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VII Inverse Problem

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

[74.1.2.1] Consider therefore briefly the problem of determining λϕ from eq. (3.2) given εω and μϕ. [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

 ∫01K⁢ω;ϕ⁢λ⁢ϕ⁢d⁢ϕ=f⁢ω, (7.1)

where

 K⁢ω;ϕ =μ⁢ϕ⁢εC⁢ω;ϕ-ε⁢ωεC⁢ω;ϕ+2⁢ε⁢ω-εB⁢ω;ϕ-ε⁢ωεB⁢ω;ϕ+2⁢ε⁢ω, (7.2) f⁢ω =-∫01εB⁢ω;ϕ-ε⁢ωεB⁢ω;ϕ+2⁢ε⁢ω⁢μ⁢ϕ⁢d⁢ϕ. (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 Kω;ϕ is not symmetric, define

 K1⁢ω,ω′ =∫01K⁢ω,ϕ⁢K⁢ω′,ϕ⁢d⁢ϕ, K2⁢ϕ,ϕ′ =∫01K⁢ω,ϕ⁢K⁢ϕ,ϕ′⁢d⁢ω.

[74.1.2.6] General results can be employed to solve eq. (7.1) if fω is continuous and such that 01fω2dω exists, and if 0101Kiω;ϕdωdϕ exists and Kiω;ϕ is piecewise continuous in 0ω1, 0ϕ1. [74.1.2.7] The Ki are symmetric and have eigenvalues λn2. [74.1.2.8] The normal modes called Ωni and Φni can be chosen orthonormal and satisfy

 λn⁢∫01K⁢ω;ϕ⁢Ωn⁢i⁢ϕ⁢d⁢ϕ=Φn⁢i⁢ω, λn⁢∫01K⁢ω;ϕ⁢Φn⁢i⁢ω⁢d⁢ω=Ωn⁢i⁢ϕ.

[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 fω can be written as

 f⁢ω=∑n,iΦn⁢i⁢ω⁢∫01f⁢ω′⁢Φn⁢i⁢ω′⁢d⁢ω′. (7.4)

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

 λ⁢ϕ=∑i,nλn⁢Ωn⁢i⁢ϕ⁢∫01f⁢ω′⁢Φn⁢i⁢ω′⁢d⁢ω′. (7.5)

[74.2.0.5] The eigenvalues λn2 are the solutions of Dλ2=0, where

 D⁢λ2=∑n=0∞κn⁢λ2⁢n, κ0=1,Ki0⁢x,y=0, κn=-1n⁢∫01Kin⁢x,x⁢d⁢x, Kin⁢x,y=∫01Ki⁢x,t⁢Kin-1⁢t,y⁢d⁢t+κn-1⁢Ki⁢x,y.

[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.