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, and VR is the residual gravitational fixed constraints (Han et al. 2006) or via stochastic constraints potential, mostly from the high-frequency (e.g. semi-diurnal and (Tangdamrongsub et al. 2012). The use of constrained least- diurnal) variable geopotential, such as tides. If we assume the resid- squares adjustment, though straightforward, is still a compromise ual gravitational potential VR can be reduced or corrected using a between the very high-precision range-rate data and the relative low- priori model, and also non-conservative force f can be measured by precision orbit data, which may tend to distort the estimation of in an on-board accelerometer, we arrive at the complete formulation situ geopotential observables caused by errors including orbit error. of energy equation for estimating geopotential VE, from a single The orbit error, inherited from the chosen reference orbit, would satellite, such as CHAMP, which can be expressed as contaminate the resulting gravity estimation especially at the low- 1 t ∂V t frequency band (Ditmar et al. 2012). In addition, our recent study V E = |r˙|2 + dt − f · r˙dt − V R − E 0. (2) 2 ∂t (Guo et al. 2015) has demonstrated that the previous formulation t0 t0 of the energy equation may contain a non-negligible approxima- And for estimating geopotential difference from a pair of satel- tion, which could sometimes overwhelm the time-variable gravity lites, such as GRACE, the formulation is simply the subtraction signal also at the low-frequency band. These issues limit the appli- between the equations of two single satellites: cation of the earlier developed in situ geopotential differences only E = E − E to regional gravity analysis, that is at the high-frequency band, and V12 V2 V1 arguably over regions with large temporal gravity field signals. As 1 t ∂V a result, large-scale gravity field inversion, including global grav- = |r˙ |2 + r˙ · r˙ + 12 dt 2 12 1 12 ∂t ity solution, has not been fully exploited based on previous energy t0 approach. t − · − · − R − 0 , The primary purpose of this study is to overcome these limitations (f2 r˙2 f1 r˙1) dt V12 E12 (3) by employing an improved energy balance approach to obtain a more t0 accurate estimation of in situ geopotential difference observables, where the subscripts represent the two satellites (‘1’, ‘2’) and their with the aim to preserve both the low- and high-frequency grav- difference (‘12’). It is worth mentioning that the orbit data (r and r˙) ity signal and consequently yield a full scale, that is both regional in both formulations are normally regarded as observables, which and global, gravity inversion. To achieve this goal, we develop a are usually from a precomputed reference orbit using high-low GPS novel formulation, called the alignment equation, to incorporate tracking data. GRACE improved energy balance approach 1775

2.1 A novel method to incorporate range-rate measurements into energy equation Application of energy balance approach on GRACE-type mission is much more challenging than CHAMP-type mission because energy eq. (3) is unable to explicitly contain the tracking measurements from the low–low SST system, that is range-rate measurements from KBR system in the case of GRACE. Previous studies of en- ergy balance approach usually treat range-rate measurements as a kind of redundancy observation, and adjust them simultaneously with orbit data along the orbit, by a non-linear least-squares esti- Figure 1. Geometric configuration of GRACE constellation and its relation- mation, where the energy equation is treated as either a fixed (Han ship with alignment equation. (a) The absolute position and velocity vector et al. 2006) or stochastic (Tangdamrongsub et al. 2012) constraint. for GRACE satellites with respect to the Earth’s centre of mass (CM). (b) The estimates would be the six intersatellite orbit state components The intersatellite components of position and velocity vector and the de- composition of velocity vector, illustrating the derivation of the alignment with some other empirical parameters. However, we know that the equation. uncertainty of GRACE orbits is around 1–2 cm in positions and µ –1

10–20 ms in velocities (only for dynamic orbit; for kinematic Downloaded from orbit the uncertainty in velocities is even worse; Kang et al. 2006), ρ˙ velocity vector r˙12 would become more accurate compared to the whereas the range-rate measurements have a much lower uncer- original vector r˙12 because the pitch angle of relative velocity vector µ –1 tainty of about 0.2 ms (Loomis et al. 2012) at high-frequency has been constrained by, or we can say, aligned to the range-rate. band. The use of constrained least-squares adjustment in energy The term ‘alignment’ actually means the relative velocity pitch, balance approach may be able to extract some information from the most sensitive intersatellite parameter to range-rate and most range-rate measurement, but it’s still a compromise between high- important parameter for gravity recovery, has been aligned to the http://gji.oxfordjournals.org/ precision data and low-precision data, as the (unknown) systematic range-rate measurement through the equation. The reason can be error, for example from orbit errors, would inevitably affect the further explained as follows. solved parameters, and subsequently bias the estimation of geopo- The alignment equation decomposes relative velocity vectors into tential difference observables. two components. Fig. 1 illustrates such decomposition by show- Therefore, in this study we present a new, alternative method to ing the simple geometric configuration of GRACE constellation, adjust intersatellite orbit state components using range-rate mea- where Fig. 1(a) shows the absolute position and velocity vector for surements. The motivation is based on a simple fact, which is that GRACE satellites with respect to the Earth’s centre of mass (CM), the range-rate measurements cannot be sensitive to all the intersatel- and Fig. 1(b) shows the decomposition of velocity vector. As shown at Ohio State University Libraries on December 20, 2016 lite orbit components. Previous study by Rowlands et al. (2002)has in Fig. 1(b), the intersatellite velocity is decomposed into two or- already shown that, among all the intersatellite components, the rel- thogonal directions. One is along the line-of-sight (LOS) direction, ative velocity pitch is the most sensitive to range-rate measurements, where unit vector is n1 = r12 |r12|, and the correspondent projec- and also one of the most important components for gravity recov- tion isρ ˙ . The other is orthogonal to the LOS direction and is in ery (Luthcke et al. 2006). The other two important components are the plane containing relative position vector and velocity vector, relative velocity magnitude and relative position pitch, but these are = × × | × × | where the unit vector is n2 (r12 r˙12 r12) r12 r˙12 r12 , much less sensitive to range-rate measurements compared to rela- with the correspondent projection of |r˙ |2 − ρ˙ 2 in order to main- tive velocity pitch. Based on this fact, we aim to develop a method 12 tain the same magnitude of the intersatellite velocity. to use range-rate measurements to only adjust the most sensitive Again, our goal here is to use high accurate measurement, that component, that is relative velocity pitch, and adopt the other less is the range-rate measured by the KBR, to adjust the most sensitive sensitive or insensitive components to be provided by the reference intersatellite parameter, that is the relative velocity pitch. In another orbits. words, range-rate should be used to replace the relative velocity The method is accomplished through a simple equation as fol- pitch component and form a new ‘pitch-free’ relative velocity vec- lows: tor. That is exactly what the alignment equation represents. Under ρ˙ ρ r r × r˙ × r the decomposition as eq. (4), the computation of r˙ only requires ˙ = ρ 12 + | |2 − ρ2 12 12 12 12 r˙12 ˙ r˙12 ˙ (4) four intersatellite quantities, which are range-rateρ ˙ , relative veloc- |r12| |r12 × r˙12 × r12| ity magnitude |r˙12|, LOS direction unit vector n1, and direction unit which is referred to as alignment equation throughout this pa- vector n2 that is always perpendicular to n1 and in the plane of in- per. Hereρ ˙ represents the range-rate measurement, r12 and r˙12 tersatellite position and velocity. Among the four quantities, two of are the relative position and velocity vector from reference orbit, them,ρ ˙ and n1, are totally independent of the velocity component × × ρ˙ | | r12 r˙12 r12 is the vector triple product between them, and r˙12 (as well as the position magnitude), and r˙12 is only dependent on represents the new relative velocity vector. As we can see, the new the velocity magnitude. The last one, unit vector n2, does rely on ρ˙ relative velocity vector r˙12 would be equal to the original vec- the velocity direction, but only the yaw angle. n2 does not depend tor r˙12 if there were no additional range-rate observation, that is on the relative velocity pitch at all simply because it is always per- ρ˙ = r˙12 · r12 |r12|. In that case, the alignment equation would de- pendicular to n1 in the plane of intersatellite position and velocity. grade to an identical equation, which represents an exact geometric Therefore, by using the alignment equation, the components that relationship between relative velocity direction vector and range- are needed from reference orbit are only relative velocity magni- rate measurement. That means the alignment equation itself does tude, relative position direction and relative velocity yaw, but not not contain any approximation. relative velocity pitch. The effect of relative velocity pitch from less Once we have an independent and more accurate measuring of accurate reference orbit is thus totally eliminated. The only contri- relative range-rate, such as the case of GRACE, then the new relative bution of the resulting relative velocity pitch is from range-rate, and 1776 K. Shang et al. therefore the most sensitive component to intersatellite observation, has been fully constrained by range-rate measurement through the alignment equation. After applying the alignment equation, the new relative velocity ρ˙ vector r˙12, would subsequently be used as the input of energy eq. (3). In the next step, we expect to determine geopotential difference observables solely from range-rate measurements, and meanwhile minimize the direct effect from the reference orbit to the estimates, which will be discussed in the next subsection.

2.2 Reconstruction of the reference orbit The reference orbit is another critical input to the energy eq. (3) in addition to the range-rate measurements. Generally there exist three different choices of reference orbits, namely the kinematic, reduced-dynamic or dynamic orbit, for the case of GRACE. Since Downloaded from range-rate should dominate the time-variable gravity information, the geopotential difference estimates do not rely much on the choice of the reference orbit. Therefore, it is possible to choose dynamic or reduced-dynamic orbit as the reference orbit for GRACE gravity field recovery, as long as the range-rate measurements are appro- http://gji.oxfordjournals.org/ priately used to correct or adjust the orbit data. In practice, various reference orbit data have indeed been implemented for GRACE real data analysis (Han et al. 2006; Tangdamrongsub et al. 2012). In this study, we choose to adopt a pure dynamic orbit as the ref- erence orbit. However instead of computing a dynamic orbit directly Figure 2. Residual geopotential differences from a closed-loop simulation from GPS observations, we use an alternative method to reconstruct based on eqs (5) and (6). Simulated dynamic orbit data are used as the input, the pure dynamic orbit from existing orbit data products. The similar so the residuals compared to apriorigravity field should be reduced to zero technique has been used for previous studies on GRACE (Liu et al. if the formulation is absolutely correct. (a) Residual geopotential differences 2010) and GOCE (Yi 2012). The idea is to treat the available orbit based on eqs (5) and (6). The blue line represents the residuals based on at Ohio State University Libraries on December 20, 2016 coordinates as pseudo observations, and estimate a pure dynamic eq. (5) and the red line represents the residuals based on eq. (6). (b) Zoomed orbit by fitting the orbit coordinates with respect to a complete view of (a), but for only residual geopotential difference based on eq. (6). reference model, via least-squares adjustment. Meanwhile, the ac- celerometer data are also simultaneously calibrated with respect to the pure dynamic orbit. The reference models we used in this study only 1.1 × 10−4 m2 s–2 (rms), which is just above the error level are identical to the models used by GFZ for solving the official of the formulation of energy equation (cf. Fig. 2b). Therefore, we GRACE Level-2 (L2) product Release 05 (RL05; Biancale & Bode simply choose the L1B GNV1B product as the input orbit data to 2006; Petit & Luzum 2010; Dahle et al. 2012;Mayer-Gurr¨ et al. reconstruct the pure dynamic orbit since it is readily available with 2012). We also adopt a similar strategy for calibrating accelerometer other L1B products. data, which is to estimate daily biases, and monthly scale factors. The purpose of using a reconstructed pure dynamic orbit is to The accelerometer data from GRACE Level 1B (L1B) ACC1B minimize the effect of mismodelled or unmodelled errors from the product, together with orientation data from L1B SCA1B product, pre-computed dynamic or reduced-dynamic orbit, and to suppress are used to model the non-gravitational forces. The GRACE L1B the random error from the kinematic orbit. When a pure dynamic data can be downloaded via http://podaac.jpl.nasa.gov/GRACE. orbit computed from apriorigravity model is used as the only As for the input of the reconstruction of the pure dynamic orbit, input for energy equation, the output from both eqs (2) and (3) we have tested three different highly accurate scientific orbit prod- would be inevitably reduced to the same apriorigravity model. ucts from independent institutes, which are kinematic orbit prod- That means no new geopotential information would be obtained uct from National Central University, Taiwan (T. Tseng, personal from energy equation if range-rate data were absent. However, once communication, 2014), kinematic orbit product from University the pure dynamic orbit has been aligned through the alignment of Bern (A. Jaggi,¨ personal communication, 2014), and reduced- equation by including the rang-rate measurements, the updated or- ρ˙ dynamic orbit from JPL, that is the GRACE L1B GNV1B product. bit r˙12 should contain the new time-variable gravity information We found the difference between the resulting reconstructed pure propagated solely from the rang-rate measurement, which would dynamic orbits using these orbit products negligible. For example be revealed by the energy equation afterward. One may argue that of one satellite on 2009 January 25, the direct orbit position dif- this process may also eliminate the possible contribution from GPS ference between two products from University of Bern and JPL is tracking data to gravity estimation, but considering the much higher about 2.48 cm (rms), and after reconstruction, the resulting orbit (∼50 times) noise level (both high frequency and low frequency) in difference is reduced to 0.76 cm (rms). More importantly, the sub- the orbit data as compared to the accuracy of the KBR range-rate sequent geopotential difference observables are also not sensitive data, we believe it is a reasonable trade-off. Besides, it is worth to the input orbit product because of the reconstruction process. For pointing out the same strategy is also used by Luthcke et al. (2006). the resulting geopotential differences on the same day, the differ- They applied the traditional dynamic method to solve monthly so- ence between the two estimates based on the two orbit products is lution, but also solely from range-rate measurements, and achieved GRACE improved energy balance approach 1777 comparable GRACE solutions as compared to the official GRACE is less than 1×10−4 m2 s–2, which is definitely negligible for cur- Level 2 data products. rent GRACE measurement accuracy and probably also for GRACE follow-on measurement accuracy in the future (Loomis et al. 2012). More detailed numerical comparison can be found in Guo et al. 2.3 Reformulation of energy equation (2015). The error level based on eq. (5) is about 0.02 m2 s–2 from When energy eq. (3) is applied to compute the geopotential dif- peak to peak, which is larger than the signal level from the time- ference, all the quantities and terms on the right-hand side can variable gravity field (see next Section on the resulting geopotential difference estimates using real data). All these errors caused by the be computed from data or based on reference models, except one term, t ∂V ∂t dt, the so-called ‘potential rotation term’. In the approximation would surely corrupt the geopotential difference es- t0 previous formulation developed by Jekeli (1999) timates. It seems that the dominant errors have the frequency close to 2 CPR, which might be removed by an additional 2 CPR param- 1 t V E ≈ |r˙|2 − ω (x y˙ − yx˙) − f · r˙dt − V R − E 0, (5) eters (Han et al. 2006; Tangdamrongsub et al. 2012) or even more 2 t0 parameters (Ramillien et al. 2011), but actually the errors contain much more high-frequency constituents (e.g. from ocean tides), the potential rotation term was approximated as t ∂V ∂t dt ≈ t0 which empirical parameters cannot fully absorb these errors. Also −ω − (x y˙ yx˙),wherex and y represent the first and second com- we know that 2 and higher CPR empirical parameters would heavily

ω Downloaded from ponent of the position vector, and is the nominal mean Earth’s contaminate gravity signal, especially the zonal geopotential coef- angular velocity. However, Guo et al. (2015) have already demon- ficients. The studies based on traditional orbit dynamic approach strated via simulation that this approximation of potential rotation also indicate that the empirical parameterization should be no more term exceeds the precision of GRACE observation, and suggested than 1 CPR (Tapley et al. 2004a) or even less parameters (Luthcke a more accurate formulation of energy equation as follows: et al. 2006), which is for the purpose to mitigate the systematic t 1 error of range-rate data, and also to better retain the time-variable http://gji.oxfordjournals.org/ E ≈ | |2 − · × − · − × − 0, V r˙ w (r r˙) a (r˙ w r) dt E (6) geopotential signal. Therefore, we conclude that in order to fully 2 t 0 exploit the precision of GRACE data, it is requisite to choose eq. (6) where a =∇V R + f is the acceleration of both residual geopoten- as the practical formulation of energy equation. tial acceleration and non-conservative acceleration, and w is Earth’s angular velocity of Earth-fixed frame relative to the inertial frame, with coordinates in the inertial frame. The third term of the right- 3 RESULTS AND ANALYSIS hand sides can be numerically integrated. The detailed derivation can be found in the Appendix, where we also demonstrate the equiv-

3.1 Geopotential difference estimates at Ohio State University Libraries on December 20, 2016 alence of the energy equations in both inertial frame and Earth-fixed frame (Jaggi¨ et al. 2008;Wanget al. 2012). Thus there is no differ- After the reduction of non-conservative acceleration and other tidal ence no matter in which frame the energy equation is used for the potential terms, each geometric observation from GRACE can be energy balance method. directly linked to a geophysical quantity, that is geopotential differ- The Appendix also shows that two approximations are included ence between the two positions of the twin satellites, and therefore in eq. (6). One is to assume VE to be static during the integral limits energy balance approach could provide a unique ability to extend from t0 to t, which is consistent with the normal GRACE conven- our conventional knowledge about both data processing and results tion that is estimating a mean gravity field during a certain time interpretation of GRACE. The geopotential difference estimates are interval. The other one is to assume the rates of Earth’s angular ve- able to directly sense the gravity information, without losing any locity vector is zero, that is ˙w = 0, which is negligible compared to high-frequency resolution, since each estimate is computed straight measurement noise level of GRACE (Guo et al. 2015). In contrast, from range-rate measurement for each epoch. Because of that, the eq. (5) contains too many rough approximations that would pollute geopotential difference estimates could not only be used for both the signal level of GRACE. In order to check the accuracy of the global and regional gravity recovery but also be regarded as an two formulations, a closed-loop validation is performed. We first in situ gravity representation without downward continuation (Han simulate pure dynamic orbits for two satellites as the input to the et al. 2006). Therefore, an accurate estimation of geopotential dif- energy equation. As we mentioned in the last subsection, using a ference observables is the key issue for energy balance approach pure dynamic orbit data as the input of energy equation should re- and is the most critical step for the subsequent temporal gravity duce the estimates to the apriorigravity field. If we define residual inversion.  E = E − E apriori geopotential difference observables as: V12 V12 V12 , Using the methods described in the previous section, geopotential that is the difference between the estimated values using energy differences are estimated for each day from 2003 to 2013. All the equation and the predicted values using the apriorigravity model, input data are from GRACE L1B data products, including GNV1B then theoretically the residual should be reduced to zero, and there- orbit data that are used to reconstruct the pure dynamic orbit and fore the non-zero residual would reveal the approximation level for estimate the daily accelerometer calibration parameters. Range-rate each formulation of energy equation. data from KBR1B product are then included to correct the velocity The residuals based on eqs (5) and (6) are presented in Fig. 2, components of the reconstructed orbit via the alignment eq. (4). for an integration period of 1 d. Fig. 2(a) shows the residuals based Next, energy eq. (3) is applied according to the formulation of on both equations, and we can see that the one based on eq. (6) is eq. (6), to compute the geopotential difference observables. reduced to almost zero, but the one based on eq. (5) contains a rela- The accelerometer measures the non-conservative force in three tive large non-zero residual with a dominate 2 cycle-per-revolution orthogonal directions of the Science Reference Frame (SRF), which (CPR) error, which is caused by the approximation primarily from can be transformed into inertial frame using the quaternions from the potential rotational term. A zoomed-in view of the error from the SCA1B product measured by the Star Camera Assembly. Ap- eq. (6) is presented in Fig. 2(b), and the peak-to-peak amplitude proximately, X direction is along roll axis in the anti-flight and 1778 K. Shang et al. Downloaded from http://gji.oxfordjournals.org/

Figure 3. Geopotential difference estimates (with mean gravity field model removed). (a) Ground track of 2003 July 17 with two ascending profiles highlighted with colour representing the values of the estimates. (b) Highlighted profile approximately along 60◦W longitude mostly above the land area, with predicted at Ohio State University Libraries on December 20, 2016 values from GRACE L2 solutions (CSR RL05, GFZ RL05a and JPL RL05). (c) The same but for highlighted profile approximately along 160◦W longitude mostly above the ocean area. (d) Time-series of estimates for the day of 2006 May 1 with predicted values from CSR RL05 solution. in-flight directions for the leading and trailing satellites, respec- to degree and order 60 with the same mean field removed) along the tively, Z direction is along the yaw axis and points to nadir and Y same profiles. We can clearly see that the geopotential differences direction is along pitch axis and forms a right-handed triad with estimates using our approach are similar to the predicted values X and Z. For GRACE satellite A, the means of the estimated ac- from the three official L2 solutions. Of course, the three series of celerometer biases are –1.1825, 29.4500 and –0.5527 µ s–2, and the predicted values look smooth since they are simply computed from rms values about the means are 0.0388, 1.1645 and 0.0687 µ s–2, existing models with a gravity field up to degree and order 60 only, for the X-, Y-andZ-axes, respectively. For GRACE satellite B, while the series of estimates is noisier because it is computed from the means of the biases are –0.5810, 10.9709 and −0.7616 µ s–2, range-rate measurements directly. and the rms values about means are 0.0211, 0.8170 and Fig. 3(b) illustrates the profile approximately along 60◦W longi- 0.1530 µ s–2. For the scale factors of satellite A, the means are tude, mostly above the rough land area, and Fig. 3(c) is for the profile 0.9493, 0.9448 and 0.9520, and the rms values about means are approximately along 160◦W longitude, mostly above the flat ocean 0.0769, 0.2156 and 0.1962. For the scale factors of satellite B, the area. With the ascending of the satellite pair, we can observe the es- means are 0.9358, 0.9494 and 0.9572, and the rms values about timates in Fig. 3(b) reveal the geopotential difference variation suc- means are 0.0651, 0.2065 and 0.1742. For the systematic errors cessively caused by West Antarctica, Amazon Basin, Hudson Bay in the geopotential differences, a number of empirical parameters and North Greenland, and the estimates in Fig. 3(c) mostly cover are estimated and removed from the geopotential difference ob- the Pacific Ocean. As we can see, the surface gravity change may be servables. These include daily bias, rate and 1 CPR parameters for directly inferred just from the geopotential difference profile. Since every orbital revolution, which absorb not only the systematic errors here we define the difference as the following satellite subtracting from range-rate data, but also the integral constants from the energy the leading satellite, when the satellite pair passes by a negative equation. gravity anomaly on ground, the geopotential difference profile will In Fig. 3(a), we highlight two ascending profiles of geopotential exhibit increasing values first and decreasing values next, and vice difference observables on a global map of the daily ground tracks versa for a positive anomaly. For example, in Fig. 3(b), from about of 2003 July 17. The geographical coordinates of each estimate 30◦N to North Pole, we can observe an increase-decrease-increase on the map are assigned to the middle point of two satellites and fluctuation of geopotential differences, meaning there should be a the colour represents its value with a mean reference gravity field negative gravity anomaly followed by a positive anomaly compared GIF48 (Ries et al. 2011) removed. In Figs 3(b) and (c), the two to the epoch of the mean reference field (2007 January 1), which profiles of estimates are, respectively, illustrated with respect to corresponds to the glacial isostatic adjustment (GIA) signal (neg- latitude and compared to the predicted values using GRACE L2 ative) in the Hudson Bay, and Greenland ice sheet ablation signal solutions (from CSR RL05, GFZ RL05a and JPL RL05, truncated (positive). GRACE improved energy balance approach 1779

In Fig. 3(d), we show a profile as a time-series for a different where GM is the geocentric gravitational constant and R is Earth’s day (2006 May 1) with predicted time-series from CSR L2 RL05 radius, n and m are degree and order, respectively, and nmax is the solution for comparison. Again, the time-series of the estimates maximum degree, that is 60 in this study. Here we exclude degree 0 seems very close to the predicted values from CSR solution. In and degree 1 coefficients, as GRACE range rate measurements are order to compare with previous study, we adopt the same method insensitive to these parameters. The coefficients αnm and βnm are from Han et al. (2006) to compute the correlation coefficient of defined as ⎧ ⎫ ⎧ ⎫ the time-series between the (smoothed) estimates and the predicted   ⎨ αnm ⎬ n+1 ⎨ cos (mλ2) ⎬ values. The resulting correlation coefficient is about 0.91 for that = R ¯ θ ⎩ β ⎭ Pnm (cos 2) ⎩ λ ⎭ particular day. The rms of the difference between the (smoothed) nm r2 sin (m 2) estimates and the predicted values is about 8.4 × 10−4 m2 s–2,     and for comparison, the rms of the values themselves is about n+1 λ R cos (m 1) 2 –2 0.002 m s . As for all the estimates from 2003 to 2013, the average − P¯nm (cos θ1) , (8) r λ value of the daily correlation coefficients is over 0.9, which is much 1 sin (m 1) higher than correlations of 0.5–0.8 reported in previous study by where (r1,θ1,λ1) and (r2,θ2,λ2) are denoted as the spherical co- Han et al. (2006). ordinates of the two satellites in Earth-fixed reference system, and P¯nm is the fully normalized Legendre function. Based on the least- Downloaded from squares principle, the solution of the unknowns (C¯ nm and S¯nm) can E 3.2 Global gravity solutions using geopotential difference be easily solved from a large number of observations (V12). observables The recovered monthly gravity solution is shown in Fig. 4(b) in terms of geoid undulation, using all the geopotential differences As an in situ observation type, geopotential differences have been from Fig. 4(a). For comparison, the geoid maps from other three widely used for gravity recovery, especially for regional gravity L2 solutions for the month of July 2003 are also shown in the http://gji.oxfordjournals.org/ recovery. However, the global gravity recovery using geopotential right column of Fig. 4 (Fig. 4d for CSR RL05, Fig. 4fforGFZ differences is not commonly used, even if it is more straightforward RL05a and Fig. 4h for JPL RL05). Here we did not apply any post- because of the linear relationship between geopotential differences processing techniques except that we replace the C20 coefficients and Stokes coefficients. In this subsection, we explored the possibil- using the values obtained from satellite laser ranging (Cheng et al. ity of global gravity recovery based on the geopotential difference 2013). Although the input geopotential differences in the left-hand estimates from our improved approach. column of Fig. 4(a) seem to agree with the predicted value, this Similar to the convention of official GRACE L2 product, we does not necessarily guarantee the recovered solution should be also produce monthly mean solutions for each calendar month. similar as well. In fact, the high frequency signal and noise in at Ohio State University Libraries on December 20, 2016 First, similar to Fig. 3(a), we plot all the accumulated geopotential Fig. 4(a) could still leak into the low frequency band during the difference estimates for an example month of July 2003 on the global inversion process. Therefore, after downward continuation from mapinFig.4(a). The data from descending passes are presented with satellite altitude to Earth surface, enlarged noise, that is north-to- an additional minus sign so they would not look opposite to the data south stripes, should be expected in the gravity solution, which can from ascending passes over the same region of the global map. From be seen from Fig. 4(b). Compared Fig. 4(b) and other three figures in Fig. 4(a), we can see that some regions with large gravity variation the right-hand column of Fig. 4, the geoid undulation map from our are manifested in the global map, including not only the highlighted solution (OSU) has fewer stripes than the JPL or the GFZ solution, regions in Fig. 3(a) but also some other regions like Alaska (glacier but slightly more stripes than the CSR solution, for this particular melting), Congo Basin (wet season), and Scandinavia (GIA). month. The comparison of Power Spectral Density (PSD) is shown For comparison, the other three figures in the left-hand column of in Fig. 5(a). We can see at lower degree (below degree 15), our Fig. 4 show the predicted values at the same geographical location PSD (OSU) matches other three PSDs very well, which means our from three GRACE L2 solutions completed to spherical harmonic solution contains highly consistent time-variable gravity signal. At degree and order 60 (Fig. 4c for CSR RL05, Fig. 4e for GFZ RL05a higher degree (above degree 15), our PSD shows similar noise level and Fig. 4g for JPL RL05). The most significant discrepancy be- as JPL’s PSD, which is slightly higher than CSR’s PSD but lower tween Fig. 4(a) and other three figures in the left (Figs 4c, e and than GFZ’s PSD. g) is that Fig. 4(a) apparently contains measurement noise which is inherited from each range-rate measurement, while other three figures are only predicted from a truncated gravity field model with 3.3 Secular and seasonal gravity variation from global resolution of up to degree 60. That shows the key contribution solutions of using energy balance approach, that is directly connecting the geometry measurements (range-rate) to the geophysical quantities The primary goal of GRACE mission is to map the temporal vari- (geopotential difference). Therefore, even though we plot the whole ation of Earth’s gravity field, for the purpose to understand the month estimates in the same global map, Fig. 4(a) still preserve the mass transports within the Earth system. Using the same method, in situ geopotential change for each epoch within a month, that is we generate a series of monthly gravity solutions up to degree and submonthly information, but Figs 4(c), (e) and (g) can only show order 60 from 2003 to 2013. Then we estimate the secular and sea- the predicted values from a monthly mean (static) gravity field. sonal variation based on our solution series. Fig. 6(a) shows the Next, we use the accumulated estimates to produce a monthly estimated secular variation from 2003 to 2013, in terms of equiv- E global solution. The relation of geopotential difference V12 and alent water height (EWH) change. Unlike the geoid map shown in Stokes coefficients (C¯ nm and S¯nm) can be expressed as Fig. 3, the EWH map normally contains heavier stripes caused by the amplification of high-frequency noise. Therefore, we applied a GM nmax n V E = α C¯ + β S¯ , (7) 150 km radius Gaussian smoothing (Wahr et al. 1998) to mitigate 12 R nm nm nm nm n=2 m=0 the error. For comparison, three EWH trend maps from GRACE L2 1780 K. Shang et al. Downloaded from http://gji.oxfordjournals.org/ at Ohio State University Libraries on December 20, 2016

Figure 4. Global map of both geopotential difference estimates and recovered gravity solution for the month of July 2003. Left-hand panel: geopotential differences (a) estimated from this study (OSU) and predicted from GRACE L2 products of (c) CSR RL05, (e) GFZ RL05a and (g) JPL RL05. Right-hand panel: recovered geoid undulation from (b) this study using geopotential difference estimates (OSU) and from GRACE L2 products of (d) CSR RL05, (f) GFZ RL05a and (g) JPL RL05. GRACE improved energy balance approach 1781

storage variations for Amazon Basin, from our solution series (OSU) and three GRACE L2 solution series. The unit is gigaton (Gt) converted from EWH. Apparently, a large seasonal signal of the water storage variation is apparent, which is mainly driven by the annual water redistribution. Our time-series shows an overall good agreement with other three time-series for both annual ampli- tude and phase. Fig. 6(f) shows the time-series of the ice sheet mass variation for Greenland. An additional reduction (Guo et al. 2010) is conducted for Greenland ice sheet to reduce the signal leakage from land to coastal ocean caused by Gaussian smoothing. From Fig. 6(f) we can see that, besides the secular trend signal, the mass variation of Greenland ice sheet also manifests a seasonal change, due to the ice accumulation and ablation. In order to further evaluate the noise level of our solutions, we adopt the similar method from Wahr et al. (2006), that is to fit and remove the trend (as shown in Fig. 6) as well as the annual and semi-annual signal from all the monthly solution. The resulting Downloaded from residual can be used to assess the noise level for each month. First in Fig. 5(b), we show the PSD of the residual for the month of July 2003 as the noise PSD. Comparing the noise PSD (Fig. 5b) and solution PSD (Fig. 5a), we can see the noise PSD confirmed that GFZ’s solution has higher noise at large degree for this month. http://gji.oxfordjournals.org/ Then we calculate the variance of the residual in terms of geoid for each month, and plot the time-series of the residual geoid variance Figure 5. Comparison of GRACE solution (a) power spectral density (PSD) in Fig. 7. We can see CSR’s solutions show an overall lowest noise and (b) PSD residual after removing semi-annual, annual and trend from the level. Other three solutions (OSU, GFZ and JPL) have comparable GRACE solution, for July 2003 from this study (OSU solution) and GRACE results, but all are higher than CSR’s noise level. From Fig. 7 we can L2 products (CSR RL05, GFZ RL05a and JPL RL05). see, the reason that GFZ’s PSD in July 2003 is higher than others in Fig. 5 is because GFZ’s solution shows higher noise level around solution series are presented as well in Figs 6(b)–(d), for CSR RL05, 2003. The mean of the residual geoid variance for OSU, GFZ, JPL GFZ RL05a and JPL RL05, respectively. Clear and consistent sec- and CSR are 2.02, 2.57, 2.31 and 1.42 mm, respectively. at Ohio State University Libraries on December 20, 2016 ular signals can be observed from Figs 6(a)–(d) for all the solution series, including negative trends in Greenland, Amundsen Sea Em- bayment, Antarctic Peninsula and Alaska, reflecting the mass loss from the ice sheets and glacier, and positive trends in Hudson Bay, 3.4 10-d Regional gravity analysis using geopotential West and East Antarctic, and Scandinavia, mainly due to GIA. We difference observables should note that the agreements are mostly over high-latitude and polar regions. Global gravity recovery usually truncates the solution to a maximum On the other hand, our result seems to have more stripes in degree (and order), such as degree 60 for RL05, corresponding to certain areas, especially over low-latitude or equatorial regions, the spatial resolution of about 300 km. However, it is reasonable such as Southeast Asia with the signal of great Sumatra–Andaman to expect the region with more data coverage should have a bet- earthquake of 2004 December 26, where our trend map show heavier ter resolution than the region with less data coverage. Considering stripes than CSR. One of the reasons for the discrepancy could be the near-polar orbit configuration, it is obviously that GRACE data the different reference model used in our processing, especially should have more coverage over high-latitude region than middle- the high-frequency models, such as ocean tides models, that could and low-latitude region. Therefore, it might be possible to obtain regionally impact the aliasing effect if the model is less accurate over enhanced spatial and/or temporal resolutions over certain region, es- certain area. Another reason may be caused by the data coverage of pecially near polar region, using geopotential difference observable our geopotential difference estimates. Since GRACE is in a near- derived temporal gravity field solutions. polar orbit, the geopotential difference estimates during a month In general, there is a trade-off between the spatial and tempo- should have much denser coverage over polar region than over low- ral resolution for GRACE gravity solutions, which means retriev- latitude region. After downward continuation, the region with less ing a high spatial resolution gravity field needs the compromise data coverage could have large aliasing error compared to the region of temporal resolution. For example, to get the high degree grav- with dense data coverage. Such an effect may be less significant for ity model (GGM03S, Tapley et al. 2007) up to degree and order some months with overall ‘good’ data coverage, like July 2003 in 180 (100 km resolution), 4 yr GRACE data are needed. In prac- Fig. 4. However for certain months, the ground tracks of GRACE tice, monthly GRACE solutions are usually truncated to degree 60 could drift to near-repeat orbit due to the orbital evolution, leading (300 km resolution) or so. One exception is that CSR recently to ‘poorer’ data coverage, which would result in larger stripes over generated new monthly solutions (Sakumura 2014) with spherical the equatorial region. harmonic coefficients complete to degree 96 (200 km resolution), About the seasonal variation in addition to the secular variation, but it is very challenging to identify high frequency signal beyond we choose two regions, Amazon Basin and Greenland, to gener- degree 60. Dai et al. (2014) used an innovative method to process ate the time-series of the average mass variation over both areas. the degree 96 solutions and successfully retrieved high frequency Fig. 6(e) shows the time-series of the average continental water earthquake signal up to degree 70. 1782 K. Shang et al. Downloaded from http://gji.oxfordjournals.org/ at Ohio State University Libraries on December 20, 2016

Figure 6. Secular and seasonal gravity variation. Top and middle rows: Trend map (in terms of equivalent water height rate in mm yr–1) for period from 2003 to 2013 (a) estimated from this study and from GRACE L2 products of (b) CSR RL05, (c) GFZ RL05a and (d) JPL RL05, 150 km Gaussian smoothing applied. Bottom row: Time-series of secular and seasonal mass variation (in Gigaton) for (e) Amazon Basin and (f) Greenland, additional leakage reduction applied.

On the other hand, several attempts have been made to improve the temporal resolution of GRACE. For example, GFZ routinely gener- ates weekly solutions, but only up to degree and order 30. Bruinsma et al. (2010) generate the 10-d solutions up to degree and order 50 with regularization based on traditional dynamic method. Kurten- bach et al. (2009) employed short-arc method (Mayer-Gurr¨ et al. 2007) under the principle of Kalman smoothing to conduct the daily snapshot solution, but the stochastic behaviour of the gravity field has to be considered as aprioriinformation. Kang et al. (2008)used traditional dynamic method to generate the so-called ‘quick-look’ solution with a moving-window strategy (with a window step of one day and window width of 15 d), but those solutions are also stabi- Figure 7. Residual rms (2003–2013) after removing trend, annual and semi- lized using regularization. An incomplete list of GRACE solutions annual signal from all the monthly solutions, based on the results from this with various temporal resolutions from different research groups can study (OSU) and GRACE L2 products (CSR RL05, GFZ RL05a and JPL be found at http://icgem.gfz-potsdam.de/ICGEM/TimeSeries.html. RL05). GRACE improved energy balance approach 1783 Downloaded from http://gji.oxfordjournals.org/ at Ohio State University Libraries on December 20, 2016

Figure 8. Regional 10-d solutions from July 2003 to September 2003 over Greenland ice sheet. Left three columns: 10-d geopotential difference subset in terms of data coverage map, where the colour represents the value with mean field removed. Middle three column: 10-d solutions up to degree 60 in terms of geoid undulation with mean field removed. Right three columns: monthly solutions up to degree 60 in terms of geoid with mean field removed, from this study, CSR RL05 and GFZ RL05a, respectively. Neither regularization nor post-processing is applied. Our improved energy balance approach could also greatly benefit It is interesting to note that although the temporal resolution is the regional gravity analysis since the geopotential difference data shortened to about ten days, the data coverage is still fairly dense can be easily implemented for gravity recovery with flexible spatial over Greenland for most of the subsets. And the resulting 10-d solu- and temporal resolution. In this study, we only discuss the temporal tion, in the middle three columns of Fig. 8 shows explicit geoid vari- resolution. Wechoose Greenland as the test region. In last subsection ation, surprisingly without introducing more stripes. Correspond- (cf. Fig. 6), we have already estimated the gravity variation over ingly, the geoid maps over equatorial region from 10-d solutions Greenland, but with a temporal resolution of a month and a spatial contains heavier stripes than monthly solutions because of the rela- resolution of degree 60. Here we show that for high-latitude region tive sparse data coverage. Therefore we conclude that, for the region like Greenland, the temporal resolution can be at least increased by like Greenland, 10-d geopotential difference data are sufficient to three times without losing the spatial resolution using our improved recover the time-variable gravity, with a resolution up to degree 60 energy balance approach. but without significantly increasing the error. In addition, we can To do that, we collect three months’ geopotential difference data clearly see the geoid fluctuation with each month through three 10-d from July 2003 to September 2003, and for each month we split the solutions, such as the geoid decreasing for the month of July. There- data into three separate data subsets for every 10 (or 11) d. Then fore, these 10-d solutions might also be able to reveal more detailed for each 10-d subset, we solve the gravity solution up to degree 60. submonthly temporal mass variations for the Greenland ice-sheet, Neither regularization nor post-processing is applied to these 10-d which is a subject for future studies. solutions. In Fig. 8, we present both the data and the solutions, from top to bottom for the month from July 2003 to September 2003. 4 CONCLUSIONS AND DISCUSSIONS The left three columns of Fig. 8 show the 10-d geopotential differ- ence subset in terms of data coverage map over Greenland, where Our new, improved energy balance approach is presented in this the colour represents the value with mean field removed; the middle paper for GRACE data analysis and result interpretation. Com- three columns show our 10-d solutions up to degree 60 in terms of pared to previous energy balance studies, the new approach can geoid undulation with mean field removed; and for comparison the better preserve gravity information from KBR data and reduce er- right three columns show the corresponding monthly solutions also ror, especially velocity error, from GPS data, with the help of the up to degree 60 in terms of geoid with mean field removed, from novel alignment equation, the reconstruction of reference orbit, and this study, CSR RL05 and GFZ RL05a, respectively. a more accurate formulation of energy equation. As a result, we 1784 K. Shang et al. obtain more accurate geopotential difference estimates with higher data, Tech. Rep., Potsdam, Deutsches GeoForschungsZentrum GFZ, doi: correlation (more than 0.9) with official GRACE RL05 monthly http://doi.org/10.2312/GFZ.b103–06011. solutions. Using the resulting geopotential difference estimates, we Bjerhammar, A., 1969. On the energy integral for satellites, Tellus, 21, demonstrate the global solution is feasible, and for the first time, 1–9. produce an independent series of GRACE monthly global solu- Bruinsma, S., Lemoine, J. M., Biancale, R. & Vales, N., 2010. CNES/GRGS 10-day gravity field models (release 2) and their evaluation, Adv. Space tions based on energy balance principle, with consistent secular Res., 45(4), 587–601. and seasonal time-variable gravity signals compared to other so- Cazenave, A. & Chen, J., 2010. Time-variable gravity from space and lutions based on dynamic method. Furthermore, we show that our present-day mass redistribution in the Earth system, Earth planet. Sci. improved geopotential difference data can be applied for gravity Lett., 298(3), 263–274. recovery with flexible temporal resolution, which is conducted over Chen, Y.Q., Schaffrin, B. & Shum, C.K., 2008. Continental water stor- Greenland ice sheets and achieve a higher temporal resolution at age changes from GRACE Line-of-sight range acceleration measure- 10-d interval, as compared to the official monthly solutions, without ments, in VI Hotine-Marussi Symposium on Theoretical and Com- the compromise of spatial resolution, nor the need to use regular- putational Geodesy , Vol. 132, ISSN, pp. 62–66, eds Peiliang, Xu, ization or post-processing. It is worth mentioning that our purpose Jingnan, Liu & Athanasios, Dermanis, International Association of is to only demonstrate that our method mimics the accuracy of L2 Geodesy Symposia, doi:10.1007978–3–540–74584–6. Cheng, M., Tapley, B.D. & Ries, J.C., 2013. Deceleration in the Earth’s products. We are not implying that our global solution based on en- oblateness, J. geophys. Res.: Solid Earth, 118, 740–747. Downloaded from ergy approach is better, that is more accurate and/or higher spatial Dai, C., Shum, C.K., Wang, R., Guo, J., Shang, K., Tapley, B. & Wang, resolution, than other products. L., 2014. Improved source parameter constraints for recent large under- Our new method would also benefit data analysis of the forth- sea earthquakes from high-degree GRACE gravity and gravity gradient coming GRACE follow-on mission, especially considering the pos- change measurements, in Proceedings of the AGU Fall Meeting,San sibility that the precision of range-rate data can be improved by up Francisco, CA, USA, 15–19 December 2014. to a factor of 20 (Loomis et al. 2012), but the precision of GPS Dahle, C., Flechtner, F., Gruber, C., Konig,¨ D., Konig,¨ R., Michalak, G. & http://gji.oxfordjournals.org/ tracking data may not have significant advances. In that case, the Neumayer, K.-H., 2012. GFZ GRACE level-2 processing standards doc- weighting of GPS tracking data would need to be further reduced ument for level-2 product release 0005, Tech. Rep., Potsdam, Deutsches for the traditional conventional dynamic method, which could be GeoForschungsZentrum GFZ, doi:10.2312/GFZ.b103–12020. Ditmar, P. & van Eck van der Sluijs, A.A., 2004. A technique for Earth’s more analogous to our method since we have already handle GPS gravity field modeling on the basis of satellite accelerations, J. Geod., 78, data and range-rate data separately through the alignment equation. 12–33. Therefore, our energy balance method might have a unique contri- Ditmar, P, Teixeira da Encarnac¸ao,˜ J. & Hashemi Farahani, H., 2012. Un- bution to the processing of more accurate data from next generation derstanding data noise in gravity field recovery on the basis of inter- satellite gravimetry mission to study mass transports of the Earth. satellite ranging measurements acquired by the satellite gravimetry mis- sion GRACE, J. Geod., 86, 441–465. at Ohio State University Libraries on December 20, 2016 Gerlach, C. et al., 2003. A CHAMP-only gravity field model from kine- ACKNOWLEDGEMENTS matic orbits using the energy integral, Geophys. Res. Lett., 30(20), 2037, doi:10.1029/2003GL018025. This research is supported by grants from NSF via the Belmont Guo, J.Y., Duan, X.J. & Shum, C.K., 2010. Non-isotropic Gaussian smooth- Forum/IGFA (ICER-1342644), NASA’s geodesy and cryosphere ing and leakage reduction for determining mass changes over land and (NNX12AK28G, NNX13AQ89G, and NNX11AR47G), State Key ocean using GRACE data, Geophys. J. Int., 181, 290–302. Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy Guo, J.Y., Shang, K., Jekeli, C. & Shum, C.K., 2015. On the energy inte- & Geophysics, Chinese Academy of Sciences (Y309473047, gral formulation of gravitational potential differences from satellite-to- satellite tracking, Celest. Mech. Dyn. Astr, 121(4), 415–429. SKLGED2015-5-3-E), and Natural Science Foundation of China Han, S.-C., 2003. Efficient global gravity determination from satellite-to- (NSFC, 41374020). GRACE data products are from NASA’s PO- satellite tracking (SST), Diss. PhD thesis, Geodetic and Geoinformation DAAC via JPL, CSR and GFZ. Some figures in this paper were Science, Department of Civil and Environmental Engineering and Geode- generated using the Generic Mapping Tools (GMT; Wessel & Smith tic Science, The Ohio State University, Columbus, OH, USA. 1991). The computational aspect of this work was supported in part Han, S.-C., Jekeli, C. & Shum, C.K., 2002. Efficient gravity field recovery by an allocation of computing resources from the Ohio Supercom- using in situ disturbing potential observables from CHAMP, Geophys. puter Center (http://www.osc.edu). We thank Tzupang Tseng, Na- Res. Lett., 29(16), 1789, doi:10.1029/2002GL015180. tional Central University, Taiwan, Adrian Jaggi,¨ University of Bern Han, S.-C., Shum, C.K., Jekeli, C. & Alsdorf, D., 2005. Improved estimation and Dahning Yuan, Jet Propulsion Laboratory, California Institute of terrestrial water storage changes from GRACE, Geophys. Res. Lett., of Technology for providing precise GRACE orbits. We acknowl- 32, L07302, doi:10.1029/2005GL022382. Han, S.-C., Shum, C.K. & Jekeli, C., 2006. Precise estimation of in situ edge the German Space Operations Center (GSOC) of the German geopotential differences from GRACE low-low satellite-to-satellite track- Aerospace Center (DLR) for providing continuously and nearly 100 ing and accelerometer data, J. geophys. Res., 111, B04411, doi:10.1029/ per cent of the raw telemetry data of the twin GRACE satellites. We 2005JB003719. thank two anonymous reviewers and the editor for their helpful and Jaggi,¨ A., Bock, H., Pail, R. & Goiginger, H., 2008. Highly-reduced dynamic constructive comments leading to improvement of the manuscript. orbits and their use for global gravity field recovery: a simulation study for GOCE, Stud. Geophys. Geod., 52(3), 341–359. Jekeli, C., 1999. The determination of gravitational potential differences REFERENCES from satellite-to-satellite tracking, Celestial Mech. Dyn. Astron., 75, 85– 100. Badura, T., Sakulin, C., Gruber, C. & Klostius, R., 2006. Derivation of the Jekeli, C., 2001. Inertial Navigation Systems with Geodetic Applications, CHAMP-only global gravity field model TUG-CHAMP04 applying the Walter de Gruyter. energy integral approach, Stud. Geophys. Geod., 50(1), 59–74. Kang, Z., Tapley, B., Bettadpur, S., Ries, J., Nagel, P. & Pastor, R., 2006. Biancale, R. & Bode, A., 2006. Mean annual and seasonal atmospheric Precise orbit determination for the GRACE mission using only GPS data, tide models based on 3-hourly and 6-hourly ECMWF surface pressure J. Geod., 80(6), 322–331. GRACE improved energy balance approach 1785

Kang, Z., Bettadpur, S., Nagel, P., Pastor, R., Pekker, T., Poole, S. & Tangdamrongsub, N., Hwang, C., Shum, C.K. & Wang, L., 2012. Regional Tapley, B., 2008. Quick-look gravity solutions from GRACE, in Proceed- surface mass anomalies from GRACE KBR measurements: application of ings of the AGU Fall Meeting 2008, San Francisco, CA, 15–19 December, L-curve regularization and apriorihydrological knowledge, J. geophys. 2008. Res., 117, B11406, doi:10.1029/2012JB009310. Kurtenbach, E., Mayer-Gurr,¨ T. & Eicker, A., 2009. Deriving daily snapshots Tapley, B.D., Bettadpur, S., Watkins, M. & Reigber, C., 2004a. The gravity of the Earth’s gravity field from GRACE L1B data using Kalman filtering, recovery and climate experiment: mission overview and early results, Geophys. Res. Lett., 36, L17102, doi:10.1029/2009GL039564. Geophys. Res. Lett., 31, L09607, doi:10.1029/2004GL019920. Liu, X., Ditmar, P., Siemes, C., Slobbe, D.C., Revtova, E., Klees, R., Riva, Tapley, B.D., Schutz, B. & Born, G.H., 2004b. Statistical Orbit Determina- R. & Zhao, Q., 2010. DEOS Mass Transport model (DMT-1) based on tion, Elsevier. GRACE satellite data: methodology and validation. Geophys. J. Int., 181, Tapley, B.D., Ries, J., Bettadpur, S., Chambers, D., Cheng, M., Condi, F. & 769–788. Poole, S., 2007. The GGM03 mean earth gravity Model from GRACE, Loomis, B.D., Nerem, R.S. & Luthcke, S.B., 2012. Simulation study of a in Proceedings of the AGU Fall Meeting 2007, San Francisco, CA, 10–14 follow-on gravity mission to GRACE. J. Geod., 86(5), 319–335. December, 2007. Luthcke, S.B., Rowlands, D.D., Lemoine, F.G., Klosko, S.M., Chinn, D. & Visser, P.N.A.M., Sneeuw, N. & Gerlach, C., 2003. Energy integral method McCarthy, J.J., 2006. Monthly spherical harmonic gravity field solutions for gravity field determination from satellite orbit coordinates, J. Geod., determined from GRACE inter-satellite range-rate data alone, Geophys. 77, 207–216. Res. Lett., 33, L02402, doi:10.1029/2005GL024846. Wahr, J., Molenaar, M. & Bryan, F., 1998. Time variability of the Mayer-Gurr,¨ T., Eicker, A. & Ilk, K.H., 2007. ITG-Grace02s: a GRACE Earth’s gravity field: hydrological and oceanic effects and their pos- Downloaded from gravity field derived from range measurements of short arcs, in Gravity sible detection using GRACE, J. geophys. Res., 103(B12), 30 205– Field of the Earth: Proceedings of the 1st International Symposium of the 30 229. International Gravity Field Service (IGFS), Special Issue 18, pp. 193– Wahr, J., Swenson, S. & Velicogna, I., 2006. Accuracy of GRACE mass 198, eds Kilic¸oglu, A. & Forsberg, R., Gen. Command of Mapp., Ankara, estimates, Geophys. Res. Lett., 33, L06401, doi:10.1029/2005GL025305. Turkey. Wang, X., Gerlach, C. & Rummel, R., 2012. Time-variable gravity field Mayer-Gurr,¨ T., Savcenko, R., Bosch, W., Daras, I., Flechtner, F. & Dahle, from satellite constellations using the energy integral, Geophys. J. Int., http://gji.oxfordjournals.org/ C., 2012. Ocean tides from satellite altimetry and GRACE, J. Geod., 190(3), 1507–1525. 59–60, 28–38. Wessel, P.& Smith, W.H.F., 1991. Free software helps map and display data, Meyer, U., Jaggi,¨ A. & Beutler, G., 2012. Monthly gravity field solutions EOS, Trans. Am. geophys. Un., 72(41), 441, doi:10.1029/90EO00319. based on GRACE observations generated with the Celestial Mechanics Yi, W., 2012. The Earth’s gravity field from GOCE, PhD dissertation, Approach, Earth planet. Sci. Lett., 345, 72–80. Munchen,¨ Technische Universitat¨ Munchen.¨ Pail, R. et al., 2011. First GOCE gravity field models derived by three different approaches, J. Geod., 85(11), 819–843. Petit, G. & Luzum, B., 2010. IERS Conventions (2010). IERS Technical APPENDIX: ENERGY EQUATIONS IN Note No. 36. Verlag des Bundesamts fur¨ Kartographie und Geodasie,¨ INERTIAL AND EARTH-FIXED FRAME Frankfurt am Main. at Ohio State University Libraries on December 20, 2016 Ramillien, G., Biancale, R., Gratton, S., Vasseur, X. & Bourgogne, S., A1. Preliminary 2011. GRACE-derived surface water mass anomalies by energy in- tegral approach: application to continental hydrology, J. Geod., 6, The position vector r, representing the satellite position relative 313–328. to Earth centre, can be expanded in any Cartesian coordinate Ries, J.C., Bettadpur, S., Poole, S. & Richter, T., 2011. Mean background frame s (called s-frame) as an ordered triplet of coordinates as s = s s s T i gravity fields for GRACE processing, in Proceedings of the GRACE Sci- r r1 r2 r3 , such as inertial frame (i-frame) as r , and Earth- ence Team Meeting, Austin, TX, 8–10 August 2011. fixed frame (e-frame) as re. The relation between the two coordinate Rowlands, D.D., Ray, R.D., Chinn, D.S. & Lemoine, F.G., 2002. Short-arc vectors can be described as analysis of intersatellite tracking data in a gravity mapping mission, J. i = i e, Geod., 76, 307–316. r Cer (A1) Rowlands, D.D., Luthcke, S.B., Klosko, S.M., Lemoine, F.G.R.,Chinn, D.S., i McCarthy, J.J., Cox, C.M. & Anderson, O.B., 2005. Resolving mass flux at where Ce is a transformation matrix representing the orientation i high spatial and temporal resolution using GRACE intersatellite measure- between the two frames. The time-derivative of Ce can be derived ments, Geophys. Res. Lett., 32, L04310, doi:10.1029/2004GL021908. using the angular velocity w between the two frames. Here we define i = ω ,ω ,ω T Rowlands, D.D., Luthcke, S.B., McCarthy, J.J., Klosko, S.M., Chinn, D.S., wie ( 1 2 3) as the angular velocity vector of the e-frame Lemoine, F.G., Boy, J.-P.& Sabaka, T.J., 2010. Global mass flux solutions with respect to the i-frame, with coordinates in the i-frame. The from GRACE: a comparison of parameter estimation strategies—mass cross-product of the angular velocity vector can be further written concentrations versus Stokes coefficients, J. geophys. Res., 115, B01403, as a skew-symmetric matrix doi:10.1029/2009JB006546. ⎛ ⎞ Sakumura, C., 2014. Comparison of Degree 60 and Degree 96 Monthly 0 −ω ω ⎜ 3 2 ⎟ Solutions, GRACE Technical Note 10. http://www.csr.utexas.   wi × = i = ⎜ ω 0 −ω ⎟ . edu/grace/TN10_CSR-GR-14–01.pdf, last accessed September ie ie ⎝ 3 1 ⎠

2015 −ω2 ω1 0 Schmidt, M., Han, S.-C., Kusche, J., Sanchez, L. & Shum, C.K., 2006. Regional high-resolution spatiotemporal gravity modeling from Using above notations, we can derive the time-derivative of the i GRACE data using spherical wavelets, Geophys. Res. Lett., 33, L08403, transformation matrix Ce as (Jekeli 2001) doi:10.1029/2005GL025509.     C˙ i =− i Ci = i Ci = Ci e = wi × Ci = Ci we × . Schmidt, M., Seitz, F. & Shum, C.K., 2008. Regional four-dimensional e ei e ie e e ie ie e e ie hydrological mass variations from GRACE, atmospheric flux con- Therefore, taking the time-derivative of (A1) yields vergence, and river gauge data, J. geophys. Res., 113, B10402, i = i e + ˙ i e = i e + e × e , doi:10.1029/2008JB005575. r˙ Cer˙ Cer Ce r˙ wie r (A2) 1786 K. Shang et al. which represent the transformation of the velocity vector between In e-frame, multiplying re to (A5), we have two frames. Taking another time-derivative of (A2) yields e · e =∇e E · e + e · e − e × e · e   r¨ r˙ V r˙ a r˙ 2 wie r˙ r˙ r¨i = Ci r¨e + 2Ci we × r˙e + Ci we × we × re , (A3)   e e ie e ie ie − e × e × e · e, wie wie r r˙ which represents the transformation of the acceleration vector be- = i = e = where the third term on the right-hand side is zero. Substituting the tween two frames. Here we need to assume ˙wie ˙wie ˙wie 0, which is the first assumption for energy equations. above equation into (A7) by considering the second assumption, we can rewrite eq. (A7), the time-derivative of static geopotential in e-frame, as A2. Newton’s law of motion in both frames dV E   = r¨e − ae + ˙we × ˙we × re · r˙e. (A9) Acceleration vector in i-frame must obey Newton’s law of motion, dt ie ie which says Clearly (A8) and (A9) are equivalent since they are both derived r¨i =∇i V E + ai , (A4) from the same equations and the same two assumptions; that is, we ∂ ∂ ∂ can easily show i T where ∇ = ( ∂ i ∂ i ∂ i ) represents the gradient operator in i- r1 r2 r3 E E dV i i i i i

= − · − × Downloaded from frame, V is Earth’s static as well as secular and seasonal time- r¨ a r˙ wie r i =∇i R + i dt variable geopotential, and a V f represent the sum of the   acceleration from residual geopotential VR and non-conservative = e − e + e × e × e · e r¨ a wie wie r r˙ force f. T In e-frame, Newton’s law of motion can be derived using eqs = ∇e V E · r˙e. (A3) and (A4) as

  http://gji.oxfordjournals.org/ e =∇e E + e − e × e − e × e × e , r¨ V a 2 ˙wie r˙ ˙wie ˙wie r (A5) A4. Energy equation in both frames where all quantities are represented in e-frame. The energy equation in i-frame is obtained by integrating (A8) with respect to time as t A3. Time-derivative of static geopotential in both frames E = i − i · i − i × i . V r¨ a r˙ wie r dt In i-frame, the time-derivative of VE can be expanded as t0 i = E E i Since ˙wie 0, we arrive at the formulation of energy equation

∂ at Ohio State University Libraries on December 20, 2016 E dV V i E T i V˙ = = + (∇ V ) · r˙ . (A6) in i-frame as follows: dt ∂t t E 1 i 2 i i i i i i i 0i E V = r˙ − w · r × r˙ − a · r˙ − w × r dt − E . The same is for e-frame where the time-derivative of V can be 2 ie ie expanded as t0 (A10) E ∂ E e E dV V e E T e V˙ = = + (∇ V ) · r˙ . (A7) Similarly, the energy equation in e-frame is obtained by integrating dt ∂t (9) with respect to time as E e Here we introduce the second assumption that (∂V ∂t) = 0 t   E E = e − e + e × e × e · e . since we need to assume that for a certain time interval V has to be V r¨ a ˙wie ˙wie r r˙ dt static, that is the partial derivative with respect to time in e-frame t0 e = should be zero. Since ˙wie 0 as well, we arrive at the formulation of energy Next we need to rewrite eqs (A6) and (A7) by eliminating VE equation in e-frame as follows: from the right-hand side. Combining (A6) and (A7) by considering t E 1 e 2 1 e e2 e e 0e the second assumption, we have V = |r˙ | − ˙w × r − a · r˙ dt − E . (A11) 2 2 ie t0 ∂V E i = ∇i E T i e − i . Again, (A10) and (A11) are equivalent just as (A8) and (A9). V Cer˙ r˙ ∂t Finally, it should be noticed that in both frames t a · r˙dt = t0 Substituting the above equation into (A6) by considering (A2) V R + t f · r˙dt, since residual geopotential VR (such as tides) is not t0 and Newton’s law in i-frame (A4), we can rewrite eq. (A6), the static in either frames. We mention that some previous studies have time-derivative of static geopotential in i-frame, as already derived the identical (e.g. Gerlach et al. 2003; Han 2003; Wang et al. 2012) or similar (e.g. Badura et al. 2006;Jaggi et al. dV E ¨ = r¨i − ai · r˙i − wi × ri . (A8) 2008) formulation. dt ie