Orbiting (LEO) satellites. Although this technique is less sensitive, it can provide mostly uninterrupted time series, which is particularly valuable for those months where no inter-satellite ranging measurements are available from GRACE or GRACE-FO. Furthermore, the increasing number of operational LEO satellites makes it attractive to produce combined Multi-LEO gravity field solutions that will take advantage of a large number of observations and the variety of complementary orbital configurations. At the Astronomical Institute of the University of Bern (AIUB) GPS-based kinematic orbits have been generate for various LEO satellites like GRACE and GOCE or are routinely processed for operational missions like GRACE-FO, SWARM, Sentinel or Jason. In this contribution, we will use these kinematic LEO positions to perform gravity field recovery with the Celestial Mechanics Approach and to derive monthly gravity field times series. By evaluating mass trends and changes in Greenland, Antarctica and the Amazon river basin, the time series of different LEOs are compared with respect to superior solutions based on inter-satellite ranging. Finally, we will combine the gravity field solutions of different LEO satellites either on solution level using variance component estimation or on normal equation level and investigate their individual contribution and the additional value of the combination

Reprocessed precise science orbits and gravity field recovery for the entire GOCE mission.

ESA’s Gravity field and steady-state Ocean Circulation Explorer (GOCE) orbited the Earth between 2009 and 2013 for the determination of the static part of Earth’s gravity field. The GPS-derived precise science orbits (PSOs) were operationally generated by the Astronomical Institute of the University of Bern (AIUB). Due to a significantly improved understanding of remaining artifacts after the end of the GOCE mission (especially in the GOCE gradiometry data), ESA initiated a reprocessing of the entire GOCE Level 1b data in 2018. In this framework, AIUB was commissioned to recompute the GOCE reduced-dynamic and kinematic PSOs. In this paper, we report on the employed precise orbit determination methods, with a focus on measures undertaken to mitigate ionosphere-induced artifacts in the kinematic orbits and thereof derived gravity field models. With respect to the PSOs computed during the operational phase of GOCE, the reprocessed PSOs show in average a 8–9% better consistency with GPS data, 31% smaller 3-dimensional reduced-dynamic orbit overlaps, an 8% better 3-dimensional consistency between reduced-dynamic and kinematic orbits, and a 3–7% reduction of satellite laser ranging residuals. In the second part of the paper, we present results from GPS-based gravity field determinations that highlight the strong benefit of the GOCE reprocessed kinematic PSOs. Due to the applied data weighting strategy, a substantially improved quality of gravity field coefficients between degree 10 and 40 is achieved, corresponding to a remarkable reduction of ionosphere-induced artifacts along the geomagnetic equator. For a static gravity field solution covering the entire mission period, geoid height differences with respect to a superior inter-satellite ranging solution are markedly reduced (43% in terms of global RMS, compared to previous GOCE GPS-based gravity fields). Furthermore, we demonstrate that the reprocessed GOCE PSOs allow to recover long-wavelength time-variable gravity field signals (up to degree 10), comparable to information derived from GPS data of dedicated satellite missions. To this end, it is essential to take into account the GOCE common-mode accelerometer data in the gravity field recovery