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[Crossref]

L. Tsang and J. Kong, “Wave theory for microwave remote
sensing of a half-space random medium with three dimensional
variations,” Radio Sci. 14, 359–369 (1979).

[Crossref]

K. Furutsu, “Multiple scattering of waves in a
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[Crossref]

A. Ishimaru, Wave Propagation and Scattering in Random
Media (Academic, New York, 1978); see Chap. 14, where Twersky’s theory is summarized.

V. Frisch, “Wave propagation in random
media,” in Probabilistic Methods in Applied
Mathematics, A. Barucha-Reid, ed. (Academic, New York, 1968).

V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave
Propagation (National Technical Information Source, Springfield, Virginia, 1971); Secs. 60 and 61.

It is demonstrated in Refs. 1 and 2 that the approximation of Eq. (1) by formula (2) is equivalent to the ladder approximation to the Bethe–Salpeter equation. Consequently, we have given this decomposition the name ladder approximation.

For example, when evaluating EE*〉 by taking moments of the integral form of the wave equation, we find that we need to evaluate terms of the form 〈∊(r′)E(r′)E*(r)〉, where ∊(r′) is the relative permittivity fluctuation at r′. If ∊ is a Gaussian random variable, the Novikov–Furutsu theorem (see Ref. 1) gives 〈∊(r′)E(r′)E*(r)〉=∫∫-∞∞∫d3r″〈∊(r′)∊(r″)〉×δ〈E(r′)E*(r)〉δ∊(r″),where δ/δ∊ is a variational derivative. It is readily demonstrated that δE(r)/δ∊(r″) = −k2E(r″)G(r, r″), where G(r, r″) is the random Green’s function and satisfies {∇2+ k2[1 + ∊(r)]}G(r, r′) = δ(r− r′). Therefore 〈∊(r′)E(r′)E*(r)〉=-k2∫∫-∞∞∫d3r″〈∊(r′)∊(r″)〉×〈G(r′,r″)E(r″)E*(r)+G*(r,r″)E(r′)E*(r″)〉,so we see that terms of the form of Eq. (1) must be evaluated.

It is demonstrated in Sec. 37 of Ref. 1 that the representation in formula (4) is valid provided that the spatial scale size of the inhomogeneities is large in comparison with the signal wavelength. In particular, it is shown that formula (4) approximately includes multiple small-angle scatterings. Clearly, however, formula (4) cannot be used in the limit when the inhomogeneities are small in comparison with a wavelength, because both formulas (3) and (4) assume that individual backscatterings are negligible. This is equivalent to Twersky’s assumption (see Ref. 3, Section 14-1) that there are no propagation paths that go through a scatterer more than once.

It is not necessary to assume that ϕ has zero mean. When ϕ has a nonzero mean we can simply lump it in with G0(…) in Eq. (7). That is, G(r′, r″)〉 = [G0(r′, r″)exp(i〈ϕ〉)]exp(−½〈ϕ2(r′, r″)〉), where ϕ is now the fluctuation of the phase about the mean value 〈ϕ〉.

V. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), Chap. 6.

The geometric-optics method gives valid results for the real and imaginary parts of the phase over propagation paths less than l02/λ, where l0is the scale size of the smallest inhomogeneities and λ is the wavelength. For propagation paths that are large in comparison with l02/λ, the geometric-optics method gives incorrect results for the imaginary part of ϕ but gives solutions for the real part of ϕ that are in agreement with results obtained by both the method of smooth perturbations and the Markov approximation method. Consequently, geometric optics gives useful results for the real part of ϕ even for large propagation lengths (i.e., paths large compared with l02/λ).

It is shown is Sec. 39 [see formula (23)] of Ref. 1 that the ray paths can be approximated by straight lines if n2〉 ≪ 1.

These conclusions do not hold in the backscatter cone (i.e., in a cone of angle L/R≪ 1 centered about θ+ γ= 0 in Fig. 1). However, this is unimportant because (1) it represents only a small small fraction of the total integral over ρ and ρ′, and (2) backscattering usually is insignificant in the limit when l is larger than a wavelength, as we have assumed.

For the case when kl 1 we expect that a sufficient condition for Ei(r″)Ej*(r)Gkl(r′, r″)〉 ≃ 〈Ei(r″)Ej*(r)〉 〈(Gkl(r′, r″)〉 is that 〈n2〉 ≪ 1, where Ei is the i th component of the vector E, Gij is the ij component of the Green’s tensor, etc. (We did not need to consider this tensor case when kl≫ 1 because cross polarization is then unimportant.) Probably for kl sufficiently small in comparison with unity the ladder approximation is valid even for 〈n2〉 1, but we have been unable to prove this rigorously.