# Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications

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## Abstract

**:**

## 1. Introduction

**MUPAGE**[24] is a fast Monte Carlo generator of bundles of atmospheric muons for underwater/ice neutrino telescopes. It is based on parametric formulas obtained from Monte Carlo simulations of cosmic ray showers generating muons in bundle, with constraints from measurements of the muon flux with underground experiments. The range of validity extends from 1.5 km to 5.0 km of water vertical depth, and from 0° up to 85° for the zenith angle.**Matrix Cascade equation MCeq**[25,26,27,28,29] uses numerical equation cascades to study fluxes. It is a complete Monte Carlo calculation scheme, capable of calculating neutrino, electron and muon fluxes up to 100 TeV, with a statistical accuracy of about a few percent. All particles have their own cascades of equations that represent the evolution of the energy spectrum as a function of atmospheric depth.**PARMA**(PHITS-based Analytical Radiation Model in the Atmosphere) allows to estimate the terrestrial cosmic ray fluxes of neutrons, protons and ions, muons, electrons, positrons and photons almost anywhere on Earth and in the Earth’s atmosphere [30]. The model is based on analytical numerical functions whose parameter values are adjusted to reproduce the EAS results. The accuracy of the EAS simulation has been well verified with various experimental data.**CRY**(Cosmic-ray Shower Library) generates showers distributions for three observation levels (sea level, 2100 m and 11,300 m) and for primary particles from 1 GeV to ${10}^{5}$ GeV according to Hagmann et al. [31], and secondary particles from ${10}^{-3}$ to ${10}^{5}$ GeV. The showers are generated in a specific area (maximum size $300\times 300$ ${\mathrm{m}}^{2}$) from pre-computed tables as explained in Hagmann et al. [32] and primary protons are produced at an altitude of 31 km in the 1976 US atmosphere [33]. In this generator, the east–west effect is not taken into account but the latitude dependence with the geomagnetic cutoff and the CR spectrum modulation are provided. It is possible to set the type of secondary particles to be studied, the altitude, the latitude, the date, the number of particles and the size of the surface of interest. The date allows to take into account the solar modulation described by Papini et al. [34]. It is possible to use pre-calculated tables from GEANT4 to take into account the configuration of the detector [35]. CRY has limitations when you investigate multi-track events in a cosmic ray experiment or identifying background events in muography.**CORSIKA**(COsmic Ray SImulations for KAscade) [36,37] is a Monte Carlo code for simulating atmospheric particle showers initiated by high-energy cosmic ray particles. Primary particles (protons or light nuclei) are tracked in the atmosphere until they interact, decay or are absorbed. All secondary particles are explicitly followed along their trajectories and their parameters are stored when they reach an observational level.

## 2. Materials and Methods

#### 2.1. Standard Use Cases

#### 2.2. Simulation’s Configuration

#### 2.2.1. Primaries at the Top of the Atmosphere

**Number of simulated showers and energy ranges**

**Slope of the primary spectra**

**Particles angles trajectories**

#### 2.2.2. Hadronic Interaction Models

#### 2.2.3. Particles Energy Thresholds

#### 2.2.4. Detector Geometry and Observation Levels

#### 2.3. External Parameters

#### 2.3.1. Earth’s Magnetic Field

#### 2.3.2. Atmosphere Parameters

_{2}, O

_{2}and Ar with volume fractions of 78.1%, 21.0% and 0.9%, respectively. The density variation of the atmosphere with altitude is modeled by 5 layers. The state of the atmosphere is described by the density of the air at each altitude level. This parameter is calculated by converting the relative humidity into saturation vapor pressure with the Magnus formula [61]. We computed the parameters and altitudes of the layer boundaries from ERA5 data, the latest climate reanalysis produced by the ECMWF (European Centre for Medium-Range Weather Forecasts), which combines large amounts of meteorological observations with estimates made from advanced modeling and data assimilation systems [62]. The choice of ERA5 is guided by the abundance of pressure levels and the availability of data at higher altitudes.

#### 2.3.3. Physical Validity Range

#### 2.4. Normalization Issues

**geometric correction factor**related to the solid angle $d\Omega $ adapted to the type of detector (see [47]). For example, in the case of a flat detector: $d\Omega =sin\left(\theta \right)cos\left(\theta \right)\phantom{\rule{0.277778em}{0ex}}d\theta \phantom{\rule{0.277778em}{0ex}}d\psi $ so:

## 3. Results

#### 3.1. Comparison with Analytical Models

**1.0**± 0.2; on the whole range, the fluxes have the same intensity. Right panel shows the flux ratio for different zenith angles $\theta $ in the range 0 to 90°, when considering the same energy ranges. The models fit well on intermediate energy ranges (10–${10}^{2}$ GeV in dark blue) except at high zenith angles; however, their intensity is different at high energy (${10}^{2}$–${10}^{4}$ GeV, in light blue and green). The 1–10 GeV is range not shown: the ratio starts around 4 at 0° and increases with the zenith angle following a power law because the Tang flux is unstable at low energy and especially at high zenith angles. Our CORSIKA relative fluxes are plotted in Figure 6 only up to 2000 GeV because beyond this point our fluxes are too unstable. The part of the most energetic muons is more important than the low energetic ones at high zenith angles $\theta $ and the opposite occurs at low angles. Low-energy muons are important for calibration and high-energy muons for conducting tomography experiments; however, analytical models are known to be poorly adapted to small and large energies, because few measurements are available for their fitting equations. The CORSIKA model instead is probably more reliable over the whole energy range. Furthermore, analytical models are not extrapolated for all zenith angles and they do not take into account geodesics parameters, a limitation overcome by the CORSIKA approach.

#### 3.2. Comparison with Real Data

#### 3.3. Geodesic and Atmospheric Factors

#### 3.3.1. Altitude Effects

**1.17**± 0.01, which means that there are 17% more muons at 1000 m than at sea level in overall. Right panel shows the flux ratio for different zenith angles $\theta $ in the range 0 to 90°, when considering the same energy ranges. It shows that at an elevation of 1000 m the flux ratio is much higher for low energies (1 to 10 and 10 to 100 GeV, in dark and light blue), and less important at high zenith angle (80 to 90°). For higher energies, the flux ratio remains more constant at each angle (in green and yellow).

#### 3.3.2. Geomagnetic Field Effects

**0.95**± 0.01; the median value, corresponding to a small effect of about 5%. As expected the ratio tends to one for energy ranges greater than 10 GeV but affects the low-energy particles that are more deflected towards the poles. The right panel shows the flux ratio with respect to the zenith angle $\theta $ for fixed energy ranges. At low energies (1 to 10 GeV in dark blue) the flux is higher for ${B}_{x}$ = 45 µT until 60° and between 85° and 90°, and lower between 60 and 85°. This probably arises from the fact that high-angles particles have to cross a larger section of the atmosphere and therefore may start their travel with higher energies, making them less sensitive to the geomagnetic field’s effect. Note that this sample dominates over the total integrated flux ratio (in black), which follows more or less the same behavior. As expected, at high energies, typically above 10 GeV, the flux ratio remains more constant with the zenith angle (in light blue, green and orange). Note that this high energy sample is usually not used in scattering mode; therefore, one has to pick corrections for the geomagnetic effects to perform absolute measurement with scattering muography.

#### 3.3.3. Atmospheric Thermodynamics Effects

**Atmosphere simple parametrization**

**Atmosphere properties**

**0.90**± 0.26, the median value, which means that the flux is 10% higher in Oimiakon (Russia) than in Dallol (Ethiopia); however, this statement is only valid for the integration over the total energy range. Right panel shows the flux ratio for different zenith angles $\theta $ in the range 0 to 90°, when considering the same energy ranges. It shows that in Dallol the flux is higher for high energies (100 to 10${}^{4}$ GeV, in green and yellow), and lower in Oimiakon. For lower energies, the flux ratio is higher in Dallol (1 to 100 GeV, in light and dark blue), and lower in Oimiakon. These effects increase with the zenith angle $\theta $. Statistical errors of fluxes simulations are fixed to 1 $\sigma $.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**CORSIKA’s coordinate system used [37].

**Figure 4.**Different components of the magnetic field from NOAA. H is the magnetic north (${B}_{x}$) and Z the vertical component of the magnetic field.

**Figure 5.**(

**Left panel**) Histogram of flux ratio values computed over the zenith angle range (details are shown in the right figure) for several energy ranges from 10 to ${10}^{4}$ GeV (in color) and for the whole energy range (in black). (

**Right panel**) Flux ratio as a function of the zenith angle $\theta $ between (1) CORSIKA and (2) Tang et al. [19], for the same energy ranges as the left panel. The latter is the reference flux (ratio denominator). Med corresponds to the median value in each energy range.

**Figure 6.**Differential relative muon fluxes (Flux/Mean Flux) simulated with CORSIKA and plotted as a function of the zenith angle and for several muon energy levels (different colors).

**Figure 7.**Integrated muon fluxes I plotted as a function of the zenith angle $\theta $. The crosses represent the measured data for different inclinations of the detector. The red line is a linear fit using $co{s}^{2}\left(\theta \right)$ on the total measured flux. The black line represents the simulated flux with CORSIKA.

**Figure 8.**(

**Left panel**) Histogram of flux ratio values computed over the zenith angle range (details are shown in the right figure) for several energy ranges from 1 to ${10}^{4}$ GeV (in color) and for the whole energy range (in black). (

**Right panel**) Flux ratio as function of the zenith angle $\theta $ in Lyon, France at (1) 1000 m of elevation and at (2) sea level, for the same energy ranges as the left panel. The second is the reference flux (ratio denominator). Same atmosphere and magnetic field for both fluxes were considered. Med corresponds to the median value in each energy range.

**Figure 9.**Comparison of two simulations, with same atmosphere and altitude, and different magnetic fields. The magnetic parameters are (1) ${B}_{x}$ = 15 µT, ${B}_{z}$ = 20 µT and (2) ${B}_{x}$ = 45 µT, ${B}_{z}$ = 20 µT. The latter is the reference flux (denominator). (

**Left panel**) Histogram of flux ratio values computed over the zenith angle range (details are shown in the right figure) for different energy ranges from 1 to 10${}^{4}$ GeV (in color) and for the whole energy range (in black). (

**Right panel**) Flux ratio dependence on the zenith angle $\theta $ between two different magnetic field fluxes for the same energy ranges as in the left panel. Med corresponds to the median value in each energy range.

**Figure 10.**(

**Top**) Temperature and (

**Bottom**) density profiles for altitudes from 0 to 50 km of extreme atmospheres: Oimiakon (in blue) and Dallol (in red) and a moderate one, Lyon (in green).

**Figure 11.**Relative humidity profiles for altitudes from 0 to 50 km of extreme atmospheres: Oimiakon (in blue) and Dallol (in red) and a moderate one, Lyon (in green).

**Figure 12.**(

**Left panel**) Histogram of flux ratio values computed over the zenith angle range (details are shown in the right figure) for several energies range from 1 to 10,000 GeV (in color) and for the whole energy range (in black). (

**Right panel**) Flux ratio as a function of the zenith angle theta of two extreme temperature atmosphere for the same energy ranges as left panel. The different atmosphere conditions are (1) Dallol (Ethiopia) (2) Oimiakon (Russia), 12/30/20, with constant geomagnetic field and altitude. The latter is the reference flux (ratio denominator). Med corresponds to the median value in each energy range.

E Threshold | Hadrons | Muons | Electrons | Photons |
---|---|---|---|---|

1. [57] | 0.1 GeV | 0.1 GeV | 0.1 MeV | 0.1 MeV |

2. [58] | 0.2 GeV | 0.2 GeV | 0.1 MeV | 0.1 MeV |

3. [59] | 0.05 GeV | 0.05 GeV | 0.003 GeV | 0.003 GeV |

4. [52] | 300 MeV | 100 MeV | 250 keV | 250 keV |

5. [60] | 0.05 GeV | 50 keV | 50 keV | 50 keV |

6. This study | 0.05 GeV | 0.01 GeV | 0.001 GeV | 0.001 GeV |

**Table 2.**Maximum observed variations comparing two CORSIKA fluxes for the studied effects: magnetic field, altitude and state of the atmosphere (extreme and seasonal effect) for several muon energy ranges.

1–10,000 GeV | 1–10 GeV | 1000–10,000 GeV | |
---|---|---|---|

Magnetic field | 5 (±4)% | 6 (±1)% | 0.2 (±1)% |

Altitude:- 1000 m/0 m- 5000 m/0 m | 15 (±1)% 115 (±10)% | 17 (±2.5)% 106 (±25)% | 7 (±2.7)% 2 (±1)% |

Atmosphere:- Extreme- Seasonal | 10 (±7)% 8 (±1)% | 13 (±10)% 9 (±3)% | 5 (±10)% 2 (±10)% |

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## Share and Cite

**MDPI and ACS Style**

Cohu, A.; Tramontini, M.; Chevalier, A.; Ianigro, J.-C.; Marteau, J. Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications. *Instruments* **2022**, *6*, 24.
https://doi.org/10.3390/instruments6030024

**AMA Style**

Cohu A, Tramontini M, Chevalier A, Ianigro J-C, Marteau J. Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications. *Instruments*. 2022; 6(3):24.
https://doi.org/10.3390/instruments6030024

**Chicago/Turabian Style**

Cohu, Amélie, Matias Tramontini, Antoine Chevalier, Jean-Christophe Ianigro, and Jacques Marteau. 2022. "Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications" *Instruments* 6, no. 3: 24.
https://doi.org/10.3390/instruments6030024