NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-25-67-2018A correlation study regarding the AE index and ACE solar wind data for
Alfvénic intervals using wavelet decomposition and reconstructionGuarnieriFernando L.fernando.guarnieri@gmail.comTsurutaniBruce T.VieiraLuis E. A.HajraRajkumarhttps://orcid.org/0000-0003-0447-1531EcherEzequielMannucciAnthony J.https://orcid.org/0000-0003-2391-8490GonzalezWalter D.Instituto Nacional de Pesquisas Espaciais – INPE, São José
dos Campos, SP, BrazilJet Propulsion Laboratory, California
Institute of Technology, Pasadena, CA, USALaboratoire de
Physique et Chimie de l'Environement et de l'Espace, CNRS, Orléans,
FranceFernando L. Guarnieri (fernando.guarnieri@gmail.com)31January2018251677618July20174August20178November20179November2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://npg.copernicus.org/articles/25/67/2018/npg-25-67-2018.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/25/67/2018/npg-25-67-2018.pdf
The purpose of this study is to present a wavelet interactive
filtering and reconstruction technique and apply this to the solar wind
magnetic field components detected at the L1 Lagrange point ∼ 0.01 AU
upstream of the Earth. These filtered interplanetary magnetic field (IMF)
data are fed into a model to calculate a time series which we call
AE∗. This model was
adjusted assuming that magnetic reconnection associated with
southward-directed IMF Bz is the main mechanism transferring energy into
the magnetosphere. The calculated AE∗ was compared to the observed AE (auroral electrojet) index using cross-correlation
analysis. The results show correlations as high as 0.90. Empirical removal of
the high-frequency, short-wavelength Alfvénic component in the IMF by
wavelet decomposition is shown to dramatically improve the correlation
between AE∗ and the observed AE index. It is envisioned that this
AE∗ can be used as the main input for a model to forecast
relativistic electrons in the Earth's outer radiation belts, which are
delayed by ∼ 1 to 2 days from intense AE events.
Introduction
Around solar maximum the major causes of geomagnetic storms and space
weather disturbances on Earth are interplanetary coronal mass ejections
(ICMEs), especially magnetic clouds (MCs), and their sheath or shocked fields
(Gonzalez et al., 2007; Echer et al., 2011). However, during the
descending and minimum solar cycle phases, high-speed solar wind streams
(HSSs) become the dominant interplanetary–heliospheric structure, causing
geomagnetic activity on Earth (Sheeley et al., 1976; Tsurutani and Gonzalez,
1987; Tsurutani et al., 1995, 2006; Guarnieri, 2006; Guarnieri et al., 2006;
Kozyra et al., 2006; Turner et al., 2006; Gonzalez et al., 2007; Echer et
al., 2011; Hajra et al., 2013). The features within the HSS causing
geomagnetic activity are large-amplitude Alfvén waves (Belcher and
Davis, 1971; Tsurutani et al., 1982, 1990, 2011a, b; Echer et al., 2011;
Hajra et al., 2013), the southward component of which leads to intermittent
magnetic reconnection between the wave magnetic fields and the magnetopause
fields, allowing the transfer of energy and momentum from the solar wind to
the magnetosphere (Dungey, 1961).
One of the most widely used indices to estimate the energy input to the
magnetosphere and ionosphere is the geomagnetic auroral electrojet (AE) index.
The index is the maximum deviation of the horizontal components of
geomagnetic field variations from a set of globally distributed ground-based
magnetometers located in and near the auroral zone in the Northern
Hemisphere. The AE index represents the overall disturbances in both
eastward and westward ionospheric electrojets located at ∼ 100 km
altitude (Sugiura and Davis, 1966). Thus, substorms and injection events
causing magnetotail plasmasheet injections into the midnight sector of the
magnetosphere with concomitant particle precipitation into the auroral zone
ionosphere may intensify both electrojets, leading to AE index increases.
The Alfvén waves causing the geomagnetic (AE index) activity have short
wavelengths, ranging from ∼ 2 × 105 to 2 × 106 km
(∼ 10 min to ∼ 2 h in duration convected in a
400 km s-1 solar wind), much smaller than ICME and MC scales. The direct use of
the IMF Bz at the L1 libration point during such structures results in poor
correlation against the AE on Earth (Tsurutani et al., 1990, 1995,
Guarnieri, 2005). It is thought that part of the problem is that Alfvén
waves are propagating in the solar wind, and what is detected at the L1 point
is not what hits the Earth's magnetosphere. Another possibility is that the
high-frequency wave power does not contribute to the solar wind energy
transfer process.
A second incentive for better understanding the relation between interplanetary
structures and the AE indices is that intense AE activity events called
HILDCAAs (Tsurutani and Gonzalez, 1987) have been shown to be indirectly
related to the production of relativistic electrons in the Earth's
magnetosphere (Hajra et al., 2014, 2015a, b). In particular Hajra (2015a)
showed that HILDCAA onsets precede relativistic ∼ 0.6 MeV electron acceleration by ∼ 1 day and
∼ 4.0 MeV electron acceleration by ∼ 2 days. These HILDCAA events
are well correlated with the presence of Alfvén waves within HSS
(Tsurutani and Gonzalez, 1987; Gonzalez et al., 1994), and the intensity of
the geomagnetic event depends on the amplitude of the negative Bz component
of the magnetic field of these waves (Guarnieri, 2005, 2006; Guarnieri et
al., 2006).
The overall picture of relativistic electron acceleration in the
magnetosphere is the following: reconnection between the southward component
of the Alfvén waves and the Earth's dayside magnetopause field (Dungey,
1961; Gonzalez and Mozer, 1974; Tsurutani et al., 1995) leads to substorms
and convection events and injections of energetic electrons into the
nightside region of the outer magnetosphere (DeForest and McIlwain, 1971;
Horne and Thorne, 1998). The energetic electron component creates electromagnetic chorus waves through
the loss cone instability (Tsurutani and Smith, 1977; Meredith et al., 2001;
Tsurutani et al., 2013). Then the chorus accelerates the high-energy
electrons to relativistic energies by resonant interactions (Inan et al.,
1978; Horne and Thorne, 1998; Thorne et al., 2005, 2013; Summers et al.,
2007; Reeves et al., 2013; Boyd et al., 2014; Hajra et al., 2015a).
The acceleration of relativistic electrons within the Earth's outer
radiation belt (Paulikas and Blake, 1979; Baker et al., 1986) is an
important physical phenomenon in space weather. These electrons are also
known as “killer electrons” for their hazardous effects to orbiting
spacecraft (Wrenn, 1995; Horne, 2003). Recent studies by Hajra et al. (2013,
2014, 2015a, b) indicate the probability that magnetospheric
relativistic electron acceleration may be predicted more than 1 day in
advance using ground-based observations of auroral activities during HSS.
This paper describes a correlation analysis using a technique of wavelet
decomposition and selective reconstruction applied in both IMF solar wind
data and AE index. The results of this technique may allow us, in the
future, to develop a more complete model to forecast the occurrence of
relativistic electrons during periods with Alfvénic fluctuations in the
interplanetary solar wind.
Methodology
Interplanetary magnetic field and solar wind parameters obtained from the
ACE (Advanced Composition Explorer) spacecraft (Stone et al., 1998) were used in this work. This data set
has ∼ 1 min resolution, and we have used the level 2
processed data. IMF vector data used in this work are in the GSM coordinate
system. The ACE spacecraft is located at the L1 libration point,
∼ 1.5 million km from the Earth, orbiting a region
around the Sun–Earth line. The data are available online at
http://www.srl.caltech.edu/ace.
The geomagnetic activity was observed through the AE indices (Sugiura and
Davis, 1966) and Dst index (Rostoker, 1972). These indices are available
through the World Data Center for Geomagnetism, Kyoto
(http://wdc.kugi.kyoto-u.ac.jp). The AE and Dst indices have 1 min and
1 h time resolutions, respectively. The AE index was used to identify the
periods of enhanced auroral electrojet activity, while the Dst index was
used only to ensure that the analysed intervals were not occurring during
the main phases of magnetic storms.
Long-lasting AE events occurred between years 1998 and 2001
(Guarnieri, 2005).
YearEventStartEndDuration (min)Date and time (UT)Date and time (UT)1998124 Apr 18:0327 Apr 06:053603222 Jul 21:0925 Jul 12:2537971999129 Apr 11:203 May 11:165757217 Aug 22:5220 Aug 12:003669331 Aug 15:322 Sep 20:303179410 Oct 20:0014 Oct 17:385619523 Oct 13:2125 Oct 20:57333767 Nov 17:0010 Nov 04:47358873 Dec 10:006 Dec 00:1537362000127 Jan 18:1031 Jan 03:15486625 Feb 16:018 Feb 05:333693324 Feb 00:0327 Feb 22:105648424 May 10:0026 May 18:0733682001111 May 14:0414 May 10:514128
For this study, we used a set of 14 geoeffective interplanetary HSS events,
previously identified by Guarnieri (2005) as the longest-lasting elevated AE
index events spanning 1998–2001. Table 1 shows a listing of these events
with the year, event start date and time, event end date and time, and event
duration (in minutes). The Alfvénicity of the solar wind for these
intervals was verified using the classical technique proposed by Belcher and
Davis (1971) (i.e. these elevated AE intervals are associated with
high-speed stream solar wind origin).
A filtering process adapted from the Meyer wavelet decomposition and
reconstruction was employed in this technique. This procedure allows a
decomposition of the signal into bands with periods in multiples of 2n
of the data cadence (1 min), with n= 1, 2, 3, … Each decomposed band
is called a “detail” and represented by Dn, where n represents the
decomposition level. Table 2 shows each decomposition level Dn, the level
n, the associated period (2n), and the period range in minutes. More
details about this technique can be found in Meyer (1993) and Kumar and
Foufoula-Georgiou (1997).
Decomposition levels and the corresponding periods range for each
level used in the wavelet decomposition technique.
The last level indicated in Table 2 (A10) contains all the periodicities
longer than 1024 min (∼ 17.1 h), and it can be considered as the
residual of the decomposition process. Further, this level contains the
average value of the data series. We choose this level to
terminate the decomposition since
details of higher orders are so smoothed that they are not useful for the AE
evaluation.
If two or more details are taken and added the time series, this will result in
an “approximation”, which can be considered as a band pass filter. Taking
the A10 level and adding it to D10 will result in approximation A9, and
so on (An- 1 = An+ Dn). In this way, the A0 level will be exactly the same as the
original signal, since it contains all the decomposition levels. The
reconstruction is an interactive process that can be started and stopped at
any decomposition level.
A computer routine was developed to adjust the parameters of the empirical
equations for the calculated AE. After the adjustments, the model was fed
with the filtered IMF and solar wind data time series. The calculated AE for
each event was compared to the real geomagnetic index observed to check the
correlation among them.
Results and discussion
Preliminary tests were performed using the cross-correlation analyses
between the AE index and several interplanetary parameters, such as |B|, Bx, By, Bz, Vsw, and Np, as was done previously by
Guarnieri (2005). The Bz magnetic field component was found to be the parameter most
related to the auroral activity. For this reason, the attempts to adjust a
function describing the AE index were mostly focused on this IMF component.
Wavelet decomposition of the AE index for event 1_1998. The right side
shows the “details” (bands) and the left side shows the “approximations”
(reconstructions).
The wavelet decomposition technique was applied in both the AE index and the
IMF Bz component. Figure 1 shows an example of wavelet decomposition and
reconstruction for the AE index. The top right panel shows the AE time
series. The panels in the right side are the “details”, identified by D1
to D10 (see Table 2 for the corresponding range of each detail). Periods
longer than 1024 min plus the average value of the data series are in
the A10 level, shown in the bottom panel, left side. The panels in the left
side are the “approximations”, which can be viewed as a cumulative sum of
details.
Since the high-speed solar wind events are characterized by enhanced AE
activity, the higher approximation levels (such as A8, A9, and A10) present
increasingly averaged values.
Wavelet decomposition on the interplanetary magnetic field Bz
component for event 1_1998. The right panels show the details (bands)
and the left panels show the approximations (reconstructions).
Figure 2 shows the wavelet decomposition and reconstruction for the solar
wind magnetic field Bz component. The sequence of panels is the same as
Fig. 1. Again, the A10 level represents a typical characteristic during
high-speed solar wind streams: a small but continuous negative Bz average
value. This characteristic was already observed by Guarnieri (2005), through
the calculation of average values of Bz during HSS intervals.
Comparing Figs. 1 and 2, an anti-correlation in level A10 is clearly
observable. This same behaviour is also present in other approximation
levels. This anti-correlation shows that AE activity is driven by the -Bz
on these long timescales during the Alfvénic intervals.
Previous work (Guarnieri, 2005) had observed that high frequencies in the
signal could hide or decrease the correlation between solar wind parameters
and geomagnetic indices, because of the presence of noisy, turbulent
activity. So, an approximation level has to be chosen in order to avoid these
high frequencies and, at the same time, be able to represent the
particularities of the signal. With a computer routine, each reconstruction
level was tested and it was found that the correlation is high up to the
level A3 (starting from A10). Levels A0, A1, and A2 include most of the high
frequencies that reduce the correlation and do not significantly improve the
signal characterization. In this work, we used reconstructions from A10 to
A3, meaning that only periods longer than 8 min were used in the model. This
decision, as well the other assumptions that were used to develop the
following empirical equations, were based on several analyses reported in
Guarnieri (2005). In that work, Guarnieri used the classical
cross-correlation technique, power spectrum, and multi-taper analysis to
correlate the Bz and AE, and the results were only significant when
periods shorter than 8 or 16 min (depending on the technique) were removed.
The studies were performed in both ACE and IMP-8 data. There was no clear
correlation employing unfiltered data and even the lag between the two time
series showed inconclusive results. Progressively removing the high
frequencies lead to correlation
increases and the lag between time series becomes more consistent. The values
obtained for correlation were in the range from ∼ 0.5 to ∼ 0.8.
The lowest values were related to events with the presence of “patches” of
different periodicities in Bz. This behaviour exposes a limitation of the
classical correlation techniques in dealing with time-located periodicities.
Once we verified the good correlation between Bz and AE, an empirical model
was developed to estimate a time series (here called AE∗) based on
interplanetary Bz and compare it against the observed AE index.
The first assumption for this empirical model is that reconnection is the
main physical process transporting energy from the solar wind into the
magnetosphere (and later to the auroral region). In this way, when Bz is
negative we would have energization of the auroral electrojets. Positive Bz
intervals implies that there is no energy input from the solar wind to the
magnetosphere, and thus for these intervals a decay function is used to estimate
the auroral current decay.
Comparison between the calculated AE∗ (dashed blue line) and
the real AE index (solid red line) for event 1_1998.
The calculation and modelling process starts with the interplanetary Bz measured
at L1 (Bz_interp), shifted in time to take into account the interplanetary structure
travel time from the L1 libration point to the Earth, using the solar wind
velocity as a proxy. Bz is the shifted time series and δ is the delay
applied:
Bz(t)=Bz_interp(t+δ).Bz is then decomposed and reconstructed up to the approximation level desired
to eliminate high frequencies, creating the Bz∗ time series (approximation for
the shifted Bz).
The field change is calculated as follows:
ΔBz(t)∗=Bz(t)∗-Bz(t-1)∗.
The first item of ae∗ is
assumed to be -Bz∗ (where ae∗ represents the calculated index
before scale and baseline adjustments).
If Bz(t)∗≤Bz(t-1)∗, then the Bz is
getting smaller or more negative, leading to energization:
ae(t)∗=ae(t-1)∗+ε⋅ΔBz(t)∗.
If Bz(t)∗ > Bz(t-1)∗, then the following is true:
ae(t)∗=aet-1∗exp(-γBz(t)∗).
Finally, the scale and baseline are adjusted in the calculated series:
AE∗=α+(β⋅ae∗),
where AE∗ is the approximation for the AE index time series. A computational
routine was developed to adjust the parameters α, β,
ε, and γ. The best results were achieved with α= 70 nT, β= 150, ε=-0.3333,
and γ= 1. The delay δ (in Eq. 1) depends on the solar wind velocity
and the shifting method employed. It is in the range of 30–70 min.
Figure 3 shows a comparison between this calculated AE∗ (blue line) and the
observed AE index (red line). The reconstructions were created up to the
level A4, meaning that only periods longer than 8 min are present.
There is a good correlation between the two series (calculated and
observed), although there are still some scale problems. However, some
particularities of the real signal were represented very well by the
calculated signal. This event was chosen due to the presence of unambiguous
features, such as those occurring at day ∼ 116
and the big peak just after day 116.5. These features were used to test the
accuracy of the model under unusual conditions.
Correlation coefficients between the calculated (AE∗) and the
observed (AE index) series for each event and in each wavelet decomposition
approximation level.
A comparison among all the events and calculated time series using
different approximation levels is shown in Table 3. The data shown in this
table are correlation coefficients between the calculated AE∗ and the
observed AE index.
Considering the A3 approximation level, all the events have correlation
coefficients higher than 0.7. With the exclusion of events ev1_2000 and ev3_2000, all
of the remaining 12 events have
correlation coefficients higher than 0.85. Event 3_1999 has correlation
coefficients larger than 0.958 for all the approximation levels.
Regarding events ev1_2000 and ev3_2000, the
low correlation coefficients observed led us to reanalyse the data plots to
understand what would be the main difference between these events and the
remaining ones. The Bz data for these two events are basically
high-frequency oscillations around a ∼ 0 nT value, without the
longer-period excursions to negative Bz values typically associated with AE
energization. These high frequencies are exactly those mostly removed by the
wavelets filtering technique we employed here, leading to a weak correlation
against the AE index. Similar results were found by Guarnieri (2005)
when analysing these exact two events, but using different techniques, and reaching
the same conclusion.
If one tries to apply Eqs. (1) to (5) to unfiltered Bz data, this may
result in a very poor correlation coefficient between the estimated AE∗ and
the observed AE index. The interactive filtering process using wavelet
decomposition allows us to effectively remove the high-frequency components
that have poor predictive value, thus obtaining higher correlation values.
These observations lead to several possible scenarios, as shown in Fig. 4. Correlated activity between interplanetary structures and
geomagnetic indices are usually related to large interplanetary structures,
such as interplanetary coronal mass ejections or long-period Alfvén
waves. These structures appear in interplanetary data as low-frequency waves. Due to the sizes of these structures, the Earth's magnetosphere may react as a whole, and so the geomagnetic
indices give us a good idea of the global magnetosphere energization.
Schematic showing the possible causes for correlated and
uncorrelated events between interplanetary parameters and geomagnetic
indices.
However, there are events with uncorrelated activity, usually
those with high frequencies present in the interplanetary data, which are
related to medium- and short-period Alfvén waves, that may miss
impingement on the magnetosphere. There are also long-period waves that can
be uncorrelated due to internal chaotic processes inside the magnetosphere.
When these high-frequency events are removed by filtering IMF Bz and AE
index data, we are able to reach high correlation values such as those shown
on Table 3.
Future work can implement a forecasting model for the AE index for such
periods with high-amplitude Alfvénic fluctuations. One has to use the
real-time data from a spacecraft around L1 and verify the Alfvénicity
through the technique employed by Belcher and Davis (1971). Once the
Alfvénicity is verified, the data series would be fed in through the
wavelet filtering and the AE∗ evaluation equations. This would give us a
forecasted AE with a delay of only a few minutes, or almost a “nowcast”.
However, the main result would be the relativistic electron forecasting.
Since Hajra et al. (2013, 2014, 2015a, b) have shown the probability
that magnetospheric relativistic electron acceleration may be predicted by
more than 1 day in advance using ground-based observations of auroral
activities during high-speed solar wind streams, the ability to obtain an
equivalent auroral index may lead to a more complete model which also
includes the forecasting of relativistic electrons. It is important to note
that data from other spacecraft or different processing levels can be
employed in this forecast. One just has to take in to account the
propagation delay according to the spacecraft position.
Conclusions
A method to calculate an AE∗ time series based on IMF data measured at L1 in
the presence of interplanetary Alfvén waves was developed. This method
employs an Alfvénicity check and a wavelet decomposition technique,
which is applied to the interplanetary magnetic field Bz component. This
calculated AE∗ was shown to be highly correlated with the observed AE index.
The correlation coefficients between the calculated and observed series can
reach values over 0.90, depending on the resolution and the level of details
assumed.
Future works can use data from the any other spacecraft located around L1 to
feed the model and obtain the AE∗, and this AE∗ feeds a routine to predict
the occurrence of relativistic electrons, giving advanced notice by more
than a day. This work will hopefully be completed within the next few years.
Final comments
Although correlations between AE∗ and AE as high as 0.90 have been
indicated in this paper, the causes for the lack of a high correlation in
some intervals are still in question. One
possibility is that there is high-frequency turbulence in the solar wind
which is not geoeffective, a point mentioned previously. However there is
another, recently discussed possibility: local generation of Alfvén
waves in the interplanetary medium (Tsurutani et al., 2018). If
Alfvén waves are generated between ACE and the Earth, this would
naturally reduce the correlation between AE∗ and AE. This may account
for intervals where AE∗ and AE are not well correlated.
Both of these explanations
are possible. In the future we will probe which of these two (or other
possibilities) are factors, and which are the dominant ones.
ACE spacecraft data is available at
http://www.srl.caltech.edu/ace (ACE, 2007). Geomagnetic data (AE and Dst indices
are available at http://wdc.kugi.kyoto-u.ac.jp/ (World Data Center for Geomagnetism, 2007).
The authors declare that they have no conflict of
interest.
This article is part of the special issue “Nonlinear Waves and
Chaos”. It is a result of the 10th International Nonlinear Wave and Chaos
Workshop (NWCW17), San Diego, United States, 20–24 March 2017.
Acknowledgements
The author Fernando L. Guarnieri would like to thank FAPESP (Brazil) projects
04/14784-4 and 2007/01449-4. Ezequiel Echer would like to thank the Brazilian
CNPq PQ 302583/2015-7 agency for financial support. The work of Rajkumar
Hajra was supported by ANR under the financial agreement ANR-15-CE31-0009-01
at LPC2E/CNRS. The authors would also like to acknowledge the ACE team for
providing the data used in this work (http://www.srl.caltech.edu/ace),
and the World Data Center for Geomagnetism, Kyoto, for the geomagnetic
indices. Portions of this research were performed at the Jet Propulsion
Laboratory, California Institute of Technology under contract with NASA.
Sponsorship of the Heliophysics Division of NASA's Science Mission
Directorate is gratefully acknowledged. Edited
by: Annick Pouquet Reviewed by: Francesca Di Mare and one
anonymous referee
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