assume there is no contamination of low momentums due to deflection in the mountain, and no cross talk between the adjacent angular bins in all detectors for simplicity. The elements can be written as:
(2.7)
where is the accidental error of di.
In equations 2.1 and 2.2, we described d i and as the density length; this can be replaced by the number of muons N i such that:
(2.8)
The elements of matrix A ij are different between equations 2.1 and 2.8. In equation 2.1, the elements of Aij can be calculated from the topology, size, and shape of the voxels that are defined. To calculate the elements of matrix , we need to consider the relationship between the density length X i and number of observed muons N i . This can be written as X i = f i (N i ) or N i = g i (X i ) by using the function f i and its inverse function g i . The functions f i and g i are not linear. We then linearize N i around with small variation δX i :
(2.9)
Given that N i can be written as and , δ N i becomes:
(2.10)
Introducing a small variation from the initial density value , δN i and δX i can be given as:
(2.11)
(2.12)
From equations 2.10 and 2.12 we obtain:
(2.13)
Finally, by comparing equations 2.11 and 2.13, the relationship between and is derived as:
(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 , , and using equation 2.8, . Hence, when we merge the bins in angular space, we sum in the same way as the muon numbers. In the case that the mountain ridge line crosses the
