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Nonlin. Processes Geophys., 20, 455–466, 2013
Advances
in
Annales
www.nonlin-processes-geophys.net/20/455/2013/
doi:10.5194/npg-20-455-2013
Geosciences
Geophysicae
© Author(s) 2013. CC Attribution 3.0 License.
Chemistry
Chemistry
1 , R. De Marco1 , and P. De Michelis2
G.
Consolini
and Physics
and Physics
1 INAF
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Discussions
Intermittency and multifractional
Brownian character of
Atmospheric
geomagnetic time series Atmospheric
– Istituto di Astrofisica e Planetologia Spaziali,Discussions
00133 Rome, Italy
Nazionale di Geofisica e Vulcanologia, 00143 Rome, Italy
2 Istituto
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Atmospheric
Atmospheric
Correspondence to: G. Consolini ([email protected])
Measurement
Measurement
Received: 30 October 2012 – Revised: 24 April Techniques
2013 – Accepted: 1 June 2013 – Published: 11 July 2013
Techniques
Discussions
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and criticality during geomagnetic storms and substorms,
Abstract. The Earth’s magnetosphere exhibits a complex behavior in response to the solar wind conditions.
This
beBiogeoscienceswhich are the most important manifestations of the magneogeosciences
tospheric activity (e.g., Klimas et al., 1996, 2000; Consolini
havior, which is described in terms of mutifractional BrowDiscussions
and Chang, 2001; Dobias and Wanliss, 2009). These studies
nian motions, could be the consequence of the occurrence
have clearly evidenced that the magnetospheric evolution is
of dynamical phase transitions. On the other hand, it has
characterized by a near-criticality dynamics, which manifests
been shown that the dynamics of the geomagnetic signals is
also characterized by intermittency at the smallest temporal
Climateboth in the scale-invariant nature of magnetotail relaxation
Climate
events and fractal properties of geomagnetic time series.
scales. Here, we focus on the existence of a possible relaof
the
Past
It has been found that the fractal character of several geotionship
in
the
geomagnetic
time
series
between
the
multiof the Past
Discussions
magnetic time series, for example those relative to the geofractional Brownian motion character and the occurrence
of
magnetic indices AE, Dst and SYM-H, changes during magintermittency. In detail, we investigate the multifractional nanetic storms and substorms (Uritsky and Pudovkin, 1998;
ture of two long time series of the horizontal intensity of the
Earth’s magnetic field as measured at L’Aquila
Geomagnetic
Wanliss,
2005; Balasis et al., 2006; Wanliss and Dobias,
Earth System
Earth System
Observatory during two years (2001 and 2008), which cor2007; Dobias and Wanliss, 2009). In particular, it has been
Dynamics
respond to different conditions of solar activity. We
propose
noticed that the self-similarity properties of these time series
Dynamics
Discussions
a possible double origin of the intermittent character of
the
change when an intense magnetospheric/geomagnetic activsmall-scale magnetic field fluctuations, which is related to
ity event approaches. For instance, Balasis et al. (2006) have
both the multifractional nature of the geomagnetic field and
shown the occurrence of an interesting transition from the
Geoscientific
Geoscientific
anti-persistent to the persistent character in the fluctuations
the intermittent character of the disturbance level.
Our results
nstrumentation
Instrumentation
suggest a more complex nature of the geomagnetic
response
of Dst index, which closely corresponds to a transition from a
to
solar
wind
changes
than
previously
thought.
Methods and
Methods andregime of quiet to one of high magnetospheric activity, characterized by the occurrence of intense geomagnetic storms.
Data Systems
Data SystemsA similar behavior has been also documented in the case of
Discussionsthe auroral electrojet (AE) index (e.g., Uritsky and Pudovkin,
1 Introduction
Geoscientific1998). This result suggests that the magnetospheric dynamGeoscientific
ics may arise from a combination of solar wind changes
The Earth’s magnetosphere, which
is the region
of space
Model
Development
and internal magnetospheric activity. However, variations of
Development
where the geomagnetic field is confined by the solar Discussions
wind,
the self-similarity features of geomagnetic indices have also
can be described as a nonequilibrium nonlinear complex sysbeen observed in the distribution functions of the smalltem. Its complex dynamics, which results from the superposcale increments (Consolini and De Michelis, 1998). These
of the internal dynamics and the external
driving dueandchanges, which usually arise from nonequilibrium dynamiHydrology
Hydrologysition
and
to the changes of solar wind conditions, has been widely incal phase transitions occurring commonly at the onset of and
Earth
System
Earth System
vestigated during the past two decades. Several
studies
have
during magnetic storms and substorms, have been interpreted
been devoted to examine the occurrence of chaos, turbulence
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Sciences
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Discussions
Ocean Science
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cean Science
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Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.
456
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
and discussed in a more general scenario, involving a near
forced and/or self-organized criticality (FSOC) dynamics
(Consolini, 1997; Uritsky and Pudovkin, 1998; Chang, 1999;
Chang et al., 2003; Consolini and Chang, 2001; Consolini
and De Michelis, 2002; Wanliss and Weygand, 2007; Wanliss
and Uritsky, 2010).
Another observable property of the magnetospheric dynamics, which is documented in literature, is the intermittent character of geomagnetic time series, i.e., the anomalous scaling of small-scale increments (fluctuations) and the
presence of a multifractal character (see, e.g., Consolini et
al., 1996; Consolini and De Michelis, 1998; Kovács et al.,
2001; Wanliss et al., 2005; Consolini and De Michelis, 2011,
etc.). This intermittent character seems to be a common property of magnetospheric dynamics that is generally observed
in both the time series of geomagnetic indices and in situ
magnetospheric and ground-based magnetic field measurements. The origin of this intermittent and multifractal character has been interpreted as a direct consequence of dynamical changes and turbulent dynamics. In this framework,
Dobias and Wanliss (2009) recently noticed that the intermittent character observed during magnetic substorms and
storms displays similar features when it is investigated in
terms of fractal processes. This observation suggests that during magnetic storms and substorms there are common dynamical processes at the basis of the critical behavior of the
Earth’s magnetospheric dynamics in response to solar wind
changes (Dobias and Wanliss, 2009).
The aforementioned different features seem reasonably to
suggest that the nature of the geomagnetic time series, and in
particular of the geomagnetic indices, may be similar to those
of multifractional Brownian motion (mBm) (Muniandy and
Lim, 2001), where the scaling features change in time. In this
case, we can legitimately ask what the link is between the observed intermittent and multifractal character of geomagnetic
signal and its possible multifractional Brownian nature.
This paper is organized as follows. At first, a brief summary of the main properties of fractional and multifractional
Brownian motions are described. Following this, the relationship between intermittency and mBm in geomagnetic time
series is analyzed. Finally, the obtained results are summarized and the implications of the findings are discussed.
2
Fractional and multifractional Brownian motion:
some general features
Fractional Brownian motion (fBm), also called fractal Brownian motion, was introduced by Mandelbrot and Van Ness
(1968) as a natural extension of ordinary Brownian motion.
A fBm has the property of being statistically self-similar,
which means that any portion of it can be viewed, from a
statistical point of view, as a scaled version of a larger part
of the same process. Unlike the classical (also known as ordinary or standard) Brownian motion, which is characterized
Nonlin. Processes Geophys., 20, 455–466, 2013
by independent increments, the fBm has a non-Markovian
character, which is manifested in the correlation between past
and future increments leading to persistent or anti-persistent
behavior.
According to Mandelbrot and Van Ness (1968) the simplest way of defining a fBm BH (t) is based on a modified
fractional Weyl integral,
nR
h
i
1
1
0
BH (t) = 1 1 −∞ (t − s)H − 2 − (−s)H − 2
0 H+2
(1)
o
Rt
1
×dB(s) + 0 (t − s)H − 2 dB(s) ,
where 0(x) is the Euler Gamma function, and H is the Hurst
exponent (sometimes also called Hölder exponent), which
varies in the interval (0, 1). The main features of the fBms
are (i) a zero mean displacement,
hBH (t)i = 0,
(2)
where h i denotes the ensemble average and (ii) a timeincreasing variance of increments,
h[δBH (s)]2 i ∝| s |2H ,
(3)
where δBH (s) = BH (t + s) − BH (t) is the increment at the
timescale s. Classical Brownian motion, which is characterized by time-independent increments, is recovered for
H = 12 . Conversely, the increments are negatively correlated (i.e., to a positive/negative fluctuation it follows a negative/positive fluctuation with a higher probability) when
0 < H < 1/2, and they are positively correlated when 1/2 <
H < 1. Moreover, the increments of fBm are stationary and
self-similar with the parameter H :
n
o
1
{δBH (s)} = λ−H δBH (λs) ,
(4)
1
where {X} = {Y } denotes that X and Y have the same finite
distribution functions.
The fBms, being self-similar processes, allow us to conveniently describe irregular signals, which arise from several
real situations. However, the pointwise regularity of a fBm is
the same along its path, and this property is sometimes undesired, restricting the application field. For this reason, in
the second half of the 1990s the definition of fBm, characterized by a global value of H , was generalized about the case
in which H is no longer independent of time, but a function of it. Consequently, in the generalization of fBm, i.e.,
mBm (Peltier and Lévy-Vehel, 1995; Benassi et al., 1997),
the Hurst exponent H depends on time, and all the features of
fBms are valid only locally. Among the different definitions
of mBms one of them, based on the fractional Weyl integral
representation, is easily obtained from Eq. (1) considering,
contrary to what happens for fBm, a time-dependent Hurst
exponent H (t). Thus, the increments of a mBm are no longer
stationary and the process is no longer globally self-similar. It
www.nonlin-processes-geophys.net/20/455/2013/
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
457
1.0
H = 0.3
H(t)
0.8
0.6
0.4
0.2
H = 0.5
XH(t)(t)
BH(t)
0.0
0.8
0.4
0.0
-0.4
H = 0.7
t
Fig. 2. A sample of mBm (lower panel), generated according to
the procedure described in Muniandy and Lim (2001) with a timet
Fig. 2. A sample of mBm (lower panel), generated according to the procedure described in Muniandy and
varying Hurst/Hölder exponent H (t) (upper panel). Note the non(2001) with a time
varying Hurst/Hölder
H(t) (upper panel). Note the non-stationary character o
stationary
character ofexponent
the fluctuations.
Fig. 1. Three samples of fBm characterized by different values of
fluctuations.
the Hurst exponent H . The fBm samples are generated from the
. 1. Three samples of fBm characterized by different values of the Hurst exponent H. The fBm samples are
algorithm described in Benassi et al. (1997) with the same sequence
Figure so
1 shows some samples of fBms, characterized by
eratedfrom theofalgorithm
et al. (1997)the
with
the same sequence
random
numbers,
random described
numbers,insoBenassi
as to emphasize
differences
inshows
termsofof
the
non-Gaussian
shape of the PDFs for the set of fBms and mBm, reported in Fi
different
values
of the Hurst exponent, while Fig. 2 displays
fluctuations.
o emphasize the differences in terms of fluctuations.
1 and 2. More
exactly, of
Figure
4 reports
the PDFs
the smallest
(τ = 1) increm
a sample
mBm.
These samples
areforgenerated
fromscale
algo-
rithms
described
in Benassi
et al. (1997)
Muniandy
δBH (τ ) and δX
), computed
using synthetic
signalsand
of more
than 5and
· 105 points. Th
H(t) (τ
ere h i denotes
ensemble
average,
and ii)
a time-increasing
variance
is the
possible
to use
a simpler
representation
of a mBm
XHof(t)increments
(t)
Asfor
it can
be seen from
the plots, thenature of the m
135 crements have Lim
been (2001),
rescaledrespectively.
to unit variance
comparison.
The nonstationary
2
based on a Riemann–Liouville fractional integral (RLMBM)
BH (s)] i ∝| s(Muniandy
|2H ,
and Lim, 2001),
fBm and mBm fluctuations have a different character. Indeed,
increments (fluctuations)(3)
manifests itself by a significant departure of its PDF from the Gaus
while the fBm fluctuations are stationary in amplitude, the
shape. The mBm
PDF
is clearly leptokurtotic
and itsnonstationary
shape depends character,
on the increment time s
mBm
fluctuations
exhibit a clearly
Zt
ere δBH (s) = BH (t + s) − BH (t) is the increment
at the time scale s. Classical Brownian moand becomeby
highly
relevant during
time
intervals
which
1
1
τ , as clearly demonstrated
the dependence
of the those
kurtosis
κ of
the mBm
increments δXH(t
(t −
XH (t) (t) = bytime independent
s)H (t)− 2 dB(s),
n, which is characterized
increments,
is recovered for(5)
H = 12 . are
Conversely,
characterized
by
very
low
Hurst/Hölder
exponent
values.
1
on the timescale τ (see Figure 5). This means that we are not capable of constructing a m
0 H (t) +
These
two properties (nonstationarity and dependence on H
increments are negatively correlated 2(i.e.,0 to a positive/negative fluctuation it follows
a nega140 scale-invariant shape of the increment PDFs at different time scales τ , using a single scaling e
values) of the mBm fluctuations are confirmed by looking at
e/positive fluctuation
withuse
a higher
probability)
0 < H <of
1/2,
they
are positively correwhere we
the notation
XH (t)when
(t), instead
BHand
(t),
to in2 (t) (computed
nent.
This is the
intermittent
of mBm, which
the counterpart o
the evidence
trend of ofthethelocal
variancecharacter
σloc
on aismovdicate
a
multifractional
Brownian
motion.
ed when 1/2 < H < 1. Moreover, the increments of fBm are stationary
and
self-similar
with
the
window of 101 points in the case of a mBm synthetic
nonstationaritying
of H(t).
Starting from this definition, a direct scheme to generate a
signal) as a function of the local Hurst exponent (Fig. 3) and
mBm has been implemented (Benassi et al., 1997; Muniandy
analyzing simultaneously the shape of the probability den This scheme uses uncorrelated unit variance
∆ and Lim, 2001).
3 Data Description
and
BH (s)} = λ−H δBH (λs) ,
(4)Analysis
sity functions
(PDFs) of fBm and mBm increments, δBH (τ )
noise, i.e.,
and δXH (t) (τ ), respectively. Indeed, Fig. 3 clearly shows a
∆
2 geomagnetic
ere {X} = {Y } denotes that jX and Y have the same finite distributionTofunctions.
investigate the
link between
intermittent
character σofloc
series and their
decrease
of thethe
local
signal variance
(t) with thetime
local
X ηi Hurst/Hölder
exponent
values,
while
Fig.
4
shows
the
nonXH (tjself-similar
wallow
(6) irregular
√
The fBms, being
processes,
us to conveniently
describe
signals,
145 tifractional
nature we
consider two sets of data relative to the geomagnetic measurements o
j −i+1 1t,
) (tj ) =
1t
Gaussian shape of the PDFs for the set of fBms and mBm,
i=1
horizontal
intensty
of the Earth’s magnetic field recorded during two distinct years:
ich arise from several real situations. However, the pointwise regularity
of a fBm
is the(H)
same
reported in Figs. 1 and 2. More exactly, Fig. 4 reports the
where
1t
=
1/(N
−
1),
N
equals
the
length
of
the
series,
η
2001
thefield.
yearfor
2008,
correspond
of maximum
solar acti
i and PDFs
ng its path, and this property is sometimes undesired, restricting year
the application
For
the which
smallest
scale (τto=periods
1) increments
δBHand
(τ )minimum
and
is a unit variance Gaussian white noise with zero mean,
and
respectively.
Data
refer
to
magnetic
field
measurements
recorded
at
L’Aquila
Geomagnetic
O
δX
(τ
),
computed
using
synthetic
signals
of
more
than
s reason, in the second half of the 90s the definition of fBm, characterized by a globalHvalue
(t) of
wi is a weight function (see Muniandy and Lim, 2001 for
5 points. The increments have been rescaled to unit
5
×
10
vatory,
which
of INTERMAGNET program, whose objective is to establish a gl
was generalized
the defined
case in which
H is no longer independent
on time,
butisapart
function
moreabout
details)
as
variance for comparison. The nonstationary nature of the
"
#1/2
mBm increments (fluctuations) manifests itself by a signifiti2H − (ti − 1t)2H
1
6 Gaussian shape. The mBm
cant departure of its PDF from the
4
wi =
.
(7)
2H 1t
PDF is clearly leptokurtotic and its shape depends on the in0(H + 21 )
crement timescale τ , as clearly demonstrated by the depenWe emphasize that synthetic mBm generated by means of
dence of the kurtosis κ of the mBm increments δXH (t) (τ )
RLMBM approximation (Eq. 6) results from the addition of
on the timescale τ (see Fig. 5). This means that we are not
weighted random values.
capable of constructing a master scale-invariant shape of the
ameter H
www.nonlin-processes-geophys.net/20/455/2013/
Nonlin. Processes Geophys., 20, 455–466, 2013
0
10
-2
σ
458
2
loc(t)
10
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
-4
10
-6
10
200
-8
Kurtosis
10
-10
10
0.0
0.2
0.4
0.6
0.8
1.0
H(t)
150
100
50
ig. 3. Plot of the local variance
2
σloc
(t)
versus the local Hurst/Hölder exponent H(t)
0
10
0
1
10
10
2
10
3
4
10
τ
2 (t) versus the local
Fig. 3. Plot of the local variance σloc
Hurst/Hölder exponent H (t).
Fig. 5. Dependence of the kurtosis of the synthetic mBm increments
δXHof(t)the
(τ kurtosis
) on theoftimescale
τ . mBm increments δXH(t) (τ ) on the timescale τ .
Fig. 5. Dependence
the synthetic
10
24.4
of co-operating high-quality digital magnetic observatories.
2001 Solar maximum
Therefore,
we
make
use
of
magnetic
field
recordings
ob24.2
tained by a permanent observatory fulfilling the same inter1
24.1
national standard of those used for constructing most of the
24.0
23.9
commonly used geomagnetic indices. In particular, the data
0.1
23.8
used in this standard are characterized by a sampling rate of
23.7
1 min and a1-04-2001
resolution of
0.1 nT, which
is higher1-10-2001
than the 1-12-2001
1-02-2001
1-06-2001
1-08-2001
usual
resolution
of
geomagnetic
indices,
allowing
a better
24.4
0.01
investigation
of
small-scale
fluctuations.
Furthermore,
these
24.3
mBm
2008 Solar
data, which are obtained from a geomagnetic observatory
at minimum
fBm H = 0.7
24.2
◦
0
◦
0
fBm
H
=
0.5
mid-latitude (Lat. 42 23 N, Long. 13 19 E) located in the
24.1
0.001
fBm H = 0.3
24.0
center of Italy, well represent the characteristic scenario in
23.9
the evolution of geomagnetic field, and the selected magnetic
-4
-2
0
2
4
23.8
field component is reflective of the Earth’s space environδBH/σ; δXH/σ
23.7
ment providing
important1-06-2008
information1-08-2008
about the state
of ge- 1-12-2008
1-02-2008
1-04-2008
1-10-2008
omagnetic activity. Indeed, the competing balance between
Fig. 4. Comparison among the PDFs of variance-normalized increthe Earth’s intrinsic magnetic field and solar wind dynamical
ments,
δBthe
) and
), of the mBm
and fBms
in (τ ), of the mBm
H (τ
H (t) (τ
ig. 4. Comparison
among
PDFs
of δX
variance
normalized
increments,
δBH (τreported
) and δXH(t)
changes drives much of the variations of the Earth’s space
previous figures. The shape of the function for the fBms is the same
nd fBms reported in previous figures. The shape of the function for the fBms is the same for all
H values.
environment
that are observed as magnetic field fluctuations
for all H values.
Fig. 6. The geomagnetic field horizontal intensity H, as recorded at L’Aquila geomagnetic Observatory (
at short timescales at ground level in our time series. Thus,
in 2001 (upper panel) and 2008 (lower panel).
our dataset can be considered to contain information on lowlatitude geomagnetic disturbances.
increment PDFs at different timescales τ , using a single scal150 network
digital
magnetic
observatories.
Therefore, we make
Figure 6high-quality
reports the two
datasets
of geomagnetic
measureing exponent. This is the evidence of the intermittent
charac- of co-operating
ments
during the
two activity
periods. At
first glance,
we im-the same inte
ter of mBm, which is the counterpart
of the nonstationarity
of magnetic field
recordings
obtained
by a permanent
observatory
fulfilling
7
mediately notice the different character and amplitude of the
of H (t).
tional standard of those used for constructing most of the commonly used geomagnetic ind
magnetic field fluctuations, which is due to the different soIn particular, lar
theactivity
data used
in this
standard
are characterized
by a the
sampling
level.
From
a statistical
point of view,
fluc- rate of 1 mi
tuations
H −ishHi)
of than
thesethe
two
datasets
are characand a resolution
of 0.1(δH
nT, =
which
higher
usual
resolution
of geomagnetic ind
3 Data description and analysis
2
2
terized
by a large
thefluctuations.
variances, σFurthermore,
5.3, data, which
155 allowing a better
investigation
of ratio
smallof
scale
2001 /σ2008 ∼ these
To investigate the link between the intermittent character of
and non-Gaussian distribution functions (here not ◦shown),
obtained from a geomagnetic observatory at1mid latitude (Lat.1 42 23’ N, Long. 13◦ 19
geomagnetic time series and their multifractional nature,
we
which show a more skewed (γ2001 ' −2.8 and γ2008 ' −0.9,
consider two sets of data relative to the geomagnetic meawhere γ 1 is the sample skewness) and kurtotic (κ2001 ' 18
surements of the horizontal intensity (H) of the Earth’s magand κ2008 ' 6, where κ is the sample
kurtosis) character in
8
netic field recorded during two distinct years: the year 2001
the case of the 2001 dataset. Furthermore, there is a slight
and the year 2008, which correspond to periods of maxidifference in the mean value of the horizontal intensity H of
mum and minimum solar activity, respectively. Data refer
the Earth’s magnetic field (1hHi ∼ 90 nT), which is due to
to magnetic field measurements recorded at L’Aquila Gethe secular variation of the field.
omagnetic Observatory, which is part of INTERMAGNET
program, whose objective is to establish a global network
[µT]
[µT]
H
Probability density function
24.3
H
Nonlin. Processes Geophys., 20, 455–466, 2013
www.nonlin-processes-geophys.net/20/455/2013/
τ
5. Dependence
of theand
kurtosis
synthetic mBm
increments
G. ConsoliniFig.
et al.:
Intermittency
mBmofinthegeomagnetic
time
series δXH(t) (τ ) on the timescale τ .
459
24.4
2001 Solar maximum
24.3
[µT]
24.2
24.1
H
24.0
23.9
23.8
23.7
1-02-2001
1-04-2001
1-06-2001
1-08-2001
1-10-2001
1-12-2001
24.4
24.3
2008 Solar minimum
[µT]
24.2
24.1
H
24.0
23.9
23.8
23.7
1-02-2008
1-04-2008
1-06-2008
1-08-2008
1-10-2008
1-12-2008
Fig. 6. The geomagnetic field horizontal intensity H, as recorded at L’Aquila Geomagnetic Observatory (Italy) in 2001 (upper panel) and
2008 (lower panel).
Fig. 6. The geomagnetic field horizontal intensity H, as recorded at L’Aquila geomagnetic Observatory (Italy)
To fully characterize the differences among the fluctuaTo analyze the multifractional character of the geomagin 2001 (upper panel) and 2008 (lower panel).
tions in the two datasets, we evaluate the kurtosis κ of the
netic time series, the first step of our analysis is to compute
magnetic field increments, δH(τ ) = H(t + τ ) − H(t), at difthe power spectral density S(f ) of the magnetic field fluctuaferent
τ . Kurtosis
is indeed
measure
tions by
applying
the
standard
Fourier
spectral
analysis.
Fig150 network of co-operating high-quality digital magnetictimescales
observatories.
Therefore,
weamake
useof whether
the data-fluctuation PDFs are peaked or flat relative to a norure 7 shows the PSDs obtained for the two selected time inmagnetic
recordings
a permanent
observatory
fulfilling
the same
mal distribution.
That
is, datasets
with internahigh kurtosis have
tervals. Both of
PSDs
exhibit field
the same
behaviorobtained
at low andby
high
PDFs
increments)
that are
characterized by
frequencies displaying
characteristic
in corre- most
tional standard
of thosefrequencies
used for constructing
of of
thefluctuations
commonly(or
used
geomagnetic
indices.
a distinct peak near the most probable value and heavy tails in
spondence with the daily variation (f ∼ 0.69 × 10−3 min−1 )
In particular, the data used in this standard are characterized by a sampling rate of 1 minute
comparison with the Gaussian distribution. Conversely, low
and its first harmonics. In addition, the PSDs at low frequen−1 )nT,
a resolution
0.1
is higher
the usual
resolution
oftop
geomagnetic
datasets
show flat
PDFs. In ourindices,
case (see Fig. 8),
cies (typicallyand
below
f ∼ 0.006 of
min
are which
consistent
with a thankurtosis
−α ) characterized by a specpower 155
law spectrum
(S(f
)
∼
f
we
get
high
values
of
kurtosis,
which
depend
onare
the timescale
allowing a better investigation of small scale fluctuations. Furthermore, these data, which
tral exponent α ∼ 1.0/1.3, while at the highest frequencies
τ in both the selected periods. This dependence is more eviobtained
from
geomagnetic
observatory
latitude
42◦ 23’
Long.
13◦ 19’to E)
(above f ∼ 0.006
min−1
) weaobserve
a steeper
power lawat mid
dent
in 2001(Lat.
than 2008.
It isN,also
interesting
observe that
spectrum, with a spectral exponent equal to α ∼ 2.3. This rethe highest values of kurtosis associated with short timescale
sult suggests that the character of large- and small-scale flucincrements are concentrated in the year 2001. This suggests
tuations may be due to different physical processes. Further- 8 that they are related to the higher solar activity level, which
more, taking account of the relationship existing in the case
causes the occurrence of large geomagnetic storms.
of fBm between the Hurst exponent H and the slope α of the
The second step of our analysis to investigate the multiPSD (α = 2H + 1) existing for a fBm, we can estimate the
fractional character of our magnetic field time series is to
value of the average H exponent for the two datasets. In both
evaluate the local Hurst/Hölder exponent Hloc (t). The evalcases we obtain a value of H ∼ 0.65, which suggests that
uation of local Hurst exponent is a nontrivial target, and for
overall character of the signals is consistent with persistent
this reason different approaches have been proposed in the
motions. The observed H value, the 1/f nature of the specpast years (e.g., Peng et al., 1994; Peltier and Lévy-Vehel,
tra and the persistent character at these timescales may be the
1995; Muniandy and Lim, 2001; Chen et al., 2002; Alessio
counterpart of a possible critical dynamics (Woodard et al.,
et al., 2002; Carbone et al., 2004; Keylock, 2010). One of
2007) in the magnetospheric response to solar wind changes
the most accurate, fast and simple methods for nonstandard
as discussed in several works (see, e.g., Consolini and Chang,
Gaussian multifractional Brownian motions is the detrend2001). Nevertheless, at the present stage on the basis of our
ing moving average (DMA) technique introduced by Alessio
analysis this has to be considered only as a speculation; other
et al. (2002) and Carbone et al. (2004). This method, which
possible explanations exist for the observed persistent charis based on the analysis of the scaling features of the loacter, for instance the contribution of regular variations (e.g.,
cal standard deviation around a moving average, is quite
the daily variation Sq ) to magnetic field measurements.
simple and seems to be more accurate than other methods
www.nonlin-processes-geophys.net/20/455/2013/
Nonlin. Processes Geophys., 20, 455–466, 2013
τ [min]
Fig. 8. The dependence of the kurtosis κ of the magnetic field increments (δH(τ )) on the timescales τ fo
460
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
two selected periods.
7
1.0
10
6
10
0.8
4
H(t)
10
2
PSD S(ƒ) [nT min]
5
10
3
10
0.6
0.4
2
10
0.2
1
10
0
10
0.0
2001
2008
-1
10
t
-2
10
-4
10
-3
10
-2
10
-1
ƒ [min ]
-1
10
0
Fig. 9. A comparison between the actual local Hurst exponent
10
(black between
line) and
Hloc (t)
estimated
using (black
the DMA
technique
Fig. 9. A comparison
the actual
local
Hurst exponent
line) and
Hloc (t) estimated usin
(green line). The average discrepancy between the true and DMAestimated local Hurst exponent is about 10 %.
DMA technique (green line). The average discrepancy between the true and DMA estimated local Hurst e
is about 10%.
Fig. 7. The power spectral density S(f ) relative to the twonent
selected
time series. The vertical dashed line is at about f = 0.006 min .
−1
about f = 0.006
minand
. Solid
andlines
dashed
are power-law
best fits.
Solid
dashed
arelines
power
law best fits.
low-pass behavior of the moving average in comparison to polynomial filtering (Smith, 2
Fig. 7. The power spectral density S(f ) relative to the two selected time series. The vertical
−1 dashed line is at
is done using a moving window of 801 points and studying
215 Carbone et al., 2004). Anyway, we demand the reader to the literature on this matter (Ale
the scaling
features for n ∈ [2, 100]. The choice of a time
kurtosis κ of the magnetic field increments, δH(τ ) = H(t + τ ) − H(t), at different timescales
τ.
al., 2002; Smith,
2003;
Carbone
et al.,has
2004)
fordone
a more
discussion
1000of whether the data-fluctuations PDFs areetpeaked
window
of
801
points
been
to detailed
ensure an
optimalon this point.
Kurtosis is indeed a measure
or
flat
relative
2001
DMA technique
consists in evaluating
the scaling features
of exponent.
the quantityFigure
σDM A9(n) defined as
noise/signal
ratio in determining
the local
to a normal distribution. That is, data sets with high kurtosis 2008
have PDFs of fluctuations (or
Kurtosis
v
shows the comparison between the actual H (t) and the estimax
X
mated1 HlocN(t)
using the DMA
2 technique for a short interval.
t
σ
(n)
=
[x(t) − x̄n (t)]
DM A
in comparison with the Gaussian distribution. Conversely, low kurtosis data
sets showThe
flat top
average
precision in
estimation of the Hloc (t) is about
Nmax
− n t=n
u tails
increments) are characterized
100 by a distinct peak near the most probable value and heavy
u
PDFs. In our case, we get high values of kurtosis, which depend on the timescale τ 10%
in both
(i.e., h| Hloc (t) − H (t) | /H (t)i ' 0.1).
10
where
x̄
(t)
is
the
on a focusing
moving time
of length
different
n
Furthermore,
ourwindow
present
study n,
onforthe
possi-values of the
the selected periods (see Figure 8). This dependence is more evident in 2001 than 2008.
Itaverage
is
ble
relationship
the multifractionality
andof inter220 window
in thetimescale
interval
[n,Nmax ], between
and in calculating
the scaling exponent
the σDM A (n), whi
also interesting to observe that the highest values of kurtosis associated
with short
0
1
2
3
4
mittency,
we
analyze
the
fluctuations
in
the
high-frequency
H
10
10
10
10
10
increments are concentrated in the year 2001. This suggests that they are
related to scale
the higher
expected
as σDM A (n) ∼ n .
domain, i.e., above the spectral break at about f = 6 ×
10−3 min−1 , which corresponds to a temporal scale lower
The second step of our analysis to investigate the multifractional character of our magnetic
than 100/200 min.
Fig. 8. The dependence of the kurtosis κ of the magnetic field in11
field8.time
series
is to of
evaluate
local
Hurst/Hölder
H(δH(τ
The
evaluation
ofτFigures
local
Fig.
The dependence
the kurtosis
κ
oftimescales
the
magneticτ field
increments
on the
timescales
for the 10 and 11 show the results of the local Hurst
loc (t).))periods.
crements
(δH(τ
)) the
on the
forexponent
the
two selected
two
selected
periods.
Hurst
exponent
is a non trivial target and for this reason, different approaches have been proposed
exponent Hloc (t), obtained using the DMA technique on
a 2001;
moving window of 801 points, in the case of the two
in the past years (e.g., Peng et al., 1994; Peltier and Lévy-Vehel, 1995; Muniandy and Lim,
(Carbone et al.,
1.0 2004). Indeed, the DMA algorithm is charselected
Chen et al., 2002; Alessio et al., 2002; Carbone et al., 2004; Keylock, 2010). One of the
most time intervals. We notice that the average value
acterized by the better low-pass behavior of the moving avhHloc (t)i of local Hurst exponents is always higher than 0.5
0.8
accurate, fasterage
and simple
methods
for to
non-standard
Gaussian
multifractional
Brownian motions is
in comparison
polynomial
filtering
(Smith, 2003;
(hHloc (t)i2001 = [0.75 ± 0.16] and hHloc (t)i2008 = [0.80 ±
the detrending
moving
average
(DMA)
technique
introduced
by
Alessio
et
al.
(2002)
and Carbone et
Carbone et al.,
0.16]), suggesting that the mean character of geomagnetic
0.62004). Nevertheless, we refer the reader to the
literature
thisismatter
et of
al.,the2002;
al. (2004). This
method,on
which
based on(Alessio
the analysis
scalingSmith,
features2003;
of the local standard
time series is well in agreement with that of a persistent
0.4
Carbone
et al.,
2004)isfor
a more
discussion
this than
deviation around
a moving
average,
quite
simpledetailed
and seems
to be moreonaccurate
other as already observed in geomagnetic index studies
mBm,
point.
The
DMA
technique
consists
in
evaluating
the
scaling
(e.g.,
Dobias and Wanliss, 2009), and with what is found in
methods (Carbone et al., 2004).
Indeed, the DMA algorithm is characterized by the better
0.2
features of the quantity σDMA (n) defined as
the case of a near-criticality dynamics (Woodard et al., 2007).
v
0.0
However, we emphasize that the structure of the fluctuation
u
N
max
u
X
1
10
field is sometimes not in agreement with a simple mBm being
σDMA (n) = t
(8)
[x(t)t− x̄n (t)]2 ,
the value of the local Hurst exponent higher than 1. FurtherNmax − n t=n
more, during periods characterized by intense disturbances
where
x̄
(t)
is
the
average
on
a
moving
time
window
of
(see, e.g., the period near the end of March 2001) the characn
Fig. 9. A comparison between
the actual local Hurst exponent (black line) and Hloc (t) estimated using the
length n, for different values of the time window in the inter of the fluctuation field seems to display a less persistent
DMA technique (green line). The average discrepancy between the true and DMA estimated local Hurst expoterval [n, Nmax ], and in calculating the scaling exponent of
character. We underline that this behavior refers to fluctunent is about 10%.
the σDMA (n), which is expected to scale as σDMA (n) ∼ nH .
ations characterized by timescales below 100 min, and that
To test the reliability of this method, we apply the local
a possible origin of such less persistent character may be
low-pass behavior
of theof
moving
in comparison
filtering
2003;
estimation
Hurstaverage
exponent
to the casetoofpolynomial
a synthetic
sig- (Smith,
related
to the rapid and impulsive variations of a web-like
5 points.
Carbone et al.,
Anyway,
we demand
thecase,
reader
the literature
onlocthis
(Alessio of filamentary currents flowing in the ionosphere.
nal2004).
of ∼ 5×10
In this
thetoevaluation
of H
(t) matter
structure
H(t)
solar activity level, which causes the occurrenceτof[min]
large geomagnetic storms.
et al., 2002; Smith, 2003; Carbone et al., 2004) for a more detailed discussion on this point. The
DMA technique
consists
in evaluating
the scaling
features
of the2013
quantity σDM A (n) defined as,
Nonlin.
Processes
Geophys.,
20,
455–466,
www.nonlin-processes-geophys.net/20/455/2013/
v
u
NX
max
u
1
2
σDM A (n) = t
[x(t) − x̄n (t)]
(8)
Nmax − n t=n
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
461
(µT)
24.3
H
24.0
Hloc(t)
23.7
1.5
1.0
0.5
0.0
1-01-2001
1-04-2001
1-07-2001
1-10-2001
1-01-2002
Date
Fig. 10. The geomagnetic field H component, as measured at L’Aquila Geomagnetic Observatory (Italy) in 2001 (upper panel), and the
corresponding local Hurst
Hloc (t) field
(lower
panel), computed
usingat the
DMAgeomagnetic
technique.Observatory (Italy) in
Fig. 10.exponent
The geomagnetic
H component,
as measured
L’Aquila
2001 (upper panel) and the corresponding local Hurst exponent Hloc (t) (lower panel), computed using the
DMA technique.
24.3
(µT)
To test the reliability of this method, we apply the local estimation of Hurst exponent to the case of
H
a synthetic signal of ∼ 5 · 105 points. In this case, the evaluation of Hloc (t) is done using a moving
24.0
window of 801 points and studying the scaling features for n ∈ [2,100]. The choice of a time
225
window of 801 points has been done to ensure an optimal noise/signal ratio in determining the
local exponent. Figure 9 shows the comparison between the actual H(t) and the estimated Hloc (t)
23.7
using the DMA
1.5 technique for a short interval. The average precision in estimation of the Hloc (t) is
Hloc(t)
about 10% (i.e., h| Hloc (t) − H(t) | /H(t)i ' 0.1).
Furthermore,
focusing our present study on the possible relationship between the multi1.0
230
fractionality and intermittency, we analyze the fluctuations in the high frequency domain, i.e.,
above the0.5
spectral break at about f = 6 · 10−3 min−1 , which corresponds to a temporal scale
lower than 100 ÷ 200 min.
Figures0.0
10 and 11 show the results of the local Hurst exponent Hloc (t), obtained using the DMA
1-01-2008
1-04-2008
1-07-2008
1-10-2008
1-01-2009
technique on a moving window of 801 points, in the case of the two selected time intervals. We notice
235
Date
that, the average value hHloc (t)i of local Hurst exponents is always higher than 0.5 (hHloc (t)i2001 =
[0.75 ±field
0.16]H
andcomponent,
hHloc (t)i2008
[0.80 ± 0.16]),
suggesting
that the mean
character of(Italy)
geomagnetic
Fig. 11. The geomagnetic
as =measured
at L’Aquila
Geomagnetic
Observatory
in 2008 (upper panel), and the
corresponding local Fig.
Hurst
exponent
H
(t)
(lower
panel),
computed
using
the
DMA
technique.
loc
time11.
series
is
well
in
agreement
with
that
of
a
persistent
mBm,
as
already
observed
in
geomagnetic
The geomagnetic field H component, as measured at L’Aquila geomagnetic Observatory (Italy) in
2008 (upper panel) and the corresponding local Hurst exponent Hloc (t) (lower panel), computed using the
technique.
Clearly, this pointDMA
requires
a different and more accurate12 values of Hloc for 2001. The observed difference (δhHloc i =
hHloc i2008 − hHloc i2001 ) is δhHloc i = 0.05 in central value.
analysis, which is outside the aim of this work.
(e.g.ofDobias
and Wanliss,
2009),
with what
found inathe
of a near-character of the fluctuAs the third stepindices
in thestudies
analysis
the mBm
character
of and This
pointistowards
lesscase
persistent
ations
during
the
period
of
solar
as a result of the
geomagnetic time criticality
series, wedynamics
compute(Woodard
the probability
distriet al., 2007). However, we emphasize that the structure maximum,
of the
more
frequent
occurrence
of
perturbed
periods.
bution functions
of
the
local
Hurst
exponents
H
(t).
These
loc
240 fluctuation field is sometimes not in agreement with a simple mBm being the value of the local Hurst
PDFs are reported in Fig. 12 for both the selected time peTo better clarify the link between the intermittency and
exponent higher than 1. Furthermore, during periods characterized by intense disturbances (see e.g.,
riods. The two PDFs are completely equivalent and consisthe multifractional nature of geomagnetic small-scale flucthe shape,
period near
the endonly
of March,
2001)
the character oftuations,
the fluctuation
field seems tothe
display
a less
we investigate
relationship
between the local
tent with a Gaussian
showing
a small
difference
persistent
underlinetowards
that this behavior
to fluctuations
bythe
timescales
in the central value,
hHloc i,character.
which We
is shifted
lower refers
Hurst
exponentcharacterized
Hloc (t) and
local variance of the signal
below 100 min, and that a possible origin of such less persistent character may be related with
245
the rapid and impulsive variations of a web-like structure of filamentary currents flowing in
www.nonlin-processes-geophys.net/20/455/2013/
Nonlin. Processes Geophys., 20, 455–466, 2013
the ionosphere. Clearly, this point requires a different and more accurate analalysis, which is
outside the aim of this work.
As the third step in the analysis of the mBm character of geomagnetic time series, we com-
462
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
a window, characterized by a width of the same magnitude
of the estimated precision, i.e., 1Hloc /Hloc ∼ 0.1. In addi2.5
tion, for each Hloc -conditioned set of increments we evaluate
the standard variation σ (δX | Hloc ) of the selected dataset.
2.0
In this way we can study the dependence of σ (δX | Hloc ) on
1.5
Hloc .
1.0
Figure 15 shows the conditioned PDFs P (y | Hloc ) of
the
standard-deviation-normalized small-scale (1 min) incre0.5
ments for the two selected time periods. A clearly different
0.0
dependence of the shape of conditioned PDFs is observable
0.0
0.5
1.0
1.5
in the two datasets. In detail, while for the year 2008 the
Hloc
shape of the conditioned PDFs looks quasi-independent of
the value of Hloc , for the year 2001 a clear dependence is
Fig. 12. The PDFs of the local Hurst exponent Hloc computed for
recovered. Indeed, the conditioned PDFs relative to the year
the
two
selected
periods
by
means
of
the
DMA
technique.
ig. 12. The PDFs of the local Hurst exponent Hloc computed for the two selected periods by means of DMA
2001 increase their leptokurtotic character with the decrease
chnique.
of Hloc . This result is the signature of a different physical na2 (t) for the two different datasets. The local variance is
σloc
ture of small-scale increments relative to periods characterharacter of theevaluated
fluctuations
during
the period
of solar
a resultofof theized
morebyfreusing
a moving
window
of maximum,
the same as
number
a lower value of Hloc . A possible explanation of this
points
as that used
in DMA analysis, i.e., Npoints = 801. In
different character of 2001/2008 conditioned PDFs could be
uent occurrence
of perturbed
periods.
addition, to reduce the effect of local trend at scales larger
the different level of solar activity and geomagnetic disturTo better clarify the link between the intermittency and the multifractional nature of geomagnetic
than the investigated ones, before evaluating the local varibances. Indeed, the more leptokurtotic character of the conmall scale fluctuations
investigate
relationship
Hurst exponent
Hloc (t)PDFs at low H values in the year 2001 has to be
ance, wewedetrend
the the
signal
in eachbetween
windowthebylocal
a moving
ditioned
2
nd the local variance
of
the
signal
σ
(t)
for
the
two
different
datasets.
The
local
variance
average technique onloca scale larger than 101 points, i.e., on
related tois the less persistent nature of very large fluctuations
timescales
larger than
maximum
timescale
occurring
valuated using
a moving window
of thethe
same
number of
points of investigated.
that used in DMA analysis,
i.e., during disturbed periods.
Figure
13
shows
the
occurrence
plot
of
the
dependence
16 we report the conditioned PDFs P (y | Hloc ) obIn
Fig.
Npoints = 801. In addition, to reduce the effect of local trend at scales larger than the investigated
of the local Hurst exponent Hloc , computed by the DMA
tained in the case of a synthetic mBm with a time-dependent
nes, before evaluating
the local
variance,
we detrend local
the signal
in eachσwindow
by a moving
average
2
technique,
on the
corresponding
variance
local
Hurst exponent Hloc (t) ∈ [0.1, 0.9]. The PDFs do not
loc for the
echnique on scale
larger
than
101
points,
i.e.
on
timescales
larger
than
the
maximum
timescale
two time intervals here considered. Hloc shows a tendency
exhibit any substantial dependence of the shape on the value
2
of Hloc . Furthermore, the shape of the conditioned PDFs
nvestigated. to increase with the increasing of σloc , although this positive
correlation
is
more
pronounced
in
2008
(Pearson
coefficient
is remarkably
similar to that obtained evaluating the condiFigure 13 shows the occurrence plot of the dependence of the local Hurst exponent H
loc , comr ∼ 0.57) than in 2001 (Pearson coefficient r2∼ 0.21). Howtioned PDFs for the year 2008, which is characterized by a
uted by DMA technique, on the corresponding local variance σloc for the two time intervals here
ever, this trend is different and opposite to that observed in
very low level of geomagnetic disturbance. Figure 17 pro2
onsidered. Hloc
a tendency
increase
with theincreasing
of σloc
, although
positive
theshows
synthetic
mBm to
signal
generated
using the algorithm
de- this
vides
a direct confirmation of this last point. In this figure we
scribed
in Sect. 2,
in Fig. 14r ∼
(Pearson
coeffireport the
orrelation is more
pronounced
in and
2008displayed
(Pearson coefficient
0.57) than
in 2001 (Pearson
co- conditioned PDFs P (y | Hloc ) with Hloc = 0.5 in
cient
r
∼
−0.88).
the
case
fficient r ∼ 0.21). Anyway, this trend is different and opposite to that observed in the syntheticof the two selected geomagnetic time series and the
The substantial discrepancy, observed between actual gesynthetic mBm.
mBm signal generated using the algorithm described in Section 2, and displayed in Figure 14
omagnetic time series and synthetic mBms, suggests that the
In the framework of fully developed fluid turbulence the
Pearson coefficient
r ∼ −0.88).
intermittency
of geomagnetic time series cannot be simply
departure from the Gaussian shape of the PDFs of smallscalemBms,
velocity longitudinal increments is sometimes modeled
described
as aobserved
consequence
theirgeomagnetic
mutifractional
nature.
The substantial
discrepancy,
betweenof
actual
time series
and synthetic
2 and H
bya means
This
different of
correlation
character
between
σloc
loc
uggests that the
intermittency
geomagnetic
time series
cannot be
simply described
as
conse- of a superposition of Gaussian distribution charsuggests the existence of a double origin of intermittency, 2 acterized by variances distributed according to a log-normal
uence of their mutifractional nature. This different correlation character between σloc
and Hloc
which is related to both their multifractional feature and flucdistribution
(Castaing et al., 1990). Analogously, we may
uggests the existence
a double origin of intermittency, which is related to both theirthink
multifracthat the non-Gaussian shape of the small-timescale intuation of
amplitudes.
crements for fixed value of H of the geomagnetic time series
clarify amplitudes.
this possible different origin of the intermitonal feature and To
fluctuation
may ofbethe
modeled using a similar approach. For instance, we
tency,
we
can
investigate
the
PDFs
of
the
small-scale
increTo clarify this possible different origin of the intermittency we can investigate the PDFs
can write
ments δH(τ ) (here τ = 1 min) of the geomagnetic field meamall-scale increments δH(τ ) (here τ = 1 min) of the geomagnetic field measurements, conditioned
!
surements, conditioned to the local Hurst exponent Hloc , and
Z
2
δB
1
compare them with the corresponding conditioned PDFs of
exp − 2 dσ 2 , (9)
P (δB | H ) = p(σ 2 | H ) √
2σ
the small-scale increments of14synthetic mBm. In what fol2π σ 2
lows, we identify these small-scale increments as δX, and the
corresponding standard-deviation-normalized increments as
where p(σ 2 | H ) is the probability distribution function of
y = δX/σ for convenience. Furthermore, because the accuthe variances conditioned to the Hurst exponent. Conseracy of the local Hurst index estimated using the DMA techquently, the PDFs of the increments of the actual geomagnique is within 10 %, to properly evaluate the PDFs of the
netic field measurements can be written as follows:
small-scale increments for each choice of Hloc we consider
3.0
PDF P(Hloc)
2001
2008
Nonlin. Processes Geophys., 20, 455–466, 2013
www.nonlin-processes-geophys.net/20/455/2013/
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
-4
0.0
-2
-3
0.5
-1
2
-2
2
2
-1
0
2008
Log [σloc ] [nT ]
-3
2001
2
2
2
-2
Log [σloc ] [nT ]
-1
Log [σloc ] [nT ]
2
2001
-3
-4
1.0
1.5
-4H
loc
0.0
0.5
0.0
1.5
0.4
-1
0.3
0.2
-2
0.1
0.0
-3
0.5
1.0
0.5
2008
1.0
Hloc-4
0.5
0.4
0.3
0.2
0.1
0.0
1.5
0.0
0.5
Hloc
1.0
Occurrence [%]
2
0
0
Occurrence [%]
Log [σloc ] [nT ]
0
463
1.5
Hloc
Fig. 13. The occurrence
of occurrence
the dependence
of the
localdependence
Hurst exponent
, computed
DMA technique,
loc local
Fig. 13.plot
The
plot
of the
of H
the
Hurstbyexponent
Hloc , computed by the DMA technique, on the corresponding
2
2
n the corresponding
variance
theyear
year 2001
andand
the year
2008 (right
local local
variance
σlocσloc
forforthe
2001(left
(leftpanel)
panel)
the year
2008 panel),
(right respecpanel), respectively. The solid white line is a linear fit of the
2 H and
Fig.line
13.isThe
occurrence
plot
of thea dependence
of the
local
Hurst
exponent
, computed
DMA technique,
denoting
positive
correlation
between
log
Hloc . Theby
significance
of this positive correlation is more
ively. The solidcorresponding
white
a scatter
linear
fitplot,
of the
corresponding
scatter
plot,
denoting
a positive
correlation
10 [σ
loc ] loc
2
2∼ 0.21).
relevant
for
2008
(r
∼
0.57)
than
2001
(r
andthe
Hloc
. Anyway, the
significance
thisfor
positive
correlation
is more
between log10 [σloc ] on
corresponding
local
varianceofσloc
the year
2001 (left
panel)relevant
and thefor
year 2008 (right panel), respec-
008 (r ∼ 0.57) thantively.
2001 (rThe
∼ 0.21).
solid white line is a linear fit of the corresponding scatter plot, denoting a positive correlation
Hloc = 0.5
Hloc = 0.6
Hloc = 0.7
Hloc = 0.8
Hloc = 0.9
2
-5
-6
2
-7
-8
0.0
P(y | Hloc)
-1
-4
Log [σloc ] [nT ]
2
2
Log [σloc ] [nT ]
2
] and Hloc . Anyway, the significance of this positive correlation is more relevant for
between log10 [σloc
-1
10
2008 (r ∼ 0.57) than 2001 (r ∼ 0.21).
-2
Year 2001
1
-3
-2
-3
0.1
0.01
-4
0.001
-5
-60.5
-4
1.0
-2
0
2
4
y
Hloc
-7
-8dependence of the local Hurst
Fig. 14. The occurrence plot of the
Hloc = 0.5
Hloc = 0.6
Hloc = 0.7
Hloc = 0.8
Hloc = 0.9
P(y | Hloc)
Fig. 14. The occurrence
plotHofloc
the
local
Hursttechnique,
exponent Hloc
by DMA technique, 1.0
0.0
exponent
, dependence
computed of
bythethe
DMA
on, computed
the0.5
corre1
Year 2008
2
2
sponding
variance
mBm
The
solid
n the corresponding
local local
variance
σloc forσsynthetic
mBm signal.
Thesignal.
solid white
line
is a linear fit of the
Hloc
loc for synthetic
white plot,
line denoting
is a linear
fit ofnegative
the corresponding
plot,
denoting
orresponding scatter
a clear
correlation (r scatter
∼ −0.88)
between
log [σ 2 ] and Hloc .
2 ] and10 loc
0.1
a clear negative correlation (r ∼ −0.88) between log10 [σloc
The used color palette is the same of Figure 13.
Hloc . The used color palette is the same of Fig. 13.
Fig. 14. The occurrence plot of the dependence of the local Hurst exponent H , computed by DMA technique,
loc
0.01
on the corresponding
local
for synthetic mBm
signal. PDFs
The solid
o the local Hurst exponent
Hloc , and compare
corresponding
conditioned
of thewhite line is a linear fit of the
2
corresponding
denoting
a clear
negativethese
correlation
(r ∼increments
−0.88) between
mall-scale increments
of Z
syntheticscatter
mBm.plot,
In what
follows,
we identify
small-scale
0.001log10 [σloc ] and Hloc .
-4
-2
0
The used color
palettedeviation
is the same
of Figure 13.
s δX, and the corresponding
standard
normalized
increments as y = δX/σ for conve2
variance
loc
them
withσthe
P (δB) =
(10)
P (δB | H )p(H )dH,
nience. Furthermore, being the accuracy of the local Hurst index evaluated using DMA technique
2
4
y
being
theHurst
distribution
of the
local Hurst
expoto) the
local
Hloc
, and
compare
themfor
with
thechoice
corresponding
conditioned
PDFs of the
within 10%, top(H
properly
evaluate
the exponent
PDFs offunction
the
small-scale
increments
each
of H
loc The
Fig.
15.
Hloc -conditioned
probability distribution funcnents.
Fig. 15. The Hloc -conditioned probability distribution functions P (y | Hloc ) of the standard deviation norma
tions P (y
| Hloc
) of the standard-deviation-normalized
small-scale
small-scale increments of synthetic mBm. In what follows, we identify
these
small-scale
increments
Last but not least, the more leptokurtotic character ized
of the
small-scale(1(1min)
min) increments
for different
values ofvalues
Hloc in of
the range
panel refe
increments
for different
Hloc [0.5,0.9].
in the Upper/lower
range
280
as δX, and
the of
corresponding
deviation normalized
increments
as y = δX/σ
forrefers
conve15 standard
conditioned
PDFs
increments
of the geomagnetic
field2001/2008,
[0.5,respectively.
0.9]. Upper/lower
panel
to year 2001/2008, respectively.
to year
during
high Furthermore,
solar activitybeing
periods
reflect
thelocal
different
nience.
the could
accuracy
of the
Hurst index evaluated using DMA technique
nature of the physical processes responsible for the geomagwe consider aincrements
window, characterized
by a width
ofloc
the same magnitude of the estimated precisio
within 10%, to properly evaluate the PDFs of the small-scale
for each choice
of H
netic disturbances. In detail, we can imagine that during these
avalanching nature of magnetospheric
dynamics (Consolini
i.e.,
∆Hloc /H
0.1. In 2001;
addition,
for each
-conditioned
set periods
of increments
loc ∼
loc2003)
periods the nature of the geomagnetic perturbations and
flucand
Chang,
Chang
et H
al.,
during
char-we evaluate th
285 standard
σ(δX
|
H
)
of
the
selected
dataset.
In
this
way
we
can
study
the dependence o
tuations may acquire a multiplicative character, which
mani- variation
loc
acterized by the occurrence of geomagnetic substorms and
15
fests in a significant departure of the conditioned PDFsσ(δX
from| Hloc ) storms.
on Hloc .
the shape obtained in the case of standard multifractional
Figure 15 shows the conditioned PDFs P (y | Hloc ) of the standard deviation normalized smal
Brownian motions. This point could be a consequence of the
scale (1 min) increments for the two selected time periods. A clearly different dependence of th
shape of conditioned PDFs is observable in the two datasets. In detail, while for the year 2008 th
www.nonlin-processes-geophys.net/20/455/2013/
290
Nonlin. Processes Geophys., 20, 455–466, 2013
shape of the conditioned PDFs looks quasi-independent on the value of Hloc , for the year 2001
clear dependence is recovered. Indeed, the conditioned PDFs relative to the year 2001 increase the
leptokurtotic character with the decrease of Hloc . This result is the signature of a different physic
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
1 Synthetic mBm
Synthetic mBm
0.1
Hloc = 0.4
HHloc ==0.5
0.4
loc
HHloc
= 0.6
loc = 0.5
HHloc
==0.7
0.6
0.1
Hloc = 0.7
| H) loc)
P(yP(y
| Hloc
1
σ(δX | Hloc)/σ(δX | 0.5)
464
loc
0.01
0.01
0.001
0.001
-4
-2
0
2
4
-4
-2
y0
y
2
4
10
1
0.1
0.01
0.2
2001
2008
mBm
0.4
0.6
0.8
1.0
Hloc
Fig. 16. The Hloc -conditioned probability distribution funcFig.normal18. Plot of the dependence of the standard deviation σ (δX |
Fig. 16. The Hloc -conditioned probability distribution functions P (y | Hloc ) of the standard
Fig. 18. deviation
Plot of the
dependence of the standard deviation σ(δX | Hloc ) on Hloc . The standard deviatio
tions P (y | Hloc ) of the standard-deviation-normalized small-scale
Hloc ) on Hloc . The standard deviation σ (δX | Hloc ) has been scaled
zed small-scale
(1-conditioned
min) increments
increments
for
values
of H
in
the
range
in
the
case
synthetic
Fig.
16. The H(1
probability
distribution
functions
P (y
H
) [0.3,0.7]
of the
standard
normalσ(δX
| Hdeviation
has
been
scaled to that corresponding
to for
Hlocconvenience.
= 0.5 for convenience.
Dashed lines are e
loc
loc
loc ) of
min)
fordifferent
different
values
ofloc
Hloc
in| the
range
[0.3,
0.7]
to that corresponding
to Hloc = 0.5
Dashed lines
case
mBm
generated
to the
algorithm
mBm
generatedin
according
toof
thesynthetic
algorithm
described
in Section
ized small-scale
(1the
min)
increments
for different
values
of Haccording
[0.3,0.7]
in the case are
of synthetic
guides.
loc2.in the range
eye guides.
Sect.
2.
mBm generateddescribed
according in
to the
algorithm
described in Section 2.
10
| H) loc)
P(yP(y
| Hloc
10
1
study has been carried on comparing actual recorded signals relative to different solar activity leve
4 Summary and conclusions
2001
with synthetic mBms. The main results of our analysis can be summarized as follows:
2008
2001
mBm
We have focused our present work on the possible link
2008
340
1. the intermittent character of geomagnetic time series cannot be simply related to their mult
between the intermittent character and the multifractional
mBm
fractional
features;
nature
of the Earth’s magnetic field fluctuations on small
Hloc = 0.5
Hloc = 0.5
1
0.1
0.1
0.01
0.01
0.001
timescales. This study has been carried out by comparing actual recorded signals relative to different solar activity levels
ing thewith
spectrum
of localmBms.
Hurst exponents.
the geomagnetic
activity
synthetic
The mainWhen
results
of our analysis
canlevel increases,
decrease
in
the
persistent
character
of
geomagnetic
fluctuations
(increments)
is observed;
be summarized as follows:
2. the high variability of solar wind activity level affects the multifractional character by chan
0.001
-4
-2
0
2
-4
-2
y0
2
y
4
4
345
3. the Hloc -conditioned
PDFs of the
geomagnetic
small-time scale
increments
1. The intermittent
character
of geomagnetic
time
series show a clea
be simply related
to their
multifractional
dependence cannot
on the geomagnetic
activity level.
Indeed,
the shape offeaconditioned PDF
Fig. 17. Comparison of the Hloc -conditioned probability distribution functions P (y | Hloc
of the standard
tures.
is )independent
on the Hloc during solar minimum, while a clear dependence of th
Fig. 17. Comparison of the Hloc -conditioned probability distrideviation
(1-conditioned
min) increments
for Hlocdistribution
= 0.5 in thefunctions
case of the
selected
Fig. 17. normalized
Comparisonsmall-scale
of the Hloc
probability
P two
(y | H
the
standard
loc ) ofgeomagnetic
shapevariability
on the Hloc
found
during
solar level
maximum.
bution functions P (y | Hloc ) of the standard-deviation-normalizedconditioned-PDF
2. The high
of issolar
wind
activity
affectsThis point is
ime seriesnormalized
and of
the synthetic
generated
according
to
algorithm
described
in
Section
2. geomagnetic
deviation
small-scale
(1 min)
increments
0.5
in
thethe
two
selected
locthe
small-scale
(1mBm
min)
increments
forforHHloc
==0.5
in the
thecase
caseofof
two
direct consequence
of the previouscharacter
point as discussed
in thethe
previous
Section;
the multifractional
by changing
spectrum
time series and selected
of the synthetic
mBm generated
according
in Section 2.
geomagnetic
time series
and to
ofthe
thealgorithm
syntheticdescribed
mBm gener-
of local Hurst exponents. When the geomagnetic activ-
atedPDFs
according
algorithm
Sect.
350
theless
observed
shape
conditioned-PDF
during
solarpersistent
minimumcharacter
resembles that of mBm
he conditioned
at lowtoHthe
values
in thedescribed
year 2001inhas
to2.
be related
to4.the
persistent
ity
levelofincreases,
a decrease
in the
while
considerable
difference
is observed (increments)
in the case of solar
maximum. This behavio
the conditioned
PDFs
at low H values
in the
year disturbed
2001 has to
be related to the
lessa persistent
of geomagnetic
fluctuations
is observed.
nature
of very large
fluctuations
occurring
during
periods.
is evident
for the conditioned-PDFs with Hloc = 0.5, which are strongly affected by th
nature
of very
fluctuations
during
periods.in the
In Figure
16 large
we
the conditioned
PDFs
P (ydisturbed
| Hthe
obtained
case
of a synthetic
loc ) small-scale
To report
conclude
theoccurring
characterization
of
incre3. The Hloc -conditioned PDFs of the geomagnetic smallgeomagnetic
disturbance
level.
ments
of conditioned
Fig.
18(t)
the dependence
In Figure
16
we
report
the
conditioned
PDFs
(y
| shows
H∈loc
) obtained
the case
of exhibit
a synthetic
mBm
with a time
dependent
local
HurstPDFs,
exponent
HPloc
[0.1,0.9].
TheinPDFs
do not
any
timescale increments show a clear dependence on the
of the
standard
deviation
σ (δXof|H
H
on[0.1,0.9].
Hloc forThe
the
studloc. )Furthermore,
mBm withdependence
a time
dependent
Hurst
∈
PDFs
dothe
notconditioned
exhibit
any
substantial
of the local
shape
on theexponent
value
H
the
shape
of
geomagnetic
activity level. Indeed, the shape of conloc
loc(t)
To outline some possible implications of our findings we briefly discuss the scenario emer
ied cases. A different behavior is recovered from the different
ditioned PDFs is independent of the Hloc during solar
substantial
dependence
of thetoshape
on the value
of Hloc .the
Furthermore,
thePDFs
shapefor
of the conditioned
PDFs
is remarkably
similar
that obtained
evaluating
conditioned
year 2008,
cases here investigated. In particular, we detect a355
small
ingposfrom our analysis
on the multifractional
of geomagnetic
fluctuations. It is know
minimum,
while a clearcharacter
dependence
of the conditioned
PDFs is
is characterized
remarkably
similar
to that
evaluating
conditioned
PDFs
the year
2008,
which
by a very
low obtained
level of
geomagnetic
disturbance.
17for
provides
a direct
itive correlation
between
σ (δX
| Hloc )the
and
Hloc forFigure
the
year
that
the Earth’s magnetospheric
dynamics
is
characterized
by
a
persistent
dynamic correl
PDF shape on the Hloc is found during solar maximum.
which is characterized
by
a in
very
low
leveltwo
of geomagnetic
disturbance.
Figure
provides
a direct
while
other
casesthe
(year
2001 and
synthetic
confirmation
to2008,
this last
point.
Inthe
this
figure
we
report
conditioned
PDFs
P (y |17
Hloc
) with H
=
loc
ThisWanliss,
point is2009)
a direct
the previous
point 2002; Wan
tions (e.g. Dobias and
and consequence
near-criticalityof
dynamics
(e.g. Consolini,
mBm
signal)
we
that the
deviation
(δX |PH(yloc
) loc ) with Has
confirmation
this
point.
In find
this
figure
we standard
report
conditioned
|H
=
0.5
in the caseto
of
thelast
two
selected
geomagnetic
timethe
series
and theσPDFs
synthetic
mBm.
loc discussed
in the previous section.
and H are clearly anti-correlated. The observed different
behavior may suggest that the origin of intermittency in geomagnetic time series could be 17
due to a more complex nature
resulting from different competing
17 processes, which are both
of an internal magnetospheric origin and directly driven by
solar wind disturbances. We will return to this point in the
next section.
0.5 in the case of theloc
two selected geomagnetic time series and the synthetic mBm.
19
4. The observed shape of conditioned
PDF during solar
minimum resembles that of mBm, while a considerable
difference is observed in the case of solar maximum.
This behavior is evident for the conditioned PDFs with
Hloc = 0.5, which are strongly affected by the geomagnetic disturbance level.
To outline some possible implications of our findings, we
briefly discuss the scenario emerging from our analysis on
Nonlin. Processes Geophys., 20, 455–466, 2013
www.nonlin-processes-geophys.net/20/455/2013/
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
the multifractional character of geomagnetic fluctuations. It
is known that the Earth’s magnetospheric dynamics is characterized by persistent dynamic correlations (e.g., Dobias
and Wanliss, 2009) and near-criticality dynamics (e.g., Consolini, 2002; Wanliss and Uritsky, 2010), which are correlated features in self-organized critical systems in stationary
conditions also away from the critical point (Woodard et al.,
2007). Furthermore, the persistent character of the magnetospheric dynamics has been shown to undergo dynamical
changes during magnetic storms and substorms (e.g., Wanliss, 2005; Balasis et al., 2006; Wanliss and Dobias, 2007).
These points suggest that the more complex observed relation between the multifractional nature and the intermittent
character of the geomagnetic fluctuations (see, e.g., Figs. 12,
13 and 14) may be the consequence of the nonstationarity
conditions of the solar wind driving. This idea is corroborated by the high temporal variability of the increment variance, which has to be related to the variation in the solar
wind magnetosphere coupling due to the changes of the solar
wind conditions. In contrast, it seems to be very difficult to
relate the nonstationary character of the solar wind driving
to the less persistent character of geomagnetic field fluctuations during disturbed periods, which seems to be a peculiar property of ground-based magnetic field measurements
with respect to geomagnetic indices, which show an increase
of persistent dynamical correlations during disturbed periods
(e.g., Dobias and Wanliss, 2009). A possible explanation of
this different nature could be found in an inherent stochastic
nature of the current patterns at the ionospheric level. Indeed,
the less persistent nature of the geomagnetic fluctuations observed during magnetic storms could be due to the superposition of temporal and spatial fluctuations. The ground-based
magnetic field measurements are the result of measurements
that occur in different places on Earth; for this reason, they
feel the effect of the local spatial structure of the ionospheric
current system patterns. This is the main difference between
the ground-based magnetic field measurements and the geomagnetic indices which differently provide global information on magnetospheric dynamics. Thus, we propose that the
turbulent nature of the solar wind driving and the complex
ionospheric current spatial pattern could be responsible for
the more complex multifractional character of the geomagnetic fluctuations observed in our work, especially regarding
the different dependence of signal variance on Hurst exponent with respect to simple synthetic mBms. Clearly, the relation between these two different sources of the ground-based
magnetic field disturbances is something that needs to be better investigated and clarified using much longer datasets and
numerical simulations.
Acknowledgements. The results presented in this paper rely on the
data collected at Geomagnetic Observatory of L’Aquila (Italy). We
thank the Italian Istituto Nazionale di Geofisica e Vulcanologia
(INGV), for supporting its operation, and INTERMAGNET for
promoting high standards of geomagnetic observatory practice
www.nonlin-processes-geophys.net/20/455/2013/
465
(http://www.intermagnet.org). This work was supported by the
Italian Space Agency (ASI) under the ASI-INAF agreement no.
I/022/10/0.
Edited by: J. Wanliss
Reviewed by: A. C.-L. Chian and one anonymous referee
References
Alessio, E., Carbone, A., Castelli, G., and Frappietro, V.: Secondorder moving average and scaling of stochastic time series, Eur.
Phys. J. B, 27, 197–200, 2002.
Balasis, G., Daglis, I. A., Kapiris, P., Mandea, M., Vassiliadis, D.,
and Eftaxias, K.: From pre-storm activity to magnetic storms: a
transition described in terms of fractal dynamics, Ann. Geophys.,
24, 3557–3567, doi:10.5194/angeo-24-3557-2006, 2006.
Benassi, A., Jaffard, S., and Roux, D.: Elliptic Gaussian random
processes, Rev. Mat. Iberoamericana, 13, 19–90, 1997.
Carbone, A., Castelli, G., and Stanley, H. E.: Time-dependent Hurst
exponent in financial time series, Physica A, 344, 267–271, 2004.
Castaing, B., Gagne, Y., and Hopfinger, E. J.: Velocity probability
density functions of high Reynolds number turbulence, Physica
D, 46, 177–200, 1990.
Chang, T.: Self-organized criticality, multi-fractal spectra, sporadic
localized reconnections and intermittent turbulence in magnetotail, Phys. Plasma, 6, 4137–4145, 1999.
Chang, T., Tam, S. W. Y., Wu, C. C., and Consolini, G.: Complexity, forced and/or self-organized criticality, and topological
phase transitions in space plasmas, Space Sci. Rev., 107, 425–
445, 2003.
Chen, Z., Ivanov, P. Ch., Hu, K., and Stanley, H. E.: Effect of nonstationarities on detrended fluctuation analysis, Phys. Rev. E, 65,
041107, doi:10.1103/PhysRevE.65.041107, 2002.
Consolini, G.: Sandpile cellular automata and magnetospheric dynamics, in: Cosmic physics in the year 2000, edited by: Aiello,
S., Iucci, N., Sironi, G., Treves, A., and Villante, U., SIF Conference Proc. 58, 123–126, 1997.
Consolini, G.: Self-Organized Criticality: A new paradigm for the
magnetotail dynamics, Fractals, 10, 275–283, 2002.
Consolini, G. and Chang, T.: Magnetic field topology and criticality
in geotail dynamics: relevance to substorm phenomena, Space
Sci. Rev., 95, 309–321, 2001.
Consolini, G. and De Michelis, P.: Non-Gaussian distribution functions of AE-index fluctuations: Evidence for time intermittency,
Geophys. Res. Lett., 25, 4087–4090, 1998.
Consolini, G. and De Michelis, P.: Fractal time statistics of AEindex burst waiting times: evidence of metastability, Nonlin.
Processes Geophys., 9, 419–423, doi:10.5194/npg-9-419-2002,
2002.
Consolini, G. and De Michelis, P.: Rank ordering multifractal analysis of the auroral electrojet index, Nonlin. Processes Geophys.,
18, 277–285, doi:10.5194/npg-18-277-2011, 2011.
Consolini, G., Marcucci, M. F., and Candidi, M.: Multifractal Structure of Auroral Electrojet Index Data, Phys. Rev. Lett., 76, 4082–
4085, 1996.
Dobias, P. and Wanliss, J. A.: Intermittency of storms and substorms: is it related to the critical behaviour?, Ann. Geophys.,
27, 2011–2018, doi:10.5194/angeo-27-2011-2009, 2009.
Nonlin. Processes Geophys., 20, 455–466, 2013
466
G. Consolini et al.: Intermittency and mBm in geomagnetic time series
Keylock, C. J.: Characterizing the structure of nonlinear systems using gradual wavelet reconstruction, Nonlin. Processes Geophys.,
17, 615–632, doi:10.5194/npg-17-615-2010, 2010.
Klimas, A. J., Vassiliadis, D. V., Baker, D. N., and Roberts, D. A.:
The organized nonlinear dynamics of magnetosphere, J. Geophys. Res., 101, 13089–13114, 1996.
Klimas, A. J., Valdivia, J. A., Vassiliadis, D., Baker, D. N., Hesse,
M., and Takalo, J.: Self-organized criticality in the substorm phenomenon and its relation to localized reconnection in the magnetosphere plasma sheet, J. Geophys. Res., 105, 18765–18780,
2000.
Kovács, P., Carbone, V., and Vörös, Z.: Wavelet-based filtering of
intermittent events from geomagnetic time-series, Planet. Space
Sci., 49, 1219–1231, 2001.
Mandelbrot, B. B. and Van Ness, J. W.: Fractional Brownian Motions, Fractional Noises and Applications, Soc. Indust. Appl.
Math. Rev., 10, 422–437, 1968.
Muniandy, S. V. and Lim, S. C.: Modeling of locally selfsimilar processes using multifractional Browninan motion
of Riemann-Liuouville type, Phys. Rev. E, 63, 046104,
doi:10.1103/PhysRevE.63.046104, 2001.
Peltier, R. F. and Lévy-Vehel, J.: Multifractional Brownian motion:
definition and preliminary results, INRIA Preprint No. 2645,
1995.
Peng, C. K., Buldryrev, S. V., Havlin, S., Simons, M., Stanley, H. E.,
and Goldberg, A. L.: Mosaic organization of DNA nucleotides,
Phys. Rev. E, 49, 1685–1689, 1994.
Rambaldi, S. and Pinazza, O.: An accurate fractional Browninan
motion generator, Physica A, 208, 21–30, 1994
Nonlin. Processes Geophys., 20, 455–466, 2013
Smith, S. W.: Digital filters, in: Digital Signal Processing: A
prractical Guide for Engineers and Scientists, Elsevier Science,
Burlington, MA, 261–343, 2003.
Uritsky, V. M. and Pudovkin, M. I.: Low frequency 1/f -like fluctuations of the AE-index as a possible manifestation of selforganized criticality in the magnetosphere, Ann. Geophys., 16,
1580–1588, doi:10.1007/s00585-998-1580-x, 1998.
Wanliss, J. A.: Fractal properties of SYM-H during quiet
and active times, J. Geophys. Res., 110, A03202,
doi:10.1029/2004JA010544, 2005.
Wanliss, J. A. and Dobias, P.: Space storms as a phase
transition, J. Atmos. Sol. Terr. Phys., 69, 675–684,
doi:10.1016/j.jastp.2007.01.001, 2007.
Wanliss, J. A. and Uristky, V.: Understanding bursty behavior in
midlatitude geomagnetic activity, J. Geophys. Res., 34, L04107,
doi:10.1029/2009JA014642, 2010.
Wanliss, J. A. and Weygand, J. M.: Power law burst lifetime distribution of the SYM-H index, Geophys. Res. Lett., 34, L04107,
doi:10.1029/2006GL028235, 2007.
Wanliss, J., Anh, V. V., Yu, Z.-G., and Watson, S.: Multifractal modeling of magnetic storms via symbolic dynamics analysis, J. Geophys. Res., 110, A08214, doi:10.1029/2004JA010996, 2005.
Woodard, R., Newman, D. E., Sánchez, R., and Carreras, B. A.:
Persistent dynamic correlations in self-organized critical systems away from their critical point, Physica A, 373, 215–230,
doi:10.1016/j.physa.2006.05.001, 2007.
www.nonlin-processes-geophys.net/20/455/2013/