Abstract

A new analytical expression for the optical transfer function of multiple-scattering media such as clouds, mists, and dust aerosols is given in terms of their microphysical characteristics. The geometrical optics approximation is used to find local optical parameters of a scattering medium, including the simple approximation of the phase function, which is the key to the solution of the problem considered here. The optical transfer function is taken within a small-angle approximation of the radiative transfer theory. A comparison with Monte Carlo data shows a fairly satisfactory accuracy of our analytic formulas.

© 1994 Optical Society of America

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References

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  1. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), Chap. 6, p. 186.
    [CrossRef]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  3. L. Bissonnette, “Imaging through fog and rain,” Opt. Eng. 31, 1045–1052 (1992).
    [CrossRef]
  4. P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Monte-Carlo calculations of modulation transfer function of an optical system operating in a turbid medium,” Appl. Opt. 32, 2813–2824(1993).
    [CrossRef] [PubMed]
  5. A. S. Drofa, A. L. Usachev, “About vision in cloud medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 933–938 (1980).
  6. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  7. K. S. Shifrin, “Scattering of light in a turbid medium,” NASA Rep. TTF-447 (NASA, Washington, D.C., 1951).
  8. W. J. Glantshing, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [CrossRef]
  9. E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 24, 695–701 (1987).
  10. E. P. Zege, A. A. Kokhanovsky, Extinction, Scattering and Absorption of Light by Spherical Particles (Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus, 1992).
  11. L. S. Dolin, “Light beam scattering in a turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 471–478 (1964).
  12. F. Davis, “Gamma function and closed functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1964), p. 80–118.
  13. E. P. Zege, A. A. Kokhanovsky, “To determination of large particles sizes under multiple light scattering in a medium,” Opt. Spektrosk. 72, 220–226 (1991).
  14. L. P. Volnistova, A. S. Drofa, “Quality of image transfer through a light scattering media,” Opt. Spektrosk. 61, 116–121 (1986).
  15. K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).

1993 (1)

1992 (1)

L. Bissonnette, “Imaging through fog and rain,” Opt. Eng. 31, 1045–1052 (1992).
[CrossRef]

1991 (1)

E. P. Zege, A. A. Kokhanovsky, “To determination of large particles sizes under multiple light scattering in a medium,” Opt. Spektrosk. 72, 220–226 (1991).

1987 (1)

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 24, 695–701 (1987).

1986 (1)

L. P. Volnistova, A. S. Drofa, “Quality of image transfer through a light scattering media,” Opt. Spektrosk. 61, 116–121 (1986).

1981 (1)

1980 (1)

A. S. Drofa, A. L. Usachev, “About vision in cloud medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 933–938 (1980).

1964 (1)

L. S. Dolin, “Light beam scattering in a turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 471–478 (1964).

Bissonnette, L.

L. Bissonnette, “Imaging through fog and rain,” Opt. Eng. 31, 1045–1052 (1992).
[CrossRef]

Bruscaglioni, P.

Chen, S.-H.

Davis, F.

F. Davis, “Gamma function and closed functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1964), p. 80–118.

Deirmendjian, D.

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

Dolin, L. S.

L. S. Dolin, “Light beam scattering in a turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 471–478 (1964).

Donelli, P.

Drofa, A. S.

L. P. Volnistova, A. S. Drofa, “Quality of image transfer through a light scattering media,” Opt. Spektrosk. 61, 116–121 (1986).

A. S. Drofa, A. L. Usachev, “About vision in cloud medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 933–938 (1980).

Glantshing, W. J.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Ismaelli, A.

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), Chap. 6, p. 186.
[CrossRef]

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), Chap. 6, p. 186.
[CrossRef]

Kokhanovsky, A. A.

E. P. Zege, A. A. Kokhanovsky, “To determination of large particles sizes under multiple light scattering in a medium,” Opt. Spektrosk. 72, 220–226 (1991).

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 24, 695–701 (1987).

E. P. Zege, A. A. Kokhanovsky, Extinction, Scattering and Absorption of Light by Spherical Particles (Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus, 1992).

Shifrin, K. S.

K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).

K. S. Shifrin, “Scattering of light in a turbid medium,” NASA Rep. TTF-447 (NASA, Washington, D.C., 1951).

Usachev, A. L.

A. S. Drofa, A. L. Usachev, “About vision in cloud medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 933–938 (1980).

Volnistova, L. P.

L. P. Volnistova, A. S. Drofa, “Quality of image transfer through a light scattering media,” Opt. Spektrosk. 61, 116–121 (1986).

Zaccanti, G.

Zege, E. P.

E. P. Zege, A. A. Kokhanovsky, “To determination of large particles sizes under multiple light scattering in a medium,” Opt. Spektrosk. 72, 220–226 (1991).

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 24, 695–701 (1987).

E. P. Zege, A. A. Kokhanovsky, Extinction, Scattering and Absorption of Light by Spherical Particles (Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus, 1992).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), Chap. 6, p. 186.
[CrossRef]

Appl. Opt. (2)

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (1)

E. P. Zege, A. A. Kokhanovsky, “Integral characteristics of light scattering by large spherical particles,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 24, 695–701 (1987).

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (1)

A. S. Drofa, A. L. Usachev, “About vision in cloud medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 933–938 (1980).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

L. S. Dolin, “Light beam scattering in a turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 471–478 (1964).

Opt. Spektrosk. (1)

E. P. Zege, A. A. Kokhanovsky, “To determination of large particles sizes under multiple light scattering in a medium,” Opt. Spektrosk. 72, 220–226 (1991).

Opt. Eng. (1)

L. Bissonnette, “Imaging through fog and rain,” Opt. Eng. 31, 1045–1052 (1992).
[CrossRef]

Opt. Spektrosk. (1)

L. P. Volnistova, A. S. Drofa, “Quality of image transfer through a light scattering media,” Opt. Spektrosk. 61, 116–121 (1986).

Other (7)

K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).

F. Davis, “Gamma function and closed functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1964), p. 80–118.

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

K. S. Shifrin, “Scattering of light in a turbid medium,” NASA Rep. TTF-447 (NASA, Washington, D.C., 1951).

E. P. Zege, A. A. Kokhanovsky, Extinction, Scattering and Absorption of Light by Spherical Particles (Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus, 1992).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), Chap. 6, p. 186.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

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Figures (4)

Fig. 1
Fig. 1

OTF calculated at τ = 1, λ = 0.7 μm, n = 1.33, κ = 0, and μ = 6 based on the proposed Eqs. (1) and (29) for a 0 = 4 μm (solid curve), a 0 = 8 μm (long-dashed curve), and a 0 = 50 μm (short-dashed curve), and based on Monte Carlo calculations for a 0 = 4 μm (asterisks).

Fig. 2
Fig. 2

Plot of the cloud C1 model of −ln S(ω*) versus τ at λ = 0.45 μm based on Eqs. (1) and (29) for ω* = 0 (solid curve), ω* = 3 (long-dashed curve), ω* = 12 (medium-dashed curve), and ω* = 50 (short-dashed curve), and on Monte Carlo calculations for the same values of ω (asterisks).

Fig. 3
Fig. 3

OTF calculated with the proposed Eqs. (1) and (29) at τ = 1, λ = 0.7 μm, n = 1.33, μ = 6, and a 0 = 4 μm, for x = 10−2 (solid curve), κ = 10−3 (short-dashed curve), and κ = 10−4 (long-dashed curve).

Fig. 4
Fig. 4

OTF calculated with the proposed Eqs. (1) and (29) at τ = 1, λ = 0.7 μm, n = 1.53, κ = 0.008, and μ = 2 for a 0 = 30 μm (solid curve), a 0 = 10 μm (short-dashed curve), and a 0 = 5 μm (long-dashed curve).

Tables (3)

Tables Icon

Table 1 PSD Moments M j and Values of Z j = M ¯ j /M j for the Most Frequently Used PSD’sa

Tables Icon

Table 2 Values of α and β as Functions of the Real Part of the Refractive Index n

Tables Icon

Table 3 The Angular Scattering Coefficient σ(θ) (km−1) for the Model Cloud C1 Provided by Mie Calculations6 and Eq. (A10) with Relative Percent Discrepancies Δ

Equations (48)

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S ( ω * ) = exp [ - τ + η ( ω * ) l ] ,
τ = ɛ l ,             η ( ω * ) = 0 1 σ ( ω * y ) d y , σ ( ω * y ) = 1 2 0 σ ( θ ) J 0 ( ω * y θ ) θ d θ .
η ( ω * ) = i = 1 3 η i ( ω * ) ,
η i ( ω * ) = N 0 π a 2 f ( a ) η i ( ω * , a ) d a ,
η i ( ω * , a ) = 1 2 0 1 d y 0 θ J 0 ( ω * y θ ) q i ( θ ) d θ ,
q 1 ( θ ) = 4 J 1 2 ( θ ρ ) / θ 2 ,             q 2 ( θ ) = exp ( - α θ ) , q 3 ( θ ) = T ( 0 ) exp ( - β θ 2 - c ) ,
η 1 ( ω * , a ) = [ 2 π arccos b - 2 ( 2 + b 2 ) ( 1 - b 2 ) 1 / 2 3 π b ] × U + ( 1 - b ) + 4 3 π b ,
η 2 ( ω * , a ) = 1 2 α 2 [ 1 + ( ω * / α ) 2 ] 1 / 2 , η 3 ( ω * , a ) = T ( 0 ) e - c 4 ω * ( π / β ) 1 / 2 erf [ ω * / 2 ( β ) 1 / 2 ] .
η 1 ( ω * , a ) = { 1 - 2 b π ( 1 - b 2 12 ) ,             b < 1 4 3 π b ,             b 1
η 1 ( ω * ) = 4 N 3 π δ 0 δ π a 3 f ( a ) d a + N δ π a 2 f ( a ) × [ 1 - 2 δ π a ( 1 - δ 2 12 a 2 ) ] d a ,
η 2 ( ω * ) = π N M 2 2 α 2 [ 1 - ( ω * / α ) 2 ] 1 / 2 ,
η 3 ( ω * ) = T ( 0 ) N π M 2 * 4 ω * ( π / β ) 1 / 2 erf [ ω * / 2 ( β ) 1 / 2 ] ,
M j = 0 a j f ( a ) d a ,             M j * = 0 a j e - c f ( a ) d a .
η 1 ( ω * ) = N [ π ( M 2 - M ¯ 2 ) - 2 δ ( M 1 - M ¯ 1 ) + δ 3 6 ( M - 1 - M ¯ - 1 ) + 4 3 δ M ¯ 3 ] ,
η 1 = N π M 2 ( 1 - ω * / π ρ 21 ) ,             ρ 21 = k M 2 / M 1 ,
η 1 = 8 k N M 3 / 3 ω * .
η 2 = N π M 2 2 α ω * ,             η 3 = N π M 2 * T ( 0 ) 4 ω * ( π / β ) 1 / 2 ,
η 2 = N π M 2 2 α 2 [ 1 - 1 2 ( ω * α ) 2 ] , η 3 = π N M 2 * T ( 0 ) 4 β [ 1 - 1 12 β ( ω * ) 2 ] .
η ( ω * ) = { 1 - Z 2 - 2 δ a 12 π ( 1 - Z 1 ) + δ 3 6 π a - 12 ( 1 - Z - 1 ) + 4 a 32 3 π δ Z 3 + 1 2 α 2 [ 1 + ( ω * / α ) 2 ] 1 / 2 + T ( 0 ) γ 4 ω * × ( π / β ) 1 / 2 erf [ ω * / 2 ( β ) 1 / 2 ] } π N M 2 ,
a i j = M i / M j ,             Z j = M ¯ j / M j ,             γ = M 2 * / M 2 .
M j * = k = 0 ( - 1 ) k ( 2 α ) k k ! M k + j ,
M 2 * = M 2 ( 1 - 2 α a 32 ) ,
S ( ) = exp ( - τ ) ,
t ( τ , μ 0 = 1 ) = exp { - τ [ 1 2 - 1 4 α 2 - M 2 * M 2 T ( 0 ) 8 β ] } ,
t ( τ , μ 0 = 1 ) = exp [ - τ ( 1 - Λ * ) ] ,
Λ * = 1 2 [ 1 + 1 2 α 2 + T ( 0 ) M 2 * 4 β M 2 ] .
Λ * = 1 2 + 1 4 α 2 + T ( 0 ) 8 β .
Λ * = 1 2 + 1 4 α 2 ,
η ( ω * ) = Σ { 1 + 8 ρ 32 3 π ω * P ( μ + 4 , Δ ) - P ( μ + 3 , Δ ) - 2 Δ π ( μ + 2 ) [ 1 - P ( μ + 2 , Δ ) ] + Δ 3 6 π μ ( μ + 1 ) ( μ + 2 ) [ 1 - P ( μ , Δ ) ] + 1 2 α 2 [ 1 + ( ω * / α ) 2 ] 1 / 2 + T ( 0 ) γ 4 ω * × ( π / β ) 1 / 2 erf ( ω * / 2 β ) } ,
P ( n , y ) = 1 - exp ( - y ) j = 0 n - 1 y j j ! .
P ( θ ) = 4 π N k 2 σ 0 I ( a , θ ) f ( a ) d a ,             0 f ( a ) d a = 1.
I ( a , θ ) = j = 1 I j ( a , θ ) .
I 1 ( a , θ ) = ρ 2 J 1 2 ( ρ θ ) θ 2 I 2 ( a , θ ) = ρ 2 R ( θ ) 4 , I 3 ( a , θ ) = ρ 2 T ( θ ) 4 exp [ - c Δ ( θ ) ] .
R ( θ ) = 1 2 j = 1 2 [ N j 2 ( 1 - q 2 ) 1 / 2 - ( n 2 - q 2 ) 1 / 2 N j 2 ( 1 - q 2 ) 1 / 2 + ( n 2 - q 2 ) 1 / 2 ] 2 ,
T ( θ ) = ( 2 n n 2 - 1 ) 4 ( n q - 1 ) 3 ( n - q ) 3 ( 1 + q 4 ) 2 q 5 ( 1 + n 2 - 2 n q ) 2 ,
Δ ( θ ) = n - q ( 1 + n 2 - 2 n q ) 1 / 2 ,             q = cos θ 2 , N 1 = 1 ,             N 2 = n 2 .
R ( θ ) = exp ( - α θ ) ,             T ( θ ) = T ( 0 ) exp ( - β θ 2 - c ) , Δ ( θ ) = 1 - 2 n - 1 4 ( n - 1 ) θ 2 ,
T ( 0 ) = 1 ( n - 1 ) 2 ( 2 n n + 1 ) 4 ,
I ( a , θ ) = ρ 2 4 j = 1 3 q j ,
q 1 = 4 J 1 2 ( θ ρ ) θ 2 ,             q 2 = exp ( - α θ ) , q 3 = T ( 0 ) exp ( - β θ 2 - c ) ,
σ ( θ ) = N π M 2 × [ D ( θ ) + exp ( - α θ ) + γ T ( 0 ) exp ( - β θ 2 - c ) ] ,
γ = M 2 * / M 2 ,             M 2 = 0 a 2 f ( a ) d a , M 2 * = 0 a 2 exp ( - c ) f ( a ) d a ,
D = 4 0 a 2 f ( a ) J 1 2 ( θ ρ ) d a θ 2 0 a 2 f ( a ) d a ,             σ ( θ ) = σ P ( θ ) .
M 2 = a 0 2 ( 1 + 1 μ ) ( 1 + 2 μ ) , γ = 1 ( 1 + 4 χ ρ 0 / μ ) μ + 3 ,             ρ 0 = k a 0 ,
D ( θ ) = ( ρ 0 μ ) n = 0 ( - n ) n Γ ( 2 n + 3 ) Γ ( 2 n + 5 + μ ) Γ ( μ + 3 ) Γ 2 ( n + 2 ) Γ ( n + 3 ) n ! ( θ ρ 0 2 μ ) 2 n .
J 1 2 ( z ) = ( z 2 ) 2 n = 0 A n z 2 n , A n = ( - 1 ) n Γ ( 2 n + 3 ) 4 n Γ 2 ( n + 2 ) Γ ( n + 3 ) n ! .
P ( 0 ) = π N M 2 ( ρ 0 / μ ) 2 ( μ + 4 ) ( μ + 3 ) / σ ,
P ( 0 ) = ( 1 + 3 / μ ) ( 1 + 4 / μ ) ρ 0 2 / 2.

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