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# VII Inverse Problem

[74.1.1.1] Up to now the direct problem was discussed which consists in finding for given and from the nonlinear eq. (3.2). [74.1.1.2] The inverse problem is to determine or from a knowledge of . [74.1.1.3] This is most important for applications such as well logging. [74.1.1.4] 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.

[74.1.2.1] Consider therefore briefly the problem of determining from eq. (3.2) given and . [74.1.2.2] A general theoretical discussion can be given based on the observation that eq. (3.2) is now linear. [74.1.2.3] It can be written as

 (7.1)

where

 (7.2) (7.3)

[74.1.2.4] Equation (7.1) is a linear Fredholm integral equation of the first kind. [74.1.2.5] Because the kernel is not symmetric, define

[74.1.2.6] 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 , . [74.1.2.7] The are symmetric and have eigenvalues . [74.1.2.8] The normal modes called and can be chosen orthonormal and satisfy

[74.2.0.1] These modes are used to solve eq. (7.1). [74.2.0.2] If (7.1) has any solution, then the inhomogeneity can be written as

 (7.4)

[74.2.0.3] It is assumed that the two sets and have been made orthonormal. [74.2.0.4] Then the solution to eq. (7.1) is given as

 (7.5)

[74.2.0.5] The eigenvalues are the solutions of , where

[74.2.0.6] These brief remarks about the inverse problem are intended to outline the general characteristics of the problem. [74.2.0.7] A more detailed discussion must await the availability of experimentally observed local porosity distibutions.[27]