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Figs. 2.5–2.10 show that the resolution generally improves as the number of directions increases. For the FBP method, the exact solution for the reconstruction comprises contributions from all β directions in Fig. 2.2 and equation 2.16. Hence, this improvement in resolution is consistent with the basic theory of FBP. For the linear inversion method, the number of independent linear equations increases as d ⁱ increases with the number of observation directions. The number of ρ ^j values interpolated from the constraints of the covariance matrix decreases and the independent information increases, which then more clearly reproduces the input image.

For the FBP method, the error bar length is almost the same in Fig. 2.7, whereas the errors become smaller as the elevation increases (Fig. 2.8). This is because the random error is correlated with the number of muons passing through the voxel. At a higher elevation, the muon flux increases because the path length decreases.

Neither the linear inversion nor FBP methods tend to reproduce the high‐frequency components of the input image (Figs. 2.7 and 2.10). In addition to this systematic error, there is also some artificial noise in the FBP results, which is especially evident in the uniform density region of Fig. 2.9f–g. Such noise is similar to that reported in various FBP studies (e.g., Kolehmainen et al., 2008; Schofield et al., 2020). The noise tends to decrease as the number of observation directions increase (Fig. 2.9), but is not completely eliminated even in the case of 16 directions.

Focusing on the 4‐directional case, the structure of the checkerboard pattern in Fig. 2.5 cannot be recognized by the linear inversion or FBP methods. In contrast, in Fig. 2.9, it is possible to qualitatively recognize the two‐cylinder structure with the linear inversion method, whereas it is difficult to recognize the cylinders with the FBP method. This implies that the linear inversion method is suitable when the expected density structure inside the volcano is rather simple, and the number of available muon detectors is small. As such, the linear inversion method appears to be more suitable than the FBP method where there are ≤16 observational directions, if the procedure to fix the ambiguity of the parameters σ _ρ and L ₀ in equation 2.6 is used.

Figure 2.7 Reconstructed E–W density profiles along the dashed line in Fig. 2.5d with a checkerboard structure using (a) linear inversion and 4‐directional; (b) FBP and 4‐directional; (c) linear inversion and 8‐directional; (d) FBP and 8‐directional; (e) linear inversion and 16‐directional; (f) FBP and 16‐directional; and (g) FBP and 32‐directional muographic images. Each dashed line represents the input density model, and thus the difference between the data points and these lines corresponds to the error of the reconstruction. The error bars in the FBP images imply only random error, which was calculated from equation 2.20.

Figure 2.8 Reconstructed vertical density profile with a checkerboard structure along the dashed line in Fig. 2.6d. The horizontal axis is the elevation, and the vertical axis is the reconstructed density. The dashed line is the input density.

Figure 2.9 Reconstructed density images with the two‐cylinder 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; and (g) FBP and 16‐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. 2.10.

Figure 2.10 Reconstructed E–W density profile with the two‐cylinder structure along the dashed line in Fig. 2.9d. (a) Linear inversion and 4‐directional; (b) FBP and 4‐directional; (c) linear inversion and 8‐directional; (d) FBP and 8‐directional; (e) linear inversion and 16‐directional; and (f) FBP and 16‐directional data. Each dashed line represents the input density model, and thus the difference between the data points and these lines corresponds to the systematic error of the reconstructions. The error bars in the FBP images are the random error calculated from equation 2.20.