On the Analysis of Geomagnetic Jerks Using Wavelet Transformation
E. Chandrasekhar and S. Adhikari
Department of Earth Sciences Indian Institute of Technology Bombay Powai, Mumbai – 400076 E-mail: esekhar@iitb.ac.in
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What are Geomagnetic Jerks? Why study them?
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What are Geomagnetic Jerks?
Differential fluid-flow on the surface of the Earth‟s outer core sometimes results in generating some sudden changes in the slope of geomagnetic secular variation field (secular acceleration). Such abrupt changes, distinctly seen in the East-West (Y-) component of the geomagnetic field and having a life-span of about a year are known as Geomagnetic Jerks (GJs).
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dY/dt
Secular variation
d2Y/dt2
Secular acceleration
Time (t)
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Why study GJs?
One of the primary interests in investigation of Geomagnetic Jerks is :
identification of geomagnetic jerks originating within the core, due to differential fluid motion and having a short time constant, generated acute interest in the proper understanding of lower mantle conductivity. This is seen
as stimuli to revisit lower mantle conductivity estimates by the theory of electromagnetic induction through the mantle.
Therefore, the present study forms a necessary platform to understand
the morphology of GJs in a broader perspective the constraints on the role of the deep mantle conductivity distribution on the geomagnetic Jerk phenomena.
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Why study GJs? ………contd.
Since Indian magnetic observatories have been maintaining longperiod geomagnetic data for decades, not much research has been done in understanding the GJ phenomena using Indian data.
Therefore, the ongoing analysis of GJs is the first of its kind
in Indian geomagnetic research.
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Why Wavelet Analysis?
The occurrence of GJs in the decadal variation of magnetic intensity is irregular rather than periodic. In other words, they sometimes occur as a global feature (seen at almost all the world magnetic observatories) and sometimes a local one (seen only at a few closely located observatories). Some hemispherical differences in the time of their occurrences were also noted (Backus, 1983; Adhikari et al., 2007).
This has lead to difficulties in correctly understanding their spatiotemporal behaviour.
the terms ‘global’ and ‘local’ are purely relative and it is not clearly known; how local is local? and how global is global?
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Conventional spectral analysis tools fail to provide any useful information towards better understanding of such diverse
spatio-temporal behaviour of the geomagnetic field.
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Narrow window good time resolution poor freq. resolution.
Wide window good freq. resolution, poor time resolution.
(Polikar, 2001)
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Wavelet analysis facilitates to detect the time-varying frequency components in a non-stationary signal with required resolution. Since GJs are high frequency signals occurring at
irregular times in non-stationary time series of geomagnetic secular field variations, use of wavelet analysis is ideally suited to detect them.
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(a) The standard basis in the time domain, i.e., Dirac delta functions, can very well localize the process in the time domain but not at all in the frequency domain. (b) In the case of Fourier bases we get exact localization in frequency but none in time, which is depicted by long horizontal cells
(c) If we were to apply a moving window to the signal and take the Fourier transform at every location in an attempt to localize the presence of a feature, we get a short time or windowed Fourier transform. This partitions the entire timefrequency plane with rectangular cells of equal size.
(d) In wavelet transformation, higher frequencies (small scale features) are well localized in time, but their frequency localization is poor. Whereas, lower frequencies (large scale features) have a very good frequency localization and poor time localization.
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Inference: With wavelets, one can see both Trees and Forest!!
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Objectives
To demonstrate the time-frequency localization of occurrence of geomagnetic jerks in the geomagnetic secular variations records of about 70 years. To identify optimal wavelet(s) that may best detect the well reported world-wide geomagnetic jerks that had occurred in 1969, 1978 and 1992. To check whether the above three GJs, which have been recorded in most of the worldwide observatories, are also present in the magnetic data of Indian observatories. To find out the occurrences of any local GJs in ABG data, which have not been recorded at other magnetic observatories.
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Previous studies of GJs
Spherical harmonic analysis (SHA):
Malin and Hodder (1982) Nevanlinna (1985). Kathey Whaler (1987)
Multivariate Data Analysis:
Wavelet Analysis: Statistical Time Series Model:
Wavelet transform coupled with Monte Carlo simulations:
Alexandrescu et al. (1995) Nagao et.al (2002)
Nenadic & Burdick (2005)
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Wavelet Analysis: Basics
A wavelet by definition, is a small wave having finite length in time (in technical terms, known as compactly supported) and represented as a real-valued function, say, ψ(t), satisfying the following properties : (i) that it integrates to zero (i.e., having zero average)
(t )dt 0
(1)
(ii) is square integrable (having finite energy). i.e.,
2 (t )dt 1
(2)
Ψ Є L2(R). Equation (1) implies that the function ψ(t) must be oscillatory (so that the sum of its positive and negative excursions cancel out in averaging) and equation (2) implies that ψ(t) must be finite in length. The wavelet function, ψ(t), signifying the time-frequency localization is defined as
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Wavelet Analysis ……..contd.
, s (t )
1 t s s
(3)
where s > 0 indicates the scale and τ indicates the translation parameter. Here, s is analogous to frequency, in the sense that high scales (low frequencies) provide overall information from the signal and low scales (high frequencies) provide detailed information from the signal. The
translation parameter, τ, is linked to the location of the window as this window is slided over the signal and thus apparently refers to time information in the transformed domain.
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If two functions f(t) and g(t) are square integrable in R (i.e., f(t), g(t) Є L2(R) then their inner product is given by
f (t ), g (t ) f (t ) g * (t )dt
R
(4)
According to (4), the CWT can be defined as the inner product of the mother wavelet ψ(t) and the signal, f(t), given by
CWT t ( , s )
1
s
t f (t ) s
dt
(5)
Equation (5) explains that the wavelet transformation gives a measure of the similarity between the signal and the wavelet function.
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Station Name
IAGA Code RSV ESK NGK ABG
Latitude Longitude 55.48°N 55.32°N 52.07°N 18.64°N 12.46°E 3.20°W 12.68°E 72.87°E
Geomag. Latitude 55.18°N 57.80°N 51.88°N 10.19°N
Geomag. Longitude 99.52°E 83.96°E 97.75°E 145.06°E
Data Range 1930-81 1941-99 1930-97 1930-99
Data
Rude Skov Eskdalemuir Niemegk Alibag
Geographical location of magnetic observatories, whose data were used in the present study
Europe
INDIA
Continuous records of global magnetic observatory data of about 7 to 8 decades with a sampling interval of one month were procured from World Data Centers (WDCs) (Kyoto, Japan and Boulder, USA).
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Initial data processing methods like, corrections for missing values, jumps, filtering, etc. were made prior to further analysis
(nT) (nT)
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Perform multiscale decomposition of the signal using an appropriate wavelet basis.
The choice of “mother wavelet” is a key to any meaningful wavelet analysis.
We have applied continuous wavelet transformation using those wavelets, that have
High compact support Higher order vanishing moments.
and
(This facilitates to aptly identify jerks, as jerks appear as a very narrow, high frequency component in the data).
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Comparison of the important properties of some 6 wavelets
Properties
regularity
db4
0.8
sym4
arbitrary
gaus3
infinitely
coif2
arbitrary
bior5.5
3 & 4 at knots
db8
1.6
compact support symmetry
vanishing moments
yes antisymmetric
4
yes Near symmetric
4
no antisymmetric
arbitrary
yes Near symmetric
4
yes symmetric
5
yes antisymmetric
8
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……..and those of another 6 wavelets
Properties haar coif1 meyr dmey gaus1 gaus2
regularity compact support
symmetry
Not continuous yes
symmetric
arbitrary yes
Near symmetric 2
infinitely no
symmetric
infinitely no
symmetric
infinitely no
antisymmetric
infinitely no
symmetric
vanishing moment
1
arbitrary
arbitrary
arbitrary
arbitrary
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Detect singularities (Jerks) based on their causal nature.
Since geomagnetic jerks manifest the second- time derivative of the geomagnetic field, they can be referred to as singularities and can be
defined as some α-derivative of the signal (Alexandrescu et al., 1995). So, Let us define the time-varying magnetic field f(t) as
f(t) = β(t-t0) + n(t)
0 (t t0 ) (t t0 )
α
(6)
Where n(t) is the noise term as a function of time and β(t-t0) is defined as
t t0 t t0
Since the singularity is included in the signal, WA can detect it and can also tell about the time of its occurrence. The above condition suggests the causal behaviour of the geomagnetic Jerk.
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The Wavelet transform of equation (6) is given by
Wf (t , s) W [(t t0 )]( t , s ) Wn (t )
The above equation can be simplified to obtain an equation of the form (Alexandrescu et al., 1995)
ln r j ( s ) ln( s ) constant
(7)
where rj denotes the ridge function, defined as the absolute value of the wavelet transform along a obtained Line of Maxima (LoM). s > 0 is the dilation parameter and α represents regularity of the geomagnetic jerk.
Equation 7 represents a straight line when plotted on a log-log scale, whose slope gives the regularity (α) of the singularity.
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If we want to study the αth derivative of the signal, f (t), the wavelet should have cancellations (vanishing moments) up to order α in order that it does not react to the lower order variations of f (t). That means, we should have (Farge, 1992)
In other words,
Rn
(t )t d t 0
GJs are usually detected by those wavelets whose number of
vanishing moments are greater than the regularity (α) of the GJ.
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Lines of Maxima (LoM)
The lines of maxima are the curves formed by joining the points of local
maxima at different scales (Mallat and Hwang, 1992). Local maxima or local minima in the WA of a signal, f(t), at any time t designates that the first derivative of the WA of the signal at time t is zero. Local maxima form if the
value of the coefficients of CWT of the signal at time t is more than the values of the coefficients of CWT of the signal at time t+Δt (right neighborhood of t0) and t-Δt (left neighborhood of t0). The lines of maxima represent discontinuities and the time at which the lines of maxima intersect scale =1 is the time of occurrence of the discontinuity.
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local maxima
CWT coef
Left neighborhood
Right neighborhood
t t-Δt t+Δt
Time
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LoM represent the discontinuities present in the analyzing signal
not all discontinuities represent GJs.
Therefore, from equation (7), it can be understood that GJs represent
only those LoM, whose logarithm of ridge function (ln (rj)) bears a
linear regression with ln (s).
However, in practice, because of the presence of noise in the data, the ridge functions deviate from their straight line behaviour.
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Types of Ridge Functions
Type 1:
Display positive linear trend (α > 0) for most of the dilation range Display wobbly pattern (yet α > 0) and may arise due to the presence of harmonics present in the data. These are the ones that do not fall in either of the above two types. They are typically due to random noise present in the data.
Type 2:
Type 3:
The different types occur due to the presence of singularities, random noise, harmonic components, etc. in the data
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Comparison of the important properties of 6 wavelets showing all the LoM
Properties
regularity
db4
0.8
sym4
arbitrary
gaus3
infinitely
coif2
arbitrary
bior5.5
3 & 4 at knots
db8
1.6
compact support symmetry
vanishing moments
yes antisymmetric
4
yes Near symmetric
4
no antisymmetric
arbitrary
yes Near symmetric
4
yes symmetric
5
yes antisymmetric
8
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Comparison of the important properties of 6 wavelets that did not show any LoM
Properties haar coif1 meyr dmey gaus1 gaus2
regularity compact support
symmetry
Not continuous yes
symmetric
arbitrary yes
Near symmetric 2
infinitely no
symmetric
infinitely no
symmetric
infinitely no
antisymmetric
infinitely no
symmetric
vanishing moment
1
arbitrary
arbitrary
arbitrary
arbitrary
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Ridge functions for the year 1952 at ABG observatory using db4, sym4, coif2, bior5.5. All these wavelets showed type 1 ridge functions, indicating a possibility of occurrence of a local GJ in ABG in the year 1952. Db4, sym4 and bior5.5 showed a regularity of 1.6, while coif2 showed a regularity of 1.8.
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Comparison of 1940 and 1960 Ridge functions of gaus3 wavelet with ridge functions of other wavelets at same year. While gaus3 showed type 1 ridge functions, db4 in 1943 showed type 3 and coif2 in 1960 showed type 2 ridge functions. As evident from other wavelets there are no GJs in 1943 or 1960 in ABG data, but still gaus3 showed „GJ-like‟ ridge function.
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What does this imply?
Two possible interpretations!!
Either the local GJs of 1943 and 1960 may be unique in the sense that only the Gaus3 wavelet could detect it !! (I don‟t know what do I mean by unique and in what way?) OR
Since most other wavelets could not detect it, Gaus3 wavelet may in fact be misleading!! In identifying such a “Jerk-like” feature
This raises a fundamental question……
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Is it possible to identify a single wavelet (or a few wavelets)
that can always detect a jerk?
In other words
Can we attribute a wavelet or a couple of wavelets, only for
detection of Geomagnetic Jerks?
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Thank You!
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