Abstract

It is shown with the help of an operational formulation of scattering processes (in linear media) that the Rytov approximation, although limited to weak scattering, contains all orders of single and multiple scatter, unlike the Born approximation, which neglects multiple scatter. This explains the often observed superiority of the former vis-à-vis the latter, when compared with empirical data. These results are quite general and apply to scattering from random interfaces as well as volumes, including combinations of surface and volume interactions. It is also noted that probability density functions (pdf’s) consisting of Gauss (or Rayleigh) scatter and non-Gaussian Class A components, under various conditions, sometimes involving intensity fluctuations, provide essentially exact pdf’s for all intensities and orders of multiple scatter. [IEEE J. Ocean. Eng. 24, 261 (1999); Proceedings of the Third International Conference on Theoretical and Computational Acoustics (World Scientific, Singapore, 1999), p. 679].

© 1999 Optical Society of America

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References

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  1. D. Middleton, “New physical–statistical methods and models for clutter and reverberation: the KA-distribution and related probability structures,” IEEE J. Ocean Eng. 24, 261–284 (1999).
    [CrossRef]
  2. D. Middleton, “The first and higher order probability densities and distributions of ocean acoustic reverberation from combined surface, bottom, and volume interactions,” in Proceedings of the Third International Conference on Theoretical and Computational Acoustics, 1997 (World Scientific, Singapore, 1999), pp. 679–694.
  3. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Secs. 8.4.3, 8.4.4, and the comments in Sec. 8.9; cf. p. 458 and Refs. 8–48 and 49 therein.
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media, I and II (Academic, New York, 1978), Sec. 17-2-2.
  5. J. M. Rytov, A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radio Physics 3, Wave Propagation Through Random Media (Springer-Verlag, New York, 1989); cf. Sec. 1.7.
  6. J. B. Keller, “Accuracy and validity of the Born and Rytov approximations,” J. Opt. Soc. Am. 59, Part 1, pp. 1003–1004 (1969).
  7. H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. 73, 500–502 (1983).
    [CrossRef]
  8. U. Frisch, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Barucha-Reid, ed. (AcademicNew York, 1968), Vol. 1, pp. 75–198.
  9. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, I and II (McGraw-Hill, New York, 1953), Sec. 7.3.
  10. Corresponding to the dominant (far) E-field component of EM radiation.
  11. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va. 22151, 1971), cf. Chap. 5.
  12. D. Middleton, “Canonical and quasi-canonical non-gaussian noise models of Class A interference,” IEEE Trans. Electromagn. Compat. 25, 96–106 (1983).
  13. D. Middleton, “Channel modeling and threshold signal processing in underwater acoustics: an analytic overview,” IEEE J. Ocean Eng. 12, 4–28 (1987).
    [CrossRef]

1999 (1)

D. Middleton, “New physical–statistical methods and models for clutter and reverberation: the KA-distribution and related probability structures,” IEEE J. Ocean Eng. 24, 261–284 (1999).
[CrossRef]

1987 (1)

D. Middleton, “Channel modeling and threshold signal processing in underwater acoustics: an analytic overview,” IEEE J. Ocean Eng. 12, 4–28 (1987).
[CrossRef]

1983 (2)

H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. 73, 500–502 (1983).
[CrossRef]

D. Middleton, “Canonical and quasi-canonical non-gaussian noise models of Class A interference,” IEEE Trans. Electromagn. Compat. 25, 96–106 (1983).

1969 (1)

Clifford, S. F.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, I and II (McGraw-Hill, New York, 1953), Sec. 7.3.

Frisch, U.

U. Frisch, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Barucha-Reid, ed. (AcademicNew York, 1968), Vol. 1, pp. 75–198.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Secs. 8.4.3, 8.4.4, and the comments in Sec. 8.9; cf. p. 458 and Refs. 8–48 and 49 therein.

Hill, R. J.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, I and II (Academic, New York, 1978), Sec. 17-2-2.

Keller, J. B.

Kravtsov, A.

J. M. Rytov, A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radio Physics 3, Wave Propagation Through Random Media (Springer-Verlag, New York, 1989); cf. Sec. 1.7.

Middleton, D.

D. Middleton, “New physical–statistical methods and models for clutter and reverberation: the KA-distribution and related probability structures,” IEEE J. Ocean Eng. 24, 261–284 (1999).
[CrossRef]

D. Middleton, “Channel modeling and threshold signal processing in underwater acoustics: an analytic overview,” IEEE J. Ocean Eng. 12, 4–28 (1987).
[CrossRef]

D. Middleton, “Canonical and quasi-canonical non-gaussian noise models of Class A interference,” IEEE Trans. Electromagn. Compat. 25, 96–106 (1983).

D. Middleton, “The first and higher order probability densities and distributions of ocean acoustic reverberation from combined surface, bottom, and volume interactions,” in Proceedings of the Third International Conference on Theoretical and Computational Acoustics, 1997 (World Scientific, Singapore, 1999), pp. 679–694.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, I and II (McGraw-Hill, New York, 1953), Sec. 7.3.

Rytov, J. M.

J. M. Rytov, A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radio Physics 3, Wave Propagation Through Random Media (Springer-Verlag, New York, 1989); cf. Sec. 1.7.

Sung, C. C.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va. 22151, 1971), cf. Chap. 5.

J. M. Rytov, A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radio Physics 3, Wave Propagation Through Random Media (Springer-Verlag, New York, 1989); cf. Sec. 1.7.

Yura, H. T.

IEEE J. Ocean Eng. (2)

D. Middleton, “New physical–statistical methods and models for clutter and reverberation: the KA-distribution and related probability structures,” IEEE J. Ocean Eng. 24, 261–284 (1999).
[CrossRef]

D. Middleton, “Channel modeling and threshold signal processing in underwater acoustics: an analytic overview,” IEEE J. Ocean Eng. 12, 4–28 (1987).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

D. Middleton, “Canonical and quasi-canonical non-gaussian noise models of Class A interference,” IEEE Trans. Electromagn. Compat. 25, 96–106 (1983).

J. Opt. Soc. Am. (2)

Other (8)

D. Middleton, “The first and higher order probability densities and distributions of ocean acoustic reverberation from combined surface, bottom, and volume interactions,” in Proceedings of the Third International Conference on Theoretical and Computational Acoustics, 1997 (World Scientific, Singapore, 1999), pp. 679–694.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Secs. 8.4.3, 8.4.4, and the comments in Sec. 8.9; cf. p. 458 and Refs. 8–48 and 49 therein.

A. Ishimaru, Wave Propagation and Scattering in Random Media, I and II (Academic, New York, 1978), Sec. 17-2-2.

J. M. Rytov, A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radio Physics 3, Wave Propagation Through Random Media (Springer-Verlag, New York, 1989); cf. Sec. 1.7.

U. Frisch, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. T. Barucha-Reid, ed. (AcademicNew York, 1968), Vol. 1, pp. 75–198.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, I and II (McGraw-Hill, New York, 1953), Sec. 7.3.

Corresponding to the dominant (far) E-field component of EM radiation.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va. 22151, 1971), cf. Chap. 5.

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Tables (1)

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Table 1 Scatter Models and Their PDF’s

Equations (61)

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{(L^(0)-Qˆ)α(Q)(R, t)=-GT+(i.c.s+b.c.s)};
GT=G^TSin=h^TSin,h^T
=-hT(τ, t|ξ)Sin(t-τ, ξ)dτ,
{α(Q)=(1ˆ-η^)-1αH},
αH(R, t)=-M^GT, η^(M^Qˆ),
M^(R, t|R, t)
=-dtdRdθg(R, t|R, t)()R,t,θ
andL^(0)g=-δttδRR,
α(Q)(R, t)=k=1η^(k)αH+αH=αH+k=1α(k),
{X(t)=Rˆα(Q)(R, t)},
Rˆ=VRhR(t-τ, t|η)()R(η),τdτ,
Xvol(Q)(t)=Rˆ[α(Q)(R, t)=αH(R, t)+dtdRg(R, t|R, t)×Q(R, t)α(Q)(R, t)],
X(Q)(t)surf.=RˆdtΣ{α(Q)(R, t)nˆg(R, t|R, t)-g(R, t|R, t)nˆα(Q)(R, t)}ΣdΣ,
η^,S(R0)dtΣnˆ{Ro g()}dΣ,
η^,Σ(To)=dtΣ[()n^2To g-To gn^2()]dΣ,
2-1aoR [1+aR(R, t)] t-1co2 [1+(R, t)] 2t2×α(Q)(R, t)=-GT+(i.c.s+b.c.s),
L^(0)2-1aoR t-1co2 2t2;
Q^V=aRaoR t+co2 2t2;
g=Fs-1Fk-1{(k2+s/aoR+s2/co2)-1}.
Xtotal(Q)(t)=Xvol(Q)(t)+Xsurf(Q)(t)+Xbot(Q)(t),
η^=η^,S+η^,V+η^,B,
α(Q)=α(Q)(R, t|Sin)=L(Sin|R, t)
α(Q)(|Sin(1)+Sin(2))=α(Q)(|Sin(1))+α(Q)(|Sin(2)),
η^(k)=(MˆQˆ)(k)=(MˆQˆ)(MˆQˆ)(MˆQˆ).
α(Q)(R, t)1αH+ηˆαH=α(0)+ηˆα(0);
[α(0)αH].
α(Q)(R, t)α(0)(R, t)exp[ψ(R, t)],
orlog α(Q)=log α(0)+ψ(R, t),
log α(Q)=log[α(0)+H^α(0)],
log α(Q)=log α(0)+log[1+H^α(0)/α(0)]
α(k)=(H^)kα(0),
ψ(R, t)=ψ0(R, t)+ψ1(R, t)+=l=0ψl(R, t)[=log α(Q)]
ψ(R, t)=ψ=ψ0+ψ1orlog α(Q)ψ0+ψ1.
ψ0=log α(0)orα(0)=exp(ψ0),
l=1ψl=log[1+H^α(0)/α(0)]=log(1+z)=z-z2/2+=ψ1+ψ2+ ,
ψ1=[H^α(0)]/α(0),|[H^α(0)]/α(0)|2,
α(Q)|Rytov=α(0) exp(ψ1)=α(0) exp{[H^α(0)]/α(0)}.
α(Q)|Rytov=α(0)(1+ψ1+ψ12/2!+)
α(0)(1+ψ1)
=α(0)+H^α(0)=α(Q)(exact);
|ψ1|2orH^α(0)=k=1η(k)α(0)2.
w1(E|Sin)R=E exp(-E2/2ψc)ψc,E0;
(ψc=E2¯/2=averageintensity),
w1(E|Sin)R=22ψ¯c1/2Γ(β+1) Eβ+12ψ¯cβ+1β+1×Kβ2(β+1)Eψ¯c1/2,
(E0, β>-1),
 w1(E|Sin)A+Rexp[-A1()]m=0A1()mm!×E2ψσ^m2 exp(-E2/4ψσ^m2),
2σ^m2(m/A+ΓA)/(1+ΓA),
ΓAσG2/Ω2A(),ψc=σG2+Ω2A(),
w1(E|Sin)R+A
=(m) 2ψ¯c1/2σ^mΓ(β+1) E2σ^m β+1ψ¯cβ+1β+1
×Kβ2(β+1)E(σ^m2ψ¯c)1/2,
α(Q)|Rytov=α(0)(1+ψ1)exp(ψ0+ψ1),
|ψ1|1/2.
ψ1ψR-ψ0=k=1α(k)/|α(0)|+i(Φ-ΦOR)Ψ+iΔΦ,
w1(E)Rytov1(2πσΨ2)1/2E
×exp{-[log(E/EOR)-Ψ¯]2/2σΨ2};
σΨ2=Ψ2¯-Ψ¯2,
E=Rˆk=1α(k);EOR=Rˆ|α(0)|,
andE=EOR exp(Ψ).
w1(E)Rytov1(2πσΨ2)1/2E exp{-[log(E/2I0)+σΨ2]2/2σΨ2};
Ψ2¯=σΨ2(1+σΨ2).

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