## Markov chain formalism for polarized light transfer in plane-parallel atmospheres, with numerical comparison to the Monte Carlo method |

Optics Express, Vol. 19, Issue 2, pp. 946-967 (2011)

http://dx.doi.org/10.1364/OE.19.000946

Acrobat PDF (1606 KB)

### Abstract

Building on the Markov chain formalism for scalar (intensity only) radiative transfer, this paper formulates the solution to polarized diffuse reflection from and transmission through a vertically inhomogeneous atmosphere. For verification, numerical results are compared to those obtained by the Monte Carlo method, showing deviations less than 1% when 90 streams are used to compute the radiation from two types of atmospheres, pure Rayleigh and Rayleigh plus aerosol, when they are divided into sublayers of optical thicknesses of less than 0.03.

© 2011 OSA

## 1. Introduction and overview

1. L. W. Esposito and L. L. House, “Radiative transfer calculated by a Markov chain formalism,” Astrophys. J. **219**, 1058–1067 (1978). [CrossRef]

2. L. W. Esposito, “An ‘adding’ algorithm for the Markov chain formalism for radiation transfer,” Astrophys. J. **233**, 661–663 (1979). [CrossRef]

3. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. **16**(4), 527–610 (1974). [CrossRef]

4. 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]

2. L. W. Esposito, “An ‘adding’ algorithm for the Markov chain formalism for radiation transfer,” Astrophys. J. **233**, 661–663 (1979). [CrossRef]

*i*) the advantage of the adding/doubling method of easy physical interpretation and high numerical accuracy in calculating the atmosphere of large optical depth, (

*ii*) the advantage of the successive orders of scattering method in separating the contribution of a specific scattering order numerically [5

5. Q. L. 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]

7. P. W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express **17**(4), 2057–2079 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2057. [CrossRef] [PubMed]

8. J. H. Hannay, “Radiative transfer: exact Rayleigh scattering series and a daylight formula,” J. Opt. Soc. Am. A **26**(3), 669–675 (2009). [CrossRef]

*iii*) the advantage of the discrete source method [9

9. F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere–I. Theory,” J. Quant. Spectrosc. Radiat. Transf. **47**(1), 19–33 (1992). [CrossRef]

11. F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT, an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. **61**(1), 105–122 (1999). [CrossRef]

1. L. W. Esposito and L. L. House, “Radiative transfer calculated by a Markov chain formalism,” Astrophys. J. **219**, 1058–1067 (1978). [CrossRef]

## 2. Scalar radiative transfer

*τ*

_{0}is illuminated by a monochromatic solar flux π

*F*

_{0}. The atmosphere is divided into

*N*sublayers so that each one has its optical thickness Δ

*τ*=

_{n}*τ*

_{n}_{+1}−

*τ*(1 ≤

_{n}*n*≤

*N*with

*τ*

_{N}_{+1}=

*τ*

_{0}), single scattering albedo ω

_{0}(

*n*) and phase function

*P*(cosΘ,

*n*) which is normalized in such a way that its integral over all directions is 4π. The scattering angle Θ is defined bywhere

*u*

_{0}= cos

*θ*

_{0}and

*u*= cos

*θ*denote the direction of the incident and scattered light, respectively, with respect to the downward increasing optical depth,

*ϕ*describe the azimuthal plane of the incident and diffuse light.

*P*

^{(}

^{m}^{)}) for cosine modes only:

*u*and specify the incidence to be upwelling or downwelling according to the sign of

*u*

_{0}, we define

*μ*= |

*u*| and

*μ*

_{0}= |

*u*

_{0}|, getting the following four regimes [3

3. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. **16**(4), 527–610 (1974). [CrossRef]

*I*

_{r}(

*τ*= 0,

*μ*,

*ϕ*) and transmitted field

*I*

_{t}(

*τ*=

*τ*

_{0},

*μ*,

*ϕ*) are resolved once

*I*

^{(0)}(

*τ*,

*μ*) and

*I*

^{(}

^{m}^{)}(

*τ*,

*μ*) are determined.

### 2.1 Initial scattering in the atmosphere

*μ*

_{0}) and the outgoing scattered light (direction cosine

*μ*), the contribution of first-order scattering to the diffusely reflected intensity from the upper boundary of the

_{i}*n*

^{th}layer (

*τ*=

*τ*) has the Fourier series

_{n}^{th}layer (

*τ*=

*τ*

_{n}_{+1}) has the Fourier series

*μ*to the scattered intensity in one direction in the next order of scattering, a quadrature weight

_{i}*w*is introduced for

_{i}*μ*on the interval 0 ≤

_{i}*μ*≤ 1.

_{i}### 2.2 Intermediate scattering in the atmosphere

*n*in the direction

*μ*travels upward to the layer

_{i}*n'*for the next order of scattering. The reduced flux for scattering in

*n'*

^{th}layer is:where on the basis of the uniform source distribution in each layer the first term stands for the intensity of the light leaving the upper boundary of

*n*

^{th}layer of optical thickness Δ

*τ*, the second term indicates the remaining light intensity reaching the lower boundary of

_{n}*n'*

^{th}layer after traveling an optical depth (

*τ*−

_{n}*τ*

_{n'}_{+1}), and the last term indicates the attenuated light flux in the

*n'*

^{th}layer. Equation (16) has the following analytical form:In a similar way, we can calculate the attenuated light flux in the

*n'*

^{th}layer for the downward travelling light (

*n*<

*n'*)In addition to the upward and downward propagation, partial light flux remains in the same layer (

*n*=

*n'*) for scattering:Knowing the reduced flux, the scattered intensity from n

*'*

^{th}layer in all directions can be calculated from its Fourier series which satisfieswhere the B-matrix

*n*

^{th}layer in direction

*μ*(for the second order of scattering

_{i}^{T}) and the A-matrix

*n'*-th layer and direction

*μ*, which is contributed by the incident light from

_{j}*n*

^{th}layer and direction

*μ*,For upwelling light

_{i}*w*is introduced to define a transition factor

_{j}### 2.3 Scattering to the boundary

*τ*= 0) and from the bottom of the atmosphere (BOA, or

*τ*=

*τ*

_{0}) are contributed by both downward- and upward-traveling light. Assuming they come from layer

*n'*in direction

*j*, and are redirected to emerge in direction

*μ*

_{e}through scattering, we then analyze their contribution to the radiation fields at the BOA and TOA.

#### 2.3.1 Downwelling light

*n'*

^{th}layer partially leaves the TOA through diffuse reflection and partially leaves the BOA through diffuse transmission. The diffuse scattering occurs in all layers

*n*with

*n*≥

*n'*so that the Fourier series of the emergent intensities are:for diffuse reflection at the TOA andfor diffuse transmission at the BOA, where the subscripts of

*R*and

*T*on the left hand side of the above two equations mean the light leaving the atmosphere in the direction

*μ*

_{e}is contributed by the scattering of the light coming from

*n'*

^{th}layer in direction

*μ*, and the subscript (

_{j}*n*, e) of each series term on the right hand side of the two equations means the scattering into the direction

*μ*

_{e}occurs in

*n*

^{th}layer.

*n*

^{th}layer to the emergent intensities is calculated by integrating over

*τ*bounded by

*τ*and

_{n}*τ*

_{n}_{+1}. In terms of the Fourier series we have where the kernels of the above integrals

*n'*

^{th}layer to the diffusely emergent light through reflection (denoted by the subscript “r”) and transmission (denoted by the subscript “t”) from/through

*n*

^{th}layer, respectively. Denoting

*n'*

^{th}layer, multiplication with matrices

*n*=

*n'*, Eq. (25) and (26) becomeandrespectively, where

#### 2.3.2 Upwelling light

*n'*

^{th}layer leaves the TOA through diffuse transmission and partially leaves the BOA through diffuse reflection. The diffuse scattering occurs in all layers

*n*with

*n*≤

*n'*so that the Fourier series of the emergent intensities are,andIntegrating the scattering over

*τ*of the

*n*

^{th}layer bounded by

*τ*and

_{n}*τ*

_{n}_{+1}givesandwhere the kernels of the above integrals

*n'*

^{th}layer to the diffusely emergent light through transmission (denoted by the subscript “t”) and reflection (denoted by the subscript “r), from/through the

*n*

^{th}layer, respectively. Denoting

*n'*

^{th}layer, multiplication with the matrices

*n*=

*n'*), Eq. (39) and (40) becomeandrespectively. Substitution of Eq. (33) into Eq. (46) and Eq. (34) into Eq. (45) givesand

### 2.4 Radiative transfer through multiple scattering

*Q*,

*R*, and

*T*matrices, the radiative transfer through different orders of scattering can be achieved in matrix form after invoking the orthogonality relation [12

12. J. V. Dave, “Intensity and polarization of the radiation emerging from a plane-parallel atmosphere containing monodispersed aerosols,” Appl. Opt. **9**(12), 2673–2684 (1970). [CrossRef] [PubMed]

*n*) and the propagating direction (

*μ*

_{i}); “0” denotes the initial light state in the atmosphere and “e” denotes the light leaving the atmosphere in direction

*μ*

_{e}. Change of the state of light from “I” to “J” is represented in the sequence “JI”. The first term on the right side of the above equation represents the diffusely emergent light experiencing two scattering events, the second term represents the emergent light experiencing three scattering events, and so on. In the form of an infinite series summation, all orders of multiple scattering are accounted for. If a specific order of scattering is not explicitly expressed and counted, Eqs. (49)-(50) have an equivalent but more compact form: where

**E**is the identity matrix.

### (2.5) Total radiation field

*µ*

_{0},

*ϕ*

_{0}), which can be estimated from

*F*

_{0}, τ

_{0}, and µ

_{0}, has to be included for the total field. Our development in this section is parallel to that of Refs [1

1. L. W. Esposito and L. L. House, “Radiative transfer calculated by a Markov chain formalism,” Astrophys. J. **219**, 1058–1067 (1978). [CrossRef]

2. L. W. Esposito, “An ‘adding’ algorithm for the Markov chain formalism for radiation transfer,” Astrophys. J. **233**, 661–663 (1979). [CrossRef]

## 3. Vector radiative transfer

**P**(cosΘ,

*n*) with two rotational matrices about angles

*i*

_{1}and

*i*

_{2}[13

13. J. W. Hovenier, “Symmetry relations for scattering of polarized light in a slab of randomly oriented particles,” J. Atmos. Sci. **26**(3), 488–499 (1969). [CrossRef]

*μ*= |

*u*| four scattering regimes in a similar form as Eqs. (3)–(6) and Eqs. (7)–(10) can be distinguished for the Mueller matrix as well as its series, with the only difference of replacing the scalar phase function “

*P*” by the Mueller matrix “

**P**”. In the same manner of expansion, the series form of the polarized light intensity described by the Stokes vector [I, Q, U, V]

^{T}is:where the

*m*

^{th}Fourier series component of the stokes vector

### 3.1 Polarized radiative transfer through multiple scattering

*m*= 0 only the cosine mode is needed in calculation so that The

*n*

^{th}term on the right hand side of Eqs. (62)-(65) means the contribution of diffuse light after experiencing (

*n*+ 1) scattering events. Difference of the vector and scalar radiative transfer case in calculating the

**Π**vector and the

**Q**,

**R**, and

**T**matrices is expressed here. For calculating the contribution of the downwelling incident light to the diffuse field at the TOA and BOA, each element of the

**R**-matrix in Eqs. (62) and (64) becomes a 4×4 cell which is calculated by Eq. (35) after replacing the phase function series

**T**-matrix in Eqs. (63) and (65) becomes a 4×4 cell which is calculated by Eq. (36) after replacing

**R**in Eqs. (62) and (64) are calculated by Eq. (43) after replacing phase function series

**T**of Eqs. (63) and (65) are calculated by Eq. (44) after replacing

**Π**vector becomes

### 3.2 “Chain-to-chain adding” strategy

*μ*≤ 1 is required, leading to a large set of linear equations. To save computation time, an “adding” algorithm was proposed in Ref [2

**233**, 661–663 (1979). [CrossRef]

3. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. **16**(4), 527–610 (1974). [CrossRef]

## 4. Illustrative numerical calculations

### 4.1 Implementation of the Monte Carlo method

*N*layers with their respective single scattering albedo and phase function; it then tracks histories in optical units, assuming the layers are internally uniform. This version of the Monte Carlo model was benchmarked extensively against the high-precision values provided by Kokhanovsky et al. [17

17. A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, and B. Mayer, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. **111**(12-13), 1931–1946 (2010). [CrossRef]

^{8}Monte Carlo histories were traced using the code that ingests discrete layers; this leads to relative uncertainties on the order of 10

^{−4}for well-sampled signals but this precision deteriorates for poorly populated signals (typically, the 3

^{rd}and 4

^{th}components of the Stokes vector). Radiances were computed using the local estimation technique [18, 19] at the same ordinates as used by the Markov chain model in the principal and perpendicular planes.

### 4.2 Separate orders of scattering in a pure Rayleigh atmosphere

_{0}= 0.5 and single scattering albedo ω

_{0}= 1. The atmosphere is divided into 20 layers. In Markov chain formalism, contributions of single and higher-order scattering are calculated by Eqs. (54)-(55) after replacing the phase function series

*ϕ*-

*ϕ*

_{0}= 0° and 180°) is given in Figs. 1 -4 , showing agreement to be better than 99%. In these figures, only the

*I*and

*Q*components are plotted since the

*U*and

*V*components vanish in the principal plane. Moreover, positive viewing zenith angles are used for the

*ϕ*-

*ϕ*

_{0}= 0° plane while their negative counterparts are used for

*ϕ*-

*ϕ*

_{0}= 180°. Note that, although the Markov chain formalism is presented for the vertically inhomogeneous atmosphere in the current paper, by setting the Mueller matrix

**P**(cosΘ,

*n*) and single scattering albedo ω

_{0}(

*n*) to be constant for all sublayers, the formalism reduces to its counterpart for the homogeneous atmosphere [1

**219**, 1058–1067 (1978). [CrossRef]

### 4.3 Vertically stratified aerosol-and-Rayleigh atmosphere

_{0}= 1.22934. The atmosphere is composed of Mie aerosols (optical depth 1.0074) of refractive index

*n*= 1.38+10

^{−8}i and Rayleigh molecules (optical depth 0.2286). For the aerosols, a lognormal distribution with parameters

*s*= 1 and

*a*

_{0}= 0.1 μm is assumed:This aerosol model was used in a recent intercomparison of aerosol retrieval algorithms described in Ref [20

20. A. A. Kokhanovsky, J. L. Deuzé, D. J. Diner, O. Dubovik, F. Ducos, C. Emde, M. J. Garay, R. G. Grainger, A. Heckel, M. Herman, I. L. Katsev, J. Keller, R. Levy, P. R. J. North, A. S. Prikhach, V. V. Rozanov, A. M. Sayer, Y. Ota, D. Tanré, G. E. Thomas, E. P. Zege, A. A. Kokhanovsky, J. L. Deuzé, D. J. Diner, O. Dubovik, F. Ducos, C. Emde, M. J. Garay, R. G. Grainger, A. Heckel, M. Herman, I. L. Katsev, J. Keller, R. Levy, P. R. J. North, A. S. Prikhach, V. V. Rozanov, A. M. Sayer, Y. Ota, D. Tanré, G. E. Thomas, and E. P. Zege, “The inter-comparison of major satellite aerosol retrieval algorithms using simulated intensity and polarization characteristics of reflected light,” Atm. Meas. Techn. **3**, 909–932 (2010). [CrossRef]

*a*≤ 30 μm, which is discretized to 40000 quadrature points. To highlight molecular scattering, we selected the MISR blue wavelength of 446.4 nm [21

21. D. J. Diner, J. C. Beckert, T. H. Reilly, C. J. Bruegge, J. E. Conel, R. A. Kahn, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, J. Muller, R. B. Myneni, P. J. Sellers, B. Pinty, and M. M. Verstraete, “Multi-angle Imaging Spectro-Radiometer (MISR) instrument description and experiment overview,” IEEE Trans. Geosci. Rem. Sens. **36**(4), 1072–1087 (1998). [CrossRef]

*θ*

_{0}= 60°. This atmospheric structure, as well as the lower-boundary and illumination conditions, were prescribed in Ref [20

20. A. A. Kokhanovsky, J. L. Deuzé, D. J. Diner, O. Dubovik, F. Ducos, C. Emde, M. J. Garay, R. G. Grainger, A. Heckel, M. Herman, I. L. Katsev, J. Keller, R. Levy, P. R. J. North, A. S. Prikhach, V. V. Rozanov, A. M. Sayer, Y. Ota, D. Tanré, G. E. Thomas, E. P. Zege, A. A. Kokhanovsky, J. L. Deuzé, D. J. Diner, O. Dubovik, F. Ducos, C. Emde, M. J. Garay, R. G. Grainger, A. Heckel, M. Herman, I. L. Katsev, J. Keller, R. Levy, P. R. J. North, A. S. Prikhach, V. V. Rozanov, A. M. Sayer, Y. Ota, D. Tanré, G. E. Thomas, and E. P. Zege, “The inter-comparison of major satellite aerosol retrieval algorithms using simulated intensity and polarization characteristics of reflected light,” Atm. Meas. Techn. **3**, 909–932 (2010). [CrossRef]

*I*and

*Q*components of the reflection field in the principal plane (

*ϕ*-

*ϕ*

_{0}= 0° and 180°) while Figs. 7 -8 give the

*I*and

*Q*components of the transmission field in the same plane. The

*U*and

*V*components vanish identically in the principal plane. In the perpendicular plane

*ϕ*-

*ϕ*

_{0}= ±90°, however, all components of the reflected and transmitted fields appear, as demonstrated in the panels of Figs. 9 -16 . Using 45 Gauss-Legendre ordinates on the interval 0 <

*μ*< 1 (equivalently, 90 streams), Figs. 5-8 and Figs. 9-16 show the disagreement between the Markov chain and Monte Carlo methods to be less than 1%. Using finer (coarse) sublayer thickness and more (less) Gauss-Legendre ordinates on

*μ*’s interval can further reduce (increase) the disagreement, but costing (saving) in computation time. For example, the maximum error of

*Q*for diffuse reflection and transmission in the principal and perpendicular planes increases to about 4% when 23 Gauss-Legendre ordinates are used but 75% computation time is saved.

## 5. Conclusion and outlook

22. P. Y. Deschamps, F. M. Breon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: Instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Rem. Sens. **32**(3), 598–615 (1994). [CrossRef]

23. M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueler, B. A. Fafaul, J. E. Hansen, R. J. Hooker, T. Itchkawich, H. B. Maring, and L. D. Travis, “Accurate monitoring of terrestrial aerosols and total solar irradiance: Introducing the Glory mission,” Bull. Am. Meteorol. Soc. **88**(5), 677–691 (2007). [CrossRef]

24. D. Crisp, R. Atlas, F. Breon, L. Brown, J. Burrows, P. Ciais, B. Connor, S. Doney, I. Fung, and D. Jacob, “The Orbiting Carbon Observatory (OCO) mission,” Adv. Space Res. **34**(4), 700–709 (2004). [CrossRef]

_{2}based on differential optical absorption spectroscopy (DOAS) in the near-IR. This calls for DOAS in both CO

_{2}and O

_{2}bands, and the later is significantly affected by aerosol scattering, therefore by polarization [25

25. V. Natraj, R. J. D. Spurr, H. Boesch, Y. Jiang, and Y. Yung, “Evaluation of errors in neglecting polarization in the forward modeling of O2 A band measurements from space, with relevance to CO2 column retrieval from polarization sensitive instruments,” J. Quant. Spectrosc. Radiat. Transf. **103**(2), 245–259 (2007). [CrossRef]

26. V. Natraj and R. J. D. Spurr, “A fast linearized pseudo-spherical two orders of scattering model to account for polarization in vertically inhomogeneous scattering-absorbing media,” J. Quant. Spectrosc. Radiat. Transf. **107**(2), 263–293 (2007). [CrossRef]

## Acknowledgements

## References and links

1. | L. W. Esposito and L. L. House, “Radiative transfer calculated by a Markov chain formalism,” Astrophys. J. |

2. | L. W. Esposito, “An ‘adding’ algorithm for the Markov chain formalism for radiation transfer,” Astrophys. J. |

3. | J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. |

4. | K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. |

5. | Q. L. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. |

6. | S. Y. Kotchenova, E. F. Vermote, R. Matarrese, and F. J. Klemm Jr., “Validation of a vector version of the 6S radiative transfer code for atmospheric correction of satellite data. Part I: path radiance,” Appl. Opt. |

7. | P. W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express |

8. | J. H. Hannay, “Radiative transfer: exact Rayleigh scattering series and a daylight formula,” J. Opt. Soc. Am. A |

9. | F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere–I. Theory,” J. Quant. Spectrosc. Radiat. Transf. |

10. | R. J. D. Spurr, “VLIDORT: A linearized pseudo-spherical vector discrete ordinate radiative transfer code for forward model and retrieval studies in multilayer multiple scattering media,” J. Quant. Spectrosc. Radiat. Transf. |

11. | F. M. Schulz, K. Stamnes, and F. Weng, “VDISORT, an improved and generalized discrete ordinate method for polarized (vector) radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. |

12. | J. V. Dave, “Intensity and polarization of the radiation emerging from a plane-parallel atmosphere containing monodispersed aerosols,” Appl. Opt. |

13. | J. W. Hovenier, “Symmetry relations for scattering of polarized light in a slab of randomly oriented particles,” J. Atmos. Sci. |

14. | C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. |

15. | J. W. Hovenier, and C. V. M. van der Mee, “Basic Relationships for Matrices Describing Scattering by Small Particles”, in Light Scattering by Nonspherical Particles, pp. 61–85, M. Mishchenko, J. W. Hovenier and L. Travis, (eds.), Academic Press, San Diego, 2000. |

16. | A. B. Davis and F. Xu, “Monte Carlo modeling of polarized light transfer in vertically varying plane-parallel atmospheres, with application to lofted aerosol layer detection using O |

17. | A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, and B. Mayer, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. |

18. | G. Marchuk, G. Mikhailov, N. Nazaraliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics, Springer-Verlag, New-York, NY (1980). |

19. | K. F. Evans, and A. Marshak, “Numerical Methods,” in 3D Radiative Transfer in Cloudy Atmospheres, pp. 243–281, A. Marshak and A. B. Davis (eds.), Springer, Heidelberg, Germany (2005). |

20. | A. A. Kokhanovsky, J. L. Deuzé, D. J. Diner, O. Dubovik, F. Ducos, C. Emde, M. J. Garay, R. G. Grainger, A. Heckel, M. Herman, I. L. Katsev, J. Keller, R. Levy, P. R. J. North, A. S. Prikhach, V. V. Rozanov, A. M. Sayer, Y. Ota, D. Tanré, G. E. Thomas, E. P. Zege, A. A. Kokhanovsky, J. L. Deuzé, D. J. Diner, O. Dubovik, F. Ducos, C. Emde, M. J. Garay, R. G. Grainger, A. Heckel, M. Herman, I. L. Katsev, J. Keller, R. Levy, P. R. J. North, A. S. Prikhach, V. V. Rozanov, A. M. Sayer, Y. Ota, D. Tanré, G. E. Thomas, and E. P. Zege, “The inter-comparison of major satellite aerosol retrieval algorithms using simulated intensity and polarization characteristics of reflected light,” Atm. Meas. Techn. |

21. | D. J. Diner, J. C. Beckert, T. H. Reilly, C. J. Bruegge, J. E. Conel, R. A. Kahn, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, J. Muller, R. B. Myneni, P. J. Sellers, B. Pinty, and M. M. Verstraete, “Multi-angle Imaging Spectro-Radiometer (MISR) instrument description and experiment overview,” IEEE Trans. Geosci. Rem. Sens. |

22. | P. Y. Deschamps, F. M. Breon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, “The POLDER mission: Instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Rem. Sens. |

23. | M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueler, B. A. Fafaul, J. E. Hansen, R. J. Hooker, T. Itchkawich, H. B. Maring, and L. D. Travis, “Accurate monitoring of terrestrial aerosols and total solar irradiance: Introducing the Glory mission,” Bull. Am. Meteorol. Soc. |

24. | D. Crisp, R. Atlas, F. Breon, L. Brown, J. Burrows, P. Ciais, B. Connor, S. Doney, I. Fung, and D. Jacob, “The Orbiting Carbon Observatory (OCO) mission,” Adv. Space Res. |

25. | V. Natraj, R. J. D. Spurr, H. Boesch, Y. Jiang, and Y. Yung, “Evaluation of errors in neglecting polarization in the forward modeling of O2 A band measurements from space, with relevance to CO2 column retrieval from polarization sensitive instruments,” J. Quant. Spectrosc. Radiat. Transf. |

26. | V. Natraj and R. J. D. Spurr, “A fast linearized pseudo-spherical two orders of scattering model to account for polarization in vertically inhomogeneous scattering-absorbing media,” J. Quant. Spectrosc. Radiat. Transf. |

**OCIS Codes**

(030.5620) Coherence and statistical optics : Radiative transfer

(290.4210) Scattering : Multiple scattering

(290.5850) Scattering : Scattering, particles

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Scattering

**History**

Original Manuscript: October 14, 2010

Revised Manuscript: December 1, 2010

Manuscript Accepted: December 20, 2010

Published: January 7, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Feng Xu, Anthony B. Davis, Robert A. West, and Larry W. Esposito, "Markov chain formalism for polarized light transfer in plane-parallel atmospheres, with numerical comparison to the Monte Carlo method," Opt. Express **19**, 946-967 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-946

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