Geophysical Journal International Geophys. J. Int. (2015) 203, 1773–1786 doi: 10.1093/gji/ggv392 GJI Gravity, geodesy and tides GRACE time-variable gravity field recovery using an improved energy balance approach Kun Shang,1 Junyi Guo,1 C.K. Shum,1,2 Chunli Dai1 and Jia Luo3 1Division of Geodetic Science, School of Earth Sciences, The Ohio State University, 125 S. Oval Mall, Columbus, OH 43210, USA. E-mail: [email protected] 2Institute of Geodesy & Geophysics, Chinese Academy of Sciences, Wuhan, China 3School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Accepted 2015 September 14. Received 2015 September 12; in original form 2015 March 19 Downloaded from SUMMARY A new approach based on energy conservation principle for satellite gravimetry mission has been developed and yields more accurate estimation of in situ geopotential difference observ- ables using K-band ranging (KBR) measurements from the Gravity Recovery and Climate Experiment (GRACE) twin-satellite mission. This new approach preserves more gravity in- http://gji.oxfordjournals.org/ formation sensed by KBR range-rate measurements and reduces orbit error as compared to previous energy balance methods. Results from analysis of 11 yr of GRACE data indicated that the resulting geopotential difference estimates agree well with predicted values from of- ficial Level 2 solutions: with much higher correlation at 0.9, as compared to 0.5–0.8 reported by previous published energy balance studies. We demonstrate that our approach produced a comparable time-variable gravity solution with the Level 2 solutions. The regional GRACE temporal gravity solutions over Greenland reveals that a substantially higher temporal resolu- at Ohio State University Libraries on December 20, 2016 tion is achievable at 10-d sampling as compared to the official monthly solutions, but without the compromise of spatial resolution, nor the need to use regularization or post-processing. Key words: Satellite geodesy; Geopotential theory; Time variable gravity. method, various alternative approaches have also been proposed 1 INTRODUCTION and implemented, such as mascon approach (Rowlands et al. 2005, Launched in March 2002, the Gravity Recovery and Climate Ex- 2010), short-arc approach (Mayer-Gurr¨ et al. 2007; Kurtenbach periment (GRACE) mission (Tapley et al. 2004a) has been map- et al. 2009), celestial mechanics approach (Meyer et al. 2012), ac- ping Earth’s time-variable gravity field for more than a decade and celeration approach (Ditmar & van Eck van der Sluijs 2004;Chen achieving remarkable and even transformative scientific advances et al. 2008;Liuet al. 2010) and the energy balance approach (Jekeli (Cazenave & Chen 2010). From the data collected by the K-band 1999; Han et al. 2006; Ramillien et al. 2011; Tangdamrongsub et al. ranging (KBR) low–low satellite-to-satellite tracking (SST) as well 2012). The last approach is the focus of this paper. as the high-low GPS tracking, monthly mean gravity field mod- Energy balance approach, also known as energy integral ap- els, known as the Level-2 data products, have been routinely esti- proach, can be traced back to the 1960s (e.g. Bjerhammar 1969), in mated by the University of Texas Center for Space Research (CSR), the early era of satellite geodesy. The basic idea of this approach is GeoForschungsZentrum (GFZ) Helmholtz-Centre Potsdam Ger- to explore the possibility of applying the principle of energy conser- man Research Centre for Geosciences, NASA’s Jet Propulsion Lab- vation, that is the constant sum of kinematic energy and potential oratory (JPL) and others. The estimation approach used by the above energy, to the SST data for direct measuring of Earth’s gravity three agencies and others (e.g. Luthcke et al. 2006; Bruinsma et al. field. The concept was investigated again by Jekeli (1999)atthe 2010) to generate these solutions is the so-called dynamic method onset of the Decade of Geopotential Missions, and he developed based on the dynamic orbit determination and geophysical parame- the first practical formulation to explicitly express the relationship ter recovery principle (Tapley et al. 2004b). Dynamic method treats between geopotential and satellite data in inertial frame (later called both KBR and GPS tracking as observations but with different the energy equation), with conceived application for the forthcom- weights, and simultaneously estimates for the state parameters in- ing satellite gravimetry missions, Challenging Minisatellite Pay- cluding the gravity coefficients, orbits and others in a least-squares load (CHAMP) and GRACE. Shortly after, Visser et al. (2003) solution. Because of the non-linear relationship between observa- similarly derived the energy equation but in the Earth-fixed frame. tions and the state parameters, linearization of both the dynamical Since then, a renewed interest of using energy balance approach to equation of motion and the observation-state equations are required estimate Earth’s static and time-variable gravity field was aroused during the estimation process. Besides the conventional dynamic during the last decade, especially for application using the data from C The Authors 2015. Published by Oxford University Press on behalf of The Royal Astronomical Society. 1773 1774 K. Shang et al. Earth satellite gravimetry missions, such as CHAMP (e.g. Han et al. range-rate observations into energy equation, together with a 2002; Gerlach et al. 2003; Badura et al. 2006), GRACE (e.g. Han method to reconstruct the related reference orbit. In addition, a et al. 2006; Ramillien et al. 2011; Tangdamrongsub et al. 2012)and more rigorous formulation of energy equation (Guo et al. 2015) Gravity Field and Steady-State Ocean Circulation Explorer (GOCE; is applied to model the in situ geopotential difference observables, e.g. Pail et al. 2011). which is requisite for the reduction of the GRACE measurements One of the major advantages of energy balance approach is that for gravity field inversion. it can be utilized to estimate the in situ geopotential observables The outline of this paper is as follows: Section 2 starts with a (for a single satellite) or geopotential difference observables (for a brief description of the general idea of energy balance formalism, pair of satellites), which means that the geopotential or geopoten- followed by a detailed description of the methodology about our tial differences can be computed at the satellite altitude, and then improved approach. Section 3 presents the numerical results using used to solve for the Earth’s gravity field. Similarly, acceleration the new energy approach, on the validation of the accuracy of the approach (Ditmar & van Eck van der Sluijs 2004) can also generate resulting improved in situ geopotential difference observables, and in situ observables, but in the form of acceleration. In contrast, con- on using the observables to demonstrate monthly and 10-d temporal ventional dynamic method normally cannot provide this kind of in gravity recovery. Finally, conclusions and discussions are summa- situ observables, where the geometric measurements, that is KBR rized in Section 4. and GPS tracking, have to be applied to directly solve gravity field, that is Stokes coefficients. The in situ geopotential observables, Downloaded from as a quantity with explicit geophysical interpretation, can serve as 2 METHODOLOGY an intermediate product between the satellite measurements and final gravity solutions, since the estimation procedure is more ef- The orbit data, both positions and velocities, are dominated by ficient because of the linear relationship between the observables the gravitational perturbations and other forces, and thus can be and gravity coefficients. More importantly, the in situ geopoten- regarded as observations and used for gravity recovery after proper http://gji.oxfordjournals.org/ tial difference observables would greatly benefit the time-variable reduction of other forces, which is the basic concept of the energy gravity recovery missions, such as GRACE, since the epoch-wise balance approach. The energy equation, a mathematical expression observables can support flexible spatial and temporal resolutions, of this concept, can be formulated in Earth-centred inertial frame (Jekeli 1999) for a single satellite as leading to regional solutions with possibly retrieving more local gravity information (Han et al. 2005; Schmidt et al. 2006, 2008; 1 t ∂V t = | |2 + − · − 0, Tangdamrongsub et al. 2012). V r˙ dt f r˙dt E (1) 2 t ∂t t However, appropriate application of energy balance approach 0 0 on GRACE-type mission data for highly accurate geopotential es- where V is the total gravitational potential (for unit mass), r (implicit timation is still a demanding task. One of the most challenging in V)andr˙ are the orbit position and velocity in inertial frame, f is at Ohio State University Libraries on December 20, 2016 the non-conservative force, t ∂V ∂t dt is the so-called potential problems is how to efficiently extract the gravity signal sensed by t0 the essential measurements from SST, that is KBR range-rate mea- rotation term, and E0 is an integral constant. The total gravitational surements, which the energy equation does not explicitly contain. potential V can be decomposed into two parts V = VE + VR,where Previous researchers attempted to adjust range-rate and orbit data VE is the geopotential, including the Earth’s mean, secular, seasonal simultaneously via a non-linear least-squares estimation with either and other variable components,