Introduction
The behaviour of natural systems like
the ionosphere is a function of changes that occur in the underlying dynamics
that exists in such a system. These underlying dynamics however can sometimes
be complex and non-linear due to superposition of different changes in the
dynamical variables that constitute it. However, there is no totally
deterministic system in nature, because all natural systems exhibit a mixture
of both stochastic and deterministic properties. Although few natural systems
have been found to be low-dimensional deterministic in the sense of the
theory, the concept of low-dimensional chaos has been proven to be fruitful
in the understanding of many complex phenomena (Hegger et al., 1999). The
degree of determinism or stochasticity in most natural systems is dependent
on how much the system can be influenced by external factors, the nature of
these external factors etc. The ionosphere, like every other natural system, possesses
its intrinsic dynamics, and it can also be influenced by other external
factors. The typical characteristics of a dynamical system like the
ionosphere are expected to naturally show the interplay between determinism
and stochasticity simply because of the fact that the ionosphere, which has
an inherent internal dynamics, is also influenced by the influx of stochastic
drivers like the solar wind, since it is influenced by external dynamics like
every other natural system. This has made pure determinism impossible in the
ionosphere, a situation that is common to every natural system and its
surroundings.
The intensity of the solar wind coming into the ionosphere varies with the
solar activity and this can sometimes result in geomagnetic storms and
substorms driven in high-intensity plasma wind at enormous speed, and it
serves as a major stochastic driver leading to storms. The solar wind is
driven from the Sun into the ionospheric system during the quiet and storm
and during relatively quiet periods of each month of the year. However, other
processes include various factors like local time variations of the neutral
winds, ionization processes, production–recombination rates, photoionization
processes, plasma diffusion and various electrodynamics processes
(Unnikrishnan, 2010). The mesosphere and the lower thermospheric dynamics as
reported by Kazimirovsky and Vergasova (2009) and the influence of gravity
waves as reported by Sindelarova (2009) can also be of great influence on the
internal dynamics of the ionosphere.
Therefore, it is of great importance to study the chaoticity and dynamical
complexity of the ionosphere and its variations in all geophysical
conditions. However, a good number of investigations have been carried out on
the concept of chaos in the upper atmosphere before now, which includes the
study on magnetospheric dynamics and the ionosphere. The study of chaos in
magnetospheric index time series such as AE and AL were initially
carried out by Vassiliadis et al. (1990), Shan et
al. (1999) and Pavlos et al. (1992). These previous efforts made by the
aforementioned researchers has led to the development of the concept of
investigating and revealing the chaoticity and the complex dynamics of the
ionosphere, and as a result, studies on the chaoticity of the ionosphere have
been conducted, by some investigators like Bhattacharyya (1990), who studied
chaotic behaviour of ionospheric diversity fluctuation using amplitude and
phase scintillation data, and found the existence of low-dimension chaos.
Also, Wernik and Yeh (1994) further revealed the chaotic behaviour of the
ionospheric turbulence using scintillation data and numerical modelling of
scintillation at high latitude. They showed that the ionospheric turbulence
attractor (if it exists) cannot be reconstructed from amplitude scintillation
data, and their measured phase scintillation data adequately reproduce the
assumed chaotic structure in the ionosphere. Also, Kumar et al. (2007)
reported the evidence of chaos in the ionosphere by showing the chaotic
nature of the underlying dynamics of the fluctuations of the total electron
content (TEC) power spectrum, indicating exponential decay and the calculated
positive value of the Lyapunov exponent. This is also supported by the
results of the comparison of the chaotic characteristics of the time series
of variations of TEC with the pseudochaotic characteristic of the
coloured noise time series. Xuann et
al. (2006) studied chaos properties of ionospheric TEC using TEC data from
1996 to 2004, and analyse the possibility of predicting it by using chaos.
They found the presence of chaos in the TEC measured in the study area, as
indicated by the positive Lyapunov exponent computed from their data. The
correlation dimension was 3.6092 from their estimation. They were also able
to show that the TEC time series can be predicted using chaos.
Also, Unnikrishnan et al. (2006a, b) have analysed the deterministic chaos at
mid-latitudes and Unnikrishnan (2010) and Unnikrishnan and Ravindran (2010)
analysed some TEC data from some Indian low-latitude stations for quiet
periods and major storm periods, and found in their results the presence of
chaos, which was indicated by a positive Lyapunov exponent, and they also
inferred that storm periods exhibit lower values compared to quiet periods.
The dynamical complexity of magnetospheric processes and the ionosphere have
been studied by a number of researchers. Balasis et al. (2008), Kazimirovsky
et al. (2006) and Kazimirovsky and Vergasova (2009) investigated the
dynamical complexity of the magnetosphere by using Tsallis entropy as a
dynamical complexity measure in Dst time series; Balasis et
al. (2009) also investigated the dynamical complexity in Dst
further by considering different entropy measures. Coco et al. (2011), using
the information theory approach, studied the dynamical changes of the polar
cap potential which is characteristic of the polar region ionosphere by
considering three cases: (i) steady interplanetary magnetic field (IMF)
Bz > 0, (ii) steady IMF Bz < 0 and (iii) a double rotation
from negative to positive and then positive to negative Bz. They
observed a neat dynamical topological transition when the IMF Bz turns
from negative to positive and vice versa, pointing toward the possible
occurrence of an order–disorder phase transition, which is the counterpart
of the large-scale convection rearrangement and of the increase of the global
coherence. Further studies on the chaotic behaviour and non-linear dynamics
of the ionosphere over the low-latitude African region is however required
for the improvement of our understanding and characterization of its
dynamical behaviour. Recently, Ogunsua et al. (2014) studied comparatively
the chaoticity of the equatorial ionosphere over Nigeria using TEC data,
considering five quietest day classifications and five most disturbed day
classifications. They were able to show the presence of chaos as indicated
the positive Lyapunov exponents and also were able to show that Tsallis
entropy can be used as a viable measure of dynamical complexity in the
ionosphere, with portions showing lower values of Tsallis entropy indicating
lower dynamical complexity, with a good relationship with Lyapunov exponents.
They found a phase transition from a higher dimension during quiet days to a
lower dimension during storm.
The low-latitude region where Nigeria is situated is known as the equatorial
anomaly region; this region is known for the equatorial ionization anomaly,
which is due to the fountain effect. Off the Equator, the E region electric
field maps along the magnetic field up to the F region
altitude in the low latitude; this eastward
electric field (E) interacts with magnetic field B in the F region during the
day. This results in the electrodynamic lifting of the F region plasma over
the Equator, known as EXB
drift. The uplifted plasma over the Equator moves along the magnetic line in
response to gravity, diffusion and pressure gradients and, hence, the
fountain effect. The fountain effect that is controlled by the EXB drift
shows the dynamics of the diurnal variation equatorial anomaly (Abdu, 1997;
Unnikrishnan, 2010). There is a reduction in the F region ionization density
at the magnetic Equator and also much enhanced ionization density at the two
anomaly crests within ±15∘ of the magnetic latitude north and
south of the Equator (Rama Rao et al., 2006). The equatorial ionization
anomaly and other natural processes which include various ionization
processes and recombination, influx of solar wind, photoionization processes
and so many other factors that occur due to variations in solar activities,
have a great influence on the systems of the ionosphere, due to their effects
on internal dynamics of the ionosphere. This portrays the ionosphere as a
typical natural system with continuous interaction with its external
environment, which led to the study of the influence of the Sun on the
ionosphere (Ogunsua et al., 2014).
The ionosphere possesses a significant level of non-linear variations that
requires more investigation, which can be studied and characterized using a
non-linear approach like the chaoticity and dynamical complexity for the
study of its dynamics. The need to study the daily variation in the dynamical
complexity of the ionosphere arises from the established knowledge and
understanding which shows that the ionosphere is a complex system with so
many variations that can arise from various dynamical changes that can be due
to various changes in different processes that contribute to the behaviour
and nature of the ionosphere. Rabiu et al. (2007) affirmed that
characterizing the ionosphere is of utmost importance due to the numerous
complexities associated with the region. The scale of these numerous
complexities interestingly changes at times from one day to another.
The concept of chaos as previously applied to ionospheric and magnetospheric
studies on quiet and stormy conditions is limited. Most investigations have
been based on quiet and storm conditions only for all studies carried out,
and none of the previous works involved the use of quiet and disturbed day
classification of geophysical conditions until recently by Ogunsua et
al. (2014), where we considered the comparative use of Lyapunov exponents and
Tsallis entropy as proxies for the internal dynamics of the ionosphere. This
is the main reason for the consideration of day-to-day variation of these
parameters in this work.
Coordinates of the GPS stations.
Station
Geographic coordinates
Dip latitude
name
Long. (∘ E)
Lat. (∘ N)
(∘ N)
Birnin Kebbi
4∘12′ E
0.62∘ N
12∘32′ N
Torro
9∘04′ E
-0.82∘ N
10∘03′ N
Yola
12∘30′ E
9∘12′ N
-1.39∘ N
Lagos
3∘23′ E
6∘27′ N
-3.07∘ N
Enugu
7∘30′ E
6∘26′ N
-3.21∘ N
Methods of data analysis and results
Time series analysis
A time series can be seen as a numerical account that describes the state of
a system from which it was measured. A given time series, Sn, can be
defined as a sequence of scalar measurement of a particular quantity taken as
a series at different portions in time for a given time interval
(Δt). The time series describes the physical appearance of an entire
system, as seen in Fig. 1. However, it may not always describe the internal
dynamics of that system. A system like the ionosphere possesses a dominant
dynamics that can be seen as diurnal, so the data should be treated so as to
be able to see its internal dynamics. The measured TEC time series were
plotted to see the dynamics of the system. A typical plot of TEC usually has
a dominant dynamics (see Fig. 1) which may be seen as the diurnal behaviour;
however, it can also be seen that there is a presence of fluctuations (which
appear to be non-linear) in the system as a result of the internal dynamics
of the ionosphere and space plasma system, due to different activities in the
ionosphere. Therefore, there is a need to minimize the influence of the
diurnal variations since we are more interested in the non-linear internal
dynamics of the system in this study; to do so, the TEC time series was
detrended by carrying out the following analysis.
Since for the given daily data of 1 min sampling time there are 1440 data
points per day, there exists a time series ti where
i = 1, 2, 3 … 1440 represents the observed time series, and there
also exists a set of ui where i = 1, 2, 3 … 1440, such that
the diurnal variation reduced time is given by
Ti=ti-uj,
where i = 1, 2, 3, …, j = mod(i, 1440) if mod(j,
1440) ≠ 0 and j = 1440 if d(j,
1440) = 0. This method will give the detrended time series represented by
Ti obtained from the original TEC data as shown in Fig. 2. This method
is similar to that used by Unnikrishnan et al. (2006) and
Unnikrishnan (2010); the further explanations of the dynamical results can be
found in Kumar et al. (2007). The detrended time series were subjected to
further analyses for the phase space reconstruction and also to obtain the
values of Lyapunov exponents, correlation dimension, Tsallis entropy and the
implementation of a surrogate data test.
Phase space reconstruction and non-linear time series analysis
The study of chaoticity and dynamical complexity in a dynamical system
requires a non-linear approach, due to the fact that systems described by
these phenomena can be referred to as non-linear complex systems. The
magnetosphere and the ionosphere are good examples of such systems. To be
able to study such phenomena, some non-linear time series analysis can be
carried out on the time series data describing such a system. The detrended
time series of TEC measurement is subjected to some non-linear time series
data analysis to obtain the mutual information and false nearest neighbours,
embedding dimension and delay coordinates for the phase space reconstruction,
and the evaluation of other chaotic quantifiers, namely, Lyapunov exponents,
correlation dimension, recurrence analysis, and entropy.
The detrended time series plot for TEC measured at Lagos.
![]()
The phase space reconstruction helps to reveal the multidirectional aspect of
the system. The phase space reconstruction is based on the embedding theorem,
such that the phase space is reconstructed to show the multidimensional
nature as follows:
Yn=sn-(m-1)τsn-(m-2)τ…sn-τsn,
where Yn are vectors in phase space. The proper choice of
embedding dimension (m) and delay time (τ) is essential for phase
space reconstruction (Fraser and Swinney, 1986; Kennel et al., 1992).
Average mutual information against time delay for TEC measured at Yola.
![]()
Fraction of false nearest neighbours against embedding dimension for
TEC measured at Yola.
![]()
If the plot showing the time-delayed mutual information shows a marked
minimum, that value can be considered to be an optimal time delay; Fig. 3
shows the mutual information plotted against the time delay. Likewise, the
minimal embedding dimension, which corresponds to the minimum number of false
nearest neighbours, can be treated as the optimum value of the embedding
dimension in Unnikrishnan et al. (2006) and Unnikrishnan (2010). A plot of
the fraction of false nearest neighbours against the embedding dimension can
be seen in Fig. 4. It was observed that for all the daily detrended TEC time
series, the choice of τ ≥ 30 and m ≥ 4 values of delay and
embedding dimension above these values is suitable for analysis of data for
all stations. The choice of τ = 30 and m = 5 was mostly used
to analyse the dynamical aspects for all the stations. The reconstructed
phase space trajectory is shown in Fig. 5.
Lyapunov exponents
The Lyapunov exponent has been a very important quantifier for the
determination of chaos in a dynamical system. This quantifier is also used
for the determination of chaos in time series, representing natural systems
like the ionosphere and magnetosphere (Unnikrishnan, 2008, 2010). A positive
Lyapunov exponent indicates divergence of trajectory in one dimension, or
alternatively an expansion of volume, which can also be said to indicate
repulsion or attraction from a fixed point. A positive Lyapunov exponent
indicates that there is evidence of chaos in a dissipative deterministic
system, where the positive Lyapunov exponent indicates divergence of
trajectory in one direction or expansion of value, and a negative value shows
convergence at trajectory or contraction of volume along another direction.
The largest Lyapunov exponent (λ1) can be used to determine the
rate of divergence as indicated by Wolf et al. (1985), where
λ1=1tlnΔx(t)x(0)=1t∑i=1tlnΔxtiΔxti-1.
The Lyapunov exponent was computed for the TEC values measured from different
stations. The evolution in state space was scanned with τ = 30 and
m = 5, as shown in Fig. 6. The day-to-day variations of the Lyapunov
exponent were computed for the entire year so as to study the annual trend of
variation. This was implemented using the method introduced by
Rosenstein (1993) and Hegger et al. (1994); both algorithms use very similar
methods. Lyapunov exponents were also computed for varying time delays at a
constant embedding dimension and also for varying embedding dimension, to
check for the stability with changes in trajectory. These can be seen in
Fig. 6b and c. The day-to-day values of Lyapunov exponents plotted for the
Enugu station and for the Toro station are shown in Fig. 7a–b. The plots of
the day-to-day values show the transient variation of the ionosphere and a
wavelike yearly pattern.
The delay representation of the phase space reconstruction of the
detrended TEC.
![]()
(a) Lyapunov exponent computed and its evolution computed
as the state space trajectory scanned with τ = 30, m = 5 for
detrended time series measured at Yola with the largest Lyapunov exponent
equal to 0.1347. (b) Lyapunov exponent computed for a different time
delay with a constant embedding dimension. (c) Lyapunov
exponents (LE) computed for different embedding dimensions (m = 3, 4, 5
and 6) at a constant time delay.
![]()
(a) The transient variations of Lyapunov exponents for 365 days of 2011
for detrended TEC measured at Enugu. (b) The transient variations of
Lyapunov exponents for 334 days (1 January–30 November) of 2011 for detrended
TEC measured at Toro.
![]()
Correlation dimension
Another relevant method to study the underlying dynamics or internal dynamics
of a system is to evaluate the dimension of the system. The correlation
dimension gives a good approximation of this as suggested by Grassberger and
Procaccia (1983a, b). The correlation dimension is preferred over the box counting
dimension because it takes into account the density of points on the
attractor (Strogatz, 1994. The correlation dimension D is defined as
D=lnC(r)lnr.
The term C(r) is the correlation sum for radius (r) where, for a small
radius (r), the correlation sum can be seen as C(r) ∼ rd for
r → 0. The correlation sum is dependent on the embedding dimension (m) of the
reconstructed phase space and is also dependent on the length of the time
series N as follows:
C(r)=2N(N-1)∑i=1N∑j=i+1NΘr-|yi-yj|,
where Θ is the Heaviside step function, with Θ(H) = 0 if
H ≤ 0 and Θ(H) = 1 for H > 0.
The correlation dimension was computed using the Theiler algorithm approach,
with the Theiler window (w) at 180. The Theiler window was chosen to be
approximately equal to the product of m and τ. A similar approach to
the computation of the correlation dimension was used by Unnikrishnan and
Ravindran (2010) to determine the correlation dimension of detrended TEC data
for some stations in India, which lies within the equatorial region, like
Nigeria. Ogunsua et al. (2014) also used similar methods for some detrended
TEC from Nigerian stations.
The correlation dimensions for data taken for the quietest day of
October 2011 and the most disturbed day of October 2011 from the Birnin Kebbi
GPS TEC measuring station were represented by Fig. 8a and b respectively. The
correlation dimension saturates at m ≥ 4 for the quietest day of the
month and at m ≥ 5 for the most disturbed day. In this illustration,
the most disturbed day of this month falls within the storm period of
October 2011. The classification of days into quiet and disturbed days in the
month of October 2011 enables us to compare the quiet and storm periods
together while comparing the quiet days with some relatively disturbed days.
(a) The correlation dimension of the detrended TEC for the
quietest day of October at Birnin Kebbi, which saturates at m ≥ 4 and
τ = 39. (b) The correlation dimension of the detrended
TEC for the most
disturbed day of October at Birnin Kebbi, which saturates at m ≥ 5
and τ = 34.
![]()
Computation of Tsallis entropy and principles of non-extensive Tsallis entropy
Entropy measures are very important statistical techniques that can be used
to describe the dynamical nature of a system. The Tsallis entropy can be used
to describe the dynamical complexity of a system and also to understand the
non-linear dynamics like chaos which may exist in a natural system. The use
of entropy measures as a method to describe the state of a physical system
has been employed in information theory for decades. The computation of
entropy allows us to describe the state of disorderliness in a system; one
can generalize this same concept to characterize the amount of information
stored in more general probability distributions (Kantz and Shrieber, 2003;
Balasis et al., 2009). The concept of information theory is basically
concerned with these principles. The information theory gives us an important
approach to time series analysis. If our time series, which is a stream of
numbers, is given as a source of information such that these numbers are
distributed according to some probability distribution, then transitions
between numbers occur with well-defined probabilities. One can deduce the
same average behaviour of the system at a different point and for the future.
The term entropy is used in both physics and information theory to describe
the amount of uncertainty or information inherent in an object or system
(Kantz and Schrieber, 2003). The state of an open system is usually
associated with a degree of uncertainty that can be quantified by the
Boltzmann–Gibbs entropy, a very useful uncertainty measure in statistical
mechanics. However, Boltzmann–Gibbs entropy cannot describe non-equilibrium
physical systems with large variability and a multifractal structure such as
the solar wind (Burgala et al., 2007; Balasis et al., 2008). One of the
crucial properties of the Boltzmann–Gibbs entropy in the context of
classical thermodynamics is extensivity, namely, proportionality with the
number of elements of the system. The Boltzmann–Gibbs entropy satisfies this
prescription if the subsystems are statistically (quasi-)independent, or
typically if the correlations within the system are essentially local. In
such cases the system is called extensive. In general, however, the situation
is not of this type and correlations may be far from negligible at all
scales. In such cases, the Boltzmann–Gibbs entropy is non-extensive (Balasis
et al., 2008, 2009). These generalizations above were proposed by
Tsallis (1988), who was inspired by the probabilistic description of
multifractal geometries. Tsallis (1988, 1998) introduced an entropy measure
by presenting an entropic expression characterized by an index q which
leads to non-extensive statistics,
Sq=k1q-11-∑i=1Wpiq,
where pi are the probabilities associated with the microscopic
configurations, W is their total number, q is a real number, and k is
Boltzmann's constant. The value q is a measure of the non-extensitvity of
the system: q → 1 corresponds to the standard extensive
Boltzmann–Gibbs statistics. This is the basis of the so-called non-extensive
statistical mechanics, which generalizes the Boltzmann–Gibbs theory. The
entropic index q characterizes the degree of nonadditivity reflected in the
following pseudoadditivity rule:
Sq(A+B)k=Sq(A)k+Sq(B)k+(1-q)Sq(A)/kSq(B)k.
The cases q > 1 and q < 1 correspond to subadditivity (or
subextensivity) and superadditivity (or superextensivity) respectively, and
q = 1 represents additivity (or extensivity). For subsystems that have
special theory probability correlations, extensivity is not valid for
Boltzmann–Gibbs entropy in such cases, but may occur for Sq with a
particular value of the index q. Such systems are sometimes referred to as
non-extensive (Boon and Tsallis, 2005; Balasis et al., 2008, 2009). The
parameter q itself is not a measure of the complexity of the system, but
measures the degree of non-extensivity of the system. It is the time
variations of the Tsallis entropy for a given q(Sq) that quantify the
dynamic changes of the complexity of the system. Lower Sq values
characterize the portions of the signal with lower complexity. In this
presentation we estimate Sq on the basis of the concept of symbolic
dynamics and by using the technique of lumping (Balasis et al., 2008, 2009).
A comparison of Tsallis entropy with Lyapunov exponents computed for the same
set of data has been carried out in this work, to see the efficacy of the
combined usage of both parameters. This is based on the established facts
that variations in the values of Tsallis entropy can be linked with that of
Lyapunov exponents' chaotic behaviour in systems as seen in Baranger et
al. (2012), Anastasiadis et al. (2005), and Kalogeropoulos et al. (2012,
2013). Coraddu et al. (2005) showed the Tsallis entropy generalization for
Lyapunov exponents. Further details can be found in Ogunsua et al. (2014);
they were able to investigate the similarities in their response to the
complex dynamics of the ionosphere, and this informs the further use of the
two quantities as indices to study the day-to-day variation of ionospheric
behaviour in this work.
(a) The transient variations of Tsallis entropy for
365 days (1 January–30 November) of 2011 for detrended TEC measured at
Enugu. (b) The transient variations of Tsallis entropy for 334 days
(1 January–30 November) of 2011 for detrended TEC measured at Toro.
![]()
The values of these entropy measures were also computed in order to study the
dynamical complexity of the system under observation (the ionosphere). The
day-to-day values of Tsallis entropy were computed for the entire year for
different stations. The day-to-day values of Tsallis entropy plotted for the
Enugu station and for the Toro station are shown in Fig. 9a and b. The plots
of the day-to-day values show the transient variation of the ionosphere and a
wavelike yearly pattern.
Mutual information plotted against time delay for the original
detrended data (blue curve) with the mutual information for the surrogate
data (red curve) for TEC data measured at Lagos for the quietest day of
March 2011.
![]()
Non-linearity test using surrogate data
The test for non-linearity using the method of surrogate data according to
Kantz and Schreber (2003) has proven to be a good test for non-linearity in
time series describing a system. It has been accepted that the method of a
surrogate data test could be a successful tool for the identification of a
non-linear deterministic structure in experimental data (Pavlos et al.,
1999). This method involves creating a test of significance of difference
between a linearly developed surrogate and original non-linear time series to
be tested. The test is done by carrying out the computation of the same
quantity on both surrogates and the original time series and then checking
for the significance of difference between the results obtained from the
surrogates with the original data. Theiler et al. (1992) suggested the
creation of surrogate data by using Monte Carlo techniques for accurate
results. According to this method, typical characteristics of data under
study are compared with those of stochastic signals (surrogates), which have
the same auto-correlation function and the power spectrum of the original
time series. It can be safely concluded from the test of significance carried
out on the surrogate and the original data that a stationary linear Gaussian
stochastic model cannot describe the process under study provided that the
behaviour of the original data and the surrogate data are significantly
different.
In this work, 10 surrogate data were generated from the original data set.
The geometrical and dynamical characteristics of the original data were then
compared to those of the surrogates using the statistical method of
significance of difference, which can be defined as
S=αSurr-αOriginalσ,
where αSurr is the mean value of the computed quantity for
the surrogate data and αOriginal is the same quantity
computed for the original TEC data; σ is the standard deviation of the
same quantity computed for the surrogate data. The significance of difference
considered for the null hypothesis to be rejected here is greater than 2,
which enables us to be able to reject the null hypothesis that the original
TEC data describing the ionospheric system can be modelled using a Gaussian
linear stochastic model with confidence greater than 95 %.
Fraction of false nearest neighbours plotted against the time
embedding dimension for the original detrended data (blue curve) with the
mutual information for the surrogate data (red curve) for TEC data measured
at Lagos for the quietest day of 20 March.
![]()
(a) Results of surrogate data tests for Lyapunov exponents
for TEC data for the quietest days of October 2011 at Birnin Kebbi station.
(b) Results of surrogate data tests for Lyapunov exponents for TEC
data for the most disturbed days of October 2011 at Birnin Kebbi station.
Original data
Surrogate data
(a)
0.1165
0.3921 ± 0.0420
0.0931
0.2029 ± 0.0756
0.1041
0.3860 ± 0.0741
0.0498
0.2891 ± 0.0598
0.1420
0.3621 ± 0.0504
(b)
0.0579
0.3039 ± 0.0541
0.0502
0.3156 ± 0.0428
0.0786
0.2527 ± 0.0296
0.1795
0.3662 ± 0.0468
0.1038
0.3100 ± 0.0416
The surrogate data test for all stations used in this study shows the
Lyapunov exponents of the surrogate data for the selected days in October
shown in the table below; the results show that the surrogate data test for
Lyapunov exponents shows a significance of difference greater than 2 for all
the selected days for all the stations. Similar results were obtained for
mutual information, fraction of false nearest neighbours, and correlation
dimension. This result gives us the confidence to reject the null hypothesis
that the data used cannot be modelled using a linear Gaussian stochastic
model, which shows that the system is a non-linear system with some level of
determinism. Figure 10 shows the plots comparing the mutual information
plotted against the time delay for the original detrended data (blue curve)
with the mutual information for the surrogate data for TEC data measured at
Lagos for the quietest day of March 2011, while Fig. 11 compares the fraction
of false nearest neighbours for the same set of data. Table 2a shows the
values of Lyapunov exponents for both original detrended and surrogate data
for TEC measured in Lagos during the quietest days and Table 2b shows the
values of Lyapunov exponents for both original detrended and surrogate data
for TEC measured in Lagos during the most disturbed days of October 2011. The
result obtained from the surrogate data test shows that we cannot represent
the original detrended data with a linear Gaussian stochastic model.
Trend filtering using the moving average approach for the daily values
The trend of a fluctuating time series can be made clearer to reveal the
general pattern of that time series, and to make the fluctuating pattern of
the daily variation of the chaoticity and dynamical complexity measures
clearer in the work, the moving average method has been employed. The method
of moving average filtering has found its applications in geophysics
(e.g. Bloomfeld, 1992; Bloomfield and Nychka, 1992; Baillie and Chung, 2002) and
in other areas like financial time series analysis, microeconomics,
biological sciences and medical sciences. The various fields mentioned
require different trend filtering methods, depending on the structure of the
time series to be analysed. Different filtering processes that can be used to
reveal the trend include the moving average filters, exponential filters,
band-pass filtering, median filtering, etc.
Suppose we have a time series z[t] such that t = 1, 2, 3 … n,
where n could assume any value, if z[t] consists of a consistently
varying trend component that appears over a longer period of time t given
as u[t] and a more rapidly varying component v[t]. The goal of trend
filtering in any research is to estimate either of the two components (Kim et
al., 2009). The purpose of trend filtering in this work is to further reveal
the general slow varying trend that appears to be obvious in the daily
variation of the values of the chaoticity and dynamical complexity of the
ionosphere, which might appear to be obviously varying with the yearly solar
activity (a quantity with a slow varying trend). To make u[t] (which
represents the general slow varying trend) smoother and in the process reduce
v[t], we apply the moving average filter.
If we assume z[t] to be our time series representing the daily variation
of the values of the chaoticity and dynamical complexity of the ionosphere,
then our smoothing with weighting vector/filter wj will create the new
sequence uj as
u[t]=z[t]⋅w[n]=12k+1∑i=-kkx[n-1].
In this work, the Savitzy–Golay method of smoothing proposed by Savitzky and
Goley (1967), which is a generalized form of the moving average, was applied
to the trend smoothing of the daily variation of the chaoticity and dynamical
complexity of the ionosphere. In this case it performs a least square fit to
a small set of L(= 2 k + 1) consecutive data to a polynomial and
then takes the midpoint of the polynomial curve as output. The smoothed time
series in this work will now be given as
u[t]=z[t]⋅ω[n]=∑i=-kkAi⋅x[n-1]∑i=-kkAi,
where
ω[n] = An∑i=-kkAi - k ≤ n ≤ k
such that Ai controls the order of the polynomial. A similar method was
described in Reddy et al. (2010).
The smoothed daily variation and the original data and the plot of the
smoothed variation only, for the Lyapunov exponents of the detrended TEC
measured at the Enugu and Toro stations, are shown in Fig. 12a and b. The
smoothed day-to-day variation for Tsallis entropy for the detrended TEC
measured at the Enugu and Toro stations respectively is shown in Fig. 13a
and b.
Discussion
The results presented in the work reveal the dynamical characteristics of the
ionosphere. These characteristics are discussed in this section, considering
the time series treatment and phase space reconstruction; the study of chaos
uses chaotic quantifiers and the use and comparison of dynamical complexity
measures in terms of their response to the variations in ionospheric
dynamics. Also discussed is the implication of the non-linearity test using
the surrogate data and the comparison of the two quantifiers and their
viability as indices for the continuous study and characterization of the
ionosphere.
(a) Daily variation of Lyapunov exponents for TEC measured
at the Enugu station for the year 2011 showing the original data (upper
panel) and the smoothed plot of daily variation of Lyapunov exponents for TEC
measured at the Enugu station for the year 2011 (lower panel).
(b) Daily variation of Lyapunov exponents for TEC measured at the
Toro station for the year 2011 showing the original data (upper panel) and
the smoothed plot of daily variation of Lyapunov exponents for TEC measured
at the Toro station for the year 2011 (lower panel).
![]()
(a) Daily variation of Tsallis entropy for TEC measured at
the Enugu station for the year 2011 showing the original data (upper panel)
and the smoothed plot of daily variation of Lyapunov exponents for TEC
measured at the Enugu station for the year 2011 (lower panel).
(b) Daily variation of Tsallis entropy for TEC measured at the Toro
station for the year 2011 showing the original data (upper panel) and the
smoothed plot of daily variation of Lyapunov exponents for TEC measured at
the Enugu station for the year 2011 (lower panel).
![]()
The time series analysis shows the appearance of some degree of non-linearity
in the internal dynamics of the ionosphere. The time series plot in Fig. 1
shows the rise in TEC to peak at the sunlit hours of the day; however, it can
be seen that the rise to the peak exhibited by the ionosphere, which is the
dominant dynamics during the day, makes it impossible to clearly see the
internal dynamics of the system from the TEC time series plot. It can be seen
that the TEC time series curve is not a smooth curve with tiny variations,
which probably describes a part of the internal dynamics. These visible tiny
variations around the edges of the time series plot can be regarded as the
rate of change of TEC, which is a phenomenon that can describe the influence
of scintillations in the ionosphere; these variations are however more
obvious during the nighttime between the 1100th and 1440th minutes of the day
(that is, between about 18:00 and 24:00 of the day). It should be noted here
that scintillation has been described as a nighttime phenomenon associated
with spread F, and it occurs around the pre-midnight and post-midnight
periods (Vyas and Chandra, 1994; Vyas and Dayanandan, 2011; Mukherjee et al.,
2012; Bhattacharyya and Pandit, 2014). The detrended data show the internal
dynamics of the system more clearly, with a pattern similar to the values
around the night period mentioned earlier. The post-sunset values (especially
at nighttime) in Fig. 1 show a similar pattern to the detrended TEC plot in
Fig. 2. It has been established that TEC does not decrease totally throughout
the night as expected normally through the simple theory that TEC builds up
during the day, but it shows some anomalous enhancements and variations, and
this can occur over a wide range of geophysical conditions (Balan and Rao,
1987; Balan et al., 1991; Unnikrishnan and Ravindran, 2010). The delay
representation of the phase space reconstruction shows a trajectory that is
clustered around its origin, for all the stations, which can be seen as an
indication of the possible presence of chaos. The degree of closeness of
these trajectories however varies for different days from one station to
another, resulting from varying degrees of variations in stochasticity and
determinism. The varying degrees of variations in stochasticity and
determinism can be attributed to the daily variations and local time
variations of photoionization, recombination, influx of solar wind and other
factors that may influence the daily variations of TEC (Unnikrishnan, 2010).
The positive values of Lyapunov exponents indicate the presence of chaos
(Wolf et al., 1985; Rosenstein et al., 1993; Hegger et al., 1999; Kantz and
Schreiber, 2003). The presence of chaos was revealed by the positive Lyapunov
exponents computed from all stations, and this is a result of the fact that
the ionosphere is a system controlled by many parameters influencing its
internal dynamics. Because of its extreme sensitivity to solar activity, the
ionosphere is a very sensitive monitor of solar events. The ionospheric
structure and peak densities in the ionosphere vary greatly with time
(sunspot cycle, seasonally and diurnally), with geographical location (polar,
auroral zones, mid-latitudes, and equatorial regions), and with certain solar-related
ionospheric disturbances. During and following a geomagnetic storm, the
ionospheric changes around the globe, as observed from the ground site, can
appear chaotic (Fuller-Rowell et al., 1994; Cosolini and Chang, 2001;
Unnikrishnan and Ravindran, 2010). The recorded presence of chaos as
indicated by the positive values of Lyapunov exponents was found in all the
computations, for all the TEC values obtained for the selected days from all
the measuring stations used in this work. This can be expected, as it agrees
with results from previous works that show that there is a reasonable
presence of chaos in the ionosphere, even in the midst of the influence of
stochastic drivers like solar wind (Bhattacharyya, 1990; Wernik and Yeh,
1994; Kumar et al., 2007; Unnikrishnan et al., 2006a, b; Unnikrishnan, 2010).
However, the values of Lyapunov exponents vary from day to day due to
variations in ionospheric processes for different days at the same latitude,
as seen in Fig. 7a and b, with Fig. 12a and b showing the day-to-day
variation (upper panel) and the smoothed curve of the day-to-day variation
(lower panel) for the entire year. There are also latitudinal variations due
to spatial variations in the various ionospheric processes taking place
simultaneously. The ionosphere is said to have a complex structure due to
these varying ionospheric processes.
The higher values of Lyapunov exponents during months of low solar activity
(the solstices) is evidence that the rate of exponential growth in
infinitesimal perturbations in the ionosphere leading to chaotic dynamics
might be of a higher degree during most of the days of those months compared
to days of the months with high solar activities showing lower values of
Lyapunov exponents (Unnikrishnan, 2010; Unnikrishnan and Ravindran, 2010).
The results of the correlation dimension values computed are within the range
of 2.7 to 3.2, with the lower values occurring mostly during the storm
periods. The lower dimension during the storm periods compared to the quiet
days may be due to the effect of a stochastic driver that occurs during
geomagnetic storms on the internal dynamics of the ionosphere, like strong
solar wind and solar flares; this could have been as a result of the fact
that the internal dynamics must have been suppressed by the external
influence. The restructuring of the internal dynamics of the ionosphere might
be responsible for low-dimension chaos during storms and also the lower
values of other measures like the Lyapunov exponents. The relatively
disturbed day however might have a higher dimension so long as it is not a
storm period, and sometimes a relatively disturbed day of the month might be
a day with storm, and in this case there is usually a lower value of
chaoticity and sometimes lower values of correlation dimension as well. The
lower value of chaoticity and dimension in the ionosphere during storms
indicates a phase transition from higher values during the quiet periods to
lower values during storm periods, which may be due to the modification of
the ionosphere by the influx of high-intensity solar wind during the storm
period (Unnikrishnan et al., 2006a, b; Unnikrishnan, 2010; Unnikrishnan and
Ravindran, 2010).
The surrogate data test shows a significance of difference greater than 2 for
all the computed measures, which enables rejection of the null hypothesis
that the ionospheric system can be represented with a linear model for all
the data used from the stations. However, it was discovered that the lower
significance of difference corresponds to the lower values of Lyapunov
exponents during storms and extremely disturbed periods (see Table 2a and b).
This may be due to the rise in stochasticity during the storm period as a
result of a drop in values of computed quantities like Lyapunov exponents.
Our ability to reject the null hypothesis for all stations however shows the
presence of determinism and confirms that the underlying dynamics of the
ionosphere is mostly non-linear. This further validates the presence of chaos
since the surrogate data test for non-linearity shows that our detrended TEC
is not a Gaussian (linear) stochastic signal (Unnikrishnan, 2010). However,
for some of the data sets, the saturation of the average mutual information
observed at delay close to unity (as seen in Fig. 3) and the saturation
Lyapunov exponent in the first time step (Fig. 6) indicate that there could
be a high influence of stochasticity in the system.
The Tsallis entropy was able to show the deterministic behaviour of the
ionosphere considering its response during storm periods compared to other
relatively quiet periods, as the rapid drop in values of Tsallis entropy
during storms shows that there is a transition from higher complexity during
quiet periods to lower complexity during storms; this response in the values
of the Tsallis entropy is similar to the response of Lyapunov exponent values
during storms. This reaction to storms shown by the values of the Tsallis
entropy computed for TEC was also described by the reaction of the Tsallis
entropy computed for Dst during storm periods (Balasis et al.,
2008, 2009). A closer observation of the day-to-day variability within a
month shows that the values were much lower for storm periods compared to the
nearest relative quiet period. For example, the storm that occurred on
25 October resulted in lower values of Lyapunov exponents and the Tsallis
entropy compared to relatively quiet days close to it. The reaction to storms
may be due to the influence of a stochastic driver like strong solar wind
flowing into the system as a result of solar flare or coronal mass ejections
(CMEs) that produce the geomagnetic storms. Although there is always an
influence of corpuscular radiation in the form of solar wind flowing from the
Sun into the ionosphere, the influence is usually low for days without storms
compared to days with geomagnetic storms as a result of solar flares, CMEs,
etc. (Unnikrishnan et al., 2006a, b; Unnikrishnan, 2010; Ogunsua et al.,
2014).
The presence of chaos and high variations in the dynamical complexity, even
in quiet periods in the ionosphere, may be due to the internal dynamics and
inherent irregularities of the ionosphere which exhibit non-linear
properties. However, this inherent dynamics may be complicated by external
factors like geomagnetic storms. This may be the main reason for the drop in
the values of Lyapunov exponents and Tsallis entropy during storms. According
to Unnikrishnan et al. (2006a, b), geomagnetic storms are extreme forms of
space weather, during which external driving forces, mainly due to solar
wind, subsequent plasmasphere–ionosphere coupling, and related disturbed
electric field and wind patterns, will develop. This in turn creates many
active degrees of freedom with various levels of coupling among them, which
alters and modifies the quiet time states of the ionosphere during a storm
period. This new situation developed by a storm may modify the
stability/instability conditions of the ionosphere, due to the superposition
of various active degrees of freedom.
The observation from the day-to-day variability of the Lyapunov exponent and
Tsallis entropy also shows irregular patterns for all stations. These
irregular variations might be due to the same factors mentioned before
(i.e. internal irregularities due to so many factors described and also due
to variation in the influx of the external stochastic drivers). The
day-to-day variability for the entire year shows a “wavelike” pattern with
the values dropping to lower values during the equinox months, especially
during the March–April equinox. This can be seen as a form of semiannual
variation, possibly resulting from the higher energy inputs during equinoxes.
This is because solar wind is maximized at the equinoxes, which might result
in higher energy input that will eventually suppress the internal dynamics to
give lower values of chaoticity. The modification of the ionosphere as a
result of the higher energy input resulting from the maximized influx of
solar wind has been reported as being responsible for the lower values of
chaoticity when average-compared to the days of the year with lower solar
wind inputs as reported by Unnikrishnan et al. (2006, 2010) and Ogunsua et
al. (2014). The semiannual pattern has been found to be similar for different
stations, as seen in Figs. 7 and 12 and Figs. 9 and 13 for Lyapunov exponents
and Tsallis entropy respectively. Figures 9 and 13 show the smoothed curves
for Lyapunov exponents and Tsallis entropy respectively, with the drop in
values at equinoxes showing more clearly. The phase transition in chaoticity
and dynamical complexity is also responsible for the wavelike variations,
with values of Lyapunov exponents and Tsallis entropy dropping during the
equinoxial months, and this may be due to the influence of the daily influx
of the solar wind having higher values during equinoxes due to the proximity
of the Earth to the Sun during this period compared to the solstice months.
The wavelike pattern observed has been described as a result of the
self-organized critical (SOC) phenomenon, a phenomenon which has been found
to exist in both the magnetosphere and the ionosphere or the space plasma
system in general, due to coupling between the two systems, since the
magnetosphere couples the ionosphere tightly to the solar wind (Lui, 2002).
Many literatures have shown the existence of chaos in the SOC in the
magnetosphere (Chang et al., 1992, 1998, 1999; Consolini et al., 1996;
Chapman et al., 1998; Freeman and Watkins, 2002; Koselov and Koselova, 2001;
Uritsky et al., 2003; Chang et al., 1992). The existence of SOC in the space
plasma system involving both the ionosphere the the magnetosphere was
described by Lui (2002) and Chang et al. (2002, 2004).
The variation along the latitude also shows the inconsistency and complexity
of the ionospheric processes. This is the reason why, for the same day of the
month, the values of Lyapunov exponents vary from one station to another.
Lyapunov exponents however appear to respond better to changes in solar
activities compared to Tsallis entropy, with more distinct results. This may
be due to the fact that Tsallis entropy is not only a measure of complexity,
but also that a measure of disorderliness in a system might not be as perfect
in describing chaos as Lyapunov exponents. Kalogeropoulos (2012, 2013) and Baranger et al. (2002) observed that Tsallis entropy has a
relationship that is not totally linear in all cases at different levels of
chaos with Lyapunov exponents as a measure of chaos.
There are also many variations in the internal dynamics of the ionosphere
that could lead to changes in chaotic behaviour. The variations of Lyapunov
exponents during quiet days might be as a result of different variations in
the intrinsic dynamics of the ionosphere. Differences in variation pattern at
different stations for the same quiet day might also be due to the same
reason. It can be affirmed that the ionosphere is a complex system that
varies with a short latitudinal or longitudinal interval such that even
stations with one or two degrees of latitudinal differences might record
different values on the same day for both quiet and disturbed periods, and
that the same might also occur for storm periods. This is illustrated by the
different pattern of variation of TEC recorded from different stations within
such a close range as used in this study.
These latitudinal variations in the values of Lyapunov exponents and Tsallis
entropy can be further described by the behaviour of the TEC because there
can be a more sporadic rate of change in TEC, as seen in the time series
plots as a result of irregularities in the internal dynamics of the
ionosphere, which might be as a result of plasma bubbles. Irregularities
develop in the evening hours at F region
altitudes of the magnetic Equator, in the form
of depletions, frequently referred to as bubbles. The edges of these
depletions are very sharp, resulting in large time rates of TEC in the
equatorial ionosphere, even during magnetically quiet conditions. The large
gradient of the equatorial ionization persists in the local post-sunset hours
till about 21:00 LT (DasGupta et al., 2007; Unnikrishnan and Ravindran,
2010). The TEC data for one station might experience an extremely sharp rate
of change in TEC that may be due to some plasma bubbles in that region, while
the TEC from the other station stays normal. These variations in the various
internal dynamics like plasma bubbles leading to scintillation can cause
variations in the dynamical response of the TEC. Hence, the irregular
variation in the values of the Lyapunov exponent and Tsallis entropy, even in
quiet periods for two relatively close stations, may be due to these
irregularities. This might also be responsible for the quiet days in the same
station that have lower values of Lyapunov exponents compared to higher
values recorded for disturbed days without the external influence of storms.
The variations of these chaos and dynamical complexity parameters might also
be a result of the anomalous TEC enhancements that might occur at night
(Balan and Rao, 1987; Balan et al., 1991). These effects can also be seen
more clearly in the Tsallis entropy values for the five-period window for
quiet days of January 2011, because the nighttime value is higher and also
shows a much higher series of fluctuations during this period compared to
other periods. As mentioned in Unnikrishnan and Ravindran (2010), the
irregular changes in the dynamical characteristics of TEC from the results of
Lyapunov exponents and Tsallis entropy also may be due to the collisional
Raleigh–Taylor instability, which may give rise to a few large
irregularities in L band measurements (Rama Rao et al., 2006; Sripathi et
al., 2008); all these can be seen as internal factors responsible for
variations in the dynamical response of TEC as recorded from the values of
the Lyapunov exponents and Tsallis entropy completed for days without storm,
which might be quiet or disturbed according to classification and also could
account for higher values of these qualifiers during disturbed days compared
to quiet days. During storms, however, the values were much lower.
Earlier, we (Ogunsua et al., 2014) showed the appearance and variation of
chaoticity quiet and disturbed day classification by the international most
quiet day (IQD) and internal most disturbed day (IDD) classifications, as
compared to quiet and storm periods used by Unnikrishnan (2006, 2010). We
were able establish that a relatively quiet day may be less chaotic compared
to a relatively disturbed day, unlike the result presented by
Unnikrishnan (2006, 2010) for quiet and storm periods. Also, the combined use
of both Lyapunov exponents and Tsallis entropy for the first time was found
to have a high correlation mostly above 80 %, which has stimulated the
interest in further research using the two diagnoses for the study of
ionospheric dynamics.
This work on the other hand presents the results for day-to-day variation and
has revealed a seasonal trend for both Lyapunov exponents and Tsallis
entropy, which appear wavelike in form, with troughs during the two
equinoxes. This was established for different stations used in this research
work. The results show the appearance of seasonal trends in spite of the
sporadic daily variation resulting from various changes in the internal
dynamics. The seasonal trend has provided more possible evidence of higher
energy input during equinoxes, since it reveals the effect of the annual
energy input to the ionosphere. The day-to-day response of these parameters
has also revealed the variations in the underlying dynamics of the system.
As a similarity between the present work and Ogunsua et al. (2014), the
relationship between Lyapunov exponents and Tsallis entropy can also be seen
from this work, as the two quantifiers exhibit similarities in their response
to the dynamical behaviour of the ionosphere, with phase transition at the
same periods of time for all stations. A further investigation of this
relationship shows that all the daily values of Tsallis entropy correlate
positively with the values of Lyapunov exponents, at values between 0.78
and 0.83.
The ability of these quantifiers to clearly reveal the ionospheric dynamical
response to solar activities and changes in its internal dynamics due to
other factors is a valid proof of the authenticity of the use of these
chaotic and dynamical measures as indices for ionospheric studies.