Power spectral density analysis using Burg's maximum entropy method was applied to the geomagnetic dipole field and its rate of change for the years 1901 to 1969. Both spectra indicate relative maxima at 0.015 cycles/year and its harmonics. These maxima correspond approximately to 66, 33, 22, 17, 13, 11, and 9-year spectral lines. The application of the same analysis techniques to the length-of-day (l.o.d) fluctuations for the period 1865 to 1961 reveal similar spectral characteristics. Although peaks were observed at higher harmonics of the fundamental frequency, the 22-year and 11-year lines are not attributed unambiguously to the solar magnetic cycle and the solar cycle. It is suggested that the similarity in the l.o.d fluctuations and the dipole field variations is related to the motion within the earth's fluid core during the past one hundred years

Jin, R. S.

English

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https://ntrs.nasa.gov/search.jsp?R=19750023547 2020-03-22T20:31:02+00:00Z
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FINAL TECHNICAL REPORT
O1 December 1974 to 91 August 1975
Research Title:
d
Correlation of the Earth 's Rotattim Rate and the
Secular Change of the Geomagaetic Field
Principal Investigator:
Dr. Rong-sheng Jin
Institution:
Department of Physics and Space Sciences
Florida Institute of Technology
Melbourne, Florida 32901
 
3 4 S6
Funding Organization: 5es,.4
Geophysics Branch - Code 922 ^?l- ^ a
Goddard 'Space Flight center
Nations! Aeronautics and Space Administratio
Greenbelt, Maryland 20771
Grant No.:
NSG 5038
c 41,
I. INTRODUCTION
The main objective of the nine-month research grant was to apply the
power spectral density analysis using the maximum entropy method (MEM) to the
geomagnotic dipole field for the years 1901 to 196 0 . Another objective was to
apply the spherical harmonic analysis to the geomagnetic data prior to 1900. The
first objective was accomplished. In addition, the application of the MI M spectral
analysis was extended to include the length of day fluctuations. From the early
phase of the work on the spherical harmonic analysis of the pre-1900 geomagnetic
data, it was concluded that the annual mean magnetic data covering a span of 14
or more years is needed to fit a spherical harmonic expansion of the magnetic
potential to the 7th order and 7th degree. Since the IBM 370-125 computer at
Florida Institute of Technology is too slow to be practical for this work, no further
progress was made. The necessity of gaining access tothe IBM 360-75 computer
at Goddard Space Flight Center was mentioned in the semi-annual status report[Jin, 1975.
The application of MLM spectral analysis to the dipole field of the earth
and the length of day fluctuations indicate that both spectra have common relative
maxima at 0.015 cycles/year and its harmonics. The method of analysis and re-
sults are given in the following sections of this report.
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2. RESEARCH ACCOMPLISHED AND METHODS OF ANALYSIS
2.1 Power Spectral Analysis of the Geomagnetic Dipole Field
Power spectral density analysis using the maximum entropy method was
'.	 originated by Burg 1967, 19681 - The method has been successful in estimating
the power spectral density when the periods are comparable to the original dots
length [Lacoss, 1971; Ulrych, 19721 . For a discrete stochastic process, the
MEM spectral density at a frequency f is given by:
P (f)
	
PM+1
f	 1	 aMk exp [-12 tt Icf 
(e01 i
'?
N I	 k=1
(1)
P
where:
- PM + 1 is the mean output power of the (M + 1) - point prediction - error -
filter that is used to whiten the time series;
- a.Mk Is are the prediction-error-filter coefficients that are determined from
the data;
- At is the uniform sampling rate;
- I  is the Nyquist frequency, and equals to 1/(2 At);
- and f Is the frequency between -fN
 and f  for which the power spectral
density is to be determined.
The prediction-error-filter coefficients, aMk, in Equation (1) are calculated from
the data using the Burg algorithm CUlrych and Bishop, 19751 , without requiring
either a zero or periodic extension of the data.
Tine values of the geomagnetic dipole field from 1901 to 1969 used for the
MEM power spectral analysis were determined from the Gauss-Schmidt coefficients
resulting from a spherical harmonic analysis of the annual mean magnetic data at
various geomagnetic observatoriesPin, 1973 1. The digitized annual mean values
of geomagnetic data were provided by the Geomagnetic Division, U. S. Coast and
 nr
	
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HGeodetic Survey. It is essentially the same as those given in "The Annual Mean
	
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Value of Geomagnotio Elements Since 1900" submitted in 1965 by the working group
on the analysis of the geomagnetic field to the International Association of Geomag-
	
j)	 nvtism and Aeronomy. Additional data up to 1969 aro.inoluded on the digitized tape;
however, the data for 1967, 1968 and 1969 are not complete. For the spherical
	
f	 harmonic analysis, the geomagnetic field near the earth's surface is assumed to be
mainly of internal origin and can be represented by the negative gradient of a mag-
netic potential function. The magnetic potential function V is expanded in terms
of the spherical harmonic function, i.e.
V = a	 (a )n+i 7 (gm cos m¢ + hm sin MO) Pm (0)
n=1 r	 m=0 n	 n	 n
where:
r, 0, and 0 are the geocentric radius, colatitude, and longitude;
a is the mean earth radius;
gm and hn are the Gauss-Schmidt coefficients;
and Pmn (9) are the Schmidt partially normalized Associated Legendre
functions .
The spherical harmonic expansion was truncated to the 7th order and 7th degree.
Earth oblateness was also taken into account. The least squares method was used
	
£	 in fitting the observatory data to obtain the Gauss-Schmidt coefficients. The method
	
fi	 has been well documented by Cain, et al 119671 . In order to obtain the global sec-
ular changes of the geomagnetic field, nine year data spans were used for analysis
	
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and the Gauss-Schmidt coefficients were assumed to be parabolic functions of time.
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The values of these coefficients together with their first and second time deriva-
tives were obtained for th6':aid-{mint of the nine-year span. Different nine-year
spans with mid-points s^;pft 7A!:,,id Oy four years for the period 1905 to 1965 were
	
$	 analyzed. The dipole moment M is given by H 0 a 3 where HO is tho dipole field
	
3	 at the magnetic equator and is given by:
	
(a	 HO = t(gi)2 + (h°) 2 + (hl	
1/2
)^^ 	 (i)L
The resulting dipole field H O from 1901 to 1969 is shown in Figure 1. In applying
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the MEM power spectral analysis to the dipole field, the number of prediction-
error-filter coefficients waia varied from y to 68. The number of spectral esti-
mates is chosen to be 300 separated by 1.67 x 10 -3 cycles/year. The existence
of the relative maxima in the dipole field spectra at 0.015 cycles/year and its
higher harmonics were found. The existence of tho fundamental ir:quency and its
harmonics are evident when the number of the prediction-orror-filter coefficients
exceeds 80% of the number of points in the time series of the dipole field. If the
number of the prediction-orror-filter coefficients is less than 80% of the record
length, only some of the harmonics of the 0.015 cycles/year will appear. A typi-
cal power spectra of the geomagnetic dipole field is shown in Figure 2. Since the
time series of the geomagnetic dipole field is nonstationary, and the criterion in
choosing the length of the prediotion-error-filter coefficients is lacking, wedecided
to test the 80% requirement. For that purpose, a time series composed of sine
waves with 0.015 cycles/year and four harmonics was generated. In addition, a
linear function derived from a linear fit to the geomagnetic dipole field H O was
added to the sine waves. The MEM power speetrs°. analysis was then applied to
this series by varying the number of prediction-error-coefficients used. It was
reassuring that the spectral line at the fundamental frequency cannot be found un-
less the number of prediction-error-filter coefficients again exceeds 80% of the
record length. The MEM power spectral analysis was also applied to the time rate
of change of the dipole field. The time derivative H O was determined from the HO
of nine consecutive years by solving for B in the parabolic function:
H O(t) _ A + Bt + Ct2 ,	 t = -4 to +4	 (3)
where A, B, and C are constants.
The time series of HO from 1905 to 1965 is given in figure 3. Its power
spectra is shown in Figure 4. Again the spectral peaks at 0.015 cycles/year and
'	 its harmonics are evident.
2.2 Power Spectral Analysis of the 'Length of Day Fluctuations
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are different from those calouiated from Newtonian mechanics based on a uniform
time (Ephemeris Time). The differ •.ice AT between the Ephemeris Time and the
Universal Time prior to 1950 are given by Brouwer 11952 . The values for AT
after 1960 can be found in the American Ephemeris and Nautical Almanac. The
time discrepancy pT is given by:
AT ^ Ephemeris Time - Universal Time
= 24.349 AL see.
where 24.349 is the time required for the sun's mean longitude to increase 1 sec.
of arc, and AL is the difference between the observed and the theoretical longi-
tudes in seconds of arc. The difference AL is given by AL = a  + bot +fo where
ao , b  are constants related to particular origin of time, January 1, 1900, and the
rate of change of the Ephemeris Time; and fo is the longitude discrepancy due to
(lie fluctuations and the secular changes in the earth's rotation rate [Munk and Mac-
Donald, 19601 . The so called Lo.d. fluctuations is proportional to the time deri-
vative of AT, i.e.,
l.o.d fluctuations = 2.738 x 10 -3 x d(AT)/dt sec. of Ephemeris Time
where 2.738 x 10-3 is the reciprocal of the number of mean solar days in a year,
and d(AT)/dt is the annual rate of change of AT in seconds. For the MEM spectral
analysis, the values of Sr from 1861 to 1948 were taken from Table 8a of Brouwer's
article 119521 , and those from 1948 to 1965 were taken from the American Ephem-
eris and Nautical Almanac. The derivative of AT from 1865 to 1961 were then ob-
tained from the nine point formula, similar to Equation (3). The time series of AT
and its time derivative are given separately in Figures 5 and 6.
The MEM power spectral analysis was applied to the time series of d(AT)/dt
which is proportional to the Lo.d. fluctuations. The number of prediction-error-
filter coefficients used were 20, 40, 60 and 60, and the number of spectral estimates
was 300 separated by 1.67 x 10 -3 cycles/year. The spectral peaks at 0.015 cycles/
year and its harmonics were found when the number of prediction-error-filter coef-
ficients are 40, 60 and R0. A lower frequency peak at 8.33 x 10 -3 cycles/year was
also found when the number of prediction-error-filter coefficients is 80. however, ii
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this 120-year spectral lino is considered spurious. A typical power spectra M, the
time derivative of AT is given in figure 7.
3. RESULTS AND CONCLUSIONS
The application of MEN spectral analysis to the dipole field and its deriva-
tive from 1901 to 1969 uncovered the existance of spectral peaks at 0.015 cycles per
year and its harmonics.
	
They correspond to 66, 33, 22, 17, 13, 11, 9-year spectral
lines.	 Currie 11973a; 197Jb, applied the MEN spectral analysis to separate time
series of mean yearly magnetic data at 28 geomagnetic observatories and reported
the existence of the 11-year solar cycle, the 22-year solar magnetic cycle, and an
approximately 60-year spectral line.	 Our spectral analysis on the time series of
the s,ivm.,gnetio dipole field confirms the existence of the d60 year spectral line,
however, the fact that we observed clearly the peaks at higher harmonics of the
fundamental frequency 0.015 cycles/year, the 22-year and 11-year spectral lines
x	
'^' cannot be attributed to the solar magnetic cycle and the solar cycle unambiguously.
It is interesting to note that a 11-year cycle not attributed to the solar cycle was
found in the horizontal component of the geomagnetic field [Rivin, 1974] . Further-
more, it is remarkable that the same spectral lines were also observed in the sped-
trum of the l.o.d. fluctuations from 1865 to 1961.	 The common spectral lines in
the l.o.d. fluctuations and the geomagnetic dipole field variations suggests that
xu either the cause of both phenomena is the same such as the magneto-hydrodynamic
oscillation I,. the fluid core of the earth or one is the cause of the other. 	 Which-
ever is the case, the fundamental frequency 0.015 cycles/year and its higher har-
monies should be related to the motion inside the fluid core of the earth for the past
ore hundred years. As for the phase relation between the 1.o : d. fluctuations and
the geomagnetio dipole field variations, this can only be resolved through a cross
corr9lation study of the two phenomena.
12
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4. ADDENDUM
These research results will be reported in an article "Spectral Linn Simi-
larity in the Geomagnetic Dipole Field Variations and Length of Day Fluctuations"
to be submitted to the Journal of Geophysical Research for publication. The ab-
stract is given in the Appendix. The preprints of the paper will be provided to
NASA at the time of submission for publication.
The generous support given by the Geophysics Branch of Goddard Space
Flight Center is gratefully acknowledged.
14
i	 I'	 I	 I	 I	 11
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i
REFERENCES
Brouwer, D., A Study of the Changes in the Rate of Rotation of the Earth, Astro.
J. , 57, No. 5, 125-146, 1952.
Burg, J. P. , Maximum Entropy Spectral Analysis, paper presented at the 37th
Annual International Meeting, Sec. of Explor. Geophysics, Oklahoma City,
Oklahoma, Oct. 31, 1967.
Burg, J. P. , A New Analysis Technique for Time Series Data, paper presented at
Advanced Study Institute on Signal Processing, NATO, Enschede, Nether-
lands, 1968.
Cain, Joseph C. , W. E. Danields, and ShirleyJ. Hendricks, An Evaluation of the
Main Geomagnetic Field, 1940-1962, J. Geophys. Res. , 70, 3647-3674,
1965.
Cain, Joseph C., and R.A. Langel, private communications.
Currie, R. G. , Pacific Region Anomaly in the Geomagnetic Spectrum at a 60 Years,
S. Afr. J. Sol., 69, 379-383, 1.973a.
Currie, R. G. , The ss 60 Year Spectral Line in Length of Days Fluctuations,. S. Afr.
J. Sol., 69, 180-182, 1973b.
Jin, R.S. , Dipole Moment of the Earth, 1905-1966, Transactions, American Geo-
physical Union, Vol. 54, No. 4, p. 235, 1973.
Jin, R.S., Correlation of the Earth's Rotation Rate and the Secular Change of the
Geomagnetic Field, Semi-Annual Status Report, NASA Grant No. NSG 5038,
June, 1975.
I.acoss , R. T. , Data Adaptive Spectral Analysis Methods, Geophysics, 36, 661-675,
1971.
McDonald, K. L. and R. H. Guest, AnAnalysis of the Earth's Magnetic Field from
1835 to 1965, Institute for Environmental Research Tech. Rep. 46-IESI,
Boulder, Colorado, July 1967.
Munk, W. H. , and G. J. F. MacDonald, The Rotation of the Earth, Cambridge Uni-
versity Press, London, 1960.
LI
i
,
I
ii
Rivin, Yu.R. , 11-year Periodicity in the Horizontal Component of the Geomagnetic
Field, Geomagnetism and Aeronomy, 14, 97-100, 1974. (English Translation)
Ulrych, T. J. , Maximum Entropy Power Spectrum of Truncated Sinusoids, J. Geo-
phys. Res. , 77, 1396-1400, 19',/.
15
l	 ^	 l
^ .	
l	 1	
r
^II	 r
y^	 F
P
Ulrych, T.J. and T.N. Bishop, Maximum Entropy Spectral Analysis and Autorr-
ressivo Docom position Rev. of Geo h s. and S Ph, s., Vol. 13 No. 1^'	 (	 ^	 P y	 I' .	 y	 ,	 ,
183-200, 1075.
;	 e
i l!	 I
A,
r
k
r	 u	 16
a
r.
A PPENDM
Spectral Line Similarity in the Geomagnetic Dipole field Variations
and Length of Day Fluctuations
ABSTRACT
Power spectral density analysis using Burg's maximum entropy method
was applied to the geomagnetic dipole 'geld and its rate of change for the years
1901 to 1969. Both spectra indicate relative maxima at 0.015 cycles/year and its
harmonics. These maxima correspond approxmately to 66, 33, 22, 17, 13, 11,
and 9-year spectral lines. The application of the same analysis techniques to the
length-of-day (l.o.d.) fluctuations for the period 1865 to 1961 reveal similar
spectral characteristics. Since we observe clearly peaks at higher harmonics
of the fundamental ffcquency, the 22-year and 11-year lines cannot be attributed
in an unambiguous manner to the solar magnetic cycle and the solar cycle. We
suggest that this similarity in the Lo.d. fluctuations and the dipole field varia-
tions is related to the motion within the earth's fluid core during the past one
hundred years.
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17
	
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Correlation of the earth's rotation rate and the secular change of the geomagnetic field

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