Royal Society Publishing

Recent results on modelling the spatial and temporal structure of the Earth's gravity field

P Moore , Q Zhang , A Alothman


The Earth's gravity field plays a central role in sea-level change. In the simplest application a precise gravity field will enable oceanographers to capitalize fully on the altimetric datasets 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 now redundant as we can observe temporal variability in the geoid due to mass redistribution in or on the total Earth system. Temporal variability, associated with interactions between the 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 the time 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 of satellite 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 Circulation Experiment (GRACE) our knowledge of the spatial distribution of the Earth's gravity field is taking a leap forward. Furthermore, GRACE is now providing insight into temporal variability through ‘monthly’ gravity field solutions. Prior to this data we relied on satellite tracking, Global Positioning System and geophysical models to give us insight into the temporal variability. We will consider results from these methodologies and compare them to preliminary results from the GRACE mission.

1. The Earth's gravitational field

The Earth's gravitational field at a point on or external to the Earth is described mathematically byEmbedded Image(1.1)where GM is the product of Newton's gravitational constant and the Earth's mass; Re the mean Earth radius; r, θ, λ spherical coordinates (radial distance, colatitude, longitude) of the point; Pl,m normalized Legendre polynomials; and Cl,m, Sl,m normalized spherical harmonic (Stokes') coefficients of degree l and order m. The lower limit of summation over l, namely 2, is a consequence of the first degree and order harmonics being zero on assuming that the frame origin is at the Earth's instantaneous centre of mass. Gravity field models contain a subset of the spherical harmonic coefficients typically up to some degree, lmax, say. The recent history of gravity field modelling, as summarized in §2, reveals progressive improvements with time through incorporation of additional data with improved geographical and temporal coverage. This has enabled the models to improve in accuracy and to be extended to shorter wavelengths by increasing the cutout degree lmax.

In practice, the spherical harmonics should not be considered as invariant but rather have temporal signatures, which are broad-band although dominated by quasi-secular and periodic components. Secular changes arise, for example, due to isostatic post-glacial rebound while the periodic components are typically due to annual and semi-annual mass redistribution. Recent gravity field models incorporate variability to the second degree zonal harmonic, C2,0, and degree 2 and 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 the period of the underlying data. Such data can span decades or, as for Challenging Microsatellite 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 dedicated gravity field missions. The first of these is the CHAMP launched in 2001 into a near polar orbit at an altitude of 450 km. CHAMP (Reigber et al. 2002) is a dual purpose mission to measure the Earth's gravity and magnetic fields. For the gravity field objective, the satellite carries Global Positioning System (GPS) receivers for precise positioning, star cameras for attitude control and a 3-axes accelerometer to measure surface accelerations. Procedures for recovering the gravity field are described by Reigber et al. (2004). GRACE, launched in March 2002, is a tandem satellite mission where the inter-satellite range, range-rate and range acceleration are derivable from a K-band microwave device (Tapley et al. 2004). Each GRACE satellite carries GPS receivers, accelerometers and star cameras. Both CHAMP and GRACE are providing data for recovery of the static gravity field with GRACE, in addition, providing monthly snapshots of the geopotential from which variability can be inferred. With reference to equation (1.1) the orbital altitude, 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. Embedded Image where h is altitude of the satellite above the Earth's surface. Both CHAMP and GRACE are in relatively low altitude orbits (400–500 km) to increase their sensitivity to gravity field anomalies. Even lower altitudes of say 200–250 km are preferable. However, the reduction in altitude is accompanied with an increase in atmospheric density with consequent impact on the satellite lifetime unless the orbit is periodically boosted to a higher altitude.

Geophysical applications of temporal and spatial modelling are numerous and well documented in the literature with dedicated special issues such as Space Science Reviews, volume 108, issues 1 and 2, 2003, providing an excellent summary. Papers in that issue include the impact of geoid improvements on large scale ocean circulation (Le Grand 2003) and sea-level studies (Woodworth & Gregory 2003). Other papers therein describe applications of temporal variability 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) are also significant to the proper interpretation of the datasets.

2. Gravity field models

It is informative to summarize the history of gravity field enhancements over the past 20 years. Advances in modelling have paralleled increases in computer power, availability of more accurate and varied tracking data, introduction of combination strategies and, most recently, the launch of CHAMP and GRACE. Table 1 presents a subset of the many models released over the past 20 years. For brevity, the table does not contain all models with no explicit mention of the excellent GRIM, TEG or OSU fields, for example. A comprehensive tabulation of gravity field models from the late 1960s onwards is available at the International Centre for Global Earth Models ( Rather, the summary is intended to illustrate the main advances in modelling. The table gives the date 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 orbit computations based on a covariance analysis. The latter is reliant on proper calibration of the covariances but is a useful measure of accuracy.

View this table:
Table 1

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

The first model included is GEM-L2 (Lerch et al. 1985). GEM-L2 is a satellite only gravity model complete to degree and order 20. Improvements are seen with GEM-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 was further extended to degree and order 50 and altimetry and in situ gravity anomalies were added to the satellite data. GEM-T3 typified the major advance associated with use of combined data sources.

The JGM series were released around the launch of T/P and included additional and better tracking. JGM-1 (Nerem et al. 1994b) was a recomputation of 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 represents resolution to half-wavelengths of about 290 km. Geoid computation requires determination 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 to degree and order 360 through use of surface gravity data and satellite altimetry.

To illustrate associated improvements in the geoid, table 1 also presents some results of the global r.m.s. geoid undulation commission error for given degree and order and the approximate degree at which the accumulative geoid error is 10 cm r.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. with EGM-96. The total EGM-96 r.m.s. geoid undulation commission error to degree and order 360 is 42.1 cm. Figure 1 shows the geoid undulation error for EGM-96 as computed from the covariance matrix complete to degree and order 70. A clear demarcation is evident between sea (altimetry data) and land (surface gravity data) due to the relatively high precision of the former and the inhomogeneous nature of the latter. In contrast, given the homogeneous nature of CHAMP and GRACE the errors show no discrimination between sea and land. The CHAMP gravity field model EIGEN-CHAMP03S (Reigber et al. 2004), has power to about degree 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 to degree and order 70–80 (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 in table 1 for EIGEN-GRACE01S and GGM01S (Tapley et al. 2003) are not significant, but merely a consequence of the different calibration procedures. The relative merits of CHAMP and GRACE can be seen from the accumulative geoid 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.

Figure 1

EGM-96 geoid error from covariance matrix to degree and order 70. From 01/GGM 01_Notes.pdf.

Gravity fields, such as the JGM, TEG and GRIM satellite only models, have used tracking data collected over decades or more. Figure 2 plots gravity anomalies determined from GRIM5-S1 (Biancale et al. 2000) a satellite only model complete to degree 99 and order 95 determined from over 20 years of satellite tracking and those from EIGEN-GRACE01S, an early GRACE model complete to degree and order 120, with selected harmonics to degree 140, derived from only 39 days of data. A visual comparison shows striking similarity while close inspection reveals the greater clarity of features from the GRACE model. The blurring in GRIM5-S1 is due to the difficulty in resolving individual harmonics. GRACE is providing significant new observational data to the extent that a GRACE model constructed from just 39 days of data is of higher accuracy that those derived from over 20 years of ground-based satellite tracking.

Figure 2

(a) Gravity anomaly map derived from 39 days of GRACE data (EIGEN-GRACE01S model). (b) Gravity anomaly map derived from tracking data of 30 Earth orbiting satellites over more than 20 years (GRIM5-S1). From

3. The temporal gravity field: theory

In this section, we introduce some theoretical aspects related to temporal variability in the Earth's gravity field. Surface mass redistribution will lead to corresponding temporal changes in the gravity potential due to attraction by the time varying surface mass and also the deformation (load) of the underlying solid Earth. Suppose there is a time-dependent change in the gravitational potential represented through changes, ΔClm and ΔSlm, in the spherical harmonic coefficients as follows:Embedded Image(3.1)

Total changes in the spherical harmonic coefficients of the gravity potential can be calculated (Wahr et al. 1998) via the load harmonics defined byEmbedded Image(3.2)where Δσ is defined as the change in surface density (i.e. mass/area), and ρw the density of water (1000 kg m−3) included so that Embedded Image and Embedded Image are dimensionless. The relation between Embedded Image and Embedded Image and the gravity field spherical harmonics is:Embedded Image(3.3)where ρav is the average density of the Earth (=5517 kg m−3) and Embedded Image the load Love number of degree l (Farrell 1972).

The loading associated with the mass per unit area Embedded Image in equation (3.2) deforms the elastic Earth and displaces points on the Earth's surface by distances ur, uθ, uλ in the radial, south and east directions, respectively, where (Moore & Wang 2003)Embedded Image(3.4)

In the above, g is gravity and Embedded Image and Embedded Image the Love and Shida load numbers (or load deformation coefficients) of degree l (Farrell 1972).

In addition to the Earth's 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 the Earth and atmosphere system. Thus, by definition, we require Embedded Image in equation (3.3). The contribution of non-zero surface loading coefficients Embedded Image and Embedded Image for m=0, 1 is equivalent to a displacement Embedded Image in the position of the satellite tracking stations where (Trupin et al. 1992)Embedded Image(3.5)

The forcing effects of surface loading described above are, in principle, observable in space geodetic techniques, which allow the static and temporal gravity field to be investigated through several independent methodologies. These include estimates of gravity field coefficients from precise orbit determination using satellite tracking, from deformation studies of the elastic Earth due to loading using change in position of satellite tracking stations or GPS receivers and GRACE. In addition, degree 2 harmonics can also be recovered from Earth rotation data (e.g. Chen et al. 2000; Chen & Wilson 2003; Gross et al. 2004).

4. Temporal variability: satellite tracking and GPS

As stated previously, temporal variability within the Earth's gravitational field is a response to the redistribution of mass on the Earth's surface and in its interior. The temporal signatures are broad-band in spectrum with intra-annual and longer time-scale variability superimposed on the inter-annual and secular trends. In particular, surface mass change in the atmosphere, oceans, hydrosphere and cryosphere are dominated by seasonal variations while processes such as isostatic glacial recovery and sea-level change give rise to long-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 (Crétaux 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 received particular 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 and degree harmonics in good agreement with geophysical models of surface mass redistribution.

With recent improvements in SLR tracking technology and quality assurance it has been possible to undertake more detailed studies of the temporal variability in the lower order and degree harmonics. Most notably, Nerem et al. (2000) recovered annual variability for degrees 2–4 inclusive from SLR tracking of Lageos1&2 over a 6 year period and used the results to discriminate between several geophysical models. Results therein established a higher correlation with a particular hydrological model but were unable to distinguish between models of the atmosphere and oceans. This was partly due to the smaller contribution of ocean mass and the likelihood of better agreement between competing models for atmospheric surface pressure at the very long wavelengths. Temporal variability to higher spatial resolution is an objective of GRACE, which is expected to produce annual variability to degree and order 40 (half-wavelength 500 km) over the lifetime of the mission (Wahr et al. 1998). Despite the outstanding promise of GRACE, satellite tracking has an important complementary role enabling analysts to determine variability in a consistent manner over a long time frame. Furthermore, satellite based results provide a mechanism for validation of the GRACE temporal variability.

The long-time series of SLR tracking of Lageos has recently established an apparent short-term reversal in the secular change in the Earth's oblateness coefficient Embedded Image (Cox & Chao 2002). On removing seasonal signatures, the variability within J2 for the period 1980–1998 is dominated by a negative secular trend associated with isostatic post-glacial rebound. After 1998, the trend was reversed with a maximum in 2000–2001 before returning to the value and trend prior to 1998. This signature indicates a pronounced global-scale mass redistribution within the Earth system from high to low latitudes reflecting an equator ward mass transport large enough to offset the ongoing isostatic recovery. Attempts to analyse the change have involved oceanic mass distribution and melting of sub-polar glaciers (Dickey et al. 2002; Chao et al. 2003). Figure 3 plots our recent time series of J2 determined from Lageos 1&2 for the period 1998–2004 (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. The detrended signals, along with a six month boxcar average, show the increase after 1998 prior to its return to the long-term mean in 2002.

Figure 3

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

The inclusion of lower altitude satellites such as Starlette (altitude ca 800 km), Stella (ca 800 km) and Ajisai (ca 1200 km) is required to compensate for the insensitivity of Lageos (ca 6000 km) to temporal variability beyond degree 4 due to the attenuation of the gravity field with height. Even then the orbits are only sensitive to temporal variability at the very low degrees, say 2–6. Furthermore, temporal variability from CHAMP seems to require the use of constraints although singular value decomposition has established that some signal is discernible (Moore et al. 2005). The alternative, as described in §3, is to use vertical deformation from a global distribution of GPS receivers. In contrast to SLR, GPS is highly sensitive to local (high degree) effects. Although the GPS global coverage is far more complete than say SLR, the high degree effects and incomplete coverage over oceanic areas leads to aliasing of the low degree harmonics. Various other systematic errors may also contaminate the GPS time series 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 to estimate daily coordinates of 166 IGS stations using the point positioning mode of the JPL software GIPSY-OASIS II (Zumberge et al. 1997). The resultant vertical deformation cleared of secular trends is plotted in figure 4 for four GPS sites for the period 1998–2003. As the GPS coordinates were recovered relative to the ITRF2000 reference frame degree one harmonics were removed in the data for figure 4 (and also figure 5) by using the geocentre motion inferred from Lageos (Moore & Wang 2003). All four stations exhibit strong seasonal signatures. Also plotted are the annual signals derived from a combination of geophysical models for atmospheric and ocean mass and land hydrology. For these we used (Moore & Wang 2003)

  1. atmospheric pressure for January 1989–March 2002 supplied by the National Center for Environmental Prediction (NCEP) Reanalysis project;

    Figure 4

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

    Figure 5

    Vertical annual amplitude and phase of IGS sites. The amplitude/ phase is defined as Acos(ωtϕ) where t (days) is relative to January 1. Arrows represent the amplitude with phase measured counterclockwise from the east.

  2. TOPEX altimetry for January 1993–December 2000 and global climatology (Levitus & Boyer 1994)

  3. soil moisture and snow mass from NCEP CDAS-1 (Climate Data Assimilation Service) for 1993–2000.

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

Within this comparison the global mass distribution from geophysical data was handled differently between continental and oceanic areas. Over the continents the total mass is simple to model being the sum of the individual continental hydrology and atmosphere masses. However, over the oceans the surface closely approximates an inverted barometer (IB) with an increase of 1 mb in pressure associated with a decrease of 1 cm in ocean height. In our approach we elected to infer bottom pressures for the water mass alone and then to combine with atmospheric pressure. In detail, the T/P altimetry was first corrected for all geophysical effects as given on the altimetric records but with the IB correction replaced by a modified correction (Hendrick et al. 1996). In this modification the IB correction was determined relative to the mean atmospheric pressure over the oceans rather than the constant value used on the altimetric records. Given the incompressibility of seawater the oceans respond to variation about the global ocean mean with a change in the mean having no impact on the height of the oceans. A global oceanic mean was calculated over each T/P cycle and used with the atmospheric pressure inferred from the IB correction on the altimetric record to estimate the modified IB correction. The seasonal sea-level changes were subsequently corrected for the steric effect using monthly climatology (Levitus & Boyer 1994). Climatological data of temperature and salinity were used to infer seasonal changes in sea water density which were combined with the derived T/P sea-surface heights to infer changes in bottom pressure of the water mass. The total mass distribution over the oceans was derived by adding atmospheric pressure. For consistency with the IB correction ECMWF pressure was used over the oceans.

Figure 5 presents the amplitude and phase of the vertical deformation averaged over the period 1998–2003 from a global network of 166 IGS sites. Using equation (3.4) this deformation can be inverted to recover the lower order and degree harmonics. However, comparisons with SLR+CHAMP (Moore et al. 2004) showed excessive annual variability in the degree 2–6 harmonics indicating aliasing from higher order effects. The obvious next step was to use a combination strategy for SLR, CHAMP and GPS (Moore et al. 2004) to capitalize on the signatures in each data source. This has the advantage of providing additional data for degrees 5 and above to compensate for the insensitivity of Lageos in the SLR solution and to provide global constraints on the GPS solution, which is relatively unconstrained over oceanic areas. In the combined approach we used the normal equations for annual and semi-annual variations in the degree 2–6 harmonics from SLR and CHAMP (Moore et al. 2005) and combined with similar normal equations from GPS. For GPS, the vertical deformation was now used to form normal equations for sinusoids at the annual and semi-annual frequencies in the lower order and degree harmonics (cf. equation 3.4). The procedure differed from that used to construct figures 4 and 5 in that the degree one harmonics were also estimated to allow for geocentre motion (cf. equation 3.5) in the GPS time series. The normals from SLR+CHAMP and GPS were weighted heuristically to capitalize on the strengths of the respective data types but with particular emphasis placed on maintaining solutions close to the degree 2 and 3 results from SLR+CHAMP. Further details of the computations, weighting strategy and comparisons against geophysical data are given in Moore et al. (2004).

Table 2 compares results for the amplitude and phase of the annual signal in the lower order zonal harmonics from the combination approach, SLR studies and geophysical models. The results show reasonable agreement in amplitude and phase for all harmonics except J4. Given the magnitude of the signals, possible aliasing between the harmonics, omission and commission errors and absorption of non-gravitational seasonal signals (e.g. tides) the level of agreement is encouraging and reflective of that obtainable by using SLR data or a combination of SLR and GPS. A plot of the annual variation in the geoid undulations from the combination solution and for the geophysical data is given in figure 6.

View this table:
Table 2

Comparison of the annual variations in degree 2–6 normalized zonals, Cn,o. (Amplitude/phase defined by A cos(ωtϕ), where t is (days) relative to January 1, ϕ degree and ω annual frequency. (A) SLR+CHAMP+GPS (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).)

Figure 6

Annual variation in the geoid for 6×6 field (C20 removed): (a) and (b) SLR+CHAMP+GPS solution; (c) and (d) geophysical data; (e) and (f) GRACE. The upper plots are for 1 January with the lower corresponding to 1 April.

5. Temporal variability: GRACE

GRACE monthly solutions of the gravity field are being made available to the scientific community as Level-2 products by the Center for Space Research (CSR), University of Texas, and GeoForschungsZentrum (GFZ), Potsdam. This study used the CSR monthly gravity field solutions ( At the time of study the community had access to 19 monthly solutions (Bettadpur 2004a,b) covering April 2002–April 2004 derived typically from 26 to 31 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 GRACE solutions are determined relative to background models of time variability associated with solid earth, ocean and pole tides, atmospheric mass and a barotropic ocean model. Ocean tides in the CSR solutions are modelled by CSR4.0. High-frequency mass redistribution due to the atmosphere and oceans is removed in the GRACE processing by inclusion of the ECMWF operational atmospheric model and the so-called PPHA barotropic model (Flechtner 2003). High-frequency variability is converted to spherical harmonics at 6 h intervals with intermediate epochs obtained by linear interpolation. Each monthly gravity field solution is associated with a monthly average of the combined atmospheric/barotropic background gravity field model. The gravitational effects of the background atmospheric and barotropic models can be reinstated by addition to provide the total gravity field. Given increasing errors at higher degrees the user is cautioned against usage of harmonics beyond degree and order 90–100 in the monthly fields and advised to employ some smoothing or truncation of the shorter wavelengths.

To illustrate the neccessity of using some form of smoothing or truncation of the high degree and order terms, geoid undulations were computed from the monthly solutions for two consecutive months, January and February, in 2003. Figure 7 shows the differences between these geoid heights with the legend representing differences in metres. The true variation should be at the millimetre level. The pattern clearly reflects the GRACE near polar orbit with the N–S trackiness resulting from deficiencies in the data and tidal modelling (Han et al. 2004). Spatial averaging (Wahr et al. 1998) can be used to smooth the GRACE estimates with meaningful mass estimates recovered using a Gaussian averaging kernel of radius 500–1000 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 monthly solutions to obtain the monthly fields that represent the total mass. These were subsequently used in two different analyses. The first is a study of the annual global change 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 was incorporated into equation (3.1) with the geoid height estimated on a regular geographical grid. The time series of geoidal heights at each grid point was fitted by a constant to obtain the annual mean and a sinsoid of period 1 year to recover the annual signal. Thus, if Embedded Image denotes the geoid height recovered from each monthly field at point with colatitude θ and longitude λ thenEmbedded Image(5.1)where t is the time in days from January 1; T=365.25; Embedded Image the mean geoid height and Ac and As the annual geoidal variation on 1 January and 1 April respectively. Figure 8 plots the spatial distribution of Ac and As. Vestiges of the N–S trackiness is still evident over the oceans, particularly in the cosine component. This can be further reduced by increasing the averaging radius to say 1000 km. Even at 500 km averaging there is little discernible signal over the oceans where zonal signatures are expected. The dominant signals are due to continental hydrology and the re-instated atmospheric pressure. Results such as these have demonstrated the power of GRACE to recover continental water storage in large catchment areas such as the Mississippi and Amazon and the area draining into the Bay of Bengal. These and other studies suggest that for continental hydrology Gaussian averaging over about 500 km is meaningful and that the later GRACE monthly fields yield errors 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 the potential for recovering the annual signal, for comparisons against geophysical models and for intra-annual studies over the lifetime of the GRACE mission.

Figure 7

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

Figure 8

Annual variation in geoid recovered from GRACE using spatial averaging with radius 500 km. (a) Cosine component corresponding to 1 January. (b) Sine component corresponding to 1 April.

In the second study, annual variability was recovered in the GRACE harmonics for the 6×6 field. This study also excluded consideration of J2=−C2,0 as the GRACE results (see below) exhibited anomalous variability in, for example, the first few monthly solutions. Figure 6 shows the annual variation from GRACE alongside that from the SLR+CHAMP+GPS combination solution and the geophysical model. The correlations and r.m.s. differences between the geophysical model, combination solution and GRACE are summarized in table 3. These results show that the agreement between SLR+CHAMP+GPS and the geophysical data is high for a 4×4 field but decreases substantially on extending to 6×6. A similar trend is observed with the combination solution and GRACE. As Lageos1&2 have little power beyond degree 4 the additional harmonics are recovered from the other three less accurate orbits and GPS. On the other hand, the agreement between the geophysical data and GRACE is maintained showing no significant loss of accuracy as the field is extended from 21 harmonics to 45. The behaviour of the agreement confirms the power of GRACE for mass redistribution studies. It is important to emphasize that the combination solution was derived from 1998 to 2003, GRACE from 2002 to 2004 and the geophysical results from data collected over a decade or more. Given the different time-scales we would expect some small differences between the annual signals.

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Table 3

Correlation and r.m.s. (mm) difference of annual geoid between combination solution (S+C+G), geophysical data (GPH) and GRACE for 4×4 and 6×6 gravity field (C20 removed).

The individual GRACE harmonics can be verified by comparison against SLR. Figure 9 plots the Lageos1&2 J2 solution recovered at 15 day intervals over 2002–2004.5. No annual or semi-annual signal has been removed. Also plotted are the 19 monthly solutions from the CSR GRACE fields with the background models re-instated. The early problems with J2 are evident but the later values agree with Lageos. 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.

Figure 9

Comparison of J2 (unnormalized) from GRACE and Lageos.

6. Conclusions

Satellite tracking, supplemented with altimetry and surface gravity data, has been the basis of gravity field models over the past 20–30 years. However, recent studies have shown that the long-wavelength static gravity field recovered from a few months of GRACE data is superior to the previous 20 years' effort. Satellite tracking such as SLR and DORIS and vertical deformations from GPS have contributed to our knowledge of temporal variability for the long wavelength field including the degree 1 terms, the so-called geocentre variability. These results can be compared against mass distributions supplied by geophysical data for atmospheric and ocean mass and land hydrology and also provide a bench-mark for GRACE. Although the temporal variability from GRACE has not as yet achieved the pre-launch baseline (Wahr et al. 2004), the early results are providing a spatial resolution unobtainable with conventional means. Further improvements in the accuracy of the temporal fields are inevitable, a consequence of the strenuous efforts being made by the science 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-annual variability. Over oceans GRACE has the potential for assimilation of mass redistribution into ocean models while on land GRACE can resolve total water column which, combined with precipitation and run-off data, may permit estimation 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.)


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


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