Time Variations In Gravity And Inferences On The Earth's Structure And Dynamics

TIME VARIATIONS IN GRAVITY AND INFERENCES ON THEEARTH’S STRUCTURE AND DYNAMICS

JACQUES HINDERER1 and DAVID CROSSLEY21Institut de Physique du Globe de Strasbourg (CNRS- ULP UMR 7516) 5, rue Descartes, 67084Strasbourg, France, e-mail: [email protected];2Department of Earth and Atmospheric

Sciences, Saint Louis University, 3507 Laclede Ave., St. Louis, MO 63103, USA

Abstract. This paper reviews how the study of the surface gravity changes is able to provide usefulinformation on the Earth’s structure and global dynamics. The spectral range which is observablewith superconducting gravimeters is broad and goes from the seismic frequency band to periodslonger than one year. We first investigate the seismic and sub-seismic bands with a special attentionpaid to the gravity detection of core modes in the liquid core and to the Slichter mode of translationof the solid inner core. In the tidal bands, we show how accurate measurements allow us to inferconstraints on various phenomena such as mantle (an-)elasticity, as well as ocean and atmosphericloading. The observation of the Free Core Nutation resonance in the diurnal frequency band is re-viewed and indirectly suggests an increase in the ellipticity of the core-mantle boundary with respectto its hydrostatic value. A similar resonance is also theoretically predicted in the diurnal band for therotation of the solid inner core (Free Inner Core Nutation) but we show that its detection is muchmore difficult because of the small amplitude and lack of a nearby tidal frequency. Oceanic andatmospheric loading mechanisms induce gravity changes over a wide spectral range and we presentsome recent progress in this field. Finally, because superconducting gravimeters have high calib-ration stability and small long-term instrumental drift, they can easily measure long period gravityvariations due to polar motion and hydrogeology.

Keywords: gravity, geodynamics, fluid core, tides, superconducting gravimeter

1. Introduction

The elasto-gravitational deformation of the Earth and subsequently the verticalgravity changes measured at the Earth’s surface are caused by many geophysicalphenomena: seismic free oscillations, possible gravito-inertial core modes includ-ing the translational mode of the solid inner core (Slichter triplet), lunisolar tides,Earth’s rotation changes, atmospheric and oceanic loading, and hydrogeology. Thespectrum of the gravity changes varies from short periods (microseismic noisearound 1–10 sec) to periods longer than 1 year (435 day Chandlerian contribu-tion from the Earth’s polar motion). Figure 1 from Crossley and Hinderer (1995)summarizes the spectrum which is observable with superconducting gravimeters(SG) thanks to their high sensitivity (close to 1 nanogal or 10−11 m.s−2) and verylow instrumental drift (less than 10 microgal per year).

Surveys in Geophysics21: 1–45, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. Surface gravity spectrum indicating the wide spectral range (from 1 sec to several year periods) observable with superconducting gravimeters(from Crossley and Hinderer 1995).

TIME VARIATIONS IN GRAVITY 3

Below a one-hour period (more precisely below 54 mm which is the eigenperiodof the fundamental spheroidal mode0S2), we have the seismic frequency bandwhere the seismic normal modes are located. These modes are dominated by elasticrestoring forces and are excited by earthquakes. The retrieval of the eigenfrequency,eigenfunction and damping is very useful for the knowledge of the Earth’s internalstructure in density and (an-)elastic parameters (rigidity and compressibility). TheEarth’s rotation and its elliptical shape also act to split the free modes (Dahlenand Sailor, 1979) and precise splitting observations are of primary importance indetermining radial and lateral heterogeneities.

For periods above one hour, we reach the so-called subseismic band where thetheoretically predicted core gravito-inertial modes are located. For these modes, theelastic restoring force is replaced by a combination of inertial (due to the rotation)and gravitational (of Archimedian type) effects. Therefore the detection of coremodes is very important for knowing the density stratification of the liquid coreor, more specifically, the stability of this stratification through the Brunt-VäissäläfrequencyN(r), which is poorly constrained from seismological studies (Kennett,1998). A special case in dynamics isSm1 (m = −1, 0, 1), the Slichter triplet, thatis predominantly a rigid translation of the Earth’s solid inner core. The detectionof this particular mode would strongly constrain the density contrast at the innercore boundary (ICB) which is poorly constrained (Masters and Shearer, 1990). Inprinciple, there is no upper bound in period (e.g., Crossley and Rochester, 1980) forthe core gravity modes but, as we will see later, the gravity noise spectrum makesdetection more and more difficult at periods longer than a day.

The tidal frequency bands range from 6 hours (quar-diurnal tides) to long-periodzonal tides (18.6 year) and include the large amplitude diurnal and semi-diurnalbands. The tidal response is therefore a unique tool for investigating the Earth’selastic transfer function over a wide spectral range from the seismic periods toalmost secular changes. At tidal frequencies, there is in addition to the Earth’s solidtides a contribution coming from the oceanic tide (and also a smaller one from theatmosphere) which perturbs the amplitude and phase of the solid tide. Thereforethe observations of inland gravity changes are a useful tool to indirectly senseocean tides; moreover accurate gravity loading observations are able to reveal thedynamic (linear) response of the oceans as well as non-linear tides. In the diurnalband, some of the tides are perturbed by the resonance effect caused by the FreeCore Nutation (FCN) which is a rotational eigenmode of the coupled solid Earth-fluid core system. The modification in the amplitudes and phases of the diurnaltides can be used to retrieve useful information on the rotational eigenmode itself,namely its period, damping and strength. This information is shown to be veryvaluable for the knowledge of core-mantle coupling and suggests a non-hydrostaticcontribution to the flattening of the core-mantle boundary (CMB). A similar reson-ance process is theoretically predicted when introducing the rotation of a slightlytilted inner core. This second mode, called by analogy the Free Inner Core Nuta-tion (FICN), leads, however, to a much smaller resonant enhancement and will be

4 JACQUES HINDERER AND DAVID CROSSLEY

difficult to detect unambiguously in the future. Nevertheless, its detection wouldbe useful for determining the density contrast, the elasticity and shape of the ICB.

The Earth’s rotation (length-of-day and polar motion) leads also to gravity con-tributions over a wide spectral range from sub-daily to Chandlerian periods. Wewill show that superconducting gravimeters indeed exhibit the long-term signaturedue to polar motion. The atmosphere is also a broadband source of gravity changesby loading the Earth, with both the meteorological continuum and organisedplanetary-scale waves of seasonal and diurnal periods contributing to this loadingprocess. Finally, other environmental contributions to gravity can be important forsome stations and help in understanding hydrogeological problems.

2. Superconducting Gravimeters

Many of the phenomena mentioned in the introduction could not be observedbefore the existence of superconducting gravimeters (SGs) with their low instru-mental drift. The SG is a device that uses magnetic levitation of a superconductingsphere instead of a mechanical spring in the case of a classical gravimeter (see thereview by Goodkind, 1999). The levitation force results from a current trapped ina coil which becomes superconducting at very low temperatures when operatedin liquid helium (evaporation temperature 4.2 K). Prothero and Goodkind (1968,1972) introduced the first prototype and the first commercial instruments weresold in 1980 by GWR Instruments based in San Diego, California to the RoyalObservatory of Belgium (Brussels) and Institut fur Angewandte Geodaesie (Frank-furt, Germany), respectively. They were followed by instruments installed in China(Wuhan) and France (Strasbourg). There are currently about 17 SGs in operationworldwide; these participate in a project called GGP (Global Geodynamics Pro-ject), operating in a network that began recording on July 1, 1997 (see Crossleyand Hinderer, 1995, and Crossley et al., 1999 for reviews). Figure 2 shows thepresent geographical distribution of this SG network.

The SG is a relative gravimeter which needs to be calibrated, i.e., one hasto know the conversion factor between the relative gravity change (in microgal)and the feedback voltage. A precise calibration factor is needed for a geophysicalinterpretation of surface gravity changes; for instance, all the applications inferredfrom the accurate observations of tidal gravimetric amplitude factors (e.g., latitudedependence, mantle anelasticity, heat flow correlation).

There are three principal methods to calibrate an SG:− the gravitational attraction of a large mass, often in a ring geometry (e.g.,

Varga et al., 1995), which has been shown by Achilli et al. (1995), to give a0.3% accuracy;

− the inertial acceleration of a moving platform (Richter, 1995; Richter et al.,1995a) which leads to a claimed high accuracy better than 0.1%;

− the parallel recording with an absolute gravimeter (AG).


Figure 2. GGP (Global Geodynamics Project) network of superconducting gravimeters indicatingthe geographical location of the stations in operation end of 1999 (BO Boulder USA; CA CantleyCanada; BE Brussels Belgium; MB Membach Belgium; ME Metsahovi Finland; ST StrasbourgFrance; WE Wettzell Germany; Je Jena Germany; BR, Brasimone Italy; VI Vienna Austria; WUWuhan China; ES Esashi Japan; MA Matsushiro Japan; KY Kyoto Japan; SY Syowa Japan; BABandung Indonesia; CB Canberra Australia; SU Sutherland South Africa).

In this last method, the accuracy varies according to the type of AG and to thelength of the recording. In a single experiment with a JILAG type instrument theaccuracy is about 1% (Hinderer et al., 1991a) and this improves to 0.3% usingrepeated experiments with the same instrument (Hinderer et al., 1995a; Amalvictet al., 1998). From a one-month record with a FG-5 (Francis, 1996), which is oneof the new generation of absolute gravimeters (Niebauer et al., 1995), the accuracyreaches 0.1%. An example of such a comparative study in Strasbourg in 1991 isgiven in Figure 3 where the circles represent the individual absolute measurementswith a JILAG-5 instrument, taken every 15 sec, and the full line of the (calibrated)gravity changes measured by SG T005. The superposition can also be done us-ing hourly mean values of absolute gravity (Hinderer et al., 1998a) and leads tosimilar results because the smaller number of samples is compensated by smalleruncertainties (in general in

√n wheren is the number of samples to be averaged).

The previous paragraph addressed the problem of the amplitude calibration buta complete calibration of the instrument also means to know the transfer function inphase. In the tidal frequency band, the internal lag of the feedback system of the SGis very low, but lags ranging to more than thirty seconds (according to the type ofelectronics) are due to the low-pass filter applied to the tidal output. To accurately


Figure 3.Calibration experiment by parallel registration of AG (JILAG5) and SG (T005) in Stras-bourg. The circles represent the single values of the absolute gravity observations (roughly every 15sec) and the full line is the (calibrated) output of the superconducting gravimeter.

determine this phase lag for periods ranging from a few seconds to several hours,there is the possibility of injecting a nominal voltage (step function or sinusoid)and of measuring the instrument’s response (see Wenzel, 1995; Van Camp et al.,2000).

It is worth emphasizing here the complementary character of the SG and AG.A schematic gravity power spectrum, Figure 4 (Lambert et al., 1995), shows thatthe drift term of the SG leads to more noise in the low frequency band (decadal tomonthly periods) with respect to the AG (being absolute and therefore in principledrift-free). On the other hand, the AG cannot measure more often than about 10sec (duration of a single drop), so there is aliasing of higher frequencies such as themicroseismic background noise which increases the AG noise level with respect toSG measurements. It is obvious that a combination of the gravity spectra (AG +SG) is therefore optimal in order to cover the whole spectral range; in particular,combined SG/AG measurements are extremely useful in recovering long periodgravity effects as recently shown by van Dam and Francis (1998).

A typical SG record is shown in Figure 5a from the instrument (SG T005)located near Strasbourg, France for 1987–1996. The largest gravity changes are


Figure 4.Compared gravity power spectra from absolute gravimeter (AG) and superconducting gra-vimeter (SG) indicating the complementary character of the two instruments operating at a same site(from Lambert et al., 1995).

of course the luni-solar tides with their multiple amplitude modulation (on thistimescale the most visible one is the semi-annual modulation). Also indicated is theinstrumental drift (assumed to be a decreasing exponential function) superimposedon the tides. For this instrument (and many of the SGs), the drift is quite large atthe beginning of the record but tends to become flat with time. The drift behaviourdepends on the technical specifications of the meter (Warburton and Brinton, 1995)but major instrumental improvements were made recently to some existing SGs andfor all new instruments by the manufacturer. An example is given in Figure 5b forthe same station as in Figure 5a but originating from the new compact SG C026where almost no visible drift is present.

The residual gravity signal, after removal of lunisolar tides, and correction forlocal pressure effects and polar motion contribution, derived from compact SGsis shown in Figure 6a for the Boulder station (SG C024) and in Figure 6b for theStrasbourg station (SG C026). Notice the change in scale with respect to Figure5 indicating small residual changes of only several microgal, which is typical asindicated in previous similar studies (see Hinderer et al., 1994a).

The spectrum of the residual gravity from SG C024 and C026 is shown inFigure 7a and 7b respectively displaying a Brownian noise (proportional tof −1

in amplitude orf −2 in power, wheref is the frequency) behaviour with residual


Figure 5. (a) Temporal changes in gravity as recorded by SG T005 in Strasbourg (France) during1987–1996; the instrumental drift which superimposes onto the tides has a typical shape close to adecreasing exponential function, (b) same but for a record of SG C026 at the same station exhibitinga very small drift component.


Figure 6. (a): Temporal changes in gravity residuals in Boulder (C024) after a least squares adjust-ment of the observed signal to tides, instrumental drift and local atmospheric pressure, (b) same forthe Strasbourg station (C026).


tidal lines (with, in particular, a strong diurnal component). Such a shape is typicalof SG spectra as shown by other studies (Jensen et al., 1995; Smylie et al., 1994;Hinderer et al., 1994a). Also remarkable is the fact that, for frequencies larger than5 or 6 cycles per day, the noise level is below the nanogal level (∼10−12 g, whereg is the mean surface gravity).

3. Seismic and Subseismic Frequency Bands

The seismic band, including the elastic normal modes with periods less than 54min, is well covered by broadband seismometers such as the STS-1 (Wielandt andStreckeisen, 1982; Wielandt and Steim, 1986). However the better relative gravi-meters are also able to retrieve these modes; it is the case for LaCoste-Rombergspring meters (Zürn et al., 1991) as well as for SGs (Kamal and Mansinha, 1992;Banka and Crossley, 1995, 1999; Richter et al., 1995b; Freybourger et al., 1997;Van Camp, 1999).

In fact, when these different types of instruments are operating at the samequiet site, their performances are similar in the sense that the signal to noise ratioof the seismic modes is almost identical as shown by the observations from theBolivian earthquake of June 1994 at the BFO (Black Forest Observatory) station inGermany (Richter et al., 1995b). Of particular importance for the structure of theEarth’s deep interior are the observations of the seismic core modes, where most ofthe energy is in the fluid core or in the solid inner core. Few reliable observationsof seismic core modes exist, but the spectral analysis of the SG at Esashi (Japan)after the Macquarie Ridge earthquake in May 1989 clearly identified core modesand showed the potential of a SG to act as a long-period seismometer (Imanishiet al.,.1992; see also Banka et al., 1998). Data from the IDA (International De-ployment of Accelerometers) network (formerly composed of LaCoste-Romberggravimeters but now also including long-period seismometers) have also been ana-lysed in the seismic band for spheroidal modes having most of their shear energytrapped in the solid inner core (Fukao and Suda, 1989; Suda and Fukao, 1990) withan autoregressive Sompi technique of spectral analysis (Hori et al., 1989).

However, the relative noise level of instruments changes when moving from theseismic band to the so-called subseismic (i.e., for periods above 54 min) band. Arecent study comparing SG T005 to a STS-1/Z (vertical component) has pointedout the increase in noise of the broadband seismometer with respect to the gravi-meter for periods above 50 min (Freybourger et al., 1997). Figure 8a shows thepower spectral densities (PSD) of the seismometer and gravimeter outputs (both inmicrogal2/Hz) for one month common record. Also indicated is the New Low NoiseModel (NLNM) which is a reference model proposed by Peterson (1993) fromthe analysis of a large number of long-period seismometers distributed worldwide.The difference in longer period instrument noise is significant and leads to a ratioof about 10 at a period of 6 h and about 20 at 12 h. Despite the higher noise,


Figure 7. (a) Amplitude spectrum of the residual gravity signal depicted in Figure 6a indicating atypical coloured-type frequency dependence; (b) same for the Strasbourg station (C026).


the seismometer data can however be used to retrieve large tidal components asalready shown by Pillet et al. (1994). A two-year long record of the same STS-1/Zrevealed a clear annual signal for which the most probable cause is a temperaturedependence on this type of seismometer. Although this feature acts everywhereover the whole frequency range of the instrument, it is most noticeable at longperiods because of the colored shape of the temperature spectrum.

Another example of PSD in the seismic and sub-seismic bands is given in Figure8b for the Cantley station in Canada (SG T012). This figure was obtained fromthe analysis of 5 quiet days where the raw gravity data of 1 sec sampling wereprocessed for each day by removing the tides, atmospheric pressure and long periodsignals; the 5 days having the lowest RMS time domain amplitudes were selectedand the spectra relative to each day were averaged. The plot shows that, at periodsbetween 50 and 1000 sec the SG noise is an order of magnitude higher than theNLNM, but drops below the noise model at longer period; the noticeable peak atabout 100 sec is due to the horizontal resonance period of the superconductingsphere.

A significant recent discovery is that the seismic normal modes, which werethought to be a transient process after earthquake excitation, seem to be continu-ously excited at all times. The most probable forcing is the turbulent pressurefluctuations in the atmosphere as shown by recent studies (Kobayashi and Nishida,1998; Lognonne et al., 1998). The first evidence for this phenomenon came froma three year record of the SG operated at Syowa station, East Antarctica (Nawaet al., 1998); it was clear from a time-frequency spectrogram that numerous linesat the nanogal level were coinciding with fundamental spheroidal eigenfrequencieswhich cannot be associated with known seismic events. Very quickly, this discoverywas confirmed by other data sets from seismometer or gravimeter global networkssuch as IRIS (Tanimoto et al., 1998) and IDA (Suda et al., 1998). It is clear that thelow noise level existing in SG record in the 2–7 mHz frequency band will allowthe GGP network to contribute effectively to this research.

The subseismic frequency band is of particular interest in geophysics becausethis is the spectral range where the theory predicts the so-called gravity-inertialmodes in the liquid core, where the restoring force is a combination of grav-ity (Archimedian) and rotation rather than elastic (see e.g., Aldridge and Lumb,1987; Crossley and Rochester, 1980). The frequency of these modes stronglydepends therefore on the square of the local Brunt-Väissälä frequencyN2(r) =−g(r)ρ−1(r)

dρ(r)

dr− g2(r)ρ(r)k−1(r) wherer is the radius,g the gravity,ρ the

density andk the compressibility. WhenN2(r) > 0, the stratification is stable andcore oscillatory modes may exist; whenN2(r) < 0, the core is unstable (convect-ive), and forN2(r) = 0, the core is in a neutral equilibrium state (also known asAdams-Williamson condition).N2(r) is weakly constrained from seismology anddifferent Earth models exhibit differentN2(r) variations around zero in the core(see Masters and Shearer, 1990; Kennett, 1998).


Figure 8. (a) Noise comparison between a superconducting gravimeter (SG T005) in Strasbourg(station J9) and the vertical component of a broadband seismometer (STS-1/Z) at Echery, Vosges (70km away from Strasbourg) in the seismic and subseismic frequency bands (from Freybourger et al.,1997); NLNM (New Low Noise Model) is a reference model proposed by Peterson (1993) from theanalysis of global seismological data; (b) Power spectral density of 5 quiet days at Cantley, Canada(SG T012).


A special case in Earth dynamics is theS11 spheroidal mode that corresponds

to a translation of the Earth’s solid inner core inside the liquid outer core. Thismode is often named in honor of Slichter (1961) who was the first to suggest thismotion. The period ofS1

1 is several hours (for most Earth models) so the Coriolisacceleration term in the equations of motion splits the mode into a triplet of distinctfrequencies.

Considerable theoretical effort has been attempted to establish the theoreticalspectrum of core modes, and many questions still remain on their anticipatedeigenfrequencies. One long standing early debate was on the validity of the so-called subseismic approximation first proposed by Smylie and Rochester (1981)(Crossley, 1993; Crossley et al., 1991; Smylie et al., 1992; Crossley and Rochester,1992; Rochester and Peng, 1973; Wu and Rochester, 1994; Rieutord, 1991).

The excitation of core modes is problematic because near-surface earthquakesare able to excite these modes only to a small level in surface gravity (Crossley,1988; Crossley et al., 1991). Even the large, deep earthquake in Bolivia, June 1994of magnitude 8.2 leads to a change close to 0.02 nanogal for the Slichter mode(Jensen et al., 1994); this is presently unobservable even with high quality SGrecords (Hinderer and Crossley, 1993; Hinderer et al., 1994a).

Also problematic is the damping of core modes and Slichter triplet. In oneEarth model (PREM), seismic anelasticity in the core compressibility has beenshown to be the primary source of attenuation (Crossley et al., 1991) and is muchmore efficient than magnetic (Crossley and Smylie, 1975; Buffett, 1992) or viscous(Friedlander and Siegmann, 1983) effects. Typical decay times are found to belarge of the order of one year while they are of the order of days for the elasticseismic modes. A small attenuation serves to partially offset the weak excitationand there would still be hope to detect these modes from long records if they areexcited in a phase-coherent manner in time. For the Slichter triplet in particular,there is a possible thermodynamic mechanism involving phase changes at the ICBthat might damp during the motion of the inner core; some preliminary studies areinconclusive suggesting very weak damping (Wu and Rochester, 1994).

The first claim for detection of core modes in gravity data was made by Mel-chior and Ducarme (1986), who found a 15 nanogal peak at 13.8 hour in theBrussels SG data. The existence of this peak was not confirmed by other instru-ments despite several attempts (Zürn et al., 1987; Mansinha et al., 1990; Cumminset al., 1991; Florsch et al., 1991). Rather than a core mode, this signal was finallyidentified to be of external origin from a regional atmosphere-ocean interaction thatwas localised in Brussels but not elsewhere (Dehant et al., 1993a). The Slichtertriplet has been the target of several observational campaigns for more than twodecades (Jackson and Slichter, 1974; Rydelek and Knopoff, 1984) but without anyclear success.

This situation changed with the claim by Smylie (1992) that he had detected theSlichter triplet in a stack of 4 SG records from Europe. This claim was based on theclose coincidence of the three observed eigenfrequencies with his own theoretical


predictions (Smylie et al., 1992). Observationally the signal to noise ratios of thethree peaks was very weak, although Smylie et al. (1994) ruled out a possible con-tamination from the atmosphere through an analysis of the pressure records. Onceagain, this detection (which would have important consequences for the densitycontrast at the ICB as well as for the viscosity properties at the same interface)could not be confirmed by other studies that either re-examined the same data setsbut with different analysis techniques, or that used other SG data sets (Jensen etal., 1995; Hinderer et al., 1995b). In a stack involving two years of simultaneousdata from the SGs in Strasbourg (France) and Cantley (Canada), none of the peaksof the triplet could be distinguished from noise despite the fact that the noise levelwas almost a factor 10 lower than in the stack of Smylie (1992). One possibledefense of the claim is that the observing periods differ in the two stacks and thatthe Slichter triplet might have been excited in a fortuitous way in the Smylie (1992)stack. However, new arguments for the presence of the same triplet in additionalSG records were given just recently from a multi-station experiment by Courtier etal. (2000).

More generally, the problem of the identification of weak harmonic signals ina random noise has been treated from a statistical point of view by Florsch et al.(1995a). The probability law for a spectral line in the presence of Gaussian whitenoise is called the Rice-Nakagami distribution and is shown on Figure 9. For largevalues of SNR, the law tends to become Gaussian and is well retrieving the correctwave amplitude (set to unity; however, for small values of SNR, the law is thewell-known Rayleigh law with a variance 2σ 2 and one can see that, for increasingvalues ofσ (increasing noise content with respect to the fixed wave amplitude) thelocation of the maximum of the distribution shifts to the right meaning a largeramplitude estimate and causing hence a bias in spectral analysis.

As a consequence, a typical gravity spectrum will exhibit several peaks aboveeven a highly (statistically) significant level (such as 4 standard deviations) as in-dicated on Figure 10. Of course, the stacking possibilities which are now possiblewith the GGP network of SGs will provide an extremely powerful tool to discrim-inate planetary-scale organised signals from local random noise, especially if oneis able to assume the spherical harmonic structure of the surface gravity (Cumminset al., 1991).

4. Tidal Frequency Bands

As can be seen in Figure 1, the main lunisolar tides range in frequency from4 cycle/day to 1 cycle/year, although there exist even lower frequencies such as(1/18.6) cycle/year for the Bradley term related to the retrogradation of the lunarnode on the ecliptic (Melchior, 1983; Warburton and Goodkind, 1977). From thesubseismic band to almost secular motions, the tides therefore provide a uniquesource of information to test the Earth’s rheology over a wide frequency range.


Figure 9.Rice-Nakagami distribution expressing the probability law of a spectral line in the presenceof noise. The wave amplitude is set to unity andσ is the parameter of the Rayleigh law describingthe random noise (from Florsch et al., 1995a). The horizontal axis represents the wave amplitude andindicates the shift to the right (larger amplitude) according to increasing values of the parameterσ .

Figure 10.Gravity spectrum of the French SG T005 between 1 and 5 hour period. The threshold ashigh as 4σ (standard deviation) is exceeded by several spectral lines. Apart the harmonics of the solarday S1 the number of random noise excursions is in fair agreement with the statistical prediction.Note that the vertical scale is in nanogal (10−3 microgal) (from Florsch et al., 1995a).


The gravity tidal response of the Earth1gn(a) (at the surfacer = a) to anexternal potentialVn is expressed by a gravimetric amplitude factor and phase delayfor each degreen, called respectivelyδn andκn:

−an1̃gn(a) = δ̃nVn = δn exp(iκn)Vn (1)

where:

δ̃n = 1+ 2

nh̃n − n+ 1

nk̃n

andh̃n, k̃n are (complex-valued) volume Love numbers of degreen.For an elastic, spherical, non-rotating Earth’s model. the gravimetric factors are

independent of the azimuthal numberm of the potential; the Earth’s ellipsoidalshape and rotation act to couple terms of different ordersm (Wahr, 1981) that leadsto a small latitude dependence ofδn (Wang, 1994).

For a pure elastic model, the phase lagκn is zero for alln and δn is close to1.16 (Dehant and Ducarme, 1987) meaning that elasticity increases by 16% thegravity change with respect to a rigid Earth (for degree two). When anelasticity istaken into account, a phase lag appears which is usually very small (less than 0.01degree) for rheological models extrapolated from the seismic bands (Dehant andZschau, 1989), and there is also a small increase in the amplitude factors them-selves (Wahr and Bergen, 1986). The effects are frequency-dependent and becomelarge for increasing periods; this is why it is important to measure accurately thelong-period zonal tides that were previously almost hidden in the noise of classicalspring gravimeter records because of the large mechanical drift.

The problem of inferring anelastic properties from the solid Earth tides ob-viously involves ocean tidal loading corrections (and to a much smaller extentatmospheric loading contributions), all of them acting at precisely the same fre-quencies (see e.g., Baker et al., 1991). A further perturbation of the gravimetricfactors should result from lateral heterogeneities in the mantle which further com-plicate the coupling structure. The question of identifying large scale effects of thistype in the gravimetric amplitude factors is still debated (e.g., Rydelek et al., 1991;Melchior, 1995).

4.1. QUAR-DIURNAL TIDES

Because the lunisolar tidal potential of degreen is inversely proportional todn+1,whered is the distance between the Moon or Sun and the Earth, the major contribu-tion to the solid Earth gravity tides is from the degreen = 2, but degree 3 and even4 cannot now be neglected. For instance, a previous CTE potential (Cartwright andTayler, 1971; Cartwright and Edden, 1973) used 505 waves; a second generation oftides improved the CTE development by increasing the number of waves to 1200or more (Tamura, 1987; Xi Quiwen, 1987, 1989; Merriam, 1992a). More recently,


a further improvement in theoretical tidal developments required a substantial in-crease of the number of individual waves to 6499 waves (Rossbeek, 1996) and12935 waves (Hartmann and Wenzel, 1995).

Additional terms coming from degree 3 are much smaller than the classical tidesof degree 2 but they are readily observed in good quality records. Furthermore, arecent study (Meichior et al., 1996) using 12 years of the Brussels SG was able toseparate components of degree 3 from degree 2 in the lunar potential differing infrequency by the lunar perigee period of 8.8 year (see also Hinderer et al., 1998b).

Besides the separation of terms due to the effect of the variation of the lunarperigee, it has been also possible to separate terms depending on the lunar node(18.6 years). The most interesting ones are two companions of waveK1 which areresponsible for the 18.6 year nutation. The differential liquid core resonance effect(see section 4.3) betweenK1 and its companions is visible and in agreement withtheoretical models (Ducarme and Melchior, 1998).

As we have seen in the previous section, the instrumental sensitivity needed todetect core modes is at or below the nanogal which is in principle a level the SGsshould reach (Warburton and Brinton, 1995). There are to our knowledge at leasttwo confirmations of the ability of the SG to reach such an accuracy. One was givenby Merriam (1992b) from the observation of gravity changes on the Canadian SGinduced by small microbarometric pressure changes in the atmosphere. The secondcomes from the observation of the degree 4, order 4 tides at a period close to 6hours in the French and Canadian data sets (Florsch et al., 1995b). Figure 11 showsthat the four largest theoretically predicted tides of about a nanogal amplitude (M4

is the largest with 3.8 nanogal) are indeed visible in the observed record takenfrom the Canadian SG at Cantley even though the signal to noise ratio is marginalfor the two smaller tides. These quar-diurnal waves are less well observed in theStrasbourg data (see Florsch et al., 1995b) because of a larger noise content, buta cross spectral analysis between the two records shows clear evidence for theexistence of these tides. However further work is required to dismiss non-linearoceanic contributions to these solid Earth tides (see also Merriam, 1995).

4.2. SEMI-DIURNAL TIDES

The semi-diurnal frequency band reveals features of the oceanic behaviour whichare not directly observable by other techniques. As pointed out recently by Merriam(1995), with the help of the response function method used in oceanography, airaccurate SG record in this band is able to exhibit the dynamic linear response of theocean with frequency, as well as non-linear tides in shallow seas. Figure 12 showsthe gravity load tide admittances (in amplitude and phase) as inferred from theCanadian SG. The slight dependence (defined by the dashed line joining the welldefined tides with small uncertainty) is known to be the signature of the dynamicocean response (see Platzman, 1981) and some of the points located outside thisline could be identified as non-linear tides in shallow seas (Bay of Fundy) by a


Figure 11.Identification of quar-diurnal tides (degree 4, order 4) of a few nanogal amplitude in SGdata indicating the high sensitivity of the cryogenic instruments.


Figure 12.Semi-diurnal ocean tide gravity effects at Cantley (Canada) (from Merriam, 1995). Thedotted line which can be drawn between some reference tides shows a smooth admittance function.The anomalous tides around 29 degrees/hour could be identified as anomalous ocean tides by tidalgauge measurements in the Bay of Fundy.

comparison with tidal gauge data. Because these tides are very small in amplitude,SGs therefore demonstrate the ability to sense ocean tides (indirectly as a result ofconvolving the tides with the gravity Green’s function) to a level of a few mm; thisis not reached by either tide gauges or even satellite altimetry.

4.3. DIURNAL TIDES AND CORE RESONANCES

The diurnal frequency band is also of primary importance for investigating oceanloading effects and a dynamic response can be found similar to that described abovefor the semi-diurnal band (Merriam, 1995). But the main interest is due to thepresence of a resonance in the solid Earth tides. It is well known that there is acoupling between the liquid core and the solid mantle caused by pressure effectsacting on the core-mantle boundary (CMB) (Hide, 1969; Hinderer et al., 1990)and the rotation of the coupled core-mantle system exhibits in addition to the freeChandlerian motion of 435 day period (in a rotating reference frame linked to the


Earth) a rotational mode of nearly diurnal period called NDFW (Nearly diurnalfree wobble) or FCN (Free core nutation) (e.g., Toomre, 1974); notice that thislater mode is nearly diurnal in a rotating Earth frame but is long period in space.

The lunisolar tides of diurnal period have a degree 2 order 1 spatial distributionwhich is able to excite this eigenmode. The resulting change in gravity caused bythe potentialV2 (at the surfacer = a) is then for every frequencyσ :

1̃g(σ, a) = −[δ̃2+ S̃

σ − σ̃f cn

]2V2

a(2)

where S̃ is the strength term having the dimension (time)−1 and depending onseveral geodynamical parameters, andσ̃f cn the FCN eigenfrequency:

σ̃f cn = −�[

1+ A(αc − β̃)Am

+K − iK ′]

(3)

where AAm

is the ratio of the Earth to mantle equatorial moments of inertia and�

the Earth’s rotational frequency. The pressure coupling is responsible for the termin core flatteningαc (Wahr and De Vries, 1989). The elastic deformation underthe CMB pressure conditions leads to the compliance termβ̃ which is complex ifanelasticity is taken into account (Wahr and Bergen, 1986);K andK ′ stand forpartly dissipative visco-magnetic coupling torques between the core and mantle(Sasao et al., 1980).

For tidal waves of frequencyσ close toσfcn, there is a resonance in the tidalgravimetric factors which has been clearly observed in precise gravity records formany years (e.g., Melchior, 1983). The high quality of tidal measurements avail-able with SGs improves the constraints on the period and damping of the NDFW(Neuberg et al., 1987; Defraigne et al., 1994, 1995) and on the degree 2 pressuregravimetric factor which is present in the expression of the strengthS̃ defined inEquation (2) (Hinderer et al., 1991c). The connection between tides and the Earth’snutations in space (see e.g., Melchior and Georis, 1968) leads to an identical res-onance due to the FCN in the lunisolar nutations which is indeed well observed(e.g., Gwinn et al., 1986; Herring et al., 1986). Moreover the diurnal resonanceeffect perturbs the tidally-induced elastic response of the Earth not only in gravitybut in all possible deformational aspects (radial and transverse displacement, tilt,strain). Frequency-dependent Love and Shida numbers retrieved from VLBI datagive a FCN resonance period of about 426 sidereal days (Haas and Schuh, 1996).An example of the FCN resonance in gravimetry is given by Figure 13 from atwo-year record of SG T005 in Strasbourg. One can see that far away from theresonance, the resonant load is zero and that the resonance process modifies theamplitude and phase of the wavesψ1 φ1, K1 andP1 according to their proximityto the FCN eigenfrequency.


Figure 13.Free core nutation resonance observed in tidal gravimetry. The full line in the real andimaginary resonant loads is the fitted model to the gravity data (2 year of the French SG T005).

Apart the resonant modification of the tides and associated nutations, anotherpossibility would be to observe the eigenmode directly and to infer its eigenperiodand damping. The detection in VLBI of the motion in space of the freely excitedFCN has been claimed recently by Herring et al. (1991) with an amplitude of lessthan a milliarcsec and again an eigenperiod close to 430 days (see also Jiang andSmylie, 1995). The gravity change induced by the rotational potential (Hindererand Legros, 1989) associated with the corresponding wobble in a geographic framewould be negligibly small of the order of 0.01 nanogal.


A summary of values for the FCN eigenperiod and quality factor retrieved fromSG data is given in Table 1; we have also added some theoretically predicted values(for Earth’s models in the hydrostatic equilibrium) as well as the result from VLBI(Very Long Baseline Interferometry) inferred from the nutations and tidal displace-ments. Also indicated is the result from a stack involving the IDA (InternationalDeployment of Accelerometers) network.

The principal result is that there is a significant discrepancy between the ob-served eigenperiod and the theory and all measurements converge in this respect.This has been attributed to a non-hydrostatic contribution to the flattening of theCMB (Neuberg et al., 1990; Wahr and De Vries, 1989) and the current opinionis that this excess flattening may be due to the dynamics of the overlying mantle(Hager et al., 1985; Forte et al., 1994). It is worth noting that the CMB deformationgenerated by mantle heterogeneities indeed leads to an increase in ellipticity of theright order of magnitude (Greff and Legros, 1996; Defraigne et al., 1996); on thecontrary, ellipticity values inferred from seismological studies (e.g., Morelli andDziewonski, 1987) do not.

Another important result is that the damping seen by gravimetric tides is sys-tematically larger than the one from VLBI (roughly by a factor 10). There is nodefinitive argument to explain the discrepancy inQ but different ocean loadingcorrections in the two methods might be partly responsible (see Merriam, 1994)for the Cantley data; see also Florsch and Hinderer, 1998). Notice that the dampingseen by the SG would imply a large ohmic dissipation in a conducting layer at thebase of the mantle in terms of magnetic coupling (Buffett, 1992) or viscosity values106 times larger than those suggested from laboratory experiments simulating theCMB thermodynamic conditions (for a review of core viscosity estimates see Lumband Aldridge, 1991) in terms of viscous coupling.

Other contributions to the FCN eigenfrequency, from either the presence of airoceanic or atmospheric surface layer (Legros and Amalvict, 1989) or from that ofthe solid inner core (see e.g., Legros et al., 1993), are found to be small.

There are several models to express the complex gravimetric factor in the FCNresonance. One is the near-resonant approximation (Zürn et al., 1986; Florsch etal., 1994) given by Equation (2) which uses in addition to the period, damping andstrength, the motion of a reference gravimetric (non-resonant) factor. A secondmodel is the damped harmonic oscillator which physically simulates the resonanceprocess (Neuberg et al., 1987). Both models are shown to be equivalent in the caseof the FCN resonance.

From the numerical point of view, there are several methods which have beenintroduced to retrieve the FCN parameter values from the resonant loads. The firstis a linearized least squares estimation based on the Marquardt (1963) algorithm;this slightly overdetermined system is used with 8 observables (4 tidal waves eachproviding a complex gravimetric factor) to determine 4 parameters (the real andimaginary eigenfrequencies, and real and imaginary strengths) (see Neuberg et al.,1987; Zürn et al., 1986; Richter and Zürn, 1988; Defraigne et al., 1994, 1995).


TABLE I

Summary of FCN eigenperiodT and quality factorQ. The three first resultsare theoretical predictions related to elastic (Wahr, 1981; Sasao et al., 1980)or inelastic (Wahr and Bergen, 1986) Earth’s models. The next results involveVLBI observations of lunisolar nutations (Herring et al., 1986), of the FCNeigenmode itself (Jiang and Smylie, 1995) or of tidal displacement (Haas andSchuh, 1996). We have also reported the results from a stack involving grav-ity data from the IDA network (Cummins and Wahr, 1983). Finally, differentresults dealing with SG data are given (B: Brussel; BH: Bad Homburg; JAP: 3Japanese SG; J9: Strasbourg; CSGI: Cantley)

FCN period and quality factor

Author Method T Q

Wahr Theory 466.9 ∞1981 elastic

Sasao et al. Theory 465 ∞1980 elastic

Wahr and Bergen Theory 473.8 78000

1986

Herring et al. VLBI 434.6± 0.6 22000–100000

1986 nutations

Jiang and Smylie VLBI 431± 6 1900–2100

1995 FCN

Haas and Schuh VLBI 426 1± 20

1996 tides

Cummins and Wahr IDA gravity 428± 12 3300–37000

1993 stack

Neuberg et al. Stacked gravity 431.0± 6.0 2800± 500

1987 B + BH SG

Sato et al. Stacked gravity 436.7± 15.0 3200–∞1994 JAP SG

Defraigne et al. Stacked gravity 425± 6 2400–8300

1994 B + BH + J9 SG

Florsch et al. J9 SG 430.7± 1.0 1700–2500

1994

Merriam CSGI SG 430± 6 5500–10000

1994

Hinderer et al. Stacked gravity 429± 8 7700–∞1995c J9 and CSGI SG


Being linearised this method searches for the mean value of the parameters andrequires starting values to begin with in the search.

If one ignores one of the waves, Florsch et al. (1994) showed that an analyticalsolution can be found in the case where there is exactly the same number of un-knowns (eigenmode, strength and reference gravimetric factor) as data (3 tides);however the solution fluctuates according to the chosen triplet of tidal waves.

Hinderer et al. (1994b) investigated an approach based on the generalized non-linear inverse method by Tarantola and Valette (1982) allowing the introductionof a priori information on the parameters to be determined. This method searchesfor the maximum probability values in the parameter space. Finally, a Bayesianapproach was performed (Florsch et al., 1995c; Florsch and Hinderer,.1998, 2000)which allows a full description of the a posteriori distribution of the solution thanksto a systematic exploration of the parameter space. In this case, the solution of theinverse problem is given by the histograms of the parameters themselves ratherthan a specific value with its uncertainty. An important point which comes outnicely from computing the joint distribution between parameters is the estimateof all possible cross-couplings existing between different parameters in the in-version; this correlation possibility was already pointed out in previous studies(Cummins and Wahr, 1993; Defraigne et al., 1994). For instance, the real partsin the numerator are strongly correlated to the real parts in the denominator (andsimilarly for the imaginary parts) while there is almost no correlation between realand imaginary parts; one should not omit this fact when interpreting the results interms of geophysics. Finally, last but not least, the processing of SG gravity data(from raw data to the resonant loads) has a non-negligible impact on the retrievalof the FCN parameters (Hinderer et al., 1995c).

Besides the FCN, there is another free mode predicted by the theory of therotation of a (slightly tilted) solid inner core coupled to the outer core and mantle.This mode, called Free Inner Core Nutation (FICN), was computed independentlyby De Vries and Wahr (1991), Mathews et al. (1991a,b) and Dehant et al. (1993b).They assumed gravitational and inertial (pressure) coupling torques between theinner core and the rest of the Earth, as well as contributions to the inertia tensorsdue to the elastogravitational deformation. The mode is found to be nearly diurnalprograde and lies again in the diurnal tidal band (in the vicinity of the progradeannual waveS1). Very similarly to the FCN, the tidal gravimetric factor is modifiedto:

1̃g(σ, a) = −[δ̃2+ B̃

σ − σ̃f icn

]2V2

a(4)

whereσ̃f icn is the FICN eigenfrequency and̃B another constant term of dimension(time)−1; the strengthB̃ involves a degree 2 gravimetric factor relative to boundaryconditions at the ICB. As shown by Figure 14 from Legros et al. (1993), the largesteffect concerns the prograde annual waveS1 and is found to be a few parts in 104


Figure 14. Theoretically predicted double resonance in gravimetry (after Legros et al., 1993). Inaddition to the well known FCN resonance, there is a new rotational eigenmode called FICN (FreeInner Core Nutation) which appears in the diurnal frequency band in the vicinity of the solar waveS1.

(De Vries and Wahr, 1991). The strength weakness of this second resonance and theuncertainty related to the atmospheric forcing at theS1 frequency (e.g., Warburtonand Goodkind, 1978) makes any gravimetric observation difficult to interpret interms of inner core properties. However its detection would bring useful informa-tion on the ICB flattening and density jump (still controversial from seismologicalstudies), as well as on elastic properties of the solid inner core.

4.4. LONG-PERIOD TIDES

The small instrumental drift of the superconducting gravimeters permits us to ex-plore the long-period part of the gravity spectrum shown on Figure 1. Zonal tidalsignals can hence be investigated for periods ranging from weeks to years (seee.g., Richter, 1990; Sato et al., 1997a; Dittfeld, 1998). An example of long-periodgravimetric amplitude factors is given in Figure 15 from Hinderer et al. (1998b).No record is yet long enough to retrieve the Bradley term of 18.66 year directly butindirect observations showing the 18.6 year modulation already exist (Rydelek etal., 1982). The annual term is also very difficult to interpret at the moment becausethere are many contributions at this frequency (Hinderer and Legros, 1991): theSa


Figure 15.Long-period gravimetric amplitude factors as derived from a tidal analysis of a 3000 dayrecord of SG T005 in Strasbourg (after Hinderer et al., 1998b).

tide of gravitational origin, the annual term in the polar motion, the annual termin the global atmospheric pressure field (Dong et al., 1996) in addition to otheryearly signals which can be site-dependent (e.g., tilt of the building, temperature).Therefore there is a mixing in the annual gravity signal (see Schwahn, 1998) which,in general, leads to strongly abnormal gravimetric amplitude and phase factorswhen expressed in terms of the tideSa alone. It is clear that an accurate estimateof zonal gravity tides would provide meaningful constraints on the Earth’s transferfunctions (mantle anelasticity) as well as on the long-period non-equilibrium tidesin the oceans. Such a study would then be complementary to the observation ofzonal tides in the length-of-day variations (Chao et al., 1995).

5. Oceanic Contribution

As already mentioned in section 4.2, there is always at every tidal frequency acontribution to the solid tide; this can be more or less important depending onthe location of the station (roughly speaking on the distance to the closest shore).The loading tide can reach several percent of the solid tide at some places but is


in general small in Europe for the diurnal component (e.g., 0.5% for waveO1 inStrasbourg). New ocean tide models became recently available from the analysisof the TOPEX-POSEIDON satellite altimetric data (Andersen et al., 1995). Thesenew maps have led to an increased effort to assess them in terms of loading effecton inland measured gravity (Francis and Melchior, 1996; Llubes and Mazzega,1996; Agnew, 1995; Goodkind, 1996). Even if some models (including the hy-drodynamical model by Le Provost et al., 1994) show better agreement betweenobserved and modeled gravity loads, no definitive conclusion should be drawn.Schwiderski’s (1980) models still serve as the standard in tidal research. It hasbeen suggested that new stations should be established along coastal regions (forinstance along the Atlantic coast in Western Europe) to better validate the oceantide models.

Another new aspect of gravity research concerns the non-tidal oceanic contri-bution. Indeed, altimeter data can also be used to evaluate the gravity loading dueto sea level changes over a wide spectrum as shown by Fukuda and Sato (1997).These authors have computed the gravity effects induced by sea level changes asderived from ERS-1 and TOPEX-POSEIDON altimeter data at the 18 stations ofthe GGP network. This effect reaches several microgal for some stations, with astrong annual component. It is therefore suggested that high-precision long termSG observations which are sensitive to mass changes will be potentially useful inthe crucial problem in altimetry of the thermal expansion of seawater.

6. Atmospheric Contribution

It is known that the second major cause of gravity changes after tides is atmo-spheric pressure, which acts everywhere and on a wide spectral range. In mostgravity studies, the local atmospheric pressure contributions are removed using abarometric admittance close to−0.3 microgal/millibar (Warburton and Goodkind,1978). Such a single factor has a flat response in the frequency domain and ithas been shown that the introduction of a frequency-dependent admittance (or aconvolution filter in the time domain) is better able to account for local effects(see Crossley et al., 1995). However this procedure is only valid for local effectsand is clearly inadequate to model regional or global atmospheric loads (see e.g.,Merriam, 1992b; Mukai et al., 1995; Boy et al., 1998a,b; Kroner and Jentzsch,1999).

Figure 16 derived from Boy et al. (1998a) shows the difference in the gravityload from atmospheric pressure in Strasbourg computed either from the local pres-sure alone (using the admittance approach with a factor−0.3 microgal/millibar)or from a modeling of the global pressure field (1.125 degree by 1.125 degree,1 value/6 hour) of ECMWF (European Center for Medium Range Weather Fore-cast) assuming a static response of the oceans to air pressure changes (Gegout andCazenave, 1993; Gegout et al., 1998) rather than a classical IB (inverted barometer)


response (see e.g., Dickman and 1988). One can notice that most of the time thelarge amplitude changes can be identified on both gravity records, but there arealso discrepancies which are pointed out in a 100 day zoom in the lower part of thegraph. The applicability of one or the other approach to correct gravity tidal signalsis still under investigation; a first result shows that the global approach decreasesthe discrepancy between observations and theory for the long-periodic tides (Boyet al., 1998a) while the influence on diurnal tides is in general small.

There are two major cases where a global approach is needed: the annualSa andthe diurnalS1 pressure waves (and their harmonics) driven by solar heating andknown to be of planetary extent (Haurwitz and Cowley, 1973; Volland, 1988; Hsuand Hoskins, 1989). In this case, the classical correction obviously fails. The atmo-spheric loading strongly perturbs theSa andS1 gravimetric factors of gravitationalorigin (see Legros and Hinderer, 1991) leading to abnormal values in amplitudeand phase. Moreover, because of seasonal amplitude modulation, other waves likeP1, K1, ψ1 andπ1 are also perturbed (see Legros et al., 1993; Dehant et al., 1996)and could possibly alter the FCN retrieval from gravimetric data. However theimprovement due to a global modeling of the gravity effects associated with solarwave S1 is disappointing most probably because of the deficiency of the staticapproximation of the ocean response in the diurnal frequency band (Boy et al.,1998a).

In addition to these harmonic signals, there are other atmospheric processesof large spatial extension with a specific gravity signature. A very interestingphenomenon is the so-called 40–50 day oscillation which has been identified innumerous geophysical signals involving the Earth-atmosphere system (Earth’s ro-tation, atmospheric angular momentum) (Langley et al., 1981; Eubanks et al., 1985,1988; Marcus et al., 1994). There is a coupling mechanism between the atmosphereand the solid mantle caused by the large scale pressure field (and associated velo-city field). It is therefore reasonable to expect a loading contribution in gravity fromthis pressure field and there is some evidence of such a signal in SG records.

7. Earth’s Rotation Effects

The gravity effect induced by changes in the Earth’s rotation (either in wobble or inaxial spin) derives from a centrifugal potential. The dominant spherical harmoniccontribution can be restricted to the degree two (e.g., Hinderer et al., 1982); thegravity change resulting from a polar motion of componentsX(t) andY (t), in theconventional terrestrial frame, at a station of colatitudeθ and longitudeλ, is (e.g.,Wahr, 1985; Hinderer et al., 1991b):

1g(θ, λ, t) = δ2�2a sin 2θ[X(t) cosλ− Y (t) sinλ] (5)

This contribution is a maximum at mid-latitudes (θ = 45◦) and can reach about3 µgal for a polar motion amplitude of|X + iY | ≈ 7.10−7, as it is the case for


Figure 16.Atmospheric gravity loading showing the differences between a local correction (baro-metric admittance approach) and a global modeling from ECMWF surface pressure data assuming astatic ocean response to air pressure changes; the lower graph is a zoom of these two approaches ona 100 day time span (after Boy et al., 1998a).


the mean value of the Chandlerian component of 435 sidereal day period (Lam-beck, 1980). There is also an annual term in the polar motion which causes agravity fluctuation of similar amplitude leading to a beating phenomenon of thetwo components. The gravity changes caused by variations in the axial rotationrate (length-of-day changes) are much smaller than those due to polar motion.

Long-period gravity changes of rotational origin have been indeed observedwith superconducting gravimeters (Richter and Zürn, 1988) because of the lowinstrumental drift of these instruments with respect to spring gravimeters. Thereare nowadays many examples in the literature where a clear identification of thepolar motion signature is shown (Richter et al., 1995b; Sato et al., 1997b; vanDam and Francis, 1998; Neumeyer and Dittfeld, 1997; Harnish et al., 1998). Ina very recent study (Loyer et al., 1999), the authors were even able to retrieve agravimetric amplitude factor of 1.2± 0.1 for the Chandlerian component thanks tothe length of the record (3000 days of SG T005 in Strasbourg) which fully allowsthe separation of the annual component from the Chandlerian one; however theuncertainty is still too high to be exploited in terms of knowledge of anelasticproperties of the Earth at long periods. There is a great hope to use a stack ofSG records from the GGP network to strongly increase the determination of thelong-period rotational gravimetric factorδ2.

Figure 17a (from Richter et al., 1995b) shows the polar motion effect on gravityas observed by a SG at the Richmond station in Florida, USA. One easily seesthat the long term fluctuation is in close agreement to the theoretically predictedchange; there is of course also a higher frequency content in the observed gravitythat might be caused by environmental effects. Another example of polar motionsignature is given in Figure 17b for the Boulder station (C026), where the residualgravity with instrumental drift removed superimposes on the predicted signal fromthe IERS polar motion. Both data sets have zero mean, but otherwise are not fittedto each other in any way.

In addition to long-term wobble changes, there are also rapid motions withtimescales between several weeks and several months (Eubanks et al., 1988) whichwould lead to gravity changes (with similar period) smaller than aµgal and moredifficult to identify. Finally, let us mention that extensive VLBI campaigns allow tomonitor the Earth’s rotation at a much higher sampling and subdaily changes havebeen detected (Freedman et al., 1994; Herring and Dong, 1994). The correspondingeffects in gravity are also currently investigated (Aldridge and Cannon, 1993).

8. Environmental Effects

Many other physical phenomena can contribute to the temporal evolution of theEarth’s gravity. A major environmental effect is the influence of subsurface waterstorage either in a saturated (ground water) or unsaturated (soil moisture) context.In some stations, the gravity changes can easily reach several microgals depending


Figure 17a.Polar motion effect in gravity as observed in Boulder, Colorado (C024).

on the location of the station and the characteristics of the aquifer(s) (Lambertand Beaumont, 1977; Mäkinen and Tattari, 1990; Delcourt-Honorez, 1995; Peteret al., 1995; Goodkind, 1986). In fact, usually, there are several components oftenrelated each other, which are entering the hydrological budget: effective cumulativeprecipitation (rainfall, snow melt), soil moisture and water table level; the complex-ity of the system sometimes only allows a moderately successful modeling of theobserved gravity changes. An example of a correlation between rainfall events andgravity residuals is shown on Figure 18 for the Boulder station, Colorado. In Figure18a, the observed rainfallr is converted to heighth of the (pseudo) groundwatertable using a hydrological model involving two time constants described in Figure18b, and finally the surface gravity effectg is estimated using a Bouguer slab effect(Crossley and Su Xu, 1998, Crossley et al., 1998; see also van Dam and Francis,1998).

Bower and Courtier (1998) could also provide a hydrological model incorpor-ating rainfall, snowmelt, evapotranspiration which explains approximately 90%of residual variance in gravity observed by SG T012 at Cantley (Canada) during1990–1993. Another clear correlation between gravity changes and ground waterlevel fluctuations was shown by Mukai (1997) for the Kyoto station in Japan wheretwo SG were run in parallel from 1992 to 1996, as well as by Richter et al. (1995b)for the Miami station, Florida.


Figure 17b.Polar motion effect in gravity as observed in Richmond, Florida, USA (from Richter etal., 1995b).

9. Conclusions

High quality observations of surface gravity changes with superconducting gravi-meters are shown to be able to provide a better knowledge of the Earth’s globaldynamics as well as its internal structure. In the high frequency band (seismicband), for periods less than one hour, the SGs have comparable performances tospring gravimeters or broadband seismometers to retrieve the elastic normal modes.In the subseismic band, for periods larger than one hour and less than the tidalperiods, SGs are superior to long-period seismometers to detect the gravity-inertialcore modes. However there is presently no clear identification of such modes in SGrecords, one of the problems in the search being the uncertainty in their theoreticaldescription (eigenfrequencies, excitation, damping). A special case of core modesis the degree one order one Slichter triplet related to the translation of the solidinner core. Despite the effort done in analysing several SG records, no confirmationof an initial claim for detection from a stack of four records in Europe could befound. The investigation of the tidal bands provides useful information on the (an-)elastic transfer function of the Earth with the help of tidal gravimetric amplitude


Figure 18a.Correlation between gravity, ground water and rainfall at Boulder, Colorado (C024)(from Crossley and Su Xu, 1998).


Figure 18b.Conversion of rainfallr to heighth of pseudo groundwater table using two time constantsτ1 andτ2; estimated gravity changeg using a Bouguer slab effect.

and phase factors. We show that quar-diurnal tides could be identified in highquality SG records clearly proving the ability of the cryogenic instruments to re-trieve extremely weak amplitude signals (at the nanogal level). In the semi-diurnalband, accurate gravity measurements are sensitive to small loading contributionsgenerated by the linear dynamic ocean response as well as by non-linear tides;these measurements are therefore valuable to sense ocean height changes of verysmall amplitude which are presently not observable by tidal gauges or even satel-lite altimetry. The diurnal tidal gravimetric factors are perturbed by the resonancecaused by the FCN (rotation of the liquid core) and its observation allows to retrieveseveral parameters directly related to properties of the core-mantle boundary. Animportant consequence is the need for a slight increase in the flattening of the CMBfrom accurate observations of the FCN eigenperiod. A second resonance (FICN)predicted by the theory of the rotation of a tilted inner core is yet unobserved anda lot of efforts will be required because of the much weaker strength of this modeas compared to the FCN. Moreover perturbations of atmospheric origin on thetides in the vicinity of the eigenmode further complicate the FICN detection. Inthe long-period band previously hidden in standard gravity records before the useof the cryogenic instruments (because of strong instrumental drift effects), there


are broadband signals due to polar motion, oceanic and atmospheric contributionsof large spatial extension, and hydrogeological effects. The observation of thesesignatures in gravity helps to understand the rheological behaviour of the Earth atlong periods (mantle anelasticity) as well as the interactions of the oceanic andatmospheric layers with the solid Earth (loading effects), especially at the yearlyand diurnal periods in the atmosphere. In most of the problems mentioned above,the separation between planetary scale phenomena of geophysical interest and localnoise contributions of random character is always a major problem. The GlobalGeodynamics Project (GGP) which started in the summer of 1997 a common six-year observing period with a network of 17 SG stations located worldwide (Europe,Japan, United States, Canada, China, Indonesia, Australia, Antarctica) should be akey to solve this problem.

Acknowledgements

J.H. would like to thank all the members of the Gravimetry team in Strasbourg forall their effort in maintaining the SG station since 1987. This study was supportedby a CNRS-INSU grant IT (Intérieur de la Terre). Both J.H. and D.C. acknow-ledge support from a NSF–CNRS cooperative grant which helped in finalizingthis review paper. We also thank two anonymous referees for their remarks andsuggestions which helped in improving this paper.

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Time Variations In Gravity And Inferences On The Earth's Structure And Dynamics

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