Tarantola, 2005):
We now consider the meaning of the covariance matrices used in equations 2.3 and 2.4 in the field of probability theory. If is a set of probability variables, then the covariance matrix
is defined as follows:
where V(x i ) is the deviation of x i and Cov(x i , x j ) = E[(x i − E[x i ])(x j − E[x j ])] (E[x i ] is the expected value of the probability variable x i ) and qualitatively reflects the correlation between x i and x j .
When solving for using equations 2.3 and 2.4, the systemis typically under‐determined, and thus every correlation Cov(ρ i , ρ j ) cannot be fixed. As such, we introduce another assumption into the density model. For the simplicity and effectivity for an under‐determined system, we may constrain the relationship between the i th and j th voxels as follows:
where σ ρ is the density contrast deviation, l(i, j) is the distance between the i th and j th voxel, and L 0 is the correlation length. Equation 2.6 assumes that the internal density is continuous on a spatial scale L 0, and that the density contrast is typically within σ ρ . However, equation 2.6 is just one possible example, and it is possible to assume different constraints in the model based on the expected structure. In this case, ρ 0, σ ρ , and L 0 are a priori parameters.
We now consider the matrix elements of in the simplest case. This matrix is also represented like equation 2.5, but the covariance between d i and d j (i ≠ j) is zero, if we 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:
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