Abstract

The atmosphere is often divided into several homogeneous layers in simulations of radiative transfer in plane-parallel media. This artificial stratification introduces discontinuities in the vertical distribution of the inherent optical properties at boundaries between layers, which result in discontinuous radiances and irradiances at layer interfaces, which lead to errors in the radiative transfer simulations. To investigate the effect of the vertical discontinuity of the atmosphere on radiative transfer simulations, a simple two layer model with only aerosols and molecules and no gas absorption is used. The results show that errors larger than 10% for radiances and several percent for irradiances could be introduced if the atmosphere is not layered properly.

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (1)

J. Xie and X. Xia, “Long-term trend in aerosol optical depth from 1980 to 2001 in north China,” Particuology 6(2), 106–111 (2008).
[CrossRef]

2005 (1)

M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. 110(D15), D15201 (2005), doi:.
[CrossRef]

2004 (3)

Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3-4), 243–259 (2004).
[CrossRef]

Q. Min, E. Joseph, and M. Duan, “Retrievals of thin cloud optical depth from a multifilter rotating shadowband radiometer,” J. Geophys. Res. 109(D2), D02201 (2004), doi:.
[CrossRef]

M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:.
[CrossRef]

2002 (1)

M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. 59(3), 524–543 (2002).
[CrossRef]

1997 (1)

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997).
[CrossRef]

1996 (1)

Q. Min and L. Harrison, “Cloud Properties Derived From Surface MFRSR Measurements and Comparison With GOES Results at the ARM SGP Site,” Geophys. Res. Lett. 23(13), 1641 (1996).
[CrossRef]

1994 (2)

1993 (1)

1992 (1)

F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 477–489 (1992).
[CrossRef]

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[CrossRef]

1990 (1)

1988 (1)

1987 (1)

Alexandrov, M.

M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:.
[CrossRef]

M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. 59(3), 524–543 (2002).
[CrossRef]

Berndt, J.

Cairns, B.

M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:.
[CrossRef]

M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. 59(3), 524–543 (2002).
[CrossRef]

Carlson, B.

M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. 59(3), 524–543 (2002).
[CrossRef]

Carlson, B. E.

M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:.
[CrossRef]

Castaño, D. J.

De Haan, J. F.

F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 477–489 (1992).
[CrossRef]

Deuze, J. L.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997).
[CrossRef]

Duan, M.

M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. 110(D15), D15201 (2005), doi:.
[CrossRef]

Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3-4), 243–259 (2004).
[CrossRef]

Q. Min, E. Joseph, and M. Duan, “Retrievals of thin cloud optical depth from a multifilter rotating shadowband radiometer,” J. Geophys. Res. 109(D2), D02201 (2004), doi:.
[CrossRef]

M. Duan and Q. Min, “A polarized radiative transfer model based on successive order of scattering method,” Adv. Atmos. Sci. Doi: (in print).

Evans, K. F.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[CrossRef]

Gordon, H. R.

Harrison, L.

Herman, M.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997).
[CrossRef]

Hovenier, J. W.

F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 477–489 (1992).
[CrossRef]

Jayaweera, K.

Joseph, E.

Q. Min, E. Joseph, and M. Duan, “Retrievals of thin cloud optical depth from a multifilter rotating shadowband radiometer,” J. Geophys. Res. 109(D2), D02201 (2004), doi:.
[CrossRef]

Kuik, F.

F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 477–489 (1992).
[CrossRef]

Lacis, A.

M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. 59(3), 524–543 (2002).
[CrossRef]

Lacis, A. A.

M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:.
[CrossRef]

Li, J.

M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. 110(D15), D15201 (2005), doi:.
[CrossRef]

Marshak, A.

M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:.
[CrossRef]

Michalsky, J.

Min, Q.

M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. 110(D15), D15201 (2005), doi:.
[CrossRef]

Q. Min, E. Joseph, and M. Duan, “Retrievals of thin cloud optical depth from a multifilter rotating shadowband radiometer,” J. Geophys. Res. 109(D2), D02201 (2004), doi:.
[CrossRef]

Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3-4), 243–259 (2004).
[CrossRef]

Q. Min and L. Harrison, “Cloud Properties Derived From Surface MFRSR Measurements and Comparison With GOES Results at the ARM SGP Site,” Geophys. Res. Lett. 23(13), 1641 (1996).
[CrossRef]

M. Duan and Q. Min, “A polarized radiative transfer model based on successive order of scattering method,” Adv. Atmos. Sci. Doi: (in print).

Morcette, J.-J.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997).
[CrossRef]

Stamnes, K.

Stephens, G. L.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[CrossRef]

Tanre, D.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997).
[CrossRef]

Teillet, P. M.

Tsay, S. C.

Vermote, E. F.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997).
[CrossRef]

Wang, M.

Wiscombe, W.

Xia, X.

J. Xie and X. Xia, “Long-term trend in aerosol optical depth from 1980 to 2001 in north China,” Particuology 6(2), 106–111 (2008).
[CrossRef]

Xie, J.

J. Xie and X. Xia, “Long-term trend in aerosol optical depth from 1980 to 2001 in north China,” Particuology 6(2), 106–111 (2008).
[CrossRef]

Adv. Atmos. Sci. (1)

M. Duan and Q. Min, “A polarized radiative transfer model based on successive order of scattering method,” Adv. Atmos. Sci. Doi: (in print).

Appl. Opt. (6)

Geophys. Res. Lett. (2)

M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:.
[CrossRef]

Q. Min and L. Harrison, “Cloud Properties Derived From Surface MFRSR Measurements and Comparison With GOES Results at the ARM SGP Site,” Geophys. Res. Lett. 23(13), 1641 (1996).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997).
[CrossRef]

J. Atmos. Sci. (1)

M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. 59(3), 524–543 (2002).
[CrossRef]

J. Geophys. Res. (2)

M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. 110(D15), D15201 (2005), doi:.
[CrossRef]

Q. Min, E. Joseph, and M. Duan, “Retrievals of thin cloud optical depth from a multifilter rotating shadowband radiometer,” J. Geophys. Res. 109(D2), D02201 (2004), doi:.
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (3)

Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3-4), 243–259 (2004).
[CrossRef]

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[CrossRef]

F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 477–489 (1992).
[CrossRef]

Particuology (1)

J. Xie and X. Xia, “Long-term trend in aerosol optical depth from 1980 to 2001 in north China,” Particuology 6(2), 106–111 (2008).
[CrossRef]

Other (2)

A. Berk, L. S. Bernstein, and D. C. Robertson, “MODTRAN: A moderate resolution model for LOWTRAN 7,” GL-TR-89–0122, Phillips Laboratory, ADA214337 (1989).

G. E. Thomas, and K. Stamnes, Radiative transfer in the atmosphere and ocean. (Cambridge, 1999), P160.

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of the two-layer model.

Fig. 2
Fig. 2

Scattering phase functions

Fig. 3
Fig. 3

Horizontal radiances (left three panels) and relative errors (right three panels) at z = 2km computed from downward radiance in the upper layer (2km + ) and upward radiance in the lower layer (2km-) with the two layer model. The true value is derived from the improved SOSVRT using a continuous IOP profile. The aerosol optical depths for layer 1 and 2 are 0.147 and 0.253, respectively, and the Rayleigh scattering optical depths for layer 1 and 2 are 0.246 and 0.070, respectively. The IOPs of each layer is given by Eqs. (20)(22), and the cosine of the solar zenith angle was μ 0 = 0.6.

Fig. 4
Fig. 4

Relative differences in upward radiances at TOA and downward radiances at the surface produced by the 2-layer model for three different values of the aerosol single scattering albedo ωa . The optical properties of the two layers are the same as shown in Fig. 3.

Fig. 5
Fig. 5

, Irradiances (fluxes) (left axis) and corresponding errors (right axis) versus aerosol single scattering albedo. The cosine of the solar zenith angle is μ 0 = 0.6

Fig. 6
Fig. 6

, Maximum error of radiance and irradiances versus the number of layers used in the simulation.

Equations (25)

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μ d I ( τ , μ , ϕ ) d τ = I ( τ , μ , ϕ ) + J ( τ , μ , ϕ )
J ( τ , μ , ϕ ) = ω 4 π π F 0 exp τ / μ 0 P ( τ , μ , ϕ ; μ 0 , ϕ 0 ) + ω 4 π 0 2 π 1 1 I ( τ , μ ' , ϕ ' ) P ( τ , μ , ϕ ; μ ' , ϕ ' ) d μ ' ϕ '
τ = z β ( x ) d x
I ( τ , μ , ϕ ) = I ( 0 , μ , ϕ ) e τ / μ + 0 τ J ( t ) e ( τ t ) / μ d t / μ
I ( τ , μ , ϕ ) = I ( b , μ , ϕ ) e ( b τ ) / μ + τ b J ( t ) e ( t τ ) / μ d t / μ
I ( τ , 0 , ϕ ) = I ( τ , 0 , ϕ )
J = ω 4 π exp ( τ / μ 0 ) P ( τ , μ , ϕ ; μ 0 , ϕ 0 ) F 0
I 1 ( τ , μ , ϕ ) = I 1 ( 0 , μ , ϕ ) e τ / μ + ω 1 P 1 4 μ 0 F 0 μ 0 μ ( e τ / μ 0 e τ / μ )
I 1 ( τ , μ , ϕ ) = I 1 ( b , μ , ϕ ) e ( b τ ) / μ + ω 2 P 2 4 μ 0 F 0 μ 0 + μ ( e τ / μ 0 e b / μ 0 e ( b τ ) / μ )
I 1 ( τ , 0 , ϕ ) = ω 1 P 1 4 F 0 e τ / μ 0
I 1 ( τ , 0 , ϕ ) = ω 2 P 2 4 F 0 e τ / μ 0
I 1 ( τ , 0 , ϕ ) I 1 ( τ , 0 , ϕ )
J n ( τ , μ , ϕ ) = 1 4 π 0 2 π 1 1 ω ( τ ) I n 1 ( τ , μ ' , ϕ ' ) P ( τ , μ , ϕ ; μ ' , ϕ ' ) d μ ' d ϕ '
I n ( τ , 0 , ϕ ) = J n ( τ , 0 , ϕ ) = 1 4 π 0 2 π 1 1 ω 1 P 1 I n 1 ( τ , μ ' , ϕ ' ) d μ ' d ϕ '
I n ( τ , 0 , ϕ ) = J n ( τ , 0 , ϕ ) = 1 4 π 0 2 π 1 1 ω 2 P 2 I n 1 ( τ , μ ' , ϕ ' ) d μ ' d ϕ '
I n ( τ , 0 , ϕ ) I n ( τ , 0 , ϕ )
β a = β a , 0 exp ( z / 2 )
β m = β m , 0 exp ( z / 8 )
τ m = 0 .008735 λ 4 .08
Δ τ x = Δ τ a , x + Δ τ m , x
ω x = ( ω a Δ τ a , x + Δ τ m , x ) / Δ τ x
P x = ( ω a Δ τ a , x P a + Δ τ m , x P m ) / ω x Δ τ x
Δ τ a , x = z 1 z 2 β a , 0 e z / 2 d z = τ a ( e z 1 / 2 e z 2 / 2 )
Δ τ m , x = z 1 z 2 β m , 0 e z / 8 d z = τ m ( e z 1 / 8 e z 2 / 8 )
I 1 ( τ , μ , ϕ ) = I 2 ( τ , μ , ϕ )

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