Abstract

Calculations have been carried out for first- and second-order backscattering from water clouds illuminated by a continuous 0.9-μ beam with a finite divergence angle. In the single-scattering calculations several cloud types were used, while only an approximation to fair weather cumulus clouds was used for double scattering. It was found that the intensity and hence the reflectivity varied with the transceiver-cloud distance for both orders of scattering. Second-order backscattering also varied with field of view. From these results a criterion is suggested for determining when the plane parallel atmosphere theories can be used with finite beams.

© 1972 Optical Society of America

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References

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  1. G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 415 (1968).
    [CrossRef] [PubMed]
  2. E. Bauer, Appl. Opt. 3, 197 (1964).
    [CrossRef]
  3. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  4. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, 1965).
  5. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  6. L. W. Carrier, G. A. Cato, K. J. von Essen, Appl. Opt. 7, 1209 (1967).
    [CrossRef]
  7. W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
    [CrossRef]
  8. J. V. Dave, Rep. 320-3237, IBM Scientific Center, Palo Alto, Calif. (1968).
  9. H. Conroy, “Molecular Schroedinger Equation, VIII. A New Method for Evaluation of Multidimensional Integrals,” Mellon Institute, Pittsburgh, Pa. (1967).
  10. H. Margenau, G. M. Murphy, The Mathematics of Physics and Engineers (Van Nostrand Co., New York, 1957).
  11. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 869 (1968).
    [CrossRef] [PubMed]

1968

1967

L. W. Carrier, G. A. Cato, K. J. von Essen, Appl. Opt. 7, 1209 (1967).
[CrossRef]

1964

Bauer, E.

Carrier, L. W.

L. W. Carrier, G. A. Cato, K. J. von Essen, Appl. Opt. 7, 1209 (1967).
[CrossRef]

Cato, G. A.

L. W. Carrier, G. A. Cato, K. J. von Essen, Appl. Opt. 7, 1209 (1967).
[CrossRef]

Conroy, H.

H. Conroy, “Molecular Schroedinger Equation, VIII. A New Method for Evaluation of Multidimensional Integrals,” Mellon Institute, Pittsburgh, Pa. (1967).

Dave, J. V.

J. V. Dave, Rep. 320-3237, IBM Scientific Center, Palo Alto, Calif. (1968).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Irvine, W. M.

W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
[CrossRef]

Kattawar, G. W.

Margenau, H.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Engineers (Van Nostrand Co., New York, 1957).

Murphy, G. M.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Engineers (Van Nostrand Co., New York, 1957).

Plass, G. N.

Pollack, J. B.

W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
[CrossRef]

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

von Essen, K. J.

L. W. Carrier, G. A. Cato, K. J. von Essen, Appl. Opt. 7, 1209 (1967).
[CrossRef]

Appl. Opt.

Icarus

W. M. Irvine, J. B. Pollack, Icarus 8, 324 (1968).
[CrossRef]

Other

J. V. Dave, Rep. 320-3237, IBM Scientific Center, Palo Alto, Calif. (1968).

H. Conroy, “Molecular Schroedinger Equation, VIII. A New Method for Evaluation of Multidimensional Integrals,” Mellon Institute, Pittsburgh, Pa. (1967).

H. Margenau, G. M. Murphy, The Mathematics of Physics and Engineers (Van Nostrand Co., New York, 1957).

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, 1965).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

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Figures (12)

Fig. 1
Fig. 1

Geometry for first-order backscattering.

Fig. 2
Fig. 2

Geometry for second-order backscattering.

Fig. 3
Fig. 3

Deirmendjian’s analytical cloud models: 1, cumulus; 2, corona; and 3, mother-of-pearl.

Fig. 4
Fig. 4

Experimentally determined cloud models reported by Carrier: 1, stratus I; 2, stratus II; 3, stratocumulus; 4, altostratus; 5, fair-weather cumulus; and 6, cumulus congestus.

Fig. 5
Fig. 5

Dependence of single-scattering intensity on the optical depth and transceiver-cloud distance: 1, 10 m; 2, 50 m; 3, 100 m; 4, 500 m; and 5, limiting case of d → ∞.

Fig. 6
Fig. 6

Graph of intensity for first-order backscattering from Deirmendjian’s cloud models as a function of transceiver-cloud distance and concentration: 1, 100 particles/cm3; 2, 150 particles/cm3; 3, 200 particles/cm3; and 4, 300 particles/cm3.

Fig. 7
Fig. 7

Intensity as a function of transceiver-cloud distance for first-order backscattering from Carrier’s cloud models: 1, stratus I; 2, stratus II; 3, stratocumulus; 4, altostratus; 5, fair-weather cumulus; and 6, cumulus congestus.

Fig. 8
Fig. 8

Second-order backscattered intensity at d = 100 m for various matched transmitter-receiver systems. Dashed line represents single-scattering contribution.

Fig. 9
Fig. 9

Dependence of second-order backscattered intensity at d = 100 m on FOV for the divergence angles: 1, 1 mrad; 2, 5 mrad; and 3, 10 mrad. Single-scattering contribution is represented by dashed line.

Fig. 10
Fig. 10

Effect of transmitter divergence angle on strength of glory about the irradiated cloud surface area at d = 100 m for the conditions: 1, α = 1 mrad; 2, α = 5 mrad; 3, α = 10 mrad; and 4, no-glory (single-scattering) case.

Fig. 11
Fig. 11

Second-order backscattered intensity for α = γ = 5 mrad as a function of transceiver-cloud distance at cloud optical depths of 0.25 and 4.0. Single-scattering intensity is represented by dashed lines for comparison.

Fig. 12
Fig. 12

Dependence of second-order backscattered intensity for α = γ = 5 mrad on optical depth at transceiver-cloud distances of 10 m and 100 m. Dashed lines represent single-scattering case.

Tables (1)

Tables Icon

Table I Constants for Deirmendjian’s Particle Size Distribution Equation

Equations (25)

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d H = H 0 F ( ψ ) d V / r 2 ,
H 0 = ( P / Ω r 2 ) exp { β [ r ( d / cos θ ) ] } ,
d H d = [ P F ( π ) / Ω r 2 ] exp { 2 β [ r ( d / cos θ ) ] } sin θ d θ d ϕ d r .
d P d = P d I ( A / d 2 ) ,
d P d = A cos θ d H d .
d I = [ d 2 F ( π ) / Ω r 2 ] exp { 2 β [ r ( d / cos θ ) ] } cos θ sin θ d θ d ϕ d r .
I = d 2 F ( π ) Ω 0 2 π d ϕ 0 α / 2 exp ( 2 β d / cos θ ) × cos θ sin θ [ d / cos θ ( d + D ) / cos θ e 2 β r d r r 2 ] d θ ,
E n ( a ) = 1 e a t t n d t ,
d / cos θ ( d + D ) / cos θ exp ( 2 β r ) r 2 d r = cos θ d 1 exp [ ( 2 β d / cos θ ) z ] z 2 d z cos θ d + D 1 exp 2 β [ ( d + D ) / cos θ ] z z 2 d z .
I = 2 π d 2 F ( π ) Ω 0 α / 2 exp ( 2 β d / cos θ ) cos 2 θ sin θ { 1 d E 2 ( 2 β d cos θ ) 1 d + D E 2 [ 2 β ( d + D ) cos θ ] } d θ .
I = d 2 F ( π ) exp ( 2 β d ) { 1 d E 2 ( 2 β d ) 1 d + D E 2 [ 2 β ( d + D ) ] } .
I d = d F ( π ) exp ( 2 β d ) E 2 ( 2 β d ) .
E 2 ( x ) = ( e x / x ) [ 1 ( 2 / x ) + ( 3 / x 2 ) + ] .
I d = [ F ( π ) / 2 β ] [ 1 exp ( 2 β D ) ] .
I d , D = F ( π ) / 2 β .
H 1 = ( P / Ω r 1 2 ) exp { β [ r 1 ( d / cos θ 1 ) ] } .
d H 2 = [ H 1 F ( θ 2 ) d V 1 / r 2 2 ] exp ( β r 2 ) .
d H d = [ d H 2 F ( ξ ) d V 2 / r 3 2 ] exp { β [ r 3 ( d / cos θ 3 ) ] } .
d I = [ d 2 F ( θ 2 ) F ( ξ ) / Ω cos 2 θ 3 r 3 2 ] exp ( β { r 1 + r 2 + r 3 d [ ( 1 / cos θ 1 ) + ( 1 / cos θ 3 ) ] } ) sin θ 1 d θ 1 d ϕ 1 d r 1 sin θ 2 d θ 2 d ϕ 2 d r 2 ,
r 3 = ( r 1 2 + r 2 2 + 2 r 1 r 2 cos θ 2 ) 1 2 ,
ξ = sin 1 [ ( r 1 / r 3 ) sin θ 2 ] .
I = d 2 exp ( 2 β d ) Ω 0 2 π d ϕ 1 0 α / 2 d d + D 0 2 π 0 π 0 r 2 max F ( θ 2 ) F ( ξ ) r 3 2 × exp [ β ( r 1 + r 2 + r 3 ) ] θ 1 sin θ 2 d r 2 d θ 2 d ϕ 2 d r 1 d θ 1 ,
r 3 = d / cos θ 3 , r 3 = ( d + D ) / cos θ 3 , or θ 3 = γ / 2 ,
β = ρ min ρ max σ ( ρ ) n ( ρ ) d ρ
F ( θ ) = ρ min ρ max i ave ( θ , ρ ) k 2 n ( ρ ) d ρ ,

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