VII Inverse Problem

[74.1.1.1] Up to now the direct problem was discussed which consists in finding ε⁢ω for given μ⁢ϕ and λ⁢ϕ from the nonlinear eq. (3.2). [74.1.1.2] The inverse problem is to determine α⁢ϕ or μ⁢ϕ from a knowledge of ε⁢ω. [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 α⁢ϕ from ε⁢ω and μ⁢ϕ in view of the fact that the local porosity distribution μ⁢ϕ can be observed much more easily than the local percolation probabilities.

[74.1.2.1] Consider therefore briefly the problem of determining λ⁢ϕ from eq. (3.2) given ε⁢ω and μ⁢ϕ. [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

where

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

[74.1.2.6] General results can be employed to solve eq. (7.1) if f⁢ω is continuous and such that ∫01f⁢ω2⁢d⁢ω exists, and if ∫01∫01Ki⁢ω;ϕ⁢d⁢ω⁢d⁢ϕ exists and Ki⁢ω;ϕ is piecewise continuous in 0≤ω≤1, 0≤ϕ≤1. [74.1.2.7] The Ki are symmetric and have eigenvalues λn2. [74.1.2.8] The normal modes called Ωn⁢i and Φn⁢i 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 f⁢ω can be written as

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

[74.2.0.5] The eigenvalues λn2 are the solutions of D⁢λ2=0, 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]