Gravity measurements from space Satellites have mapped the gravity fields of the Earth,

LIMESTONE POROSITY

5.5 Gravity measurements from space Satellites have mapped the gravity fields of the Earth,

Moon, Mars, and Venus for fundamental constraints on planetary properties and evolution. Satellites map the grav- ity field in either passive or active survey modes. Passive measurements include tracking a satellite’s orbit relative to the ground or to other satellites, and satellite-hosted gravimeter observations, whereas active measurements use satellite-generated altimetry (i.e. electromagnetic or laser pulses) to image the geoid in oceans. Variations in both sets of measurements can be related to gravity field anomalies.

Gravity anomalies from geological features that mea- sure less than several hundred kilometers are highly atten- uated at satellite elevations. As a result they are not observed in passive measurements. Active measurements, on the other hand, provide higher-resolution data over the sea surface, but these data are restricted to off-shore marine environments in contrast to the broader coverage

116 Gravity data acquisition

including over the land surface obtained by passive mea- surements. The resolution of active measurements is dic- tated primarily by the footprint of the altimeter on the ocean surface that is a very close approximation of the geoid. For example, Geosat (1985–1989) and ERS-1 (1991–1996) and ERS-2 (1995–2003) satellite altimetry data have a footprint of about 7×7 km², so that the marine altimetry data have mapped gravity anomalies at wavelengths of about 15–20 km and larger for roughly 70% of the Earth’s surface. The shorter-wavelength resolu- tion limit largely results from waveform retracking studies which have improved the extraction of the ocean surface signals in the altimetry data (e.g.Andersenet al., 2010).

5.5.1 Passive measurements

Tracking the satellite in its orbit can determine the basal harmonic components of the planetary gravity field. Satel- lite orbits are mostly a function of the lower-order grav- ity harmonics and the effects of solar wind, lunar tidal forces, and other perturbing parameters (e.g. Kaula, 1966;Reigber, 1989). Depending on altitude, the satel- lite is primarily affected by the gravity harmonics through degree 20 or 30 with the power of the higher-order har- monics tapering off to near zero through degree 70.

A model for predicting a satellite’s location (C) may be calculated froma prioriestimates of the above-mentioned parameters (Pi) by

C= n

i=1

Pi. (5.11)

As shown in Figure 5.18, differencing the calculated orbital location with the actual observed location (O) of the satellite yields a residual that is a function of both the errors in the model parameters (Pi) and the observations (eobs). Transformations between the Earth-centered and satellite-centered frames of reference must be determined, as must the relationships between the orbital path and the various parameters used to predict the orbital path.

Knowing these relationships, the difference between the actual location (as determined from observation) and the predicted location may be expressed by a well- established observation equation (e.g. Kaula, 1966;

Torge, 1989;Reigber, 1989) according to the general formula

(O−C)= n

i=1

Pi+eobs. (5.12)

An a priorimodel is assumed to be close to the ideal parameters, so that the desired corrections are small and

j = 1 j = 2 j = 3

j = 1 j = 2 j = 3

Calculated ( C) Observed(O)

Anomalous mass with positive density contrast

(O – C )_j

Ground track coordinates

Free-air gravity

g _FAA

Orbital displacement

0

FIGURE 5.18 Generalized gravity field measurement by ground-based tracking of a satellite. Adapted fromv o n F r e s eet al.

(1999c). The orbital displacement (O−C)jat locationjis the difference between the calculated satellite orbital height (C) determined froma priorigravity field parameters and its observed height (O), which reflects the actual Earth with an anomalous mass of positive density contrast. Corrections (Pi) to thea priorivalues are determined using Equation 5.12 to generatea posteriori parameters, which are used to estimate the free-air gravity anomalies (gFAA).

may be linearized. With multiple (j) observations, this result becomes a linear system of equations that may be solved by least squares for modifications (Pi) to the parameters. This approach, however, involves tedious satellite tracking calculations, and will also be limited over regions where satellite tracking data are sparse.

Additional refinements to this method have been offered (e.g. Lerch, 1991; Anderson and Cazenave, 1986;Lemoine et al., 1998) that include schemes for weighting Equation 5.12 and adapting it for the gravity inversion of satellite-to-satellite tracking data.

Satellite-to-satellite data in High–Low and Low–Low (e.g. NASA’s GRACE mission) tracking configurations (Figure 5.19) provide more continuous coverage for determining the gravity field. In addition, the use of GPS for tracking satellites, such as in the TOPEX/Poseidon (1992–2006) and GRACE (2002−) missions, yields exceptionally good orbit determinations. However, these data still limit the solution of the gravity field to the lower-order components that predominate at satellite altitudes owing to attenuation of the signal.

The accuracy of the passive satellite gravity field esti- mates also is greatest for the lowest-order components and decreases with increasing order. The GRACE mission,

5.5 Gravity measurements from space 117

j = 1 j = 2 j = 3

j = 1 j = 2 j = 3

Calculated (C) Observed (O)

Anomalous mass with positive density contrast

Initial increase in satellite displacement

Satellite-to-satellite tracking (low–low configuration)

(O – C)_j

g

_FAA

Free-air gravity

Orbital displacement

0

FIGURE 5.19 Generalized gravity field measurement by satellite-to-satellite tracking in the Low–Low configuration. Adapted fromv o n F r e s eet al.(1999c). The fundamental observation at locationjis the orbital displacement (O−C)jbetween the satellites in the orbital plane. The calculated distance (C) is the difference between satellite positions in the absence of an anomalous mass as determined froma priorigravity field parameters. An anomalous mass with positive density contrast will cause the observed displacement (O) to increase atj=1as the leading satellite is pulled forward, then decrease atj=2as the trailing satellite surges forward while the leading satellite decelerates, and finally increase back to the initial displacement at j=3as the satellite pair moves beyond the anomalous mass effect.

These changes in the displacement may be used in Equation 5.12 to refine the gravity field parameters and thereby estimate the free-air gravity anomalies (gFAA).

operating at altitudes of roughly 450–500 km, for example, has produced the most precise set of gravity field obser- vations to date. However, the repeatability of the shortest 500 km wavelength components is no better than about 5 mGal, whereas the wavelength components in excess of several thousand kilometers are coherent or repeatable to within several microgals.

The GOCE mission (2009−) hosts the first and to date only spaceborne gravity gradiometer. The gradiometer consists of six accelerometers arranged to form three gra- diometer arms mounted on an ultra-stable diamond struc- ture or cage. The gradiometer measures the application of electrostatic forces that maintain a levitated proof mass at the center of the cage. These results give the acceleration difference between each pair of identical accelerometers along each arm that is about 50 cm in length. The three arms are orthogonal to each other with one aligned along the satellite’s trajectory, another perpendicular to the tra-

jectory, and the third one pointed to the Earth’s center.

The differential accelerations are combined with the star tracker and GPS data to estimate the gravity gradient com- ponents and perturbing angular accelerations. A primary objective of the GOCE mission is to use the gravity gra- dient tensor observations in combination with the satel- lite tracking data to resolve the Earth’s gravity field com- ponents over wavelengths from a few thousand to about 200 km with an accuracy of 1 mGal (e.g.Drinkwater et al., 2003).

5.5.2 Active measurements

Satellite altimeters map significantly higher-order com- ponents of the Earth’s anomalous gravity field over the oceans than observed at satellite altitudes. They are truly measuring gravity at the Earth’s surface. The altimetry determines the distance from the satellite to the surface of the ocean (Figure 5.20) which serves as a proxy indicator of the Earth’s geoid undulation. The geoid represents the effect of the disturbing potential at the Earth’s surface, and therefore includes the higher-order gravity components.

By remotely measuring this surface with radar, remark- ably detailed and accurate gravity information has been retrieved for exploration of the Earth’s oceanic crust.

The accuracy and resolution of altimetry-derived grav- ity anomalies are significantly limited by errors in deter- mining the satellite orbits. Geodetic satellites commonly operate in inclined polar orbits where half of the ground tracks ascend from south to north across the Earth to inter- sect the other half that descend from north to south. The orbital mismatches at the cross-over points can be used to reduce satellite orbit errors significantly for optimal altimetry estimates of the gravity field (e.g.Kim, 1996;

Scharroo and Visser, 1998). The areal density of cross-over points increases markedly towards the poles where the dense track coverage results in enhanced spatial resolution of the height variation of the sea surface. For example, Geosat data, which cover the southern oceans to 72^◦ S, have an along-track data interval of roughly 7 km, whereas the spacing between tracks at 60^◦S is only about 2–3 km.

The observed altimetry data (ρobs) must first be screened and corrected for errors caused by the wet and dry troposphere, atmospheric pressure, the ionosphere, and other factors (Figure 5.20). The corrected altimetry data (ρ) can then be differenced with orbital elevations (H) determined from tracking and orbital models to gener- ate sea surface height (SSH) variations. Corrected SSHs are generated by applying models for other surficial fac- tors such as the static sea surface topography (SSST) and

118 Gravity data acquisition

ρ H

SSH

SSST

DSST

N

Satellite orbit

Dynamic sea surface topography Static sea surface topography Reference ellipsoid Geoid

H = SSH + ρ, where ρ = ρ_obs + ρ_cor + e_obs + e_cor

SSH = N + DSST + SSST + e_DSST + e_SSST

FIGURE 5.20 Generalized gravity field measurement by satellite altimetry. Adapted fromv o n F r e s eet al.(1999c). Altimetry observations (ρobs) with errorseobsare corrected for various atmospheric effects (ρcor) with errorsecor. The corrected altimetry data (ρ) then are differenced with the orbital heights (H) to generate sea surface heights (SSH). Removal of the dynamic sea surface topography (DSST) and static sea surface topography (SSST) with errorseDSSTandeSSSTleaves the desired geoid undulations (N) that are converted into free-air gravity anomalies (gFAA) by the fundamental equation of geodesy (Equation 5.14).

performing a cross-over adjustment to reduce the dynamic Sea Surface Topography (DSST) and orbit errors arising in the determination ofH.

Models for correcting theρobsare generally available with the altimetry data (e.g. Kim, 1996), so that the geoidal components can be readily recovered. However, the corrected SSHs contain residual errors, and thus do not accurately reflect the geoid undulations at this level of processing.

The common approach for extracting gravity anoma- lies from the corrected SSHs is to determine along-track vertical deflections from the descending and ascending orbits at the cross-over points. These vertical deflections can be converted directly into gravity anomalies with- out the necessity of computing the geoid (e.g.Haxby et al., 1983;Sandwell, 1992;McAdooandMarks, 1992;Kim, 1996;SandwellandSmith, 1997). How- ever, the effort to evaluate geoid undulations from the corrected SSHs facilitates the imposition of strong geo- logical constraints such as the depths of possible sources and enhanced reduction of residual orbital errors in the altimetry-derived gravity estimates (e.g. Kim and von Frese, 1993;von Freseet al., 1999c).

In this geoid-mapping approach, the two sets of sub- parallel ascending and descending tracking orbits are pro- cessed for separate geoid estimates. Lithospheric signals that are larger than the spacing between the ground tracks of the subparallel orbits are coherent between two neigh- boring tracks, and hence may be extracted with spec-

tral correlation filters. Inversely transforming the cor- relative wavenumber components from each data track yields geoid undulation estimates where non-correlative features are suppressed, involving presumably crustal effects that are small compared with track spacing and non-lithospheric effects from temporal and spatial varia- tions of ocean currents, measurement and data reduction errors, etc.

Separate geoids then can be produced from gridding the correlation filtered ascending and descending data tracks.

Typically, the geoid estimates exhibit strong washboard effects that parallel the orbital tracks and reflect errors in orbit determination, along-track data processing, and other non-geologic effects (e.g.Kim, 1996;Kim et al., 1998a). In the spectrum of each map, the track-line noise is restricted predominantly to two of the quadrants with the other two being relatively uncontaminated. The pair of clean quadrants in one map is orthogonal to the clean pair in the other map because the track orientations of the ascending and descending data sets are different. Hence, a new spectrum can be constructed by combining the two pairs of clean quadrants to yield geoid undulation estimates where track-line noise is severely suppressed.

The resultant grid of geoid undulations (N) can be used to estimate free-air gravity anomalies (gFAA) because it is a function of the disturbing potential (T) as expressed in Bruns’ formula (HeiskanenandMoritz, 1967)

T =N×gN, (5.13)

5.5 Gravity measurements from space 119

100 50

0

-50

28 30 32 34 36 38 40

gFAA (mGal)gFAA (mGal)

Longitude (degree east) 100

50

0

28 30 32 34 36 38 40

Longitude (degree east) Mean-0.63RMS

3.40 STD 3.34Max

6.0 Min -18.7 CC

0.9929

Mean-1.09RMS 3.04 STD

2.83Ma x 8.5 Min

-13.0 CC 0.9959

BGR SHIP OSU WCF

BGR SHIP OSU WCF

-50 -100

-100

FIGURE 5.21 Free-air gravity anomaly (gFAA) comparisons between ship measurements (BGR SHIP) and wavenumber correlation filtered Geosat altimetry-derived estimates (OSU WCF) along west–east running tracks that cross the Gunnerus Ridge in the Larsen Sea, Antarctica, near65^◦S(top) and67^◦S(bottom). Adapted fromv o n F r e s eet al.(1999c). The statistics of their differences, which are given in the upper left corner of each profile comparison, include the mean, root-mean-square (RMS), standard deviation (STD), maximum (Max), and minimum (Min) values. The coefficient of correlation (CC) between the observed and estimated anomaly profiles is also given.

where g_N is normal gravity given by the International Gravity Formula for a standard Earth of homogeneous mass. Typically, geoid undulations are used to predictgFAA

through the fundamental equation of geodesy gFAA= −∂T

∂r −2T

a_e = −∂(N×gN)

∂r −2(N×gN) a_e ,

(5.14) whereae is the Earth’s mean radius, and the radial (r) gradient ofNis estimated by the spectral filter.

Figure 5.21 gives examples of the remarkable capacity of satellite altimetry to estimate marine gravity anoma- lies (von Frese et al., 1999c). Both west–east running profiles compare Geosat altimetry-derived free-air grav- ity anomaly estimates with ship observations across the Gunnerus Ridge in the Larsen Sea, off Japan’s Syowa

Station on the coast of Queen Maud Land, East Antarctica.

The Geosat estimates accurately map detailed features of the marine gravity field, even for regions such this where the sea surface is relatively stormy and contaminated by floating ice. These results are quite close to the theoretical accuracy limit of 3 mGal for the Geosat-derivedgFAAfound by a coherency analysis of hundreds of exact repeat orbits (Sailor and Driscoll, 1993). In practice, however, errors in estimating satellite altimetry-derivedg_FAArange more typically around 4–6 mGal, owing to the effects of water depth, sea state, ocean currents, sea ice, and other factors on local sea surface conditions (e.g.Andersen et al., 2010).

5.5.3 Satellite gravity mapping progress

Gravity observations at satellite altitudes provide impor- tant and relatively newly available boundary conditions for interpreting variations in the near-surface gravity field.

In practice, a model satisfying just a single altitude slice of the gravity field effectively accounts for field behavior only to vertical distances within roughly a station interval or so of the observation surface because of measurement errors. However, the model that jointly satisfies the sets of satellite-altitude and near-surface observations offers an enhanced picture of how the gravity field may vary through the unsurveyed altitudes between the boundary conditions. Thus, the jointly constrained model provides more insight into the altitude behavior of the field for inter- preting anomalous subsurface mass variations than the sin- gle altitude model can, with or without implementing the classical Poisson or far-field constraint of zero effect at infinite distance (e.g. Appendix A.4.3).

Satellite-altitude gravity surveys of the Earth have been carried out to date by geodesists for mapping the geoidal undulations and the temporal variations of the very low- order gravity components that may constrain temporal changes of the crustal waters and ice sheets. Current state-of-the-art satellite gravity data are being collected by the GRACE and GOCE missions at orbital altitudes of about 450–500 km and 270 km, respectively. The satellites operate in near-Sun-synchronous orbits in the dawn–dusk meridian plane of the Earth to maximize solar charging of batteries. Each orbit takes roughly 96 minutes to complete, by which time the Earth has rotated through an equatorial distance of about 2,667 km.

The near-polar orbits are inclined relative to the Earth’s equatorial plane at 89^◦ for GRACE, and at 96.7^◦ for GOCE. Thus, the spacing between ground tracks decreases greatly towards the poles where they become tangential to the margins of the pole-centered holes or coverage gaps

120 Gravity data acquisition

of diameters 2^◦and 13.4^◦for the respective GRACE and GOCE missions. In addition, the ground tracks for the GRACE and GOCE satellites roughly repeat every 61 revolutions/4 days and 467 revolutions/29 days, respec- tively. Thus, the track coverage from the GRACE and GOCE missions varies from track intervals of essentially zero kilometers at the margins of the polar holes to maxi- mum intervals at the equator of about 44 and 6 km, respec- tively.

The orbital r-radial, θ-colatitude, and ϕ-longitude coordinates of each satellite altitude gravity observation are accurately located by on-board GPS receivers with on-board star cameras giving the attitude of the observa- tion for interpreting the measured gravity component. The GRACE gravity data are inferred from dual one-way range changes between the two satellites measured by a K-band microwave ranging system with a precision of about 1μm per second, whereas the GOCE gradiometry data are col- lected at intervals of about 47 km along the mission tracks.

Because the observations have essentially no sensitivity for gravity wavelengths smaller than the orbital altitudes above the Earth’s surface, the along-track gravity data are averaged in bins that are one or two degrees on a side andr km thick. The bin averaging ignores the effects of the altitude variationsrin the observations that may range from a few kilometers to several tens of kilome- ters. It consists simply of computing an initial average and standard deviation, deleting all outliers with values equal to or greater than the mean plus or minus three standard deviations. From the remaining values, a new bin average and standard deviation divided by the square root of the number of data points are computed for the final gravity estimate and its standard error, respectively.

The binning process yields a grid of gravity estimates at the mean altitude of the bins that is transformed for analysis and interpretation into a set of spherical harmonic coeffi- cients. These coefficients give the amplitudes of orthogo- nal sine and cosine functions that span the Earth’s spheri- cal surface at the average radiusa_e(≈6,378.1 km) which closely approximate the bin-averaged gravity values when weighted for the altitude difference of the grid relative to aeand summed at the grid points.

The spherical harmonic model is based on the Cartesian-to-spherical coordinate transformation in Equa- tion 3.82. Because Laplace’s equation (Equation 3.6) in spherical coordinates can be solved by separation of vari- ables, the gravitational potential may be written as

U(r, θ , ϕ)= ξ r

∞ n=0

ae

r nl

m=0

P_n,m[sin(θ)]

×[Cn,mcos(mϕ)+Sn,msin(mϕ)], (5.15)

whereξ(≈3.986 004 415×10¹⁴m³/s²) is the product of bigGand the Earth’s mass, and the second summation, which is independent ofr, involves the spherical harmonic coefficientsCn,m, Sn,mof degreenand orderm, and the associated Legendre functions Pn,m[sin(θ)] of degree n and orderm(e.g.Picket al., 1973;Presset al., 2007;

Blakely, 1995).

In general, the gravitational spherical harmonic model consists of a set of constants that specify ξ and a_e and the C_n,m, S_n,m coefficients. However, the mod- eled gravitational variations have wavelengths of the Earth’s circumference divided bymin longitude and by (n− m) in latitude where−n≤m≤n. Thus, satellite gravity measurements at 400 km altitude, for example, may be reliably modeled only for orders m≤100 (= [2π×ae≈40,000]/400) because they observe wave- lengths no smaller than about 400 km, assuming negligible measurement errors.

In 2008, the US National Geospatial Intelligence Agency (NGA) released the Earth Gravitational Model EGM2008 (Figure 5.22). It is the most comprehen- sive model to date of the Earth’s satellite, airborne, marine, and terrestrial gravity observations (http://earth- info.nga.mil/GandG/wgs84/ gravitymod/egm2008). This gravitational model is complete to spherical harmonic degree and order 2,159 and contains additional coefficients extending to degree 2,190 and order 2,159. Thus, the model accounts for anomaly wavelengths of about 18–20 km and longer, but only where the constraining gravity observa- tions provided commensurate or higher-wavelength reso- lution. Over Antarctica, for example, the gravity field is basically constrained by satellite altitude observations so that the EGM08 coefficients are most reliable for modeling the Antarctic gravity field only at satellite altitudes. Thus, care is required in applying these coefficients for subsur- face investigations, owing to the uneven spatial coverage and wavelength resolution of the gravity surveys used in producing the spherical harmonic model.

Spherical harmonic models of satellite altitude gravity observations for the Moon, Mars, and Venus that funda- mentally constrain the subsurface properties of these mass- differentiated planetary bodies have also been constructed.

The gravity observations are based on the tracking of the satellites by NASA’s Deep Space Network (DSN), which is an international network of antennas supporting interplan- etary spacecraft missions and radio and radar astronomy observations for the exploration of the solar system and the universe. The DSN currently consists of three deep-space communications facilities placed about 120^◦apart around the world, at Goldstone in California’s Mojave Desert and near Madrid, Spain, and Canberra, Australia. This configuration of antennas permits constant observation of