i . The functions f i and g i are not linear. We then linearize N i around (2.9)
Given that N i can be written as
Introducing a small variation from the initial density value
From equations 2.10 and 2.12 we obtain:
Finally, by comparing equations 2.11 and 2.13, the relationship between
(2.14)
There are some advantages to using equation 2.8 instead of equation 2.1. For example, we can define the relationship between the i th and k th bins (Fig. 2.1). If N' k is the number of muons after merging angular bins, then, by comparing N′ k = ∑ i N i ,
Figure 2.1 Schematic of merging angular bins in generating a muographic image. Horizontal and vertical axes represent relative azimuth and elevation angle. Minimum angular bins constructed from the dotted gray lines (index i) belong to larger bins defined by the solid black squares (index k). The dashed black line shows the mountain ridge line.
The reconstructed density image obtained with equation 2.3 depends on the values of the parameters σ ρ and L 0 in equation 2.6. One solution to find the best‐fit parameters was described in Nishiyama et al. (2014a), which is to search for the parameters in a forward simulation by minimizing the discrepancy Φ between the assumed input density model
One clear advantage of this linear inversion method compared to the method described in the next section is the higher spatial resolution that can be achieved when combined with gravity observation
