Abstract We propose an unconstrained approach torecover regional time-variations of surface mass anomaliesusing Level-1 Gravity Recovery and Climate Experiment(GRACE) orbit observations, for reaching spatial resolu-tions of a few hundreds of kilometers. Potential differ-ences between the twin GRACE vehicles are determinedalong short satellite tracks using the energy integral method(i.e., integration of orbit parameters vs. time) in a quasi-iner-tial terrestrial reference frame. Potential differences residualscorresponding mainly to changes in continental hydrologyare then obtained after removing the gravitational effects ofthe known geophysical phenomena that are mainly the sta-tic part of the Earth’s gravity field and time-varying contri-butions to gravity (Sun, Moon, planets, atmosphere, ocean,tides, variations of Earth’s rotation axis) through ad hocmodels. Regional surface mass anomalies are restored frompotential difference anomalies of 10 to 30-day orbits onto1◦ continental grids by regularization techniques based on
G. Ramillien (B)GRGS, DTP, CNRS, UMR 5562, Observatoire Midi-Pyrénées,14, Avenue Edouard Belin, 31400 Toulouse Cedex 01, Francee-mail: [email protected]
R. BiancaleGRGS, CNES, Observatoire Midi-Pyrénées,14, Avenue Edouard Belin, 31400 Toulouse Cedex 01, France
S. GrattonINPT-IRIT, University of Toulouse, 118, Route de Narbonne,31062 Toulouse, France
X. VasseurCERFACS, 42, Avenue Gaspard Coriolis,31057 Toulouse, France
S. BourgogneNOVELTIS, Parc Technologique du Canal,2, Avenue de l’Europe, 31520 Ramonville, Saint-Agne, France
singular value decomposition. Error budget analysis has beenmade by considering the important effects of spectrum trun-cation, the time length of observation (or spatial coverage ofthe data to invert) and for different levels of noise.
Since its launch, in March 2002, and placement onto a quasi-polar orbit at 400–500 km altitude, the Gravity Recoveryand Climate Experiment (GRACE) mission has provideda global mapping of the time-variable component of theEarth’s gravity field (Tapley et al. 2004a,b; Schmidt et al.2006). The unprecedented precision provided by this mis-sion, on the centimeter-level of geoid height, enables thedetection of tiny changes of gravity mainly due to mass redis-tributions of air and water inside the Earth’s fluid envelops(atmosphere, oceans and continental waters) from monthlyto decade timescales. Besides, GRACE has also proved itsability to detect the crustal readjustments associated to theSumatra-Andaman earthquake of December 2004 (Han et al.2006; Panet et al. 2007).
So far, several organisms, including the Center for SpaceResearch at University of Texas (UTCSR), Jet Propul-sion Laboratory (JPL), GeoForschungsZentrum (GFZ) andGroupe de Recherche en Géodésie Spatiale (GRGS), useGRACE Level-1 measurements of orbital parameters (i.e.,positions, velocities, GPS tracking data, K-Band Range mea-surements, on-board three-axis accelerometer data) to pro-duce time series of global gravity solutions in terms of Stokescoefficients (i.e., normalized spherical harmonic coefficients
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of the geo-potential). For instance, the spatial resolution ofthe global solutions provided by GRGS lies around 400 kmat the Earth’s surface as their spectrum is truncated at the50th degree (Lemoine et al. 2007a).
These solutions are corrected from the gravitational effectsof atmosphere mass variations and ocean tides and the resid-uals should represent mainly the continental hydrology com-ponent (Wahr et al. 1998; Dickey et al. 1997; Schmidt et al.2006). Several previous studies have demonstrated the abil-ity of GRACE to provide realistic estimates of continentalwater changes in different parts of the globe (Rodell andFamiglietti 1999, 2001; Wahr et al. 2004; Ramillien et al.2004), as well as to estimate the mass balance of ice sheets(Velicogna and Wahr 2005, 2006a,b; Ramillien et al. 2006a),changes in vertical fluxes (Rodell et al. 2004; Syed et al.2005; Ramillien et al. 2006b; Swenson and Wahr 2006). SeeRamillien et al. (2008) and Schmidt et al. (2008) for completereviews of GRACE applications on continental hydrologyand glaciology.
However, to address more complete studies and deeplyexploit the GRACE data, the global spherical harmonics rep-resentation of the Earth’s gravity field remains to be lim-ited by: (1) the north–south striping effect that degradesthe solutions in the high-frequency domain, (2) the alias-ing of the estimate, in the regional averaging process, byother signals coming from jointing geographical zones (i.e.,spectrum “leakage” of significant regional gravity variationover the whole sphere) (Awange et al. 2009) and (3) Gibb’seffects from spectrum truncations (i.e., omission errors). Inthis sense, a spatial resolution of ∼ 400 km is not enough toundertake detailed studies on water mass fluxes at smallerscales (i.e., inside continental basins). To compensate forthese shortcomings that are related to the global spheri-cal harmonics representation of gravity signals, alternativeapproaches have been proposed in recent years.
Instead of globally describing the geo-potential—usingorthogonal Legendre functions—the regional methods pro-pose to determine spatio-temporal variations of the gravityfield in a particular geographical area. The advantages ofconsidering such techniques to study objects of small dimen-sions such as hydrological sub-basins and mountains glaciersare the obvious decrease of energy leakage (with the lim-itations this effect imposes) from adjacent regions and thepossible gain in spatial resolution. Moreover, specific objectsand associated geometries of surface mass concentrations canbe defined by considering the satellite track data only pres-ent in the region under-study, and finally estimate the equiv-alent-water heights of the surface mass elements by linearinversion. These regional methods offer the advantage of notrequiring empirical post-processing low-pass filtering whichsmooth the global solutions and thus remove small-scale fea-tures (<400 km) of the continent hydrology. The number ofparameters to fit from satellite data can also be highly reduced
in this manner: for instance, for a spatial resolution of 100 kmand for geographical regions of 80◦ × 80◦ such as SouthAmerica, we need to adjust more than 40,000 spherical har-monic coefficients, as compared to around 6,400 equivalent-water heights by considering a regional approach. Using aquasi-homogeneous distribution of surface elements ensuresthere is no longer a singularity at the poles, which is thecase for classical geographical grids and orthogonal Legen-dre functions.
Numerous developments have already been made on suchmethods (Lemoine et al. 1998; Jekeli 1999; Garcia 2002; Hanet al. 2003; Han 2004; Han et al. 2005; Rowlands et al. 2002,2005; Luthcke et al. 2006). Originally, Muller and Sjogren(1968) used this kind of approach to characterize the massexcess distribution on or beneath the surface of the Moon.In this study, equivalent-water heights of mass elements areestimated independently from one another, whereas someauthors, in previous studies, had preferred to introduce a pri-ori constraints such as space and time correlation matricesin order to stabilize the linear inversion of GRACE satellitedata (e.g., Lemoine et al. 1998, 2007a,b; Rowlands et al.2005, 2010; Luthcke et al. 2006).
It is possible to locally describe the variations of thegravity field by introducing regional basis of orthogonal func-tions such as band-limited spherical harmonics (i.e., “mas-cons”) (Lemoine et al. 1998), spherical splines (Keller 2004;Eicker et al. 2007; Eicker 2008), as well as spherical wave-lets (Fengler et al. 2007). Later, Slepians functions have alsobeen used to make regional decompositions of the gravityfield (Han et al. 2008a,b). In this part, the mechanical energyconservation of an orbiting satellite has already been estab-lished earlier for CHAMP satellite data (Földvàry et al. 2003;Kusche and van Loon 2003; Howe et al. 2003; Ilk and Löcher2003; Löcher and Ilk 2005, 2007) and “Low-Low” satel-lites configuration (Jekeli 1999). So far, regional solutionshave been computed by Ohio State University (Garcia 2002;Han et al. 2003; Han 2004; Han et al. 2005) and NASAGoddard Space Flight Center (Rowlands et al. 2002, 2005;Lemoine et al. 2007b; Han et al. 2008a,b). These groups haveachieved higher stemporal resolutions of 15 and 10 days,respectively. Times series of GRACE-based regional solu-tions, the so-called “mascons”, are now available for tiles of4◦ × 4◦ in various geographical regions at NASA’s web site:http://grace.sgt-inc.com (Lemoine et al. 1998; Luthcke et al.2006).
By developing completely and implementing a regionalmethod for estimating surface water mass concentrationsfrom inter-satellite K-band range rate (KBRR) measure-ments with an accuracy of 0.1 µm/s (or 10 µm after inte-gration vs. time), the main challenges are: (1) the access toa better temporal (∼10 days) and spatial resolution of thesesurface gravity sources—at least of 200 km—and (2) thepossibility to face out inherent disadvantages of using the
classical global representation of time-varying gravity fieldin spherical harmonics (i.e., energy leakage, spectrum trun-cation error, north–south striping due to orbital geometryresonances). There are no differences between a classicalunconstrained spherical harmonic solution and the “mas-cons” approach when no constraints are applied (Rowlandset al. 2010). Instead of developing another “mascons”-typeapproach, equivalent to a regional spherical harmonics rep-resentation, we propose here a regional approach using noexplicit basis of regional orthogonal functions and a min-imum of a priori information on surface distribution ofmass to be restored from Level-1 GRACE data. Sphericalharmonics are implicitly involved using the Legendre poly-nomials for developing the inverse of the Cartesian dis-tance. This enables us to introduce elastic Love numbersto take compensation of mass by the Earth’s surface intoaccount.
The main steps of our regional approach presented in thenext section are: (1) the determination of the along-trackdifferences of potential anomaly that are only caused bychange in continental water storage using the energy integralproperties (i.e., assuming the constant compensation betweenkinetic and potential energies in time for each GRACE satel-lite) and then removing the potentials of known phenom-ena (i.e., atmosphere and oceanic masses including tides,the Moon, the Sun and the other planets attractions, polaraxis variations) to extract continental hydrology signals; (2)the decomposition of the chosen geographical region intoa homogeneous distribution of jointing elementary surfacetiles, following algorithms proposed by Eicker (2008); (3)the estimation of the equivalent-water heights at the tile cen-ters by robust inversion of the residuals potential anomaliesover the considered region, and based on matrix regulariza-tion techniques.
Firstly, the energy conservation is validated with GRACEorbits simulated through different input potential models, fordemonstration purposes. Secondly, an inverse regularizationscheme has been applied to be restored the 1◦ resolutionsurface water mass anomalies over an entire continent (e.g.,South America). More than the spatial coverage of the satel-lite tracks (10–30 days), the importance of the level of trun-cation of the eigenvalue spectrum (by keeping at least 99%of the signals energy) to reach higher spatial resolutions isdemonstrated. Finally, we investigate the impact of quasi-random noise in the KBRR residuals on the recovery, thisrepresents another source of error that degrades the regionalwater mass anomaly estimates. Tests of recovering surfacewater mass anomaly from realistic along-track KBRR resid-uals have been made by adding noise in our simulations toestablish an error budget. Finally, inversions of dynamicalorbits adjusted by GINS, a software for dynamical orbitogra-phy from real Level-1 GRACE data for 10 days are presentedto complete the demonstration.
2 Methodology
2.1 Inertial earth-centred reference frame
In order to describe the motion of any orbiting satellite, acoordinate system needs to be defined. The motion of a satel-lite follows the mechanical laws that remains to be valid inan inertial reference frame. The xy-plane of such a frame isthe plane of the Earth’s equator, and the origin is the centerof our planet. The x-axis is directed to the vernal equinox ofJ2000.0 (i.e., Julian Date of 12 h, the 1 January 2000). Thez-axis coincides with the mean terrestrial rotation axis. Inreality, this latter axis moves circularly with a period about26,000 years (i.e., precession) and has faster movements from14 days to 18.6 years (i.e., nutation). Because of the acceler-ation acting on the centre of the Earth, this is a quasi-inertialsystem and relativistic effects should be taken into account.However, we consider that the reference frame remains to bepurely inertial to simplify our demonstrations.
2.2 System
We consider artificial satellites orbiting around the Earth’scenter of mass. In the specific case of the GRACE mission,there are two satellites on the same orbital plane: A and B.Each satellite is assumed to be represented by a materialpoint which position and velocity vectors in this frame arer = (x, y, z) and r = (x, y, z), respectively.
2.3 External forces acting on the system
We distinguish two types of forces: (1) the gravitationalaccelerations that are conservative and derive from a scalarfunction named “potential”, and (2) the non-conservativeforces (i.e., atmospheric drag and solar pressure effects oneach satellite). In the following, the physical quantities areexpressed per unit of mass.
2.4 Fundamental equation of dynamics
According to the second Newton’s law, in the inertial frame S,the acceleration of a material point results from the sum ofall the forces acting on it, and in particular the gravitationalg that derives from a scalar potential V , and f the non-grav-itational ones:
r = g + f (1)
The three-axis accelerometers are shielded from f , so theyare only affected by g. Since total accelerations relativelyto the satellite are detected, by difference the accelerometermeasurements correspond to the non-gravitational forces.
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2.5 Energy integral equation
As a dynamic system, the satellite changes continuously itsenergy level by the existence of dissipating forces, so that the“energy integral” along short satellite tracks is considered foreach epoch (Visser et al. 2003). It is based on the simple factthat the gain of gravitational potential of the satellite is equalto the loss of kinetic energy—or vice versa—along its tra-jectory, once the effects of dissipative actions are removed.In the following, we consider perturbations of the gravita-tional potential V, such as V = V0 + δV where V0 is the apriori known potential and δV the incremental gravitationalpotential.
By multiplying Eq. 1 by the velocity r and then integrat-ing between times t0 and t , we make different energy termsappear:
• the kinetic energy:
T =t∫
t0
r rdt = 1
2(r)2 + T0 (2)
• the work produced by the non-gravitational forces:
W =t∫
t0
r f dt + W0 (3)
where T0 and W0 are constants of integration. By consideringthe total derivative of the scalar potential V versus time:
dV
dt= ∂V
∂t+ r .∇V (4)
we get:
r .∇V = dV
dt− ∂V
∂t(5)
Thus:t∫
t0
r.gdt =t∫
t0
r .∇V dt = V −t∫
t0
∂V
∂tdt (6)
where the last term of this equation is the “rotation” potentialdefined by:
� =t∫
t0
∂V
∂tdt. (7)
Jekeli (1999) proposed a numerical approximation of thislatter term by considering �, the angular velocity of theEarth:
� ≈ �(x y − yx). (8)
We found a more precise computation of � for hydrologi-cal applications by considering the static part of the gravi-tational potential VT, that represents nearly 99% of the totalfield measured by GRACE during the period of observation,using Eq. 4:
� ≈ VT −t∫
t0
r∇VTdt. (9)
For each GRACE satellite located at a radial distance r fromthe center of the Earth and at geocentric longitude λ andcolatitude θ , the potential due to the solid Earth is evaluatedusing:
VT(r, λ, θ) = G M
R
N∑n=0
n∑m=0
(R
r
)n+1
{Cnm cos(mλ)
+Snm sin(mλ)} Pnm(cos θ) (10)
where Cnm and Snm are the fully normalized Stokes coeffi-cients (i.e., dimensionless harmonic coefficients of the geo-potential) of degree n and order m, and Pnm(cos θ)Pnm isthe associated Legendre function. G and R are the gravita-tional constant (∼6.674288679 × 10−11 m3 kg−1 s−2) andthe mean Earth’s radius (∼6,371 km), respectively. M is thetotal mass of the Earth (∼5.9736 × 1024 kg). N is the maxi-mum degree of the decomposition and defines somehow thespatial resolution (i.e., half the minimum wavelength). Thea priori gravity field model we consider for VT is the recentEIGEN-GL04C combined model based on GRACE, LAG-EOS and surface data to degree and order N = 180.
Finally, the complete energy integral equation of the satel-lite along its trajectory is:
T = V + W −�+ E0 (11a)
where E0 is the sum of all the constants of integration versustime.
By defining the “reduced” kinetic energy, we have:
δT ∗ = T − W +�− E0 − V0 (11b)
The latter equation describes the difference between theobserved kinetic energy and the kinetic energy accordingto our a priori knowledge, while V0 can be referred as themodeled dynamic variations of potential.
The simplified “drag-free” energy equation is given by:
δV = δT ∗ (12)
where the upper-script symbol “∗” is related to the kineticenergy residuals corrected from known gravitational poten-tials, provided in the following, and the work of non-conservative forces effects f .
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GRACE-derived surface water mass anomalies 317
2.6 Time-varying potentials
The a priori known gravitational potentials represented by V0
including the solid Earth’s gravity field must be removed inthe reduced energy integral equation (Eq. 12) for extract-ing the contribution of non-modeled mass transfers, thusmainly the continental water storage variations. The a prioriknown gravitational potentials have also been used to vali-date numerically the conservation of the mechanical energyin several orbit simulations:
• Gravitational potentials caused by the atmospheric andoceanic mass variations are computed using 3-D pressuregrids of 6 h from ECMWF and the barotropic MOG2Docean model (Carrère and Lyard 2003), respectively.
• The potential of the direct tides due to the Sun and theMoon (see Lambeck 1988, p. 132):
V P(r, λ, θ) = G MP
r
3∑n=2
kn
( r
rP
)n+1Pn0(cosψ) (13)
where MP and rP are the mass and the position of the per-turbing body (i.e., the Sun or the Moon), respectively.ψ is the angular distance between the center of mass(r ′, λ′, θ ′) of the considered body and the point of obser-vation (r, λ, θ), and kn is the elastic nth degree Lovenumber. The 3-D perturbations due to the Sun, the Moonand six other main planets can be implemented followingthe DE403 ephemerides from JPL (Standish et al. 1995).
• The Solid Earth tides and associated pole variations arederived from IERS 2003 (McCarthy and Petit 2003). Asit is not part of the IERS 2003 conventions, geocen-ter motion is not considered. The induced effects of theSun and the Moon on the Earth’s oblateness J2 are alsoconsidered by the following equation:
V2(r, λ, θ) = GmP
(1
rP0
− �r .�rP0
r3
)+ mP
MU2 (14a)
where �rP0 is the vector between the satellite and the Sun
(or the Moon) and U2 a second degree term given by:
U2 = G M
ae
(ae
r
)3C20 P20(cos θ) (14b)
with ae the semi-major axis of the ellipsoidal Earth(∼6,378.136 km).
• The potential due to the indirect polar tides:
V polar(r, λ, θ) = �2 R2
3k2
(R
r
)3
(xpolar cos λ
+ypolar sin λ)P21(cos θ) (15)
where (xpolar, ypolar) is the instantaneous position of thez-axis in the equatorial plane with respect to its meanlocation over time. k2 is the second degree Love number(∼0.303).
• The gravitational potential due to indirect atmosphericand oceanic tides:
V tides = 4πGρw R
γ
N∑n=2
1 + k′n
2n + 1
(R
r
)n+1
×n∑
m=0
Pnm(cos θ)∑
l
−∑+χ±
l,nm (16a)
here, the plus and minus symbols represent “prograde”(i.e., westward) and “retrograde” (i.e., eastward) waves,respectively, γ the mean gravity (∼ 9.80 m/s2) andl the wave number. ρw is the mean water density(∼ 1,000 kg/m3). The ocean tides and associated polevariations are computed in the FES2004 model that usesatmospheric pressure and wind stress as forcing terms (LeProvost et al. 1994; Desai 2002). The spatial variation ofthe atmospheric or oceanic pressure (in Pascal) is givenby:
χ±l,nm = C±
l,nm sin(θl(t)± mλ− ε±l,nm
)(16b)
where the amplitude C±l,mn and phase ε±l,nm are simply
related to different daily, semi-daily and long-term tidalwaves, and θl(t) is computed from the co-tidal maps of abarotropic ocean and atmosphere.
2.7 Residual potential differences between the two GRACEsatellites
We consider the differences of gravitational potentialbetween A and B versus time using the previous “drag-free”orbit equation (Eq. 12):
δVAB = δVB − δVA = δT ∗AB = δT ∗
B − δT ∗A . (17)
Many studies have shown that energy integral is sensi-tive for velocity errors induced by numerical differentiation(Földvàry 2007a,b; Reubelt 2009). Thus, to avoid the useof numerical differentiation in the classical energy integralapproach, the present method derives velocities and positionsby numerically integrating the a priori known acceleration ofa dynamic model. The difference of kinetic energy betweenthe two GRACE vehicles can be expressed as a function ofthe KBRR residuals α∗
AB :
δT ∗AB = 1
2
( ˙r2B − ˙r2
A
)= rAB ( ˙rB − ˙rA)= rAB
(α∗
ABu AB +vAB)
(18)
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Fig. 1 Different types ofdivisions (grids) of the terrestrialsphere into juxtaposed surfaceelements. Reuter (1982) and thetriangular tessellation offer themost homogeneous distributionsof tiles (Eicker 2008), especiallynear the north and south poles
where α∗AB represents the KBRR residuals in µm/s obtained
after the orbit computation using the a priori known cor-recting forces. u AB is the unit vector along the line-of-sight(L-O-S) of the GRACE vehicles and vAB is a complementaryvector in the perpendicular plane of the L-O-S, respectively,and rAB is their instantaneous mean velocity given by:
rAB = 1
2(rA + rB). (19)
An approximation of Eq. 18 can be found in Jekeli (1999) ifthe perpendicular vector vAB is assumed to be small enoughto be neglected. Residuals potential anomalies δVAB aredefined as the differences between potential observations(or pseudo-observations from orbit simulation) from Eq. 17and the sum of the known potentials that we can model byconsidering Eqs. 10 and 13–16. In the simulation mode forvalidating the energy integral of the satellites using pseudo-observations, these residuals should be very small. In thecase of real orbits, δVAB represents the contributions of thenon-modeled phenomena to the gravity field, and mainly con-tinental hydrology variations.
2.8 Estimation of regional surface mass anomalies
The region of interest consists of juxtaposed regular tilesof unknown equivalent-water heights δh j to be determinedfor a given period �t = t − t0 by inverting the observedsatellite potential anomalies between the two GRACE vehi-cles A and B. These thin mass elements are supposed to be theunique sources causing the observed gravitational potentialanomalies.
The center of the j th element of mass is located at colati-tude θ j and longitude λ j and its elementary surface is δS j . Ifthe mean water density ρw is taken as a reference for surfacemass changes (i.e., supposing a constant density material),we have:
For a “classical” geographical grid corresponding to rectan-gular tiles, the j th elementary surfaces are simply:
δS j = R2δθδλ sin θ j (21)
where δθ and δλ are the dimensions in degrees of an ele-mentary surface of the grid, in radians, along latitude andlongitude directions, respectively.
Other surface geometries have been successfully imple-mented to determine sets of centers (θ j , λ j ) and elementarysurfaces δS j over regions on the terrestrial sphere (see Fig. 1),such as a “triangular” icosahedron-type decomposition intoquasi-homogeneous surface elements, Reuter’s, Driscoll’s,etc. (see Eicker 2008 for details on the different algorithmsfor subdividing an unit sphere).
According to the first Newton’s law, the difference of grav-itational potential between the two GRACE satellites A andB and due to continental hydrology is related to surface massanomalies δm j as follows:
δVAB = G∑
j
(1
L Bj
− 1
L Aj
)δm j . (22a)
The distance operators for the two GRACE vehicles aregiven by:
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GRACE-derived surface water mass anomalies 319
L A,Bj =
(R2 + r2
j − 2r j R cosψ A,Bj
)1/2. (22a)
The Al-Kashi’s formula for calculating angular distancesbetween points in spherical triangles gives:
cosψ A,Bj =cos θ j cos θA,B +sin θ j sin θA,B cos(λ j −λA,B).
(23)
As the inverse of the distance L is the generating functionfor the Legendre polynomials, we have another expressionfor this inverse of the Cartesian distance:
1
L A,Bj
= 1
R
∞∑n=0
(R
r j
)n+1
Pn
(cosψ A,B
j
). (24)
Interestingly, this latter expression enables us to input theelastic Love number knof degree n in Eq. 22a for consider-ing instantaneous compensation of the Earth’s surface.
Equation 22a can be written as a simple linear system ofequations Y = �X to be solved, where the coefficients ofthe design matrix are given by:
�i j = GρwδS j
(1
L B,ij
− 1
L A,ij
). (25)
and X = {δh} j=1,...,P is the vector containing the equivalent-water heights to be restored in , and Y = {δVAB}i=1,...,N isthe vector of the along-track potential difference anomaliesobserved over this region.
As it is often the case for estimating sources from observedpotentials, the linear system cannot be inverted using a classi-cal least-squares-based strategy since the problem is ill-posed(i.e., the condition number of the design matrix � is high).
Two main approaches can be considered to regularize theproblem: (1) methods based on physics, and (2) purely alge-braic methods, with the former one to be favoured wheneverpossible as it relies on the knowledge of the properties thatthe solution should satisfy in theory. For instance, it couldhave limited energy in some regions, or satisfy some bal-ance equations. In this case, the regularization takes the formof the so-called background quadratic penalty term, so thatthe parameter estimation can be written as a minimizationproblem:
MinimumX
(‖Y − �X‖2
D−1 + ‖X − X0‖2C−1
)(26)
where the symmetric and positive definite matrices D and Crepresent error covariance matrices for the observations andthe a priori X0 terms, respectively. In our case, C is a diagonalmatrix modelling uncorrelated errors with a standard devi-ation up to 10 cm. This pessimistic range of error has beenobtained by the analysis of series of Level-2 GRACE solu-tions, as noise is clearly present in the global solutions overthe oceanic areas. Equivalent-water heights to be restored areassumed not to be correlated to avoid any artificial smoothing
of the solution. Since we assume that the equivalent-waterheights X represent only the continental hydrology, we haveadded a background term Z to force the solution vector Xto be small enough over the oceans (e.g., <1% of the signalsenergy over the continents).
The problem posed by Eq. 22a is not appropriately stabi-lized and thus it needs an additional regularization. For doingso, the L-curve Tikhonov approach has been implemented(Hansen and O’Leary 1993), where the balance between theaccuracy of the solution and the residual norm is based onsome heuristics that have proved successful on a broad classof problems of this type:
MinimumX
(‖Y − �X‖2D−1 + ‖Z(X − X0)‖2
C−1
+ κ2‖Z(X − X0)‖2C−1) (27)
where κ is a regularization parameter, for finding a compro-mise between: (1) the solution norm:
ξκ = ‖X‖2 (28a)
and (2) the residual norm:
ζκ =(‖Y − �X‖2
D−1 + ‖Z(X − X0)‖2C−1
) 12
(28b)
where the discrete Picard condition holds that is, when theFourier coefficients of the observed minus predicted term inthe left singular vector decay faster to zero than the singularvalues (Hansen and O’Leary 1993). Our task consists of spot-ting the corner at which the curvature of the L-curve becomesinfinite. We therefore look for the regularization parameterκ for which the curvature:
κ →◦ζκ
◦◦ξκ − ◦
ξκ
◦◦ζκ
[(
◦ζκ)2 + (
◦ξκ)2
]3/2 (29)
is maximal using a scalar nonlinear function minimizationalgorithm based on the golden section search with parabolicinterpolation (Forsythe et al. 1977). The upper script sym-bols “◦” and “◦◦” represent the first and second derivativesversus κ , respectively.
For our purpose, and following Hansen (1997), we arerather going to consider the curvature of the L-curve in alog–log scale, which often allows for an easier detection ofthe corner. As a final remark, the computation of the log ofthe curve points for each κ , as required by the minimizer,is a computationally expensive part of the algorithm, sinceit implies solving Eq. 22a for each κ . In our case, we haveoptimized the computation by calculating first the singularvalue decomposition (SVD) of the matrix:
[D−1/2�
C−1/2 Z
]=
I∑i=1
Ui Si V Ti (30)
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320 G. Ramillien et al.
where U and V are two unitary matrices, I is the numericalrank of the matrix and S is a diagonal matrix that containsthe singular values σi .
This enables us to compute the L-curve quantities bysimple vector operations as expressed by the following equa-tions:
ζ 2κ =
I∑i=1
(κ2
σ 2i + κ2
U Ti Y
)2
(31a)
and
ξ2κ =
I∑i=1
(σi
σ 2i + κ2
U Ti Y
)2
. (31b)
Once the optimal regularization parameter (or equivalently,the optimal rank I0 of the singular value truncation) is deter-mined by the L-curve analysis, the SVD solution is computedusing:
X − X0 =I∑
i=1
σi
σ 2i + κ2
(U Ti Y )Vi . (32)
where S−1I0
is the diagonal matrix which elements are theinverses of the singular values up to I0.
3 Numerical applications
For this purpose, the Géodésie par Intégrations NumériquesSimultanées (GINS) software for orbitography developed byGRGS in Toulouse, France (Lemoine et al. 2007a; Bruinsmaet al. 2010) was used for GRACE orbit simulations by inte-gration a set of known accelerations (or equivalently, a prioriknown potentials), considering non-conservative forces (i.e.,“drag-free” orbits) from a starting ephemerid. A dynamicalleast-squares orbit adjustment is routinely used with back-ground gravity models. Ten or 5-s sampled orbits are gener-ated for daily periods and for each GRACE vehicle and/orsimulated orbits are obtained by combination of simulatedmeasurements of the two GRACE satellites. These data con-tain the lists of successive adjusted positions, velocities, andtabulated values of a priori known potentials which have beenderived by spatial integration of gravitational accelerations.In other words, the mathematical expressions of the potentialshave been derived analytically from the expressions of theknown accelerations implemented in GINS. Successive esti-mated velocities and positions along the tracks are used afteradjustment of the dynamic orbit by GINS to evaluate termsof Eqs. 18 and 19. KBRR residuals are obtained as the differ-ence between observed KBRR—given in the Level-1B datafiles—and the GINS-modelled KBRR values using knownaccelerations/potentials (see Sect. 2.6). As KBRR residu-als still contain systematic errors, such as unrealistic oscil-lations near the revolution period along the satellite track,
empirical parameters for each 24 h orbit are also fitted andremoved in the KBRR data (i.e., bias and bias-rate values and2 terms per revolution, plus extra double terms for the half,third and fourth of this period) (Bruinsma et al. 2010). Thus,each run of GINS provides the KBRR residuals α∗
AB afterremoving the sum of all the a priori known forces includingthe static field, which corresponds to the mixing of gravitysignals of unmodelled geophysical phenomena includingcontinental hydrology signals, errors in the correcting grav-itational models plus noise. If our simulations of GRACEorbits were made using completely known accelerations/potentials, we should have logically α∗
AB ≈ 0. In the caseof real orbits, this is not the case as KBRR residuals containat least three sources (in order of magnitudes): measurementerrors, model errors of the applied corrections and acceler-ations from non-modelled sources, such as the gravitationaleffects of continental hydrology.
As a model of perturbation of surface forces for conti-nental hydrology is needed in our simulations to prove theapplicability of the presented method, we consider regional1◦ geographical maps of continental total water storagechange provided by the WGHM global hydrology model(Döll et al. 2003). We choose South America [90◦W–30◦W;60◦S–20◦N] in our simulations and inversions, as this regionof the world experiences the most important seasonal cycleamplitudes of several hundreds of millimeters of equivalent-water height. These reference maps have been used to gen-erate “reference” GRACE orbits and KBRR residuals α∗
ABthrough the GINS software and for 10 to 30-day periods.Alternatively, Eq. 22a can also be used to compute more rap-idly potential differences between the two GRACE vehicles,once the geometries of surface mass concentrations are given.
3.1 Validation of the energy conservation alongthe satellite track
Validation of the energy integral equation consists of ver-ifying that the difference between: (1) the GINS-derived“model” potentials from integration of given 3D acceler-ations, and (2) the potentials obtained by computing thekinetic energy variations from velocities, is close to zero.An example of validating all the gravitational potentialsalong the orbits is presented in Fig. 2, showing the poten-tial differences between the two GRACE vehicles are tiny.Model potentials have been taken one-by-one to check theenergy conservation versus time. After a Lagrange interpola-tion over three points, the five-value Bode’s formula is usedto compute the numerical integration of the orbit parame-ters of Eqs. 2–9. Root-mean squared (RMS) differences aregenerally <10−8 m2/s2 for each GRACE satellite, whereasthe RMS difference between the two vehicles remains to bearound 10−10 m2/s2 due to rounding errors in the integrationprocess, compared to the machine precision is 10−12 m2/s2.
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GRACE-derived surface water mass anomalies 321
Fig. 2 Validation of the energyintegral estimates usingsimulated GRACE orbits.Potential anomaly errors foreach GRACE vehicle computedas the differences between theenergy integral-derived andmodel potentials. The latter aredetermined analytically fromintegration of knownaccelerations through the GINSsoftware
Fig. 3 Example of a simulatedcase of restoring water massvariations for August 2006 overSouth America: a potentialdifference anomalies along the30-day GRACE orbit (note thechange of the sign of thepotential difference anomaliesbetween ascending anddescending tracks for the samesurface mass anomaly), b the 1◦regional solution obtained afterL-curve regularization (see alsoFig. 4) and corresponding to∼99.9% of the singular valuespectrum, c differences betweenthe regional solution and thereference WGHM grids
These very small differences prove that the energy integral istheoretically verified with an acceptable accuracy for detect-ing continental hydrology variations, that are typically inthe range of 10−4–10−1 m2/s2 according to GRACE orbitsimulations using continental hydrology model outputs.Therefore, the perturbation of the gravity field caused bycontinental hydrology change would clearly appear in theKBRR residuals α∗
AB of real GRACE orbits after correctionof the a priori known models of potential.
3.2 Inversion of simulated error-free potential anomalies
Validation consists of comparing the gridded values restoredfrom GINS-simulated along-track KBRR residuals over
South America, with the reference monthly WGHM grids.For example, in August 2006, the water mass anomalyreaches +315 mm of equivalent-water height in the Ori-noco drainage basin, whereas an important geographicallyspread negative anomaly of −180 mm (i.e., loss of watermass) appears in the southern part of the Amazon basin.By considering this reference regional grid, the correspond-ing amplitudes of the along-track potential difference anom-alies are of ±5.7 × 10−3 m2/s2, as presented in Fig. 3a.The number of tiles to be restored is 61 × 81 = 4,941onto a simple geographical grid using the 38,729 on-trackpotential differences computed previously. Once the corre-sponding 38,729 × 4,941 Newtonian matrix is built fromthe satellites and tiles positions (see Eqs. 22a, 27), the
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Fig. 4 L-curve analysis in the case of water mass determination overSouth America, and the main inflexion point (star symbol) adjusted asthe best numerical compromise between norms of the solution and theresiduals with potential anomaly observations
search for the optimal regularization parameter κ is made,as illustrated in Fig. 4. The pseudo-inverse linear operator isbuilt and applied to the vector of potential difference resid-uals. Figure 3b presents the corresponding 99.9% spectrumenergy solution of the main continental hydrology variationsfor South America with realistic amplitudes (from −132 to+202 mm of equivalent-water height with a spatial variabil-ity of ∼ 39 mm RMS). The errors of recovery (i.e., differ-ences between the estimated and reference WGHM maps forAugust 2006) are of 21 mm RMS in terms of equivalent-waterheight for the complete region, they are presented in Fig. 3c.This error map exhibits small and medium scale undulationsthat correspond to information missing in the regional solu-tion considering 99.9% of the SVD energy spectrum.
3.3 Inversion of real GRACE orbits
Five-second sampled daily orbit (positions, velocities) arerestored for several days using GINS and from Level-1GRACE data. By correction of the potential models in theorbit adjustment process, the KBRR residuals α∗
AB are alsoprovided by GINS in a separate file. The “drag-free” energyintegral equation (Eq. 18) is then used to derive the corre-sponding potential difference residuals. An example of 5-ssampled potential difference residuals for several satelliterevolutions of beginning of the 11 August 2009 is presentedin Fig. 5. This plot shows variations of ±1.5 × 10−2 m2/s2
with variability of ∼1.93 × 10−3 m2/s2 RMS, and revealsthe presence of an important short-wavelength noise in theKBRR residuals that consequently contaminates the poten-tial differences (∼1.21 × 10−3 m2/s2 RMS). Each shorttrack of potential difference anomalies extracted over a largeregion (i.e., the Amazon basin) is corrected by a linear trendnot to invert unrealistic long-wavelength signals. The designmatrix � is computed by considering a 1◦ geographical gridover South America and using Eq. 22a. Figure 6 presentsthe regional solutions obtained considering increasing num-bers of singular values in the regularization and without low-pass filtering the KBRR data before inversion. Besides theimprovement of the spatial resolution (i.e., gain of details),the regional solution reveals the amplification of unrealis-tic medium-wavelength structures, especially seen over theoceans (Fig. 6c).
4 Discussion
Tests with GRACE orbits simulated with perturbing surfaceforces from continental hydrology have shown that the 1◦
Fig. 5 Potential anomaliescomputed using KBRR residualsof the GINS-derived 5-ssampling GRACE orbit on the11 August 2009. These residualsshould mainly correspond tocontinental hydrology change(not modelled in GINS), errorsin a priori known models andnoise. Dark line is for the valuesafter 1,000-s low-pass filtering
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GRACE-derived surface water mass anomalies 323
Fig. 6 Real case of restoringwater mass variations overSouth America 10-day KBRRresiduals (11–21 August 2009),partly presented in Fig. 6, andfor different levels of thesingular value truncation: a99%, b 99.9% and c 99.99%
Fig. 7 RMS error of recoveryversus the level of singular valuetruncation while estimating a 1◦grid over South America forAugust 2006 from a 30-dayorbit. The corresponding spatialresolutions of the regionalsolutions are in grey
regional solution can be fully recovered numerically, with anexcellent spatial resolution, in absence of noise in the KBRRdata to invert. However, in the case of real adjusted GRACEorbits and KBRR residuals, the solution is clearly degradedby unrealistic signals in the data.
Close to the 5–10 s sampling rate frequency, the spuri-ous noise has magnitudes that can reach several times theamplitudes of the continental hydrology signature. The high-frequency part of the noise is related to KBR measure-ment errors appearing during the data acquisition. A purephysical reason is that because of upward continuation ofgravity anomalies, the potential anomaly created by surfacewater mass distribution and measured at satellite altitudecorresponds to band-limited signals and thus remains quitesmooth. Besides, errors of longer wavelengths from the a
priori known models cannot be excluded, as these modelsare far to be completely perfect, or they do not model non-periodic phenomena.
Impact of the singular value truncation on the solutionaccuracy for a simulated error-free orbit of 30 days (e.g.,August 2006) is illustrated in Fig. 7. Errors of recoveryincrease linearly with the power spectrum of the solutionor, equivalently, with the number of omitted singular values.More singular values from SVD analysis need to be retainedto gain small-scale details in the solution. The correspondingspatial resolutions are derived from the decay of the powerspectra by 2D Fourier analysis of the solutions. The spa-tial resolution improves logically with the number of singu-lar values kept for regularization to build the pseudo-inverseoperator. Without adding noise, the difference of considering
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Fig. 8 Amplification of theRMS error of recovery withpseudo-random noise pollutingthe starting KBRR data in thecase of 10- and 30-day GRACEorbits
shorter orbits (i.e., 10 days) than 30-day data to recover 1◦sampled grids remains to be small and <1 mm of equivalent-water height RMS.
Quasi-random noise has been artificially added to along-track potential difference anomalies to study the stability ofthe inversion process versus the singular value cutting. Theresults considering realistic noise levels of 10−3 m2/s2 and10−4 m2/s2 amplitudes are presented in Fig. 8. The presenceof noise makes the omission error increase exponentially withthe spatial resolution. As the pseudo-inverse operator is lin-ear, the error by singular values omission increases regularlywith the amplitude of the noise, and it can be deduced by asimple scaling of these error curves. Inverting noisy 30-dayorbits yields more accurate results than considering 10-dayorbits, as having three times of along-track KBRR obser-vations reinforces the signals-over-noise ratio. With a bettercoverage of satellite tracks (i.e., 30 days), the error of singularvalue omission gets reduced by a factor ∼ 2, when trunca-tion of the singular value spectrum is more than 10−5%. Fromthese last results, one may argue that long periods of obser-vation (at least 30 days) are always better than shorter onesfor precise determination of water mass anomalies. However,due to sub-monthly variability of continental hydrology, anacceptable compromise between spatial and temporal reso-lutions needs to be found (Pail 2000; Freeden and Schreiner2008). In the case of the “mascons” approach, a discussionof the trade-off between temporal and spatial constraints canbe found in Rowlands et al. (2005).
Besides low-pass filtering, the noisy KBRR residualsbefore inversion, we have also tried to constrain the regionalsolution by considering a priori information in the inversion.Since unrealistic structures from noise are clearly amplifiedover the oceanic areas (see Fig. 6c), using a geographicalmask has been tested to force the recovered equivalent-water
heights over the oceans not to exceed 10% of the total sig-nals power, thus making the solution purely continental. Theimprovement of the amplitudes of the regional solution overcontinents is statistically small (∼0.2 mm RMS) at the levelof 0.1–0.001% of singular value truncation. If the 99.9%solution looks more realistic over South America, this isclearly not the case for the oceans, while the 99% solu-tion appears reasonable over oceanic areas but too smoothover land. Obviously, introducing such numerical constraintsover oceans does not solve the problem of the noise overcontinental areas.
Various dimensions of surface tiles have been tested in theinversion of simulated GRACE orbits, from 8◦ × 8◦ down to1◦ × 1◦, and similar amplitudes of equivalent-water heightshave been recovered by energy integral analysis. The spatialresolution of the gridded solution remains to be limited bythe intrinsic resolution of the starting KBRR data to invert.In other words, there is no significant gain in consideringtiles of dimensions smaller than 2◦ squares in the proposedinversion. Particularly for real GRACE orbits, regional solu-tions are compared to the 10-day “stabilized” global solu-tions provided and regularly updated by GRGS (http://grgs.obs-mip.fr/index.php) (Lemoine et al. 2007a; Bruinsma et al.2010), and presents roughly comparable patterns of sur-face water mass variations for the same period of 11–20August 2009 (Fig. 9). The differences observed between theCNES/GRGS spherical harmonic solution and the localizedapproach developed in the present study may originate from:(1) the differences between the energy integral method com-pare to the classical equation-of-motion approach as used forthe spherical harmonic solution, and (2) the higher numberof parameters estimated compared to a degree-50 sphericalharmonic solution. However, the choices of the shape and thelocalizations of the centres of the surface tiles have significant
Fig. 9 Comparison ofcontinental water change overSouth America for August 2009according to different sources:a regional solution computedonto a 1◦ geographical grid;b regional solution computedonto triangular surface elementsof ∼10,000 km2 and c“stabilized” 10-day GRACEsolution from GRGS for thesame period (August 2009).Elastic compensation of thesurface water masses has beentaken into account in thecomputation of a and bsolutions
Fig. 10 Ten-day regionalsolutions during the year 2009considering a SVD spectrumtruncation of ∼99.9% andgeographical tiles for surfacedensity of 2◦ × 2◦. Thecorresponding calendar intervalsare indicated at the top of themaps
consequences on the final result, as illustrated by the solutionspresented in Fig. 9a (i.e., triangular) and Fig. 9b (i.e., geo-graphical) that were obtained using the same along-track dif-ferences of potential. In the first case, all triangular elementshave a constant surface of ∼104 km2 on the terrestrial sphereby construction, while for the classical geographical grid, thesurfaces of the tiles vary from 1.2×104 km2 at equator downto the half of this surface at 60◦S due to the cosine of latitudefactor. Consequently, the number of unknown water heightsto adjust and the weighting of the potential differences in the
regularization process can be different, and thus the equiva-lent-water heights estimates. Conceptually, it would be betterto work with a homogeneous distribution of surface elementsat regional scale, such as the triangular-type tessellation, thatoffers the advantage of having no numerical singularities atpoles.
As an example of time-variations of water mass, Fig. 10presents 10-day regional solutions showing the importantseasonal signals of continental water storage, especiallyinside the Amazon basin.
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5 Conclusion
An unconstrained regional approach to refine the regionaltime-variations of the gravity field measured by GRACE hasbeen developed and used to obtain a spatial resolution of a fewhundreds of kilometers, and thus to access to detailed surfacewater mass patterns. The two steps of the proposed method(i.e., potential variations using along-track KBRR residualsand linear inversion by regularization and no time and spa-tial a priori smoothing constraints) have been numericallyvalidated with GINS software-based GRACE orbits. Elasticcompensation of the Earth’s surface due to the load of thewater mass change can be also considered in the inversionprocess. An interesting advantage of the regional approachis the possibility of dealing with mass elements of variouspre-defined or adapted shapes and sizes.
While inverting GRACE orbits for continental hydrology,the choice of regularization parameter to reach optimal spa-tial resolution is critical, it needs to be optimally determinedusing classically the L-curve criteria, or noise level analysis.Unfortunately, the recovery of regional solutions from realorbits remains to be degraded by the amplification of unreal-istic medium-wavelength signals in the KBRR residuals withthe spatial resolution, as it can be seen over the oceanic areas.It is true that inverting true GRACE orbit is more complexthan a simple simulation with adding white noise. There aremany systematic errors (and not quasi-random) left in theGRACE processing, in particular leading to the large north–south striping which cannot be explained by any mis-mod-eling of geophysical signals (atmosphere, ocean, tides, etc.).This point would require more investigation in the future.Constraints of considering an oceans/continent geographicalmask can be input in the inversion to reduce these effectsover the oceans, and thus reinforce the signals on the con-tinents. A special effort for removing this type of artefactsby filtering directly the KBRR data needs to be made in thefuture. Another following of this study would be to producetime series of regional maps for continental water storagechange for validation with independent datasets, and use ofthese regional solutions for analysing the redistribution ofsurface masses in response to climatic variations.
Acknowledgments We would like to thank the three anonymous ref-erees for their scientific comments and many valuable suggestions thathelped us to improve the manuscript. We are also grateful to Dr. LuciaSeoane for having produced the time series of 10-day regional solutionsfrom Level-1 GRACE data. Her post-doctoral research was supportedby the ADTAO project of the RTRA/STAE foundation.
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