[22.214.171.124] Up to now the direct problem was discussed which consists in finding for given and from the nonlinear eq. (3.2). [126.96.36.199] The inverse problem is to determine or from a knowledge of . [188.8.131.52] This is most important for applications such as well logging. [184.108.40.206] Particularly important is the problem of determining from and in view of the fact that the local porosity distribution can be observed much more easily than the local percolation probabilities.
[220.127.116.11] Consider therefore briefly the problem of determining from eq. (3.2) given and . [18.104.22.168] A general theoretical discussion can be given based on the observation that eq. (3.2) is now linear. [22.214.171.124] It can be written as
[126.96.36.199] Equation (7.1) is a linear Fredholm integral equation of the first kind. [188.8.131.52] Because the kernel is not symmetric, define
[184.108.40.206] General results can be employed to solve eq. (7.1) if is continuous and such that exists, and if exists and is piecewise continuous in , . [220.127.116.11] The are symmetric and have eigenvalues . [18.104.22.168] The normal modes called and can be chosen orthonormal and satisfy
[22.214.171.124] It is assumed that the two sets and have been made orthonormal. [126.96.36.199] Then the solution to eq. (7.1) is given as
[188.8.131.52] The eigenvalues are the solutions of , where
[184.108.40.206] These brief remarks about the inverse problem are intended to outline the general characteristics of the problem. [220.127.116.11] A more detailed discussion must await the availability of experimentally observed local porosity distibutions.