Phil. Trans. R. Soc. a 2006 Moore 1009 26

7/28/2019 Phil. Trans. R. Soc. a 2006 Moore 1009 26

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doi: 10.1098/rsta.2006.1751, 1009-10263642006Phil. Trans. R. Soc. A

P Moore, Q Zhang and A Alothmantemporal structure of the Earth's gravity fieldRecent results on modelling the spatial and

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Recent results on modelling the spatial andtemporal structure of the Earths gravity field

BY P. MOORE*, Q. ZHANG AND A. ALOTHMAN

School of Civil Engineering and Geosciences, Newcastle University, ClaremontTower, Newcastle upon Tyne NE1 7RU, UK

The Earths gravity field plays a central role in sea-level change. In the simplest applicationa precise gravity field will enable oceanographers to capitalize fully on the altimetricdatasets collected over the past decade or more by providing a geoid from which absolute

sea-level topography can be recovered. However, the concept of a static gravity field is nowredundant as we can observe temporal variability in the geoid due to mass redistribution inor on the total Earth system. Temporal variability, associated with interactions betweenthe land, oceans and atmosphere, can be investigated through mass redistributions with,for example, flow of water from the land being balanced by an increase in ocean mass.Furthermore, as ocean transport is an important contributor to the mass redistribution thetime varying gravity field can also be used to validate Global Ocean Circulation models.

This paper will review the recent history of static and temporal gravity field recovery,from the 1980s to the present day. In particular, mention will be made of the role ofsatellite laser ranging and other space tracking techniques, satellite altimetry and in situ

gravity which formed the basis of gravity field determination until the last few years.With the launch of Challenging Microsatellite Payload and Gravity and CirculationExperiment (GRACE) our knowledge of the spatial distribution of the Earths gravityfield is taking a leap forward. Furthermore, GRACE is now providing insight intotemporal variability through monthly gravity field solutions. Prior to this data werelied on satellite tracking, Global Positioning System and geophysical models to give usinsight into the temporal variability. We will consider results from these methodologiesand compare them to preliminary results from the GRACE mission.

Keywords: gravity field; temporal variation; gravity and circulation experiment

1. The Earths gravitational field

The Earths gravitational field at a point on or external to the Earth is describedmathematically by

Vr;q;lZGMr

1CXNlZ2

Rer

lXlmZ0

Clm cos mlCSlm sin mlPl;mcos q( )

; 1:1

where GM is the product of Newtons gravitational constant and the Earths mass;Re the mean Earth radius; r, q, l spherical coordinates (radial distance, colatitude,

Phil. Trans. R. Soc. A (2006) 364, 10091026

doi:10.1098/rsta.2006.1751

Published online 22 February 2006

One contribution of 20 to a Theme Issue Sea level science.

* Author for correspondence ([email protected]).

1009 q 2006 The Royal Society

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longitude) of the point; Pl,m normalized Legendre polynomials; and Cl,m, Sl,mnormalized spherical harmonic (Stokes) coefficients of degree land order m. Thelower limit of summation over l, namely 2, is a consequence of the first degree andorder harmonics being zero on assuming that the frame origin is at the Earthsinstantaneous centre of mass. Gravity field models contain a subset of the spherical

harmonic coefficients typically up to some degree, lmax, say. The recent history ofgravity field modelling, as summarized in 2, reveals progressive improvementswith time through incorporation of additional data with improved geographicaland temporal coverage. This has enabled the models to improve in accuracy and tobe extended to shorter wavelengths by increasing the cutout degree lmax.

In practice, the spherical harmonics should not be considered as invariant butrather have temporal signatures, which are broad-band although dominated byquasi-secular and periodic components. Secular changes arise, for example, dueto isostatic post-glacial rebound while the periodic components are typically dueto annual and semi-annual mass redistribution. Recent gravity field models

incorporate variability to the second degree zonal harmonic, C2,0, and degree 2and order 1 harmonics, C2,1 and S2,1. Given our recognition of variant harmonics,static gravity field solutions are to be interpreted as time averaged over theperiod of the underlying data. Such data can span decades or, as for ChallengingMicrosatellite Payload (CHAMP) and Gravity and Circulation Experiment(GRACE) (see below), just a few months or years.

The importance of the gravity field has motivated the launch of dedicatedgravity field missions. The first of these is the CHAMP launched in 2001 into a nearpolar orbit at an altitude of 450 km. CHAMP (Reigber et al. 2002) is a dual purposemission to measure the Earths gravity and magnetic fields. For the gravity fieldobjective, the satellite carries Global Positioning System (GPS) receivers forprecise positioning, star cameras for attitude control and a 3-axes accelerometer tomeasure surface accelerations. Procedures for recovering the gravity field aredescribed by Reigber et al. (2004). GRACE, launched in March 2002, is a tandemsatellite mission where the inter-satellite range, range-rate and range accelerationare derivable from a K-band microwave device (Tapley et al. 2004). Each GRACEsatellite carries GPS receivers, accelerometers and star cameras. Both CHAMPand GRACE are providing data for recovery of the static gravity field withGRACE, in addition, providing monthly snapshots of the geopotential fromwhich variability can be inferred. With reference to equation (1.1) the orbitalaltitude, h, of a satellite is a measure of its ability to provide gravity field data due

to the attenuation of the gravitational potential with altitude, i.e. Re=ReChlwhere h is altitude of the satellite above the Earths surface. Both CHAMP andGRACE are in relatively low altitude orbits (400500 km) to increase theirsensitivity to gravity field anomalies. Even lower altitudes of say 200250 km arepreferable. However, the reduction in altitude is accompanied with an increase inatmospheric density with consequent impact on the satellite lifetime unless theorbit is periodically boosted to a higher altitude.

Geophysical applications of temporal and spatial modelling are numerous andwell documented in the literature with dedicated special issues such as SpaceScience Reviews, volume 108, issues 1 and 2, 2003, providing an excellent

summary. Papers in that issue include the impact of geoid improvements on largescale ocean circulation (Le Grand 2003) and sea-level studies (Woodworth &Gregory 2003). Other papers therein describe applications of temporal variability

P. Moore and others1010

Phil. Trans. R. Soc. A (2006)



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to studies of ocean mass (Nerem et al. 2003) and continental water storage(Swenson & Wahr 2003). In addition, comments on tidal aliasing (Knudsen 2003)and error characteristics of dedicated gravity field missions (Schrama 2003) arealso significant to the proper interpretation of the datasets.

2. Gravity field models

It is informative to summarize the history of gravity field enhancements over thepast 20 years. Advances in modelling have paralleled increases in computer power,availability of more accurate and varied tracking data, introduction of combinationstrategies and, most recently, the launch of CHAMP and GRACE. Table 1 presentsa subset of the many models released over the past 20 years. For brevity, the tabledoes not contain all models with no explicit mention of the excellent GRIM, TEG orOSU fields, for example. A comprehensive tabulation of gravity field models fromthe late 1960s onwards is available at the International Centre for Global EarthModels (http://icgem.gfz-potsdam.de/ICGEM/ICGEM.html). Rather, the sum-

mary is intended to illustrate the main advances in modelling. The table gives thedate the field was created, the underlying data sources, maximum degree and order,and the estimated accuracy of radial positioning of TOPEX/Poseidon (T/P) in

Table 1. Historical perspective of selected gravity field models. (Satellite: satellite tracking only;combination: satellite tracking, altimetry and gravimetry.)

gravitymodel date data

maximumdegree

TP radialerror (cm)

r.m.s. geoiderror (cm)

degree of10 cm

accumulativegeoid error

GEM-L2 1982 satellite 20 65.4GEM-T1 1988 satellite 36 25.0GEM-T3 1994 combination 50 6.8JGM-1 1993 combination 70 3.4JGM-2 1994 combination 70 2.2JGM-3 1994 combination 70 1.1 53.8 (degree 70)EGM-96 1996 combination 360 0.9 19.0 (degree

70), 42.1

(degree 360)

18

E-CH03Sa 2004 CHAMP(3 year)

120 withselectedharmonics todegree 140

5.0 (degree 50) 45

E-GR01Sb 2003 GRACE(39 day)

120 withselectedharmonics todegree 140

1.0 (degree55), 2.0(degree 80)

100

GGM01S 2003 GRACE(111 day)

120 2.0 (degree70), 6.0

(degree 90)

120

aEIGEN-CHAMP03S.bEIGEN-GRACE01S.

1011Earths gravity field


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orbit computations based on a covariance analysis. The latter is reliant on propercalibration of the covariances but is a useful measure of accuracy.

The first model included is GEM-L2 (Lerch et al. 1985). GEM-L2 is a satelliteonly gravity model complete to degree and order 20. Improvements are seen withGEM-T1 (Marsh et al. 1988), as the resolution of the model was increased up to

degree and order 36, and with GEM-T3 (Nerem et al. 1994a) when the model wasfurther extended to degree and order 50 and altimetry and in situ gravityanomalies were added to the satellite data. GEM-T3 typified the major advanceassociated with use of combined data sources.

The JGM series were released around the launch of T/P and includedadditional and better tracking. JGM-1 (Nerem et al. 1994b) was a recomputationof GEM-T3 to degree and order 70 while JGM-2 (Nerem et al. 1994b) and JGM-3(Tapley et al. 1996) included increasing quantities of Satellite Laser Ranging(SLR), Doppler Orbitography and Radiopositioning Integrated by Satellite(DORIS) and GPS tracking of T/P. A 70!70 field is usually sufficient for

satellite orbit determination but, in terms of the geoid, this only representsresolution to half-wavelengths of about 290 km. Geoid computation requiresdetermination to higher degree and order. For example, EGM-96 (Lemoine et al.1998), which used the normal equations from JGM-3, extended the field out todegree and order 360 through use of surface gravity data and satellite altimetry.

To illustrate associated improvements in the geoid, table 1 also presents someresults of the global r.m.s. geoid undulation commission error for given degree andorder and the approximate degree at which the accumulative geoid error is 10 cmr.m.s. Improvement from JGM-3 to EGM-96 is illustrated by the reduction in r.m.s.geoid error (cutoff at degree 70) from 54 cm r.m.s. with JGM-3 to 19 cm r.m.s. withEGM-96. The total EGM-96 r.m.s. geoid undulation commission error to degreeand order 360 is 42.1 cm. Figure 1 shows the geoid undulation error for EGM-96 ascomputed from the covariance matrix complete to degree and order 70. A cleardemarcation is evident between sea (altimetry data) and land (surface gravitydata) due to the relatively high precision of the former and the inhomogeneousnature of the latter. In contrast, given the homogeneous nature of CHAMP andGRACE the errors show no discrimination between sea and land. The CHAMPgravity field model EIGEN-CHAMP03S (Reigber et al. 2004), has power to aboutdegree 50 with r.m.s. geoidal error of 5 cm. The improved sensitivity of the inter-satellite ranging with GRACE is evident in a r.m.s. geoidal error of about 2 cm todegree and order 7080 (300 km resolution), increasing to 6 cm for the 90!90

field (200 km resolution). The small differences between the r.m.s. geoid errors intable 1 for EIGEN-GRACE01S and GGM01S (Tapley et al. 2003) are notsignificant, but merely a consequence of the different calibration procedures. Therelative merits of CHAMP and GRACE can be seen from the accumulativegeoid error which reaches 10 cm with EGM-96 at about 1200 km half-wavelength(degree 18). The same level is achieved at 450 km half-wavelength (degree 45)with EIGEN-CHAMP03S, at 200 km half-wavelength (degree 100) with EIGEN-GRACE01S and at 166 km half-wavelength (degree 120) with GGM01S.

Gravity fields, such as the JGM, TEG and GRIM satellite only models, haveused tracking data collected over decades or more. Figure 2 plots gravity

anomalies determined from GRIM5-S1 (Biancale et al. 2000) a satellite onlymodel complete to degree 99 and order 95 determined from over 20 years ofsatellite tracking and those from EIGEN-GRACE01S, an early GRACE model





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complete to degree and order 120, with selected harmonics to degree 140, derivedfrom only 39 days of data. A visual comparison shows striking similarity whileclose inspection reveals the greater clarity of features from the GRACE model.The blurring in GRIM5-S1 is due to the difficulty in resolving individualharmonics. GRACE is providing significant new observational data to the extent

that a GRACE model constructed from just 39 days of data is of higher accuracythat those derived from over 20 years of ground-based satellite tracking.

3. The temporal gravity field: theory

In this section, we introduce some theoretical aspects related to temporal varia-bility in the Earths gravity field. Surface mass redistribution will lead to corres-ponding temporal changes in the gravity potential due to attraction by the timevarying surface mass and also the deformation (load) of the underlying solid Earth.

Suppose there is a time-dependent change in thegravitational potential representedthrough changes, DClm and DSlm, in the spherical harmonic coefficients as follows:

DVr; q; lZGMr

XNlZ0

XlmZ0

Rer

lDClm cos mlCDSlm sin mlPl;mcos q: 3:1

Total changes in the spherical harmonic coefficients of the gravity potential canbe calculated (Wahr et al. 1998) via the load harmonics defined by

DClm

DSlm

( )Z

1

4Rprw

2p0

dl

p

0Dsq; lPl;m

cos ml

sin ml

( )cos q dq; 3:2

where Ds is defined as the change in surface density (i.e. mass/area), and rw thedensity of water (1000 kg mK3) included so that DClm and DSlm are dimensionless.The relation between DClm and DSlm and the gravity field spherical harmonics is:

DClm

DSlm

( )Z

3rwrav

1Ck0l2lC1

DClm

DSlm

( ); 3:3

where rav is the average density of the Earth (Z5517 kg mK3) and k0l the load

Love number of degree l (Farrell 1972).The loading associated with the mass per unit area Dsq; l in equation (3.2)

deforms the elastic Earth and displaces points on the Earths surface by distances

ur, uq, ul in the radial, south and east directions, respectively, where (Moore &Wang 2003)

urZGM

gRe

XNlZ1

XlmZ0

h0lPl;mcos qDClm cos mlCDSlm sin ml

uqZGM

gRe

XNlZ1

XlmZ0

l0lvPl;m

vqcos qDClm cos mlCDSlm sin ml

ulZGM

gRe cos qXN

lZ

1X

l

mZ

0

l0lPl;mcos qKmDClm sin mlCmDSlm cos ml

9>>>>>>>>>>=>>>>>>>>>>;

: 3:4

In the above, g is gravity and h0l and l0l the Love and Shida load numbers (or

load deformation coefficients) of degree l (Farrell 1972).





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In addition to the Earths surface deformation, another displacement, the so-called geocentre motion, is a consequence of the origin of the inertial frame for

orbit determination being defined as the instantaneous centre of mass of theEarth and atmosphere system. Thus, by definition, we require k0lZK1 inequation (3.3). The contribution of non-zero surface loading coefficients DClmand DSlm for mZ0, 1 is equivalent to a displacement Xg; Yg; Zg in the positionof the satellite tracking stations where (Trupin et al. 1992)

XgZReffiffiffi

3p

1Kh0lC2l

0l

3

!rw

ravDC11

YgZR

effiffiffi3p 1Kh0lC2l0l

3 ! rw

ravDS

11

ZgZReffiffiffi

3p

1Kh0lC2l

0l

3

!rw

ravDC10

9>>>>>>>>>>=>>>>>>>>>>;

:3:5

The forcing effects of surface loading described above are, in principle,observable in space geodetic techniques, which allow the static and temporalgravity field to be investigated through several independent methodologies. Theseinclude estimates of gravity field coefficients from precise orbit determinationusing satellite tracking, from deformation studies of the elastic Earth due to

loading using change in position of satellite tracking stations or GPS receivers andGRACE. In addition, degree 2 harmonics can also be recovered from Earthrotation data (e.g. Chen et al. 2000; Chen & Wilson 2003; Gross et al. 2004).

Figure 1. EGM-96 geoid error from covariance matrix to degree and order 70. From http://www.csr.utexas.edu/grace/gravity/ggm 01/GGM 01_Notes.pdf.



http://www.csr.utexas.edu/grace/gravity/ggm01/GGM01_Notes.pdfhttp://www.csr.utexas.edu/grace/gravity/ggm01/GGM01_Notes.pdfhttp://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://www.csr.utexas.edu/grace/gravity/ggm01/GGM01_Notes.pdfhttp://www.csr.utexas.edu/grace/gravity/ggm01/GGM01_Notes.pdf


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4. Temporal variability: satellite tracking and GPS

As stated previously, temporal variability within the Earths gravitational field isa response to the redistribution of mass on the Earths surface and in its interior.The temporal signatures are broad-band in spectrum with intra-annual and

50

(a)

(b)

0

50

50

0

50

0 50

120 90 60 30 0 30 60 90 120

100 150 200 250 300 350

0 50

100 80 60 40 20 0 20 40 60 80 100

100 150 200 250 300 350

gravity anomaly (mGal)

Figure 2. (a) Gravity anomaly map derived from 39 days of GRACE data (EIGEN-GRACE01Smodel). (b) Gravity anomaly map derived from tracking data of 30 Earth orbiting satellites overmore than 20 years (GRIM5-S1). From http://op.gfz-potsdam.de/grace/results/.



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longer time-scale variability superimposed on the inter-annual and seculartrends. In particular, surface mass change in the atmosphere, oceans,hydrosphere and cryosphere are dominated by seasonal variations whileprocesses such as isostatic glacial recovery and sea-level change give rise tolong-term secular or quasi-secular signatures. Studies of temporal variability

have included inferences from satellite orbital perturbations using SLR (e.g.Nerem et al. 1993), from DORIS to SPOT and TOPEX/Poseidon (Cretaux et al.2002) and from deformation studies using GPS (Blewitt et al. 2001; Wu et al.2003; Gross et al. 2004). SLR tracking to passive geodetic satellites has receivedparticular attention given the 20 year time series of observations to Lageos1&2.Lageos has been used to detect seasonal changes (Dong et al. 1996; Cheng &Tapley 1999) and contributed to studies of geopotential zonal rates (Cheng et al.1997). Several of these papers present variations of the zonal and lower order anddegree harmonics in good agreement with geophysical models of surface massredistribution.

With recent improvements in SLR tracking technology and quality assuranceit has been possible to undertake more detailed studies of the temporalvariability in the lower order and degree harmonics. Most notably, Nerem et al.(2000) recovered annual variability for degrees 24 inclusive from SLR trackingof Lageos1&2 over a 6 year period and used the results to discriminate betweenseveral geophysical models. Results therein established a higher correlation witha particular hydrological model but were unable to distinguish between models ofthe atmosphere and oceans. This was partly due to the smaller contribution ofocean mass and the likelihood of better agreement between competing models foratmospheric surface pressure at the very long wavelengths. Temporal variability

to higher spatial resolution is an objective of GRACE, which is expected toproduce annual variability to degree and order 40 (half-wavelength 500 km) overthe lifetime of the mission (Wahr et al. 1998). Despite the outstanding promise ofGRACE, satellite tracking has an important complementary role enablinganalysts to determine variability in a consistent manner over a long time frame.Furthermore, satellite based results provide a mechanism for validation of theGRACE temporal variability.

The long-time series of SLR tracking of Lageos has recently established anapparent short-term reversal in the secular change in the Earths oblatenesscoefficient J2ZKC2;0 (Cox & Chao 2002). On removing seasonal signatures, the

variability within J2 for the period 19801998 is dominated by a negative seculartrend associated with isostatic post-glacial rebound. After 1998, the trend wasreversed with a maximum in 20002001 before returning to the value and trendprior to 1998. This signature indicates a pronounced global-scale massredistribution within the Earth system from high to low latitudes reflecting anequator ward mass transport large enough to offset the ongoing isostatic recovery.Attempts to analyse the change have involved oceanic mass distribution andmelting of sub-polar glaciers (Dickey et al. 2002; Chao et al. 2003). Figure 3 plotsour recent time series ofJ2 determined from Lageos 1&2 for the period 19982004(Moore et al. 2005). The Lageos data exhibits a strong seasonal signature, which

has been fitted by sinusoids at the annual and semi-annual periodicities. Thedetrended signals, along with a six month boxcar average, show the increase after1998 prior to its return to the long-term mean in 2002.





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The inclusion of lower altitude satellites such as Starlette (altitude ca800 km),Stella (ca 800 km) and Ajisai (ca 1200 km) is required to compensate for theinsensitivity of Lageos (ca 6000 km) to temporal variability beyond degree 4 dueto the attenuation of the gravity field with height. Even then the orbits are onlysensitive to temporal variability at the very low degrees, say 26. Furthermore,temporal variability from CHAMP seems to require the use of constraintsalthough singular value decomposition has established that some signal isdiscernible (Moore et al. 2005). The alternative, as described in 3, is to usevertical deformation from a global distribution of GPS receivers. In contrast to

SLR, GPS is highly sensitive to local (high degree) effects. Although the GPSglobal coverage is far more complete than say SLR, the high degree effects andincomplete coverage over oceanic areas leads to aliasing of the low degreeharmonics. Various other systematic errors may also contaminate the GPS timeseries from direct mismodelling or aliasing (Penna & Stewart 2003). However,temporal variability to degree and order 6 has been recovered by Wu et al. (2003).

For GPS a 4-year dataset from January 1999 to December 2002 was used toestimate daily coordinates of 166 IGS stations using the point positioning modeof the JPL software GIPSY-OASIS II (Zumberge et al. 1997). The resultantvertical deformation cleared of secular trends is plotted in figure 4 for four GPS

sites for the period 19982003. As the GPS coordinates were recovered relative tothe ITRF2000 reference frame degree one harmonics were removed in the datafor figure 4 (and also figure 5) by using the geocentre motion inferred from Lageos(Moore & Wang 2003). All four stations exhibit strong seasonal signatures. Alsoplotted are the annual signals derived from a combination of geophysical modelsfor atmospheric and ocean mass and land hydrology. For these we used (Moore &Wang 2003)

(i) atmospheric pressure for January 1989March 2002 supplied by theNational Center for Environmental Prediction (NCEP) Reanalysis

project;(ii) TOPEX altimetry for January 1993December 2000 and globalclimatology (Levitus & Boyer 1994)

1998 2000 2002 2004

year year

5.0

1.0

3.0

7.0

(a) (b)

D

J2

( 1.0

1

0

10)

Lageos1&2

fitting

1998 2000 2002 20045.0

0

5.0

10.0Lageos1&2 seasonalvariation removed

6-month mean

Figure 3. (a) J2 (unnormalized) from Lageos1&2 with annual and semi-annual fit (solid line). (b) J2from Lageos with annual and semi-annual periodicities removed with six month boxcar average(solid line).





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(iii) soil moisture and snow mass from NCEP CDAS-1 (Climate DataAssimilation Service) for 19932000.

Other sources of geophysical data were examined with the above giving thebest overall agreement to the satellite results.

Within this comparison the global mass distribution from geophysical datawas handled differently between continental and oceanic areas. Over thecontinents the total mass is simple to model being the sum of the individualcontinental hydrology and atmosphere masses. However, over the oceans thesurface closely approximates an inverted barometer (IB) with an increase of 1 mbin pressure associated with a decrease of 1 cm in ocean height. In our approachwe elected to infer bottom pressures for the water mass alone and then tocombine with atmospheric pressure. In detail, the T/P altimetry was first

corrected for all geophysical effects as given on the altimetric records but withthe IB correction replaced by a modified correction (Hendrick et al. 1996). In thismodification the IB correction was determined relative to the mean atmosphericpressure over the oceans rather than the constant value used on the altimetricrecords. Given the incompressibility of seawater the oceans respond to variationabout the global ocean mean with a change in the mean having no impact on theheight of the oceans. A global oceanic mean was calculated over each T/P cycleand used with the atmospheric pressure inferred from the IB correction on thealtimetric record to estimate the modified IB correction. The seasonal sea-levelchanges were subsequently corrected for the steric effect using monthly

climatology (Levitus & Boyer 1994). Climatological data of temperature andsalinity were used to infer seasonal changes in sea water density which werecombined with the derived T/P sea-surface heights to infer changes in bottom

4

2

0

2

4(a) (b)

(c) (d)

(cm)

4

2

0

2

4

(cm)

1997 1998 1999 2000 2001 2002 2003

1997 1998 1999 2000 2001 2002 2003 1997 1998 1999 2000 2001 2002 2003

2000 2001 2002 2003

Figure 4. Daily site vertical deformation estimated from GPS with annual estimates fromgeophysical models (solid line). (a) ALIC (Alice Springs, Australia); (b) BAHR (Bahrain); (c)IRKT (Irkutsk, Russia); (d) NKLG (NKoltang, Gabon).





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pressure of the water mass. The total mass distribution over the oceans wasderived by adding atmospheric pressure. For consistency with the IB correctionECMWF pressure was used over the oceans.

Figure 5 presents the amplitude and phase of the vertical deformation averagedover the period 19982003 from a global network of 166 IGS sites. Using equation(3.4) this deformation can be inverted to recover the lower order and degreeharmonics. However, comparisons with SLRCCHAMP (Moore et al. 2004)showed excessive annual variability in the degree 26 harmonics indicating aliasingfrom higher order effects. The obvious next step was to use a combination strategyfor SLR, CHAMP and GPS (Moore et al. 2004) to capitalize on the signatures ineach data source. This has the advantage of providing additional data for degrees 5and above to compensate for the insensitivity of Lageos in the SLR solution and toprovide global constraints on the GPS solution, which is relatively unconstrainedover oceanic areas. In the combined approach we used the normal equations forannual and semi-annual variations in the degree 26 harmonics from SLR andCHAMP (Moore et al. 2005) and combined with similar normal equations from

GPS. For GPS, the vertical deformation was now used to form normal equationsfor sinusoids at the annual and semi-annual frequencies in the lower order anddegree harmonics (cf. equation 3.4). The procedure differed from that used toconstruct figures 4 and 5 in that the degree one harmonics were also estimated toallow for geocentre motion (cf. equation 3.5) in the GPS time series. The normalsfrom SLRCCHAMP and GPS were weighted heuristically to capitalize on thestrengths of the respective data types but with particular emphasis placed onmaintaining solutions close to the degree 2 and 3 results from SLRCCHAMP.Further details of the computations, weighting strategy and comparisons againstgeophysical data are given in Moore et al. (2004).

Table 2 compares results for the amplitude and phase of the annual signal inthe lower order zonal harmonics from the combination approach, SLR studies andgeophysical models. The results show reasonable agreement in amplitude and

60

0

60

60

0

60

0 60 120

10 mm

180 240 300 360

0 60 120 180 240 300 360

Figure 5. Vertical annual amplitude and phase of IGS sites. The amplitude/ phase is defined asAcos(utKf) where t (days) is relative to January 1. Arrows represent the amplitude with phasemeasured counterclockwise from the east.





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phase for all harmonics except J4. Given the magnitude of the signals, possiblealiasing between the harmonics, omission and commission errors and absorptionof non-gravitational seasonal signals (e.g. tides) the level of agreement isencouraging and reflective of that obtainable by using SLR data or a combinationof SLR and GPS. A plot of the annual variation in the geoid undulations from thecombination solution and for the geophysical data is given in figure 6.

5. Temporal variability: GRACE

GRACE monthly solutions of the gravity field are being made available to thescientific community as Level-2 products by the Center for Space Research (CSR),University of Texas, and GeoForschungsZentrum (GFZ), Potsdam. This study usedthe CSR monthly gravity field solutions (http://podaac.jpl.nasa.gov/grace/ ).At the time of study the community had access to 19 monthly solutions(Bettadpur 2004a,b) covering April 2002April 2004 derived typically from 26 to31 days of data although three solutions were obtained from 13, 18 and 22 days.The monthly fields are generally complete to degree and order 120. The GRACEsolutions are determined relative to background models of time variability

associated with solid earth, ocean and pole tides, atmospheric mass and abarotropic ocean model. Ocean tides in the CSR solutions are modelled by CSR4.0.High-frequency mass redistribution due to the atmosphere and oceans is removedin the GRACE processing by inclusion of the ECMWF operational atmosphericmodel and the so-called PPHA barotropic model (Flechtner 2003). High-frequencyvariability is converted to spherical harmonics at 6 h intervals with intermediateepochs obtained by linear interpolation. Each monthly gravity field solution isassociated with a monthly average of the combined atmospheric/barotropicbackground gravity field model. The gravitational effects of the backgroundatmospheric and barotropic models can be reinstated by addition to provide the

total gravity field. Given increasing errors at higher degrees the user is cautionedagainst usage of harmonics beyond degree and order 90100 in the monthly fieldsand advised to employ some smoothing or truncation of the shorter wavelengths.

Table 2. Comparison of the annual variations in degree 26 normalized zonals, Cn,o.(Amplitude/phase defined by A cos(utKf), where t is (days) relative to January 1, f degreeand u annual frequency. (A) SLRCCHAMPCGPS (this work); (B) Cheng et al. (1997), SLR data;(C) Nerem et al. (2000), Lageos 1&2; (D) geophysical model: atmosphere (NCEP); ocean (T/P);hydrology (CDAS-1).)

degree

amplitude (1.0!10K10) phase (degree)

A B C D A B C D

2 0.73 1.25 1.06 0.79 291.1 320 305.2 322.63 1.49 2.16 1.38 1.53 202.4 161 163.1 213.74 0.27 1.07 0.26 0.24 1.1 158 178.5 269.35 1.12 1.12 1.01 43.3 332 79.66 0.48 0.26 0.18 112.3 157 53.0



http://podaac.jpl.nasa.gov/grace/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://rsta.royalsocietypublishing.org/http://podaac.jpl.nasa.gov/grace/


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To illustrate the neccessity of using some form of smoothing or truncation of thehigh degree and order terms, geoid undulations were computed from the monthlysolutions for two consecutive months, January and February, in 2003. Figure 7shows the differences between these geoid heights with the legend representingdifferences in metres. The true variation should be at the millimetre level. The

pattern clearly reflects the GRACE near polar orbit with the NS trackinessresulting from deficiencies in the data and tidal modelling (Han et al. 2004). Spatialaveraging (Wahr et al. 1998) can be used to smooth the GRACE estimates withmeaningful mass estimates recovered using a Gaussian averaging kernel of radius5001000 km (Wahr et al. 2004). As a demonstration of this capability we re-instated the background gravity field models for each of the 19 near monthlysolutions to obtain the monthly fields that represent the total mass. These weresubsequently used in two different analyses. The first is a study of the annual globalchange derived by using spatial averaging with radius 500 km (Wahr et al. 1998).The second used harmonics from the long-wavelength 6!6 field with an annual

variation fitted to each harmonic. In the first example, spatial averaging wasincorporated into equation (3.1) with the geoid height estimated on a regulargeographical grid. The time series of geoidal heights at each grid point was fitted bya constant to obtain the annual mean and a sinsoid of period 1 year to recover theannual signal. Thus, if Ngq; l; t denotes the geoid height recovered from eachmonthly field at point with colatitude q and longitude l then

Ngq; l; tZNq; lCAc cos2pt=TCAs sin2pt=T; 5:1where t is the time in days from January 1; TZ365.25; Nq; l the mean geoidheight and Ac and As the annual geoidal variation on 1 January and 1 Aprilrespectively. Figure8 plots the spatial distribution ofA

c

and As

. Vestiges of the NStrackiness is still evident over the oceans, particularly in the cosine component.This canbe further reduced by increasing theaveraging radius to say 1000 km.Evenat 500 km averaging there is little discernible signal over the oceans where zonalsignatures are expected. The dominant signals aredueto continental hydrology andthe re-instated atmospheric pressure. Results such as these have demonstrated thepower of GRACE to recover continental water storage in large catchment areassuch as the Mississippi and Amazon and the area draining into the Bay of Bengal.These and other studies suggest that for continental hydrology Gaussian averagingover about 500 km is meaningful and that the later GRACE monthly fields yielderrors some 30% smaller than the early fields. A detailed comparison of the

variability in figure 8 is beyond the scope of this review but the figures illustrate thepotential for recovering the annual signal, for comparisons against geophysicalmodels and for intra-annual studies over the lifetime of the GRACE mission.

In the second study, annual variability was recovered in the GRACEharmonics for the 6!6 field. This study also excluded consideration ofJ2ZKC2,0as the GRACE results (see below) exhibited anomalous variability in, forexample, the first few monthly solutions. Figure 6 shows the annual variationfrom GRACE alongside that from the SLRCCHAMPCGPS combinationsolution and the geophysical model. The correlations and r.m.s. differencesbetween the geophysical model, combination solution and GRACE are

summarized in table 3. These results show that the agreement between SLRCCHAMPCGPS and the geophysical data is high for a 4!4 field but decreasessubstantially on extending to 6!6. A similar trend is observed with the





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combination solution and GRACE. As Lageos1&2 have little power beyond

degree 4 the additional harmonics are recovered from the other three lessaccurate orbits and GPS. On the other hand, the agreement between thegeophysical data and GRACE is maintained showing no significant loss of

Figure 7. GRACE: February 2003January 2003 geoid height differences (m).

Figure 8. Annual variation in geoid recovered from GRACE using spatial averaging with radius500 km. (a) Cosine component corresponding to 1 January. (b) Sine component corresponding to1 April.

Figure 6. Annual variation in the geoid for 6!6 field (C20 removed): (a) and (b) SLRCCHAMPCGPS solution; (c) and (d) geophysical data; (e) and (f) GRACE. The upper plots are for 1 Januarywith the lower corresponding to 1 April.





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accuracy as the field is extended from 21 harmonics to 45. The behaviour of theagreement confirms the power of GRACE for mass redistribution studies. It is

important to emphasize that the combination solution was derived from 1998 to2003, GRACE from 2002 to 2004 and the geophysical results from data collectedover a decade or more. Given the different time-scales we would expect somesmall differences between the annual signals.

The individual GRACE harmonics can be verified by comparison against SLR.Figure 9 plots the Lageos1&2 J2 solution recovered at 15 day intervals over20022004.5. No annual or semi-annual signal has been removed. Also plotted arethe 19 monthly solutions from the CSR GRACE fields with the background modelsre-instated. The early problems with J2 are evident but the later values agree withLageos. The anomalous value in January 2004 is from a field recovered from just

13 days of data. However, no explanation is offered for the final (April 2004) value.

6. Conclusions

Satellite tracking, supplemented with altimetry and surface gravity data, has beenthe basis of gravity field models over the past 2030 years. However, recent studieshave shown that the long-wavelength static gravity field recovered from a fewmonthsofGRACEdataissuperiortotheprevious20yearseffort.Satellitetrackingsuch as SLR and DORIS and vertical deformations from GPS have contributed to

our knowledge of temporal variability for the long wavelength field including thedegree 1 terms, the so-called geocentre variability. These results can be comparedagainst mass distributions supplied by geophysical data for atmospheric and oceanmass and land hydrology and also provide a bench-mark for GRACE. Although thetemporal variability from GRACE has not as yet achieved the pre-launch baseline(Wahr et al. 2004), the early results are providing a spatial resolution unobtainablewith conventional means. Further improvements in the accuracy of the temporalfields are inevitable, a consequence of the strenuous efforts being made by thescience teams at CSR and GFZ to reduce the systematic errors in the GRACE data.

GRACE is already providing excellent science. The GRACE monthly

solutions are beginning to provide insight into the inter-annual and intra-annualvariability. Over oceans GRACE has the potential for assimilation of massredistribution into ocean models while on land GRACE can resolve total water

Table 3. Correlation and r.m.s. (mm) difference of annual geoid between combination solution(SCCCG), geophysical data (GPH) and GRACE for 4!4 and 6!6 gravity field (C20 removed).

4!4 6!6

cosine sine cosine sine

cor. r.m.s. cor. r.m.s. cor. r.m.s. cor. r.m.s.

GPH/SCCCG 0.89 0.81 0.82 0.81 0.70 1.37 0.68 1.18GPH/GRACE 0.89 0.81 0.82 0.78 0.86 0.95 0.80 0.89GRACE/SCCCG 0.81 1.06 0.80 0.76 0.59 1.67 0.71 1.08





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column which, combined with precipitation and run-off data, may permitestimation of variability in sub-surface storage.

The authors wish to thank the Natural Environmental Research Council for financial support(grant NER/A/S/2000/00612.)

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