A matrix approach to inversion of inhomogeneous attenuation profile through a symmetric layered object (Q1902248)

The authors consider the reconstruction problem for the attenuation coefficient of a symmetric layered object when the number of layers and their geometry are known. The ray source position is fixed and a set of receptors is placed on a receiving plane. The model of this reconstruction problem is a system of linear equations \(Sk = B\). Here \(k\) is the unknown vector of the layer attenuation coefficients, \(B\) is the vector of (logarithmic) measurements and the matrix coefficient \(S_{ij}\) denotes the distance passing through by the \(i\)-th ray in the \(j\)-th layer. The resolving procedure is based on the least squares formula \(k = (S^TS)^{-1} S^TB\). However the invertibility of the matrix \(S^TS\) and its conditioning are not investigated in the paper. Two numerical results for a layered spheroid and a sphere composed of two different halves are obtained.