The Energy Balance Approach
The energy balance approach to geopotential modeling from satellite-to-satellite tracking offers the advantage of emulating an in-situ measurement system, analogous to satellite-borne gravity gradiometry, but in terms of differences in potential. Although the theoretical background is well known, based on the conservation of energy in celestial mechanics, its application to geopotential determination from satellite tracking had to await the capability for accurate, independent, kinematic orbit determination. This is now possible with Global Navigation Satellite Systems (GNSS), such as GPS. The precision tracking between two co-orbiting satellites brings additional short-wavelength information to the in-situ measurement. These tutorial notes derive the exact relationships between the gravitational potential and the orbital state vectors from the energy balance perspective, both in the inertial frame and in the Earth-fixed frame. Also derived are the equations that specifically incorporate the range-rate between co-orbiting satellites. Particular attention is given to the rotation potential and the temporal dependence of the potential due to various sources, including tidal variations, Earth’s orientation and deformation, and terrestrial mass fluxes. A detailed analysis of magnitudes then leads to possibly acceptable approximations. It also shows for a satellite configuration such as GRACE (Gravity Recovery and Climate Experiment) that the radial component of the relative velocity between two co-orbiting satellites is as important in magnitude as the along-track component. Thus, the measured inter-satellite range-rate, for example, cannot be used alone to determine the potential difference. However, it is also shown that the short-wavelength content of the potential resides more in the along-track component than in the radial component, which demonstrates the significance of the range-rate measurement. A simple error analysis of the system identifies the requirements for state-vector accuracies (both for position and velocity) in relation to the range-rate accuracy. The observational equations for satellite-to-satellite tracking are derived and related to both global and regional geopotential modeling. The discussion concludes with a sample of published case studies that have demonstrated the energy balance approach and achieved tangible results in large-scale hydrological mass flux monitoring.
KeywordsGlobal Navigation Satellite System Global Navigation Satellite System Inertial Frame Energy Balance Equation Ocean Tide
These lecture notes would not have been completed without the stimulating, helpful, and much appreciated discussions with colleagues, specifically Dr. Junyi Guo, Dr. Kun Shang, and Mr. Nlingi Habana (in particular, who helped with the exercises) of the Division of Geodetic Science, School of Earth Sciences, at the Ohio State University, as well Prof. Shin Chan Han (University of Newcastle), Prof. Srinivas Bettadpur (CSR, University of Texas), and Prof. Jakob Flury and Dr. Majid Naeimi of the Institut für Erdmessung, Leibniz Universität Hannover. Particular gratitude from the author goes to the sponsor of these lectures, the Wilhelm and Else Heraeus Foundation.