ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus GmbHGöttingen, Germany10.5194/angeo-33-405-2015Online NARMAX model for electron fluxes at GEOBoyntonR. J.r.boynton@sheffield.ac.ukBalikhinM. A.BillingsS. A.Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S1 3JD, UKR. J. Boynton (r.boynton@sheffield.ac.uk)27March201533340541114August201413January20156March2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/33/405/2015/angeo-33-405-2015.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/33/405/2015/angeo-33-405-2015.pdf
Multi-input single-output (MISO) nonlinear autoregressive moving average with
exogenous inputs (NARMAX) models have been derived to forecast the >0.8 MeV
and >2 MeV electron fluxes at geostationary Earth orbit (GEO). The NARMAX
algorithm is able to identify mathematical model for a wide class of
nonlinear systems from input–output data. The models employ solar wind
parameters as inputs to provide an estimate of the average electron flux for
the following day, i.e. the 1-day forecast. The identified models
are shown to provide a reliable forecast for both >0.8 and >2 MeV
electron fluxes and are capable of providing real-time warnings of when the
electron fluxes will be dangerously high for satellite systems. These models,
named SNB3GEO >0.8 and >2 MeV electron flux models, have been
implemented online at http://www.ssg.group.shef.ac.uk/USSW/UOSSW.html.
Magnetospheric physics (energetic particlestrapped; solar wind–magnetosphere interactions)Introduction
The configuration of the magnetic field in the region of the terrestrial
radiation belts allows for charged particles to be trapped. As such, the
radiation belts contain energetic electron from tens of keV to several MeV.
The population of the energetic electrons can vary by large amounts in very
short timescales . High fluxes of the energetic electrons
can cause problems for modern technological systems and can be hazards for
humans in space. Satellites in both low Earth orbit and geostationary Earth
orbit (GEO) have an increased probability of suffering onboard satellite
system malfunctioning, which can result in permanent hardware damage
. By powering down certain systems that are at risk, the
effects of the energetic particles can be mitigated. However, this will
require prior knowledge of high energetic electron populations that are
dangerous to satellites. Therefore, models are needed to forecast when large
fluxes of highly energetic electrons will occur.
Although the radiation belts were discovered more than half a century ago
, during the very first in situ space measurements, there still
lacks a comprehensive physical model of the solar wind interaction with the
terrestrial magnetosphere and the dynamics of the radiation belts.
investigated the relationship between the disturbances within
the magnetosphere and radiation belts, which was found to be very complex.
About half of the magnetic storms led to a significant increase in electron
fluxes, a quarter resulted in a decrease while the final quarter of the
disturbances had no effect on the electron fluxes. A similar study, performed
by , showed that while a southward interplanetary magnetic field
(IMF) gave rise to 49 CME magnetospheric disturbances, only 21 of these
resulted in increases of energetic electrons. Although, in stream interaction
regions associated with disturbances, they recorded an 83 % probability of
an electron flux increase.
Despite the complexity of the particle acceleration within the radiation
belts, there are a number of proposed models that explain the dynamics
. Some models assume that a combination of radial
diffusion and low-altitude ULF waves lead to a recirculation effect,
repeatedly accelerating the particles . The
diffusion of trapped energetic electrons from the cusp into radiation belts
has been suggested by . , and all advocate electron
cyclotron heating by whistler waves. An enhanced earthward transport from the
tail to GEO by repetitive substorm injections was put forward by
. proposed the more exotic explanation of the
Jovian shock increasing the electron population in the Earth's radiation
belts. The review by includes all these mechanisms for
acceleration. The main candidates are based on either local diffusion due to
particle interactions with waves or radial diffusion
.
An alternative approach to developing a model based on first principles is
to deduce a forecasting model for the radiation belt electron fluxes directly
from data . employed linear
prediction filters (LPFs) to characterize and to predict the 3–40 MeV electron
measurements using solar wind velocity and geomagnetic indices such as Kp and
AE as inputs. The use of LPFs allowed the authors to analyse how the inputs
influence the output, reporting a 27-day periodicity and a peak lag at 2–3 days.
Since this study, most of the data analysis research into energetic
electrons has been accomplished by neural networks (NNs). The inputs to the
NNs are often the geomagnetic indices such as the Dst index and the daily sum of
the global geomagnetic index, Kp . NNs have
provided results that are significantly more accurate than those from LPFs;
however, the NNs are much more difficult to interpret than the LPFs. They result
in a complex array of neurons, each with an activation function, linked
through a maze of other neurons by a set of weights. This makes the
relationship between the input and output very hard to understand.
Nonlinear autoregressive moving average with exogenous inputs (NARMAX) models
have the advantage of providing accurate results and at the same time are
very easy to interpret. The NARMAX algorithm was initially developed for
complex engineering and biological systems but has since been employed in
many other fields, such as space weather. In solar–terrestrial physics, the
NARMAX methodology was first employed to develop forecasting models for the
Dst index using solar wind inputs . NARMAX has also been
used to model the electron fluxes at GEO . Recently, the NARMAX
approach has been used to identify the inputs of a natural dynamical system
for cases when there is an absence of knowledge. The error reduction ratio
(ERR), which is the basis of the NARMAX model structure selection, was used
by to analyse how the previously proposed coupling functions
influence the Dst index. This study produced a new coupling function that
was then used as an input to model the Dst index in a following paper
. This technique has recently been applied to various energies
of the electron flux ranging from 24.1 keV to 3.5 MeV, obtaining some
unexpected results . found that for 1.8–3.5 MeV
electrons, the solar wind density had the most influence. The following
study by reported an increasing influence in density from
∼1 until 1.8 MeV, above which it became the most important control
parameter for the electron fluxes at GEO. They also identified a quantitative
timescale of the electron flux enhancement as a function of energy that
allowed to argue that local diffusion is not dominant at GEO.
Since there are time delays between the solar wind velocity increases and the
electron flux enhancements at GEO, it is possible to quantitatively estimate
the 1-day-ahead electron fluxes from solar wind parameters. The work by
indicates that this is possible for energies above 270 keV,
where they report that the velocity of the previous day is the parameter that
has the most influence on the fluxes.
The main aim of this study was to develop two NARMAX models for the >0.8 and >2MeV
electron fluxes, measured at GEO by the GOES spacecraft,
which are able to provide an accurate online forecast for 1 day ahead. One
of the best online forecast models was implemented by the National Oceanic
and Atmospheric Administration's Space Weather Prediction Center (NOAA-SWPC);
however, even this forecast is very far from perfect. As such, one of
the goals of this paper was to produce a model that would give a more
accurate estimate than the model by NOAA-SWPC. This was achieved by
validating the model on an interval of data, in other words, to see how the
model forecasts would have performed during this interval.
MethodologyNARMAX model
The NARMAX approach, first developed by , is one of
the most advanced data-based modelling techniques. It is a black box
methodology that can automatically derive a model from solely input–output
data. A multi-input single-output (MISO) NARMAX model was used to represent
the dynamics of the electron fluxes at GEO. The general MISO NARMAX model can
be represented by Eq. () :
y(t)=F[y(t-1),…,y(t-ny),u1(t-1),…,u1(t-nu1),…,um(t-1),…,um(t-num),…,e(t-1),…,e(t-ne)]+e(t),
where y, u and e are the output, input and noise respectively, m is
the number of inputs to the system and ny, nu1, ... ,num are the
maximum time lags of the output and the m inputs respectively. F[⋅]
is some nonlinear function and can be expanded in terms of polynomials,
rational functions, B-Splines, radial basis functions etc. The noise terms of
the NARMAX model allow the capture of noise entering the nonlinear system
internally, resulting in coloured noise, rather than just an additive white
noise
There are three stages to identify a NARMAX model. The first stage, model
structure detection, is to identify the most significant model terms by
evaluating all the possible combinations of the past inputs, past outputs and
past noise values with the use of the ERR. An advantage of the ERR in
selecting input terms is that ERR is independent of the possible nonlinear
and correlated noise . The second stage is parameter estimation,
where the coefficients are calculated for each of the selected model terms in
the structure detection stage. The inclusion of the noise terms can eliminate
bias in estimating the coefficients. The final stage is model validation,
using methods such as the correlation tests or model
performance analysis. The full description of the NARMAX algorithm is very
complex and, as such, it is beyond the scope of this paper. A complete
description of the NARMAX algorithm can be found in the study of
.
The NARMAX algorithm requires both input and output data for the system to
deduce a model. In this study, the output for each of the two models is the
daily averaged >0.8MeV electron flux and the >2MeV electron flux. Both
of these measurements are taken from the Geostationary Operational
Environmental Satellite (GOES) at GEO and are supplied by the National
Oceanic and Atmospheric Administration (NOAA) National Weather Service (NWS)
Space Weather Prediction Center.
As discussed in the Introduction, previous data-based models have used
geomagnetic indices and the solar wind velocity to forecast the electron
fluxes at GEO. The recent study by analysed the solar wind
control parameters for a range of electron flux energies. They found that for
energies <1.8MeV, the solar wind velocity was the most important solar
wind parameter. However, the ERR results show that other solar wind
parameters also play a minor role, such as density, and the z component of
the magnetic field in GSM coordinates. For higher energies (>1.8MeV) the
solar wind density is reported to have the most control of the electron
fluxes but with the velocity still playing an important role. As such, the
solar wind velocity v, density n, z component of the IMF Bz and Dst
index were considered inputs for the models. Along with these parameters, the
fraction of time that the solar wind remains southward within each day,
τBs, was also included. This parameter was calculated from the 1 minBz by finding the time within each 24 h period that the IMF was
southward and dividing it by the total time within the day. These data were
from the Advanced Composition Explorer (ACE) spacecraft positioned at the L1
Lagrange and supplied by the OMNI website for training the model.
Model training
The NARMAX algorithm was then employed to obtain the two models for both the
>0.8MeV electron flux and the >2MeV electron flux. This was achieved
using sections of the data to train the models. These sections had to
contain continuous equally sampled data, i.e. no data gaps within the
interval for both input and output data.
The >0.8MeV electron channel was used for the first time on the GOES 13,
which only became operational on 14 April 2010. Therefore, there were not
many data to train the model on, and the training data for the >0.8MeV model
were chosen to start on 10 April 2010 and end on 31 December 2010.
The >2MeV electron channel has been in use since GOES 6, and data
are available from the late 1980s. Therefore, there were many more choices for the
>2MeV model training interval. This was chosen to start on 11 July 2004
and end on 11 October 2005.
The final models for both energies only included terms of past output, v,
n and τBs, implying that both the Dst index and Bz have a
negligible influence on the electron fluxes. The models were then validated
on separate validation data, which will be discussed in Sect. .
Model forecast showing measured electron flux in blue and the model
estimate in red for (a)>0.8MeV electron flux, (b)>2MeV electron flux
from 1 January 2011 to 31 May 2011.
Model performance analysis
Model performance analysis was used to validate the model and test whether the
model would be accurate enough for real-time online forecasts of the 1-day-ahead electron flux at GEO. This was achieved using past data intervals to
investigate how accurate the 1-day forecasts would have been compared
to the electron flux observed by the GOES spacecraft.
Electron flux data from GOES 13 were used to evaluate the performance of the
model. GOES 13 became the primary GOES satellite for the Energetic Proton
Electron and Alpha Detector (EPEAD) on 14 April 2010. Thus, the period of
data to analyse the >2MeV electron flux model was from 14 April 2010 to 30 June 2012.
The previous GOES satellites measured the channel for >2MeV electrons but
not >0.8MeV electrons; therefore, the >0.8MeV electron channel was used
for the first time on GOES 13. As mentioned above, the >0.8MeV electron
flux model was trained on data up to 31 December 2010, while the
>2MeV
model was trained on data from 2005. Employing the data from 14 April 2010 to
31 December 2010 to evaluate the >0.8MeV model would be bad practice.
Therefore, the period of data to analyse the >0.8MeV model could not be
the same as the >2MeV model and was instead from 1 January 2011 to 30 June 2012.
Therefore, both models had different training and validation data.
The 1-day forecasts were calculated using the >0.8 and
>2MeV
models on the validation intervals mentioned above. Figure
depicts the measured (blue) and the model 1-day forecast (red) for
the period between 1 January 2011 and 31 May 2011. Figure 1a shows the >0.8MeV
model and Fig. 1b the >2MeV model. The >2MeV model has a visibly
better performance than the >0.8MeV model; the >0.8MeV model tends to
overshoot when an increase in flux occurs. An issue with both these models
(and other electron flux models) is that the model has a tendency to lag the
measured flux. An example of this can be seen in Fig. on 2 March 2011
for the >0.8MeV model and 1 April 2011 for the >2MeV model.
In addition to simply inspecting the figures showing the difference between
the forecast and observed electron flux at GEO, the statistics of how the
model 1-day forecast relates to the measurement needs to be
calculated. Here, the performance of the models were statistically analysed
using the prediction efficiency (PE) (Eq. ), and the correlation
coefficient (CC) (Eq. ). These statistics are common in analysing
the performance of models and have been used by
, , and to name a few.
EPE=1-∑t=1Ny(t)-y^(t)2∑t=1Ny(t)-y¯(t)2,ρyy^=∑t=1Ny(t)-y¯(t)y^(t)-y^¯(t)∑t=1Ny(t)-y¯(t)2∑t=1Ny^(t)-y^¯(t)2,
where EPE is the PE, ρ is the CC, y(t) is the output at time t,
y^ is the estimated output from the model and N is the length of the
data. The PE is a quantitative measure of the normalised error, where a high PE
(unity) indicates a low error and a low PE (≤0) shows that the
error of the model is equivalent to (or greater than) the variance of the
measured output and, thus, is very inaccurate. The CC is often employed as an
indicator of a model's performance and shows the linear dependence between the
measured and the forecast, unity showing high dependence and zero none.
The >0.8MeV model resulted in a PE of 0.700 and CC of 0.847 between 1 January 2011
and 30 June 2012, while the >2MeV model was found to have a
PE of 0.786 and CC of 0.894 between 14 April 2010 and 30 June 2012. Therefore,
both models show a high dependence between the measured and estimate, and an
error much smaller than the measured electron flux variance. The lower PE for
the >0.8MeV model can be partly attributed to the overshoot observed in
Fig. , which increases the error of the forecast.
Comparison with NOAA-SWPC electron flux model
The aim of this study was to derive electron flux models that provide a high
accuracy for the 1-day forecast and implement them online in real time.
The CC and the PE show that the models are accurate; however, since a high
accuracy is relative, the >2MeV electron flux model was compared to
another online electron flux model. This model was the NOAA-SWPC model based
on the work by . This was the final hurdle that the model needed
to pass before it was deemed suitable for real-time online implementation.
The SWPC provides a forecast of the >2MeV electron flux at the website
http://www.swpc.noaa.gov/products/relativistic-electron-forecast-model. The forecast is calculated from the LPF model
by , which, as described in Sect. , employs the
solar wind velocity, Kp index and AE index. The model has multiple modes:
three that use ACE data, which estimate the 1-day-ahead, 2-day-ahead and
3-day-ahead electron flux at GOES. Only the 1-day-ahead estimate is
considered here, since in this study the aim was to forecast 1 day in
advance. The NOAA-SWPC provide statistics of the prediction efficiency for
the previous year at
http://services.swpc.noaa.gov/text/relativistic-electron-fluence-statistics.txt.
Between 29 August 2011 and 28 August 2012 the PE of the SWPC model was
67.8 %. The prediction efficiency of the SNB3GEO >2MeV electron flux
model was calculated for the same time period and came to 77.5 %. Therefore,
the NARMAX model had a 9.7 % higher PE than the NOAA-SWPC for the same period
of time.
Conclusions
The main aim of this study was to produce two electron flux models at GEO:
for energies of >0.8 and >2MeV. Both of these models have been shown
statistically to provide an accurate 1-day forecast for the
validation period used in this study, as shown by the high prediction
efficiencies and correlation coefficients between the 1-day forecasts
and the measured fluxes.
The NARMAX >2MeV model was compared to the NOAA-SWPC electron flux model
. The PE for the NARMAX electron flux model was shown to be
∼10 % higher than the SWPC model for the same period, illustrating that
these models have the potential to provide an accurate real-time forecast for
the following day's electron flux.
Thus, the goal to implement the models online to deliver a real-time forecast
for the next day using the real-time data provided by the NOAA NWS Space
Weather Prediction Center has been achieved and the online forecast of
SNB3GEO electron flux models can be found at
http://www.ssg.group.shef.ac.uk/USSW/UOSSW.html.
Acknowledgements
The authors would like to acknowledge the financial support from EPSRC and
ERC. The authors would also like to thank the OMNIWeb service for providing
the past solar wind data and the NOAA NWS Space Weather Prediction Center for
the use of the real-time data from both ACE and GOES. Topical Editor V. Fedun thanks two anonymous referees
for their help in evaluating this paper.
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