New Method of Assessment of the Integral Fluence of Solar Energetic (> 1 GV Rigidity) Particles from Neutron Monitor Data
Abstract
A new method to reconstruct the highrigidity part (≥ 1 GV) of the spectral fluence of solar energetic particles (SEP) for GLE events, based on the worldwide neutron monitor (NM) network data, is presented. The method is based on the effective rigidity \(R_{\mathrm{eff}}\) and scaling factor \(K_{\mathrm{eff}}\). In contrast to many other methods based on derivation of the bestfit parameters of a prescribed spectral shape, it provides a true nonparametric (viz. free of a priori assumptions on the exact spectrum) estimate of fluence. We reconstructed the SEP fluences for two recent GLE events, #69 (20 Jan. 2005) and #71 (17 May 2012), using four NM yield functions: (CD00 – Clem and Dorman in Space Sci. Rev.93, 335, 2000), (CM12 – CaballeroLopez and Moraal in J. Geophys. Res.117, A12103, 2012), (Mi13 – Mishev, Usoskin, and Kovaltsov in J. Geophys. Res.118, 2783, 2013), and (Ma16 – Mangeard et al. in J. Geophys. Res.121, 7435, 2016b). The results were compared with full reconstructions and direct measurements by the PAMELA instrument. While reconstructions based on Mi13 and CM12 yield functions are consistent with the measurements, those based on CD00 and Ma16 ones underestimate the fluence by a factor of 2 – 3. It is also shown that the often used powerlaw approximation of the highenergy tail of SEP spectrum does not properly describe the GLE spectrum in the NMenergy range. Therefore, the earlier estimates of GLE integral fluences need to be revised.
Keywords
Cosmic rays Solar1 Introduction
While galactic cosmic rays (GCRs) always bombard the Earth’s atmosphere, with the intensity being somewhat modulated by solar activity in the course of the 11year solar cycle (see, e.g. a review by Potgieter, 2013), sometimes sporadic fluxes of solar energetic particles (SEPs) can impinge on the Earth’s atmosphere, as caused by solar eruptive events like solar flares or coronal mass ejections (e.g., Vainio et al., 2009; Desai and Giacalone, 2016). During SEP events, fluxes of lowerenergy particles (below several hundred MeV) can get enhanced over the GCR background by many orders of magnitude during several to tens of hours. It is important to study such events for different reasons, from purely academic, viz. studying solar eruptive events and probing the inner heliosphere, to very practical ones, since these fluxes pose serious radiation hazards for spacebased technologies and even to highlatitude commercial jet flights (Gopalswamy, 2018; Shea and Smart, 2012).
Variability of SEPs is continuously monitored by space missions, such as GOES (Geostationary Operational Environmental Satellites), SoHO (SolarHeliospheric Observatory), etc. over the last several decades. However, due to natural limitations, most space missions are able to measure mainly the lowenergy range of particles, ≤ 100 MeV. A few instruments can detect higher energies, GOES/HEPAD (High Energy Proton and Alpha Detector) can extend the energy range to 700 MeV (P10 channel) and to the integrated flux above 700 MeV (P11 channel). In addition, the SoHO/EPHIN (Electron Proton and Helium Instrument) can measure SEPs up to 500 MeV (Kühl et al., 2017). Two missions are/were able to measure more energetic particles in space, PAMELA (Payload for Antimatter Matter Exploration and Lightnuclei Astrophysics, Adriani et al., 2014) was in operation June 2006 through January 2016, while AMS02 (Alpha Magnetic Spectrometer, Aguilar et al., 2017) is in operation since 2011. Despite their excellent performance and sensitivity, both these missions are not well suited for SEP monitoring because of their low orbits, whose major fraction is located inside the geomagnetic field and is thus protected from lowenergy cosmic particles. Therefore, the only type of detectors able to continuously monitor the energy range above several hundred MeV is a groundbased neutron monitor (NM; see Simpson, 2000). On one hand, NM is an energyintegrating detector unable to directly measure the particle energy/rigidity spectrum. On the other hand, there is the worldwide network of NMs, located in different places with different geomagnetic rigidity cutoffs, which makes it possible to roughly assess the spectrum of energetic particles during SEP events. The key here is the knowledge of the yield function of a NM that quantifies the response of a NM to a monoenergetic unit flux of primary energetic particles on the top of the atmosphere (e.g., Clem and Dorman, 2000). Usually, the spectrum of SEPs is reconstructed parametrically, so that the bestfit parameters of a prescribed SEP spectral shape are defined by fitting the modeled responses of several NMs to the measured ones (e.g., Cramp et al., 1997; Vashenyuk, Balabin, and Stoker, 2007; Mishev, Kocharov, and Usoskin, 2014), explicitly considering also the SEP pitchangle anisotropy, which can be large in the initial impulsive phase of the event. This method, while allowing for estimate of the timevariable spectral and angular distributions of SEPs during the events, is very laborious and not always stable, and may lead to large uncertainties (Bütikofer and Flückiger, 2015), mostly due to differences in NM yield functions. Of course, the NMbased estimates can be made only for hardspectrum SEP events, which can initiate atmospheric cascades and be detected by groundbased NMs. This class of events is called GroundLevel Enhancements or GLEs (Poluianov et al., 2017). At present there are known 72 such events (a list can be found at https://gle.oulu.fi ).
For practical applications, it is often sufficient to know not the peak flux and its temporal/angular distributions, but the integral fluence (flux integrated over the entire event). Determination of the event fluence is more robust and is usually done under an assumption of the isotropic distribution of SEP particles near Earth. A detailed method for that was proposed by Tylka and Dietrich (2009), who fitted a power law in rigidity tail of the Bandfunction spectral shape to the measured NM responses for most of the GLEs (see Raukunen et al., 2018, – called R18 henceforth). However, this method is parametric, viz. based on an explicit assumption of the powerlaw spectral shape. In addition, it uses an outdated yield function of Clem and Dorman (2000).
Here we propose a further development of the method by Tylka and Dietrich (2009), by introducing the effective rigidity of a NM, which enables one to make a nonparametric (viz. free of explicit assumptions of the spectral shape) reconstructions of the GLE integral fluence, based on the data from the NM network.
2 Assessment of the Integral SEP Fluence from NM Data
2.1 General Approach
The method described above contains two important simplifications: first, it ignores the angular distribution of SEPs, and second, it is based on a prescribed spectral shape (power law in rigidity). While the former one is reasonable as the integral fluence is largely defined for the main isotropic phase of the event, the latter assumption makes the method parametric (viz., not the spectrum per se is estimated but parameters of a prescribed shape, without validation whether this shape is applicable) and may lead to a significant systematic uncertainty.
2.2 NM Yield Function
The NM is a groundbased detector, where secondary nucleonic particles produced in the atmospheric cascade are detected instead of the primary energetic cosmic rays. Thus, the process of the atmospheric cascade needs to be properly modeled in order to study the cosmicray variability. Successful efforts in modeling the cosmicray induced atmospheric cascade were made over 60 years (e.g., Debrunner and Brunberg, 1968), but only during the last decades it became possible, thanks to the improving computer performance and development of appropriate fulltarget MonteCarlo packages, to conduct detailed simulations. This led to the concept of the yield function (YF) of a NM defined as the response of the detector (in terms of counts) to the unit flux of primary cosmicray particles outside the Earth’s atmosphere and magnetosphere (e.g., Clem and Dorman, 2000).

CD00 (Clem and Dorman, 2000) YF was computed numerically as a first detailed MonteCarlo simulation, using the FLUKA (FLUktuierende KAscade or Fluctuating Cascade – Fassò et al., 2001; Ballarini et al., 2006) package, of the cosmicray induced atmospheric cascade in the atmosphere;

CM12 (CaballeroLopez and Moraal, 2012) YF was empirically constructed based on latitudinal surveys of a NM, and thus defined only for the rigidities below 15 GV, it was extended to higher energies/rigidities theoretically;

Mi13 (Mishev, Usoskin, and Kovaltsov, 2013) YF was computed using the PLANETOCOSMICS GEANT4 simulation tool (Desorgher et al., 2005, 2009), considering, for the first time, the finite lateral size of the atmospheric cascade and the NM’s electronic dead time;

Ma16 (Mangeard et al., 2016b,a) YF was also computed using the FLUKA package (release 2011; see Böhlen et al., 2014).
We note that YF is usually defined for the intensity of primary energetic particles, \(J\) (see Equation 5), while SEPs are typically presented via the omnidirectional flux/fluence \(F\). For the isotropic case, the two quantities are related as \(F=4\pi \cdot J\) (Grieder, 2001, Chapter 1.6).
2.3 NM Effective Rigidity for GLE
The NM is an energyintegrating detector that cannot directly measure the differential energy spectrum of cosmic rays. However, it can record the integral spectrum. The ideal integral particle detector would have a steplike YF (viz., zero below the threshold energy \(E_{\mathrm{th}}\), and constant above it). The response of such an ideal detector is directly proportional to the integral flux of primary particles with energy above this threshold energy \(E_{\mathrm{th}}\). While the YF of NM is not ideal, it is close to that (very sharp, nearly steplike rise, especially for nonpolar NMs followed by a gradual increase roughly proportional to the energy; see Figure 1a). This makes it possible to define the effective energy/rigidity of a NM for the given typical spectrum of primary particles, GCR or GLE. The effective energy/rigidity, \(E_{\mathrm{eff}}/R_{\mathrm{eff}}\), is such that the integral flux of primary particles above this threshold is nearly proportional to the count rate of the detector, viz. an analog of \(E_{\mathrm{th}}\).
The concept of the effective energy/rigidity for NM has been used in application to study GCR variability (e.g., Alanko et al., 2003; Asvestari et al., 2017). The same concept was applied also to such integral ‘detectors’ as production of cosmogenic isotopes in the Earth’s atmosphere (Kovaltsov et al., 2014; Asvestari et al., 2017) and lunar rocks (Poluianov, Kovaltsov, and Usoskin, 2018). Such an effective energy/rigidity was recently introduced for the sealevel polar NMs for detection of GLEs (Koldobskiy, Kovaltsov, and Usoskin, 2018b).
Values of the effective rigidity \(R_{\mathrm{eff}}\) (in GV) with the fullrange uncertainties as a function of cutoff rigidities \(P_{\mathrm{c}}\) (rows) for different atmospheric depths (columns) as given in the top line. Computations were done for the Mi13 YF. See Figure 2.
\(P_{\mathrm{c}}\) (GV)  700 g cm^{−2}  800 g cm^{−2}  900 g cm^{−2}  1000 g cm^{−2} 

0  \( 1.31_{0.07}^{+0.03} \)  \( 1.34_{0.08}^{+0.04} \)  \( 1.38_{0.09}^{+0.04} \)  \( 1.43_{0.11}^{+0.05}\) 
1  \( 1.39_{0.06}^{+0.03} \)  \( 1.42_{0.07}^{+0.03} \)  \( 1.45_{0.09}^{+0.04} \)  \( 1.50_{0.10}^{+0.04}\) 
2  \( 2.28_{0.10}^{+0.03} \)  \( 2.30_{0.11}^{+0.03} \)  \( 2.33_{0.12}^{+0.04} \)  \( 2.35_{0.14}^{+0.04}\) 
3  \( 3.28_{0.14}^{+0.03} \)  \( 3.30_{0.15}^{+0.04} \)  \( 3.32_{0.17}^{+0.04} \)  \( 3.34_{0.18}^{+0.05}\) 
4  \( 4.26_{0.18}^{+0.04} \)  \( 4.28_{0.19}^{+0.05} \)  \( 4.29_{0.21}^{+0.05} \)  \( 4.30_{0.23}^{+0.05}\) 
5  \( 5.24_{0.20}^{+0.04} \)  \( 5.25_{0.23}^{+0.04} \)  \( 5.27_{0.22}^{+0.05} \)  \( 5.27_{0.26}^{+0.05}\) 
6  \( 6.22_{0.21}^{+0.04} \)  \( 6.23_{0.23}^{+0.05} \)  \( 6.24_{0.25}^{+0.05} \)  \( 6.25_{0.25}^{+0.05}\) 
7  \( 7.20_{0.21}^{+0.04} \)  \( 7.21_{0.23}^{+0.05} \)  \( 7.22_{0.24}^{+0.05} \)  \( 7.22_{0.27}^{+0.05}\) 
8  \( 8.18_{0.21}^{+0.04} \)  \( 8.19_{0.22}^{+0.04} \)  \( 8.20_{0.23}^{+0.05} \)  \( 8.20_{0.26}^{+0.05}\) 
9  \( 9.16_{0.20}^{+0.03} \)  \( 9.17_{0.21}^{+0.04} \)  \( 9.18_{0.21}^{+0.05} \)  \( 9.18_{0.24}^{+0.04}\) 
10  \( 10.14_{0.20}^{+0.03} \)  \( 10.15_{0.21}^{+0.03} \)  \( 10.16_{0.21}^{+0.04} \)  \( 10.16_{0.23}^{+0.04}\) 
\(P_{\mathrm{c}}\) (GV)  700 g cm^{−2}  800 g cm^{−2}  900 g cm^{−2}  1000 g cm^{−2} 

0  \(0.1(4.2_{0.9}^{+0.6})\)  \(0.1(9.7_{2.1}^{+1.5})\)  \(2.3_{0.5}^{+0.4}\)  \(5.4_{1.4}^{+1.1}\) 
1  \(0.1(3.7_{0.7}^{+0.5})\)  \(0.1(9.0_{1.9}^{+1.3})\)  \(2.1_{0.5}^{+0.3}\)  \(4.9_{1.2}^{+0.9}\) 
2  \(0.1(1.1_{0.2}^{+0.1})\)  \(0.1(2.5_{0.4}^{+0.2})\)  \(0.1(5.6_{1.1}^{+0.6})\)  \(1.4_{0.3}^{+0.2}\) 
3  \(10^{2}(4.7_{0.8}^{+0.4})\)  \(0.1(1.0_{0.2}^{+0.1})\)  \(0.1(2.3_{0.4}^{+0.2})\)  \(0.1(5.2_{1.1}^{+0.5})\) 
4  \(10^{2}(2.7_{0.4}^{+0.2})\)  \(10^{2}(5.7_{1.0}^{+0.5})\)  \(0.1(1.3_{0.2}^{+0.1})\)  \(0.1(2.8_{0.5}^{+0.3})\) 
5  \(10^{2}(1.8_{0.3}^{+0.1})\)  \(10^{2}(3.7_{0.6}^{+0.3})\)  \(10^{2}(8.0_{1.3}^{+0.7})\)  \(0.1(1.7_{0.3}^{+0.2})\) 
6  \(10^{2}(1.3_{0.2}^{+0.1})\)  \(10^{2}(2.7_{0.4}^{+0.2})\)  \(10^{2}(5.7_{0.9}^{+0.4})\)  \(0.1(1.2_{0.2}^{+0.1})\) 
7  \(10^{2}(1.0_{0.1}^{+0.1})\)  \(10^{2}(2.1_{0.3}^{+0.1})\)  \(10^{2}(4.3_{0.6}^{+0.3})\)  \(10^{2}(9.4_{1.4}^{+0.7})\) 
8  \(10^{3}(8.5_{0.9}^{+0.4})\)  \(10^{2}(1.7_{0.2}^{+0.1})\)  \(10^{2}(3.5_{0.4}^{+0.2})\)  \(10^{2}(7.5_{0.9}^{+0.5})\) 
9  \(10^{3}(7.4_{0.7}^{+0.3})\)  \(10^{2}(1.5_{0.1}^{+0.1})\)  \(10^{2}(3.0_{0.3}^{+0.2})\)  \(10^{2}(6.3_{0.7}^{+0.3})\) 
10  \(10^{3}(6.5_{0.4}^{+0.3})\)  \(10^{2}(1.3_{0.1}^{+0.1})\)  \(10^{2}(2.6_{0.2}^{+0.1})\)  \(10^{2}(5.5_{0.5}^{+0.3})\) 
It is important that the effective rigidity and scaling factor are defined robustly for a wide range of the geomagnetic rigidity cutoffs, and they are independent of the exact SEP spectrum, in a reasonable range of parameters. Thus, for each GLE and each NM, one can estimate, using Equation 6, the integral fluence \(F(>R_{ \mathrm{eff}})\) of SEPs, and a set of such NMs with different values of \(R_{\mathrm{eff}}\) makes it possible to perform a nonparametric reconstruction (i.e., one without an explicit assumption on the spectral shape) of the event’s integral spectrum.
3 Test of the EffectiveRigidity Method
Results of the GLE #69 analysis: NM name (fourletter abbreviations are given according to the list at http://gle.oulu.fi ); Geomagnetic cutoff rigidity \(P_{\mathrm{c}}\); Atmospheric depth; Effective rigidity \(R_{\mathrm{eff}}\); Scaling factor \(K_{\mathrm{eff}}\) (for the standard 1NM64 counter); GLE integral increase \(X\); Estimated SEP integral fluence \(F_{\mathrm{SEP}} (>R_{\mathrm{eff}})\). The values correspond to the Mi13 yield function.
NM name  \(P_{\mathrm{c}}\) [GV]  Depth [g cm^{−2}]  \(R_{\mathrm{eff}}\) [GV]  K [cm^{2} count]^{−1}  X [% hr]  \(F_{\mathrm{SEP}} (>R_{\mathrm{eff}})\) [protons cm^{−2}] 

SOPO  0.1  701.0  \(1.31^{+0.03}_{0.07}\)  \((4.25^{+0.57}_{0.87})\cdot 10^{1}\)  1287  \((1.77^{+0.25}_{0.34} )\cdot 10^{6}\) 
SNAE  0.69  896.8  \(1.38^{+0.04}_{0.09}\)  \(2.23^{+0.40}_{0.52}\)  385.1  \((7.67^{+1.35}_{1.82} )\cdot 10^{5}\) 
BRBG  0.01  983.4  \(1.42^{+0.04}_{0.11}\)  \(4.72^{+0.91}_{1.23}\)  295.6  \((6.85^{+1.31}_{1.80} )\cdot 10^{5}\) 
APTY  0.65  996.7  \(1.43^{+0.05}_{0.11}\)  \(5.23^{+1.03}_{1.36}\)  362  \((8.48^{+1.67}_{2.24} )\cdot 10^{5}\) 
MCMD  0.3  1004.2  \(1.43^{+0.05}_{0.11}\)  \(5.67^{+1.11}_{1.52}\)  780.7  \((1.89^{+0.37}_{0.51} )\cdot 10^{6}\) 
MWSN  0.22  1005.1  \(1.43^{+0.05}_{0.11}\)  \(5.73^{+1.12}_{1.54}\)  364.1  \((8.85^{+1.72}_{2.41} )\cdot 10^{5}\) 
TERA  0.01  1005.5  \(1.43^{+0.05}_{0.11}\)  \(5.76^{+1.13}_{1.55}\)  885.1  \((2.15^{+0.42}_{0.59} )\cdot 10^{6}\) 
FSMT  0.3  1015.4  \(1.44^{+0.05}_{0.11}\)  \(6.14^{+1.23}_{1.64}\)  363.5  \((8.83^{+1.77}_{2.39} )\cdot 10^{5}\) 
CALG  1.08  900.1  \(1.45^{+0.04}_{0.09}\)  \(5.76^{+1.14}_{1.52}\)  355.6  \((7.57^{+1.19}_{1.66} )\cdot 10^{5}\) 
NAIN  0.3  1027.2  \(1.45^{+0.05}_{0.11}\)  \(1.90^{+0.30}_{0.41}\)  457.3  \((9.63^{+1.97}_{2.60} )\cdot 10^{5}\) 
NRLK  0.63  1035.0  \(1.45^{+0.05}_{0.11}\)  \(7.81^{+1.58}_{2.21}\)  284.4  \((5.63^{+1.14}_{1.57} )\cdot 10^{5}\) 
OULU  0.81  1009.2  \(1.45^{+0.05}_{0.1}\)  \(7.62^{+1.54}_{2.13}\)  305.8  \((8.12^{+1.62}_{2.11} )\cdot 10^{5}\) 
THUL  0.3  1031.1  \(1.45^{+0.05}_{0.11}\)  \(6.69^{+1.38}_{1.80}\)  392.6  \((6.04^{+1.23}_{1.66} )\cdot 10^{5}\) 
CAPS  0.35  1041.1  \(1.46^{+0.05}_{0.11}\)  \(7.29^{+1.48}_{2.01}\)  220.9  \((7.15^{+1.49}_{1.95} )\cdot 10^{5}\) 
INVK  0.3  1038.8  \(1.46^{+0.05}_{0.11}\)  \(6.98^{+1.43}_{1.89}\)  241.6  \((7.59^{+1.59}_{2.08} )\cdot 10^{5}\) 
TXBY  0.48  1039.7  \(1.46^{+0.05}_{0.11}\)  \(7.69^{+1.55}_{2.16}\)  206.3  \((5.14^{+1.08}_{1.41} )\cdot 10^{5}\) 
KERG  1.14  1019.4  \(1.56^{+0.04}_{0.1}\)  \(4.98^{+0.88}_{1.21}\)  309.5  \((5.50^{+0.95}_{1.32} )\cdot 10^{5}\) 
YKTK  1.65  1040.5  \(1.97^{+0.04}_{0.12}\)  \(3.01^{+0.39}_{0.70}\)  179.9  \((1.68^{+0.22}_{0.38} )\cdot 10^{5}\) 
KGSN  1.88  1019.8  \(2.16^{+0.04}_{0.13}\)  \(1.78^{+0.22}_{0.39}\)  146.4  \((9.98^{+1.21}_{2.20} )\cdot 10^{4}\) 
MGDN  2.1  1010.9  \(2.45^{+0.04}_{0.15}\)  \(1.29^{+0.15}_{0.28}\)  122.1  \((5.88^{+0.64}_{1.32} )\cdot 10^{4}\) 
KIEL  2.36  1000.4  \(2.65^{+0.04}_{0.16}\)  \((8.74^{+0.96}_{1.87})\cdot 10^{1}\)  103.6  \((3.92^{+0.43}_{0.84} )\cdot 10^{4}\) 
NWRK  2.4  1028.1  \(2.66^{+0.04}_{0.16}\)  \(1.09^{+0.12}_{0.24}\)  124.3  \((4.89^{+0.56}_{1.06} )\cdot 10^{4}\) 
MOSC  2.43  1011.1  \(2.75^{+0.04}_{0.16}\)  \((9.15^{+0.99}_{1.99})\cdot 10^{1}\)  103.2  \((3.62^{+0.39}_{0.78} )\cdot 10^{4}\) 
LARC  2.72  999.0  \(3.04^{+0.04}_{0.18}\)  \((6.36^{+0.65}_{1.36})\cdot 10^{1}\)  59.1  \((1.57^{+0.16}_{0.33} )\cdot 10^{4}\) 
CLMX  3  692.2  \(3.18^{+0.03}_{0.14}\)  \((4.36^{+0.34}_{0.70})\cdot 10^{2}\)  80.6  \((1.22^{+0.10}_{0.20} )\cdot 10^{4}\) 
NVBK  2.91  1018.6  \(3.24^{+0.05}_{0.19}\)  \((6.34^{+0.66}_{1.34})\cdot 10^{1}\)  52.2  \((1.21^{+0.13}_{0.25} )\cdot 10^{4}\) 
IRK2  3.64  813.1  \(3.89^{+0.04}_{0.18}\)  \((7.65^{+0.66}_{1.34})\cdot 10^{2}\)  26.2  \((2.96^{+0.28}_{0.53} )\cdot 10^{3}\) 
LMKS  3.84  736.7  \(4.07^{+0.04}_{0.18}\)  \((3.83^{+0.30}_{0.64})\cdot 10^{2}\)  23.1  \((2.17^{+0.19}_{0.37} )\cdot 10^{3}\) 
JUN1  4.49  655.0  \(4.65^{+0.04}_{0.17}\)  \((1.52^{+0.11}_{0.22})\cdot 10^{2}\)  21.4  \((1.36^{+0.12}_{0.20} )\cdot 10^{3}\) 
JUNG  4.49  655.0  \(4.65^{+0.04}_{0.17}\)  \((1.52^{+0.11}_{0.22})\cdot 10^{2}\)  19.3  \((1.23^{+0.11}_{0.19} )\cdot 10^{3}\) 
HRMS  4.58  1032.7  \(4.79^{+0.05}_{0.24}\)  \((2.69^{+0.25}_{0.50})\cdot 10^{1}\)  7.8  \((7.04^{+1.21}_{1.68} )\cdot 10^{2}\) 
BKSN  5.7  834.4  \(5.94^{+0.05}_{0.24}\)  \((3.84^{+0.29}_{0.60})\cdot 10^{2}\)  4.7  \((2.10^{+0.49}_{0.56} )\cdot 10^{2}\) 
ROME  6.27  1028.5  \(6.44^{+0.05}_{0.27}\)  \((1.40^{+0.11}_{0.22})\cdot 10^{1}\)  1.7  \((7.72^{+5.23}_{5.34} )\cdot 10\) 
ERV3  6.94  697.1  \(7.1^{+0.04}_{0.21}\)  \((1.02^{+0.06}_{0.12})\cdot 10^{2}\)  1.5  \((4.14^{+2.82}_{2.85} )\cdot 10\) 
ERVN  6.94  813.8  \(7.11^{+0.04}_{0.24}\)  \((2.35^{+0.15}_{0.31})\cdot 10^{2}\)  0.7  \((2.10^{+3.11}_{3.12} )\cdot 10\) 
3.1 GLE #69, 20 Jan. 2005
As the first test of the method we considered GLE #69, which occurred on 20 January 2005 and was the strongest event over the last cycles and the second strongest ever directly observed. It was an impulsive event with the high peak (\({\approx} \,3350\) and 4800% for the SOPB and SOPO NMs, respectively) and was characterized by a very strong anisotropy of its initial phase (e.g., Plainaki et al., 2007; Matthiä et al., 2009). Since the effects of anisotropy and temporal evolution of fluxes cannot be considered by the effectiverigidity approach, we can only provide a rough estimate, while the exact spectral reconstruction requires a laborious and complicated method with particle tracing (e.g., Plainaki et al., 2007; Mishev and Usoskin, 2016).
For the comparison we use four NM yield functions as mentioned before. The corresponding values of the \(R_{\mathrm{eff}}\) and \(K_{ \mathrm{eff}}\) were computed for all the YFs, but shown only for Mi13 in Table 3.
First, the overall shape of the spectrum, reconstructed here, is robust for different YFs and disagrees with the Bandfunction shape (note that in this energy/rigidity range the Band function is close to a simple power law with the index \(\gamma _{2}\), following the notations of R18): while all spectra merge at the highrigidity tail of 6 – 7 GV and lowrigidity head (\({<}\,1\) GV), they have an essential excess in the midrigidity range of several GV. We note that the wide spread of lowrigidity points is caused by the anisotropic impulsive phase observed by polar NMs. This suggests that the single powerlaw tail of the Bandfunction shape is not well representative of such spectrum. Despite the discrepancy in the absolute calibration of the reconstructed spectrum, all YFbased calculations agree in that the spectrum rolls off at high rigidities, above 5 GV, with respect to a purely powerlaw highenergy tail of the Band function. In order to exclude a possibility that the rolloff is an artifact related to the ‘training’ of the method on the modified powerlaw spectral shape (Equation 7), we have tested the method also for the pure power law (\(\delta \gamma =0\)) and found that it does not have any significant effect on the values of \(R_{\mathrm{eff}}\) and \(K_{\mathrm{eff}}\). Accordingly, the rolloff of the spectrum is realistic and should be taken into account either using the modified power law (Equation 7) or the Ellison–Ramaty spectra shape (Ellison and Ramaty, 1985), which is a power law over energy with an exponential cutoff.
Second, the midrigidity excess ranges for different yield functions between 1.5 – 2 for CD00 and Ma16 YFs to 2.5 – 4 for CM12 and Mi13 YFs. Overall, the fluence reconstructed here with the use of the CD00 and Ma16 YF is comparable to that of R18 which was also based on the CD00 YF. In the case of Mi13 and CM12, the reconstructed fluence above a few GV rigidity is a factor of 2 – 3 higher than that proposed by R18. Considering that CM12 and Mi13 YFs better represent the lowenergy part (Koldobskiy et al., 2019), particularly CM12 one which was empirically constructed from the latitudinal survey data, these estimates look more realistic.
3.2 GLE #71: 17 May 2012
4 Summary
Here we presented a new fast method for assessment of the highrigidity part (above 1 GV) of the spectral fluence of SEPs for GLE events based on the data from the worldwide NM network. The method is based on the effective rigidity \(R_{\mathrm{eff}}\) and scaling factor \(K_{ \mathrm{eff}}\), calculated for each NM, where the SEP integral fluence is directly related to the NM response to the event. This method is nonparametric so that it provides an estimate of the spectrum without any explicit assumption of the spectral shape. This is an important difference to most of the other methods which fit a prescribed spectral shape into the data by finding the bestfit parameters for that shape, often without a proper validation. This method is simple and fast avoiding laborious computations needed for the full reconstruction, but it neglects the possible anisotropy of the impulsive phase of the event.
We tested the method for two recent GLE events, #69 (20 Jan. 2005), which was the second strongest observed one and had a very anisotropic SEP distribution during the impulsive phase of the event, and #71 (17 May 2012), which was a moderate, but also strongly anisotropic event, whose spectrum was directly measured by PAMELA instrument. For both events we found a good qualitative agreement between the spectral shape of the reconstructed here, on one hand, and that for fully reconstructed spectra, on the other hand. By confronting the spectra assessed here with those obtained via a full NMbased reconstruction, we estimated the uncertainty related to the anisotropy as being within 10%.
Next, we compared our present reconstructions with the directly measured spectrum for GLE #71. Only the reconstructions based on Mi13 and CM12 YFs appear quantitatively consistent with the measurements, while the results based on CD00 and Ma16 lead to an underestimate of the spectrum by a factor 2 – 3 in the lowerrigidity range. This is most probably related to the fact that the latter two YFs tend to overestimate the NM response to lowerenergy particles, as was found by Koldobskiy et al. (2019) from an analysis of GCR spectra measured directly by AMS02 and PAMELA experiments. In particular, the CD00 YF is known to fail reproducing the observed NM latitudinal surveys (Clem and Dorman, 2000; CaballeroLopez and Moraal, 2012; Mishev, Usoskin, and Kovaltsov, 2013) implying that it overestimates the sensitivity of a NM to the lowerenergy particles. Since the model by R18 is based on the CD00 YF, it may underestimate the SEP fluence by a factor of two to three. It is also shown that the powerlaw approximation of the highenergy tail, used in particular in the Bandfunction parameterization, does not properly describe the form of the GLE spectrum in the NMenergy range. Therefore, the earlier estimates of GLE integral fluences need to be revised. A proper reconstruction of the SEP integral fluence for the known GLE events is planned for a forthcoming work.
Notes
Acknowledgements
Open access funding provided by University of Oulu including Oulu University Hospital. Data of GLE recorded by NMs were obtained from the International GLE database http://gle.oulu.fi and NMDB database ( www.nmdb.eu , founded under the European Union’s FP7 programme, contract no. 213007). PIs and teams of all the groundbased neutron monitors whose data were used here, are gratefully acknowledged. This work was partially supported by the ReSoLVE Centre of Excellence (project no. 307411) and HEAIM Project (no. 314982 and no. 316223) of Academy of Finland, by the grant no. MK6160.2018.2 of the President of the Russian Federation and MEPhI Academic Excellence Project (contract 02.a03.21.0005).
Disclosure of Potential Conflicts of Interest
The authors declare that they have no conflicts of interest.
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