# Presentation: 129184

## 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](mailto:esekhar@iitb.ac.in)_

## What are Geomagnetic Jerks?   Why study them?

## What are Geomagnetic Jerks?   

**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).**_

## Time (t)

- dY/dt

- d2Y/dt2

- Secular variation

- Secular
- acceleration

## 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.

**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.**

**Why study GJs?**

## Why study GJs? .........contd.

- Since Indian magnetic observatories have been maintaining long-period 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.***

## Why Wavelet Analysis?   

**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 spatio-temporal behaviour.

***the terms ‘global’ and ‘local’ are purely relative and it is ***_***not clearly known***_***; ***
_***how local is local? and how global is global? ***_

## Conventional spectral analysis tools fail to provide any useful information towards better understanding of such diverse spatio-temporal behaviour of the geomagnetic field.

##        Wide window  

-        Wide window
- good freq. resolution, poor time resolution.

-        Narrow window
- good time resolution poor freq. resolution.

- (Polikar, 2001)

## 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.

## (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 time-frequency 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.

## Inference: 
With wavelets, one can see both Trees and Forest!!

- With wavelets, one can see both _Trees and Forest_!!

## 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.

## Previous studies of  GJs 

- Spherical harmonic analysis (SHA):  		Malin and Hodder (1982)
- 							Nevanlinna (1985).
- Multivariate Data Analysis:			Kathey Whaler (1987)
- Wavelet Analysis: 				Alexandrescu et al. (1995)
- Statistical Time Series Model: 		Nagao et.al (2002)
- Wavelet transform coupled with
- 	Monte Carlo simulations: 			Nenadic & Burdick (2005)

## Wavelet Analysis: Basics

- (*ii*) is square integrable (having finite energy). i.e.,

- (2)

*Ψ** **Є** L**2*(*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

- (1)

**Wavelet Analysis****:**** Basics**

## Wavelet Analysis ........contd.

- (3)

**Wavelet Analysis ........contd.**

## If two functions f(t) and g(t) are square integrable in R 

- (i.e., *f*(*t*), *g*(*t*) *Є** L**2*(*R*) then their inner product is given by

- Equation (5) explains that the wavelet transformation gives a measure of the similarity between the signal and the wavelet function.

- (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

- (5)

## Data

| Station
Name | IAGA Code | Latitude | Longitude | Geomag. Latitude | Geomag.
Longitude | Data Range |
| --- | --- | --- | --- | --- | --- | --- |
| Rude Skov | RSV | 55.48°N | 12.46°E | 55.18°N | 99.52°E | 1930-81 |
| Eskdalemuir | ESK | 55.32°N | 3.20°W | 57.80°N | 83.96°E | 1941-99 |
| Niemegk | NGK | 52.07°N | 12.68°E | 51.88°N | 97.75°E | 1930-97 |
| Alibag | ABG | 18.64°N | 72.87°E | 10.19°N | 145.06°E | 1930-99 |

- 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).

**Europe**

**INDIA**

## Initial data processing methods like, corrections for missing values, jumps, filtering, etc. were made prior to further analysis

**Initial data processing methods like, corrections for missing values, jumps, filtering, etc. were made prior to further analysis**

## 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                      and
Higher order vanishing moments.

(This facilitates to aptly identify jerks, as jerks appear as a very narrow, high frequency component in the data).

- We have applied continuous wavelet transformation using those wavelets, that have
- High compact support                      and
- Higher order vanishing moments.
- (This facilitates to aptly identify jerks, as jerks appear as a very narrow, high frequency component in the data).

**Perform multiscale decomposition of the signal using an appropriate wavelet basis.**

## Comparison of the important properties of some 6 wavelets

| Properties | db4 | sym4 | gaus3 | coif2 | bior5.5 | db8 |
| --- | --- | --- | --- | --- | --- | --- |
| regularity | 0.8 | arbitrary | infinitely | arbitrary | 3 & 4 at knots | 1.6 |
| compact support | yes | yes | no | yes | yes | yes |
| symmetry | antisymmetric | Near symmetric | antisymmetric | Near symmetric | symmetric | antisymmetric |
| vanishing moments | 4 | 4 | arbitrary | 4 | 5 | 8 |

## ........and those of another 6 wavelets

| Properties | haar | coif1 | meyr | dmey | gaus1 | gaus2 |
| --- | --- | --- | --- | --- | --- | --- |
| regularity | Not continuous | arbitrary | infinitely | infinitely | infinitely | infinitely |
| compact support | yes | yes | no | no | no | no |
| symmetry | symmetric | Near  symmetric | symmetric | symmetric | antisymmetric | symmetric |
| vanishing moment | 1 | 2 | arbitrary | arbitrary | arbitrary | arbitrary |

## 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
- 	Where *n*(*t*) is the noise term as a function of time and *β*(*t-t*0) is defined as
- 	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****.**

*α*

*f*(*t*) =

*β*(*t-t*0)

- + *n*(*t*)

- (6)

## 	The Wavelet transform of equation (6) is given by

	The above equation can be simplified to obtain an equation of the form 
(Alexandrescu et al., 1995)

						constant

	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. 

- 	The above equation can be simplified to obtain an equation of the form
- (Alexandrescu et al., 1995)
- 						constant
- 	where *r**j* 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.

- (7)

## 	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,

	GJs are usually detected by those wavelets whose number of vanishing moments are greater than the regularity (α) of the GJ.

- 	In other words,
- 	_GJs are usually detected by those wavelets whose number of vanishing moments are greater than the regularity (__*α*__) of the GJ._

## 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+*Δ*t* (right neighborhood of *t*0) and *t-*Δ*t* (left neighborhood of *t*0). 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.

## t+Δt

- t-Δt

- t

- Time

- local maxima

- Left neighborhood

- Right neighborhood

- CWT coef

## 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 (*r**j*)) 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.

- Adhikari et al. (2007)

## Type 1:	Display positive linear trend (α > 0) for most 		of the dilation range

Type 2:	Display wobbly pattern (yet α > 0) and may 		arise due to the presence of harmonics 			present in the data.

Type 3:	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:	Display wobbly pattern (yet *α > 0*) and may 		arise due to the presence of harmonics 			present in the data.
- Type 3:	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.

**Types of Ridge Functions**

**The different types occur due to the presence of singularities, random noise, harmonic components, etc. in the data**

## Slide 30

## Comparison of the important properties of 6 wavelets showing all the LoM

| Properties | db4 | sym4 | gaus3 | coif2 | bior5.5 | db8 |
| --- | --- | --- | --- | --- | --- | --- |
| regularity | 0.8 | arbitrary | infinitely | arbitrary | 3 & 4 at knots | 1.6 |
| compact support | yes | yes | no | yes | yes | yes |
| symmetry | antisymmetric | Near symmetric | antisymmetric | Near symmetric | symmetric | antisymmetric |
| vanishing moments | 4 | 4 | arbitrary | 4 | 5 | 8 |

## Comparison of the important properties of 6 wavelets that did not show any LoM 

| Properties | haar | coif1 | meyr | dmey | gaus1 | gaus2 |
| --- | --- | --- | --- | --- | --- | --- |
| regularity | Not continuous | arbitrary | infinitely | infinitely | infinitely | infinitely |
| compact support | yes | yes | no | no | no | no |
| symmetry | symmetric | Near  symmetric | symmetric | symmetric | antisymmetric | symmetric |
| vanishing moment | 1 | 2 | arbitrary | arbitrary | arbitrary | arbitrary |

## 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. 

- 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.

## 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. 

- 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.

## What does this imply?

- 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

- Two possible interpretations!!

- This raises a fundamental question......

## 				
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?

- _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?_

## Utilize the CWT coefficients of the identified optimum wavelet(s) and estimate their phase characteristics.

- The wavelet transformation coefficients of the optimum wavelet(s) that remain after eliminating the spurious ones (through iterative least-squares sense) will be utilized to understand their phase characteristics.
- This is to know, if all the Geomagnetic Jerks have the same phase or not, during their occurrences?
- _If so what does that mean? _
- _If not, what does that mean?_
- This has important implications on the understanding of fluid motions of the Earth's outer core at that time, during and after the occurrence of Jerks.

## Thank You!

## Shapes of some wavelets used in the present study.

**Shapes of some wavelets used in the present study.**