Abstract

At optical wavelengths, geometrical optics holds that the extinction efficiency of raindrops is equal to two. This approximation yields a wavelength-independent extinction coefficient that, however, can hardly be used to predict accurately rain extinction measured in optical transmissions. Actually, in addition to the extinct direct incoming light, a significant part of the power scattered by the rain particles reaches the receiver. This leads to a reduced apparent extinction that depends on both rain characteristics and link parameters. A simple method is proposed to evaluate this apparent extinction. It accounts for the additional scattered power that enters the receiver when one considers the forward-scattering pattern of the raindrops as well as the multiple-scattering effects using, respectively, the Fraunhofer diffraction and Twersky theory. It results in a direct analytical formula that enables a quick and accurate estimation of the rain apparent extinction and highlights the influence of the link parameters. Predictions of apparent extinction through rain are found in excellent agreement with measurements in the visible and IR regions.

© 1996 Optical Society of America

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References

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  1. A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media. 2: Poly-dispersions,” Appl. Opt. 17, 3169–3176 (1978).
  2. W. G. Tam, A. Zardecki, “Multiple scattering corrections to the Beer-Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
  3. L. R. Bissonnette, “Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 530.
  5. L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume 2 (Academic, New York, 1978), Chap. 14, pp. 253–294.
  7. C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).
  8. T. Manabe, T. Ihara, Y. Furuhama, “Inference of raindrop size distribution from attenuation and rain rate measurements,” IEEE Trans. Antennas Propag. AP-32, 474–478 (1984).
  9. J. S. Marshall, W. M. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).
  10. J. Joss, A. Waldvogel, “Raindrop size distributions and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1969).

1988

1987

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

1984

T. Manabe, T. Ihara, Y. Furuhama, “Inference of raindrop size distribution from attenuation and rain rate measurements,” IEEE Trans. Antennas Propag. AP-32, 474–478 (1984).

1982

1978

1969

J. Joss, A. Waldvogel, “Raindrop size distributions and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1969).

1949

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).

1948

J. S. Marshall, W. M. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).

Bissonnette, L. R.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 530.

Box, M. A.

Brillouin, L.

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).

Carter, D. G.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Deepak, A.

Egget, P. A.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Furuhama, Y.

T. Manabe, T. Ihara, Y. Furuhama, “Inference of raindrop size distribution from attenuation and rain rate measurements,” IEEE Trans. Antennas Propag. AP-32, 474–478 (1984).

Gibbins, C. J.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 530.

Ihara, T.

T. Manabe, T. Ihara, Y. Furuhama, “Inference of raindrop size distribution from attenuation and rain rate measurements,” IEEE Trans. Antennas Propag. AP-32, 474–478 (1984).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume 2 (Academic, New York, 1978), Chap. 14, pp. 253–294.

Joss, J.

J. Joss, A. Waldvogel, “Raindrop size distributions and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1969).

Lidiard, K. A.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Manabe, T.

T. Manabe, T. Ihara, Y. Furuhama, “Inference of raindrop size distribution from attenuation and rain rate measurements,” IEEE Trans. Antennas Propag. AP-32, 474–478 (1984).

Marshall, J. S.

J. S. Marshall, W. M. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).

Palmer, W. M. K.

J. S. Marshall, W. M. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).

Pike, M. G.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Tam, W. G.

Tracey, M. A.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Waldvogel, A.

J. Joss, A. Waldvogel, “Raindrop size distributions and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1969).

White, E. H.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Woodroffe, J. M.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Yilmaz, U. M.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

Zardecki, A.

Appl. Opt.

IEEE Trans. Antennas Propag.

T. Manabe, T. Ihara, Y. Furuhama, “Inference of raindrop size distribution from attenuation and rain rate measurements,” IEEE Trans. Antennas Propag. AP-32, 474–478 (1984).

J. Appl. Phys.

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).

J. Atmos. Sci.

J. Joss, A. Waldvogel, “Raindrop size distributions and sampling size errors,” J. Atmos. Sci. 26, 566–569 (1969).

J. Inst. Electron. Radio Eng.

C. J. Gibbins, D. G. Carter, P. A. Egget, K. A. Lidiard, M. G. Pike, M. A. Tracey, E. H. White, J. M. Woodroffe, U. M. Yilmaz, “A 500 m experimental range for propagation studies at millimetre, infrared and optical wavelengths,” J. Inst. Electron. Radio Eng. 57, 227–234 (1987).

J. Meteorol.

J. S. Marshall, W. M. K. Palmer, “The distribution of raindrops with size,” J. Meteorol. 5, 165–166 (1948).

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume 2 (Academic, New York, 1978), Chap. 14, pp. 253–294.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 530.

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of the link and raindrop scattering geometry.

Fig. 2
Fig. 2

Time series of the apparent extinction coefficient during a rain event (solid curve), along with predictions at 10.6 μm (plus signs) and at 0.63 μm (open circles).

Fig. 3
Fig. 3

Relationships between the apparent rain extinction and the rainfall rate, as measured from joint statistics (asterisks) and as predicted by Eqs. (22) and (23) (solid curves), at 10.6 μm (upper curve) and 0.63 μm (lower curve). Dashed curves are estimated relationships for which we considered theoretical extinction [Eq. (3)] (curve 1) and used single-scattering approximation [Eq. (15)] at the infrared (curve 2) and visible (curve 3) wavelengths.

Equations (24)

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P = P 0 exp ( - α L ) .
α = 10 - 6 0 σ ext ( D ) N ( D ) d D .
α = 10 - 6 0 Q ext N ( D ) π D 2 / 4 d D .
S ( θ , φ ) = x 2 1 + cos θ 2 J 1 ( x sin θ ) x sin θ ,
I diff ( θ , φ ) = I i λ 2 4 π 2 r 2 S ( θ , φ ) 2 ,
σ diff = 0 2 π 0 π I diff ( θ , φ ) I i r 2 sin θ d θ d φ ( mm 2 ) .
Q diff = σ diff π D 2 / 4 = x 2 2 0 π ( 1 + cos θ ) 2 × [ J 1 ( x sin θ ) x sin θ ] 2 sin θ d θ .
θ = arctan ( d / 2 L - z ) .
Q diff = x 2 2 θ π ( 1 + cos θ ) 2 [ J 1 ( x sin θ ) x sin θ ] 2 sin θ d θ .
Q diff = Q diff - x 2 2 0 θ ( 1 + cos θ ) 2 [ J 1 ( x sin θ ) x sin θ ] 2 sin θ d θ .
Q ext = Q ext - x 2 2 0 θ ( 1 + cos θ ) 2 [ J 1 ( x sin θ ) x sin θ ] 2 sin θ d θ .
α = 1 / L 10 - 6 0 L 0 Q ext ( D , z ) N ( D ) π D 2 / 4 d D d z .
α = α - 1 / L 10 - 6 0 L 0 { x 2 2 0 θ ( 1 + cos θ ) 2 × [ J 1 ( x sin θ ) x sin θ ] 2 sin θ d θ } N ( D ) π D 2 / 4 d D d z .
Q ext = Q ext - [ 1 - J 0 2 ( x sin θ ) - J 1 2 ( x sin θ ) ] ,
α = α - 10 - 6 0 ( 1 - tan θ t [ g ( 1 ) - g ( sin θ t ) ] ) N ( D ) π D 2 / 4 d D ,
g ( t ) = - ( 3 x 2 t 2 + 1 t ) J 0 2 ( x t ) - 4 x 2 t 3 J 1 2 ( x t ) + 3 x 2 J 0 ( x t ) J 1 ( x t ) - x 2 t 6 J 0 ( x t ) J 2 ( x t ) - x 6 J 1 ( x t ) J 2 ( x t ) .
I = exp ( - ρ σ abs L ) { exp ( - ρ σ sca L ) + q [ 1 - exp ( - ρ σ sca L ) ] } ,
q = Ω r d Ω s f ( 0 ^ , i ^ s ) 2 4 π d Ω s f ( 0 ^ , i ^ s ) 2 ,
α ( m - 1 ) = - ln ( I ) / L .
q = ( λ / 2 π ) 2 0 2 π 0 θ S ( θ , φ ) 2 sin θ d θ d φ σ sca = Q ext - Q ext Q sca .
q = Q ext - 1 / L 0 L Q ext d z Q sca .
α IR = - 1 / L ln { exp ( - α L ) + 2 q × [ exp ( - α / 2 L ) - exp ( - α L ) ] } ,
α vis = - 1 / L ln { exp ( - α L ) + q [ 1 - exp ( - α L ) ] } ,
N ( D ) = 8000 exp ( - 3.8 R - 0.167 D ) .

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