1. Wavelet Based Estimation of the Hurst Exponent for the Horizontal Geomagnetic Field at MAGDAS Equatorial Stations
G. Gopir1,2,*, N. S. A. Hamid1,2, N. Misran1,3,
A. M. Hasbi1,3 & K. Yumoto4
1Institute of Space Science (ANGKASA), Universiti Kebangsaan Malaysia, Bangi, Malaysia
2School of Applied Physics, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia
3Faculty of Engineering & Built Environment,
Universiti Kebangsaan Malaysia, Bangi, Malaysia
4Space Environment Research Center, Kyushu University, Fukuoka, Japan *
ISWI - Helwan, Egypt November 2010

2. ISWI - Helwan, Egypt - 6-10 November 2010
CONTENTS
Introduction
Objectives
Data
Methodology
Result and Discussion
Conclusion
ISWI - Helwan, Egypt November 2010

3. ISWI - Helwan, Egypt - 6-10 November 2010
INTRODUCTION
The geomagnetic field, or Earth’s magnetic field:
known to be scaling, fractal and self-affine due to modulations by the magnetosphere and lithosphere
also non-stationary and contains transients during active or disturbed periods
thus geomagnetic time series could be analyzed using wavelet to extract the fractal parameter of Hurst exponent
Here, we apply the wavelet variance analysis to calculate the Hurst exponent for the horizontal component of the geomagnetic field
observed by the global network of the Magnetic Data Acquisition System (MAGDAS)
developed and installed by the Space Environment Research Center (SERC) of Kyushu University, Japan.
ISWI - Helwan, Egypt November 2010

4. ISWI - Helwan, Egypt - 6-10 November 2010
OBJECTIVES
Use wavelet analysis to calculate the Hurst exponent and determine fractal nature of geomagnetic field
Determine whether geomagnetic field for solar quiet and active periods have significantly different Hurst exponents
Determine whether geomagnetic activity could be quantified by fractal exponent
ISWI - Helwan, Egypt November 2010

5. ISWI - Helwan, Egypt - 6-10 November 2010
DATA
Time series for horizontal component, H, of geomagnetic field measured by MAGDAS
Data from MAGDAS equatorial stations of Cebu and Davao in the Philippines (UT+9); and Langkawi in Malaysia (UT+8)
Data sampled continuously with minute and second samplings for periods of one day and one month covering solar quiet and active periods
ISWI - Helwan, Egypt November 2010

6. MAGDAS DATA AND STATIONS
Extreme values of the geomagnetic indices of Dst, Kp and Ap in nanotesla (nT) for the studied days of August 2005 (from from World Data Center for Geomagnetism, Kyoto). The days are classified as quiet (QD) or active (AD) based on the threshold values of 50, 4 and 6 nT for Dst, Kp and Ap, respectively.
ISWI - Helwan, Egypt November 2010

7. DAILY DATA – MINUTE SAMPLING
The horizontal geomagnetic field component, H, sampled at time t of every minute at the MAGDAS stations of (a) Cebu on 11 August 2005, (b) Cebu on 24 August 2005, (c) Davao on 11 August 2005, and (d) Davao on 24 August 2005
ISWI - Helwan, Egypt November 2010

8. DESCRIPTIVE STATISTICS - of daily H data, minute sampling
Descriptive statistical parameters of the horizontal geomagnetic field, H, sampled every minute at the MAGDAS stations of Cebu and Davao in the Philippines for the quiet and active days of August 2005.
ISWI - Helwan, Egypt November 2010

9. MONTHLY DATA – MINUTE SAMPLING
The horizontal geomagnetic field, H, sampled every minute at time t for February 2007 at the MAGDAS stations of (a) Davao in the Philippines and (b) Langkawi in Malaysia.
ISWI - Helwan, Egypt November 2010

10. ISWI - Helwan, Egypt - 6-10 November 2010
METHODOLOGY
Continuous wavelet transform of horizontal geomagnetic field time series
Uses Mexican hat mother wavelet
Variance of wavelet transforme determined and scaling gives Hurst exponent, or Hurst coefficient
Wavelet variance method verified by simulation of fractional Brownian motion (FBM) time series
ISWI - Helwan, Egypt November 2010

11. WAVELET VARIANCE ANALYSIS
Continuous wavelet transform, or coefficient, for time series f(t):
Here g* is complex conjugate of g, and g is mother wavelet,
With t - position or translation parameter;
a – scale or dilatation parameter
Plot of modulus square of W(t,a) in t-a plane is known as scalogram
Here use Mexican hat mother wavelet,
 Variance of W(t,a),
If time series is scaling (linear), v(a) is a power law in a,
exponent of wavelet variance
ISWI - Helwan, Egypt November 2010

12. WAVELET HURST EXPONENT AND FBM
Double log plot of v(a) versus a gives
Wavelet Hurst exponent, Hw, is defined for ,
FGN – fractional Gaussian noise
FBM – fractional Brownian motion
FBM is defined by Mandelbrot and Ness (1968),
Or use the more recent definition of Abry and Sellan (1996)
In MATLAB, just a function away
ISWI - Helwan, Egypt November 2010

13. WAVELET ANALYSIS OF DAILY DATA - average subtracted from time series
The variation of horizontal geomagnetic field component, H, from the daily average and sampled at time t of every minute for the quiet day (QD) of 11 August 2005 and for the active day (AD) of 24 August 2005 at the MAGDAS stations of Cebu and Davao in the Philippines.
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14. ISWI - Helwan, Egypt - 6-10 November 2010
DAILY DATA - SCALOGRAM
Scalograms of the Mexican hat wavelet transform for the horizontal geomagnetic field at the Cebu MAGDAS station with minute sampling on the quiet day of (a) 11 August 2005 and on the active day of (b) 24 August 2005.
ISWI - Helwan, Egypt November 2010

15. DAILY DATA – wavelet transform coefficient
The wavelet transform coefficient, W(a,t), with Mexican hat mother wavelet for the horizontal geomagnetic field of the Cebu MAGDAS station with minute sampling on the quiet day of 11 August 2005.
ISWI - Helwan, Egypt November 2010

16. DAILY DATA – wavelet variance
Log of the Mexican hat wavelet variance, V(a), versus log of scale, a for the minute sampling of MAGDAS data at (a) Cebu on 11 August 2005, (b) Cebu on 24 August 2005, (c) Davao on 11 August 2005, and (d) Davao on 24 August 2005.
ISWI - Helwan, Egypt November 2010

17. DAILY DATA – wavelet Hurst exponents
Hurst coefficients, Hw, derived from the wavelet variance analysis using the Mexican mother wavelet for the daily horizontal geomagnetic field at the MAGDAS stations of Cebu and Davao during the quiet day (QD) of 11 August 2005 and the active day (AD) of 24 August 2005.
Hw of (i.e. anti-persistent) for solar quiet day (QD)
and (persistent) for active day (AD)
for the second and minute sampled horizontal geomagnetic field
ISWI - Helwan, Egypt November 2010

18. MONTHLY DATA WAVELET ANALYSIS - average subtracted
The variation of horizontal geomagnetic field component, H, from the monthly average and sampled at time t of every minute for the quiet month of February 2007 at the MAGDAS stations of (a) Davao in the Philippines and (b) Langkawi in Malaysia.
ISWI - Helwan, Egypt November 2010

19. MONTHLY DATA - SCALOGRAM
Scalograms of the Mexican hat wavelet transform for the horizontal geomagnetic field sampled every minute in February 2007 at the MAGDAS station of (a) Davao and (b) Langkawi.
ISWI - Helwan, Egypt November 2010

20. MONTHLY DATA – wavelet transform coefficient
The wavelet transform coefficient, W(a,t), with Mexican hat mother wavelet for the horizontal geomagnetic field of the Davao MAGDAS station with minute sampling during the quiet month of February 2007.
ISWI - Helwan, Egypt November 2010

21. MONTHLY DATA – wavelet variance
Log of the Mexican hat wavelet variance, V(a), versus log of scale, a, for the minute sampling of MAGDAS data in February 2007 at (a) Davao and (d) Langkawi
ISWI - Helwan, Egypt November 2010

22. MONTHLY DATA – wavelet Hurst exponents
Hurst coefficients, Hw, derived from the wavelet variance analysis using the Mexican mother wavelet for the horizontal geomagnetic field with minute sampling at the MAGDAS stations of Davao and Langkawi during the quiet month of February 2007.
Hw of (i.e. anti-persistent) for this solar quiet month
for the minute sampled horizontal geomagnetic field
at both stations
ISWI - Helwan, Egypt November 2010

23. Wavelet Hurst exponents of SIMULATED FBM TIME SERIES
Hurst exponents, Hw, derived from the wavelet variance analysis using the Mexican mother wavelet for simulated fractional Brownian motion (FBM) time series with input Hurst exponents of H. The sample sizes N are chosen to represent the number of minutes and seconds in a day, respectively.
Hw tend to diverge at larger H (i.e. nearer 1 or more persistent) for a smaller size sample (i.e. shorter time series).
Thus the larger the sample size of the time series, the better
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24. ISWI - Helwan, Egypt - 6-10 November 2010
CONCLUSION
Calculated Hurst exponents of horizontal geomagnetic field time series at equatorial regions using wavelet variance analysis
Found that the horizontal geomagnetic field is moderately anti-persistent (wavelet Hurst exponent of 0.35 – 0.48) during solar quiet period and persistent (wavelet Hurst exponent of 0.61 – 0.75) during solar active period.
Similar trends of Hurst exponents were observed for different sampling rates and different locations
The Hurst exponent, specifically derived by the wavelet variance method, could be used to quantify geomagnetic activity.
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25. ACKNOWLEDGEMENT AND THANK YOU
We thank the Space Environment Research Center (SERC) of Kyushu University, United Nation (UN), National Aeronautic and Space Administration (NASA), Japan Aerospace Exploration Agency (JAXA) and Helwan University for financial support and organisation of ISWI Workshop. This work was also supported by the Ministry of Higher Education (MOHE) of Malaysia under grant UKM-LL-02-FRGS
SHUKRAN …
ISWI - Helwan, Egypt November 2010