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Geophys. J. Int. (1999) 136, 1^7
Determination of the gravimetric factor at the Chandler period
from Earth orientation data and superconducting
gravimetry observations
Sylvain Loyer,1, * Jacques Hinderer2 and Jean-Paul Boy2
1
2
Observatoire de Paris (DANOF/URA 1125), 61 avenue de l'Observatoire, F-75014 Paris, France
Institut de Physique du Globe (UMR 7516 CNRS-ULP), 5 rue Renë Descartes, F-67084 Strasbourg, France
Accepted 1998 May 20. Received 1998 February 9; in original form 1997 September 15
S U M M A RY
We have used more than 8 years' worth of a combination of superconducting gravimeter
(SG) data from the Strasbourg station and Very Long Baseline Interferometry (VLBI)
Earth orientation data in order to determine the gravimetric factor at the Chandler
period of 435 days. The method presented here takes advantage of the relation between
the rotational potential and the gravity variation at the Earth's surface. The length of
the data set is a critical parameter in such a determination but appears to be long
enough to separate the annual and Chandler components. Our results are in a good
agreement with the long-period theoretical gravimetric factor relative to an elastic solid
earth model and a static pole tide in the oceans. We also ¢nd that the e¡ect of the
atmosphere (computed from a global atmospheric model assuming a static response
of the oceans to air-pressure changes) is not negligible and perturbs the amplitude of the
Chandler wave observed in the gravimetric data.
Key words: Chandler wobble, Earth's rotation, superconducting gravimetry.
1
IN T ROD U C T I O N
Long-period gravity changes due to the rotational motion of
the Earth have been observed thanks to long series of superconducting gravimeter measurements (Richter & ZÏrn 1988;
De Meyer & Ducarme 1991; Richter et al. 1995). Indeed, one
of the major advantages of the cryogenic gravimeters with
respect to the classical spring meters is the small instrumental
drift due to the long-term stability of the magnetic levitation.
Despite this fact, most of the previous attempts to retrieve
the gravimetric factor using the theoretical perturbation of
rotational origin have given unrealistic results. In particular,
the gravimetric factor relative to the Chandlerian motion
of 435 days could be found with reasonable values only
when major assumptions were made about the annual wave
component, where contributions of di¡erent origin sum up
(Hinderer & Legros 1991) and lead to strong anomalies (see e.g.
Richter 1986). The major limitation of such determinations,
as noted by various authors, is the short length of time of
* Now at: GRGS/UMR 5562, Observatoire midi Pyrënëes, 18 avenue
Edouard Belin, 31400 Toulouse, France. E-mail: [email protected]
ß 1999 RAS
the gravimetric data and/or the de¢ciency of atmospheric
corrections. Either the data set is too short to separate the
annual and Chandler components that appear in the rotational
perturbation, or an incomplete atmospheric correction is
applied to the measurements leading to spurious e¡ects at
annual periods that a¡ect the results. The detection of very
long-period signals in a gravity record is known to be di¤cult
owing to the `red' character of any frequency-dependent
gravity noise spectrum (see e.g. Jensen et al. 1995).
The measured gravity signal *g involves di¡erent components
and can be split as follows:
*g~*gtides (t)z*gcor (t)z*gm (t)zn(t) ,
(1)
where *gtides is the classical solid Earth tidal part of the
signal due to the varying tidal potential, *gcor is the sum of
the perturbations that a¡ect gravity measurements (which
include instrumental drift, gravity variations induced by watertable changes, and atmospheric mass redistribution, as well as
loading e¡ects, tidal ocean loading, etc.) and n(t) is the additional measurement noise. The gravimetric perturbation due to
the variable rotational motion of the Earth, *gm , is usually also
considered as a correction to the gravimetric measurements
by applying a theoretical factor (close to the purely elastic
value of 1:16) to the non-rigid Earth response.
1
2
S. Loyer, J. Hinderer and J.-P. Boy
In this study, we consider together the rotational potential
(as an input for the Earth system) and the varying potential
measured by the relative gravimeter (output) in order to
determine the value of the Earth's transfer function at the
period of Chandler motion.
2 T H E ROTAT I O NA L G R AVI MET R IC
PE RT U R BAT IO N
Variations in the rotation of the Earth relative to a space ¢xed
frame, either in the rate of rotation or in the position of the axis
of rotation, induce varying centrifugal forces that a¡ect surface
gravity measurements. The variable part of the centrifugal
acceleration along the normal at the surface is given to ¢rst
order by
*gm ~)2 r[2m3 sin2 h{ sin 2h(m1 cos jzm2 sin j)]
(2)
(see e.g. Wahr 1985; Capitaine 1986; Hinderer & Legros 1989),
where we have introduced the geocentric coordinates of the
station, longitude j, colatitude h and radius r. The parameters
m1 , m2 and m3 are related to the coordinates of the vector of
rotation u expressed in the Earth's ¢xed frame by the usual
de¢nition, u~)(m1 , m2 , 1zm3 ), where ) is the mean angular
velocity of the Earth. The mi are small dimensionless quantities
(of the order of 10{6 ).
The gravitational variation (2) is the theoretical perturbation for a rigid Earth. The real gravimetric perturbation
induced by the centrifugal potential can be related to the
theoretical one by using the gravimetric factor d, which
describes how the Earth reacts to external forcing and depends
on the rheology of the planet. The gravimetric factor d is not
frequency-dependent if we consider a simple non-rigid earth
model (an elastic earth without £uid and without an inner
core). We must, however, allow frequency dependence in
order to take into account anelasticity e¡ects (Wahr & Bergen
1986; Dehant 1987; Dehant & Zschau 1989), the resonant
response of the £uid or solid inner core rotations (Legros et al.
1993) or even ellipticity e¡ects (Wahr 1981). The gravimetric
factor can be determined from the observations using the
following relation in the spectral domain (see e.g. Defraigne
et al. 1994):
*g¬
(p)
d¬ (p)~ observed
,
*g¬rigid (p)
(3)
valid for any frequency p; as we will see later, this expression
using complex-valued quantities allows us to introduce both a
gravimetric amplitude factor and a gravimetric phase shift.
The coordinates of the instantaneous vector of rotation
of the Earth (appearing in eq. 2) can easily be related to the
variations of the orientation matrix of the Earth in space by use
of the following relation:
0
0
B
M
ç M{1 ~@ {u3
u2
u3
0
{u1
{u2
1
C
u1 A .
(4)
0
The complete orientation matrix M between the celestial
reference frame and the terrestrial reference frame is classically
expressed with the help of ¢ve parameters, the coordinates of
the pole, xp , yp , the universal time UT 1, and the celestial pole
o¡set dt, d (McCarthy 1992):
M~R2 ({xp )R1 ({yp )R3 (UT 1)NP(dt, d) ,
(5)
where the Ri represent the rotation matrices around the iaxis and
NP is the precession^nutation matrix. With this formulation, it
can be shown that relation (4) can be reduced for the ¢rst two
components of u to
p
Pç
m~p{i ç zi e{i)t ,
)
)
(6)
with the following complex notations:
m~m1 zi m2 ,
p~xp {i yp ,
(7)
P~dt sin 0 zi d ,
and where 0 is the mean obliquity of reference. A complete
discussion on these developments can be found in Brzezinski
& Capitaine (1993). Eqs (6) and (7) can be used together in
eq. (2) to obtain the theoretical variation of the gravity induced
by variations of the instantaneous vector of rotation. The third
parameter, m3 , is two orders of magnitude lower than m1 and
m2 and can generally be neglected. Note that the usual form
of this correction involves only the ¢rst term of eq. (6); that
is, the parameter m1 (respectively m2 ) is identi¢ed with xp
(respectively {yp ). One component appearing in this part is
the famous Chandlerian oscillation, which is a free rotational
motion of the Earth with a period of 435 days in an Earth-¢xed
reference; the excitation process is partly attributed to the
atmosphere (e.g. Wahr 1983) but is still under investigation.
The third term of eq. (6) introduces quasi-diurnal periodic
variations in *gm up to 0.3 ]gal and at the same periods as the
tidal waves (Melchior 1983), so that it can be ignored for longterm-variation studies. The second term of eq. (6) relative to
the derivative of polar motion vanishes for very long periods,
but cannot be neglected for studies focusing on rapid rotational
changes, as was unfortunately done in Aldridge & Cannon
(1993). Neglecting this term is equivalent to an error of 100 per
cent (respectively 1 per cent) in *gm for a 1-day (respectively
100-day) periodic variation.
To avoid these problems, we decided to compute the
variations of the angular vector of rotation directly from eq. (4)
without any simpli¢cation. The description of the method
is given in the following section; another example of such a
calculation can be found in Bolotin et al. (1997).
3
DATA A NA LYSI S A N D R ES U LT S
3.1 Earth orientation data
We use two di¡erent sets of data to compute the variations
of the instantaneous vector of rotation of the Earth. The
¢rst, EOPC04, is a combined series of the ¢ve parameters of
orientation provided by the IERS on a one-day basis, which
covers the same period as the gravimetric measurements. The
second is a series obtained from VLBI measurements and
provided by the Nasa Goddard Space Flight Center (GSFC).
The two sets are not independent since around 90 per cent of
the VLBI determinations enter into the combined series of
the IERS.
ß 1999 RAS, GJI 136, 1^7
Gravimetric factor determination at the Chandler period
3
Figure 1. Gravity variation induced by variations of the instantaneous axis of rotation at Strasbourg (latitude 48:6220N, longitude
7:6840E) from October 1987 to March 1996.
Figure 2. Temporal gravity variation as measured by the superconducting gravimeter GWR model T005 at Strasbourg since 1997
October 1. The instrumental drift component modelled as an
exponential function is superimposed.
The ¢ve parameters of each series are ¢rst interpolated by
the use of spline functions. We then compute M and M
ç at
hourly intervals and we obtain the time-series m1 , m2 and m3
by use of eq. (4). These results are ¢nally introduced into
eq. (2) to provide the theoretical *gm shown in Fig. 1. The
curve shows essentially the beating between the annual and
Chandlerian components of the polar motion, which reaches
amplitudes of +5]gal. Also superimposed onto this longperiod feature are high-frequency changes of much smaller
amplitude corresponding to the third term of eq. (6) and
related to the precession^nutation o¡sets.
cycle and harmonics, monthly and fortnightly tides as well
as a superimposed long-term instrumental drift, which tends
to decrease with time; we modelled this last term as an
exponential function of time (1{e{(t{t0 )=q ), with a time
constant q~806 days]. A typical amplitude range for tides
is 100^200 ]gal, which is much larger than the rotational
signature we are seeking.
3.2
Superconducting gravimetric data
The gravimetric data used in this study are provided by a long
record of more than 3000 days (8.5 yr from October 1987 to
March 1996) from the superconducting gravimeter GWR
model T005 located near Strasbourg. We start from an hourly
series where di¡erent pre-analysis procedures have been
applied. The major undesired perturbations of short duration
(spikes, gaps due to stormy weather, for example, instrumental
maintenance or helium re¢ll, earthquakes, data acquisition
failures) have ¢rst been replaced with a theoretically predicted
`local' tide using simultaneously a recent tidal potential
(Tamura 1987) and mean observed gravimetric amplitude and
phase factors. The major problems were found by using
the slewrate method, where spikes in the gravity derivative
above a speci¢ed threshold are easily detected (see e.g. Crossley
et al. 1993), but no step (o¡set) correction was performed
because of the spurious consequences for the long-term gravity
signal that might occur afterwards. The smaller remaining
`spikes' appearing in the residuals of a preliminary tidal
analysis have also been removed. The detailed procedure
of these preliminary treatments of the data is presented
extensively in Florsch et al. (1991, 1995). The resulting
`cleaned' series, *g, is presented in Fig. 2. The major features
are the di¡erent modulations appearing between tidal waves
of di¡erent frequencies [yearly cycle and harmonics, daily
ß 1999 RAS, GJI 136, 1^7
3.3
Atmospheric pressure correction
The atmospheric correction used in this study was computed from a global pressure ¢eld (1.1250|1:1250, 1 sample
every 6 hr) provided by the ECMWF (European Centre for
Medium Range Weather Forecast) and is based on surfaceloading Green's functions (e.g. Merriam 1992) assuming a
static response of the oceans to air-pressure changes (Gegout
1995; Gegout & Legros 1997). This assumption is di¡erent
from both IBO (inverted barometer oceans) and NIBO
(non-inverted barometer oceans) hypotheses, which are usually
considered in studying the temporal changes in gravity due
to atmospheric loading (e.g. Gegout & Cazenave 1993). This
approach is hence completely di¡erent from the usual one,
which is only local and consists of removing a gravity series
proportional to the pressure changes measured at the station
site via the so-called barometric admittance coe¤cient (see e.g.
Crossley et al. 1995). It has been noted by various authors that
this local approach is not a satisfactory one, and, in particular,
it fails to correct the main atmospheric variations at annual
(Sa) and diurnal (S1) periods because of the large-scale
extension of the associated pressure ¢elds, and it fails in the
long-period frequency band (e.g. above a 10-day period), where
the characteristic length scale of a pressure perturbation is
large (e.g. Warburton & Goodkind 1978).
3.4
The gravimetric factor at the Chandler period
We ¢rst apply to the data a low-pass ¢lter whose frequency
response is shown in Fig. 3. All periods above 90 days are
4
S. Loyer, J. Hinderer and J.-P. Boy
Figure 3. Low-pass-¢lter characteristics used to extract the longperiod variations from the gravimetric signal.
unmodi¢ed but there is a 50 per cent attenuation for periods of
30 days. The ¢lter is symmetrical and does not induce any
phase shift.
After ¢ltering, we adopt the following simple model with
three periodic waves (annual, semi-annual and Chandler) and
an exponential drift:
X
s(t)~B0 zB1 (1{e{(t{t0 )=q )z
Ai cos [ui (t{to )zri ] ,
(8)
i
and we solve the previous equation using a least-squares ¢t
method for the following parameters: amplitude Ai and phase
i for each of the waves and B0 , B1 and q for the exponential
drift. The same procedure is applied to the two kinds of
series, *g and *gm , and to the di¡erent data sets used
(GSFC, EOPC04 for the theoretical rotational perturbation,
and the measured *g with or without the global atmospheric
correction). We can now compute the gravimetric factor de{ii
at the Chandler period according to eq. (3), and taking into
account a possible phase lag, by
Ameasured
d~
,
Atheoretical
Figure 4. Variations of the gravimetric factor de{ii according to the
length of the data used for the estimation.
respectively, indicating the beginning of the separation of the
two waves. Both phase and amplitude factors regularly
decrease with increasing time. The phase shift tends to converge to a value close to 200 and the module (bottom)
converges relatively fast to a ¢nal value close to 1.18. Our data
set exceeding 3000 days (^8:5 yr) is then long enough to
obtain uncorrelated estimations of the parameters of the
annual and Chandler terms.
Another way to check this point is to perform a spectral
analysis of the series. This is shown in Fig. 5 where the
semi-annual, annual and Chandlerian components are clearly
identi¢ed.
The ¢nal results using the longest available sets are given
in Table 1. For each experiment we display in Fig. 6 the
(9)
i~(rmeasured {rtheoretical ) ,
where the subscript `measured' indicates the gravimetric
variations measured by the superconducting gravimeter (either
corrected by the global atmospheric correction or not), and the
subscript `theoretical' indicates the theoretical perturbation
induced by the Earth's rotation (for an oceanless rigid earth
model).
The method presented in the previous section is expected to
give good results if the following hypotheses are valid:
(1) the length of the data set is long enough to separate the
Chandler and annual periods;
(2) the measured gravimetric perturbation at the Chandler
period is mainly due to the rotational potential.
The ¢rst hypothesis has been tested by varying the length
of the data used in the estimation procedure (we determine
a gravimetric factorömodule and phaseöat the Chandler
period for di¡erent lengths ranging from 1000 days to
^3100 days). The result of this experiment is presented in
Fig. 4. The phase estimation feature of the curve (top) changes
near the theoretical value of 6.4 yr~2300 days~( fa {fcw ){1 ,
where fa and fcw are the annual and Chandlerian frequencies,
Figure 5. Spectrum of the ¢ltered gravimetric data.
ß 1999 RAS, GJI 136, 1^7
Gravimetric factor determination at the Chandler period
5
Table 1. Estimates of the gravimetric amplitude factor and phase lag of the Chandler
component in gravity according to di¡erent models of Earth orientation data, atmospheric
pressure correction and long-period wave adjustment.
Orientation Atmospheric
data
correction
GSFC
GSFC
EOPC04
GSFC
GSFC
no
yes
yes
yes
yes
Estimated
waves
d
Sa Ssa Ch:
Sa Ssa Ch:
Sa Ssa Ch:
Sa Ssa Ch:zgroup Lp1
all{
0:93
1:19
1:18
1:18
1:26
Results
p
i
0:1
0:1
0:1
0:1
0:1
90
220
220
180
190
p
50
50
50
50
160
Corr: graph
in Fig: 6
(a)
(b)
(c)
* Lp group: four of the main long periodic tides (9.3 yr and 1305, 205 and 121 days).
{ Sa Ssa Ch.zthe group Lpz300, 500 and 600 days (see text for details).
Figure 6. Estimated Chandler component from the superconducting
gravimetric series using GSFC Earth orientation data. (a) Fit of Sa, Ssa
and CW; (b) same as (a) zLp group; (c) same as (b) z300-, 500- and
600-day waves (see Table 1 for more details).
agreement between the estimated Chandlerian component
and the corresponding gravimetric series. The ¢rst curve is
the Chandlerian wave estimated from the series *gm and
a¡ected by the tidal gravimetric factor de{ii. The second curve
represents the ¢ltered gravimetric data where we suppress
the non-Chandlerian terms given by the estimates. The agreement between the two curves is a check of the quality of the
estimation performed.
As expected, the use of the combined series EOPC04 in place
of the GSFC series has very little in£uence on the results. On
the other hand, the use of a global atmospheric correction
leads to a 30 per cent increase in the gravimetric factor d in the
right direction; that is, closer to the theoretically predicted
value of 1.185 (see below). Notice here that a similar correction
ß 1999 RAS, GJI 136, 1^7
of global size was also very e¤cient in correcting the longperiod zonal tidal gravimetric factors (Boy et al. 1998;
Hinderer et al. 1997).
We can reasonably suppose that the gravimetric series
contain signals at long-period tidal waves other than Sa and
Ssa even if they are not visible in the spectrum shown in Fig. 5.
Taking into account additional long-period tidal waves (9.3 yr
and 1305, 205 and 121 days), however, does not change the
results signi¢cantly. In fact, the spectrum of the residuals (not
shown) still contains energy in the band 250^650 days. We have
also performed another analysis in which we added three
more waves (600, 500 and 300 days) in order to minimize the
residuals. This last experiment gives the best visual agreement
in Fig. 6, but the correlation between the 500-day wave and the
Chandlerian component induces larger uncertainties in the
determined complex gravimetric factor. Moreover, there is no
theoretical background to justify the introduction of these last
three waves into the adjustment except the fact that a signal
at a given frequency exhibiting a time-varying amplitude can
be represented by the sum of ¢xed-amplitude signals with
di¡erent frequencies. The origin of such signals is still unclear
and we can only speculate on some possible relationship with
local environmental e¡ects; further work using gravimetric
measurements from di¡erent sites is required to decide whether
it is only a local e¡ect.
When comparing our values with the theoretically predicted
ones for the gravimetric factor at the Chandler period, several
comments can be made. First, there are at least two reasons
why this factor should di¡er from the purely elastic value of
1.16. On the one hand, mantle inelasticity (see Wahr & Bergen
1986; Dehant & Zschau 1989), which is known to a¡ect predominantly long-period deformation, slightly increases the
gravimetric amplitude factor and causes a small phase delay
(much smaller, however, than our observed phase shift). On the
other hand, there is a response in the oceans to the Chandlerian
motion (pole tide) that loads the Earth (e.g. Dahlen 1976; Wahr
1985; Dickman & Steinberg 1986; Dickman 1988), and the
resulting gravity change is a¡ected. If the pole tide is in
static equilibrium, there is no phase shift and the gravimetric
factor is increased by 0.04 in the case of global oceans without
continents (Hinderer & Legros 1989); if the oceans are nonglobal, a rough computation taking into account the degree 0,
order 0 of the ocean function, O00 ~0:697 (the proportion of
water on the Earth), shows that the increase is only 0.025,
leading to a total gravimetric factor of 1.185, which is close to
our determination. The phase shifts range from 90 to 220 with
an uncertainty of about 50. It is clear that these values are large
6
S. Loyer, J. Hinderer and J.-P. Boy
and greatly exceed the expected shifts caused by anelastic
properties of the Earth (e.g. Dehant & Zschau 1989). There is
no clear explanation for this; the only positive point is that the
phase shifts always tend to decrease with increasing data
length, as shown in Fig. 4. It is clear, however, that the large
uncertainty in our estimates of the transfer function at the
Chandler period does not allow any further discussion on
inelastic e¡ects and dynamic pole tide e¡ects.
4
C O NCLUS IO N S
We have determined the gravimetric factor at the Chandler
period from a 3000-day comparison between the theoretically
predicted rotational gravimetric perturbation and the measurements made by a superconducting gravimeter in Strasbourg.
The length of the data set is a critical parameter in such a
determination: at least 6.4 yr of continuous measurements are
needed to apply the method described in this paper. We con¢rm
that the gravimetric e¡ect at the Chandler period is mainly
induced by variations in the Earth's orientation. However, the
atmospheric contribution at this period is not negligible and a
correction involving the loading contribution from a global
atmospheric pressure ¢eld (with a static oceanic response to
air-pressure changes) leads to a modi¢cation of the gravimetric
signal by about 30 per cent. Taking advantage of the facts
that the rotational potential has a global in£uence and the
stacking procedures reduce considerably the in£uence of
local e¡ects appearing in SG measurements (e.g. Hinderer
et al. 1994, 1995), it is highly desirable to combine di¡erent
data sets of long duration from well separated stations to
improve such a study. The simultaneous use of data sets
that will become available thanks to the worldwide network
of superconducting gravimeters in the frame of the Global
Geodynamics Project (GGP) (see Crossley & Hinderer 1995)
will provide an opportunity to determine with more accuracy
the gravimetric factor at the Chandler period.
AC K NOW L E D GM E N T S
We are grateful to the European Centre for Medium Range
Weather Forecasts (ECMWF) in Reading, UK for having
supplied the atmospheric pressure data used in this study.
We thank the referees for their helpful comments on the
manuscript.
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