Muography. Группа авторов

density length, but in this paper, we first treat d ⁱ as the density length. Equation 2.1 can be represented by vectors ,, and the matrix A:

(2.2)

In many cases, multi‐directional muography based on equation 2.2 is an under‐determined system that limits the spatial and density resolution. If is the observational value of , then and form a covariance matrix of and , respectively. In addition, we also define as initial values, set the model expectation as , and denote the posterior covariance matrix as . The relationship between these parameters is (e.g., Nishiyama et al., 2014a; Tarantola, 2005):

(2.3)

(2.4)

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:

(2.5)

      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:

(2.6)

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,