Abstract

We report on a way of building bidirectional surface reflectivity into the Markov chain formalism for polarized radiative transfer through a vertically inhomogeneous atmosphere. Numerical results are compared to those obtained by the Monte Carlo method, showing the accuracy of the Markov chain method when 90 streams are used to compute the radiation from a Rayleigh-plus-aerosol atmosphere that overlies a surface with a bidirectional reflection function consisting of both depolarizing and polarizing parts.

© 2011 Optical Society of America

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References

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  1. L. W. Esposito, Astrophys. J. 233, 661 (1979).
    [CrossRef]
  2. L. W. Esposito and L. L. House, Astrophys. J. 219, 1058 (1978).
    [CrossRef]
  3. F. Xu, A. B. Davis, R. A. West, and L. W. Esposito, Opt. Express 19, 946 (2011).
    [CrossRef] [PubMed]
  4. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  5. J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
    [CrossRef]
  6. H. Rahman, B. Pinty, and M. M. Verstraete, J. Geophys. Res. 98D, 20779 (1993).
    [CrossRef]
  7. J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
    [CrossRef]
  8. R. G. Priest and S. R. Meier, Opt. Eng. 41, 988 (2002).
    [CrossRef]
  9. C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006).
    [CrossRef]
  10. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  11. A. B. Davis and F. Xu are preparing a manuscript to be called “Monte Carlo modeling of polarized light transfer in vertically varying plane-parallel atmospheres, with application to lofted aerosol layer detection using O2 spectroscopy.”

2011

2002

R. G. Priest and S. R. Meier, Opt. Eng. 41, 988 (2002).
[CrossRef]

1998

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

1993

H. Rahman, B. Pinty, and M. M. Verstraete, J. Geophys. Res. 98D, 20779 (1993).
[CrossRef]

1979

L. W. Esposito, Astrophys. J. 233, 661 (1979).
[CrossRef]

1978

L. W. Esposito and L. L. House, Astrophys. J. 219, 1058 (1978).
[CrossRef]

1969

J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
[CrossRef]

Bohren, C. F.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Clothiaux, E. E.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006).
[CrossRef]

Davis, A. B.

F. Xu, A. B. Davis, R. A. West, and L. W. Esposito, Opt. Express 19, 946 (2011).
[CrossRef] [PubMed]

A. B. Davis and F. Xu are preparing a manuscript to be called “Monte Carlo modeling of polarized light transfer in vertically varying plane-parallel atmospheres, with application to lofted aerosol layer detection using O2 spectroscopy.”

Diner, D. J.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

Esposito, L. W.

F. Xu, A. B. Davis, R. A. West, and L. W. Esposito, Opt. Express 19, 946 (2011).
[CrossRef] [PubMed]

L. W. Esposito, Astrophys. J. 233, 661 (1979).
[CrossRef]

L. W. Esposito and L. L. House, Astrophys. J. 219, 1058 (1978).
[CrossRef]

Gordon, H. R.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

House, L. L.

L. W. Esposito and L. L. House, Astrophys. J. 219, 1058 (1978).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
[CrossRef]

Knyazikhin, Y.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Martonchik, J. V.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

Meier, S. R.

R. G. Priest and S. R. Meier, Opt. Eng. 41, 988 (2002).
[CrossRef]

Myneni, R. B.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

Pinty, B.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

H. Rahman, B. Pinty, and M. M. Verstraete, J. Geophys. Res. 98D, 20779 (1993).
[CrossRef]

Priest, R. G.

R. G. Priest and S. R. Meier, Opt. Eng. 41, 988 (2002).
[CrossRef]

Rahman, H.

H. Rahman, B. Pinty, and M. M. Verstraete, J. Geophys. Res. 98D, 20779 (1993).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Verstraete, M. M.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

H. Rahman, B. Pinty, and M. M. Verstraete, J. Geophys. Res. 98D, 20779 (1993).
[CrossRef]

West, R. A.

Xu, F.

F. Xu, A. B. Davis, R. A. West, and L. W. Esposito, Opt. Express 19, 946 (2011).
[CrossRef] [PubMed]

A. B. Davis and F. Xu are preparing a manuscript to be called “Monte Carlo modeling of polarized light transfer in vertically varying plane-parallel atmospheres, with application to lofted aerosol layer detection using O2 spectroscopy.”

Astrophys. J.

L. W. Esposito, Astrophys. J. 233, 661 (1979).
[CrossRef]

L. W. Esposito and L. L. House, Astrophys. J. 219, 1058 (1978).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, IEEE Trans. Geosci. Remote Sens. 36, 1266 (1998).
[CrossRef]

J. Atmos. Sci.

J. W. Hovenier, J. Atmos. Sci. 26, 488 (1969).
[CrossRef]

J. Geophys. Res.

H. Rahman, B. Pinty, and M. M. Verstraete, J. Geophys. Res. 98D, 20779 (1993).
[CrossRef]

Opt. Eng.

R. G. Priest and S. R. Meier, Opt. Eng. 41, 988 (2002).
[CrossRef]

Opt. Express

Other

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006).
[CrossRef]

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

A. B. Davis and F. Xu are preparing a manuscript to be called “Monte Carlo modeling of polarized light transfer in vertically varying plane-parallel atmospheres, with application to lofted aerosol layer detection using O2 spectroscopy.”

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Figures (2)

Fig. 1
Fig. 1

Markov chain and Monte Carlo calculation of the I, Q, U, and V components of the radiation from an inhomogeneous atmosphere of optical thickness 0.41 and properties described in [3]. For clarity, the Q and V results are multiplied by 5 and 100, respectively.

Fig. 2
Fig. 2

Markov chain’s computation of two- dimensional radiance contributed by multiple scattering, including (a) I, (b) Q, (c) U, and (d) V for the case in Fig. 1. The azimuthal and viewing angles are labeled in black (along dashed–dotted lines) and red (on two circles), respectively, only in the first subfigure.

Equations (18)

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( 2 δ 0 m ) I M ( m ) = R ( m ) ( E Q ( m ) ) 1 Π 0 ( m ) ,
I = I S + m = 0 ( 2 δ 0 m ) [ I M , c ( m ) cos m ( ϕ ϕ 0 ) + I M , s ( m ) sin m ( ϕ ϕ 0 ) ] .
ρ r ( μ , μ 0 , ϕ ϕ 0 ) = L ( π i 2 ) ρ ( μ , ϕ ; μ 0 , ϕ 0 ) L ( i 1 ) .
ρ r ( μ , μ 0 , ϕ ϕ 0 ) = ρ r , c ( 0 ) ( μ , μ 0 ) + 2 m = 1 [ ρ r , c ( m ) ( μ , μ 0 ) cos m ( ϕ ϕ 0 ) + ρ r , s ( m ) ( μ , μ 0 ) sin m ( ϕ ϕ 0 ) ] .
[ Π ( N , i ) , c ( m ) Π ( N , i ) , s ( m ) ] = 2 w i exp ( τ 0 μ 0 ) [ μ i ρ r , c ( m ) ( μ i , μ 0 ) μ i ρ r , s ( m ) ( μ i , μ 0 ) ] μ 0 F 0 ,
W ( n , i ) , ( N , i ) = exp ( τ 0 τ n + 1 μ i ) [ 0 Δ τ n exp ( x μ i ) d x μ i ] = exp ( τ 0 τ n + 1 μ i ) [ 1 exp ( Δ τ n μ i ) ] .
W ( N , i ) , ( n , i ) = [ 1 Δ τ n 0 Δ τ n exp ( x μ i ) d x ] exp ( τ 0 τ n + 1 μ i ) = μ i Δ τ n exp ( τ 0 τ n + 1 μ i ) [ 1 exp ( Δ τ n μ i ) ] .
Q ( n , j ) , ( n , i ) ( m ) = { w j ω 0 ( n ) P ( m ) ( μ j , μ i ; n ) 2 W ( n , i ) , ( n , i ) , upwelling light 2 w j μ j ρ r , c / s ( m ) ( μ j , μ i ) W ( n , i ) , ( n , i ) , downwelling light ,
R e , ( n , j ) ( m ) = { n = n N R ( n , e ) , ( n , j ) ( m ) , n N 0 n = N ,
R ( n , e ) , ( n , j ) ( m ) = τ n τ n + 1 I r , ( n , e ) , ( n , j ) ( m ) , d ( τ ) d τ μ e , n N ,
R ( N , e ) , ( n , j ) ( m ) = 1 Δ τ n τ n τ n + 1 d x μ e exp ( τ 0 μ e ) μ e ρ r , c / s ( m ) ( μ e , μ j ) exp ( τ 0 x μ j ) = 1 Δ τ n μ j ρ r , c / s ( m ) ( μ e , μ j ) exp ( τ 0 μ e ) [ exp ( τ 0 τ n + 1 μ j ) exp ( τ 0 τ n μ j ) ] .
R e , ( n , j ) ( m ) = { n = 1 n R ( n , e ) , ( n , j ) ( m ) , n N n = 1 N 1 R ( n , e ) , ( n , j ) ( m ) , n = N .
R ( n , e ) , ( n , j ) ( m ) = τ n τ n + 1 I t , ( n , e ) , ( n , j ) ( m ) , u ( τ ) d τ μ e ,
I t , ( n , e ) , ( N , j ) ( m ) , u ( τ ) = 1 2 1 μ j exp ( τ μ e ) ω 0 ( n ) P ( m ) ( μ e , μ j ; n ) 2 exp ( τ 0 τ μ j ) .
R ( n , e ) , ( N , j ) ( m ) = 1 2 1 μ e μ j ω 0 ( n ) P ( m ) ( μ e , μ j ; n ) 2 × [ exp ( τ n + 1 τ 0 μ j τ n + 1 μ e ) exp ( τ n τ 0 μ j τ n μ e ) ] .
R ( n , e ) , ( N , j ) ( m ) = 1 2 1 μ j μ e ω 0 ( n ) P ( m ) ( μ e , μ j ; n ) 2 exp ( τ 0 μ j ) ( τ n + 1 τ n ) .
I S , N ( μ e , μ 0 , ϕ e ϕ 0 ) = exp [ τ 0 ( 1 μ e + 1 μ 0 ) ] ρ r ( μ e , μ 0 , ϕ e ϕ 0 ) μ 0 F 0 .
ρ = a [ μ μ 0 ( μ + μ 0 ) ] k 1 exp ( b cos Θ ) [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] + ξ π p ( β ) sin ( Θ / 2 ) 2 μ μ 0 ( μ + μ 0 ) [ F 11 F 12 0 0 F 12 F 11 0 0 0 0 F 33 F 34 0 0 F 34 F 33 ] S h ( μ , μ 0 ) ,

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