The geomagnetic AA* index: a study

Adv. Space Res.

0273-I 177/01$20.00+ 0.00

www.elsevier.com/locate/asr

PII: SO273-1177(01)00523-3

THE GEOMAGNETIC V.M. Silbergleit ’ Facultad de Ingenien’a,

Vol.28, No. 6, pp. 879-884,200l

0 2001 COSPAR. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain

Pergamon

AA* INDEX:

‘p2 , S.V. Gigola’

A STUDY

and C.E. D’Attellisl

de Buenos Aires, Au. Paseo Coldn 850, Piso 2, 10&G?-BuenosAires, Argentina National de Investigaciones Cientificas y Te’cnicas), Argentina.

hive&dad

2 CONICET(Consejo

ABSTRACT The temporal evolution of geomagnetic activity can be studied through the AA* index. Here, the time series of the above mentioned index is analyzed using the wavelet transform for estimating the Hurst parameter H. According to the H value obtained we confirm that the geomagnetic data, is a time series with long-range dependence characteristic, infering that the current geomagnetic activity is related to the prior activity a very long time ago (‘75 years). 0 2001 COSPAR. Published by Elsevier Science Ltd. All rights reserved.

INTRODUCTION Geomagnetic storms periods are considered as recurrent or sporadic according to their statistical behavior. Recurrent geomagnetic activity is controlled by high-speed solar wind streams originate in solar coronal holes and by the interplanetary magnetic field orientation (Russell and McPherron, 1973). Geomagnetic activity cycles are similar to sunspot cycles, though there are some important differences, as was considered by Joselyn (1996). There is an upward trend in overall geomagnetic activity since 1900, as was observed by Feynman and Crooker (1978). It is now confirmed that the seasonalities in the geomagnetic activity result from modulation in either solar wind parameters or the heliosphere-magnetosphere interaction. It was established that the annual variations of the geomagnetic activity have two origins: a) the interaction of Bz (southward component of the interplanetary magnetic field) with the magnetosphere and b) the variations of the angle between the Earth-Sun line and the geomagnetic dipole direction (Ruse11 and McPherron, 1973, Menvielle, 1996 and references therein). The modulation of the solar wind parameters in the neighborhood of the earth are related to the ll-year solar cycle and from the 27 days synodic rotation of the sun. In contrast to recurrent storms large random geomagnetic events develop at Earth as a result of aperiodic solar events and are usually associated with storms with sudden commencements. Their large intensities are related to the ejection of fast moving solar mass into interplanetary space which happens sporadically (Baker, 2000). Ruzmaikin et al. (1995) discussed the origin of the l/f low-frequency spectrum of fluctuations in the solar wind. They suggested that the causes of the “flicker noise” l/f spectrum of fluctuations observed in the solar wind is unambiguously related to the mechanism driving the wind out of the Sun. The present article examines the geomagnetic activity using the maximum average 24-hours disturbance (AA* index, baaed on linear aa indices for each major magnetic storm). Using wavelet and fractal analysis we verify that the AA* data series is the l/f-type (Beran, 1994), implying that the elementary classical statistics is not appropiate for studying this kind of data. GEOMAGNETIC aa AND AA* INDICES The aa index database begins in 1868 due to the work of Mayaud (1972). The network chosen (only two subauroral observatories) is imposed by the condition to go back as far as possible in the past (Ma,yaud, 1980). The 3-h values are not significant when they are considered one by one, only individual daily (or half daily) values are. These values represent the average amplitude, measured in nano tesla (nT), of the largest geomagnetic amplitude within the eight (or four) 3-h periods in each day (or half of a day). The geomagnetic AA* igdex is the maximum average 24-hour geomagnetic activity obtained from the aa indices

879

880

V. M. Silbergleit et al.

60.

I

50.

0

Fig. 1. AA* index values from

200

400

600

800

Months from February1911

1000

1911to 1997.

for magnetic storm periods. It is generated at World Data Center A for Solar Terrestrial Physics. According to the AA* criteria a “major geomagnetic storm” is considered in progress when the mean value of eight consecutive 3-h aa indices exceeds a threshold of 60 nT. The maximum mean value obtained is designated as AA*. These data are illustrated by Figure 1 which shows the AA* index values as ordinates and the time as abscissas. The geomagnetic data were downloaded from the National Geophysical Data Center (NGDC) in Boulder, Colorado via FTP at f tp ://ftp .ngdc .noaa. gov/STP/GEOMAGNETICDATA/INDICES/. The aa and AA* data sets provide a description of the geomagnetic activity over decades, and we use them to analyse the statistical behavior of geomagnetic activity. Previous papers using the AA* index were published by Silbergleit (1999a and 1999b). MATHEMATICAL Long-range

TOOLS

dependence

Phenomena that exhibit long-range dependence (LRD) characteristics have been found in many fields, including geophysical and economic time series, biological and speech signal, noise in electronic devices, burst errors in communications and traffic in computer networks (Abry and Veitch, 1998). In a wide-sense stationary processes the LRD phenomenon is characterized by a power-law decrease of the autocorrelation function C(r) for large lags 7, i.e., C(r) behaves as (l/~)~ with 0 < 01 < 1, r -+ 00. Also the LRD can be defined by imposing a condition on the spectral density S(w): S(w) behaves as l/lwlp, 0 < ,S < 1, when w --t 0 (Beran, 1994). LRD phenomena are also intimately connected to the properties of scale invariance, self-similarity or fractals signals, and therefore often associated to stochastic processes as fractional Brownian motion (fBm) (Mandelbrot and van Ness, 1968), which presents such characteristics. Fractional

Brownian

Motion

The fractional Brownian BH(~), t E R such that:

motion

and Gaussian

Noise of Hurst parameter N, with 0 < H < 1, is a zero mean Gaussian

1. BH(0) = 0 2. the differences

BH(t + s) - BH(~) h ave finite dimensional

normal

distributions.

process

AA* Index

881

l0g~(2~) (scale)

Fig. 2. Power Multiresolution

The fBm has two important

analysis for the AA* time series in Fig. .l.

characteristics

(Mandelbrot

and van Ness, 1968):

0 nonstationarity; 0 self-similarity. The nonstationary

character

E[BH(t)BH(S)] = ;

of fBm is evidenced

by its autocorrelation

function

given by

[Iti2H+ IS12H - It - s,2H] ,

(1)

where o2 = r(l

- 2H)-,

cos nH

(2)

and I? is the Gamma function. From this, the variance of a fBm is W[BH(t)]

= 02]t]?

(3)

These stochastic processes generalize the ordinary Brownian motion, obtained as special case for H = l/2. The fBm is a nonstationary process with stationary increments, i.e., GH,~(~) = BH(~ + T) - BH(~) is a stationary process named fractional Gaussian noise (fGn). The spectrum of fGn reads as l/]~]~~-~ (Abry and Veitch, 1998). As it was mentioned, LRD is defined by the slow power-law decrease of the autocorrelation function: 1 C(k) = Cl------k2-2H

’

k +

co,

where cl > 0 and $ < H < 1. The Hurst parameter corresponds to the classical case of short-range dependence.

H measures

long-range

dependence;

H =

i

882

V. M. Silbergleit er al.

0.2' 0

Fig. 3. Autocorrelation

An equivalent

statement

In the next section

a method

800

600

400

I 1000

L (lags in units of a month)

and its significant

for the spectrum

- l)sin(7r(l-

c2 = $T(2H

Wavelet

Function

200

values.

is

H)).

(6)

for estimating

H based on wavelets is introduced.

analysis

The wavelet transform, which is essentially a time-scale analysis, is an appropriate tool for analyzing this type of stochastic processes, and it was used recently by several authors (Flandrin, 1992; Chen and Lin, 1994; Hirchoren and D’Attellis, 1997; Hirchoren and D’Attellis, 1998). In a multiresolution framework (Chui, 1992; Daubechies, 1992), the wavelet series of a process X(t) up to the scale J is given by +C== c &)$6(2-Jt

X(t) = 2-J/2

- Ic) +

where g(t)

k j=-m

k=-w

is the wavelet function,

2-i’2

“c”

dj(rc)?j!J(2+t - Ic),

4(t) is the scaling function,

and +C=

+CQ dj(lc) = 2N2

/

x(t)$J(2-jt

(7)

k=-m

- Ic)&

--oo

aj(k)

= 2-j/2 .I

x(t)4(2-h

- Icpt;

(8)

-cc

j(scale) and k(time) are integer numbers, dj(lc) the detail or wavelet coefficients and aj(lc) the approximation coefficients. In the case of fGn process, the sequence of wavelet coefficients satisfies (Abry and Veitch, 1998) Zogz[var(dj(lc))]

= (2H - 1)j + constant.

(9)

883

AA’ Index

Using Eq. 9 the parameter H can (in a log:, - log2 plot), as illustrated the data analyzed is 1024, nine levels is a dyadic analysis, i.e. 2j with j = We observe that

war[dj(k)] =

be obtained from the slope of the variance plotted as a function of j in Figure 2 for the AA* index plotted in Figure 1. Since the length of in the multiresolution analysis are used (Remark: the multiresolution 1, . . . . 9).

E[d$)] - E2[dj(k)] = E[dj(k)],

(10)

of H depends on since dj(lc) is given by Eq. 8 and X(t) is a zero mean process. Then the estimation E[($(Ic)], and, in consequence, an estimation of the mean value of d;(k) is necessary. As it was pointed out by Beran (1994), even the most elementary of classical statistics require a revision in face of LRD. According to D’Attellis and Hirchoren (1999), it is possible to use a wavelet function such that the standard estimation of E[$(Ic)], given by

(11) where j = 1, . . . . J and nj is the number of the wavelet coefficients in the scale j, is valid. Eq. 11 is the power in the scale j, when an orthonormal multiresolution is used.

The meaning

of

RESULTS AND CONCLUSIONS In the previous sections a review of the definition of the AA* index (and its capability to characterize perturbed periods) and the mathematical tools adequate to analize this data set are presented. The AA* index values from 1911 to 1997 are illustrated by Figure 1 which also shows that the geomagnetic activity is modulated in a quite complex way by the ll-year solar cycle. The results of the wavelet analysis are depicted by stars in Figure 2. The solid straight line is the least-square approximation with slope equal to 0.86 & 0.12, from which Eq. 7 leads to H = 0.93 5 0.06. This shows that the studied data have the characteristics of LRD processes (because H is greater than 0.5 and smaller than 1). It is important to note that Eq. 4 determines only the decay of the correlations. It does not say that there are some specific lags for which C(k) is particularly large. In order to quantify the long range dependence found in the AA* times series, the autocorrelation function is shown in Figure 3. A standard method for checking for non-zero correlation is the following (Priestley, 1981): C(lc) is considered significant if ]C(lc)( > 2/,/Z As Figure 3 shows, there are significant values of the autocorrelation function for lags of 900 months; however, the first value inside de band corresponds to Ic = 40. Taking into account the above mentioned results, we infer that the current geomagnetic activity is related to the prior activity that occurred a long time ago. Both the solar wind and geomagnetic conditions independently exhibit the same long-term dependence suggesting that their states are correlated on long time scale. ACKNOWLEDGMENTS The authors are grateful to the referees for helpful comments, and also thanks to L. Morris and J. Allen of WDC-A at STP for generating the data. This work was partially supported by Facultad de Ingenieria, Universidad de Buenos Aires and CONICET, of Argentina. REFERENCES Abry, P., and D. Veitch, Wavelet analysis of long range dependent traffic, IEEE nansuctions on Information Theory, 44, 2-15, 1998. Baker, D.N., Effects of the sun on the earths environment, Journal of Atmospheric and Solar-Terrestrial Physics, 62, 1669-1681, 2000. Beran, J., Statistics for long-memory processes, Chapman & Hall, New York, NY, 1994. Chen, B.S., and C.W. Lin, Multiscale Wiener filter for restoration of fractal signals: Wavelet filter bank approach, IEEE Transactions on Signal Processing, 42, 2972-2982, 1994. Chui, C.K., An introduction to wavelets, San Diego: Academic Press, 1992.

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D’Attellis, C.E., and G.A. Hirchoren, Wavelet-based estimations in fractional Brownian motion, Latin American Applied Research, 29, 221-225, 1999. Daubechies, I., Ten lectures on wavelets, SIAM, 1992. Feynman, J., and N.U. Crooker, The solar wind at the turn of the century, Nature, 275, 626-627, 1978. Flandrin, P., Wavelet analysis and synthesis of fractional Brownian motion, IEEE Transactions on Information Theory, 38, 910-917, 1992. Hirchoren, G., and C.E. D’Attellis, On the optimal number of scales in estimation of fractal signals using wavelets and filter banks, Signal Processing, 63, 55-63, 1997. Hirchoren, G., and C. E. D’Attellis, Estimation of fractal signals using wavelets and filter banks, IEEE Transactions on Signal Processing, 46, 1624-1630, 1998. Joselyn, J.A., Geomagnetic activity during solar cycles 12-22 and the outlook for cycle 23, in STPW-V Proceedings, eds. G. Heckman, K. Marubashi, M.A. Shea, D.F. Smart and R.Thompson, 297-300, 1996. Mandelbrot, B., and J. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10,422-437, 1968. Mayaud, P.N., The aa indices: A 100 years series characterizing the magnetic activity, Journal of Geophysical Research, 77, 6870-6874, 1972. Mayaud, P.N., Derivation, meaning and use of geomagnetic indices, in Geophysical monograph, AGU, 22, pp. 76-85, 1980. Menvielle, M., Geomagnetic indices and geomagnetic activity forecasting, in STPW-V Proceedings, eds. G. Heckman, K. Marubashi, M.A. Shea, D.F. Smart and R.Thompson, 286-296, 1996. Priestley, M.B., Spectral analysis of time series, Academic Press, London, 1981. Russell, C.T., and R. L. McPherron, Semiannual variation of geomagnetic activity, Journal of Geophysical Review, 78, 92-108, 1973. Ruzmaikin, A.A., B.E. Goldstein, and E.J. Smith, On the origen of the l/f spectrum of fluctuations in the solar wind, in AIP Conference Proceedings, eds. D. Winterhalter, J. T.Gosling, S.R.Habbal, W.S.Kurth and M.Neugebauer382, 225-228, 1995. Silbergleit, V.M., Forecast of the most geomagnetically disturbed days, Earth, Planets and Space, 51, 19-22, 1999a. Silbergleit, V.M., The most geomagnetically disturbed 24 hours, Studia Geophysics et Geodaetica, 43, 194200, 1999b.