11institutetext: Institute for Astronomy, Astrophysics, Space Applications and
Remote Sensing (IAASARS), National Observatory of Athens, I. Metaxa & Vas.
Pavlou St. GR-15236, Penteli, Greece
11email<EMAIL_ADDRESS>22institutetext: Department of Physics and
Astronomy, University of Turku, 20014 Turku, Finland 33institutetext: Aboa
Space Research Oy, Tierankatu 4B, 20520 Turku, Finland 44institutetext:
European Space Research and Technology Centre (ESTEC), Space Environment and
Effects Section, Keplerlaan 1, 2200AG Noordwijk, The Netherlands
55institutetext: Dep. Física Quàntica i Astrofísica (FQA), Universitat de
Barcelona (UB), c. Martí i Franquès, 1, 08028 Barcelona, Spain
66institutetext: Institut de Ciències del Cosmos (ICCUB), Universitat de
Barcelona (UB), c. Martí i Franquès, 1, 08028 Barcelona, Spain
77institutetext: Institut d’Estudis Espacials de Catalunya (IEEC), c. Gran
Capità, 2-4, 08034 Barcelona, Spain 88institutetext: Royal Belgian Institute
for Space Aeronomy (BIRA-IASB), Avenue Circulaire 3, 1180 Uccle, Belgium

# The Probabilistic Solar Particle Event foRecasting (PROSPER) Model

Athanasios Papaioannou 11 Rami Vainio 22 Osku Raukunen 2233 Piers Jiggens 44
Angels Aran 556677 Mark Dierckxsens 88 Sotirios A. Mallios 11 Miikka Paassilta
22 Anastasios Anastasiadis 11

###### Abstract

The Probabilistic Solar Particle Event foRecasting (PROSPER) model predicts
the probability of occurrence and the expected peak flux of Solar Energetic
Particle (SEP) events. Predictions are derived for a set of integral proton
energies (i.e. E$>$10, $>$30 and $>$100 MeV) from characteristics of solar
flares (longitude, magnitude), coronal mass ejections (width, speed) and
combinations of both. Herein the PROSPER model methodology for deriving the
SEP event forecasts is described and the validation of the model, based on
archived data, is presented for a set of case studies. The PROSPER model has
been incorporated into the new operational Advanced Solar Particle Event
Casting System (ASPECS) tool to provide nowcasting (short term forecasting) of
SEP events as part of ESA’s future SEP Advanced Warning System (SAWS). ASPECS
also provides the capability to interrogate PROSPER for historical cases via a
run on demand functionality.

###### keywords:

solar flares – coronal mass ejections – solar energetic particles –
operational tool – validation – radiation storms

##  1 Introduction

Solar energetic particle (SEP) events constitute a significant component of
the near Earth radiation environment and consist of protons, electrons and
heavier ions (Vainio et al. 2009). Such events originate from particle
acceleration in solar flares (SFs) and/or shocks associated with coronal mass
ejections (CMEs) (e.g. Reames 2015; Vlahos et al. 2019). Once energetic
particles are accelerated and injected into open magnetic field lines, they
are consequently routed through the interplanetary magnetic field (IMF) (Belov
et al. 2005; Cane and Lario 2006). Subsequently, when an observer (i.e.
spacecraft) is magnetically connected to the source of the particles,
enhancements by several orders of magnitude above the pre-event background are
observed in-situ, with SEP events often observed in a broad range of solar
longitudes (Rouillard et al. 2012; Lario et al. 2016). SEPs can last from a
few hours to several days and their relative composition varies by many orders
of magnitude from event to event (Reames 2013; Desai and Giacalone 2016). The
classical paradigm divides SEP events into two categories, see e.g. Reames
(1999): those particles that are accelerated at SFs (Aschwanden 2002) are
known as impulsive events, and other particle populations that are accelerated
by near-Sun CME-driven shocks are termed as gradual (Reames 1999, 2002; Kahler
2001; Cane and Lario 2006). However, this “two class” picture does not match
the diversity and wealth of the observed SEP event properties which indicate a
more complex nature (Cane et al. 2010; Papaioannou et al. 2016; Vlahos et al.
2019).

SEP events have a direct space weather impact on electronics and humans (Baker
2004). For example, the survivability of a spacecraft is directly affected by
the total energy deposited by the passage of energetic particles. This may
result in the degradation and ultimate failure of its electronic components
due to ionisation or displacement damage mechanisms (Daly et al. 1996; Feynman
and Gabriel 2000). Effects on electronics also include single event effects
(SEEs) (Pellish et al. 2010), these appear when particles deposit sufficient
energy or charge in a sensitive region of a component, with non-permanent
(soft) errors such as bit flips, as well as, permanent (hard) errors such as
latchups (i.e. SELs) and burnouts (SEBs) taking place (Sexton 2003; Harboe-
Sørensen 2013). Furthermore, SEP events are a major threat to human
spaceflight outside the protective shield of the Earth’s magnetosphere
(Bizzarri et al. 2017; Townsend 2021) and pose a severe danger for aircrews
and passengers on polar flights (Tobiska et al. 2015; Mishev et al. 2015;
Miroshnichenko 2018). The ionizing particle radiation can lead to damage to
human cells and DNA alternations (Azzam et al. 2012). There are two basic
parameters that have a decisive role in the radiation effects in humans: the
strength of the exposure and the specific organ/tissue that was encountered.
For example, a short-term exposure to high doses of radiation depositing
energy in the eye can lead to acute radiation effects such as cataracts. Long-
term exposure to low doses of radiation deposing energy in bone marrow may
progressively lead to leukaemia or other types of cancer. The former effects
are categorized as deterministic while the latter as stochastic (Cucinotta et
al. 2003). The high-energy tail of the SEP spectrum, in which ions are
accelerated to relativistic energies, is dominant in spacecraft orbits well
within the magnetosphere. In addition, high-energy particles upon interaction
with matter, from e.g. the shielding material of a spacecraft, produce
secondaries that can enhance the radiation effect of SEP events and are of
major concern in heavily shielded environments such as human spaceflight
(Raukunen et al. 2018). Nonetheless, these high-energy particles can also
reach the Earth’s atmosphere and generate secondaries through nuclear
reactions. Consequently, these secondaries are recorded as a significant
sudden increase at ground level, as detected by e.g. neutron monitors (NMs).
Such events are termed as Ground-level enhancements (GLEs) (Bütikofer et al.
2009; Asvestari et al. 2017; Mishev et al. 2018). This results in enhanced
ionisation together with modifications of the local chemistry of the high
polar atmosphere of the Earth (Usoskin et al. 2011; Mironova and Usoskin
2014).

SEPs, along with trapped radiation in planetary environments, are responsible
for cumulative (dose) effects on spacecraft electronics and materials under
low and moderate amounts of shielding. These are modelled statistically to
produce specification (climatological) models to predict degradation over the
duration of a mission. Such models make use of fits to SEP flux data in the
form of lognormal, truncated power law, and exponential cut-off power law
distributions to derive fluxes as a function of mission duration and
confidence (Jiggens et al. (2018); Raukunen et al. (2018) and references
therein). However, these models cannot make short-term forecasts of the SEP
environment to provide warnings for operational missions, human spaceflight,
launch operators and aircraft operators.

A small fraction of all solar flares and CMEs lead to SEP events and as a
result the prognosis of SEPs is not a trivial task, since this is a highly
imbalanced problem (see the relevant discussion in Lavasa et al. 2021). The
scientific questions that one needs to address in such studies (see e.g.
Anastasiadis et al. 2019) are summarized as: If we know the characteristics of
the parent solar events, could the probability of SEP occurrence be reliably
inferred and how do the characteristics of SEP events (e.g. peak flux) map to
the characteristics of their parent solar events ? Both questions are being
largely addressed through the implementation of databases and the
establishment of empirical and/or semi-empirical statistical relations (see
e.g. Gopalswamy et al. 2003, 2004; Garcia 2004a, b; Belov et al. 2005;
Laurenza et al. 2009, 2018; Núñez 2011; Trottet et al. 2015; Dierckxsens et
al. 2015; Anastasiadis et al. 2017; Papaioannou et al. 2018b; Kahler and Ling
2018; Richardson et al. 2018). The underlying idea is to identify a proper
proxy (or combinations of proxies) that can be used for the unfolding of
patterns and relationships among the parameters of SEP events and their parent
solar events, using observational evidence at hand.In turn, such empirical
relations point to the underlying physical processes of the SEP generation
(Balch 2008). For example, Laurenza et al. (2009) provides prognosis of the
SEP occurrence based on solar flare location, size (i.e. soft X-ray (SXR)
fluence) and evidence of particle escape (i.e. radio fluence at $\sim$1 MHz);
Papaioannou et al. (2018b) provides short-term forecasts of the SEP occurrence
and the corresponding peak flux utilizing CME characteristics (e.g. width and
speed); Kahler and Ling (2018) has used the SXR peak flux ratio, following
Garcia (2004a), to establish the probability of SEP occurrence and Richardson
et al. (2018) used the CME speed and the direction relative to the observer’s
site (i.e. magnetic connection) to predict the peak intensity of protons. In
addition, the work from Posner (2007), has proven the concept of short-term
forecasting of the appearance and intensity of solar ion events by means of
relativistic electrons, making use of the higher speed of these electrons
propagating from the Sun to 1 AU. Hence, it appears that such empirical or
semi-empirical relations, can be used for the forecasting of solar radiation
storms. Given the complexity and the incomplete knowledge of the underlying
physical mechanisms at work, recent studies attempt to make use of higher
order statistical relations (see e.g. Papaioannou et al. 2018a) and machine
learning approaches (see Lavasa et al. 2021) in order to infer the probability
of SEP occurrence. Such studies make use of the complete parameter space at
hand, utilizing both solar flare and CME characteristics and have shown
promising results. In this work, the Probabilistic Solar Particle Event
Forecasting (PROSPER) Model, is presented. This model is integrated into the
Advanced Solar Particle Event Casting System (ASPECS) operational tool that
provides predictions of the probability of occurrence of SEP events, the
expected peak flux and the resulting SEP time profile (http://phobos-
srv.space.noa.gr/). PROSPER applies a novel, data-driven methodology to
predict SEP events probabilistically for a set of integral energies, namely
E$>$10, $>$30 and $>$100 MeV111PROSPER was further extended to E$>$60 MeV but
the results are not shown through the ASPECS operational system. As it is
going to be presented in detail, PROSPER takes advantage of the Bayes theorem
(Bayes and Price 1763), taking into account all observational evidence at
hand, without any bias. Bayesian approaches have been increasingly applied to
the field of solar physics (Arregui 2022) and to solar flare forecasting, in
particular (see e.g. Wheatland 2004, 2005, and reference there in). The model
comes with three modes of operation depending on the available inputs : (a)
CME characteristics (width, speed); (b) SF characteristics (longitude,
magnitude) and (c) combinations of both CME and SF characteristics. The
methodology is outlined and discussed in Section 2. The application of the
methodology to all modes of operation is detailed in Section 3. Validation
results for a set of case studies are presented in Section 4. Finally, the
results are summarised and discussed in Section 5.

##  2 Analysis

###  2.1 Data

For the development of the PROSPER model, a catalogue of SEP events that
includes 314 SEP events from 1984–2013 was utilized (e.g. Papaioannou et al.
2016). This SEP event catalogue is based on Geostationary Operational
Enviromental Satellite (GOES)/Energetic Particle Sensor (EPS) data (for
details on the data set see Sandberg et al. (2014)) and it includes key
information on the proton peak flux and the total fluence of the identified
SEP events in the three integral energy channels (namely, E $>$ 10; $>$30;
$>$100 MeV)222PROSPER will further be extended to E$>$300 MeV as part of the
ASPECS tool. It further includes the associated solar sources of the SEP
events in terms of solar flares and CME characteristics. In particular, it
includes solar SXR flux measurements, provided by GOES, recorded in the same
period including 35306 C, M and X class flare events
(ftp://ftp.ngdc.noaa.gov/STP/space-weather/solar-data/solar-features/solar-
flares/x-rays/goes/). From the initial sample of SXR flares 14818 events for
which the location was not available were excluded, leading to a sample of
20429 C, M and X class flares. At the same time, the CME identifications (i.e.
plane-of-sky speeds and angular widths - AW) are made by the Large Angle and
Spectroscopic Coronagraph (LASCO) (Brueckner et al. 1995) onboard the Solar
and Heliospheric Observatory (SoHO) in the period from 1997–2013. The initial
CME sample consists of 22143 events, which was reduced to 3693 events when
association criteria between CMEs and solar flares were applied (see details
in Papaioannou et al. (2016))333by construction, CMEs associated to far side
solar flares are not included in our sample. The CME identifications, utilized
in this study, are included in the Coordinated Data Analysis Web (CDAW) online
CME Catalog (https://cdaw.gsfc.nasa.gov/CME˙list/; Gopalswamy et al. 2009).

###  2.2 Mathematical formulation of PROSPER

In this section the general methodological approach for the implementation of
PROSPER is presented. PROSPER provides both the Probability of SEP Occurrence
$P(\rm SEP)$ (subsection 2.2.1) and the expected Peak Proton Flux (subsection
2.2.2). The first part, starts with the construction of Cumulative
Distribution Functions (CDFs) by the data, continues with the implementation
of Probability Density Functions (PDFs) and concludes with the application of
the Bayes formula, from which the $P(\rm SEP)$ is directly provided. The
second part, presents the establishment of the peak proton flux with the
construction of the relevant CDFs.

####  2.2.1 Probability of SEP occurrence

This section presents the procedure for the establishment of $P(\rm SEP)$
within PROSPER mode of operation (a), i.e. utilising CME characteristics
alone. PROSPER is applied to E$>$10, $>$30, ($>$60) and $>$100 MeV and for
three modes of operation i.e., (a), (b) and (c) in Table 1 which provides an
overview of the modes of operation for PROSPER, the inputs used, as well as,
the bins applied based on the characteristics of their parent solar events. In
particular, CMEs are grouped with respect to their angular width (see details
in Papaioannou et al. 2018b), while solar flares by their longitude. Solar
events originating in the west of the Sun, as observed by an observer on
Earth, are more likely to produce large SEPs detectable at Earth (see e.g.
Swalwell et al. 2017), especially those that lie within W20o-W80o (see Cliver
et al. 2012, and references therein). From our sample roughly half of the SEP
events ($\sim$46%) are associated with a solar flare originating west of W20o,
hence our choice of positional requirement in the algorithms. Such events are
termed as: “well connected” whereas those east of W20o as “poorly connected”.

Table 1: Details on PROSPER’s modes of operation, inputs and initial binning Mode of operation | Inputs | Bins | Resulting function
---|---|---|---
|  |  | per bin
(a) CME | speed | continuous speed |
|  | AW = 360o (Halo - H) |  $f(V_{CME})$ [3]
| & AW | 120o $\leq$ AW $<$ 360o (Partial Halo - PH) |
|  | AW $<$ 120o (Non-Halo - NH) |
(b) flare | Soft X-ray | continuous magnitude |
| flare magnitude | lon. $\geq$20o (well connected - WC) |  $f(F_{SXR})$ [2]
| & longitude | lon. $<$20o (poorly connected - PC) |
(c) flare & CME | all of the above | [H or PH or NH] & WC |  $f(V_{CME},F_{SXR})$ [6]
|  | [H or PH or NH] & PC |
Acronyms & bins |  |  |
AW = angular width | H = Halo | PH= Partial Halo | NH = Non-Halo
lon.= longitude | WC = well connected | PC = poorly connected |
with respect to the | (lon. $\geq$20o) | (lon. $<$20o) |
Sun-Earth line |  |  |

Column 1 denotes the mode of operation, column 2 declares the inputs used per
mode, column 3 details the initial bins applied to the data based on the
inputs and column 4 presents the resulting number of functions per bin. The
numbers in the brackets of column 4 denote the number of resulting functions
per mode and integral energy. These functions are the P($\rm SEP$) ones which
are going to be detailed in the following steps for each of the selections.
Hence for each integral energy (e.g. E$>$10 MeV) there are 3 $f(V_{CME})$, 2
$f(F_{SXR})$ and 6 $f(V_{CME},F_{SXR})$ functions in accordance to column 3.
PROSPER is applied independently to each of the integral energies considered.

##### Implementation of Cumulative Distribution Functions

First, we bin the data in accordance to Table 1. The empirical cumulative
distribution function (CDF) for each sub-sample was constructed by sorting the
events in the ascending order of the continuous parameter (CME speed and/or
solar flare magnitude; see also Table 1), and using the standard fractional
ordinates as probabilities. As parent solar events (i.e. parameterized by
flares and/or CMEs) can be either associated with SEP events or not, it is
possible to construct two CDFs: one for each sample, i.e. one for all parent
solar events (i.e. parameter, $\Lambda$) of the specific binning:
$P(\Lambda<\lambda)$444$\lambda$ refers to the continuous parameter of Table 1
per case and one for all parent solar events that were associated with an SEP
event in that bin, $P(\Lambda<\lambda\,|\,\text{SEP})$.

Next, the resulting (filtered) data were fit with a log-normal distribution
(see e.g. Raukunen et al. 2018), which was found to be the most robust of the
flux distributions applied to SEP fluxes for specification models, in this
work. This is given by the following equation:

$F(\lambda)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\log_{10}(\lambda)-\mu}{\sigma\sqrt{2}}\right)\right],$
(1)

where $F(\lambda)$ is the probability of $\Lambda$ being lower than $\lambda$,
$\mathrm{erf}(x)$ is the error function, and $\mu$ and $\sigma$ are the mean
and standard deviation of $\log_{10}(\lambda)$, respectively.

##### Implementation of the Probability Density Functions

These CDFs were consequently used in order to construct probabilities for
which the CME speed/flare magnitude, $\lambda$ lies within a certain range
defined by an upper and a lower limit, i.e. $\in[\lambda_{1},\lambda_{2}]$.
The resulting independent probability of observing $\alpha$, i.e. the
probability of $\Lambda$ to fall within a certain range of values is given by
the subtraction of two CDFs constructed by all data points in a sample one for
the lower limit, i.e. $\Lambda<\lambda_{1}$ and one for the upper one, i.e.
$\Lambda<\lambda_{2}$ leading to $P\left(\Lambda<\lambda_{2}\right)$ and
$P\left(\Lambda<\lambda_{1}\right)$. The subtraction of which results in:

$P(\alpha)=P\left(\Lambda\in[\lambda_{1},\lambda_{2}]\right)=P\left(\Lambda<\lambda_{2}\right)-P\left(\Lambda<\lambda_{1}\right).$
(2)

The independent probability of observing $\beta$, i.e. the probability of SEP
events in a sample, is based on the number of SEP associated events
($N_{SEP}$) and the total number of events ($N_{total}$) in the sample and is
defined as:

$P(\beta)=P(SEP)=N_{SEP}/N_{total}.$ (3)

The conditional probability of observing $\alpha$ under the condition of
observing $\beta$, $P(\alpha\,|\,\beta)$, i.e. the probability of the
$\Lambda$ to fall within a certain range of $\lambda$, under the condition
that these values refer to SEP events. This is also given as the subtraction
of two CDFs. In particular, the first CDF refers to the lower limit, i.e.
$\Lambda<\lambda_{1}\,|\,\text{SEP}$ and the second to the upper one, i.e.
$\Lambda<\lambda_{2}\,|\,\text{SEP}$ leading to
$P\left(\Lambda<\lambda_{1}\,|\,\text{SEP}\right)$ and
$P\left(\Lambda<\lambda_{2}\,|\,\text{SEP}\right)$. As a result the
conditional probability is defined as:

$P(\alpha\,|\,\beta)=P\left(\Lambda\in[\lambda_{1},\lambda_{2}]\,|\,\text{SEP}\right)=P\left(\Lambda<\lambda_{2}\,|\,\text{SEP}\right)-P\left(\Lambda<\lambda_{1}\,|\,\text{SEP}\right).$
(4)

Using the Bayes theorem (Bayes and Price 1763; Joyce 1999) one may define the
inverse conditional probability:

$P(\beta\,|\,\alpha)=P\left(\text{SEP}\,|\,\Lambda,\in[\lambda_{1},\lambda_{2}]\right)$
(5)

which is a PDF and provides the probability of SEP occurrence when $\Lambda$
$\in[\lambda_{1},\lambda_{2}]$. The Bayes formula dictates that:

$P(\beta\,|\,\alpha)=\frac{P(\alpha\,|\,\beta)P(\beta)}{P(\alpha)}$ (6)

Merging Equations 2–6, the following formula is obtained:

$P\left(\text{SEP}\,|\,\Lambda\in[\lambda_{1},\lambda_{2}]\right)=\frac{\left[P\left(\Lambda<\lambda_{2}\,|\,\text{SEP}\right)-P\left(\Lambda<\lambda_{1}\,|\,\text{SEP}\right)\right]P(\text{SEP)}}{P\left(\Lambda<\lambda_{2}\right)-P\left(\Lambda<\lambda_{1}\right)}.$
(7)

Hence, Equation 7 shows the probability to have an SEP event under the
condition that $\Lambda\in[\lambda_{1},\lambda_{2}]$. For example, in the case
of CMEs, substituting $\Lambda$ with $V_{CME}$, $\lambda_{1}$ with $V_{1}$ and
$\lambda_{2}$ with $V_{2}$, Equation 7 becomes:

$P\left(\text{SEP}\,|\,V_{\mathrm{CME}}\in\left[V_{1},V_{2}\right]\right)=\frac{\left[P\left(V_{{\rm
CME}}<V_{2}\,|\,\text{SEP}\right)-P\left(V_{{\rm
CME}}<V_{1}\,|\,\text{SEP}\right)\right]P(\text{SEP})}{P\left(V_{{\rm
CME}}<V_{2}\right)-P\left(V_{{\rm CME}}<V_{1}\right)}.$ (8)

Moreover, setting $V_{1}=V$ and $V_{2}=V+{\rm dV}$ and dividing both nominator
and denominator by $dV$ while letting ${\rm d}V\to 0$ results to:

$P\left(\text{SEP}\,|\,V_{{\rm CME}}=V\right)=\frac{P^{\prime}\left(V_{{\rm
CME}}<V\,|\,{\rm SEP}\right)}{P^{\prime}\left(V_{{\rm CME}}<V\right)}P({\rm
SEP})\\\ =\frac{f_{{\rm SEP}}(V)}{f(V)}P({\rm SEP}).$ (9)

Both Equations 8 and 9 provide the probability of occurrence of an SEP event
given a CME of speed, $V$.

In Equation 9, $f(V)=P^{\prime}(V_{{\rm CME}}<V)$ is the probability density
function (PDF) of the CME speeds for the binned sub-sample of choice (see
Table 1) and $f_{{\rm SEP}}(V)$ is the PDF for those CMEs of the bin that are
associated with SEP events. In order to construct the PDFs (i.e. $f(V)$ &
$f_{{\rm SEP}}(V)$) from the relevant CDFs, there are two ways: (a) apply
numerical differentiation (labeled as Method 1) and (b) fit the data points
with an empirical CDF and directly take the derivatives of the relevant
distributions, as a function of the CME speed ($V_{CME},[km/s]$) (labeled as
Method 2). As it can be seen in Figure 1, for the case of Halo CMEs and E$>$10
MeV SEP events, the output is equivalent. Therefore, from the empirical CDFs,
which are based on data, and the applied log-normal fits, one can directly
obtain the relevant PDFs and from the application of the Bayes formula
(Equation 9) obtain an analytical expression that answers to the question:
What is the probability that an SEP event [P(SEP)] will occur if one knows the
CME speed ($V_{CME}$) (and width) of the driving CME ?

Equation 9 also stands for the case of solar flares. In this case, the
application of the Bayes formula based on solar flare longitude and flux (F)
results in:

$P\left(\text{SEP}\,|\,F_{{\rm SXR}}=F\right)=\frac{P^{\prime}\left(F_{{\rm
SXR}}<F\,|\,{\rm SEP}\right)}{P^{\prime}\left(F_{{\rm SXR}}<F\right)}P({\rm
SEP})\\\ =\frac{f_{{\rm SEP}}(F)}{f(F)}P({\rm SEP}).$ (10)

At this point, it is noted that Equation 6 acquires very large values and
tends to infinity as $P(\alpha)\rightarrow 0$. For this reason, using some
elementary mathematical properties of sets and probabilities, Equation 6 can
be written in a form which is more stable and appropriate for numerical
calculations, as:

$P(\beta\,|\,\alpha)=\frac{P(\alpha\,|\,\beta)P(\beta)}{P(\alpha|\beta)P(\beta)+P(\alpha\,|\,\beta^{c})P(\beta^{c})},$
(11)

with $\beta^{c}$ being the complementary event of $\beta$ (Maritz and Lwin
2018). Although the denominators of Eq. 6 and Eq. 11 are equivalent, the
former $\rightarrow 0$ faster than the latter one. As a result Equation 11 has
an optimal numerical behaviour and allows one to obtain outputs at a wider
range compared to Equation 6 before the denominator of Equation 6 $\rightarrow
0$. Therefore, the formulas used in PROSPER calculations are the more complex
expressions of Equations 7–10 based on substitution of Equations 2–5 in
Equation 11.

Additionally, in the case of a solar flare and an associated CME the Bayes
formula used in PROSPER gets the general form:

$P(\beta_{i,j}\,|\,V,F)=\frac{f_{\beta_{i}}(V)f_{\beta_{j}}(F)P(\beta_{i,j})}{[f_{\beta_{i}}(V)f_{\beta_{j}}(F)P(\beta_{i,j})+f_{\beta_{i}^{c}}(V)f_{\beta_{j}^{c}}(F)P(\beta_{i,j}^{c})]}.$
(12)

where $i$ refers to CME $AW$, i.e. Halo, Partial Halo, Non Halo and $j$ to the
solar flare longitude (“well” and/or “poorly” connected); see details per
selection in Table 1). $P(\beta_{i,j}$ and $P(\beta_{i,j}^{c})$ are calculated
directly by the measurements.

####  2.2.2 Peak flux estimation

PROSPER provides a peak flux estimation for a given (user-defined) confidence
based on the $P(\text{SEP})$ described here above. First the data are
filtered, based on the $P(\rm SEP)$ that was derived in subsection 2.2.1. In
particular, once $P(\rm SEP)$ is obtained, peak flux data are filtered based
on the integral energy that the SEP event is expected to reach. That said, if
i.e. an SEP event is expected to reach E$>$100 MeV only the events from the
list with significant peak fluxes reaching E$>$100 MeV are employed across all
integral energies. The same stands for filtering the data samples based on
other integral energies based on the obtained $P(\rm SEP)$ (i.e. if an event
is expected to reach E$>$30 MeV). At any case, the presence of a high energy
population is an important filter for lower energy fluxes since it ensures
spectral coherence. Next, additional binning is applied using the available
characteristics of the parent solar events (i.e. SF magnitude & location, CME
speed & width). Once the data samples are retrieved we utilize the tabulated
peak proton flux (PPF) for the SEP events in the list employed in Papaioannou
et al. (2016) in order to derive the relevant distributions.

The filtered peak fluxes were fit with the exponential cut-off power law that
was found to be the most robust of the flux distributions applied for
specification models (consistent with findings from both Jiggens et al. (2018)
and Raukunen et al. (2018)). This is given by the following equation:

$P\left(\rm F_{p}\geq
PPF_{0}\,|\,SEP\right)=1-\frac{PPF_{0}^{-\gamma}\exp\left(\frac{x_{low}}{x_{lim}}\right)}{x_{low}^{-\gamma}\exp\left(\frac{PPF_{0}}{x_{lim}}\right)},$
(13)

with $\gamma$ being the power-law exponent, $x_{low}$ a parameter related to
the lower limit of the $F_{P}$ distribution, and $x_{lim}$ a parameter related
to the upper limit of the $F_{P}$ distribution.

Equation 13 answers the question: What is the expected probability that the
peak flux will exceed a certain threshold, ($PPF_{0}$) for a given integral
energy, provided that it is certain an SEP event will occur? From Equation 13,
one can find the expected peak flux that corresponds to a specific probability
threshold, solving Equation 13 for $PPF_{0}$ for a given value of $P\left(\rm
F_{P}\geq PPF_{0}\,|\,SEP\right)=P_{thres}$. Following this, the $P(\rm
SEP\,|\,\Lambda=\lambda_{0})$ is used to modulate the prediction of the
expected peak flux. In essence, a weighted average is employed, driven by the
obtained $P(\rm SEP\,|\,\Lambda=\lambda_{0})$. There are two components in
this averaging: the first one is the $PPF_{0}$ for a specific user defined
threshold multiplied by $P(\rm SEP\,|\,\Lambda=\lambda_{0})$ and the second
one is the background value of the specific integral energy multiplied by the
1 - $P(\rm SEP\,|\,\Lambda=\lambda_{0})$. The averaging leads to $PPF_{thres}$
and the obtained $P(\rm SEP\,|\,\Lambda=\lambda_{0})$ is used as a weight. To
this end PROSPER provides $F_{P\rm thres}$ obtained from Equation 14:

$F_{P\rm thres}=PPF_{0}\cdot P(\rm
SEP\,|\,\Lambda=\lambda_{0})+\textit{background flux}\cdot\left[1-P(\rm
SEP\,|\,\Lambda=\lambda_{0})\right]$ (14)

In the limiting cases of the binary extremes of $P(\rm
SEP|\Lambda=\lambda_{0})$, the following expression is obtained:

$F_{P\rm thres}=\begin{cases}PPF_{0}&\mbox{, }P(\rm
SEP\,|\,\Lambda=\lambda_{0})=1\\\ \textit{background flux}&\mbox{, }P(\rm
SEP\,|\,\Lambda=\lambda_{0})=0\\\ \end{cases}$ (15)

The background flux was calculated on the basis of the recorded intensity of
each integral energy of interest (i.e. E$>$10-; $>$30-; and $>$100 MeV)555data
taken from https://satdat.ngdc.noaa.gov/sem/goes/data/avg/ and corresponds to
the mean value obtained in September 2009 (e.g. during the solar minimum),
which is: 0.23, 0.122, 0.089 and 0.050 pfu, respectively (see Figure 2).
Evidently, depending on the CL, $thres$, of choice (by default in the ASPECS
tool, PROSPER provides $thres$=50%, and 90%, which were defined by the user
community), and thus the model provides $PPF_{\rm 50}$ and $PPF_{\rm 90}$.

##  3 Results

###  3.1 Coronal Mass Ejections

In the case of CMEs, PROSPER’s concept was applied across three CME AW bins
(see Table 1), for all integral energies of our database (E $>$ 10; $>$30;
($>$60) and $>$100 MeV) and for both the distribution of all CMEs speeds
($V_{CME}$) in each sub-sample, as well as, the distribution of the speeds of
CMEs related to SEPs. Figure 3 depicts the obtained empirical CDFs from the
data (i.e. $P(V_{{\rm CME}}<V)$). From top to bottom, on the left hand side,
the column refers to all Halo, all Partial Halo and all Non Halo CMEs of our
sample. Additionally, similar plots for $P(V_{{\rm CME}}<V\,|\,{\rm SEP})$ for
an integral energy of E$>$10 MeV is provided on the right hand side of the
plot, for each case, following the same labeling. Each sub-plot of Figure 3
presents the data points (in black color) for each distribution. Additionally,
the log-normal fit (from Equation 1) to these points is presented as a red-
line, with the fraction of the data (in %) represented by the fit. Finally,
the mean absolute error (MAE)666MAE=$\sum_{i=1}^{n}\,|\,y_{i}-x_{i}\,|\,/n$,
with $n$ being the number of pairs, $y$ and $x$ the fitted and the actual
value, respectively was calculated at each case and is imprinted on every
plot. Although the related results are not shown in Figure 3, it is noteworthy
that when shifting to higher energies, i.e. from E$>$10 MeV to E$>$100 MeV, a
reduction in the number of SEP events that reach higher energies is apparent.
In turn, fits for these energies rely on fewer points.

A combined representation of these CDFs, but for all integral energies of
interest (e.g. E$>$10; $>$30; ($>$60) and $>$100 MeV) is given in Figure
4777Although E$>$60 MeV is not implemented in the ASPECS tool, we present the
obtained fits for $P(\rm SEP)$ in this and consequent Figures for consistency.
Each panel provides the fitted CDFs for the $P(V_{{\rm CME}}<V)$ (black line)
and $P(V_{{\rm CME}}<V\,|\,{\rm SEP})$ distributions - the latter for each
integral energy, color coded as: E$>$10 MeV - red line; E$>$30 MeV - blue
line; E$>$60 MeV - green line and E$>$100 MeV - orange line. From top to
bottom, panels refer to Halo, Partial Halo and Non Halo CMEs, respectively.
The PDFs from the derivatives of the distributions $f(V)$ and $f_{{\rm
SEP}}(V)$ are consequently presented in Figure 5, following the same reasoning
(and labeling) as in Figure 4. It should be noted that the X-axis (i.e.
$V_{CME}$) in all panels extend to a simulated value of 10000 km/s. However,
the actual data point to an upper limit $V_{CME}$ of $\sim$3000 km/s (see
Figure 3). That said, the derived Bayes $P(\rm SEP)$ is obtained up to that
$V_{CME}$. However, we present the whole evolution of the fit and we overlay a
gray border hatched rectangle area after that limit.

The final step in this formulation is to employ the Bayes formula (Equation 9)
and retrieve the fits that reply to the question: what is the probability of
SEP occurrence from a CME with known width (AW) and speed ($V_{CME}$)? These
fits are presented in Figure 6. Each panel corresponds to one integral energy
of interest, and each line within every panel refers to Halo (blue line);
Partial Halo (red line) and Non Halo (green line) CMEs. Hence, for each
integral energy a total of 3 $f(V_{CME})$ functions are obtained (see Table
1). The abscissa of Figure 6 gives the $V_{CME}$ and the ordinate provides the
probability of SEP occurrence. Hence, for a CME with known AW, one may select
the proper fit and from the known $V_{CME}$ directly obtain the expected
probability, $P(\rm SEP)$. It can be noted that up to the limit of $V_{CME}$
$\sim$3000 km/s for any given CME speed, the largest $P(\rm SEP)$ is obtained
for Halo (blue line), followed by Partial Halo (red line) and Non Halo (green
line). Moreover, both the red and the green curve would have a similar
behaviour as the blue curve, if the X-axis would have been expanded to
$>$10000 km/s. In addition, when crossing the $V_{CME}$ $\sim$ 3000 km/s limit
it can be seen that Partial Halo (red line) and Non Halo (green line) CMEs
lead to a higher $P(\rm SEP)$ compared to Halo CMEs (blue line). This is the
output of the application of Equation 9 and it points to the possibility to
get a higher $P(\rm SEP)$ for Partial and Non Halo CMEs provided that these
CMEs are faster than any CME ever observed. Nonetheless, the application of an
upper limit for $V_{CME}$ that is inferred by the actual sample used, results
into higher probabilities for Halo CMEs up to that limit. That said, all
observed CMEs with $V_{CME}>$ 3000 km/s are treated as having a $V_{CME}$ =
3000 km/s.

As noted in subsection 2.2.2, based on user consultations, a focused
probability range from 50-90% has been selected, while taking into account, as
a filter, the probability to derive an SEP event at a respective energy. If an
SEP event is expected to reach e.g. E$>$100 MeV the data for all energies is
binned based on this energy filtering and consequently the probabilities are
derived based on CME speed and width. The obtained results for the case of CME
inputs are depicted in Figure 7. There are three different filters based on
the predicted integral energy that the particles will reach and different bins
based on the CME characteristics. In particular, panels (a) and (b) of Fig. 7
refer to SEP events that are expected to reach an integral energy of E$>$10
MeV. The red (blue) lines depict the fast (slow) bins and each of the panels
refer to Halo (panel (a)) and Not Halo (panel (b)) CMEs. In addition, panels
(c)-(f) of Fig. 7 refer to SEP events that are expected to reach an integral
energy of E$>$30 MeV. The blue (orange) lines in each of the panels present
the E$>$10 (E$>$30) MeV fits, whereas each of the panels stands for a
different selection of AW and CME speed – panel (c): Halo and slow; panel (d)
Halo and fast; panel (e) not Halo and slow and panel (f) not Halo and fast
CMEs. Finally, panel (g) of Fig. 7 represents the fits for a filtering at
E$>$100 MeV and each line corresponds to an integral energy of interest color
coded as: red (E$>$10 MeV); green (E$>$30 MeV) and blue (E$>$100 MeV). All
bins and selections are detailed in Table 2.

Table 2: Details on PROSPER’s peak flux fits for the CME bins, derived by
Equation 13. The filtering on the integral energy is first applied and then
there are two bins. One for the width of the CME (i.e. AW = 360oHalo or AW
$<$360oNot Halo) and one for the CME speed (i.e. $V_{CME}$ being $<$ or $\geq$
1250 km/s). These fits are presented in Figure 7. if an event reaches E$>$10
MeV
---
AW = 360o (Halo - H) |  | AW $<$ 360o (Not Halo) |
$V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $\geq$ 1250 km/s |  $V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $\geq$ 1250 km/s
$\gamma$ = 0.36 |  $\gamma$ = 0.26 |  $\gamma$ = 0.53 |  $\gamma$ = 0.32
$x_{low}$ = 3.68E+00 |  $x_{low}$ = 6.38E+00 |  $x_{low}$ = 2.50E+00 |  $x_{low}$ = 4.00E+00
$x_{lim}$ = 5.37E+01 |  $x_{lim}$ = 1.18E+04 |  $x_{lim}$ = 1.49E+03 |  $x_{lim}$ = 8.95E+03
if an event reaches E$>$30 MeV
AW = 360o (Halo - H) |  | AW = 360o (Halo - H) |
$V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $\geq$ 1250 km/s |  $V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $\geq$ 1250 km/s
E$>$10 MeV |  | E$>$30 MeV |
$\gamma$ = 0.30 |  $\gamma$ = 0.26 |  $\gamma$ = 1.09 |  $\gamma$ = 0.30
$x_{low}$ = 3.68E+00 |  $x_{low}$ = 6.38E+00 |  $x_{low}$ = 1.23E+00 |  $x_{low}$ = 1.24E+00
$x_{lim}$ = 5.27E+01 |  $x_{lim}$ = 1.19E+04 |  $x_{lim}$ = 1.54E+02 |  $x_{lim}$ = 2.27E+03
AW $<$ 360o (Not Halo) |  | AW $<$ 360o (Not Halo) |
$V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $\geq$ 1250 km/s |  $V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $\geq$ 1250 km/s
$\gamma$ = 0.45 |  $\gamma$ = 0.30 |  $\gamma$ = 0.58 |  $\gamma$ = 0.32
$x_{low}$ = 3.94E+00 |  $x_{low}$ = 5.72E+00 |  $x_{low}$ = 9.06E-01 |  $x_{low}$ = 6.71E-01
$x_{lim}$ = 9.96E+02 |  $x_{lim}$ = 8.55E+03 |  $x_{lim}$ = 3.23E+03 |  $x_{lim}$ = 3.51E+03
if an event reaches E$>$100 MeV
No bin
All data points
E$>$10 MeV | E$>$30 MeV | E$>$100 MeV |
$\gamma$ = 0.26 |  $\gamma$ = 0.28 |  $\gamma$ = 0.41 |
$x_{low}$ = 9.50E+00 |  $x_{low}$ = 2.30E+00 |  $x_{low}$ = 3.02E-01 |
$x_{lim}$ = 1.19E+04 |  $x_{lim}$ = 2.04E+03 |  $x_{lim}$ = 2.58E+02 |

###  3.2 Solar Flares

A similar probabilistic approach was further applied for solar flare inputs
replacing V${}_{\textrm{CME}}$ with SXR peak flux and imposing two
longitudinal source location ranges: well connected ($longitude\geq 20^{o}$),
poorly connected ($longitude<20^{o}$) in place of CME angular width. Applying
the same methodology we obtained the following fits presented in Figure 8.
These fits are similar to Figure 6 but for the SF case. Each panel corresponds
to one integral energy of interest, and each line within every panel refers to
well connected (red line) and poorly connected (blue line) SFs. As a result,
for each integral energy a total of 2 $f(F_{SXR})$ functions are obtained (see
Table 1). The X-axis of sub-panels in Figure 8 gives the $F_{SXR}$ and the
Y-axis provides the probability of SEP occurrence. Hence, for a solar flare
with known longitude, one may select the proper fit and from the magnitude of
the solar flare directly obtain the expected probability, $P(\rm SEP)$. In
this case, data point to an upper limit $F_{SXR}$ of X28 and thus the derived
Bayes $P(\rm SEP)$ is obtained up to that limit. Again, the total evolution of
the simulated fit is presented and we overlay a gray hatched area after that
limit.

In the case of the peak flux, the focused range of 50 & 90% is maintained
across all modes of operation. Figure 9 provides the outputs per filtered
integral energy and bins based on solar flare characteristics. Specifically,
Fig. 9(a) refers to a filtering of E$>$10 MeV with each of the presented fits
color coded as: red ($F_{SXR}<$ M3.0); green (M3.0 $\leq F_{SXR}<$ X1.0) and
blue ($F_{SXR}\geq$ X1.0). Moreover, Fig. 9, panels (b), (d) and (f) refer to
a filtering at E$>$30 MeV. Red (green) fits present the integral energy of
E$>$10 (E$>$30) MeV, while each of the three panels point to individual solar
flare bins. Finally, panels (c), (e) and (g) of Fig. 9 are obtained with a
filtering at E$>$100 MeV. In this case, each of the lines represent an
integral energy: red (E$>$10 MeV); green (E$>$30 MeV) and blue (E$>$100 MeV).
Again, each of these three panels have to do with a selection based on solar
flare magnitude. All of the fits depicted in Figure 9 are detailed in Table 3.

Table 3: Details on PROSPER’s peak flux fits for the SF bins, derived by
Equation 13. The filtering on the integral energy is first applied and then
there are three bins based on the magnitude of the solar flare. These fits are
presented in Figure 9. if an event reaches E$>$10 MeV
---
Flare flux $<$ M3.0 | M3.0 $\leq$ Flare flux $<$ X1.0 | Flare flux $\geq$ X1.0
$\gamma$ = 0.56 |  $\gamma$ = 0.40 |  $\gamma$ = 0.27
$x_{low}$ = 2.63E+00 |  $x_{low}$ = 2.97E+00 |  $x_{low}$ = 7.69E+00
$x_{lim}$ = 1.46E+03 |  $x_{lim}$ = 1.17E+04 |  $x_{lim}$ = 1.39E+04
if an event reaches E$>$30 MeV
Flare flux $<$ M3.0 | M3.0 $\leq$ Flare flux $<$ X1.0 | Flare flux $\geq$ X1.0
E$>$10 MeV
$\gamma$ = 0.52 |  $\gamma$ = 0.36 |  $\gamma$ = 0.28
$x_{low}$ = 5.66E+00 |  $x_{low}$ = 4.81E+00 |  $x_{low}$ = 1.29E+01
$x_{lim}$ = 1.36E+03 |  $x_{lim}$ = 1.01E+04 |  $x_{lim}$ = 1.43E+04
E$>$30 MeV
$\gamma$ = 0.76 |  $\gamma$ = 0.53 |  $\gamma$ = 0.28
$x_{low}$ = 1.13E+00 |  $x_{low}$ = 9.29E-01 |  $x_{low}$ = 1.59E+00
$x_{lim}$ = 3.27E+02 |  $x_{lim}$ = 5.72E+03 |  $x_{lim}$ = 2.29E+03
if an event reaches E$>$100 MeV
Flare flux $<$ M3.0 | M3.0 $\leq$ Flare flux $<$ X1.0 | Flare flux $\geq$ X1.0
E$>$10 MeV
$\gamma$ = 0.47 |  $\gamma$ = 0.27 |  $\gamma$ = 0.21
$x_{low}$ = 7.74E+00 |  $x_{low}$ = 1.06E+01 |  $x_{low}$ = 2.66E+01
$x_{lim}$ = 2.05E+04 |  $x_{lim}$ = 9.42E+03 |  $x_{lim}$ = 1.60E+04
E$>$30 MeV
$\gamma$ = 0.57 |  $\gamma$ = 0.36 |  $\gamma$ = 0.20
$x_{low}$ = 1.91E+00 |  $x_{low}$ = 2.74E+00 |  $x_{low}$ = 5.33E+00
$x_{lim}$ = 5.56E+02 |  $x_{lim}$ = 2.52E+03 |  $x_{lim}$ = 2.16E+03
E$>$100 MeV
$\gamma$ = 0.72 |  $\gamma$ = 0.53 |  $\gamma$ = 0.23
$x_{low}$ = 3.67E-01 |  $x_{low}$ = 3.33E-01 |  $x_{low}$ = 7.12E-01
$x_{lim}$ = 5.03E+00 |  $x_{lim}$ = 1.88E+02 |  $x_{lim}$ = 1.65E+02

###  3.3 Solar Flares & CMEs

For the case of a combination of a solar flare (with known magnitude and
longitude) and a CME (with known AW and speed) being identified Equation 12
was employed but enforcing more conditions, depending on the obtained
characteristics (e.g. flare longitudinal bin and magnitude, CME width and
velocity). In this case, all combinations were considered (e.g. Halo, Partial
halo and Non halo, together with well and poorly connected) for all respective
energies. Figure 10 depicts derived probability of SEP occurrence using both
flare and CME inputs for the case of Halo and well connected events for an
integral energy of E$>$10 MeV. Each line corresponds to a specific solar flare
class. In particular, C1.0 (black), M1.0 (blue), X1.0 (red) and X10.0
(magenta) lines are shown. Based in the above, the X-axis of Figure 10 is
extended up to $V_{CME}$ = 3000 km/s. From this plot one can derive the
probability to get an SEP event, in case the longitude of the parent solar
flare falls within the bin (lon $\geq$20o \- well connected) and the
corresponding CME is a Halo one (AW = 360o). Then from the specific magnitude
of the flare, the proper curve (fit) is selected and the speed of the CME
(X-axis on the plot) leads to the probability of SEP occurrence (Y-axis on the
plot). Similar fits have been constructed for all combinations of: (lon. x AW
x integral proton energy). As a result, Equation 12 was applied in total in 18
cases (not shown).

Figure 11 shows the obtained fits per filtered energy and bins based on all
obtained inputs. Similar to the previous cases the fits are presented in
detail in Table 4. As a rule, the dashed lines in the case of E$>$10 MeV (Fig.
11)(a) and E$>$100 MeV (Fig. 11)(b), (d), (f) present the bins with the
largest CME speed per case for a given solar flare flux bin, while the
continuous lines depict the cases with the slower CMEs for the same solar
flare bins (see details of the bins in each panel of Fig. 11 and Table 4).
Panel (a) of Figure 11 shows E$>$10 MeV fits for six different selections of
solar flares & CMEs (see detail in Table 4). Each pair of lines (i.e. solid
and dashed) colored with the same color refer to a similar solar flare bin. In
particular, these are: $F_{SXR}<$ M3.0 (red); M3.0 $\leq F_{SXR}<$X1.0 (blue)
and $F_{SXR}\geq$ X1.0 (magenta) (see panel (a)). Moreover, panels (b), (d)
and (f) of Fig. 11 represent the case of applying a filtering at E$>$100 MeV.
In all these panels each line refers to an integral energy color coded as: red
(E$>$10 MeV); blue (E$>$30 MeV) and magenta (E$>$100 MeV). However, there was
no event associated with a solar flare of magnitude $<$X3.0 for any CME speed,
reaching an integral energy of E$>$100 MeV. Therefore, the relevant panel (f)
in Figure 11 shows only the fits for events associated with $\geq$X3.0 for
each of the three CME bin (i.e. only dashed lines) (see Table 4). In addition,
no reliable fit for SEP events associated with solar flares of magnitude
$<$M6.5 and CME speed $V_{CME}$ $<$ 1350 km/s, at E$>$10 MeV, when a filter on
reaching an integral energy of E$>$100 MeV is applied, was found (see Fig.
11(b) and Table 4). Hence, there is no solid red line displayed in this panel.
Finally, panels (c)-(e), (g)-(i) and (h)-(j) present the cases with a
filtering at E$>$30 MeV. Each panel provides two fits color coded per integral
energy as: E$>$10 MeV (red) and E$>$30 MeV (blue). Each of the mentioned pairs
of panels refer to the same solar flare bin (i.e. $F_{SXR}<$ M3.0; M3.0 $\leq
F_{SXR}<$X1.0 and $F_{SXR}\geq$X1.0) while differ in the CME speed as detailed
in Table 4. As it can be seen,in two cases that an event reaches E$>$30 MeV
these seems to be a cross-over between the obtained fit at E$>$10 and E$>$30
MeV (i.e. panels (c) and (g)). However, these crossings are below a threshold
of 40%. Evidently, if a CL $<$50% is chosen then in this particular case (i.e.
E$>$30 MeV) a spectral incoherence arises. This means that in a future work an
update of the selection in PROSPER for the solar flare & CME case that will
lead to spectral coherence at all cases for lower CL should be pursued.

Table 4: Details on PROSPER’s peak flux fits for the combined solar flares & CME bins, derived by Equation 13. The filtering on the integral energy is first applied and then bins on the solar flare magnitude and CME speed are applied. These fits are presented in Figure 11. if an event reaches E$>$10 MeV | if an event reaches E$>$100 MeV
---|---
Flare flux $<$ M3.0 | M3.0 $<$ Flare flux $\leq$ X1.0 | Flare flux $>$ X1.0 | Flare flux $<$ M6.0 | M6.0 $<$ Flare flux $\leq$ X3.0 | Flare flux $>$ X3.0
$V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $<$ 1400 km/s |  $V_{CME}$ $<$ 1600 km/s | E$>$10 MeV
$\gamma$ = 0.45 |  $\gamma$ = 0.69 |  $\gamma$ = 0.22 |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1600 km/s
$x_{low}$ = 2.98E+00 |  $x_{low}$ = 3.30E+00 |  $x_{low}$ = 4.63E+00 |  $\gamma$= |  $\gamma$ = 0.48 |  $\gamma$ =
$x_{lim}$ = 3.85E+01 |  $x_{lim}$ = 6.49E+02 |  $x_{lim}$ = 4.27E+02 |  $x_{low}$ = |  $x_{low}$ = 8.70E+00 |  $x_{low}$ =
$V_{CME}$ $\geq$ 1250 km/s |  $V_{CME}$ $\geq$ 1400 km/s |  $V_{CME}$ $\geq$ 1600 km/s |  $x_{lim}$ = |  $x_{lim}$ = 8.49E+02 |  $x_{lim}$ =
$\gamma$ = 0.35 |  $\gamma$ = 0.23 |  $\gamma$ = 0.21 |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1600 km/s
$x_{low}$ = 4.23E+00 |  $x_{low}$ = 4.87E+00 |  $x_{low}$ = 2.82E+01 |  $\gamma$=0.36 |  $\gamma$ = 0.18 |  $\gamma$ =0.18
$x_{lim}$ = 1.58E+03 |  $x_{lim}$ = 8.19E+03 |  $x_{lim}$ = 1.35E+04 |  $x_{low}$ = 2.47E+01 |  $x_{low}$ = 1.85E+01 |  $x_{low}$ = 6.16E+01
if an event reaches E$>$30 MeV |  $x_{lim}$ = 2.38E+04 |  $x_{lim}$ = 1.17E+04 |  $x_{lim}$ = 1.35E+04
Flare flux $<$ M3.0 | M3.0 $<$ Flare flux $\leq$ X1.0 | Flare flux $>$ X1.0 | E$>$30 MeV
E$>$10 MeV |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1600 km/s
$V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1650 km/s |  $\gamma$=0.41 |  $\gamma$ = 0.47 |  $\gamma$ =
$\gamma$ = 0.17 |  $\gamma$ = 0.64 |  $\gamma$ = 0.26 |  $x_{low}$ = 2.96E+00 |  $x_{low}$ = 2.38E+00 |  $x_{low}$ =
$x_{low}$ = 6.14E+00 |  $x_{low}$ = 4.05E+00 |  $x_{low}$ = 7.80E+00 |  $x_{lim}$ = 9.71E+00 |  $x_{lim}$ = 1.01E+02 |  $x_{lim}$ =
$x_{lim}$ = 2.98E+01 |  $x_{lim}$ = 7.45E+02 |  $x_{lim}$ = 4.59E+02 |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1600 km/s
$V_{CME}$ $\geq$ 1250 km/s |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1650 km/s |  $\gamma$=0.24 |  $\gamma$ = 0.26 |  $\gamma$ = 0.16
$\gamma$ = 0.32 |  $\gamma$ = 0.22 |  $\gamma$ = 0.21 |  $x_{low}$ = 2.23E+00 |  $x_{low}$ = 4.97E+00 |  $x_{low}$ = 1.64E+01
$x_{low}$ = 6.06E+00 |  $x_{low}$ = 5.27E+00 |  $x_{low}$ = 2.82E+01 |  $x_{lim}$ = 4.47E+02 |  $x_{lim}$ = 3.14E+03 |  $x_{lim}$ = 1.76E+03
$x_{lim}$ = 1.50E+03 |  $x_{lim}$ = 8.04E+03 |  $x_{lim}$ = 1.35E+04 | E$>$100 MeV
E$>$30 MeV |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1600 km/s
$V_{CME}$ $<$ 1250 km/s |  $V_{CME}$ $<$ 1350 km/s |  $V_{CME}$ $<$ 1650 km/s |  $\gamma$=1.88 |  $\gamma$ = 0.77 |  $\gamma$ =
$\gamma$ = 0.29 |  $\gamma$ = 0.05 |  $\gamma$ = 0.40 |  $x_{low}$ = 4.04E-01 |  $x_{low}$ = 4.60E-01 |  $x_{low}$ =
$x_{low}$ = 1.26E+01 |  $x_{low}$ = 5.86E+00 |  $x_{low}$ = 1.33E+00 |  $x_{lim}$ = 4.66E+02 |  $x_{lim}$ = 4.67E+00 |  $x_{lim}$ =
$x_{lim}$ = 1.48E+00 |  $x_{lim}$ = 4.77E+00 |  $x_{lim}$ = 2.82E+02 |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1600 km/s
$V_{CME}$ $\geq$ 1250 km/s |  $V_{CME}$ $\geq$ 1350 km/s |  $V_{CME}$ $\geq$ 1650 km/s |  $\gamma$=0.34 |  $\gamma$ = 0.32, |  $\gamma$ = 0.26
$\gamma$ = 0.63 |  $\gamma$ = 0.33 |  $\gamma$ = 0.18 |  $x_{low}$ = 3.75E-01 |  $x_{low}$ = 2.33E-01 |  $x_{low}$ =1.18E+00
$x_{low}$ = 1.19E+00 |  $x_{low}$ = 1.08E+00 |  $x_{low}$ = 2.14E+00 |  $x_{lim}$ = 6.82E+00 |  $x_{lim}$ = 1.11E+02 |  $x_{lim}$ = 2.92E+02
$x_{lim}$ = 1.23E+03 |  $x_{lim}$ = 2.66E+03 |  $x_{lim}$ = 2.00E+03 |  |  |

##  4 Validation

Table 5 provides all inputs of the NASA CCMC SEP scoreboard campaign
events888https://ccmc.gsfc.nasa.gov/assessment/topics/SEP/campaign2020.php
that have been investigated in this part of the validation of PROSPER,
presented in this work. The purpose of this challenge is to facilitate
collaboration of as many as possible SEP modelers and to provide a
standardized validation procedure, with specific inputs being used by all
parties. Such a challenge is based on curated events and greatly facilitates
the comparison of outputs from different prediction methods. Nonetheless,
larger samples of events are needed for a more complete evaluation of these
methods, including PROSPER. The inputs (i.e. solar flare and/or CME
characteristics) have been utilized in order to achieve the PROSPER’s $P(\rm
SEP)$ per mode of operation. PROSPER peak flux outputs are compared to the
derived peak fluxes by the SEP challenge at two integral energies, i.e. E$>$10
and E$>$100 MeV. These values are presented in the last two columns of Table
5. The SEP event on 06 January 2014 was not associated with a solar flare (see
Table 5), although, a back-sided origin of the driving CME and a (probably
occulted) C2.1 solar flare999https://cdaw.gsfc.nasa.gov/CME_list/sepe/ have
been reported (Thakur et al. 2014). Moreover, the sequence of SEP events at 04
& 06 September 2017 are considered as one event leading to a single peak flux
for the E$>$10 MeV. An SEP event started on 04 September 2017 with a
significant enhancement recorded at E$>$10 MeV. However, on 06 September 2017
the strongest solar flare of solar cycle 24 was marked (i.e. X9.3) (Jiggens et
al. 2019) and an SEP event was clearly visible in higher energies (up to
E$>$100 MeV), while characterized also as a sub-GLE (Mishev et al. 2017).
Nonetheless, the E$>$10 MeV flux continued to be sustained after its
initiation on 04 September 2017 and throughout both SEP events lasting several
days. Finally, four SEP events did not exceed a threshold of 1 pfu at E$>$100
MeV and thus are presented in Table 5 as “N/A” entries. It is important to
note that in this part of our analysis, the parameters of Table 5 are taken
“as is” from the campaign event website and all comparisons and results are
driven by the parent solar event parameters and the observed peak proton
fluxes at E$>$10 & E$>$100 MeV.

Table 5: Details of the SEP events’ solar event parameters (i.e. solar flare and CME characteristics) and observed peak proton flux at E$>$10-; & E$>$100 MeV, used in the validation, provided under the SHINE/ISWAT/ESWW SEP Model Validation Challenge. SEP event | Flare information | CME information | E>10 MeV | E>100 MeV
---|---|---|---|---
Date | Onset | Mag. | Lon. | Start | Speed | Width | Peak Flux | Peak Flux
| Time | (W/m2) | (∘) | Time | (km/s) | (∘) | (pfu)
| (HH:MM) |  |  | (HH:MM) |  |  |  |
2012-03-07 | 0:02 | X5.4 | E15 | 0:24 | 2684 | 360 | 6529.8 | 69.272
2012-05-17 | 1:25 | M5.1 | W89 | 1:48 | 1582 | 360 | 255.44 | 20.445
2012-07-12 | 15:37 | X1.4 | W02 | 16:48 | 885 | 360 | 96.08 | N/A
2013-04-11 | 6:55 | M6.5 | E12 | 7:24 | 861 | 360 | 113.55 | 2.0297
2014-01-06 | - | - | - |  $\,|\,$ 8:00 | 1402 | 360 | 42.17 | 4.079
2014-01-07 | 18:04 | X1.2 | W11 | 18:24 | 1830 | 360 | 1026.1 | 4.2687
2017-07-14 | 1:07 | M2.4 | W33 | 1:25 | 1200 | 360 | 22.374 | N/A
2017-09-04 | 20:28 | M5.5 | W16 | 20:36 | 1418 | 360 | 844.38 | N/A
2017-09-06 | 11:53 | X9.3 | W34 | 12:24 | 1571 | 360
2017-09-10 | 15:35 | X8.2 | W88 | 16:00 | 3163 | 360 | 1493.5 | 68.128

###  4.1 Probability of detection

Using the associated parent solar events as input parameters (see Table 5),
derived the probability of SEP detection per event and for all three PROSPER’s
modes of operation (see Table 1). Figure 12 presents the obtained outputs.
Results are presented in red (flare mode of operation), blue (CME mode of
operation) and green (solar flare & CME mode of operation). Each panel
corresponds to each integral energy (i.e. E$>$10-; $>$30-; and $>$100 MeV).
Within the panel of E$>$10 MeV (upper panel) the threshold ($pt$) above which
the probabilistic forecast is transformed into a categorical one is over-
plotted with a dotted horizontal black line. This $pt$ was directly obtained
by Table 1 of Anastasiadis et al. (2017) and corresponds to the threshold
above which an SEP event would have been identified. Such a $pt$ is obtained
at the perfect skill of a forecasting algorithm and is usually applied for a
direct discrete analysis of SEP yes/no. However, it must be noted that PROSPER
gives a probabilistic forecast. From the inspection of the E$>$10 MeV panel in
Figure 12, it seems that the PROSPER’s mode of operation based on solar flare
input (red bars) would spot almost all of the events: $\sim$90% (i.e. 8/9
events for which flare input was available). The SEP event that would not have
been spotted by the PROPSER’s flare mode of operation is the one that occurred
on 14 July 2017. This SEP event was associated with a “well-connected” (W33o)
solar flare but with relatively modest-magnitude (M2.4) that, although led to
a probability of SEP occurrence of $\sim$20%, was lower than the $pt$ and thus
would not have been spotted by PROSPER. Moreover, the $P(\rm SEP)$ seems to be
quite successful (100%) for the CME module (i.e. 10/10 events for which CME
input was available were identified as SEP events (i.e. crossing the $pt$).
Adding to this, the 14 July 2017 event that was “missed” by the flare mode of
operation was associated with a halo CME with a speed of 1200 km/s – which
enhanced the probability of SEP occurrence to $\sim$ 58% in the CME mode and
thus pushed it above the $pt$ threshold, resulting into an identification of
the event by PROSPER. Finally, the green bars represent PROSPER’s outputs for
the solar flare & CME mode of operation. As it can be seen, the hit rate is
100% (i.e. 9/9 events for which both flare and CME inputs were available).

These outputs (i.e. $P(\rm SEP)$) stem from the fits presented in Section 3
here above. In particular, 5/9 ($\sim$66%) of the parent solar flares included
in Table 5 are “poorly connected” (i.e. longitude $<$20o) and 4/9 ($\sim$44%)
are “well connected” (i.e. longitude $\geq$ 20o). Hence, the fits used in
these cases are presented in Figure 8. Figure 12 shows that at an integral
energy E$>$10 MeV the largest differences in the achieved $P(\rm SEP)$ are due
to the magnitude of the associated solar flare. For example, the first two
events on the list are the 07 March 2012 (X5.4/E15 -“poorly connected”) and 17
May 2012 (M5.1/W89 - “well connected”) with the obtained probability being
higher for the former of the two (see Figure 12). When shifting to higher
energies (E$>$100 MeV) the differences in the obtained $P(\rm SEP)$ values
(red bars) become larger because of the additional separation of the fits due
to the longitudinal difference. Moreover, the two events on 06 & 10 September
2017 result to very high probabilities of SEP occurrence, which seem to be
preserved in each integral energy of interest. The specific SEP events are
associated with very strong ($>$X8.0) and “well connected” solar flares, which
seem to be dominant. All driving CMEs are Halo and fast, thereby all of the
$P(\rm SEP)$ values (blue bars) are obtained from the blue lines (fits) of
Figure 6, and the differences are due to the $V_{CME}$ inputs, with larger
energies requiring a higher value of $V_{CME}$ in order to lead to a larger
$P(\rm SEP)$. That said, for the same $V_{CME}$ the obtained $P(\rm SEP)$ is
lower when shifting to higher energies in all tested cases. Finally, for the
flare & CME mode of operation (green bars) one can notice that for the cases
of 12 July 2012 and 11 April 2013 that are both “poorly connected” ones,
associated with comparable CMEs (see Table 5) the difference between the
achieved $P(\rm SEP)$ is driven by the magnitude of the associated solar flare
and the former case leads to higher $P(\rm SEP)$ at all integral energies. For
the SEP event on 04 September 2017 which is also a “poorly connected” one the
$V_{CME}$ is almost a factor of $\sim$1.7 larger than the previous two cases,
although the magnitude of the associated solar flare is M5.5 (i.e. less strong
than the previous two cases). However, the obtained $P(\rm SEP)$ is higher
than both cases at all integral energies, pointing to the role of the
$V_{CME}$ in the establishment of that probability. Finally, the SEP event on
14 July 2017 which is a “well connected” one and associated with a CME of
$V_{CME}$ = 1200 km/s leads to a higher $P(\rm SEP)$ compared to the case of
the “poorly connected” SEP event at 11 April 2013 at all integral energies. In
the latter case the associated flare is stronger (M6.5 versus M2.4) but the
CME speed is slower (861 versus 1200 km/s). Thus the better magnetic
connection and the higher $V_{CME}$ seem to lead to a higher $P(\rm SEP)$.

###  4.2 Peak proton flux

Figure 13 provides scatter plots of the predicted peak flux versus the
observed peak flux at E$>$10 MeV (top row) and E$>$100 MeV (bottom row) for
each of the three PROSPER’s modes of operation. In particular the first column
corresponds to the PROSPER’s outputs based on SF input, the middle column
refers to the outputs of PROSPER utilizing CME input alone and the third
column represents outputs from PROSPER’s third mode of operation that makes
use of both SF and CME inputs. Blue circles denote the obtained outputs at a
50% CL (lower limit) and red circles present the same outputs for a 90% CL
(upper limit).

The predicted peak flux at E $>$ 10 MeV as a function of the observed peak
flux at the same energy, utilizing only SF information (upper panel on the
left hand side of Figure 13), shows a strong correlation especially in the
lower limit (i.e. 50% CL) with the correlation coefficient ($cc$) being 0.76
(lower limit) and 0.72 (upper limit). Moreover, more than half of the
predicted values seem to be very close to the dichotomous line which indicates
a perfect prediction. However, the upper limit predictions (90% CL) show an
over-forecasting being above the perfect dichotomous prediction line, with
these predictions being within one order of magnitude from the observed peak
fluxes, but not always. Nonetheless, the lower and the upper limits seem to
capture the actual peak flux quite well at E$>$10 MeV. When moving to the CME
mode of operation (column in the middle) for E$>$10 MeV, a weaker correlation
is observed with the $cc$ being 0.33 (lower limit) and 0.32 (upper limit)
respectively. The tendency for over predicting is also visible in this case
and holds for the majority of the events both at the lower and the upper
limit. At the same time, the CME mode of operation, seems not to provide
granularity since all of the points are predicted using as input halo and fast
CME and with the expectation of an SEP event reaching E$>$100 MeV. Hence the
differences are small between events, since all outputs are derived by the
fits of the bottom panel of Figure 10. As a result, those small differences
are corroborated with the small differences between the established $\rm
P(SEP)$ (see Equation 14). Moving to the results of the combined SF & CME mode
of operation (column at the right hand side), a stronger correlation is
achieved ($cc$ is 0.77 - lower limit & 0.71 - upper limit). Furthermore, the
predictions at both the lower and the upper limit seem to be very close to the
dichotomous line, with a small tendency for over-forecasting in the upper
limit predictions. Nonetheless, the predictions at the lower limit for the
strongest SEP events (i.e. those achieving the largest peak flux) at E$>$10
MeV seem to fall at the perfect dichotomous line.

In the case of E$>$100 MeV, the peak flux seems to be captured well by all
three modes of operation, with the obtained $cc$ being on average $\geq$ 0.90.
Especially with the upper limit in the case of the solar flare input (panel at
the left hand side at the bottom row) and the flare & CME (panel on the right
hand side at the bottom row) modules the agreement is quite reasonable with
most of the events predicted at the upper limit (90%) being either on the
dichotomous line or very close to it (i.e. within less then 1 order of
magnitude).

The results presented in Figure 13, underline the inherent difficulty of the
SEP characteristics prognosis. However, the usage of the upper (90%) and lower
(50%) limit in PROSPER shows promise and seems to capture the expected peak
flux of the SEP events, reasonably well.

##  5 Discussion & Conclusions

A new probabilistic model, PROSPER, that provides short-term forecasting
(nowcasting) of SEP events (i.e. probability of occurrence and corresponding
peak flux) has been presented in this work. In particular, PROSPER was
implemented based on a straightforward application of the Bayes theorem using
probability distribution functions constructed from a large database of solar
flares, CMEs and SEPs. The output of the prediction is the probability of a
solar event to lead to an SEP event, i.e. P($\rm SEP$). As a second step, the
estimated peak proton flux of the SEP event is evaluated. PROSPER has three
modes of operation based on the inputs received. There is one mode that
utilizes solar flare data (magnitude and longitude), one that makes use of CME
identifications (AW and speed) and a third one that makes use of the combined
input of solar flares and CMEs (see Table 1). The basic implementations of
PROSPER presented in this study are summarized as follows:

  * •

Probability of SEP occurrence. For the identification of $P(\rm SEP)$ in each
case, log-normal fits (Equation 1) to the data were utilized (see e.g. Figures
3 & 4) and the resulting distributions were combined under the Bayes formula
(Equation 11). These Bayes fits were obtained for different integral energies
spanning from E$>$10-; E$>$30- and E$>$100 MeV (see Figures 6, 8 and 10).

  * •

Peak fluxes. The modeling of the peak fluxes was based on exponential cut-off
fits (Equation 14) to cumulative distribution functions (CDFs) constructed by
the recorded peak fluxes of the SEP events in our sample. In this case, first
the data are filtered based on the highest expected integral energy that an
SEP event will occur (e.g. applying a thresholding on the $P(\rm SEP)$
obtained in the previous step and thus utilizing the prediction of an event
being expected to reach E$>$100 MeV, E$>$30 MeV or E$>$10 MeV). Detailed fits
are presented in Figures 7, 9 and 11 as well as in Tables 2, 3 and 4,
respectively.

The ramifications of Bayes’ rule are countless and have gained wide
recognition in data driven, Space Weather related topics. Nonetheless, to our
knowledge, PROSPER is the first SEP prediction model that allows the
estimation of P($\rm SEP$) utilizing Bayes’ theorem (see Camporeale 2019, and
references there in). PROSPER allows a direct estimation of P($\rm SEP$),
based on a previous outcome having occurred in similar circumstances and in
doing so, it takes into account all available data without any a priori bias.
As a result, the benefit of such an approach is the natural and principled way
of combining prior information with data, within a solid decision theoretical
framework, providing exact inferences that are conditional on data (Hartigan
2012).

Moreover, all PROSPER’s modes of operation were validated based on detailed
case studies of SEP events selected as part of the NASA CCMC SEP scoreboard
challenge (see Table 5). Blind tests with archived parent solar data of these
events were applied to PROSPER’s modes in order to derive the probabilities of
SEP detection per mode of operation and per SEP event. The obtained results
were discussed in a comparative manner, showing that the magnetic connectivity
(i.e. “well connected” events), strong solar flares and fast CMEs lead to
higher achieved $P(\rm SEP)$ values. The validation results of the PROSPER
models are based on a sample of 10 SEP events and are summarized as follows:

  * •

PROSPER’s module that is based on solar flare input would spot $\sim$90% (i.e.
8/9) of the events for which flare input was available.

  * •

For the CME module the hit rate is 100% (i.e. 10/10 events for which CME input
was available were identified as SEP events).

  * •

The solar flare & CME mode of operation, also has a hit rate of 100% (i.e. 9/9
events for which both flare and CME inputs were available).

Additionally, the largest differences in the achieved $P(\rm SEP)$ in lower
energies (i.e. at an integral energy E$>$10 MeV) are due to the magnitude of
the associated solar flare, $F_{SXR}$. For the higher energies (i.e. E$>$100
MeV) the differences in the obtained $P(\rm SEP)$ values become larger due to
the additional separation of the fits, which is driven by their longitudinal
difference. Finally, the solar flare & CME module revealed that the better
magnetic connection and the higher $V_{CME}$ seem to lead to a higher $P(\rm
SEP)$.

Furthermore, we included scatter plots of observed versus predicted SEP peak
fluxes to quantify the reliability of PROSPER’s predictions. Although the SEP
peak fluxes are difficult to infer based on the characteristics of their
parent solar events, the usage of an upper (90%) and lower (50%) limit in
PROSPER seemed to capture the expected peak flux of the SEP events, reasonably
well. In particular, at E $>$ 10 MeV, for the solar flare module, the
predicted peak flux, as a function of the observed peak flux shows a strong
correlation especially in the lower limit (i.e. 50% CL) with more than half of
the the predicted values being close to the dichotomous line – indicating a
perfect prediction. Nonetheless, the CME module at the same energy seems not
to capture the expected peak flux with granularity. However, the combined SF &
CME mode of operation shows that the predictions at both the lower and the
upper limit seem to be very close to the dichotomous line, with a small
tendency for over-forecasting in the 90% limit predictions. Nonetheless, the
predictions at 50% for the SEP events achieving the largest peak flux at
E$>$10 MeV seem to fall at the perfect dichotomous line. Additionally, in the
case of E$>$100 MeV, the peak flux seems to be captured well by all three
modes of operation, especially with the 90% limit in the case of the SF and
the flare & CME modules the agreement is quite reasonable with most of the
events predicted at 90% being either on the dichotomous line or very close to
it (i.e. within less then 1 order of magnitude).

The PROSPER model has been incorporated into the new operational Advanced
Solar Particle Event Casting System (ASPECS) tool providing outputs in real-
time through the web portal http://phobos-srv.space.noa.gr/. In addition,
ASPECS offers the capability to interrogate PROSPER for historical cases via a
run on demand functionality. ASPECS is the first realisation of ESA’s SEP
Advanced Warning System (SAWS) - a modular framework for forecasting solar
energetic particle (SEP) events, their characteristics and profiles.

A point that still needs to be addressed in the validation is how the model
reacts when solar events not associated with SEPs are introduced as inputs.
This is a part of an ongoing follow-up study focusing on the performance of
the ASPECS tool. On a preliminary basis, it can be commented that PROSPER
achieves categorical scores in line with the expected range for data driven
models. The range of scores are taken from Table 1 of Anastasiadis et al.
(2017) (i.e. Probability of Detection – POD $\sim$50-70%, False Alarm Rate –
FAR $\sim$30-50%). Such a range is realistic and inherently imposed due to the
imbalanced dataset that is being used for SEP prediction efforts (see the
relevant detail discussion in Lavasa et al. 2021). In fact, recently, Stumpo
et al. (2021) have shown that the major drawback in predicting the occurrence
of SEPs, i.e. P($\rm SEP$), in the framework of statistical forecasting
concepts, is the optimization of FAR which directly depends on the imbalance
of the dataset used. That means that, the greater the imbalance, the greater
the FAR is affected by the presence of false positives. Taking all of these
into account, in our next study such a thorough validation step will be
implemented.

It should be mentioned that, PROSPER’s CME and flare & CME modes of operation
are subject to the CME input data used. In particular, while developing
PROSPER, CME data from the CDAW CME catalogue have been used. It was pointed
out by Richardson et al. (2015) that different CME catalogues provide
different estimations of the same CME event, hence given that PROSPER is a
data driven model, it is inherently affected by such differences. On top of
the CDAW CME catalogue, other options include: (a) the Computer-Aided CME
Tracking (CACTUS) catalog (https://wwwbis.sidc.be/cactus/), which is compiled
using a specialized software package (Robbrecht and Berghmans 2004); (b) the
SEEDS (Solar Eruptive Event Detection System) catalog
(http://spaceweather.gmu.edu/seeds/) (Olmedo et al. 2008); (c) the CORIMP
(coronal image processing) method catalog
(http://alshamess.ifa.hawaii.edu/CORIMP/) (Byrne et al. 2012) and (d) the
Space Weather Database Of Notification, Knowledge, Information (DONKI)
developed by CCMC (https://ccmc.gsfc.nasa.gov/donki/). Therefore, a natural
next step of this study is to adapt PROSPER to each of these catalogues and
quantify differences/changes following the study by Richardson et al. (2015).

The usage of CME identifications (e.g. angular width and speed) for the
derivation of the SEP occurrence probabilities (P($\rm SEP$)) and expected
peak proton flux have been explored, already, in a few operational efforts
(e.g. Dierckxsens et al. 2015; Papaioannou et al. 2018b; Richardson et al.
2018). On top of that statistical studies have pointed out that P($\rm SEP$)
increases as a function of solar flare magnitude ($F_{SXR})$, longitude and
CME speed ($V_{CME}$) - especially when considering solar flare-CME couples
situated on the west part of the visible solar disk (i.e. Dierckxsens et al.
2015). In addition, it was recently showed that $F_{SXR}$, $V_{CME}$ and SXR
fluence have the largest potential for discriminating solar events that are
associated with SEP events (Lavasa et al. 2021), while as many as targeted
parameters need to be taken into account, in order to build a reliable and
more accurate SEP predicting system (see e.g. Papaioannou et al. 2018b).
PROSPER takes advantage of such findings and comes with three modes of
operation. Nevertheless, its performance certainly benefits from the provision
of reliable CME estimates or a proper proxy for the CME speed and width, which
would be very much desirable to obtain in near-real time.

###### Acknowledgements.

This work was supported through the ESA Contract No. 4000120480/NL/LF/hh
“Solar Energetic Particle (SEP) Advanced Warning System (SAWS)”. Athanasios
Papaioannou and Angels Aran acknowledge the support from the project
MDM-2014-0369 of ICCUB (Unidad de Excelencia “María de Maeztu”). The CME
Catalog used in this work is generated and maintained at the CDAW Data Center
by NASA and The Catholic University of America in cooperation with the Naval
Research Laboratory. SOHO is a project of international cooperation between
ESA and NASA. Athanasios Papaioannou, Rami Vainio & Anastasios Anastasiadis
acknowledge the International Space Science Institute and the supported
International Team 441: High EneRgy sOlar partICle Events Analysis (HEROIC,
http://www.issibern.ch/teams/heroic/). Finally, Athanasios Papaioannou
acknowledges support from NASA/LWS project NNH19ZDA001N-LWS. The authors would
like to thank Dr Manolis Georgoulis for stimulating discussions, Dr Katie
Whitman for fruitful collaboration and exchange of ideas as concerns the
validation of PROSPER and Mr George Vasalos for valuable technical assistance.
Furthermore, the authors would like to thank the anonymous referees for a
critical and constructive reading of the manuscript and for valuable comments
that improved the contents of the paper.

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Figure 1: An illustration of the two methods (Method 1; (blue color) and
Method 2; (red color)), as a superposition investigated in this work. Both
fits in the plots correspond to the Probability Distribution Functions (PDFs)
for the case of Halo CMEs, with respect to the CME speed $V_{CME}$. See text
for details

Figure 2: The distributions of the GOES/EPS data per integral energy of
interest for September 2009. The relevant percentiles of 68%-95% & 99.7% of
the distributions are depicted as vertical dashed lines. The mean values of
these distributions (presented as solid vertical lines) are used as the
background level per channel in Equation 14.

Figure 3: The observational Cumulative Distribution Functions (CDFs)
constructed by the database (i.e. all black points correspond to actual data).
The red line depicts the log-normal fit to the data per case. Each plot
further includes the fraction of the data explained by the fit (in [%]) [i.e.
a 100% fraction means that all data points have been used in the corresponding
fit], the mean absolute error (MAE) and the fit that was used (i.e. log-
normal). From top to bottom the column on the left hand side corresponds to
the CDFs per CME width (Halo, Partial Halo, Non-Halo) for all CMEs in each
sample. The column on the right hand side corresponds to the CDFs in case that
the CME was associated with an SEP at an energy of E$>$10 MeV, at each width
bin. Similar fits (and plots) have been constructed for all other SEP integral
energies of interest but are not shown.

Figure 4: The Cumulative Distribution Functions (CDFs) for the case of Halo
CMEs (top panel); Partial Halo CMEs (middle panel) and Non-Halo CMEs (bottom
panel). The black color represents all CMEs in the respective sample, whereas
CMEs associated with SEP events for different integral energies are colour
coded. The gray border hatched rectangle area provides the limit for the
$V_{CME}$.

Figure 5: The Probability Distribution Functions (PDFs) for the case of Halo
(top panel), Partial Halo (middle panel) and Non Halo (bottom panel) CMEs, as
derived by the CDFs, applying Method 2. The black color represents all CMEs in
the sample, whereas CMEs associated with SEP events for different integral
energies are colour coded. The gray border hatched rectangle area provides the
limit for the $V_{CME}$.

Figure 6: The curves for the estimation of the probability of SEP occurrence
when both the speed ($V_{CME}$, X-axis) and the width (Halo, Partial Halo and
Non-Halo) of a CME are known. Each panel corresponds to an integral energy
(i.e. E$>$10-; $>$30-; $>$60- and $>$100 MeV). Within each panel three fits
are presented, one per CME width - colour coded as: Halo (blue), Partial Halo
(red), Non Halo (green) CMEs. The gray border hatched rectangle area provides
the limit for the $V_{CME}$. See text for details.

Figure 7: The CDFs constructed by the database (i.e. all points correspond to
actual data) for each integral energy of choice, for the estimation of the
peak flux (Peak flux [pfu], X-axis), when both the speed and the width of a
CME are known. Lines depict the exponential cut-off fits to the data points
per case (from Equation 13). Panels: (a) and (b) correspond to a filtering
integral energy at E$>$10MeV; (c)-(f) to E$>$30 MeV and (g) to E$>$100 MeV).
See text and Table 3 for details on the bins and the obtained parameters of
the fit per case.

Figure 8: The curves for the estimation of the probability of SEP occurrence
when both the SXR magnitude and the longitude of a solar flare are known. Each
panel corresponds to an integral energy (i.e. E$>$10-;$>$30-;$>$60- and $>$100
MeV). Within each panel two fits are presented, one per longitudinal bin -
colour coded as: well connected (red), and poorly connected (blue) solar
flares. See text for details. The gray border hatched rectangle area provides
the limit for the $F_{SXR}$. This is similar to Figure 6.

Figure 9: The CDFs constructed by the database for each integral energy of
choice, for the estimation of the peak flux (Peak flux [pfu], X-axis), when
the SXR solar flare magnitude is known. Lines depict the exponential cut-off
fits to the data points per case. Panel (a) corresponds to a filtering of the
integral energy at E$>$10 MeV. The left hand column (panels (b), (d), (f))
corresponds to a filtering integral energy at E$>$30 MeV and the right hand
column (panels (c), (d), (g)) to a filtering integral energy at E$>$100 MeV.
Black color denotes the SXR flare flux bin in each panel. See text and Table 3
for details on the bins and the obtained parameters of the fit per case.

Figure 10: The curves for the estimation of the probability of SEP occurrence
when both a CME and a solar flare have taken place and thus the SXR magnitude
(X-axis) and the longitude of a solar flare, as well as, the CME speed and AW
are known. The figure presents the output of PROSPER for the case of E$>$10
MeV with Halo CMEs (AW = 360o) and well connected flares (lon $\geq$20o). Each
fit corresponds to a solar flare class, color coded as: C1.0 (black line),
M1.0 (blue line), X1.0 (red line) and X10 (magenta line). The X-axis shows the
speed of the CME (in km s-1).

Figure 11: The CDFs constructed by the database (i.e. all points correspond to
actual data) for each integral energy of choice. See text and Table 4 for
details on the bins and the obtained parameters of the fit per case. Each
panel further displays the selected bins and the displayed energies, if more
than one are included in a panel.

Figure 12: Comparison of the predicted probabilities of SEP detection per
PROSPER’s mode of operation for all cases included in Table 5, for each
integral energy of interest (i.e. E$>$10-; E$>$30-; and E$>$100 MeV). In each
panel, the red histogram represents the outputs of PROSPER based on SF data,
the blue histogram represents the outputs of the model based on CME data, and
the green histogram represents the outputs of the model based on both SF & CME
inputs. In the upper panel the horizontal line represent the threshold above
which most SEP prediction modules would issue a notification for a forthcoming
SEP event – see details in Table 1 of Anastasiadis et al. (2017).

Figure 13: Comparison of predicted with observed peak flux for the SEP events
of Table 5, based on solar flare (column at the left hand side), CME (column
in the middle) and solar flare & CME (column at the right hand side) input
data. The top row refers to an integral energy of E$>$10 MeV and the bottom to
E$>$100 MeV.