Abstract

Analytical equations for the diffused scattered light correction factor of Sun photometers are derived and analyzed. It is shown that corrections are weakly dependent on the atmospheric optical thickness. They are influenced mostly by the size of aerosol particles encountered by sunlight on its way to a Sun photometer. In addition, the accuracy of the small-angle approximation used in the work is studied with numerical calculations based on the exact radiative transfer equation.

© 2007 Optical Society of America

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  1. W. von Hoyningen-Huene, "Untersuchung von Optischen Eigenschaften einer Aerosolhaltigen Atmosphare zur Ableitung von Aerosolparameteren und ihrer Bedeutung fur Kurzwelligen Strahlungstransfer," Habilitation thesis (Leipzig University, 1975).
  2. M. Shiobara and S. Asano, "Estimation of cirrus optical thickness from Sun photometer measurements," J. Appl. Meteorol. 33, 672-681 (1988).
    [CrossRef]
  3. R. G. Timanovskaya, "Optical parameters of cirrus clouds obtained using ship observations of direct solar light," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 258-262 (1994).
  4. E. P. Zege, I. L. Katsev, I. N. Polonsky, and A. S. Prikhach, "Effect of scattered light on accuracy of measurements of thin cloud optical thickness by a Sun photometer," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 308-316 (1994).
  5. P. P. Anikin, "Optical thickness of semitransparent clouds and evaluation of cloud particle size," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 231-236 (1994).
  6. P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
    [CrossRef]
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, 1997).
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  9. A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).
  10. E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer through a Scattering Medium (Springer, 1991).
    [CrossRef]
  11. V. V. Rozanov and A. A. Kokhanovsky, "The solution of the vector radiative transfer equation using the discrete ordinates technique: selected applications," Atmos. Res. 79, 241-265 (2006).
    [CrossRef]
  12. W. von Hoyningen-Huene and P. Posse, "Nonsphericity of aerosol particles and their contribution to radiative forcing," J. Quant. Spectrosc. Radiat. Transf. 57, 651-688 (1997).
    [CrossRef]
  13. O. Dubovik and M. D. King, "A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements," J. Geophys. Res. 105, 20673-20696 (2000).
    [CrossRef]

2006

V. V. Rozanov and A. A. Kokhanovsky, "The solution of the vector radiative transfer equation using the discrete ordinates technique: selected applications," Atmos. Res. 79, 241-265 (2006).
[CrossRef]

2004

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

2000

O. Dubovik and M. D. King, "A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements," J. Geophys. Res. 105, 20673-20696 (2000).
[CrossRef]

1997

W. von Hoyningen-Huene and P. Posse, "Nonsphericity of aerosol particles and their contribution to radiative forcing," J. Quant. Spectrosc. Radiat. Transf. 57, 651-688 (1997).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, 1997).

1994

R. G. Timanovskaya, "Optical parameters of cirrus clouds obtained using ship observations of direct solar light," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 258-262 (1994).

E. P. Zege, I. L. Katsev, I. N. Polonsky, and A. S. Prikhach, "Effect of scattered light on accuracy of measurements of thin cloud optical thickness by a Sun photometer," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 308-316 (1994).

P. P. Anikin, "Optical thickness of semitransparent clouds and evaluation of cloud particle size," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 231-236 (1994).

1991

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer through a Scattering Medium (Springer, 1991).
[CrossRef]

1988

M. Shiobara and S. Asano, "Estimation of cirrus optical thickness from Sun photometer measurements," J. Appl. Meteorol. 33, 672-681 (1988).
[CrossRef]

1981

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

1975

W. von Hoyningen-Huene, "Untersuchung von Optischen Eigenschaften einer Aerosolhaltigen Atmosphare zur Ableitung von Aerosolparameteren und ihrer Bedeutung fur Kurzwelligen Strahlungstransfer," Habilitation thesis (Leipzig University, 1975).

Anikin, P. P.

P. P. Anikin, "Optical thickness of semitransparent clouds and evaluation of cloud particle size," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 231-236 (1994).

Asano, S.

M. Shiobara and S. Asano, "Estimation of cirrus optical thickness from Sun photometer measurements," J. Appl. Meteorol. 33, 672-681 (1988).
[CrossRef]

Box, M.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Dubovik, O.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

O. Dubovik and M. D. King, "A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements," J. Geophys. Res. 105, 20673-20696 (2000).
[CrossRef]

Holben, B. N.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, 1997).

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer through a Scattering Medium (Springer, 1991).
[CrossRef]

Katsev, I. L.

E. P. Zege, I. L. Katsev, I. N. Polonsky, and A. S. Prikhach, "Effect of scattered light on accuracy of measurements of thin cloud optical thickness by a Sun photometer," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 308-316 (1994).

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer through a Scattering Medium (Springer, 1991).
[CrossRef]

King, M. D.

O. Dubovik and M. D. King, "A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements," J. Geophys. Res. 105, 20673-20696 (2000).
[CrossRef]

Kokhanovsky, A. A.

V. V. Rozanov and A. A. Kokhanovsky, "The solution of the vector radiative transfer equation using the discrete ordinates technique: selected applications," Atmos. Res. 79, 241-265 (2006).
[CrossRef]

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

Livingston, J. M.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Polonsky, I. N.

E. P. Zege, I. L. Katsev, I. N. Polonsky, and A. S. Prikhach, "Effect of scattered light on accuracy of measurements of thin cloud optical thickness by a Sun photometer," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 308-316 (1994).

Posse, P.

W. von Hoyningen-Huene and P. Posse, "Nonsphericity of aerosol particles and their contribution to radiative forcing," J. Quant. Spectrosc. Radiat. Transf. 57, 651-688 (1997).
[CrossRef]

Prikhach, A. S.

E. P. Zege, I. L. Katsev, I. N. Polonsky, and A. S. Prikhach, "Effect of scattered light on accuracy of measurements of thin cloud optical thickness by a Sun photometer," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 308-316 (1994).

Ramirez, S. A.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Redemann, J.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Rozanov, V. V.

V. V. Rozanov and A. A. Kokhanovsky, "The solution of the vector radiative transfer equation using the discrete ordinates technique: selected applications," Atmos. Res. 79, 241-265 (2006).
[CrossRef]

Russel, P. B.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Schmid, B.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Shiobara, M.

M. Shiobara and S. Asano, "Estimation of cirrus optical thickness from Sun photometer measurements," J. Appl. Meteorol. 33, 672-681 (1988).
[CrossRef]

Timanovskaya, R. G.

R. G. Timanovskaya, "Optical parameters of cirrus clouds obtained using ship observations of direct solar light," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 258-262 (1994).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

von Hoyningen-Huene, W.

W. von Hoyningen-Huene and P. Posse, "Nonsphericity of aerosol particles and their contribution to radiative forcing," J. Quant. Spectrosc. Radiat. Transf. 57, 651-688 (1997).
[CrossRef]

W. von Hoyningen-Huene, "Untersuchung von Optischen Eigenschaften einer Aerosolhaltigen Atmosphare zur Ableitung von Aerosolparameteren und ihrer Bedeutung fur Kurzwelligen Strahlungstransfer," Habilitation thesis (Leipzig University, 1975).

Wang, J.

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

Zege, E. P.

E. P. Zege, I. L. Katsev, I. N. Polonsky, and A. S. Prikhach, "Effect of scattered light on accuracy of measurements of thin cloud optical thickness by a Sun photometer," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 308-316 (1994).

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer through a Scattering Medium (Springer, 1991).
[CrossRef]

Atmos. Res.

V. V. Rozanov and A. A. Kokhanovsky, "The solution of the vector radiative transfer equation using the discrete ordinates technique: selected applications," Atmos. Res. 79, 241-265 (2006).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys.

R. G. Timanovskaya, "Optical parameters of cirrus clouds obtained using ship observations of direct solar light," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 258-262 (1994).

E. P. Zege, I. L. Katsev, I. N. Polonsky, and A. S. Prikhach, "Effect of scattered light on accuracy of measurements of thin cloud optical thickness by a Sun photometer," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 308-316 (1994).

P. P. Anikin, "Optical thickness of semitransparent clouds and evaluation of cloud particle size," Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 30, 231-236 (1994).

J. Appl. Meteorol.

M. Shiobara and S. Asano, "Estimation of cirrus optical thickness from Sun photometer measurements," J. Appl. Meteorol. 33, 672-681 (1988).
[CrossRef]

J. Geophys. Res.

O. Dubovik and M. D. King, "A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements," J. Geophys. Res. 105, 20673-20696 (2000).
[CrossRef]

P. B. Russel, J. M. Livingston, O. Dubovik, S. A. Ramirez, J. Wang, J. Redemann, B. Schmid, M. Box, and B. N. Holben, "Sunlight transmission through desert dust and marine aerosols: diffuse light corrections to Sun photometry and pyrheliometry," J. Geophys. Res. 109, D08207 doi: 10.1029/2003jd004292 (2004).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

W. von Hoyningen-Huene and P. Posse, "Nonsphericity of aerosol particles and their contribution to radiative forcing," J. Quant. Spectrosc. Radiat. Transf. 57, 651-688 (1997).
[CrossRef]

Other

W. von Hoyningen-Huene, "Untersuchung von Optischen Eigenschaften einer Aerosolhaltigen Atmosphare zur Ableitung von Aerosolparameteren und ihrer Bedeutung fur Kurzwelligen Strahlungstransfer," Habilitation thesis (Leipzig University, 1975).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, 1997).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer through a Scattering Medium (Springer, 1991).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Phase function of spherical polydispersions with effective radius of 4, 6, 8, 10, and 12 μ m [lower lines as θ 0 correspond to smaller particles (see Eq. (15) starting from a ef = 4 μ m ]. Results obtained using Mie theory are shown by solid curves and the approximation is given by dotted curves. Calculations have been performed for the gamma PSD with the half-width parameter μ = 6 and λ = 0.5 μ m . The complex refraction index m = 1.52 0.008 i was used in exact numerical calculations.

Fig. 2
Fig. 2

Dependence of correction factor on (a) effective radius ( θ 0 = 0.6 ° ) and (b) scaling parameter. The input for calculations is the same as for Fig. 1.

Fig. 3
Fig. 3

Dependence of transmission function at θ = 1 ° on optical thickness according to the approximation (solid curve) and exact calculations (open circles) at a ef = 4 μ m . Other input parameters as for Fig. 1.

Fig. 4
Fig. 4

Same as Fig. 3 except in a broader range of τ.

Fig. 5
Fig. 5

Comparison of the exact theory (solid curves) and approximation (dotted curves) for different values of optical thickness (0.1, 0.3, 0.5, 1.0) and a ef = 4 μ m . Other input parameters are as for Fig. 1.

Fig. 6
Fig. 6

Same as Fig. 5 except at τ = 2 , 5 , 10 .

Fig. 7
Fig. 7

Error of the approximation at different values of τ shown in the legend and a ef = 4 μ m .

Fig. 8
Fig. 8

Dependence of correction factor on optical thickness at θ 0 = 0.6 ° and a ef = 2 , 4, 6, 8, 10, 12, 15, 20, and 30 μ m . Lower curves correspond to smaller sizes starting from a ef = 2 μ m . (The coefficients κ j have been calculated using the Mie theory for the same conditions as for Fig. 1; they are shown in Fig. 9.)

Fig. 9
Fig. 9

Coefficients h j for the cases shown in Fig. 8. Lower curves correspond to smaller sizes starting from a ef = 2 μ m .

Tables (1)

Tables Icon

Table 1 Selected Ratios of Moments for Gamma and Lognormal PSDs

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

F = Ω 0 A I d Ω ,
F = Σ Ω 0 I d Ω ,
I dir = E 0 exp ( x ) δ ( Ω 0 Ω ) ,
I dif = ω 0 E 0 p ( θ ) x exp ( x ) 4 π ,
F = Σ E 0 exp ( x ) ( 1 + f x ) ,
f = ω 0 2 0 θ 0 p ( θ ) sin θ d θ ,
F = Σ E 0 exp [ ( 1 f ) x ] ;
τ = C τ 0 ,
C = 1 1 f
i ( θ ) = ρ 4 4 Φ 2 ( θ ρ ) ,
Φ ( θ ρ ) = 2 J 1 ( θ ρ ) θ ρ ,
p ( θ ) = 2 π N 0 φ ( r ) ( i 1 + i 2 ) d r k 2 K sca ,
K sca = N 0 π r 2 φ ( r ) Q sca d r .
p ̑ ( θ ) = 2 J 1 2 ( k θ r ) θ 2 ,
y ( k θ r ) = 0 y ( k θ r ) r 2 φ ( r ) d r 0 r 2 φ ( r ) d r
p ̑ ( 0 ) = k 2 M 42 2 ,
M n = 0 r n φ ( r ) d r
p ̑ ( θ ) = k 2 M 42 2 ( 1 k 2 M 64 θ 2 4 ) .
f = 1 2 [ 1 J 0 2 ( k θ 0 r ) J 2 2 ( k θ 0 r ) ] ,
τ = τ 0 1 1 2 [ 1 J 0 2 ( k θ 0 a ) J 2 2 ( k θ 0 a ) ] ,
C = 2 1 + J 0 2 ( k θ 0 a ) + J 2 2 ( k θ 0 a ) .
μ d I ( μ , τ ) d τ = I ( μ , τ ) + ω 0 2 1 1 I ( τ , η ) p ¯ ( η , μ ) d η ,
p ¯ ( μ , η ) = j = 0 h j P j ( μ ) P j ( η )
p ( θ ) = j = 0 h j P j ( cos θ ) .
d I ( μ , τ ) d τ = I ( μ , τ ) + ω 0 2 1 1 I ( η , τ ) p ¯ ( η , μ ) d η .
I ( μ , τ ) = j = 0 ν j ( τ ) P j ( μ )
d ν j d τ = ν j + ω 0 h j 2 j + 1 ν j ,
1 1 P i ( η ) P j ( η ) d η = 2 2 j + 1 δ i j .
ν j = A j exp ( c j τ ) ,
c j = 1 ω 0 h j 2 j + 1
I ( μ , 0 ) = E 0 δ ( 1 μ ) .
δ ( 1 μ ) = 1 4 π j = 0 ( 2 j + 1 ) P j ( μ ) ,
A j = E 0 4 π ( 2 j + 1 ) .
I ( μ ) = E 0 4 π j = 0 ( 2 j + 1 ) exp ( c j τ ) P j ( μ ) .
I dif ( μ ) = E 0 4 π j = 0 ( 2 j + 1 ) [ exp ( c j τ ) exp ( τ ) ] P j ( μ ) ,
I dir ( μ ) = E 0 exp ( τ ) 4 π j = 0 ( 2 j + 1 ) P j ( μ ) .
I dif ( μ ) = E 0 τ 4 π j = 0 ( 2 j + 1 ) ( 1 c j ) P j ( μ ) ,
I dif ( μ ) = ω 0 E 0 p ( θ ) τ 4 π ,
I dif ( θ ) = E 0 exp ( τ ) 2 π j = 0 α j [ exp ( κ j τ ) 1 ] J 0 ( α j θ ) ,
κ j = ω 0 h j 2 j + 1 .
T = π I dif μ 0 E 0 .
T = exp ( τ ) 2 j = 0 α j [ exp ( κ j τ ) 1 ] J 0 ( α j θ ) .
F dif = Σ E 0 exp ( τ ) j = 0 α j [ exp ( κ j τ ) 1 ] D j ( θ 0 ) ,
D j ( θ 0 ) = 0 θ 0 J 0 ( α j θ ) sin θ d θ .
D j ( θ 0 ) = θ 0 J 1 ( α j θ 0 ) α j ,
J 0 ( s ) s d s = s J 1 ( s ) ,
F dif = Σ E 0 θ 0 ψ ( τ ) exp ( τ ) ,
ψ ( τ ) = j = 0 [ exp ( κ j τ ) 1 ] J 1 ( α j θ 0 ) .
F = Σ E 0 [ 1 + θ 0 ψ ( τ ) ] exp ( τ ) ,
F = Σ E 0 exp ( τ 0 ) ,
τ 0 = τ { 1 γ ( τ ) } ,
γ ( τ ) = 1 τ ln [ 1 + θ 0 ψ ( τ ) ] .
C = 1 1 γ ( τ ) .
ψ ( τ ) = τ j = 0 κ j J 1 ( α j θ 0 ) ,
γ = θ 0 j = 0 κ j J 1 ( α j θ 0 ) ,
γ = ω 0 2 θ 0 j = 0 h j J 1 ( α j θ 0 ) α j ,

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