Wavelet Based Estimation of the Hurst Exponent for the Horizontal Geomagnetic Field at MAGDAS Equatorial Stations

G. Gopir 1,2,* , N. S. A. Hamid 1,2 , N. Misran 1,3 , A. M. Hasbi 1,3 & K. Yumoto 4 1

Institute of Space Science (ANGKASA), Universiti Kebangsaan Malaysia, 43600 Bangi, Malaysia

2

School of Applied Physics, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Malaysia

3

Faculty of Engineering & Built Environment, Universiti Kebangsaan Malaysia, 43600 Bangi, Malaysia

4

Space Environment Research Center, Kyushu University, Fukuoka, Japan *Email: [email protected]

, [email protected]

ISWI - Helwan, Egypt - 6-10 November 2010 1

CONTENTS

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

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 - 6-10 November 2010 3

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 - 6-10 November 2010 4

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 - 6-10 November 2010 5

MAGDAS DATA AND STATIONS

Station, country Ce bu Philippine s Davao Philippine s Langkawi Malaysia Ge ographical coordinate 10.36

° N, 123.91°E 7.00°N, 125.40°E 6.30°N, 99.78°E Ge omagne tic coordinate Pe riod and re marks 2.53°N, 195.06°E 11 Aug 2005 - daily, quie t, min, se c 24 Aug 2005 - daily, active , min, se c 1.02°S, 196.54°E 11 Aug 2005 - daily, quie t, min, se c 24 Aug 2005 - daily, active , min, se c Fe b 2007 - monthly, quie t, min 2.32°S, 171.29°E Fe b 2007 - monthly, quie t, min

• 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.

Day 11 Aug 2005 24 Aug 2005 Dst 20 -216 Kp 2 9 Ap 6 110 Classification Quie t day (QD) Active day (AD)

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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 - 6-10 November 2010 7

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.

Station Date Count Minimum (nT) Maximum (nT) Range (nT) Ave rage (nT) Me dian (nT) Standard de viation (nT) Coe fficie nt of variance (%) Ske wne ss Kurtosis Ce bu Davao 11-Aug-05 24-Aug-05 11-Aug-05 24-Aug-05 1440 38909 38969 60 38929 38921 17.97

0.05

1.058

2.60

1440 38612 39036 424 38852 38823 118.62

0.31

-9.092

1.86

1440 39069 39139 70 39092 39080 21.66

0.06

21.009

2.42

1440 38757 39212 455 39003 38971 127.20

0.33

0.036

1.79

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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.

Station Count Minimum (nT) Maximum (nT) Range (nT) Ave rage (nT) Me dian (nT) Standard de viation (nT) Coe fficie nt of variance (%) Ske wne ss Kurtosis Davao 40320 39043 39230 187 39113 39101 31.5

0.08

1.30

3.92

Langkawi 40320 40404 40571 167 40467 40461 25.3

0.06

1.27

4.37

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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 - 6-10 November 2010 10

•

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 - 6-10 November 2010 11

• •

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 - 6-10 November 2010 12

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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DAILY DATA - SCALOGRAM

ISWI - Helwan, Egypt - 6-10 November 2010 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.

14

DAILY DATA –

wavelet transform coefficient ISWI - Helwan, Egypt - 6-10 November 2010 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.

15

DAILY DATA – wavelet variance

ISWI - Helwan, Egypt - 6-10 November 2010 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.

16

DAILY DATA – wavelet Hurst exponents

Hurst coefficients, H

w

, 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.

Station Ce bu Davao Sampling Minute Se cond Minute Se cond Date 11-Aug-05 (QD) 24-Aug-05 (AD) 0.48±0.02

0.47±0.02

0.36±0.01

0.38±0.01

0.75±0.01

0.73±0.01

0.61±0.02

0.62±0.01

H w

of 0.36-0.48 (i.e. anti-persistent) for solar quiet day (QD) and 0.61-0.75 (persistent) for active day (AD) for the second and minute sampled horizontal geomagnetic field ISWI - Helwan, Egypt - 6-10 November 2010 17

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 - 6-10 November 2010 18

MONTHLY DATA - SCALOGRAM

ISWI - Helwan, Egypt - 6-10 November 2010 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.

19

MONTHLY DATA –

wavelet transform coefficient ISWI - Helwan, Egypt - 6-10 November 2010 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.

20

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 - 6-10 November 2010 21

MONTHLY DATA – wavelet Hurst exponents

Hurst coefficients, H

w

, 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.

Station Davao Langkawi

H w

0.35±0.01

0.43±0.02

H w

of 0.35-0.43 (i.e. anti-persistent) for this solar quiet month for the minute sampled horizontal geomagnetic field at both stations ISWI - Helwan, Egypt - 6-10 November 2010 22

Wavelet Hurst exponents of SIMULATED FBM TIME SERIES

Hurst exponents, H

w

, 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.

H

0.4

0.5

0.6

0.7

N

1440 0.40±0.00

0.50±0.01

0.62±0.01

0.65±0.02

86400 0.41±0.01

0.50±0.00

0.60±0.01

0.71±0.01

H w

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 ISWI - Helwan, Egypt - 6-10 November 2010 23

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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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-FRGS0002-2007.

SHUKRAN … ISWI - Helwan, Egypt - 6-10 November 2010 25