Figure 1

Fits of the modified Gaisser parametrization to experimental data in the low energy regime between $\theta =0$ and $\theta =87°$. The experimental data are taken from Refs. 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38. The modified Gaisser parametrization is given by Eqs. (3, 4, 5, 6, 7, 9, 10).Reuse & Permissions

Figure 2

Average vertical muon intensity $I_{\mu }^v(h)$ versus vertical depth $h$ beneath a flat surface in standard rock. Experimental and simulated data are labeled by EX and MC, respectively. The experimental data of the flat surface overburden are taken from Refs. 8, 39 and the simulated data by Kudryavtsev et al. from Ref. 20. The number of simulated events per data point in this figure is $N={10}^6$. The set of simulated data labeled as “Present work” is generated from the modified Gaisser parametrization of the surface muon intensity in Eqs. (4, 5, 6, 7).Reuse & Permissions

Figure 3

Ratio of simulated vertical muon intensity $I_{\mu }^v(h{)}_{\mathrm{MC}}$ over the experimental vertical muon intensity $I_{\mu }^v(h{)}_{\mathrm{EX}}$ versus shallow depth $h$ beneath a flat surface in standard rock. The experimental data of the flat surface overburden are taken from Refs. 8, 39. The number of simulated events per data point in this figure is $N={10}^6$. The simulated data are generated from the modified Gaisser parametrization incorporating Eqs. (4, 5, 6, 7).Reuse & Permissions

Figure 4

A visualization of the 3D topological profile of the Ardennes Mountains over the CHOOZ site generated by $\mathtt{3}\mathtt{D}\mathtt{F}\mathtt{i}\mathtt{e}\mathtt{l}\mathtt{d}$ from a 2D contour map.Reuse & Permissions

Figure 5

A sketch of a vertical cylindrical detector. The inner volume is indicated by dotted lines which is taken to be cylindrical for Super-K and CHOOZ but spherical for KamLAND.Reuse & Permissions

Figure 6

Integrated muon intensity distribution at Super-K, KamLAND, and CHOOZ. The number of energy bins is $M=500$. The total number of simulated events is $5\times {10}^6$. The average muon energies of the three sites are quoted in the legend.Reuse & Permissions

Figure 7

Angular distribution of final muons at Super-K. The total number of simulated events is $5\times {10}^6$. The Super-K digital map defines $\varphi =0$ to be along the northerly axis and the sense of rotation to be clockwise.Reuse & Permissions

Figure 8

Angular distribution of final muons at KamLAND. The total number of simulated events is $5\times {10}^6$.Reuse & Permissions

Figure 9

Angular distribution of final muons at CHOOZ. The total number of simulated events is $5\times {10}^6$.Reuse & Permissions

Figure 10

Comparisons of the $\cos \theta$ and $\varphi$ distributions of final muons at KamLAND. The total number of simulated events of the present $\mathtt{M}\mathtt{U}\mathtt{S}\mathtt{I}\mathtt{C}$ simulation is $5\times {10}^6$. The muon fitter results [40] represent actual experimental data. The modified $\mathtt{M}\mathtt{U}\mathtt{S}\mathtt{U}\mathtt{N}$ simulation [41] is a standard rock calculation while the present simulation is an exact calculation based on the Ikenoyama rock composition. Both the muon fitter and modified $\mathtt{M}\mathtt{U}\mathtt{S}\mathtt{U}\mathtt{N}$ results are rescaled from arbitrary units to fit the present results.Reuse & Permissions

Figure 11

Comparisons of the $\theta$ and $\varphi$ distributions of final muons at CHOOZ. The total number of simulated events of the present $\mathtt{M}\mathtt{U}\mathtt{S}\mathtt{I}\mathtt{C}$ simulation is $5\times {10}^6$. “Simulated Total Flux” refers to the simulated muon flux integrated over the entire hemispherical solid angle. The total flux is binned to generate the angular distributions. “Simulated Experiment” is the simulated muon flux integrated over a limited range of solid angle described by the geometry of the RPC plates used in the cosmic ray experimental setup on the CHOOZ site. Both the experimental and simulated experimental results are rescaled such that the small angle parts of all three $\theta$ distributions agree.Reuse & Permissions

Figure 12

Average muon energy $E_{\mu }$ versus $\theta$ and $\varphi$ at KamLAND. The total number of simulated events is $5\times {10}^6$.Reuse & Permissions

Figure 13

Average muon energy $E_{\mu }$ versus $\theta$ and $\varphi$ at CHOOZ. The total number of simulated events is $5\times {10}^6$.Reuse & Permissions

Figure 14

An illustration of the binning strategy of slant depth $X$ in the $\theta \mathrm{\text{−}}\varphi$ space. The bars represent regions of the solid angle corresponding to the edges of a 3D topographical map and are blocked from the random generation of $\theta$ and $\varphi$. The black dots represent the original lattice sites from a latticized hill profile. The dotted lines partition the remaining solid angle into regions of nearest neighbors.Reuse & Permissions

Figure 15

Integrated muon flux versus muon energy on ground level. The angular integration is taken over the entire hemisphere. Experimental data support the feature that the energy spectrum for $E_{\mu }<1\mathrm{GeV}$ is almost flat [8].Reuse & Permissions

Figure 16

Integrated muon flux versus the zenith angle on ground level. The energy integration is taken over the range $0.106–{10}^6\mathrm{GeV}$. The experimental data are taken from Refs. 42, 43.Reuse & Permissions