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Figs. 2.5–2.6 show the results of the reconstructions using the FBP and linear inversion methods with 4‐, 8‐, 16‐, and 32‐directional muographic images of a horizontal cross‐section at 470 m a.s.l. and a vertical cross‐section along a northing of 150 m, respectively. For the 32 points of the linear inversion, it was not easy to obtain a result due to the long computational time. Fig. 2.7 shows the reconstructed density and its random errors due to muon statistics along an easting coordinate at 470 m a.s.l. and a northing of 150 m. Fig. 2.8 shows the density profile along the elevation direction at an easting of −50 m and northing of 150 m. In the FBP plots in Figs. 2.7–2.8, there is a difference between the center value of the plot (reconstructed density without random numbers) and the dashed line (input model), which represents a systematic error of the reconstruction. The lengths of the error bars in the FBP results represent the random error due to the observed muon statistics (equation 2.20). Fig. 2.9 shows the reconstructed images of a horizontal cross section at 470 m a.s.l., and Fig. 2.10 is the density profile along the easting coordinate in the case of two‐cylinder structure.

Figure 2.5 Reconstructed density images with a checkerboard structure of a horizontal cross‐section at 470 m a.s.l. using (a) linear inversion and 4‐directional; (b) linear inversion and 8‐directional; (c) linear inversion and 16‐directional (e) FBP and 4‐directional; (f) FBP and 8‐directional; (g) FBP and 16‐directional; and (h) FBP and 32‐directional muographic images. (d) Input density model. The solid dashed line indicates E–W line at a northing coordinate of 150 m and 470 m a.s.l., which is corresponding to the horizontal axis in Fig. 2.7.

Figure 2.6 Reconstructed density images with a checkerboard structure of a vertical cross‐section at a relative northing of 150 m using (a) linear inversion and 4‐directional; (b) linear inversion and 8‐directional; (c) linear inversion and 16‐directional; (e) FBP and 4‐directional; (f) FBP and 8‐directional; (g) FBP and 16‐directional; and (h) FBP and 32‐directional muographic images. (d) Input density model. The solid dashed line indicates elevation line at an easting coordinate of –50 m and a northing of 150 m, which is corresponding to the horizontal axis in Fig. 2.8.

For the linear inversion results, Φ(σ _ρ , L ₀) defined in equation 2.15 was calculated in the range of σ _ρ = 0.1 − 1.0 g/cm³ with a 0.1 g/cm³ pitch and the range of L ₀ = 10 − 100 m with a 10 m pitch. The σ _ρ and L ₀ values at the lowest Φ value for the checkerboard structure were 0.1 g/cm³ and 10 m in the 4‐ and 8‐directional case, and 0.1 g/cm³ and 20 m in the 16‐directional case, respectively. For the case of the two‐cylinder structure, the σ _ρ and L ₀ values at the lowest Φ value were 0.1 g/cm³ and 30 m for the 4‐directional case, and 0.2 g/cm³ and 20 m for the 8‐ and 16‐directional cases, respectively.