Geophys. J. R. ustr. Soc. (1976)46, 67-73.
Maximum Entropy Spectrum of Long-Period Polar Motion
R. 0. Vicente
Department of Applied Mathematics, Faculty of Sciences, Lisbon University, Portugal
and R. G. Currie
Magnetic Observatory, Council Scientific and Industrial Research, Hermanus 7200,
Republic of South Africa
(Received 1976 February 10)
Summary
The maximum entropy spectra of various sets of ILS-IPMS data show
a strong signal at 29.8k2.3 yr which we interpret as a long-period term
in the complex polar motion. The Q, of this signal is 7.6+ 3.4 leading to a
damping constant of 76 yr.
1. Introduction
The motion of the pole depends on the dynamics and the structure of the Earth.
Some of its components are due to the free and forced nutations while others, such as
the annual term, are a consequence of the Earth's structure generating the complex
polar motion. It is possible to forecast some components theoretically but others
have been discovered from the observations (Rochester 1973).
Using individual ILS stations, Markowitz (1970) deduced a 24-yr period for the
motion of the pole. With a longer but less homogeneous record, Rykhlova (1969)
obtained a considerably longer period of 40 yr. McCarthy (1972) found no evidence
for a 24-yr signal from observations of Washington latitude and concluded that if any
signal does exist for the ILS-IPMS data, a longer period is more appropriate.
Utilizing sophisticated signal processing techniques we present strong evidence for a
30-yr signal in ILS-IPMS data, which we classify as a long period component of the
polar motion.
The interpretation of this long period signal, considering not only dynamics but
also hydrodynamics and magnetohydrodynamics, is difficult but according to Busse
(1970) it may represent the response of the mantle to a nutation of the solid inner
core inertially coupled to the mantle via the liquid core.
2. Nutations of the Earth
The term ' nutation ' has been employed for centuries to designate periodic oscillations of the Earth's axes, and it does give a clear picture of one of the complicated
motions of the Earth. Recently, some of these nutations have been called wobbles.
Motion of the Earth relative to its centre of mass, considered as a fixed point and
the origin of the different systems of co-ordinates employed, can be described by several
67
68
R. 0. Vicente and R. G. Currie
sets of differential equations. The equation of motion of the Earth, considered as a
rigid body, can be expressed by the vector equation
showing that the time derivative of the angular moment H around the centre of mass
is equal to the vector G of the external forces. Taking a system of rectangular axes
xi(i = 1,2,3) coinciding with the principal axes of inertia of the Earth, the projection
of this vector equation on this set of axes corresponds to the well-known Euler equations of the dynamics of rigid bodies.
The motion of the Earth can be considered as a steady state of rotation perturbed
by external forces, following the well-established practice of celestial mechanics. This
way of looking at the problem is justified because the effects of the external forces,
for instance, due to the Sun and Moon, are comparatively small. Assuming that
external forces do not exist G = 0 and
that is, the angular momentum is constant and the vector H is fixed in space. This
simple result has been forgotten in many astronomical and geophysical papers
published on the subject (Vicente 1975). The case G = 0 corresponds to the free
motion of the Earth. The most important is the free Eulerian nutation with a
theoretical free period of about 10 months assuming a highly-simplified model of
Earth structure. The observed period, called Chandler’s period, is about 14 months
and corresponds to the actual behaviour of our planet. With 95 per cent coddence
Currie (1975) finds the true period to be within 433-2+0.8 mean solar days.
The case G # 0, considering the perturbations of the external forces, corresponds
to the forced motions of the Earth. Among the most important of these are the
luni-solar precession with period of 26 OOO yr and the lunar nutation with period of
18.6 yr.
The agreement of the values of the free and forced nutations, computed from
astronomical observations and calculated from theoretical Earth models, depends not
only on the models adopted for the Earth‘s structure but also on the possibility of
solving the corresponding system of differential equations. Unfortunately, there are
so many models for the internal structure of the Earth that it is difficult to choose
the most convenient one for the theoretical calculations. The adoption of an
international agreed reference model for the Earth’s structure (Vicente 1973a, b) is
therefore important.
We can classify the components of polar motion according to their periods as
short, medium, and long. The nearly diurnal nutation can be considered a short
period, the free Eulerian nutation as medium and, finally, we have the long period
ones like the 30-yr signal. It has been the accepted convention in astronomy to classify
the nutations into short period (with periods less than 35 days) and long period terms.
The annual term of the polar motion is a component due to purely geophysical
causes and cannot be forecast by the existing theories (Jeffreys & Vicente 1957). It
is therefore possible to postulate the existence of long period components due to
other purely geophysical causes.
Some authors have designated by secular motion a component of the polar
motion that they claim to have detected in the observations. This is not an adequate
name for any component because it might be a linear term or a term with a very long
period compared to the observed record length; one cannot forecast the behaviour
of such components. It is not a correct scientific procedure to speak about secular
Maximum entropy spectrum of long-period polar motion
69
motion of the pole when observations are so limited and we have already sufficient
knowledge about the complexities of polar motion.
3. Data and computations
Pilot analyses were conducted using the spline interpolated ILS-IPMS monthly
mean value data set (see Currie 1974) of Vicente BE Yumi (1969, 1970), while final
processing was done on the eight available data sets. To optimally investigate
decadal periods, the high frequency Chandler and annual terms as well as any
longer-period term should be removed prior to spectrum analysis. By conventional
filtering techniques this would entail loss of a significant portion of the record
so the predictive error filtering method of Ulrych et al. (1973) was employed to extend
the record to twice its original length (2 x 888 points). The prediction error filter
(PEF) length was finally set at 10 per cent (pc) record length as results for PEF = 5 pc
and 15pc were sensibly the same.
A 2M+ 1 (M = 100) low pass filter was applied to the extended record with a
dimensionless Nyquist frequency cut off of 1/24. We next, (1) truncated the filtered
series to original record length and retained every 24th point to yield the 38 points
shown in Fig. 1 for X and Y components, and (2) applied a M = 100 or 200 high pass
filter, the latter of whose amplitude response is shown in Fig. 3, to the record followed
by truncation and decimation as before; the 38 point records are shown in Fig. 2 where
periods near 30 yr have been reduced by a factor of 0.7 (see Fig. 3).
Spectral analyses using the maximum entropy method of Burg (see Ulrych &
Bishop 1975) were carried out on the various records with PEF = 30, 40 and 50pc
record length, and 200 estimates computed at intervals il(200) (2At) cpy where
I3o.l. Graph of splined ILS-IPMS data with a sampling interval of 2 yr.
70
R. 0. Vicente and R. G. Currie
i = 1, ..., 200 and Ar = 2 yr. The amplitude spectra of Fig. 2 data, with PEF = 40 pc
and i,,, = 100, are given in Fig. 3. The spectra have been recoloured by the inverse
response of the M = 200 filter shown; beyond 70 yr attenuation of the filter is so
severe that the rapid rise of the spectrum is due principally to the inverse response of
the filter. The spectrum below 70 yr is dominated by a strong peak at 30 yr with an
amplitude signal to noise ratio of 4.
Before discussing measured parameters of the signal, general comments are needed.
The time domain presentation of Figs 1 and 2 are grossly similar to those of
Rykhlova (1969), Markowitz (1970) and McCarthy (1972). For X, local maxima
occur near 1922 and 1948 while local minima are 1936 and 1968. Neglecting other
possible long period trends most of the power in the time series is visually estimated to
lie between 26 and 32 years. In rough qualitative fashion the X and Y series in Figs 1
and 2 appear to be out of phase by about 180". Results of spectrum analysis were
similar using the records in Fig. 1 or the records resulting when the M = 100 high pass
filter was employed; however, all the measurements reported in Table 1 were
obtained from records as illustrated in Fig. 2.
The data employed is derived from the latitude observations determined by the
ILS-IPMS chain of instruments, situated at the same latitude and observing since
1900, being an outstanding example of international co-operation. Since the service
has operated for so long and under several directors, the published results have not
been presented in a consistent system. This is an important difficulty and many
scientists, working in this field, are not well aware of the systematic errors affecting
the observations (Vicente & Yumi 1969).
The method employed in the computations shows its advantages from the fact that
it gives consistent results when applied to various sets of observations, smoothed
A
10.04
0.04 r
Year
FIG.2. Graph of splined ILS-IPMS data sampled every 2 yr after an M = 200
high pass filter was applied to remove longer period trends and the DC component.
71
Maximum entropy spectrum of long-period polar motion
Table 1
Measurements of the 30-yr period component and Q, for the indicated data sets
Vicente & Yumi (1969,1970)
(1900-1974)
Stoyko (1972)
(1 890-1966)
Iijirna (1965)
1900-1962)
Proverbio, Carta & Mazzoleni (1969)
(1 900-1969)
Fedorov ef af. (1971)
(1900-1968)
CIO /BIH (see Proverbio & Quesada 1972)
(1900-1969)
Mean
* Splined data of Vicente & Yumi (see Currie
t Monthly values 1900-1966
Q,
Period (yr)
29.5f1.8*
30.752.7
30.5f 2.2t
30.5f 2-2$
28.6f 2.4
8-8f 4*0*
7.254.4
7*8+3*0t
9.2f 4*8$
6.8f 2.8
29-75 2.4
5.652.0
28.3f 1.4
9.0f2.8
30.5 f 1.7
6.452.8
29.8f2.3
7.6f3.4
1974)
$ Tenth-yearly values 1890-1966
in quite different ways. We are aware of the lack of homogeneity of the data, but better
results can only be obtained after the entire set of observations is reduced by the same
method. There is a working group of the International Astronomical Union charged
with this task (Vicente 1971).
4. Period, Q, and amplitude
Table 1 presents measured parameters from various sets of published latitude
observations of X and Y components which have been smoothed in various ways.
Except for the Y component of Fedorov et al. (1971), a strong signal at approximately
30 yr was detected in all the data sets using maximum entropy. Long-period motion
of the pole seems to exist for the ILS-IPMS data and McCarthy (1972) says that it
has a longer period than the period derived by Markowitz (1970). However,
considering the gross uncertainties in estimating a period from time domain graphs
such as Figs 1 and 2 we designate the signal found here as a 30-yr period component
of the polar motion.
Column 3 of Table 1 presents the measurements for Q, =f,/2Af where T, = IF,
is nominally 30 yr and Af is the half-width of the peak 3db below peak amplitude
(Munk & MacDonald 1960, p. 21). As illustrated in Fig. 3, nominal Q, is 7-6f3.4
which is about nine times smaller than Q, = 72 f 20* for the Chandler period (Currie
1974) and about 6 times smaller than Qp= 44f24 for the pole tide (Currie 1975).
Q, = 7.6 corresponds to a damping constant of 76 yr indicating that this component
of the polar motion is highly damped compared to the free Eulerian nutation corresponding to the Chandler period. Indeed, unless constantly maintained, our measurements show the component would decay by e-l within 80 yr. The cross-spectrum
between X and Y components yielded a phase difference of 180” at 30 yr, corresponding to linear polarization, in agreement with the visual assessment from Fig. 2. In
contrast, the Chandler components are circularly polarized.
Estimates of amplitude using maximum entropy tend to be poor but from Fig. 3,
taking the noise level as about O”402,we obtain O”aO3. Results from all the spectra
range from 0”-01 to 0”.04.
* The Qw reported in Currie (1974)is a factor of 2 too small and has been corrected in a correction
note added in proof in Currie (1975).
72
R. 0. Vicente and R. C.C d e
I
~0.01
r 30.6 yr
10.005
i0n
)
-
0
20
40
60
80
100
0.5
8
?!
"
rn (0.00125) cycles/yeor
FIG. 3. Maximum entropy amplitude spectrum (PEF = 40 per cent) of the data
shown in Fig. 2. The transfer function of the M = 200 high pass filter shows that,
at periods of 30 yr, unit amplitudes are attenuated by a factor of 0.7.
5. Dissipation problems
Scientists are concerned about the different values obtained for Q from the
14-month variation of latitude and oceanic pole tide analyses, and, therefore, for the
damping constants. The value estimated for the 30-yr signal is substantially less, by a
factor of from 6 to 9, than Q for the Chandler period. The difference may be principally
due to the rheological behaviour of the Earth at such diverse periods.
One can also consider the hypothesis that the damping for the Chandler period, for
the corresponding induced pole tide, or for the 30-yr signal originates in different
layers of the solid and liquid parts of the Earth. Our knowledge about the causes of
damping is so inadequate that we cannot rule out the possibility that the damping
of the different components of polar motion is located at different depths of the Earth.
Hydromagnetic motions in the core are a possible explanation not only in the case of
damping but also if we admit that these motions exhibit some periodicity. Different
authors take different views on the subject and the complexities of hydromagnetic
turbulence in the core have been pointed out by Rochester et al. (1975).
A less likely hypothesis is that the values determined do not correspond to any
damping at all, but are a feature associated with the interaction of some of the several
boundary layers that our planet has and that play such a prominent role in some
phenomena that are better understood than polar motion.
References
Busse, F. H., 1970. The dynamical coupling between inner core and mantle of the
Earth and the 24-year libration of the pole, Earthquake displacementfields and the
rotation of the Earth, pp. 88-98, eds L. Mansinha, D. E. Smylie 8c A. E. Beck
D. Reidel, Dordrecht, Netherlands.
Maximum entropy spectrum of long-period polar motion
73
Currie, R. G., 1974. Period and Q, of the Chandler wobble, Geophys. J. R. astr. SOC.,
38, 179-185.
Currie, R. G., 1975. Period, Qp and amplitude of the pole tide, Geophys. J. R . astr.
SOC.,43, 73-86.
Fedorov, E. P., Karsum, A. A., Major, S. P., Panchenko, N. I., Tarady, V. K. &
Yatskiv, Ya. S., 1971. Symp. IAU No. 48, Morioka, Japan.
Iijima, S., 1965. Ann. Tokyo Astr. Obs., Vol. M,No. 4, 182.
Jeffreys, H. & Vicente, R. O., 1957. The theory of nutation and the variation of
latitude, Mon. Not. R. astr. SOC.,117, 142-173.
Markowitz, W., 1970. Sudden changes in rotational acceleration of the Earth and
secular motion of the pole, Earthquake displacementfiela5 and the rotation of the
Earth, pp. 69-81, eds L. Mansinha, D. E. Smylie & A. E. Beck, D. Reidel,
Dordrecht, Netherlands.
McCarthy, D. D., 1972. Secular and nonpolar variation of Washington latitude, in
Rotation of the Earth, pp. 86-96, eds P. Melchior & S. Yumi, D. Reidel,
Dordrecht, Netherlands.
Munk, W. H. & MacDonald, G. J. F., 1960. The rotation of the Earth, a
geophysical discussion, 323 pp, Cambridge University Press.
Proverbio, E. & Quesada, V., 1972. Homogeneous systems of polar co-ordinates,
Journee Luxembourgeois de geodinamique, Avril session.
Proverbio, E., Carta, F. & Mazzoleni, F., 1969. Contr. Oss. Astr. Milano-Merate,
No. 319.
Rochester, M.G.,1973. The Earth's rotation, Trans. Am. geophys. Un., 54,769-780.
Rochester, M. G., Jacobs, J. A., Smylie, D. E. &Chong, K. F.,1975. Can precession
power the geomagnetic dynamo?, Geophys. J. R . astr. SOC.,43, 661-678.
Rykhlova, L. V., 1969. Evaluation of the Earth's free nutation parameters from
119 years of observations, Sov. Astr. AJ, 13, 544-545.
Stoyko, A., 1972. Le mouvement du pole instantane; la variation des latitudes et des
longitudes, Vistas in astronomy, 13, 51-132.
Ulrych, T. J., Smylie, D. E., Jensen, 0. G. & Clarke, G. K. C., 1973. Predictive
filtering and smoothing of short records by using maximum entropy, J. geophys.
Res., 78,49594964.
Ulrych, T. J. & Bishop, T. N., 1975. Maximum entropy spectral analysis and
autoregressive decomposition, Rev. geophys. space Phys., 13, 183-200.
Vicente, R. O., 1971. The need for a homogeneous system of coordinates of the
pole, Rev. Fac. Ci&nc.Lisboa, 2u s., A14, 5-8.
Vicente, R. O., 1973a. The need for a standard model of the Earth's structure, Bull.
geodes., No. 107, 105-106.
Vicente, R. O., 1973b. Inversion methods for a standard Earth model, Geophys. J. R.
astr. SOC.,35, 353-355.
Vicente, R. O., 1975. Free and forced motions of the Earth, Boll. Geodes. Sci. A f l ,
34, N.2, 173-183.
Vicente, R. 0. & Yum', S., 1969. Coordinates of the pole (1899-1968) referred to the
conventional international origin, Publ. int. Latit. Obs. Mizusawa, Vol. VII, 41-50.
Vicente, R. 0. & Yumi, S., 1970. Revised values (1941-1962) of the coordinates of
the pole referred to the conventional international origin, Publ. int. Latit. Obs.
Mizusawa, Vol. W,109-1 12.