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Advances in Space Research 47 (2011) 1020–1028

GPS-only gravity field recovery with GOCE, CHAMP, and GRACE

A. Jäggi ⇑, H. Bock, L. Prange, U. Meyer, G. BeutlerAstronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

Received 23 August 2010; received in revised form 5 November 2010; accepted 8 November 2010Available online 13 November 2010

Abstract

Gravity missions such as the Gravity field and steady-state Ocean Circulation Explorer (GOCE) are equipped with onboard GlobalPositioning System (GPS) receivers for precise orbit determination (POD), instrument time-tagging, and the extraction of the long wave-length part of the Earth’s gravity field. The very low orbital altitude of the GOCE satellite and the availability of dense 1 s GPS trackingdata are ideal characteristics to exploit the contribution of GPS high-low Satellite-to-Satellite Tracking (hl-SST) to gravity field deter-mination. We present gravity field solutions based on about 8 months of GOCE GPS hl-SST data from 2009 and compare the resultswith those obtained from the CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE)missions. The very low orbital altitude of GOCE significantly improves gravity field recovery from GPS hl-SST data above degree 20, butnot for the degrees below 20, where the quality of the spherical harmonic coefficients remains essentially unchanged. Despite the limitedtime span of GOCE data used, the gravity field of the Earth can be resolved up to about degree 115 using GPS data only. Empiricallydetermined phase center variations (PCVs) of the GOCE onboard GPS helix antenna are, however, mandatory to achieve thisperformance.� 2010 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: GOCE; Kinematic positions; Gravity field recovery; Polar gap; Phase center variations (PCVs); GPS

1. Introduction

In the last decade observations from the Global Posi-tioning System (GPS) have been established as an impor-tant pillar for gravity missions. Since the launch of theCHAllenging Minisatellite Payload mission (CHAMP,Reigber et al., 1998) GPS sensors are not only used as akey tracking system for precise orbit determination(POD) but also for extracting the long wavelength partof the Earth’s static gravity field (Reigber et al., 2003).Although current gravity missions such as the GravityRecovery And Climate Experiment (GRACE, Tapleyet al., 2004) and the Gravity field and steady-state OceanCirculation Explorer (GOCE, Drinkwater et al., 2006)are equipped with other core instruments, they still makeuse of the GPS high-low Satellite-to-Satellite Tracking(hl-SST) to support the determination of the low degree

0273-1177/$36.00 � 2010 COSPAR. Published by Elsevier Ltd. All rights resedoi:10.1016/j.asr.2010.11.008

⇑ Corresponding author.E-mail address: adrian.jaeggi@aiub.unibe.ch (A. Jäggi).

spherical harmonic (SH) coefficients of the Earth’s gravityfield. In the case of GOCE these coefficients are even exclu-sively determined from GPS data as the measurements ofthe core instrument, the three-axis gravity gradiometer,are band-limited (Pail et al., 2006).

GOCE was launched on 17 March, 2009 into a sun-synchronous, dusk-dawn orbit with an initial mean altitudeof 287.91 km (mean distance from the geocenter minus theequatorial Earth radius). After a descent phase of abouthalf a year the first measurement and operational phase(MOP-1) has started on 29 September, 2009 at a mean alti-tude of 259.56 km, which corresponds to a repeat cycle of979 revolutions in 61 days. The very low Earth orbit(LEO) of the GOCE satellite is perfectly suited to exploitthe contribution of GPS hl-SST to gravity field recoveryand to compare the GOCE results with those obtainedfrom CHAMP and GRACE, which have been providingGPS hl-SST data for more than 8 years.

Section 2 shortly reviews the methods of kinematic orbitdetermination and subsequent gravity field determination

rved.

http://dx.doi.org/10.1016/j.asr.2010.11.008mailto:adrian.jaeggi@aiub.unibe.chhttp://dx.doi.org/10.1016/j.asr.2010.11.008

A. Jäggi et al. / Advances in Space Research 47 (2011) 1020–1028 1021

underlying this article. Section 3 compares gravity fieldsolutions obtained from GOCE, CHAMP, and GRACE,and discusses the impact of a high data sampling rateand the limitation due to the polar gap in the case of thesun-synchronous GOCE orbit. Section 4 has the focus onthe phase center modeling of the GOCE GPS antennaand compares the results with the experience gained fromthe analysis of GRACE data. Section 5 compares theresults with those obtained in the frame of the GOCEHigh-level Processing Facility (HPF, Koop et al., 2006).

2. Method used for GPS-only gravity field recovery

A three-step procedure is applied to gravity field recov-ery from GOCE, CHAMP, and GRACE GPS data accord-ing to the Celestial Mechanics Approach (Beutler et al.,2010). In a first step GPS observations are processed toderive kinematic LEO positions at the measurement epochstogether with the associated covariance information (seeSection 2.1). In a second step the kinematic LEO positionsare weighted according to the covariance information andserve as pseudo-observations to set up normal equations ona daily basis for the unknown gravity field coefficients in ageneralized orbit determination problem (see Section 2.2).In a third step the daily normal equations may be modifiedand are accumulated into monthly, annual, and multi-annual systems (see Section 2.3).

2.1. Step I: kinematic orbit determination

The geometric strength and the high density of GPSobservations allows for a purely geometrical approach todetermine kinematic LEO positions at the observationepochs by precise point positioning (Švehla and Rothacher,2004). The kinematic positions are determined in a stan-dard least-squares adjustment process of GPS observationstogether with all other relevant parameters without usingany information on LEO dynamics. A band-limited partof the full covariance matrix of kinematic positions maybe efficiently derived in the course of kinematic orbitdetermination.

The high-rate satellite clock corrections (Bock et al.,2009) and the final GPS orbits from the Center for OrbitDetermination in Europe (CODE, Dach et al., 2009) areused together with attitude data from the star trackers toprocess the undifferenced Level 1b GPS carrier phasetracking data of the CHAMP, GRACE, and GOCE mis-sion for kinematic orbit determination.

GRACE 30 s kinematic positions were computed at theAstronomical Institute of the University of Bern (AIUB)for various studies on LEO POD (e.g., Jäggi et al.,2009b) and for gravity field recovery with inter-satelliteK-band data (Jäggi et al., in press), whereas CHAMP 10 skinematic positions were used to exploit GPS-based gravityfield recovery (Prange et al., 2010). GOCE 1 s kinematicpositions are computed at AIUB in the frame of the GOCEHPF as part of the GOCE precise science orbit product

(Bock et al., 2007). They are provided to the user commu-nity together with a band-limited part of the covariancematrix covering four off-diagonal blocks (EGG-C, 2008).

Kinematic positions are particularly sensitive to a cor-rect modeling of the antenna phase center location as noconstraints are imposed by dynamic models on theepoch-wise estimated positions. Special care has to betaken to empirically correct for phase center variations(PCVs) of the LEO GPS antennas in order not to deterio-rate the kinematic positions and the subsequent gravityfield recovery (Jäggi et al., 2009b). If not stated differently,empirical PCVs are used for kinematic orbit determinationin this article.

2.2. Step II: generalized orbit determination

The equation of motion of a LEO satellite including allperturbations reads in the inertial frame as

€r ¼ �GM rr3þ f 1ðt; r; _r; q1; . . . ; qdÞ; ð1Þ

where GM denotes the gravity parameter of the Earth, rand _r represent the satellite position and velocity, and f1 de-notes the perturbing acceleration. The initial conditionsr(t0) = r(a,e, i,X,x,T0; t0) and _rðt0Þ ¼ _rða; e; i;X;x; T 0; t0Þat epoch t0 are defined by six Keplerian osculating ele-ments, e.g., a,e, i,X,x,T0. The parameters q1,. . .,qd inEq. (1) denote d additional parameters considered as un-knowns, e.g., arc-specific orbit parameters and generalparameters such as gravity field coefficients.

In a first step a priori orbits for gravity field determina-tion are computed on a daily basis. Based on a selected apriori force model (defined by an a priori gravity fieldmodel, ocean tide model, etc.) the kinematic positions,weighted according to the covariance information fromSection 2.1, are fitted by numerically integrating the equa-tion of motion (1) and by adjusting arc-specific orbitparameters. Efficient numerical integration techniquesare applied to solve the variational equations (Beutler,2005) in order to obtain the required partial derivatives.As accelerometer data need not necessarily be taken intoaccount to derive GPS-only gravity field solutions of highquality (Prange et al., 2009), arc-specific empirical param-eters are set up in addition to the six Keplerian osculatingelements. Constant and once-per-revolution empiricalaccelerations acting over the entire daily arcs are set upin the radial, along-track, and cross-track directions tocompensate for the main part of the unmodeled non-gravitational perturbations. In analogy to Jäggi et al.(2009a), remaining deficiencies are captured by settingup low-degree polynomials (degree 3) for the along-trackaccelerations and, in particular, by estimating additionalpseudo-stochastic pulses (instantaneous velocity changes)at predefined epochs for the radial, along-track, andcross-track directions. Pseudo-stochastic pulses do notaffect the LEO trajectory in-between the pulse epochsand are thus well suited for gravity field recovery. A

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ITG−GRACE03SGRACE B (1 year)GOCE (0.7 years)CHAMP (8 years)

Fig. 1. Square-roots of degree difference variances of gravity fieldrecoveries from GRACE B, GOCE, and CHAMP kinematic positions.

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rather short spacing of 6 min is used between subsequentpulse epochs for satellites at very low altitudes such asGOCE, whereas 15 min are used for the GRACE satel-lites flying at a much higher orbital altitude. The param-etrization allows to start gravity field recovery fromEGM96 (Lemoine et al., 1997) serving as a priori gravityfield model without the need for iterations.

Based on the computed a priori orbits r0(t) gravity fieldrecovery from kinematic positions is set up as a generalizedorbit improvement problem (Beutler et al., 2010). Theactual orbits r(t) are expressed as truncated Taylor serieswith respect to the unknown parameters pi (arc-specificorbit parameters and the unknown SH coefficients) aboutthe a priori orbits, which are represented by the a prioriparameter values pi0:

rðtÞ ¼ r0ðtÞ þXn

i¼1

@r0ðtÞ@pi

� Dpi; ð2Þ

where Dpi ¼:

pi � pi0 denote the n ¼:

6 + d corrections con-sidered as unknown for each daily arc. The partial deriva-tives with respect to all parameters allow it to set up thedaily normal equations based on kinematic positions forall parameters according to standard least-squaresadjustment.

2.3. Step III: normal equation handling

Arc-specific parameters are pre-eliminated before thedaily normal equations are accumulated into normal equa-tion systems covering longer time spans, e.g., severalmonths or years. The accumulated normal equation systemis eventually inverted in order to obtain the corrections ofthe SH coefficients with respect to the a priori gravity fieldcoefficients and the associated full covariance information.No regularizations are applied to compute the gravity fieldsolutions presented in this article.

3. GPS-only gravity field recovery from GOCE, CHAMP,and GRACE

Relying on the methods described in Section 2, gravityfield determination was performed using GPS hl-SST datafrom the GOCE, CHAMP, and GRACE satellites. StartingApril 20, 2009, GOCE 1 s GPS data were used until the endof the year 2009 to compute gravity field solutions usingdifferent processing options (details are provided in the fol-lowing sections). The GOCE GPS-only solutions based on1 s kinematic positions are compared with results obtainedfrom 30 s GRACE B GPS data covering 1 year (Jäggiet al., 2009b) and from 10 s CHAMP GPS data covering8 years (Prange, 2010).

Fig. 1 shows the square-roots of the degree differencevariances of hl-SST-only solutions up to degree 90 withrespect to the superior gravity field model ITG-GRACE03S (mainly based on ultra-precise inter-satelliteK-band data, Mayer-Gürr, 2008). The GOCE and

GRACE B solutions are based on a similar amount of dataand are thus of comparable quality at low degrees. Asexpected, a lower orbital altitude of a satellite is hardlybeneficial for improving the recovery of the very lowdegrees (Prange et al., 2009). At higher degrees, however,GOCE is significantly better than GRACE thanks to thelower altitude. Although the differences to ITG-GRACE03S are still larger than for CHAMP (the signifi-cantly larger amount of CHAMP data currently markswhat can be ultimately achieved from GPS hl-SST), thepotential of GOCE GPS hl-SST gravity field recovery isclearly indicated by the smallest slope in Fig. 1 and isexpected to give better results than CHAMP at higherdegrees in the near future when more data will be available.The omission errors beyond degree 80 are significant andindicate a sensitivity far beyond degree 90 for the GOCEsolution, even if it is currently just based on about 8months of data (see Section 3.2).

3.1. Data sampling

The 10 s sampling of the CHAMP and GRACE Level1b GPS data allows to compute kinematic positions every10 s at maximum, or, depending on the availability ofhigh-rate GPS satellite clock corrections, only every 30 s.The 1 s sampling of the GOCE Level 1b GPS data allowsit for the first time to compute kinematic positions at a1 s spacing for a gravity mission. For this purpose high-rateGPS clock corrections are computed in the frame of theGOCE HPF with a sampling of 5 s, which are linearlyinterpolated to 1 s for the GOCE kinematic orbit determi-nation without significant loss of orbit accuracy (Bocket al., 2009). Apart from the correlations induced by theclock interpolation, and the correlations caused by the car-rier phase ambiguities, kinematic positions are fully inde-pendent. Every single position contains additional gravity

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ITG−GRACE03S30−sec (epoch cov)05−sec (epoch cov)01−sec (epoch cov)01−sec (04−sec cov)

Fig. 2. Square-roots of degree difference variances of gravity fieldrecoveries from GOCE kinematic positions using different samplingintervals.

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field information and is, in principle, expected to improvegravity field recovery (Jäggi et al., 2008).

In order to study the impact of the position sampling ongravity field recovery, the original series of 1 s GOCE kine-matic positions is sampled to 5 and 30 s for a test periodstarting on April 20 and ending on November 5, 2009.Fig. 2 shows the square-roots of the degree difference vari-ances of recoveries up to degree 90 when either takingcovariance information over four off-diagonal blocks intoaccount (“04-sec cov”) for the 1 s GOCE kinematic posi-tions, or when only considering the epoch-wise covarianceinformation (“epoch cov”) for the original 1 s or the sam-pled 5 and 30 s GOCE kinematic positions. The qualityof the recovered gravity field is significantly improved forthe higher degrees when processing kinematic positionswith 5 instead of 30 s sampling, but no improvement at

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ITG−GRACE03Snmax = 110nmax = 120

Fig. 3. Square-roots of degree difference variances of gravity field recoveriesconsidering all SH coefficients (left) or excluding zonal and near-zonal terms a

all is achieved for degrees below 20 when increasing theposition sampling. This either means that ITG-GRACE03S “sees” something different for the low degrees,or that the GOCE hl-SST solutions are limited by system-atic errors showing up at low degrees first. Similar observa-tions were already reported for GRACE hl-SST solutions(Jäggi et al., 2009a). Fig. 2 shows that the recovered gravityfield is only marginally improved when the position sam-pling is further increased to 1 s, which may be partly causedby the 5 s clock corrections used for the kinematic orbitdetermination. Fig. 2 also shows that even the most correctsolution, taking covariances over four off-diagonal blocksinto account, is not able to further improve gravity fieldrecovery from GOCE hl-SST data.

3.2. Maximum resolution and polar gap

The low orbital altitude of GOCE allows to solve for SHcoefficients up to degrees higher than 90. Fig. 3 (left) showsthe square-roots of the degree difference variances of corre-sponding recoveries up to degree 110 and 120, respectively.A degradation of the unconstrained gravity field solutiondue to the GOCE orbit characteristics is, however, startingto become more pronounced when increasing the maxi-mum degree. Due to the sun-synchronous orbit the patternof the ground-tracks does not cover the entire Earth, butleaves caps around the poles of about 6.5� without datacoverage. Due to this polar gap the zonal and near-zonalterms are only weakly determined (Sneeuw and vanGelderen, 1997). Fig. 4 shows the coefficient differencesto ITG-GRACE03S up to degree 120 and confirms thatthe degradation is indeed caused by the zonal and near-zonal coefficients, implying that the degree difference vari-ances shown in Fig. 3 (left) are dominated by few weaklydetermined SH coefficients. Degree difference medians ordegree difference variances with zonal and near-zonal termsexcluded, e.g., according to the rule of thumb given by van

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from GOCE kinematic positions using different maximum degrees andccording to van Gelderen and Koop (1997) (right).

−100 −50 0 50 100

20

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Fig. 4. Coefficient differences of a GOCE hl-SST solution with respect toITG-GRACE03S (logarithmic scale).

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Gelderen and Koop (1997), are thus more convenient whenanalyzing GOCE solutions obtained without regulariza-tions. Fig. 3 (right) shows that the reduced degree differ-ence variances do not show any degradation whenincreasing the maximum degree from 110 to 120. They evensuggest that a slightly larger maximum degree could havebeen chosen.

The degradation of the zonal and near-zonal coefficientsdue to the polar gap is inherent to unconstrained GOCEsolutions, but immediately disappears if the normal equa-tions from GOCE GPS hl-SST are combined with the nor-mal equations from GRACE GPS hl-SST. Fig. 5 shows thesquare-roots of the degree difference variances of a recov-ery up to degree 120 obtained from 30 s GRACE B GPSdata covering the year 2009 and its combination withGOCE. Thanks to the almost polar GRACE orbit no deg-radation of zonal and near-zonal coefficients is seen in the

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ITG−GRACE03SGRACE BGOCE & GRACE B

Fig. 5. Square-roots of degree difference variances of a combined gravityfield recovery from GRACE B and GOCE kinematic positions.

degree difference variances of the combined solution, apartfrom degrees 2 and 3 which are dominated by the GOCEsolution. Further investigations are needed to better under-stand the not yet optimal recovery of these degrees, whichis partly caused by the rather short spacing of the pseudo-stochastic pulses.

4. Impact of PCVs

Due to different antenna types used onboard differentgravity missions (chokering antennas onboard CHAMPand GRACE, helix antennas onboard GOCE) and addi-tional error sources encountered in the spacecraft environ-ment, e.g., different near-field multipath, the impact ofsystematic errors on kinematic orbit determination andsubsequent gravity field recovery is not the same for thedifferent gravity missions. Focusing on gravity field recov-ery, the empirical PCV modeling for the LEO GPS receiverantennas is subsequently compared for the GRACE andGOCE gravity missions.

4.1. GRACE

Empirical PCVs were generated in an iterative proce-dure from reduced-dynamic carrier phase residuals from362 days of GRACE A and GRACE B GPS data collectedby the onboard chokering antennas in the year 2007 (Jäggiet al., 2009b). Fig. 6 shows the correction maps with a res-olution of 1� � 1� in an antenna-fixed coordinate system.Note that the azimuth of 0� points into the direction offlight for both satellites and that an elevation cut-off angleof 5� was used. The patterns show a patchy structure ofsystematic carrier phase variations with amplitudes of typ-ically ±1 cm. They are similar for both GRACE antennas,except for the bottom part of Fig. 6 (left) which is affectedby receiver internal cross-talk due to the active occultationantenna onboard GRACE A. For details we refer to (Jäggiet al., 2009b).

Fig. 7 shows the square-roots of the degree differencevariances of annual recoveries up to degree 120 fromGRACE A and GRACE B 30 s kinematic positions wheneither neglecting PCVs for the kinematic orbit determina-tion, or when empirically correcting for them. There is asignificant improvement of the recovered low degree SHcoefficients when PCVs are corrected for. Gravity fieldrecovery from GRACE A profits in particular due to themore pronounced systematic errors caused by the activeoccultation antenna, and eventually yields a solution com-parable to GRACE B. The slightly inferior quality of theGRACE A solution is no longer related to systematicerrors but to the higher noise of the GRACE A GPS carrierphase observations (Kroes, 2006).

4.2. GOCE

Empirical PCVs were generated in analogy to Section 4.1from 154 days of GOCE GPS data collected by the

Fig. 6. Empirical PCVs in millimeters for the trailing satellite GRACE A (left) and the leading satellite GRACE B (right) based on 362 days ofionosphere-free carrier phase residuals in the year 2007.

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ITG−GRACE03SGRACE−B (PCV not corrected)GRACE−B (PCV corrected)GRACE−A (PCV not corrected)GRACE−A (PCV corrected)

Fig. 7. Impact of empirical PCVs on gravity field recovery from GRACEkinematic positions.

Fig. 8. Empirical PCVs in millimeters for the GOCE main helix antennabased on 154 days of ionosphere-free carrier phase residuals in the year2009.

A. Jäggi et al. / Advances in Space Research 47 (2011) 1020–1028 1025

onboard main helix antenna in the year 2009 (Bock et al.,submitted for publication). Fig. 8 shows the correctionmap with a resolution of 1� � 1� in an antenna-fixed coor-dinate system. Note that the azimuth of 0� points into thedirection of flight and that an elevation cut-off angle of 0�was applied. The pattern shows a completely different andmore complicated structure as that of GRACE with signif-icantly larger amplitudes of up to ±3 cm. For details werefer to (Bock et al., submitted for publication).

Fig. 9 shows the square-roots of the degree differencevariances (zonal and near-zonal terms excluded in analogyto Section 3.2) of recoveries up to degree 120 from about 8months of GOCE 1 s kinematic positions when eitherneglecting PCVs for the kinematic orbit determination(“PCV not corrected”), or when empirically correctingfor them (“PCV corrected (RD)”). Also for GOCE a signif-icant improvement of the recovered SH coefficients can be

recognized when PCVs are corrected for. The improvementis even more pronounced than for GRACE due to the lar-ger amplitudes of the systematic variations shown in Fig. 8.Note, as well, that the SH coefficients are significantlyimproved up to the highest degree due to the considerablysmaller structures in the PCVs of the GOCE helix antennathan in the GRACE chokering antennas.

PCVs derived from reduced-dynamic carrier phaseresiduals might be affected by the gravity field model usedfor the reduced-dynamic orbit determination and indirectlybias the performed gravity field recovery. Truly indepen-dent PCVs were thus generated in an iterative procedurefrom carrier phase residuals of the kinematic GOCE orbitdetermination using the same amount of 154 days of GPSdata. Although it is not possible to generate exactly thesame PCV correction map using the kinematic residuals,

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Fig. 9. Impact of empirical PCVs on gravity field recovery from GOCEkinematic positions.

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Fig. 10. Comparison of AIUB gravity field recoveries with GPS-onlysolutions used in the frame of the GOCE HPF.

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the impact on gravity field recovery is essentially the samefor both series of GOCE 1 s kinematic positions, as shownin Fig. 9 (“PCV corrected (KIN)”).

5. Comparison with HPF solutions

Gravity field recovery in the frame of the GOCE HPF ismainly based on the measurements of the three-axis gravitygradiometer, but also on the positions obtained from theprecise science orbit product. Due to the limited measure-ment bandwidth of the gradiometer, the low degree SHcoefficients are even exclusively determined from theGPS-based GOCE orbit positions up to about degrees20–30 (Pail et al., 2006). In order to assess the quality ofthe results presented in this article, the solutions are com-pared with GPS-only solutions used in the frame of theGOCE HPF for gravity field recovery. As only the GPShl-SST contribution underlying the time-wise solution (Pailet al., in press) is based on the kinematic positions of theprecise science orbit product, and thus independent fromthe GRACE gravity field model used for orbit determina-tion (EGG-C, 2010), the comparison is currently restrictedto the GPS hl-SST solution computed at the Institute ofNavigation and Satellite Geodesy (INAS) of the GrazUniversity of Technology.

Fig. 10 shows the square-roots of the degree differencevariances of the recoveries computed at INAS and AIUBwith respect to ITG-GRACE03S. The solutions to be com-pared are based on the same set of 1 s kinematic positionscovering a period of two months (November and Decem-ber), whereas the final AIUB solution obtained from 8months of data is included for comparison as well.Fig. 10 shows that the AIUB two-months solution is betterthan the GPS hl-SST contribution computed at INAS,apart from degree 2 which is not yet of a good quality asalready mentioned in Section 3.2. For most degrees theAIUB solution is, however, about a factor of

ffiffiffi3p

better,

which is related to the energy-balance approach used atINAS. The comparison with the final solution based on 8months of data also shows that a significant qualityimprovement is expected when using longer series ofGOCE data, even if they are from the descent phase ofthe satellite.

6. Conclusions

Gravity field recovery from about 8 months of GOCEGPS hl-SST data was performed and compared to gravityfield results obtained from 8 years of CHAMP and 1 yearof GRACE GPS hl-SST data. Although the low orbitalaltitude of GOCE was hardly beneficial for improving therecovery of the very low degrees, a significantly improvedrecovery resulted for the high degrees. Based on the limitedtime span of about 8 months of GOCE GPS data, theEarth’s gravity field could be resolved up to about degree115, apart from the zonal and near-zonal SH coefficients,which are degraded due to the polar gap. Compared toCHAMP and GRACE, GOCE GPS hl-SST gravity fieldsolutions yield smallest slopes in degree difference varianceplots with respect to superior GRACE K-band-basedgravity field solutions. Although the results from 8 yearsof CHAMP data are still superior, GOCE is expected togive better GPS-based results than CHAMP at higherdegrees when more data become available.

The 1 s sampling of the GOCE Level 1b GPS dataallows it for the first time to compute kinematic positionsof a gravity mission every second and to use this densesampling of positions for gravity field determination.Gravity field solutions based on the full 1 s position sam-pling are, however, only marginally better than solutionsbased on a reduced 5 s position sampling, which may berelated to the linear interpolation of the GPS satellite clockcorrections from 5 to 1 s. For the recovery of the SH coef-ficients below degree 20 it was even found to be sufficient to

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use a position sampling of 30 s. The presence of systematicerrors, which show up at low degrees first, might explainsuch a behavior.

PCVs of the GOCE helix antennas are considerably lar-ger than PCVs of the GRACE chokering antennas. Conse-quently, their impact on gravity field estimation is morepronounced than for GRACE and has to be carefully mod-eled. PCVs determined iteratively from reduced-dynamicor kinematic carrier phase residuals were found to beslightly different, but no significant differences were foundin the resulting gravity field solutions. Their use is indis-pensable to achieve the best results using GOCE GPS hl-SST for gravity field recovery.

The comparison with GPS-only solutions used in theframe of the GOCE HPF confirms that the CelestialMechanics Approach is well suited to exploit the contribu-tion of GOCE GPS hl-SST to gravity field determination.

Acknowledgments

The authors are grateful to the Center for Orbit Deter-mination in Europe (CODE) for using GPS orbit solutionsand clock corrections. CODE is a joint venture of theAstronomical Institute of the University of Bern (AIUB,Switzerland), the Federal Office of Topography (swisstopo,Switzerland), the Federal Agency for Cartography andGeodesy (BKG, Germany), and the Institute for Astro-nomical and Physical Geodesy of the Technische UniversitätMünchen (IAPG, Germany). The authors are grateful tothe European Space Agency (ESA) for providing theGOCE data for this investigation. The generous financialsupport provided by the Swiss National Science Founda-tion and the Institute for Advanced Study (IAS) of theTechnische Universität München in the frame of theproject “Satellite Geodesy” is gratefully acknowledged.

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GPS-only gravity field recovery with GOCE, CHAMP, and GRACEIntroductionMethod used for GPS-only gravity field recoveryStep I: kinematic orbit determinationStep II: generalized orbit determinationStep III: normal equation handling

GPS-only gravity field recovery from GOCE, CHAMP, and GRACEData samplingMaximum resolution and polar gap

Impact of PCVsGRACEGOCE

Comparison with HPF solutionsConclusionsAcknowledgmentsReferences