On the Numerical Implementation of a Perturbation Method for Satellite Gravity Mapping
In 2008 P. Xu (Celest Mech Dyn Astron, 100:231–249) proposed a strictly kinematic perturbation method for determining the Earth’s gravitational field from continuous satellite tracking. The main idea is to process orbital arcs of arbitrary length, thus mi
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Abstract
In 2008 P. Xu (Celest Mech Dyn Astron, 100:231–249) proposed a strictly kinematic perturbation method for determining the Earth’s gravitational field from continuous satellite tracking. The main idea is to process orbital arcs of arbitrary length, thus minimizing superfluous parameter estimation associated with stitching together short-arc solutions, and at the same time formulating the problem in terms of standard linear parameter estimation. While the original formulation appears mathematically robust, its nested quadruple alongtrack integrations are computationally challenging. We reduce the formulation to double integrals and show that the method is numerically not feasible as originally envisaged. On the other hand, by abandoning the rigorous Gauss-Markov formalism, we show the numerical feasibility of processing multiple-day orbital arcs. The methodology lends itself to high-low and low-low satellite-to-satellite tracking, or combinations thereof, as for GRACE-like systems. Keywords
GNSS satellite tracking Gravitational field estimation Numerical orbit integration Satellite perturbation theory
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Introduction
With the modern ability to track low-Earth-orbiting satellites continually and uniformly with Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS), the standard methods to extract estimates of the Earth’s gravitational field from satellite tracking data may be re-visited. Prior to GNSS tracking, ground-based tracking created a patch-work of data as a satellite rose and set at any particular tracking station; and, accordingly, a good a priori or reference orbit was essential in stitching the observed arcs together. A perturbation theory based on
C. Jekeli () • N. Habana Division of Geodetic Science, School of Earth Sciences, Ohio State University, Columbus, OH, USA e-mail: [email protected]
Keplerian elements, for example, served to separate secular and long-period orbital variations from the more short-term resonances, which facilitated this estimation process. These techniques are still practiced today in various forms; a good review is given in the volume by Naeimi and Flury (2017). Xu (2008) proposed a radical change from this methodology specifically in view of the proven accurate tracking capabilities with GNSS. Continual and uniform high-accuracy tracking, arguably obviates piecing together short arcs and simplifies the overall problem setup as all formulations may be made with the straightforward use of Cartesian coordinates. The idea certainly has tremendous theoretical and practical appeal and in this paper we aim to elucidate this in the simplest terms, but the numerical implementation, never attempted by the originator, puts at least some limitations on the length of the orbital arc that constitutes a segment of the overall estimation process. It is this aspect of the proposed methodology that we wish to highlight and further develop in this short note.
International Association of Geodesy Symposia, https://doi.org/10.1007/1345_2019_81,
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