Draft 29 Aug 2022
Determining the timing of driver influences on 1.8-3.5 MeV electron flux
at geosynchronous orbit using ARMAX methodology
L. E. Simms1,2, M. J. Engebretson1,
G. D. Reeves3
1Department of Physics, Augsburg University,
Minneapolis, MN, USA
2Climate and Space Sciences and Engineering,
University of Michigan, Ann Arbor, MI, USA
3Space Science and Applications Group, Los Alamos
National Laboratory, Los Alamos, NM, USA
Corresponding author: L. E. Simms (simmsl@augsburg.edu)
Keypoints
1. ARMAX models show drivers of relativistic electron flux are
influential only within a few hours of flux changes
2. Contrary to simple correlation findings, influences are lower in
magnitude and act more immediately
3. Stepwise multiple regression shows less cumulative effects of drivers
in after-storm periods than simple correlation would suggest
Abstract
Although lagged correlations have suggested influences of solar wind
velocity (V) and number density (N), IMF Bz, ULF wave power, and
substorms (as measured by AE) on MeV electron flux at geosynchronous
orbit over an impressive number of hours and days, a satellite’s diurnal
cycle can inflate correlations, associations between drivers may produce
spurious effects, and correlations between all previous time steps may
create an appearance of additive influence over many hours.
Autoregressive-moving average transfer function (ARMAX) multiple
regressions incorporating previous hours simultaneously can eliminate
cycles and assess the impact of parameters, at each hour, while others
are controlled. ARMAX influences are an order of magnitude lower than
correlations. Most influence occurs within a few hours, not the many
hours suggested by correlation. Over all hours, V and N show an initial
negative impact, with longer term positive influences over the 9 (V) or
27 (N) h. Bz is initially a positive influence, longer term (6 h)
negative effect. ULF waves impact flux in the first (positive) and
second (negative) hour before the flux measurement, with further
negative influences in the 12- 24 h before. AE (representing electron
injection by substorms) shows only a short term (1 h) positive
influence. However, when only recovery and after-recovery storm periods
are considered (using stepwise regression), there are positive
influences of ULF waves and V, negative influences of N and Bz, while AE
shows no influence.
Plain Language Summary
The influence of solar wind, waves, and substorms on high energy
electrons at geosynchronous orbit can appear to occur over a number of
hours and days. However, these long duration correlations may be due to
diurnal cycles in satellite data, associations between the driving
parameters, or correlations of each variable with itself over previous
time steps. These extraneous correlations can be corrected for using
autoregressive-moving average multiple regression models including
previous hours simultaneously. Once these are controlled, the
correlations between possible driving parameters and high energy
electrons are both lower and influential only over a few hours.
Introduction
The response of high energy (MeV) electron flux at geosynchronous orbit
to various solar wind, IMF, and magnetospheric parameters has been well
studied at a daily cadence (e.g., Balikhin et al., 2011; Mathie & Mann,
2000; Reeves et al., 2011; Potapov, 2017; Sakaguchi et al., 2015; Simms
et al., 2016; 2018; Wing et al., 2016), but injection or acceleration of
electrons may occur within 24 h (e.g., Reeves et al., 1998). Daily
averages may obscure this activity, and the associations of hourly flux
with possible drivers is not as well documented.
The timing and direction of correlations between drivers and flux has
been used to infer the physical processes that result in flux changes.
For example, ULF wave-driven inward radial diffusion, resulting in
electron acceleration, is thought to require a number of days of
previous high wave activity (O’Brien et al., 2001; Friedel et al., 2002,
Osmane et al., 2022), although the diffusion itself may happen fairly
rapidly (Jaynes et al., 2018). ULF waves may also result in outward
radial diffusion, leading to electron loss, which would be reflected in
a negative correlation, possibly at a different time step (Elkington et
al., 2003). High solar wind velocity is thought to drive these ULF waves
perhaps via the Kelvin Helmholtz effect (Rostoker et al., 1998) or via
its contribution to solar wind pressure variations, with the latter
thought to be more likely due to the timing of the maximum correlation
(Takahashi and Ukhorskiy, 2007). However, both solar wind velocity and
density contribute to pressure and its variations, therefore the
correlation of number density with electron flux is also important to
consider (Balikhin et al., 2011; Borovsky & Denton, 2014; Boynton et
al., 2013; Lyatsky & Khazanov, 2008). Pressure, and therefore both
velocity and number density, may also result in rapid flux reductions
due to magnetopause shadowing (Shprits et al., 2006; Loto’aniu et al.,
2010; Staples et al., 2022; Tu et al., 2019), and there is no reason to
discount the possibility that both positive and negative effects could
be due to these same variables acting in different ways and at different
time scales. Substorms have also been proposed to influence relativistic
electron flux through the injection of seed electrons (hundreds of keV),
which provide a population that can be accelerated to high energies
between the substorm-injected energetic-electron flux in the
magnetosphere and relativistic electron fluxes (Birn et al., 1998; Hwang
et al., 2007), as well as source electrons (tens of keV), producing the
VLF waves that contribute to electron acceleration (Jaynes et al., 2015;
Friedel et al., 2002; Summers et al., 2002; Boyd et al., 2014). Substorm
activity itself, however, appears to be dependent on a southward IMF Bz
(Jaynes et al., 2015). Various magnetospheric indices such as Kp and Dst
also correlate well with flux (Borovsky & Denton, 2014; Lam, 2004;
Sakaguchi et al., 2015, Su et al., 2018), but the proposed physical
action of these indices is not as clear and they tend to correlate
highly with the parameters listed above, meaning any correlations
between flux and Kp or Dst might more likely be the result of their
correlation with other drivers rather than an actual physical
relationship with flux.
Understanding the timing of the action of these various driving
parameters on electron flux would be helpful in determining the physical
relationships between them. Previously, cross correlations (simple
correlations at each time step) have been used to study this
statistically. Integrating over a number of hours may increase the
correlations with flux (Romanova & Pilipenko, 2009). Maximum
correlations were found, when integrated, for solar wind velocity (98
h), number density (38 h), and pressure (16 h), IMF Bz (116 h), AE (140
h), a ground ULF index (123 h), Dst (106 h), and Kp (138 h), among other
possible variables (Borovksy, 2017). However, integrating or averaging
over a number of hours can obscure the details of the time dependent
interactions between predictors and flux. In addition, any moving
average of this sort will lag behind any trend that occurs in the time
series. Even a small trend will result in a moving average that is
consistently above or below actual values (Hyndman & Athanasopoulos,
2018). Lower energy electrons (<100 keV) have also been shown
to correlate with specific time lags of V and N within 24 h (Stepanov et
al., 2021).
Simple correlations can confound a number of different processes into
one number. Driving factors may be correlated among themselves, action
at one time step will be correlated with other time steps, and
co-cycling or common trends between electron flux and possible drivers
can greatly inflate the apparent correlations. The first problem can be
addressed by various means including multiple regression using all
variables (Simms et al., 2014; 2016; Potapov, 2017; Sakaguchi et al.,
2015) or conditional mutual information (Wing et al., 2016; Osmane et
al., 2022). At an hourly cadence, controlling velocity for number
density and vice versa, the peak correlation of flux with solar wind
velocity has been found at 44-56 h when solar wind density is
controlled, while the peak solar wind density correlation with flux,
when velocity is controlled, is at 7-11 h (Wing et al., 2022).
Similarly, correlations between electron flux and ULF activity peaked at
48 h (Osmane et al., 2022). However, in these studies, the effect due to
other time steps was not removed from the analysis of each other time
step. The second problem can be mitigated in a similar fashion by
including more than one time step in a multiple regression (Simms et
al., 2018).
However, common cycles and trends that inflate correlations can be dealt
with by describing the time behavior with autoregressive (AR) and moving
average (MA) terms. Using these ARMA terms (as well as differencing:
subtracting previous observations) methods, we have previously found
correlations, while statistically significant, much lower than that seen
in uncorrected lagged correlations. Electron flux-solar wind velocity
correlations peaked at 0.1 (vs. 0.7 seen in lagged correlations) between
24-48 h previous, while flux-ULF correlations were strongest at 0-1 h
(-0.2) and at 24-30 h (<0.1), again much lower than a simple
lagged correlation around 0.5 peaking at 50 h (Simms et al., 2022).
However, these analyses, over 0-96 h, included only 1 lag at a time. In
the present study, we seek to gain a fuller understanding of the timing
of each influence, corrected for both the other factors as well as other
time lags of itself, using ARMAX analyses to remove the co-cycling that
can contribute to spurious correlations in these data.
In addition, we use subsets of this data, choosing periods following
storms, to study the timing of influence during more disturbed periods.
A continuous (and long) time series is not possible for this
storm-subsetted data so we cannot analyze it using AR and MA terms.
Instead, using a flux measurement following recovery, we use stepwise
regression on predictors from previous time steps to describe the points
of highest influence.
Data and Analysis Approach
We use hourly averaged log10 electron fluxes (log
(electrons/(cm2/s/sr/keV))) in the 1.8-3.5 MeV
(“relativistic”) and 100 keV (“seed”) ranges from the Los Alamos
National Laboratory (LANL) Energetic Spectrometer for Particles (ESP)
instrument located at geosynchronous orbit (≈ 6.6 RE) on
the 1994-084 satellite. We limit the period to 25Oct1995 – 13Jun2002
when there was a minimum of missing data. Short periods of missing
values were interpolated from surrounding values. Hourly averages of the
log10 solar wind velocity (km/s) (V),
log10 number density (#/cc) (N), IMF Bz (GSM: nT) (Bz),
log10 pressure (nPa) (P), log10 AE (nT),
and Dst (nT) from the OMNIweb database are used. We use the Kozyreva et
al. (2007) log10(ULF Pc5 index) which provides an hourly
measure of Pc5 (2–7 mHz) ULF power (in nT2/Hz)
observed in the local time range from 0500 to 1500 hours by ground-based
magnetometers stationed between 60 and 70°N corrected geomagnetic
latitude (ULF).
Analyses were performed in MATLAB and SPSS. ARIMAX models were developed
using the SPSS TSMODEL procedure.
We chose a parsimonious model for
the dependent variable (relativistic electron flux), adding AR and MA
terms until the partial autocorrelation function (PACF) contained no
significant terms. Using the ARMA terms chosen by this procedure, we
then added lagged inputs as independent variables (up to 96 h prior to
the electron flux measurement). Electron flux was fit with both
autoregressive (at 1 h) and moving average terms (at both at 1 and 2 h)
and daily autoregressive and moving average terms (both at 1 d)
(Balikhin et al., 2011; Boynton et al., 2013; Pankratz, 1991; Makridakis
et al., 1998; Hyndman & Athanasopoulos, 2018; Simms et al., 2019).
In the overall analyses, we report which coefficients were statistically
significant, both at a standard alpha level of 0.05, and at a corrected
threshold using the Holm-Bonferroni method to control
the familywise
error rate. The p-value is the probability that the null hypothesis is
true. In regression, the null hypothesis is that the slope of the
relationship between dependent and independent variables (the regression
coefficient) is zero. If the p-value is less than the standard level of
0.05, this means there is only a 5 % probability that the null
hypothesis of a zero slope is true. We would thus reject this null
hypothesis and conclude that there is a relationship between the
variables (i.e., a non-zero slope). However, if many comparisons are
being made, the 5 % level of rejection means that, randomly, we will
unknowingly be accepting a false null hypothesis 5% of the time. While
this is an acceptable rate for a single comparison (the comparisonwise
error rate), when making >20 comparisons the likelihood of
finding a false relationship in the entire set of comparisons (the
familywise error rate) is nearly 100%. In order to keep these mistakes
low (i.e., to keep the family-wise error rate controlled), various
corrections can be made. In this case, we use the Holm-Bonferroni method
to decrease the threshold p-value based on the number of comparisons
being made (Holm, 1979).
We identify 206 storms in the dataset where Dst dropped below -40 nT and
where the end of recovery (a Dst rise above -30 nT) was reached before
the start of the next storm. Of these, there were 169 storms with at
least 44 h after the end of recovery before the next storm. To reduce
the number of lags to test, we average variables over every 2 h. Using
relativistic electron flux at this point, we identify the statistically
significant lag times for each parameter by performing a stepwise
regression. This reduces the lags to those that are most influential by
only entering each lag if its coefficient is statistically significant,
and only retaining that lag in the model if it does not lose
significance when another lag is entered. For entering variables, we set
the threshold p-value to 0.05. We set the threshold p-value to 0.10 for
removal of variables (Neter et al., 1985).
We compare several analysis approaches:
a. Simple cross correlations (lagged correlations) for each hour and
each predictor variable with relativistic electron flux up to 96 h
(Figure 1).
b. ARMAX analysis for each variable at each hour independently (Figure
1)
c. ARMAX analysis with all hours, combining V, N, Bz, ULF, and AE in a
single analysis (Figure 2).
d. Simple cross correlation (lagged correlations) during storm periods
to 44 h after the end of recovery (Figure 4).
e. Stepwise regression of each parameter individually during this pre
and post storm recovery period (Figure 4).
f. Stepwise regression combining V, N, Bz, ULF, and AE in a single
analysis during this pre and post storm recovery period (Figure 5).
Results
Although simple correlations of possible drivers with electron flux can
be fairly high, when common cycles and trends are accounted for using
the ARMAX methodology, the associations are much lower (Figure 1). Each
variable is analyzed separately, each hour separately, either as a
simple lagged correlation (blue line) or with AR and MA terms added to
account for common cycling (green line). Solar wind velocity has simple
correlations > 0.4 (0.42 50 h previous), ULF shows simple
correlations peaking at 0.30 (62 h), and the correlation of N with flux
is greatest in magnitude at -0.31 (6 h). Somewhat lower correlations are
seen with AE (0.26 at 61 h), Dst (-0.23 at 54 h), Bz (-0.08 at 61 h), P
(both -0.19 at 9 h and 0.15 at 96 h). Seed electrons show the highest
simple correlation, up to 0.87 1 h earlier. (Note difference in y-axis
scale for seed electrons.)
However, the introduction of ARMA terms to describe the cycling and
trends greatly reduces the association of each variable with flux. The
coefficients of the input variables of the ARMAX models are an order of
magnitude lower than the correlation coefficients. Thus, much of the
apparent correlation found in simple correlations is merely due to
co-cycling and long-term trends. (As we use standardized variables, the
single variable correlations can be compared directly to the regression
coefficients for these variables in the ARMAX models.) Once an attempt
at removing cycling and trends is made, the greatest influence of some
variables (Bz, ULF, and AE) occurs within the first few hours, in
contrast to the lagged correlations. The removal of cycling also appears
to result in strong negative associations of electron flux with V in the
first hour, but with strong positive effects 14-96 h previous. N and P
show negative associations with flux over 1-20 h previous. Seed
electrons still show a spike in positive correlation at 24 h periods,
indicating that the removal of cycles was not entirely successful and
that even these corrected coefficients are suspect. It is also possible
that the high correlation of seed and relativistic electrons is a
reflection of both these energies being driven by the same processes
rather than one driving the other. Although we include Dst in this
single variable analysis, we do not have a well-defined physical
explanation for how Dst might drive relativistic electron flux. We
therefore suspect that correlations between flux and Dst are due to a
mutual correlation with the actual drivers rather than to Dst having any
direct or indirect influence on electron. In addition, much of the Dst
effect may simply be the result of the magnetosphere rebounding so that
the radiation belt is now at the altitude of the satellite rather than
any actual increase in electron flux.
However, as all these variables and lags are intercorrelated with each
other, these individual correlations, even with cycling and trend
influences removed, are still misleading. For example, it is impossible
to know if the V-flux correlation is simply a restatement of the N-flux
correlation, given that it is known V and N are highly (negatively)
correlated with each other. These variables must be analyzed
simultaneously to assess the degree to which each correlates with flux
when other possible drivers are held constant. We therefore combine the
possible drivers (both direct and indirect) into a single ARMAX model
(Figure 2). We drop P from the model as it is so highly correlated with
N and V as to create multicollinearity problems, leaving P fighting to
explain the same variation as N and V. We also drop seed electrons,
since they may be a result and not a cause, and Dst, as it may only be
correlated with the actual drivers. For this analysis, we report which
coefficients are statistically significant, both at a standard alpha
level of 0.05 (light blue bars), and at a corrected threshold using the
Holm-Bonferroni method to control
the family-wise
error rate (dark blue bars). We use this more conservative family-wise
error rate in our conclusions.
In the combined analysis, V (Figure 2a) and N (2b) show negative
influences at 1 h. This is likely the signature of an initial pressure
pulse, which both N and V contribute to. This is preceded by positive
influence, with the effect of N being longer lasting (up to 24 or 40 h
previous). IMF Bz (c) shows a positive influence at 1 h with negative
influences at 3-8 h. ULF (d) shows two opposing rapid effects (positive
at hour 1, negative at hour 2) with continued negative influence in the
preceding hours. AE (e) is lower in magnitude than the other effects
with only a single (positive) strongly significant effect at hour 1.
Thus, most of the influence of these parameters occurs in the first few
hours with only N showing a reliably significant influence out to 24 h.
Note that most of these influences in the full ARMAX model are not only
lower than that seen in the simple lagged correlations, but they are
strongest over a shorter time frame, and may be in the opposite
direction from that seen in the simpler and uncorrected analyses.
Conclusions about direction and timing of influence based on lagged
correlations are often unsupported.
4.1 Influence during Storms
The influence of these parameters may differ during geomagnetic storms.
We perform similar analyses on a subset of identified storms over this
time period. First, we create a superposed epoch analysis of 2 h
averaged electron flux during the 206 storms in this period in which
recovery was allowed to finish without a new storm occurring (i.e., when
Dst was allowed to rise to -30 nT following the main phase). The epoch
is centered at this end of recovery marker (Figure 3). The maximum flux
does not occur during recovery, but in the hours and days after the Dst
rise above -30 nT. In this data set, the average flux during storms
(black line) rises to a peak at 44-48 h following recovery.
The number of storms available for this analysis falls steadily as the
analysis period is increased, due to the possibility of subsequent
storms occurring after each storm’s recovery period. While there are 206
storms that reach the end of recovery, there are only 184 and 169 at the
first and second peaks of flux. The lower sample size both increases the
95% confidence interval around the mean (dashed lines) and reduces the
degrees of freedom available in the error term for regression analysis.
For this reason, we limit the correlation and regression analyses to
those storms (n=169) where flux has risen to near its average maximum at
44 h after recovery, but the number of storms available for the analysis
is not as limited as it is in the later hours.
As we only use one point to measure electron flux, co-cycling
correlations should not be as much of a problem. Individual parameters
(Figure 4) back to 24 h before the end of recovery are analyzed both by
simple lagged correlation where each lag is correlated independent of
the others (top plot of each panel) and by a stepwise regression
allowing the incorporation of any statistically significant predictor
lag (the lagged predictor model in the bottom plot of each panel). We
might have used a lagged predictor model without the stepwise procedure,
entering all lags at once, however, if all lags are included, the low
number of storms (n=169) and high number of predictors (68 lags for each
variable) mean that the ability of the regression to pick out the
statistically significant lags is low. In addition, certain close lags
may tend to fight against each other to explain the same small bit of
variation, which can result in apparently significant opposing pairs of
coefficients in quick succession. In stepwise regression each lag is
entered into the analysis only if its influence is statistically
significant and only retained in the model if it does not lose
significance when another lag is entered (Neter et al., 1985).
In comparison to the more generalized lagged correlations, the lagged
predictor regression analyses give a better understanding of when the
action of each predictor occurs. Statistically significant (dark blue:
p<0.05) and nonsignificant (white: p>0.05) lags
are shown for each parameter in the regressions and the lagged
correlations. While the lagged correlations show a more familiar pattern
of many influential lags, possibly peaking at some number of hours
before the post-storm flux measurement for most parameters, when all
lags are considered for incorporation in regression, the timing of the
action of each parameter becomes much more specific.
For example, the simple lagged correlations indicate that V (4a: peak
influence of r=0.60 2-4 h after recovery) has strong, significant, and
long lasting effects. However, the lagged predictor regression analysis
shows a strong effect of V only 2 h before the end of recovery, with no
long lasting effects. The spread of effect over many hours in the lagged
correlation is only an artifact of each hour’s V being correlated with
itself at different time steps. N, and to a lesser extent Bz and Dst,
also appear to have effects lasting over many hours (in the lagged
correlations), but the stepwise regression only identifies 1 or 2 lags
at which these parameters are most strongly correlated with flux (N: 4 h
before the end of recovery and 12 h after; Bz: 18 h after recovery; Dst:
26 h after the end of recovery). ULF and AE show somewhat more
persistent effect, with 3 peaks of influence (20 h before and at 8 and
18 h after the end of recovery for ULF, and at 14 h before and at 8 and
20 h after recovery for AE). Pressure shows no significant influence in
the studied time period, whether in lagged correlation or stepwise
regression. The strongest correlation of seed with relativistic
electrons occurs at 44 h after the end of recovery, at the hour where
relativistic flux is being predicted. This suggests that this is also an
artifact, probably due to them being measured at the same location or
being simultaneously driven by the same influences. Somewhat more
trustworthy associations of seed and relativistic electrons are found at
2 h after (positive) and 4 h before (negative) the end of recovery, in
that they are not simultaneous.
As above, the intercorrelations of the possible drivers mean that single
variable analyses (as in Figure 4) may not accurately assess the
influences of each variable. We combine the 5 most likely drivers (V,N,
Bz, ULF, and AE) into one stepwise regression. In this analysis, the
effect of V appears at 18 h after recovery, N 33 h, Bz at 6 h (as well
as 20 h before the end), and ULF at 22 and 32 h after the end of
recovery as well as 10 h before the end. AE has no significant
influence. This is somewhat different from the individual stepwise
analyses, but markedly different from the simple lagged correlations. In
the stepwise analyses, there is no long term cumulative effect of any
variable, with significant influences occurring at (at most) 3 time
steps. While the individual stepwise regressions show 1-3 significant
time steps near the point of maximum lagged correlation, when combined
into a single analysis, these significant hours no longer correspond to
the maximal simple correlations. Addition of other variables changes the
response: the most marked difference is the disappearance entirely of
the AE effect.
Discussion
In previous work, simple correlations suggested influences of solar wind
velocity and number density, IMF Bz, ULF wave power, and substorms (as
measured by AE) on MeV electron flux over an impressive number of hours
and days. However, the diurnal cycle in flux measurements from
geosynchronous satellites can inflate correlations, the associations
between potential drivers may produce spurious effects, and correlations
between all previous hours of each driver may create the appearance that
it acts additively over many hours. Autoregressive-moving average
transfer function (ARMAX) multiple regressions incorporating previous
hours simultaneously can eliminate these cycles and study the impact of
each parameter, at each hour, while the others are controlled. This can
accomplish the same goal as integrating over a number of hours (Borovsky
2017) while retaining the details of which hours during the integration
period are most influential.
When studying all hours, using
ARMAX lagged correlations, we show that the impact of potential drivers
is at least an order of magnitude lower than correlations alone would
suggest for all tested parameters, with most of this influence occurring
within a few hours of the electron measurement.
This is in contrast to previous
studies which found higher correlations over all hours (e.g., Borovksy,
2017; Osmane et al., 2022; Wing et al., 2022). Much of the correlation
found between ULF and flux, for example, appears to be the result of
co-cycling parameters (Simms et al., 2022). Removing this diurnal cycle
greatly reduces the apparent influence, both in magnitude and in time.
The long, potentially cumulative effects seen in simple correlations
disappear for most variables, either in the introduction of the ARMAX
terms (Figure 1) or when variables are studied in a combined analysis
(Figure 2). However, with cycles removed, we are better able to study
the actual associations between variables, keeping in mind that our goal
is not to find the highest correlation but the one that answers the
questions of interest.
Of the five variables we study with both lagged ARMAX correlations and
combined variable models, we find the following result from over all
hours (using lagged ARMAX correlations) and from periods just following
storms (using stepwise regression):
a. Over all hours, with cycles removed, V, which commonly shows the most
impressive, positive simple lagged correlations, shows a more nuanced
influence, with the initial effect (in the first hour) being negative
(perhaps driving magnetopause shadowing), preceded by several hours of
positive influence (2-10 h previous to flux measurement, presumably the
result of driving processes that result in electron acceleration). This
contrasts with previous studies that conclude the peak correlation
occurs much further in the past, 45-65 h (simple correlations), 40-100 h
(using conditional mutual information) (both from Wing et al., 2022), or
81 h (when optimized for correlation with 4 other variables) (Borovsky
2017). The removal of cycles, which removes much of the spurious
correlation, leads to a completely different conclusion about the timing
of V influence on flux. Integrating over a number of hours, rather than
testing each lag simultaneously, obscures the complexities of the
relationship. A 98 h time integration of V may correlate better with
electron flux (Borovsky, 2017), but it averages out the opposing effects
of V at different time steps and gives the impression that there must be
long periods of high V to drive electrons to higher energies, which is
not supported by our current analysis.
In contrast, the response of electrons to V only during after storm
periods is either a positive response to a single hour at the end of
recovery (in the single variable analysis of Figure 4) or a positive
response to a single hour 18 h after the end of recovery (in the
combined analysis of Figure 5). As we do not explicitly test the flux
response (after storms) to the beginning of the storm period, it is
likely that we miss the response to an initial storm pressure pulse that
V would contribute to or magnetopause shadowing. Thus, the quick
negative response to V (or P) is only visible in the analyses using all
periods (in the first hour). In the after-storm period analysis, we are
left with only the positive response to V. This may not be a direct
response to V, but rather an indirect reaction to other processes driven
or associated with the velocity.
b. N behaves similarly to V in that the lagged, single variable ARMAX
correlations roughly follow the trend of the negative simple
correlations, albeit lower in magnitude. However, in the combined ARMAX
analysis (over all hours), N is mostly a positive influence (barring the
initial negative influence 1 h before the flux measurement), and its
effects last beyond 24 h even if the more conservative p-value cut off
is used (2-28 h previous to the flux measurement). While this covers the
range of previously found peaks (13 h if no correction for V is made,
7-11 h if V is accounted for (Wing et al., 2022), or 11 h for the
optimal N lag (Borovsky, 2017)), the influence we find is opposite in
sign to previous studies and the single variable analyses. This shows
the importance of considering the joint effects of variables rather than
studying them singly. The initial negative response, again, is likely
the response of electron flux to pressure pulses (driving the field
lines below the altitude of the LANL satellite or to more permanent
magnetopause shadowing). The longer term, positive response to N is a
surprising result of this analysis. It appears to be the result of
considering all lags at once, with only a near-term negative effect (at
a lag of 1 h). Once this lag is essentially removed, the other lags show
a positive effect. Similar to the V positive effect previous to the
negative immediate effect, this may be due to N driving processes that
increase flux. However, this does not appear to be a feature of the
after-storm response to N. After storms, N retains its negative
influence on flux.
c. In the overall analyses (all hours), Bz is mostly influential in the
few hours leading up to the flux observation: positive in the hour
immediately before and negative in the 6 h previous. However, when only
the after-storm period is considered, Bz shows only a negative influence
with 2 important time lags: 20 h before the end of recovery and 4 h
after. (We do not explicitly capture the southward Bz turn at the start
of storms.)
d. Over all hours, with cycles removed, the strongest ULF influences are
in the first (positive) and second (negative) hour before the flux
measurement, with further negative influences in the 12- 24 h before.
This is in marked contrast to the sustained, positive simple
correlations up to 96 h previous. The maximum correlation does not occur
48-50 h before as previously reported (Osmane et al., 2022), nor are
several hours or days of driving necessary to produce a rise in electron
flux (O’Brien et al., 2001). In our storm specific analyses, the ULF
appears to act positively at several key points during and after
recovery, supporting the findings of some storm case studies that the
rise in ULF power precedes the electron flux rise by several hours
(Jaynes et al., 2018; Rostoker et al., 1998), although not by several
days (Baker et al., 1998, Elkington et al., 2003). Increases could be
the result of inward radial diffusion, but ULF waves have also been
speculated to result in electron loss by outward radial diffusion which
may be the weaker negative effect we see over the 12-24 h period
preceding in the overall analysis and which does not appear in the
after-storm analysis. This is much less than the previously postulated
1-6 days (Elkington et al., 2003) or 40 h (Wing et al., 2022) for
outward radial diffusion to occur.
e. AE, which is thought to represent the lower energy electron injection
by substorms, is similarly much reduced in apparent effect, perhaps with
only a single positive influence in the hour just before flux (over all
hours). Following storm periods, the AE at first appears to have an
influence similar to that of ULF, but this effect disappears completely
in the after-storm analysis of combined variables. It is possible that
substorm levels following storms are somewhat constant, in that there is
usually the necessary activity to inject electrons. Either storms do not
vary much in their substorm activity, or all storms reach the necessary
threshold and further activity results in no further injection.
For various reasons, we do not include solar wind pressure, seed
electrons, or Dst in the combined analyses, in part because they overlap
in measuring some of the same influences we do include. Pressure shows a
lower correlation in the overall analyses than the solar wind velocity
and number density that contribute to it, and the correlation is very
low in the hours around the end of storm recovery. Beyond this, the
inclusion of a derived variable (P) with its components (V and N) can
make for difficulties in interpreting the variables that are used to
produce it.
Seed electrons present a particular problem in this analysis for several
reasons. First, both seed and relativistic electrons are measured at the
same satellite in our dataset, making them correlated in both time and
space. We can already see this might be a problem in the simple
correlations that cycle at a roughly 24 h period. The ARMAX correction
does not get rid of this problem. While it is possible that obtaining
seed electron data from another satellite might get around this problem,
the cycling of flux at any geosynchronous location (i.e., the altitude
we are studying) makes this unlikely. Second, it might be presumed that
whichever parameters drive the acceleration of electrons to relativistic
energies are the same parameters that drive the acceleration to seed
electron energies. Thus, increased seed electron flux may be only a
reflection of the same processes resulting in increases in higher energy
electron flux. This in itself would create a high correlation between
the two electron energies, even if one had no influence on the other.
Nor would it be surprising if a rise in seed electron populations
preceded that of the relativistic-energy electrons as it would take less
time for these processes to accelerate electrons to the lower seed
electron energy. While it is true that seed electrons are highly
predictive of MeV flux, this does not prove that they are influential.
Appearing first might give the appearance that seed electrons must be
present before relativistic electrons and therefore must be causal, but
it does not prove it. We have no way of uncoupling this relationship as
we cannot produce the same magnetospheric conditions with different
levels of seed electrons. For these reasons, we do not include seed
electrons in the combined analyses. It is not that we do not think seed
electrons have no influence, only that we recognize we have no way of
testing that hypothesis.
We leave Dst out of the combined analyses because it is a more
generalized index of magnetospheric activity. We have chosen to include
AE (another somewhat general index) instead. The AE, as it is a measure
of substorm activity, has the advantage of including the possible effect
of electron injection. In the after-storm period, in particular, the
inclusion of Dst is problematic as it is also being used to identify the
periods of study. This could result in false correlations that are
merely artifacts of the Dst being used in this manner.
5. Conclusions
A simple correlation of time series data can contain several elements of
information within this single number, not all of which are pertinent to
the question of whether one variable drives the other. Both co-cycling
behavior and associations with variables omitted from the analysis can
contribute to the correlation. These effects must be removed before
relying on correlational analysis to determine if two variables show an
association. In cross correlation (lag) analysis, correlations between
previous hours of the same variable can also create the illusion that a
driver might have a long and cumulative influence only because the
driver is correlated with itself at different hours. To determine the
most accurate timing of influence, time series data should have cycling
behavior removed and driver lags should be considered simultaneously,
not individually, to determine which are most influential. Similarly,
predictors should be studied concurrently to resolve the effect of each
individually.
Applying ARMAX techniques to all hours, we find that the association of
each possible driver with relativistic electron flux is an order of
magnitude lower than simple correlation. Driver associations with flux
are generally strongest within a few hours, not the many hours that have
been suggested previously. Solar wind velocity and number density show
an initial negative impact, with longer term positive influences over no
more than 27 h. Bz is initially a positive influence, with its longer
term negative effect only up to 6 h. ULF waves impact flux in the first
(positive) and second (negative) hour before the flux measurement, with
further negative influences in the 12- 24 h before. Substorms (AE) show
only a short term (1 h) positive influence.
When only after-storm periods are studied (using stepwise regression),
influences occur only at a few lags in the recovery or after recovery
period. In these after-storm hours there are positive influences of ULF
waves and V, negative influences of N and Bz. AE shows no influence in
these more disturbed periods.
Acknowledgements
Work at Augsburg University was supported by NSF grants AGS-1651263 and
AGS-2013648.
Data Availability Statement
Electron flux data were obtained from Los Alamos National Laboratory
(LANL) geosynchronous energetic particle instruments
(https://zenodo.org/record/5834856). The ULF index is available at
http://ulf.gcras.ru/plot_ulf.html. Solar wind (V) data are available
from Goddard Space Flight Center Space Physics Data Facility at the OMNI
Web data website (http://omniweb.gsfc.nasa.gov/html/ow_data.html).
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Figure Captions
Figure 1. Cross correlations (in blue; scale: -0.4 - 0.4 except seed
electrons) and ARMAX lag coefficients (in green; scale: -0.02 - 0.02
except seed electrons) of parameters with electron flux. Cross
correlations are up to 10 times greater than ARMAX coefficients,
reflecting that most simple correlation is due to co-cycling of
parameters with flux. ARMAX coefficients (with co-cycling removed) may
not peak in the same hour nor in the same direction as the cross
correlations. (Note the different scale for seed electron flux
correlations and coefficients.)
Figure 2. Coefficients of the lagged predictors (over 48 h) when all
lags are included in a simultaneous ARMAX multiple regression ncluding
V, N, Bz, ULF, and AE. Nonsignificant coefficients are white bars. Those
with p<0.05 are in light blue. Dark blue are those
coefficients that are still statistically significant when the Holm
correction for multiple comparisons is applied.
Figure 3. Relativistic electron flux (Z scores; black line) superposed
at end of recovery and averaged over available storms. Number of storms
showing end of recovery marker (> -30 nT following main
phase Dst drop) is n=206. 95% confidence interval shown as dashed
lines.
Figure 4. Coefficients from individual analyses of lagged parameters
used to predict relativistic electron flux 44 h after the end of storm
recovery. Top plot of each panel are the cross correlations; bottom plot
of each panel are the significant lags chosen by stepwise regression.
Significant coefficients (p<0.05) are blue bars.
Figure 5. Coefficients from a combined multiple regression analysis of
lagged parameters predicting relativistic electron flux 44 h after the
end of storm recovery. Significant lags chosen by stepwise regression.