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A global experimental model for gravity tides of the Earth

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Journal of Geodynamics 38 (2004) 293–306
A global experimental model for gravity tides of the Earth
Jianqiao Xu
a
,
∗
, Heping Sun
a
, Bernard Ducarme
b
a
Key Laboratory of Geodynamic Geodesy, Institute of Geodesy and Geophysics, Chinese Academy of Sciences,174 Xudong Road, Wuhan 430077, PR China
b
Belgium National Fund for Scientiﬁc Research, Royal Observatory of Belgium, 3 Av. Circulaire, B-1180 Brussels, Belgium
Received 30 November 2003; received in revised form 24 May 2004; accepted 9 July 2004
Abstract
The long-term, continuous and high-quality tidal gravity data, recorded with the superconducting gravimeters(SGs) at 19 stations in the Global Geodynamics Project (GGP), were simultaneously used to investigate the globalpattern of the tidal gravity variations. The atmospheric effects were removed from the gravity observations by usingthe simultaneous pressure records at the stations. A total of six global co-tidal models were employed to removethe loading effects of oceanic tides. The resonance parameters of the Earth’s free core nutation (FCN), as well asthe spheroidal constant components in the gravimetric factors of waves O
1
and M
2
, were accurately retrieved. Asa result, a global experimental model for gravity tides (GEMGT) was developed, considering the nearly diurnalresonance and the latitude-dependence of the gravimetric amplitude factors. The ﬁnal results indicate that the meandiscrepancy of the four main tidal waves (i.e. O
1
, K
1
, M
2
and S
2
) between the GEMGT and SG observations isless than 0.2% on average. The GEMGT is in good agreement with the theoretical models based on the inelasticnon-hydrostatic equilibrium Earth models [Dehant, V., Defraigne, P., Wahr, J., 1999. Tides for a convective Earth.J. Geophys. Res. 104, 1035–1058; Mathews, P.M., 2001. Love numbers and gravimetric factor for diurnal tides. J.Geodetic Soc. Jpn. 46 (4), 231–236] with a mean discrepancy less than 0.15%. However, the GEMGT is in closeraccordance with the theoretical model given by Mathews [Mathews, P.M., 2001. Love numbers and gravimetricfactorfordiurnaltides.J.GeodeticSoc.Jpn.46(4),231–236]forthediurnaltideswhileitisincloseragreementwithone obtained by Dehant et al. [Dehant, V., Defraigne, P., Wahr, J., 1999. Tides for a convective Earth. J. Geophys.Res. 104, 1035–1058] for the semi-diurnal tides.© 2004 Elsevier Ltd. All rights reserved.
∗
Tel.: +86 27 6888 1373; fax: +86 27 8678 3841.
E-mail address:
xujq@asch.whigg.ac.cn (J. Xu).0264-3707/$ – see front matter © 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.jog.2004.07.003
294
J. Xu et al. / Journal of Geodynamics 38 (2004) 293–306
1. Introduction
The combined gravitational attraction of the Sun and Moon on the Earth explains the orbital motion of the planet in space and the relatively small luni–solar tidal movements, inducing periodical perturbationsin the gravitational potential and deformations of the Earth (usually called the Earth’s tides). On the onehand, the tidal deformations of the solid Earth can be theoretically computed by integrating the coupledmomentum, mass conservation, Poisson’s and constitutive equations based on a real Earth model (Wahr,1981a; Dehant, 1987; Mathews et al., 1995). On the other hand, the tidal parameters of different tidaleffects, such as gravity, tilt and strain tides, can be determined by using the modern geodetic observationtechniques (Melchior, 1994; Melchior et al., 1996; Xu et al., 1997, 2000; Sun et al., 1999, 2001). In
theoretical computations, the rotation and slight ellipticity of the Earth and the inelasticity and anisotropyof the mantle have been taken into account gradually (Wahr, 1981a; Dehant, 1987; Li et al., 1996; Zhanget al., 1999; Dehant et al., 1999). The difference between the numerical results obtained in theoreticalsimulation and the tidal gravity observations is less than 0.5% (Xu et al., 2000). However, there also exist
differences between the theoretical models using different numerical integration techniques (Dehant,1987; Mathews et al., 1995; Li et al., 1996; Dehant et al., 1999; Mathews, 2001).Due to their low drift, high sensitivity and high precision, the SG gravity data have successfullycontributed to the studies on geodynamics such as the Earth’s tides (Melchior et al., 1996; Xu et al.,1997; Sun et al., 1999, 2001), oceanic and atmospheric loading effects (Warburton et al., 1975; De Meyerand Ducarme, 1991; Merriam, 1993; Sun and Luo, 1998; Xu et al., 1999a; Kroner and Jentzsch, 1999),accurate retrieval of the resonance parameters of the FCN (Merriam, 1994; Defraigne et al., 1994; Xuet al., 1999b, 2002), polar motion (Loyer et al., 1999; Xu et al., 2004), free oscillations (Van Camp,
1999; Banka and Crossley, 1999), gravity signals srcinating from the Earth’s ﬂuid outer core (Melchiorand Ducarme, 1986; Aldridge and Lumb, 1987; Smylie, 1992; Smylie and Jiang, 1993; Hinderer et al.,1995; Courtier et al., 2000). Most of the previous studies concern only gravity data recorded at individualstations. In this study, 24 continuous data sets (total 806673h) at 19 GGP stations are used to investigatethe tidal variation characteristics. A global experimental model of the Earth response to gravity tides isdeveloped.
2. Tidal gravity observations
In this study, we processed 24 long-term, continuous and high-quality tidal gravity data sequencesof a total 806673-h duration observed with the SGs at 19 GGP stations, as well as the correspondingbarometric pressure, at each site. In these observatories, four stations are located in the southern hemi-sphere(BandunginIndonesia,SutherlandinSouthAfrica,CanberrainAustraliaandSyowainAntarctica),two stations in the North American continent (Boulder in the USA and Cantley in Canada), four stationsin Asia (Esashi, Kyoto and Matsushiro in Japan and Wuhan in China) and the other nine stations inEurope (Brussels and Membach in Belgium, Brasimone in Italy, Metsahovi in Finland, Moxa, Potsdamand Wetzell in Germany, Strasbourg in France and Vienna in Austria). The basic information, e.g. theobserving period, station name, coordinates and instrument type, are listed in Table 1.We treated all the data sets using identical techniques for data preprocessing and harmonic analysis.TheT-Soft,agraphicalandinteractivesoftwareforpreprocessingEarth’stidedatadevelopedbyVauterin(1998),wasusedtoremovevisuallysomebadrecordssuchasspikes,stepsandlarge-amplitudevibrations
J. Xu et al. / Journal of Geodynamics 38 (2004) 293–306
295Table 1Basic information of the tidal gravity observations at GGP observatoriesNo. Observing period Station Country Latitude(
◦
E)Longitude(
◦
E)Instrument
a
C
(
Gal/hPa) Stdv.(
Gal)1 19971219–19991231 Bandung Indonesia
−
6
.
8964 107
.
6317 T008
−
0.3524
±
0.2426 0
.
74502 19820421–20000821 Brussels Belgium 50
.
7986 4
.
3581 T003
−
0.3467
±
0.0049 0
.
17433 19980110–20010531 Boulder USA 40
.
1308 254
.
7672 CT24
−
0.3568
±
0.0174 0
.
24694 19920801–20000201 Brasimone Italy 44
.
1235 11
.
1183 T015
−
0.3019
±
0.0304 0
.
29545 19891107–19990930 Cantley Canada 45
.
5850 284
.
1929 T012
−
0.3293
±
0.0058 0
.
14436 19970701–19991231 Canberra Australia
−
35
.
3206 149
.
0077 CT31
−
0.3392
±
0.0103 0
.
07767 19970701–19991231 Esashi Japan 39
.
1511 141
.
3318 T007
−
0.3549
±
0.0109 0
.
12868 19970701–19991231 Kyoto Japan 35
.
0278 135
.
7858 T009
−
0.3183
±
0.0382 0
.
33239 19970501–19991231 Matsushiro Japan 36
.
5430 138
.
2070 T011
−
0.4648
±
0.0085 0
.
095410 19950804–20000531 Membach Belgium 50
.
6093 6
.
0066 CT21
−
0.3286
±
0.0059 0
.
100711 19940811–20000630 Metsahovi Finland 60
.
2172 24
.
3958 T020
−
0.3654
±
0.0070 0
.
129912 20000101–20010831 Moxa Germany 50
.
6450 11
.
6160 CD34
−
0.3301
±
0.0064 0
.
053613 20000101–20010831
−
0.3340
±
0.0077 0
.
063814 19920630–19981008 Potsdam Germany 52
.
3809 13
.
0682 T018
−
0.3313
±
0.0042 0
.
085515 19870711–19960625 Strasbourg France 48
.
6223 7
.
680 T005
−
0.3128
±
0.0099 0
.
226516 19970301–19990731 CT26
−
0.3394
±
0.0074 0
.
079717 20000327–20010801 Sutherland South Africa
−
32
.
3814 20
.
8109 CD37
−
0.2866
±
0.0168 0
.
065618 20000930–20010801
−
0.2717
±
0.0227 0
.
067819 19970701–19981231 Syowa Antarctica
−
69
.
0070 39
.
5950 T016
−
0.4115
±
0.0093 0
.
110320 19970701–19990630 Vienna Austria 48
.
2493 16
.
3579 CT25
−
0.3467
±
0.0071 0
.
066221 19960728–19980923 Wettzell T103
−
0.3374
±
0.0309 0
.
263922 19981104–20010831 Germany 49
.
1458 12
.
8794 CD29
−
0.3434
±
0.0100 0
.
054423 19981104–20010831
−
0.3245
±
0.0134 0
.
073424 19971220–20000831 Wuhan China 30
.
5139 114
.
4898 CT32
−
0.3237
±
0.0107 0
.
0750
C
is the barometric efﬁciency or atmospheric pressure admittance. Stdv. stands for the standard deviation.
a
“T” refers to “large Dewar”, the ﬁrst generation SG, “CT” to “compact”, the second generation SG, and “CD” to “dualsphere”, has the lower and upper sensors and can record two gravity data sets at the same time.
caused by large earthquakes, and to interpolate some small gaps by means of an interactive procedure.A low-pass ﬁlter was used to decimate the minute sampled data series to hourly ones. The tidal gravityparameters, including amplitude factors
δ
and phase differences
ϕ
, and the local atmospheric pressureadmittance
C
were determined accurately by using the standard harmonic analysis software Eterna 3.30developed by Wenzel (1996). In the harmonic analysis, the accurate tide-generating potential with 1200
tidal waves, developed by Tamura (1987), was used.
The numerical results indicate that the standard deviation of the tidal gravity observations reaches amaximum of 0.745
Gal at Bandung, Indonesia, but is less than 0.4
Gal at any other station. The meanvalue is 0.1565
Gal.The atmospheric pressure admittance
C
, estimated for all 19 GGP stations, ranges from
−
0.272 to
−
0.465
Gal/hPa, and the average value is
−
0.3360
±
0.0036
Gal/hPa. These values are in agreementwith those obtained in previous similar studies (De Meyer and Ducarme, 1991; Merriam, 1993; Sun andLuo, 1998; Xu et al., 1999a; Kroner and Jentzsch, 1999; Ducarme et al., 2002) or theoretically expectedwith the use of the global and/or local pressure data (Niebauer, 1988; Merriam, 1992; Sun, 1995). The
atmospheric admittance and its standard deviation for each data set were listed in Table 1. The local or
296
J. Xu et al. / Journal of Geodynamics 38 (2004) 293–306
Fig. 1. Gravity residuals and their amplitude spectra recorded with the superconducting gravimeter at Wuhan in 1998. (a1) and(a2) are obtained without atmospheric correction; (b1) and (b2) with atmospheric correction using local barometric pressuredata and (c1) and (c2) with atmospheric correction using global barometric pressure data.
even global barometric pressure should be taken into account in removing the atmospheric effects uponthe gravity observations (Niebauer, 1988; Merriam, 1992; Sun, 1995). The case for Wuhan in 1998 was
chosen as an example to discuss how different the atmospheric correction of tidal gravity observationscan be, using either the hourly local pressure at the station or the 6-h-sampled global pressure data on agrid of 2.5
◦
×
2.5
◦
. Based on the atmospheric Green’s function for the preliminary reference Earth model(DziewonskiandAnderson,1981)andthestandardatmosphericmodel,thesix-hourlyglobalatmospheric
effects on gravity at Wuhan were computed and then interpolated to those with an hourly sampling rate.The numerical results indicate that atmospheric correction does not lead to signiﬁcant difference of theanalysis results in the diurnal and semi-diurnal tidal bands, using either the station pressure or the globalpressure.Duetothelowersamplingrateoftheglobalpressuredata,localatmosphericcorrectionprovidesa lower standard deviation. Kroner and Jentzsch (1999) draw a similar conclusion using the observations
fromPotsdam,Germany.Thegravityresiduals,whichwereobtainedbysubtractingthetidalgravityfromtheoriginalgravityrecords,andtheiramplitudespectrabeforeandafteratmosphericcorrectionareshownin Fig. 1. It is found that atmospheric correction signiﬁcantly reduces the gravity residuals in the whole
frequencyrange,especiallythoseinthelow-frequencyband.However,duetopoortemporalresolutionof the global pressure data, the gravity residuals after global atmospheric correction are slightly noisier thanthose after local atmospheric correction in the high-frequency band including the tidal frequency bands.Therefore, we may draw the conclusion that correction of the atmospheric effects with only the localpressure data is acceptable for this kind study, until global barometric pressure data of higher temporalresolution become available.
J. Xu et al. / Journal of Geodynamics 38 (2004) 293–306
297
In order to check the long-term stability of the instruments, each data set was separated into severalsegments with length of 2 years. Comparing the results among each data segment, it is found that differ-ences of amplitude factors
δ
of four main tidal waves (i.e. O
1
, K
1
, M
2
and S
2
) are at the order of 0.1%for various observing periods. A similar conclusion is obtained by Ducarme et al. (2002). It implies that
these tidal gravity observations are stable enough to determine accurately the global experimental modelfor gravity tides.
3. Oceanic loading correction
The oceanic loading effects are the dominant components of the observed residual given by the phasordifference between the observed and theoretically simulated vectors of each tidal wave. The loadingcorrection vectors
L
to gravity, including the amplitude
L
and the phase
λ
, can be calculated (Xu andMao,1988;Sun,1992)usingtheproceduredescribedinFarrell(1972)anddigitalmodelsofoceanictides.
Schwiderski (1980) developed the ﬁrst global model of oceanic tides (denoted as Sch80) based on theobservations recorded with a worldwide network of tidal gauges. Since 1994, thanks to the developmentof the altimeter technology, the accumulation of altimeter data and the application of the ﬁnite elementsmodeling, many global models, covering larger regions with higher spatial resolution, have becomeavailable.Usually, the oceanic models provide us only the co-tidal maps of four diurnal waves (Q
1
, O
1
, P
1
andK
1
)andfoursemidiurnalwaves(N
2
,M
2
,S
2
andK
2
).Inthisstudy,theloadingcorrectionvectorsforotherrelatively small-amplitude tidal waves (e.g. waves
1
and
1
) were interpolated by using a quadraticregression model in frequency domain. It may be written as
L
(
σ
)
·
cos(
λ
(
σ
))[
T
(
σ
)
·
R
(
σ
)]
=
α
r
+
β
r
·
σ
+
γ
r
·
σ
2
,L
(
σ
)
·
sin(
λ
(
σ
))[
T
(
σ
)
·
R
(
σ
)]
=
α
i
+
β
i
·
σ
+
γ
i
·
σ
2
,
(1)where
T
(
σ
),
L
(
σ
) and
λ
(
σ
) are the equilibrium tidal height, amplitude and phase of the ocean loadingcorrectionvectorswithfrequency
σ
,respectively;
R
(
σ
),aparameterintroducedbyWahrandSasao(1981)to describe the resonance effects of the FCN on the oceanic tide with frequency
σ
and
α
r
,
β
r
,
γ
r
,
α
i
,
β
i
and
γ
i
, the regression coefﬁcients to be determined.We adopted the theoretical model for gravity tides, obtained by Dehant et al. (1999) (denoted asDDW99) for an inelastic and non-hydrostatic equilibrium Earth, as reference model for this study. Wecomputedtheoceanicloadingcorrectionvectorsusingsixglobaloceanicmodels:Sch80,Csr3.0,Fes952,Tpxo2, Csr4.0 and Ori96. The amplitudes of the residuals before and after oceanic loading correction forwaves O
1
and M
2
are presented in Fig. 2. It is obvious that the observed residuals arise dominantly from
the oceanic loading effects. However, for the European stations, the loading effects in the diurnal bandare relatively small.As previously pointed out by Ducarme et al. (2002) and Baker and Bos (2003), our results indicate
that each oceanic model has its own local property and uncertainty at different level. For example, theloading correction vectors, based on the Ori96, can explain very well the observed residuals of gravitytides at Chinese and Japanese stations, but not at the other stations. After oceanic loading correction,

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