towards the geomagnetic North Pole, some eastern trajectories of the incoming cosmic rays are suppressed (East‐West effect). This geomagnetic cutoff is caused by the geomagnetic shielding that affects the charged cosmic rays arriving from outside the magnetosphere, and its threshold rigidities range from less than 109 V near the geomagnetic poles to about 1.6 × 1010 V for vertical particles near the equator (Smart & Shea, 1994). As a result, cosmic-ray muons from the east are found to have a weaker flux than those from the west. This East‐West geomagnetic effect is more pronounced at higher altitudes. In middle latitude regions at sea level, only primary GCRs having lower energies are more affected by the geomagnetic field, but they do not contribute to production of muons in the atmosphere. For this reason, there has been a long discussion about whether the East‐West effect is observable in the sea‐level muon flux (Hansen et al., 2005). The recent measurements with the Far detector used for the NOvA experiment showed the East‐West asymmetry of cosmic‐ray muons of ~1% (Petrova, 2019) within the zenith angular range between 30o and 70o at Ash River, MN, USA.
This East‐West geomagnetic effect strongly depends on the latitude where the cosmic‐ray muons are measured. First measured 70 years ago, the vertical flux of muons with momentum around 0.33 GeV/c at latitude 60° was 1.8 times higher with respect to the flux at the equator (Conversi, 1950). This latitudinal geomagnetic effect was more recently measured between the middle latitude region (the cutoff values of 2.7 × 109 V) and the region near the equator (the cutoff values of 1.4 × 1010 V), respectively, indicating variations by 20 and 10% in the flux of muons with energies of ~0.3 GeV/c and ~1 GeV/c and no significant variations could be seen in the muons above 3 GeV/c (Allkofer et al., 1975). In conclusion, this latitudinal effect is practically negligible in muography as long as the measurements are conducted within the mid‐latitude region. However, in the region near the equator, it can influence muons below ~5 GeV at sea level (Allkofer et al., 1975).
1.2.3 Altitudinal Atmospheric Effects in the Muon Flux
The muons that arrive at sea level are the last stage of a multi‐step cascade process. The average muon energy loss in the atmosphere is in the order of 2 GeV. Therefore, sub‐GeV muons originated in the upper atmosphere cannot reach sea level, and the muon energy spectrum varies at different atmospheric depths; in particular, in energy regions lower than 2 GeV. Based on the results of the CAPRICE 94 muon measurements (Boezio et al., 2000), the low‐energy (< 2 GeV) muon flux increases by a factor of 5 if the atmospheric depth is reduced by one half of the value at sea level (500 hPa) (Engel et al. 2001). On the contrary, the flux of muons with higher energies (3–20 GeV) increases only by a factor of 1.5 (Engel et al. 2001).
1.2.4 Seasonal Atmospheric Effect in the Muon Flux
When the time‐sequential muographic images are evaluated, it is necessary to consider the daily and seasonal variations in the muon flux. These variations are mainly caused by the spatio‐temporal variations in the interplanetary magnetic field (IMF) and the atmospheric temperature, and affect the entire energy range of the muon spectrum. Anisotropies of the IMF are related to the solar activities (11‐year and 22‐year solar cycles and large‐scale bursts of plasma and magnetic field, such as coronal mass ejections [CME]). The 11‐year and 22‐year solar cycles affect the sub‐GeV energy region of the cosmic‐ray muons, and variations of the muon flux due to this effect are small (a few percent) (de Mendonca et al., 2016). The CME also affects only the sub‐GeV energy region, but the muon flux (E > 100 MeV) can be periodically changed by 10% within a few days (Augusto et al., 2012).
The temperature variations cause barometric (tropospheric pressure) variations, and thus affect the muon count rate. Therefore, the muon count rate shows a negative temperature correlation within a day. Since daily barometric variations are generally less than one percent in the mid‐latitude region, the corresponding muon flux variations are up to a few percent. For example, the daily flux variations measured at 44o 21′N, 76 m a.s.l. was ± 2.5% for muons with energies above 0.4 GeV; this corresponds to the muon range of 200 g/cm2 in rock (Saftoiu et al., 2010).
A major contributor to the seasonal variations in muon flux is not only the tropospheric but also the stratospheric temperature (or pressure) variations. While the tropospheric temperature variations affect low‐energy muons, the stratospheric temperature variations affect high‐energy muons. The temperature variations in the stratosphere, where the mesons are copiously produced, change the MFP of the meson‐nucleon interactions in the stratosphere. As a consequence, high‐energy muons have a positive stratospheric temperature correlation. The Daya‐Bay detectors were located underground at three different depths (ranging from 250–860 hg/cm2) and recorded the seasonal variations in muon flux at each depth (Daya Bay Collaboration, 2018). There was a tendency for the seasonal variations to be more enhanced for deeper detectors: 0.5% at a depth of 250 hg/cm2 and 1% at a depth of 860 hg/cm2. At a deeper site (3,800 hg/cm2), stronger variations (~3%) were observed (Borexino Collaboration, 2019).
In conclusion, muographers have to consider the following two factors when they conduct the time‐dependent measurements: (i) The stratospheric effect in high‐energy muons must be considered. However, currently, there are no experimental data for the horizontal muons. Such data will be necessary in the near future. (ii) When muographers need to normalize the time‐sequential muographic data to the open‐sky muon count rate for the purpose of cancellation of the factors originated in the detector configuration, such as geometrical acceptance, efficiency, etc., the spatial scale of the target objects and the timescale of the measurements have to be well considered so that the aforementioned geomagnetic and atmospheric effects become negligible in their measurements.
1.2.5 Muon Flux Reduction through Matter
During the ionization process, muons frequently collide with electrons, losing a very small fraction of their energy in each collision. If the energy loss of muons in matter were only due to ionization, muons of a given energy would have an almost unique range because the number of such encounters is proportional to the densimetric length they traverse. However, for high‐energy muons, the radiative processes become more predominant, and at the same time, fluctuations within the range are enhanced by the radiative processes, namely bremsstrahlung, direct pair production, and photonuclear interactions. In these processes, muons lose a large but random fraction of their energy. For high‐energy muons (> 1 TeV), the contribution of the ionization process is small compared with the other three processes. For example, the radiative process dominates above 708 GeV in SiO2 (Groom et al., 2001).
The muon's energy loss rate depends on materials. The electrons have a tendency to be more concentrated per unit mass in lighter materials, and thus muons lose more energy in lighter materials through the ionization process. However, larger nuclei increase the possibilities of radiative processes occurring and thus, high‐energy muons lose more energy in heavier materials. For example, the 1‐GeV muon's range is 550 g/cm2 in SiO2 but muons with the same energy can penetrate only 470 g/cm2 of water (Groom et al., 2001). On the contrary, while 10‐TeV muons can penetrate 6,900 hg/cm2 in SiO2, they can penetrate 7,800 hg/cm2 in water (Groom et al., 2001).
For the calculation of the analytical range of high‐energy muons, the energy loss relationship presented by Adair and Kasha (1977) is convenient:
