Asian Transactions on Basic and Applied Sciences (ATBAS ISSN: 2221-4267) Volume 01 Issue 06
The Distribution of Non-Dipole Part of Earth's Magnetic Field
*Loay E. George
** Bushra M. Ibrahim
*Department of Computer Science / College of Science / Baghdad University
**Atmosphere and Space Science Center / Space Technology and Communication
Directorate / Ministry of Science and Technology
Abstract
In this paper, the time variation of declination angle of the non-dipole part of
geomagnetic field is investigated for the time period (1900-2010). The International
Geomagnetic Reference Field (IGRF) model is adopted as geomagnetic data source for
generating the non-dipole part. It is noticed that there are many poles associated with the
non-dipole part of the main field, and they distributed over all the globe. Also, it is
noticed that some of these poles are moving relative to each other, and some of the poles
pairs are appeared and others disappeared. A statistical analysis is conducted to find out
the time variation of the first order statistical features of the non-dipole part. The analyses
indicated that the time change of statistical parameters systematic.
Key words: IGRF, Main Geomagnetic Field, Temporal Geomagnetic Field, Geomagnetic
Modeling
1. Introduction
According to recently evaluated data, the
total energy in the Earth’s magnetic field is
decreasing rapidly. This contradicts frequent
evolutionist claims that a minor (“nondipole”) part of the field is storing up enough
energy to compensate for the large and steady
loss of energy from its main (“dipole”) part.
The non-dipole fields had not been measured
very accurately before 1976, while, this part
need to be more accurately measured than the
dipole part. However, shortly after 1967, the
non dipole measurements began to get better
[1].
The International Association of Geomagnetism and Aeronomy (IAGA) had organized
a systematic global effort to gather and publish
more accurate data on the Earth’s magnetic
field. In 1970 the association published the
International Geomagnetic Reference Field
(IGRF), which describing the dipole and non
dipole part of the magnetic field, this model is
updated every five years to accommodate the
changes of the Earth's field [2]. IGRF is used
to calculate the magnetic field intensity (F) and
all its components {i.e., the angles: declination
(D), inclination (I); and the components:
horizontal (H), vertical (Z), north (X), and east
(Y)} [3].
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Dawson and Newitt (1978) [4] made a
comparison between IGRF at 1975 with IGRF
1965 for the components X, Y, and Z, the
comparison showed large secular variation
departures over many parts of the world,
especially in Z component.
Maundt et al. (1985) [5] compared the
secular variation of IGRF model for 1965,
1975, and 1980-1985 with analytical secular
variation models, based on low degree polynomial approximations. They showed that the
main features of secular variation in Europe
are represented correctly.
Wen-Yao et al. (2000) [6] made analysis of
IGRF 1900-2000; they showed that the Earth's
main magnetic field has changed dramatically
during the 20th century. Its dipole moment has
decreased by 6.5% since 1900, the strength of
the quadrapole and octapole have increased by
95% and 74% respectively.
In this project, the IGRF is used to describe
the variation of declination angle of the nondipole part of the Earth's magnetic field from
year 1900 till 2010, and then a statistical
analysis is applied on the generated geomagnetic data. A grid of points is projected on the
entire planet, such that the grid nodes are
separated from each other by 0.25o (in latitude
and/or longitude direction).
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Asian Transactions on Basic and Applied Sciences (ATBAS ISSN: 2221-4267) Volume 01 Issue 06
2. Statistical Method
The variation of the declination angle for
the non-dipole part has been investigated for
the time interval 1900-2010, because IAGA
geomagnetic group had produced the definitively values of gauss coefficients each 5 years,
so the time step is taken 5 years. The figures
of the declination angle for this time interval
are shown in appendix.
At year 1900, it was noticed the appearance of 10 poles distributed over all the glob
area. At 1905, two new poles appeared around
the location (Lon:-162o, Lat: 58o; & Lon:-165o,
Lat: -48o). For ease the region containing these
two poles is denoted "region A". Also, it is
noticed that there is another couple of poles
approaching each other and at 1905 their
coordinates are (Lon: 0o, Lat: 38o; & Lon -8o,
Lat 56o), the region containing these two
poles is called "region B". At 1910, the pair of
poles in region-A moves apart from each other.
Also, the poles in region- B continued moving
toward each other. With time progress, the
distance between A-region poles is gradually
increased, while the poles of region-B
continually become closer, and at 1925 they
disappear.
After the distance between the poles of
region-A become high, one of the two poles
belong to this region starts approaching to
another pole (which is located near to the
region Lon: -148o, Lat: 20o); the region
containing the new pair of poles is denoted
"region C". The distance between the poles of
region C is continually decreases, and at 2005
these two poles disappear. At this year the
number of poles become 8, and they distributed
over all the globe area. At 2010 the poles of
region C begun to appear again, and the
number of poles become 10.
A statistical analysis had been conducted
on the resulted data in order to explain the
behavior of non-dipole part. The first order
statistical parameters, including histogram
moments (i.e., mean, standard deviation, skewness, and kort) had been calculated for each
year.
To determine the statistical parameters
over the globe, a grid of uniformly distributed
points was projected on the Earth surface, the
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angular distance between neighbor points was
taken  along the meridian (i.e., latitudinal
direction) and  along the longitudinal
direction. The value of angular distances ( 
&  ) depend on the number of points ( n  1 )
taken along each latitude and on the number
of points ( k ) along the longitude. Since, the
separating distance between the adjacent grid
points is variant (i.e. it become small when
moving toward poles). So, the area represented
by each grid node is varied; and specifically it
depends on the latitude value. For accurate
determination of statistical parameters the
effectiveness of the declination angle at each
node value was weighted by the area (A)
represented by that node.
For a uniformly distributed points over a
spherical grid, shown in figure (1), the geographic coordinates  ,   at the node (i, j) of the
grid are determined using the following
equations:
 i  i
 j  j
(1)
(2)
and the area (A) represented by the point i, j 
is given as:
 R 2 (  )2
if i{ 0 ,n}

4k
A( i , j ) 
2
 2R  cos( i ) if 0  i n

k
(3)
where, R represents the radius of the Earth.
As shown in equations (3) & (4), the area
of uniformly distributed points of a spherical
grid is strongly depends on the latitude value.
Figure (1) The grid of uniformly distributed
points on Earth surface
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Asian Transactions on Basic and Applied Sciences (ATBAS ISSN: 2221-4267) Volume 01 Issue 06
k 1 n
j 0 i 0
(4)
n
k  w(i )
i 0
Where, D is the declination angle, w is the
weight of the element value.
Standard deviation (  ) is a statistical
parameter that tells us how tightly all the
elements are clustered around the mean in a
set of data. Skewness (S) measures the extent to
which side the data deviate (positively or
negatively) from the mean value. The kort (K)
indicates whether the data histogram has long
tail or not. The histogram moments have been
determined using the following equations:
k 1 n
2
j 0 i 0
n
k  w(i )
S 
3
  w(i )D(i , j )  m
j 0i 0
  w(i)D(i , j )  m
4
j 0i 0
n
k  w(i )
(7)
i 0
Where,  represent the standard deviation, S
represent the skewness, and K represent the
kort.
3. Results and Discussion
Figure (2) presents the temporal variation
of declination mean value. Since 1990, its rate
of change become slow, and its value become
closer to 0o. At 1960, the variance in
declination component due to non dipole
contribution was at the highest values relate to
its values at other epochs of the century.
Figure (3) shows the temporal variation of the
standard deviation of declination angle, the
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-12
-18
(6)
n
k  w(i )
i 0
K4 
-9
Figure (2) The temporal variation of declination
mean value
3
k 1 n
-6
Year
i 0
k 1 n
0
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
-3
-15
(5)
101
StdDev of Declination
2 
  w(i)D(i , j )  m
100.8
100.6
100.4
100.2
100
99.8
99.6
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
Year
Figure (3) The temporal variation of standard
deviation value of declination
Skew of Declination
m
  w(i) D(i , j )
presented results indicate insignificant
variation (1%) is occurred in standard
deviation values.
Figure (4) shows that the skewness continually decreases, and after 1942 it becomes
negative. This indicates that the biasing
toward higher values is altered to be toward
lower values (relative to the mean). Also,
within the time interval lay between the two
epochs 1940 & 1954, the rate of decrease in
skewness was relatively high.
Figure (5) presents the kort variation versus
time, the results refer that variation in kort is
small (around 2%).
The kort vaiation indicates that the
direction of the inclination angle of the nondipole component was inverted.
Mean of Declination
The weighted mean (m) can be found from
adding up all the nodes values multiplied by
their weights w (i.e., equivalent to area of
representation; A) and then divide the result
by the sum of weights values, as shown in the
following equation:
50
40
30
20
10
0
-101900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
-20
-30
-40
-50
Year
Figure (4) The time variation of skewness
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Asian Transactions on Basic and Applied Sciences (ATBAS ISSN: 2221-4267) Volume 01 Issue 06
the geodynamic origins for this non dipole
behavior. Also, other geomagnetic models
could be investigated to find out the degree of
compatibility between the presented results in
this paper with the results of other models.
Kort of Declination
119.5
119
118.5
118
117.5
117
116.5
116
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
Year
Figure (5) The time variation of Kort
4. Conclusions
The declination angle for the non-dipole
part showed that there are 10 poles distributed
over all the globe area at year 1900. In 1995,
the number of poles increased by 2. And, at
2005 the number of poles reduced to 8, and at
2010 the number of poles returns 10 poles.
The statistical analyses showed that the
mean value increases with time. While, the
dispersion measures (i.e., standard deviation
and kort) didn't show significant variations.
While the skewness variation show a
systematic decrease, which indicates the
continual change of declination values from
higher to lower values (relative to the mean).
All the results printed in this paper
depended on IGRF model and they concerned
with main component of the geomagnetic
field. Further studies are needed to find out
5. References
[1] Hurnphreys, D. R., "The Earth's magnetic
field is still losing energy", CRSQ, in
press, 2002.
[2] Humphreys, D. R., "The Earth's magnetic
field: Closing a Loophole in the case for
its Youth", Creation Matters, Vol. 7, N. 2,
2002.
[3] Geological Survey of Canada, "Magnetic
Components", April 2008.
[4] Dawson E., L. R. Newitt, "IGRF Comparisons", Physics of the Earth and Planetary
Interiors, Vol. 16, Issue 2, 1978.
[5] Maundt W., E. Ritter, A. Best, "Test of
IGRF (1980) main field and secular
variations in Europe", Physics of the Earth
and Planetary Interiors, Vol. 37, Issue 1,
1985.
[6] Wen-yao XU, WEl Zigang, MA Shizhuang, "Dramatic variations in the Earth's
main magnetic field during the 20th
century", Chinese Science Bulletin, Vol.
45, No. 21, November 2000.
Appendix
(Declination Spatial Distribution)
(A) In 1900
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(B) 1905
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(C) In 1910
(D) In 1915
(E) In 1920
(F) In 1925
(G) In 1930
(H) In 1935
(I) In 1940
(J) In 1945
(K) 1950
(L) In 1955
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(M) In 1960
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(N) In 1965
(O) In 1970
(P) In 1975
(R) In 1980
(S) In 1985
(T) In 1990
(U) In 1995
(V) In 2000
(W) In 2005
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(X) In 2010
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