# Results from The Latin American Giant Observatory Space Weather Simulation
Chain

###### Abstract

The Space Weather program of the Latin American Giant Observatory (LAGO)
Collaboration was designed to study the variation of the flux of atmospheric
secondary particles at ground level produced during the interaction of cosmic
rays with the air. This work complements and expands the inference
capabilities of the LAGO detection network to identify the influence of solar
activity on the particle flux, at places having different geomagnetic rigidity
cut-offs and atmospheric depths. This program is developed through a series of
Monte Carlo sequential simulations to compute the intensity spectrum of the
various components of the radiation field on the ground. A key feature of
these calculations is that we performed detailed radiation transport
computations as a function of incident direction, time, altitude, as well as
latitude and longitude. Magnetic rigidity calculations and corrections for
geomagnetic field activity are established by using the MAGNETOCOSMICS code,
and the estimation of the flux of secondaries at ground level is implemented
by using the CORSIKA code; thus we can examine the local peculiarities in the
penumbral regions with a more realistic description of the atmospheric and
geomagnetic response in these complex regions of the rigidity space. As an
example of our calculation scheme, we report some result on the flux at ground
level for two LAGO locations: Bucaramanga-Colombia and San Carlos de
Bariloche-Argentina, for the geomagnetically active period of May 2005\.

Laboratorio Detección de Partículas y Radiación (CNEA,CONICET,UNCUYO), San
Carlos de Bariloche, Argentina.

Instituto de Tecnologías en Detección y Astropartículas (CNEA,CONICET,UNSAM),
Buenos Aires, Argentina.

Escuela de Física, Universidad Industrial de Santander, 680002 Bucaramanga,
Colombia.

Departamento de Física, Universidad de Los Andes 5101 Mérida, Venezuela.

M. Suárez-Durán<EMAIL_ADDRESS>
An extended rigidity cutoff is calculated under quiet (secular) & transient
conditions for the geomagnetic field.

Penumbra region is reinterpreted as a probability function of the arrival
direction.

The effect of the geomagnetic field in the flux of Galactic Cosmic Rays can be
estimated from changes on the flux of secondary particles at ground level.

##  1 Introduction

Solar activity has a strong influence on the modulation of the flux of
galactic cosmic rays (GCRs) arriving at Earth, whose transport through the
heliosphere is one of the topics of major interest in space physics,
presenting several unresolved questions. Space weather physics is experiencing
a fast growing interest nowadays because of evidence that environmental
conditions in the near-Earth space have direct and indirect impacts on
technology and the global economy (Schrijver et al., 2015).

One of most puzzling modulation of the cosmic rays flux at ground level is
called Forbush Decrease (FD): a rapid reduction in the observed galactic
cosmic ray intensity followed by a slow exponential-like recovery (Usoskin et
al., 2008). This phenomenon initially was reported by S.E. Forbush –and almost
simultaneously by V.F. Hess & A. Demmelmair– in 1937 (see e.g. Forbush, 1937;
Hess and Demmelmair, 1937). Later, in the 1950’s, the works of Simpson, Fonger
& Treiman (Simpson et al., 1953), and Singer (Singer, 1954), showed that FDs
are related to the solar activity interacting with the interplanetary medium.
More recently, Lockwood showed a dependence of the magnitude of the FD upon
the vertical cutoff rigidity (Lockwood, 1971).

FDs can be classified in two groups: recurrent and non-recurrent FD. While the
first group (Lockwood, 1971) have a symmetric profile and are well associated
with co-rotating high speed solar wind streams (Cane, 2000), non-recurrents FD
have sudden onsets with a maximum depression within a day, and a more gradual
recovery (Cane, 2000; Belov et al., 2014) , and are related to the interaction
of GCRs, Interplanetary Coronal Mass Ejections (ICME) and a perturbed
geomagnetic field (it is perturbed by its interaction with the same ICME).

Understanding these very complex phenomena depends upon: in situ measurements
of the interaction of GCRs-ICME; tracking the propagation of GCRs through the
Geomagnetic Field (GF hereafter); taking into account their interaction with
the atmosphere, and also the variation of the particles produced by the latter
interaction (hereafter secondaries) at ground level.

As the secondaries are produced by the interaction of GCRs with the atmosphere
(hereafter denoted as primaries), the modulation of secondary particles needs
to be monitored and carefully corrected by taking into account several
atmospheric factors that could modify the flux of secondaries at Earth’s
surface (the Atmospheric density profile is one of those factors because this
is proportional to the absorption of secondaries particles) (The Pierre Auger
Collaboration, 2011; Dasso et al., 2012; H. Asorey for the LAGO Collaboration,
2013; The Pierre Auger Collaboration, 2015; H. Asorey and S. Dasso and L. A.
Núñez and Y. Pérez and C. Sarmiento-Cano and M. Suárez-Durán for the LAGO
Collaboration, 2015). Furthermore, a series of completed and detailed
simulations are needed to characterize the expected flux at the detector
altitude. This kind of simulation must take into account several important
factors such as GF conditions, i.e., estimations of the magnetic rigidity of
the GCRs, the interaction of GCRs with atmosphere, the variations in
atmospheric depth and the detector response.

Direct solar wind observations using spacecraft can provide insight of the
interplanetary magnetic field; but its global structure can not be completely
monitored by on-board measurements, because they can only detect local
variations along the trajectory of the probe within the solar wind. On the
other hand, ground observatories with detectors spread on very large areas,
registering indirectly low energy GCRs, can provide important, alternative and
complementary information about the broader structure of the interplanetary
magnetic fields and its influence on the GCRs flux. In this context, a global
network of both type of observatories combines different techniques to monitor
the development of FDs at different geomagnetic latitudes and rigidity
cutoffs, will enrich future studies on the detection of solar modulation of
GCRs (see e.g. Abbasi et al., 2008; The Pierre Auger Collaboration, 2015).

The Latin American Giant Observatory (LAGO) (Allard et al., 2008; Asorey et
al., 2015) has developed a program to understand the influence of the space
weather phenomena on the flux of GCRs. This program, called LAGO Space Weather
(LAGO-SW) (H. Asorey for the LAGO Collaboration, 2013), includes a precise
simulation that takes into account the geomagnetic corrections and a detailed
measurement of the modulation on the flux of secondaries, and evaluates if
this modulation have possible causal correlations with space weather
phenomena, like FDs (H. Asorey and S. Dasso and L. A. Núñez and Y. Pérez and
C. Sarmiento-Cano and M. Suárez-Durán for the LAGO Collaboration, 2015).

Nowadays, computational capabilities allow the extension of the usual
approach, which is to consider only the components of the GCR flux locally and
include geomagnetic effects by an effective rigidity cutoff for vertical
primaries (Masías-Meza and Dasso, 2014). The detailed simulations described in
this work, include these effects over different arrival directions during
dynamic events affecting the geomagnetic field and atmospheric conditions. We
have generalized previous attempts by including not only secular, but also
transient variations of the directional geomagnetic rigidity cutoff.

This paper is organized as follows: in section 2, the Latin American Giant
Observatory and its space weather program are briefly described; then, in
Section 3, we present our space weather simulation chain, focusing on
geomagnetic corrections of the primary flux; in Section 4 we discuss our main
results in secular conditions and later under geomagnetic disturbances for two
LAGO sites: Bucaramanga-Colombia and San Carlos de Bariloche-Argentina.
Finally, in section 5 some finals remarks and future projects are considered.

##  2 The LAGO Space Weather Program

The Latin American Giant Observatory (LAGO) is an extended astroparticle
observatory on a continental scale, promoting training and research in
astroparticle physics in Latin America covering three main areas: search for
the high energy component of gamma rays bursts (GRBs) at high altitude sites,
space weather phenomena, and background radiation at ground level (Asorey et
al., 2015).

The LAGO detection network consists of ground-level water-Cherenkov particle
detectors (WCDs), spanning over several sites, located at significantly
different latitudes and various altitudes –from Mexico to the Patagonia and
from mean sea level up to more than 5000 meters of altitude. After the
installation of new detectors at the Antarctica Peninsula (Dasso et al.,
2015), LAGO will cover a large range of geomagnetic rigidity cutoffs and
atmospheric absorption/depths (I. Sidelnik for the LAGO Collaboration, 2015).
The current/planned distribution and status of the LAGO detection network is
shown in Figure 1. This network of detectors is operated by the LAGO
Collaboration: a non-centralized and distributed collaborative network of more
than 80 scientists from institutions of te Latin American countries
(Argentina, Bolivia, Brazil, Chile, Colombia, Ecuador, Guatemala, Mexico, Peru
and Venezuela) and Spain. The LAGO Collaboration is using WCDs in all sites,
due to their proved reliability, high detection, low cost and efficiency of
the detection of all components present in atmospheric extensive showers
(Asorey et al., 2015; I. Sidelnik for the LAGO Collaboration, 2015; A. Galindo
for the LAGO Collaboration, 2015; Dasso et al., 2015).

Figure 1: Left panel indicates the geographical distribution and altitudes of
LAGO water Cherenkov detectors: operational are represented by blue triangles;
in deployment by orange squares and finally, red circles indicating planned
sites. The right panel shows the vertical rigidity cutoff at each LAGO site.

As explained in Asorey et al. (2015), the LAGO scientific and academic
objectives are organized in different programs. The Space Weather LAGO Program
is devised to study variations in the flux of secondary particles at ground
level and its relation to the heliospheric modulation of GCRs. The LAGO
detector network determines the flux of secondary particles in different bands
of deposited energy in the detector, by using pulse shape discrimination
techniques. This is what we have called the multi-spectral analysis technique
(H. Asorey and S. Dasso and L. A. Núñez and Y. Pérez and C. Sarmiento-Cano and
M. Suárez-Durán for the LAGO Collaboration, 2015). The total energy threshold
for the detection of secondary particles are $\simeq 0.4$ MeV for gammas,
$\simeq 0.8$ MeV for electrons and $\simeq 160$ MeV for muons.

By combining all the data measured at different locations, LAGO provides
simultaneous and detailed information of the temporal evolution of the
secondary flux at different geomagnetic locations. This can help to get a
better understanding of the small and large spacetime scales of the
disturbances produced by different space weather phenomena (H. Asorey and S.
Dasso and L. A. Núñez and Y. Pérez and C. Sarmiento-Cano and M. Suárez-Durán
for the LAGO Collaboration, 2015).

Any attempt to estimate the expected flux of secondaries at any detector of
the LAGO network should be based on a detailed simulation that takes into
account all possible sources of flux variation. This complex approach
comprises processes occurring at different spatial and time scales, following
this conceptual scheme:

GCR Flux | $\xrightarrow{\mathrm{Heliosphere}}$ | Modulated Flux | $\xrightarrow{\mathrm{Magnetosphere}}$ | Primaries $\rightarrow\cdots$
---|---|---|---|---
$\cdots\rightarrow$ Primaries | $\xrightarrow{\mathrm{Atmosphere}}$ | Secondaries | $\xrightarrow{\mathrm{Detector}}$ | Signals.

As it can be easily appreciated, the above simulation pipeline considers three
important factors with different spatial and time scales: the geomagnetic
effects, the development of the extensive air showers in the atmosphere, and
the detector response at ground level. According to the above scheme, in this
work we focused in stages covering from the modulated flux to the flux of
secondary particles at ground level.

Our Simulation Chain can be depicted in three main consecutive blocks:

  1. 1.

The effects of GF on the propagation of charged particles, contributing to the
background radiation at ground level, are characterized by the directional
rigidity cutoff, $R_{\mathrm{C}}$, at each LAGO site and calculated using the
MAGNETOCOSMICS code (Desorgher, 2004) applying the backtracking technique (see
e.g. Masías-Meza and Dasso, 2014). The Geomagnetic Field at any point on Earth
is determined by using the International Geomagnetic Field Reference, version
11 (International Association of Geomagnetism and Aeronomy, Working Group
V-MOD, 2010) at the near-Earth GF ($r<5R_{\oplus}$) –$r$ distance from Earth
center and $R_{\oplus}$ is the Earth radius ($6371$ km)– and through the
Tsyganenko Magnetic Field model version 2001 (TSY01 hereafter) (Tsyganenko,
2002) to describe the outer GF ($r>5R_{\oplus}$).

  2. 2.

The second step of the chain is based on the CORSIKA code (Heck et al., 1998).
Extensive air showers produced during the interaction of cosmic rays with the
atmosphere are simulated with extreme detail to obtain a very comprehensive
set of secondaries at ground level.

  3. 3.

Finally, a GEANT4 model (Agostinelli et al., 2003) of the detector response to
the different types of secondary particles is being implemented (Otiniano et
al., 2015; Vargas et al., 2015; Calderón et al., 2015) and will be reported in
the near future.

##  3 The space weather simulation chain

The propagation of charged particle through the GF has been studied since the
60s and was focused on understanding how the penumbra region changes with the
geographical position (see e.g. Shea et al., 1965; Carmichael et al., 1969;
Smart and Shea, 2012). In this section we shall discuss our novel approach to
understand the penumbra region and our proposal for a new method to calculate
the magnetic rigidity as a function of time. We shall also describe in detail
how geomagnetic effects on the low energy flux of primaries can be infered
from observations of secondary particles at ground level by means of the
following procedure:

  1. 1.

To find a magnetic rigidity function,
$R_{\mathrm{m}}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},t,\theta,\phi)$, at a
particular geographical position –i.e. latitude (Lat), longitude (Lon) and
altitude above sea level (Alt)–, time ($t$) and arrival direction of the
particle –i.e. zenithal ($\theta$) and azimuthal ($\phi$) angle;

  2. 2.

To calculate the flux of primaries at the top of the atmosphere ($\approx 112$
km a.s.l. (Above Sea Level)), filtered by the magnetic rigidity function
$R_{\mathrm{m}}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},t,\theta,\phi)$;

  3. 3.

To estimate the flux of secondaries at ground level produced by the
interactions of the impinging GCRs with the atmosphere.

The following subsections will develop all details for the above-mentioned
actions.

###  3.1 Magnetic rigidity as function of time

The direction of the velocity of a GCR ($\hat{I}=\vec{v}/|\vec{v}|$) changes
along the particle trajectory inside the dynamic GF,
$\mathbf{\vec{B}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},t)}$, according to the
equation

$\frac{d\hat{I}}{ds}=\frac{1}{R_{\mathrm{m}}}\left(\hat{I}\times\mathbf{\vec{B}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},t)}\right)\,,$
(1)

where $s$ is the path length along the particle trajectory and
$R_{\mathrm{m}}=pc/Ze$ the magnetic rigidity; with $p$ as the particle
momentum, $c$ the light speed, $Z$ the atomic number and $e$ the electric
charge of the electron. The variation of $\hat{I}$ is weighted by
$R_{\mathrm{m}}$ and therefore a GCR is able to arrive at some specific
geographical point –under some configuration of
$\mathbf{\vec{B}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},t)}$ associated with
the trajectory of the particle arriving to a particular position, i.e Latitude
($\mathrm{Lat}$), Longitude ($\mathrm{Lon}$), Altitude ($\mathrm{Alt}$)– if
its $R_{\mathrm{m}}$ has the right value. Thus, we can write $R_{\mathrm{m}}$
as

$R_{\mathrm{m}}\equiv
R_{\mathrm{m}}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},t)\,.$ (2)

Following standard definitions (see e.g. Cooke et al., 1991), particles with
allowed $R_{\mathrm{m}}$ will reach at certain geographical position, while
those with forbidden $R_{\mathrm{m}}$ will not. With these considerations,
three different ranges of $R_{\mathrm{m}}$ can be defined:

  * •

Forbidden range: a continuous range which goes from zero to the first allowed
value of $R_{\mathrm{m}}$, say $R_{\mathrm{L}}$;

  * •

Allowed range: cotaining all the rigidities above a certain $R_{\mathrm{m}}$
value, say $R_{\mathrm{U}}$, for which all the rigidities containted in this
range are allowed.

  * •

Penumbra range: the range ($R_{\mathrm{L}}<R_{\mathrm{m}}<R_{\mathrm{U}}$)
connecting the allowed and forbidden ranges.

The penumbra is characterized by a single, effective, rigidity value (Shea et
al., 1965; Smart and Shea, 2009), which is used to establish whether a GCR
arrives, or not, at the particular geographical point. This value is called
the rigidity cutoff ($R_{\mathrm{C}}$) and can be defined as

$R_{\mathrm{C}}=R_{\mathrm{U}}-\sum_{k=R_{\mathrm{L}}}^{R_{\mathrm{U}}}\Delta{R_{k}}^{\mathrm{allowed}}\,,$
(3)

where $\Delta{R_{k}}$ is the resolution of the $R_{\mathrm{m}}$ calculation.
Strictly speaking, $R_{\mathrm{U}}$ and $R_{\mathrm{L}}$ depend on time, the
arrival direction, the geographical position and the altitude; thus, we should
consider that, at a geographical point,

$R_{\mathrm{m}}=R_{\mathrm{m}}\left(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},t,\theta,\phi\right)\,,$
(4)

It is important to note that definition (3) has the implicit assumption that
all the particle trajectories have the same contribution in the penumbra
region, i.e., the flat GCR spectrum approximation, according to (Dorman et
al., 2008). In this approximation, the very complex problem of allowed
trajectories in the penumbra region is simply replaced by an effective cutoff,
only calculated for vertical primaries. We have refined this approximation by
considering the penumbra not as a sharp cutoff, but as a relatively smooth
transition between the forbidden and the allowed regions. In our approach, we
extend the concept of the effective rigidity cutoff assuming that it can be
approximated by a cumulative probability function (CDF).

The next subsection outlines the method we have implemented to calculate
$R_{\mathrm{m}}$ and to characterize the penumbra region as a cumulative
probability function (CPF).

####  3.1.1 Magnetic Rigidity Calculation

We performed the $R_{\mathrm{m}}$ calculation by the backtracking technique
(see e.g. Masías-Meza and Dasso, 2014), via the MAGNETCOSMICS (MAGCOS) code,
with a resolution of $0.01$ GV, considering two conditions: secular and
dynamic geomagnetic field effects. For secular conditions we used the
configuration of the geomagnetic field on $6$ UTC April 26th 2005, because at
this time the registered Disturbance Storm Time Index (Dst index hereafter,
https://www.ngdc.noaa.gov/stp/geomag/dst.html) was zero with a variability of
$0.79$ nT from 0 UTC of April 26th to 12 UTC of the same day
(http://wdc.kugi.kyoto-u.ac.jp/dst_final/200504/index.html). For the dynamic
GF contribution, we calculated the $R_{\mathrm{m}}$ according to the GF
configuration for each hour of May 2005, setting the parameters: Dynamic
pressure, Dst index, interplanetary magnetic fields components $B_{x}$ and
$B_{z}$, and Tsyganenko’s parameters for model TSY01: G1 and G2(Tsyganenko,
2002).

These parameters were taken from the Virtual Radiation Belt Observatory
(Weigel et al., 2009); $R_{\mathrm{m}}$ values were calculated for zenith
angles from $0^{\circ}$ to $90^{\circ}$, with $\Delta\theta=15^{\circ}$ and
azimuth angles (for each $\theta$) from $0^{\circ}$ to $360^{\circ}$ with
$\Delta\phi=15^{\circ}$, for both site.

####  3.1.2 Interpreting the Penumbra Region

Instead of the standard simplifying assumption for the penumbra region we
build a cumulative probability function (CPF), valid from $R_{\mathrm{L}}$ to
$R_{\mathrm{U}}$, which replaces the usual concept of $R_{\mathrm{C}}$,
(defined in equation (3)). We denote this CPF as $P(R_{\mathrm{m}}(\theta))$,
which represents the probabilty of the cosmic ray arriving at some
geographical position with zenith angle $\theta$, at time $t$, with
$R_{\mathrm{m}}$; we take into account the following considerations:

  * •

the backtracking technique performed by MAGCOS is a deterministic method,
which implies that it is not possible to calculate a statistical set of
$R_{\mathrm{m}}$ values for a specific arrival direction, i.e., pair of
($\theta$,$\phi$);

  * •

for each zenith angle we consider $24$ uniform ranges in azimuth with an
angular amplitude of $15^{\mathrm{o}}$ each one, i.e. for each zenith angle we
have $24$ different penumbra regions; and

  * •

for each $\theta$, the associated set of $R_{\mathrm{m}}$ in the $24$ penumbra
regions has a global minimal value ($R_{\mathrm{Lmin}}$ hereafter) and a
global maximum value ($R_{\mathrm{Umax}}$ hereafter).

Accordingly, it is possible to come up with a frequentist approach, assuming a
probability function defined as:

$\wp(R_{\mathrm{m}}(\theta))=\frac{\\#R_{\mathrm{m_{\mathrm{allowed}}}}(\theta)}{\\#R_{\mathrm{m_{\mathrm{tot-
allowed}}}}(\theta)}\,,$ (5)

where, for each $\theta$, we have averaged over the azimuth angle, $\phi$,
within each penumbral range; the fraction of the number ($\\#$) of allowed
$R_{\mathrm{m}}$ values ($R_{\mathrm{m_{allowed}}}(\theta)$) over the total
number of $R_{m}$ values calculated for the $\theta$’s set
($R_{\mathrm{m_{tot-allowed}}}(\theta)$).

Equation (5) implies that the domain interval for $R_{\mathrm{m_{tot-
allowed}}}$ is

$\mathcal{D}(R_{\mathrm{m}_{\mathrm{tot-
allowed}}})=\left\\{R_{\mathrm{m}}(\theta):\left(R_{\mathrm{m}}\geq
R_{\mathrm{Lmin}},\,R_{\mathrm{m}}\leq R_{\mathrm{Umax}}\right)\right\\}\,.$
(6)

Thus, from (5) and (6) we define the cumulative distribution function for a
GCR, arriving at the observation point with rigidity $R_{\mathrm{m}}(\theta)$
as

$P(R_{\mathrm{m}}(\theta))=\sum_{\mathcal{R}_{m}=\mathrm{R_{Lmin}}}^{\mathcal{R}_{m}=\mathrm{R_{Umax}}}\wp(\mathcal{R}_{m}(\theta))\,.$
(7)

Notice that equation (7) implies that a GCR with
$R_{\mathrm{m}}>R_{\mathrm{Umax}}$ has a probability of $1$ to arrive at the
observation point through the zenith angle $\theta$; meanwhile a GCR with
$R_{\mathrm{m}}<R_{\mathrm{Lmin}}$ has $0$ probability to arrive at the same
point with the same angle.

Currently, the usual $R_{\mathrm{C}}$ is interpreted as a unique value in the
penumbra region, which separates only two possibilities for an incoming
particle with a zenith angle $\theta$: arriving or not arriving. If a charged
particle has a $R_{\mathrm{m}}~{}>~{}R_{\mathrm{C}}$ then it is considered to
arrive at the geographical point, in the opposite case, if
$R_{\mathrm{m}}~{}<~{}R_{\mathrm{C}}$, it does not arrive. With our approach,
it is clear that a GCR, with $R_{\mathrm{m}}$ and zenith angle $\theta$, can
reach at the geographical point if $P(R_{\mathrm{m}}(\theta))~{}=~{}1$,
whereas with $P(R_{\mathrm{m}}(\theta))~{}=~{}0$ will not. But, if the
$R_{\mathrm{m}}$ belongs to the penumbra region, it does not meet any of the
above criteria because $P(R_{\mathrm{m}}(\theta))$ is between 0 and 1. To set
this value of $P(R_{\mathrm{m}}(\theta))$ in terms of arriving or not
arriving, i.e., 0 or 1, we implement a Metropolis Monte Carlo algorithm as
follows: for a $P(R_{\mathrm{m}}(\theta))$ value, different from $1$ and $0$,
we calculate a random number: $0~{}<~{}P_{\mathrm{temp}}~{}<~{}1$. Then,

  * •

if $P(R_{\mathrm{m}}(\theta))\geq P_{\mathrm{temp}}$, then
$P(R_{\mathrm{m}}(\theta))=1$; otherwise

  * •

if $P(R_{\mathrm{m}}(\theta))<P_{\mathrm{temp}}$, then
$P(R_{\mathrm{m}}(\theta))=0$.

Therefore, we interpret the rigidity cutoff $R_{\mathrm{C}}$ as function of
the cumulative distribution function, i.e.,

$R_{\mathrm{C}}=R_{\mathrm{C}}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},\theta,P(R_{\mathrm{m}}(\theta)))\,.$
(8)

Now, from the dynamic magnetic rigidity definition we perform the same type of
calculations but including the time ($t$) dependence, by evaluating equation
(4) for different conditions at particular moments.

After applying the same procedure, we obtained the dynamic rigidity cut-off of
a site, as

$R_{\mathrm{C}}=R_{\mathrm{C}}(\mathrm{Lat},\mathrm{Lon},\mathrm{Alt},\theta,t,P(R_{\mathrm{m}}(\theta),t))\,,$
(9)

where $P(R_{\mathrm{m}}(\theta),t)$ represents the cumulative distribution
function calculated under GF conditions at the moment $t$, i.e.

$P(R_{\mathrm{m}}(\theta))=\sum_{\mathcal{R}\mathrm{m}=\mathrm{R_{\mathrm{Lmin}}}}^{\mathcal{R}m=\mathrm{R_{\mathrm{Umax}}}}\wp(\mathcal{R}m(\theta),t)\,.$
(10)

At this point, we shall introduce three types of rigidity cut off labeled by
three different indexes i.e,
$R_{\mathrm{C}}\rightarrow\mathrm{R}_{\mathrm{C}(i)}\,$:

  * •

$i=0$: for the standard definition of rigidity cutoff, i.e. equation (3).

  * •

$i=1$: for rigidity cutoff –secular conditions, i.e. typical time scales
greater than one year– as a function of the cumulative distribution function,
i.e. equation (7) and (8).

  * •

$i=2$: for rigidity cutoff –dynamics conditions– as a function of the
cumulative distribution function for the UTC time-stamp, i.e. equation (9) and
(10).

With these two new types of rigidity cut-off, we shall redefine, in the next
sections, several physical parameters associated with
$\mathrm{R}_{\mathrm{C}(i)}$.

Figure 2: These two plots show the representation of the penumbra region. For
both, we have calculated –according with the method presented in section
3.1.1– the magnetic rigidity for a cosmic ray arriving at Bucaramanga,
Colombia with a zenith angle of $15^{\circ}$, azimuth angle of $300^{\circ}$
and $t$ corresponding to May $13^{\mathrm{th}}$, $2005$. In the left plot, we
show the standard representation of the penumbra region, i.e. equation (3),
where the violet bars represent the intervals for allowed rigidities. The
right plot displays our interpretation of the penumbra region averaged for all
the azimuth angles, i.e., equations (5) and (7): the cosmic ray probability of
reaching Bucaramanga, Colombia with $\theta=15^{\circ}$, on May
$13^{\mathrm{th}}$ of 2005, versus magnetic rigidity $R_{\mathrm{m}}$.

In the Figure 2, we show an example of the refinement for the estimation of
the magnetic rigidity at Bucaramanga-Colombia, on May 13th 2005\. In the left
plot, we display the results of the standard method to calculate the rigidity
cutoff (equation (3)). It is clear that even in those “not allowed zones”
(between $10.2$ GeV and $11.2$ GeV) there are several trajectories that could
contribute to the flux at the observation point. This could be particularly
important when it is needed to determine the background flux at high altitude
sites, such as the LAGO site of Mount Chacaltaya at $5250$ m a.s.l., or even
for the determination of the expected flux of secondaries impacting aircrafts
(Pinilla et al., 2015; Asorey et al., 2017). In the same Figure 2 (right
plot), we illustrate our new method, displaying $P(R_{\mathrm{m}}(\theta))$
for different magnetic rigidities, considering a primary with
$\theta=15^{\circ}$.

With our method in determining the local directional rigidity cutoff, it is
possible to refine the calculation of the flux of particles at any observation
point while taking into account GF disturbances, either in long time scales
(secular conditions) on during short term transient phenomena.

###  3.2 Estimation of the Primary Flux filtered by
$\mathrm{R}_{\mathrm{C}(i)}$

The second step in our simulation chain is to estimate GCR flux arriving at
some geographical point ($112$ km a.s.l., $\mathrm{Lat},\mathrm{Lon}$) in the
area $\mathrm{d}S$, during time $\mathrm{d}t$, in the solid angle –$\theta$ is
the zenith angle– $\mathrm{d}\Omega=2\pi\sin(\theta)\mathrm{d}\theta$, within
the energy interval $\mathrm{d}E$ and with minimum allowed primary momentum of

$p_{\mathrm{min}}=\frac{Ze}{c}\mathrm{R}_{\mathrm{C}(i)}\,,$ (11)

with $Z$ the atomic number. Equation (11) allows us to filter primaries with
insufficient $R_{\mathrm{m}}$ to arrive at the point
($\mathrm{Lat},\mathrm{Lon},\mathrm{Alt}$).

We estimated the GCR flux, $\Phi$, at an altitude of $112$ km a.s.l., in
accordance with the Linsley atmospheric model (NOAA, 1976); e.g. at this
altitude the mass overburden vanishes (Heck et al., 1998), and we approximate
$\Phi$ by a simple power law of the form:

$\Phi(E,Z,A,\Omega)=\frac{\mathrm{d}N(E)}{\mathrm{d}S\,\mathrm{d}\Omega\,\mathrm{d}t\,\mathrm{d}E}\simeq
j_{0}(Z,A)\left(\frac{E}{E_{0}}\right)^{\alpha(E,Z,A)}\,,$ (12)

where the spectral index ($\alpha(E,Z,A)$) can be considered constant with
respect the energy, i.e. $\alpha(E,Z,A)~{}\approx~{}\alpha(Z,A)$, from
$10^{11}$ eV to $10^{15}$ eV (Letessier-Selvon and Stanev, 2011) and $E_{0}$
has a value of $10^{12}$ eV. For each type of GCR considered, $\alpha(Z,A)$,
is individualized by its mass number (A) and atomic number (Z). Finally,
$j_{0}(Z,A)$ is the normalization parameter. Both, the spectral indices and
$j_{0}$, have been obtained from the compilation produced by (Wiebel-Sooth et
al., 1998).

We calculated $\Phi$ using the fact that multiple observations have confirmed
that at low energies ($E\lesssim 5.5\times 10^{19}$ eV) the GCR flux can be
considered isotropic (see e.g. Abraham et al., 2007) and, in this case,
equation (12) is integrated to obtain the expected number of primaries for
every nuclei $(Z,A)$ as:

$N(Z,A,\theta)=\left(\pi\Delta S\Delta
t\sin^{2}(\theta)\right)j_{0}(Z,A)\frac{\left(E/E_{0}\right)^{\alpha(Z,A)+1}}{\alpha(Z,A)+1}\Big{|}_{E_{\min}}^{E_{\max}}\,,$
(13)

with $E_{\max}-E_{\min}\equiv\Delta E$ as the energy gap, which, in our case,
varies from a few GeV ($E_{\min}$) up to $10^{6}$ GeV ($E_{\max}$) (Asorey, H.
for The Pierre Auger Collaboration, 2011). It is clear that the first factor
depends only on the zenith angle $\theta$, and so,
$\mathcal{N}(\theta)\equiv\sin^{2}(\theta)\pi\Delta S\Delta t$. Thus, equation
(13) can be expressed as

$N(Z,A,\theta)=\mathcal{N}(\theta)j_{0}(Z,A)\frac{\left(E/E_{0}\right)^{\alpha^{\prime}(Z,A)}}{\alpha^{\prime}(Z,A)}\Big{|}_{E_{\min}}^{E_{\max}}\,,$
(14)

where $\alpha^{\prime}(Z,A)=\alpha(Z,A)+1$. For the calculation of
$N(Z,A,\theta)$ we have used: $\Delta S~{}=~{}1$ m2, $\Delta t=86400$ s, i.e.,
at least one day of the primary flux per square meter for each primary in the
range $1\leq Z\leq 26$, for $\theta$ from $0^{\circ}$ to $90^{\circ}$. The
$N(Z,A,\theta)$ will then be filtered via $E_{\mathrm{\min}}$ according to
equation (11).

This means that we can identify three different kinds of primary fluxes, one
per each different GF conditions ${R_{\mathrm{C}}(i)}$:

  * •

$\Phi_{(0)}$ for $R_{\mathrm{C}(0)}$, i.e. $p_{\mathrm{\min}}c=Ze\times
R_{\mathrm{C}(0)}$.

  * •

$\Phi_{(1)}$ for $R_{\mathrm{C}(1)}$, i.e. $p_{\mathrm{\min}}c=Ze\times
R_{\mathrm{C}(1)}$.

  * •

$\Phi_{(2)}$ for $R_{\mathrm{C}(2)}$, i.e. $p_{\mathrm{\min}}c=Ze\times
R_{\mathrm{C}(2)}$.

Thus, the number of particles given by (14) will be susceptible to
$R_{\mathrm{m}}$ corrections by the modification of the local rigidity cutoff,
and it can be re-written as

$N_{(i)}=\mathcal{N}(\theta)j_{0}(Z,A)\frac{\left(E/E_{0}\right)^{\alpha^{\prime}(Z,A)}}{\alpha^{\prime}(Z,A)}$
(15)

Here, the subindex $i$ of any quantity denotes the type of effect included.

As value of $E_{\max}$ we have used $10^{6}$ GeV, because at these energies
the flux is so low that it can not affect the distribution of the secondary
background at ground level. It is important to stress that, for a given point,
$E_{\min(i)}$ depends on the primary $Z$, theta arrival direction $(\theta)$
and time, i.e., $E_{\min(i)}\equiv E_{\min(i)}(Z,\theta,t)$.

When GF corrections are calculated with our method, the new expected primary
flux $\Phi_{(1)}$ obtained from equation (15) will depend on the arrival
direction of each primary (Dorman et al., 2008).

###  3.3 Estimation of flux of secondary particles at ground level corrected
by the Geomagnetic Field

The following step of the simulation chain is the correction for the effect of
the Geomagnetic Field on the flux of secondary particles at ground level. As
was mentioned in section 2, one of the main objectives of this simulation
chain is to calculate the expected flux of secondaries at the detector level
at any site of the LAGO network. Once the primary flux $\Phi_{(i)}$ is
calculated, the second step is to determine the interactions of those
primaries with the atmosphere. This simulation step is performed with the
CORSIKA code (Heck et al., 1998) (Currently, CORSIKA v7.3500, compiled using
the following options: QGSJET-II-04 (Ostapchenko, 2011); GHEISHA-2002; EGS4;
curved and external atmosphere, and volumetric detector). The local
geomagnetic field values $B_{x}$ and $B_{z}$ needed by CORSIKA to account for
GF effects during particles propagation in the atmosphere are obtained from
the IGRF-11 model. Secondary particles are tracked to the lowest energy
threshold allowed by CORSIKA for each type (currently, $E_{s}\geq 5\times
10^{-2}$ GeV for $\mu^{\pm}$ and hadrons (excluding $\pi^{0}$), and $E_{s}\geq
5\times 10^{-5}$ GeV for $e^{\pm},\pi^{0}$ and $\gamma$) to get the most
comprehensive distribution of secondaries at each site. In this work, the
atmosphere at each site was simulated by using profiles of the applicable
MODTRAN atmospheric model (Kneizys et al., 1996) provided with CORSIKA. For
the Bucaramanga site we use a tropical profile and for San Carlos de Bariloche
a midlatitude summer profile. Currently, the LAGO collaboration is developing
and validating a method to obtain the local atmospheric profiles for each LAGO
site during different weather conditions based on the Global Data Assimilation
System (GDAS) (NOAA, 2009), as differences have been observed between generic
MODTRAN models and balloon measurements at the planed LAGO site in Antarctica
(Dasso et al., 2015).

A large number of primary showers need to be simulated (typical values are of
several billions of showers for 24 h of flux at each site). A set of local
clusters have been deployed and tuned for this particular calculation. These
are maintained at some institutions of the LAGO Collaboration. This simulation
chain has been also integrated into a dedicated Virtual Organization,
lagoproject, as part of the European Grid Infrastructure (EGI,
http://www.egi.eu) activities. The Grid implementation of CORSIKA was deployed
with two “flavors”, which run by using GridWay Metascheduler
(http://www.gridway.org/doku.php) (Huedo et al., 2001) or with a second
approach through a Catania Science Gateway interface (Barbera et al., 2011).
Massive calculations can be executed with the former, via the Montera
(Rodriguez-Pascual et al., 2013), the GWpilot (Rubio-Montero et al., 2015a) or
the GWcloud (Rubio-Montero et al., 2015b) frameworks.

In the Science Gateway approach a user can seamlessly run a code on different
infrastructures by accessing a unique web-based entry point with an identity
provision. Users only have to upload the input data or invoke a PID
(persistent identifier or reference to a digital set of files) and click on
the run icon. The final result will be retrieved whenever the job has ended.
The underlying infrastructure is absolutely transparent to the user and the
system decides on which sites and computing platform the code will be compiled
and run (Rodriguez-Pascual et al., 2015; Asorey et al., 2016).

To deal with the computational complexity introduced by the refinement
described in the previous subsection, we built a library for each site
containing the simulated particles starting from a very low momentum primary
threshold of $\sim(350\times Z)$ MeV$/c$ (i.e. $1$ GeV of total energy for
protons). Each secondary impinging the detector is tagged with information
from its parent primary particle, which allows the calculation of its magnetic
rigidity $R_{\mathrm{m}}$. Then, because each secondary at the ground comes
from some primary impinging at the atmosphere, from the $R_{\mathrm{C}(i)}$
obtained for each condition according to equations (11) and (15), we are able
to determine if each secondary would reach the detector under that particular
GF condition.

##  4 Results for the LAGO sites of Bucaramanga, Colombia and San Carlos de
Bariloche, Argentina

###  4.1 Magnetic Rigidity

As we mentioned before, we applied our simulation chain to the location of two
LAGO sites: Bucaramanga, Colombia (BGA) and San Carlos de Bariloche, Argentina
(BRC). Results for the standard rigidity cutoff calculation,
$R_{\mathrm{C}(0)}$, are displayed in Figure 3 for each site. As expected,
there is a strong dependence between $R_{\mathrm{C}(0)}$ and the arrival
directions at both cities, which induces a noticeable decrease in the number
of GCRs producing secondary particles at ground level. For Bucaramanga, it is
interesting to mention the oddity of the behavior of the rigidity cutoff for
large azimuthal ($250^{\circ}\lesssim\phi\lesssim 300^{\circ}$) and zenithal
($45^{\circ}\lesssim\theta\lesssim 90^{\circ}$) angles. We have backtracked
several incoming trajectories and discovered that this anomaly in the rigidity
cutoff seems to be associated with the deflection of particles with low
$R_{\mathrm{m}}$, whose trajectories cross zones with high gradients of the GF
(Suárez-Durán, 2015).

Figure 3:  Calculation of $R_{\mathrm{C}(0)}$ (GF secular conditions, equation
(3)) at the atmosphere edge ($112$ km a.s.l.), as a function of the azimuth
angle $\phi$ and for different zenith angles $\theta$. This calculation is
presented for two LAGO sites: Bucaramanga, Colombia, left, and San Carlos de
Bariloche, Argentina, right. The strong dependence between $R_{\mathrm{C}(0)}$
and the arrival directions at both cities, is evident.

The cumulative probability distributions (7), as functions of the magnetic
rigidity for various zenith angles, are displayed in Figure 4. There, lower
magnetic rigidities are associated with particle trajectories having small
zenith angles. Notice that both plots are qualitatively different and this
probably is evidence of the complexity of the GF present at the two very
distinct latitudes.

Figure 4: Magnetic rigidity calculated on the penumbra region as a cumulative
distribution function, according to equation (7) for different arriving zenith
angles at the sites of Bucaramanga (left) and San Carlos de Bariloche (right).

###  4.2 Primary Flux Corrected by GF

As was explained in section 3.1.2, once the penumbral CDFs is obtained, it is
possible to refine the calculation of the expected primary flux $\Phi_{(i)}$
and the corresponding flux of background secondaries $\Xi_{(i)}$ at ground
level. In Figure 5 the GCR flux $\Phi_{(0)}$ and $\Phi_{(1)}$ are displayed
for both LAGO sites: BGA and BRC. Only those primaries producing secondaries
at ground level are shown. In both cases, the influence of the GF corrections
is only significant at lower energies, $\mathbf{\sim 15}$ GeV. As expected,
instead of a sharp cutoff as in the standard case, a smooth cutoff is
observed, corresponding to the different rigidities cutoff in the different
regimes, and the flux of primaries is affected according with figure 4, i.e.,
a bigger effect for BGA (rigidity up to $\sim 50$ GV) than BRC (rigidity up to
$\sim 30$ GV).

Figure 5:  Expected GCRs flux at 112 km a.s.l. producing secondaries at ground
level with only the standard corrections ($\Phi_{(0)}$, blue crosses) and with
the proposed method in secular conditions ($\Phi_{(1)}$, green squares), for
Bucaramanga (left) and San Carlos de Bariloche (right). The influence of the
GF corrections is evident at the low energy limits.

The primary flux interacts with the atmosphere and produces the secondary flux
$\Xi_{(i)}$ at the ground level. These interactions were simulated by CORSIKA
obtaining a very comprehensive distribution of particles at the detector
level. To estimate the response of the WCD to each type of secondary particle,
this flux is analyzed using a detailed Geant4 simulation of the LAGO detector
will be described and the first preliminary results were showed by Otiniano et
al. (2015).

###  4.3 Secondary Flux Corrected by GF

Figure 6 displays the simulated spectra of secondaries $\Xi_{(1)}$ (under
secular conditions) at both cities. A noticeable peak for the distribution of
secondary neutrons and protons are evident at both sites. At these low
altitudes, a muon hump is also visible in the distribution spectra, and this
is typically used as a calibration point for WCD (see e.g. Etchegoyen et al.,
2005; H. Asorey for the LAGO Collaboration, 2013).

Figure 6: The expected secondary spectra at detector level $\Xi_{(1)}$ is
shown for Bucaramanga (BGA, left) and San Carlos de Bariloche (BRC, right),
for different type of secondaries: photons (purple), electrons and positrons
(blue), muons (orange), pions (red), neutrons (pink), protons (gold), total
spectrum of secondaries (green).

By defining the flux percentage difference, $\Delta\Xi_{i-j}$, it is possible
to get a better understanding of the energy range where the geomagnetic
corrections are more important, specially when dynamic variations are
considered. Thus

$\mathrm{Difference}_{\%}\equiv\Delta\Xi_{i-j}=100\left(\frac{\Xi_{(i)}-\Xi_{(j)}}{\Xi_{(i)}}\right)\%\,,$
(16)

where $i,j$ are the indices corresponding to the configuration of the GF
introduced in the section 3.1.2. To evaluate the impact of this new method,
the differences between cases $(0)$ (standard calculation) and $(1)$ (new
method), as a function of the secondary particles momentum, are illustrated in
Figure 7. The presence of a peak at $\sim 500$ MeV/c is evident for both sites
in the $\Delta\Xi_{0-1}$ distribution, located between $100$ MeV/c and $\sim
3$ GeV/c. For lower energies, the difference is a bit larger at BGA than at
BRC, as we expected after figure 3. When we explored in more detail the
particle component of the secondaries at these momenta, we found that these
differences are dominated by secondary neutrons (Suárez-Durán, 2015), where
the diminution is of the order of $35\%$. This result, in our simulation,
agree with the fact that variations in the flux registered by Neutron Monitors
are a proxy of the changing conditions in the near-earth space environment.
For energies higher than $\sim 10$ GeV, corrections are not important.

Figure 7: Percentage Flux Difference of secondaries at the ground between the
flux calculated for for the standard definition of rigidity cutoff (equation
(3)) and for rigidity cutoff secular conditions (equation (8)), i.e.
$\Delta\Xi_{0-1}$: BGA (left) and BRC (right). It clearly peaks at $\sim 500$
MeV/c and is framed between $100$ MeV/c and $3$ GeV.

Finally, this tool allows the study of the impact of dynamic conditions of the
GF in the distribution of secondary particles by comparing secular conditions
of the GF with the evolution of the GF states as a function of time,
$\Delta\Xi_{1-2}$. This is because the calculations performed by this method
are focusing on the background of the GCR flux, e.g., we did not consider the
solar particle event during the geomagnetic storm of May 13-17, 2005, but just
the influence of the state of the GF, for this UTC-time, over the background
of GCR flux. This is shown in Figure 8, where the time evolution of
$\Delta\Xi_{1-2}$ is displayed at both sites for May of 2005. We have selected
this particular month because the strong geomagnetic storm on May 13-17, 2005,
generated intense disturbances in the GF (see e.g. Adekoya et al., 2012; Bisi
et al., 2010; Galav et al., ). Three particular cases are shown: the total
flux secondary particles, $\Delta\Xi_{1-2}$, the muon flux
$\Delta\Xi_{1-2}^{\mu}$ and the neutron flux $\Delta\Xi_{1-2}^{n}$. It is
clear that, beside the time coincidence of the flux variations, it is more
significant at Bucaramanga than at Bariloche, and that the neutron flux at
ground level is the most affected component by the GF activity, which
reinforces the known sensitivity of this particular constituent for the
observation of geomagnetic disturbances (see e.g. Belov et al., 2005).

As reference, on the background of each sub-plot in figure 8 (gray line) the
flux of neutrons at ground level is shown, registered by two Neutron Monitors
(NM) with similar rigidities to both sites; i.e. $R_{\mathrm{C}(0)}$ of
$10.75$ GV for NM of ESOI and $8.28$ GV for NM of Mexico. As reference, the
$R_{\mathrm{C}(0)}$ for BGA is $11$ GV and for BRC is $8.1$ GV. Both NM show a
decrease between $300$ and $400$ in elapsed UTC time, that is in coincidence
with our simulation results. Because we have simulated the effect of the GF
under GCR flux, i.e., we do not simulate solar particle events, it is
possible, with our approach, to estimate the contribution of the GF topology
to a Forbush decrease event.

Figure 8: Time evolution of the expected flux of secondaries during dynamic
conditions of the GF, $\Delta\Xi_{1-2}$ for May of 2005 at Bucaramanga (left)
and at San Carlos de Bariloche (right). On background, in gray, is presented
the neutron flux is shown, registered by two Neutron monitors with similar
$R_{\mathrm{C}(0)}$: to BGA, ESOI with $R_{\mathrm{C}(0)}=10.75$ GV (left); to
BRC, Mexico with $R_{\mathrm{C}(0)}=8.28$ GV (right). The data for the neutron
monitor was taken from http://www.nmdb.eu. The first row corresponds to the
total flux of particles at ground level, while in the second row we illustrate
the evolution of the muon flux $\Delta\Xi_{1-2}^{\mu}$. The third one displays
the neutron flux $\Delta\Xi_{1-2}^{n}$. Note the scale difference on the
y-axis for each plot.

##  5 Final remarks

In this paper, we have presented the LAGO space weather chain of simulations
devised to obtain precise calculations of secondary particle flux at ground
level that can be used at every geographical position. It takes into account
geomagnetic corrections for both secular (long term phenomena with typical
time scale larger than a year) and transient events (with typical time scales
of hours to days). We shall consider all calculations performed without this
new method as a first approximation to the more precise determination of the
real flux of secondaries particles, calculated when the effects of the
geomagnetic field are fully considered.

Variations of the flux of secondaries at two LAGO sites with different
latitude/longitude (Bucaramanga, $7^{\circ}\,8^{{}^{\prime}}$N,
$73^{\circ}\,0^{{}^{\prime}}$W and San Carlos de Bariloche
$41^{\circ}\,9^{{}^{\prime}}$N, $71^{\circ}\,18^{{}^{\prime}}$W) were
calculated for both secular geomagnetic conditions and under transient events,
during the geomagnetic active month of May 2005. Our simulations show that the
secondary flux is sensitive to the latitude and that the secondary neutrons at
the ground level are the most affected flux component due to variations of the
geomagnetic field during space weather phenomena. While our calculation relies
on the isotropy of the GCR flux, it is important to note that during certain
FDs, small anisotropies in the flux of primaries could be induced by the
configuration of the incoming magnetic cloud and the disturbances of the
geomagnetic field during these particular events. Actually a $\sim 1\%$
anisotropy in the flux of secondary muons was observed at ground level during
the Forubush decrease of December 13, 2006 (Kane, 2006; Fushishita et al.,
2010). However, since our WCD are not sensitive to the arrival direction of
secondary particles, we will not be able to detect such small effect while the
total flux of secondaries remains constant.

Several dedicated clusters and a Grid-based implementation have been deployed
for these calculations. A dedicated Virtual Organization, lagoproject, part of
the European Grid Infrastructure (EGI, http://www.egi.eu) activities has been
created, and available tools for Grid have been adapted and implemented to run
CORSIKA in a absolutely transparent way to the user.

The standard definition of the penumbra region for magnetic rigidities
generates a complex structure of particle trajectories: permitted, prohibited
and quasi-trapped orbits, which does not allow to derive all values for the
$R_{\mathrm{m}}$ (Smart et al., 2006). Currently, calculations of rigidity
cutoff tend not to consider the effects involved in the penumbra, and always
use a single effective value (equation (3)) to account and characterize all
the complexity of involved effects (Smart and Shea, 2009). In this paper the
concept of rigidity cutoff $R_{\mathrm{C}}$ has been generalized as a time
dependent function of the cumulative probability distribution (see equation
(10)). With this refinement, at the penumbra region, we can obtain a non-
vanishing probability to have an incoming particle (with a zenithal angle
$\theta$) contributing to the flux of primaries at the observation point.

Combining the data measured at different locations of the LAGO detection
network, with those obtained from the detailed simulation performed by this
space weather chain, we are now capable of providing a better understanding of
the temporal evolution and of the small and large scales disturbances of the
space weather phenomena.

###### Acknowledgements.

The authors thanks the enlightening suggestions and criticism from the
anonymous referees and also from the AGU Space Weather Editorial Office which
have helped to make this work clearer and more focused. We appreciate the
support of Vicerrectoría Investigación y Extensión Universidad Industrial de
Santander for its permanent sponsorship, and acknowledge the financial support
of Departamento Administrativo de Ciencia, Tecnología e Innovación of Colombia
(COLCIENCIAS) under contracts FP44842-051-2015 and FP44842-661-2015. The
authors acknowledge the support of COLCIENCIAS, CONICET and MINCyT for funding
bilateral cooperation Argentina-Colombia, grant AR:CO-15/02 CO:729-2015. HA
and MSD acknowledge the support from Innóvate Perú, grant 398-PNICP-PIBA-2014.
Significant parts of the calculations needed for this work was performed with
the computational support of the Centros de Supercomputación y Cálculo
Científico de la Universidad Industrial de Santander. We also acknowledge the
NMDB database (www.nmdb.eu), founded under the European Union’s FP7 programme
(contract no. 213007) for providing the data. Neutron monitor of the Emilio
Segre Observatory is supported by collaboration ICRC-ESO (Tel Aviv University
and Israel Space Agency, Israel) and University “Roma Tre” with IFSI-CNR
(Italy). Neutron monitor data of Mexico City is provided by the Cosmic Ray
Group of the Geophysical Institute at the Universidad Autónoma de México
(UNAM). The authors are grateful to the LAGO and Pierre Auger Observatory
Collaboration members (http://lagoproject.org/collab.html) for their
continuous engagement and support.

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