Abstract

A vector radiative transfer model has been developed for coupled atmosphere and ocean systems based on the Successive Order of Scattering (SOS) Method. The emphasis of this study is to make the model easy-to-use and computationally efficient. This model provides the full Stokes vector at arbitrary locations which can be conveniently specified by users. The model is capable of tracking and labeling different sources of the photons that are measured, e.g. water leaving radiances and reflected sky lights. This model also has the capability to separate florescence from multi-scattered sunlight. The δ - fit technique has been adopted to reduce computational time associated with the strongly forward-peaked scattering phase matrices. The exponential - linear approximation has been used to reduce the number of discretized vertical layers while maintaining the accuracy. This model is developed to serve the remote sensing community in harvesting physical parameters from multi-platform, multi-sensor measurements that target different components of the atmosphere-oceanic system.

© 2009 Optical Society of America

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2008

2007

S. Y. Kotchenova and E. F. Vermote, "Validation of a vector version of the 6S radiative transfer code for atmospheric correction of satellite data. Part II. Homogeneous Lambertian and anisotropic surfaces," Appl. Opt. 46, 4455-4464 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-20-4455.
[CrossRef] [PubMed]

J. Lenoble, M. Herman, J.L. Deuzé, B. Lafrance, R. Santer, D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107, 479-507 (2007).
[CrossRef]

T. Suzuki, T. Nakajima, M. Tanaka, "Scaling algorithms for the calculation of solar radiative fluxes," J. Quant. Spectrosc. Radiat. Transfer 107, 458-469 (2007).
[CrossRef]

K. F. Evans, "SHDOMPPDA: A radiative transfer model for cloudy sky data assimilation". J. Atmos. Sci. 64, 3858-3868 (2007).
[CrossRef]

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

2006

2005

M. Herman, J.L. Deuzé, A. Marchand, B. Roger, and P. Lallart, "Aerosol remote sensing from POLDER/ADEOS over the ocean: improved retrieval using a nonspherical particle model," J. Geophys. Res. 110, D10S02 (2005).
[CrossRef]

M. Duan and Q. Min, "A semi-analytic technique to speed up successive order of scattering model for optically thick media," J. Quant. Spectrosc. Radiat. Transfer 95, 21-32 (2005).
[CrossRef]

T. Greenwald, R. Bennartz, C. O’Dell, and A. Heidinger, "Fast computation of microwave radiances for data assimilation using the Successive Order of Scattering method," J. Appl. Meteorol. 44, 960-966 (2005).
[CrossRef]

A. K. Heidinger, C. O’Dell, R. Bennartz, and T. Greenwald, "The successive-order-of-interaction radiative transfer model. part I: model development," J. Appl. Meteorol. Climatol. 45, 1388-1402 (2005).
[CrossRef]

2004

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

2003

B. Cairns, E. E. Russell, J. D. LaVeigne, and P. M. W. Tennant. "Research scanning polarimeter and airborne usage for remote sensing of aerosols," Proc SPIE 5158, 33-44 (2003).
[CrossRef]

2001

2000

C. E. Siewert, "A discrete-ordinates solution for radiative-transfer models that include polarization effects," J. Quant. Spectrosc. Radiat. Transfer 64, 227-254 (2000).
[CrossRef]

Y. -X. Hu, B. Wielicki, B. Lin, G. Gibson, S. -C. Tsay, K. Stamnes, and T. Wong, "™ -Fit: A fast and accurate treatment of particle scattering phase functions with weighted singular-value decomposition least-squares fitting," J. Quant. Spectrosc. Radiat. Transfer 65, 681-690 (2000).
[CrossRef]

1999

F. M. Schulz, K. Stanes, and F. Weng, "VDISORT: An improved and generalized discrete ordinate method for polarized vector radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 61, 105-122 (1999).
[CrossRef]

L. Roberti and C. Kummerow, "Monte Carlo calculations of polarized microwave radiation emerging from cloud structures," J. Geophy. Res. 104, 2093-2104 (1999).
[CrossRef]

B. Bulgarelli, V. B. Kisselev, and L. Roberti, "Radiative Transfer in the Atmosphere-Ocean System: The Finite-Element Method," Appl. Opt. 38, 1530-1542 (1999), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-9-1530.
[CrossRef]

1998

K. F. Evans, "The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

1997

M. I. Mishchenko and L. D. Travis, "Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight," J. Geophys. Res. 102, 16989-17013 (1997).
[CrossRef]

J. L. Haferman, T. F. Smith, and W. F. Krajewski, "A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles," J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

1996

E. P. Zege, and L. I. Chaikovskaya, "New approach to the polarized radiative transfer problem," J. Quant. Spectrosc. Radiat. Transfer 55, 19-31 (1996).
[CrossRef]

Q. Liu and E. Ruprecht, "Radiative transfer model: matrix operator method," Appl. Opt. 35, 4229-4237 (1996).
[CrossRef] [PubMed]

1995

1994

Z. Jin and K. Stamnes, "Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system," Appl. Opt. 33, 431-442 (1994).
[CrossRef] [PubMed]

A. Sánchez, T. F. Smith, and W. F. Krajewski, "A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method," Atmos. Research 33, 283-308 (1994).
[CrossRef]

1993

1992

D. M. O’Brien, "Accelerated quasi Monte Carlo integration of the radiative transfer equation," J. Quant. Spectrosc. Radiat. Transfer 48, 41-59 (1992).
[CrossRef]

A. Kylling and K. Stamnes, "Efficient yet accurate solution of the linear transport equation in the presence of internal sources: the exponential-linear-in depth approximation," J. Comput. Phys. 102, 265-276 (1992).
[CrossRef]

W. M. F. Wauben and J. W. Hovenier, "Polarized radiation of an atmosphere containing randomly-oriented spheroids," J. Quant. Spectrosc. Radiat. Transfer 47, 491-504 (1992).
[CrossRef]

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. Transfer 47, 19-33 (1992).
[CrossRef]

F. Weng, "A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere-II. Application," J. Quant. Spectrosc. Radiat. Transfer 47, 35-42 (1992).
[CrossRef]

1991

M. I. Mishchenko, "Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles," J. Quant. Spectrosc. Radiat. Transfer 46, 171-181 (1991).
[CrossRef]

1990

C. V. M. Van der Mee, J.W. Hovenier, "Expansion coefficients in polarized light transfer," Astron. Astrophys. 228, 559-568 (1990).

1989

J. L. Deuzé, M. Herman, R. Santer, "Fourier series expansion of the transfer equation in the oceanatmosphere system," J. Quant. Spectrosc. Radiat. Transfer 41, 483-494 (1989).
[CrossRef]

G. W. Kattawar and C. N. Adams, "Stokes vector calculations of the submarine light field in an atmosphereocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization," Limnol. Oceanogr. 34, 1453-1472 (1989).
[CrossRef]

R. D. M. Garcia and C. E. Siewert, "The FN method for radiative transfer models that include polarization effects," J. Quant. Spectrosc. Radiat. Transfer 41, 117-145 (1989).
[CrossRef]

1988

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, "Numerically stable algorithm for discrete-ordinate method radiative trasfer in multiple scattering and emitting layered media," Appl. Opt. 27, 2502-2509 (1988).
[CrossRef] [PubMed]

A. Morel, "Optical modeling of the upper ocean in relation to its biogenous matter content (Case I Waters)," J. Geophys. Res. 93, 10479-10768 (1988).
[CrossRef]

T. Nakajima and M. Tanaka, "Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation," J. Quant. Spectrosc. Radiat. Transfer 40, 51-69 (1988).
[CrossRef]

1987

R.B. Myneni, G. Asrar, and E. T. Kanemasu, "Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA)," Agric. For. Meteorol. 39, 1-12 (1987).
[CrossRef]

1986

R. D. M. Garcia and C. E. Siewert, "A generalized spherical harmonics solution for radiative transfer models that include polarization effects," J. Quant. Spectrosc. Radiat. Transfer 36, 401-423 (1986).
[CrossRef]

1985

1984

1983

T. Nakajima, M. Tanaka, "Effect of wind-generated waves on the transfer of solar radiation in the atmosphereocean system," J. Quant. Spectrosc. Radiat. Transfer 29, 521-537 (1983).
[CrossRef]

J. W. Hovenier and C. V. M. van der Mee, "Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere," Astron. Astrophys. 128, 1-16 (1983).

1982

C. E. Siewert, "On the phase matrix basic to the scattering of polarized light," Astron. Astrophys. 109, 195-200 (1982).

1981

1977

W. J. Wiscombe, "The Delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions," J. Atmos. Sci. 34, 1408-1422 (1977).
[CrossRef]

1973

1971

J. E. Hansen, "Multiple scattering of polarized light in planetary atmospheres. part II. Sunlight reflected by terrestrial water clouds," J. Atmos. Sci. 28, 1400-1426 (1971).
[CrossRef]

1970

C. N. Adams and G. W. Kattawar, "Solutions of the equation of radiative transfer by an invariant imbedding approach," J. Quant. Spectrosc. Radiat. Transfer 10, 341-366 (1970).
[CrossRef]

1968

1959

I. Kuˇsˇcer and M. Ribariˇc, "Matrix formalism in the theory of diffusion of light," Opt. Acta 6, 42-51 (1959).
[CrossRef]

1956

I. M. Gel’fand, and Z. Y. Sapiro, "Representation of the group of rotations of 3-dimensional space and their applications," Amer. Math. Soc. Transl. 2, 207-316 (1956).

1954

C. Cox and W. Munk, "Statistics of sea surface derived from sun glitter," J. Mar. Res. 13, 198-227 (1954).

1942

V. A. Ambartzumian, "A new method for computing light scattering in turbid media," Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 3, 97-104 (1942).

1941

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

1939

A. Hammad and S. Chapman, "The primary and secondary scattering of sun light in a plane-stratified atmosphere of uniform composition," Philos. Mag. 28, 99-110 (1939).

1862

G. G. Stokes, "On the intensity of the light reflected from or transmitted through a pile of plates," Proc. Roy. Soc., London  11, 545-556 (1862).

Adams, C. N.

G. W. Kattawar and C. N. Adams, "Stokes vector calculations of the submarine light field in an atmosphereocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization," Limnol. Oceanogr. 34, 1453-1472 (1989).
[CrossRef]

C. N. Adams and G. W. Kattawar, "Solutions of the equation of radiative transfer by an invariant imbedding approach," J. Quant. Spectrosc. Radiat. Transfer 10, 341-366 (1970).
[CrossRef]

Ambartzumian, V. A.

V. A. Ambartzumian, "A new method for computing light scattering in turbid media," Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 3, 97-104 (1942).

Asrar, G.

R.B. Myneni, G. Asrar, and E. T. Kanemasu, "Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA)," Agric. For. Meteorol. 39, 1-12 (1987).
[CrossRef]

Baum, B.

Y. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Bennartz, R.

T. Greenwald, R. Bennartz, C. O’Dell, and A. Heidinger, "Fast computation of microwave radiances for data assimilation using the Successive Order of Scattering method," J. Appl. Meteorol. 44, 960-966 (2005).
[CrossRef]

A. K. Heidinger, C. O’Dell, R. Bennartz, and T. Greenwald, "The successive-order-of-interaction radiative transfer model. part I: model development," J. Appl. Meteorol. Climatol. 45, 1388-1402 (2005).
[CrossRef]

Bulgarelli, B.

Cairns, B.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

J. Chowdhary, B. Cairns, and L. D. Travis, "Contribution of water-leaving radiances to multiangle, multispectral polarimetric observations over the open ocean: bio-optical model results for case 1 waters," Appl. Opt. 45, 5542-5567 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ao-45-22-5542.
[CrossRef] [PubMed]

B. Cairns, E. E. Russell, J. D. LaVeigne, and P. M. W. Tennant. "Research scanning polarimeter and airborne usage for remote sensing of aerosols," Proc SPIE 5158, 33-44 (2003).
[CrossRef]

Carder, K. L.

Catchings, F. E.

Chaikovskaya, L. I.

Chami, M.

Chapman, S.

A. Hammad and S. Chapman, "The primary and secondary scattering of sun light in a plane-stratified atmosphere of uniform composition," Philos. Mag. 28, 99-110 (1939).

Charlock, T. P.

Chowdhary, J.

Cox, C.

C. Cox and W. Munk, "Statistics of sea surface derived from sun glitter," J. Mar. Res. 13, 198-227 (1954).

Deuzé, J. L.

J. L. Deuzé, M. Herman, R. Santer, "Fourier series expansion of the transfer equation in the oceanatmosphere system," J. Quant. Spectrosc. Radiat. Transfer 41, 483-494 (1989).
[CrossRef]

Deuzé, J.L.

J. Lenoble, M. Herman, J.L. Deuzé, B. Lafrance, R. Santer, D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107, 479-507 (2007).
[CrossRef]

M. Herman, J.L. Deuzé, A. Marchand, B. Roger, and P. Lallart, "Aerosol remote sensing from POLDER/ADEOS over the ocean: improved retrieval using a nonspherical particle model," J. Geophys. Res. 110, D10S02 (2005).
[CrossRef]

Dilligeard, E.

Duan, M.

M. Duan and Q. Min, "A semi-analytic technique to speed up successive order of scattering model for optically thick media," J. Quant. Spectrosc. Radiat. Transfer 95, 21-32 (2005).
[CrossRef]

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

Evans, K. F.

K. F. Evans, "SHDOMPPDA: A radiative transfer model for cloudy sky data assimilation". J. Atmos. Sci. 64, 3858-3868 (2007).
[CrossRef]

K. F. Evans, "The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

Fafaul, B. A.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

Fischer, J.

Fry, E. S.

Garcia, R. D. M.

R. D. M. Garcia and C. E. Siewert, "The FN method for radiative transfer models that include polarization effects," J. Quant. Spectrosc. Radiat. Transfer 41, 117-145 (1989).
[CrossRef]

R. D. M. Garcia and C. E. Siewert, "A generalized spherical harmonics solution for radiative transfer models that include polarization effects," J. Quant. Spectrosc. Radiat. Transfer 36, 401-423 (1986).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, and Z. Y. Sapiro, "Representation of the group of rotations of 3-dimensional space and their applications," Amer. Math. Soc. Transl. 2, 207-316 (1956).

Gentili, B.

Gibson, G.

Y. -X. Hu, B. Wielicki, B. Lin, G. Gibson, S. -C. Tsay, K. Stamnes, and T. Wong, "™ -Fit: A fast and accurate treatment of particle scattering phase functions with weighted singular-value decomposition least-squares fitting," J. Quant. Spectrosc. Radiat. Transfer 65, 681-690 (2000).
[CrossRef]

Gordon, H. R.

Grassl, H.

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Greenwald, T.

A. K. Heidinger, C. O’Dell, R. Bennartz, and T. Greenwald, "The successive-order-of-interaction radiative transfer model. part I: model development," J. Appl. Meteorol. Climatol. 45, 1388-1402 (2005).
[CrossRef]

T. Greenwald, R. Bennartz, C. O’Dell, and A. Heidinger, "Fast computation of microwave radiances for data assimilation using the Successive Order of Scattering method," J. Appl. Meteorol. 44, 960-966 (2005).
[CrossRef]

Haferman, J. L.

J. L. Haferman, T. F. Smith, and W. F. Krajewski, "A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles," J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

Hammad, A.

A. Hammad and S. Chapman, "The primary and secondary scattering of sun light in a plane-stratified atmosphere of uniform composition," Philos. Mag. 28, 99-110 (1939).

Hansen, J. E.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

J. E. Hansen, "Multiple scattering of polarized light in planetary atmospheres. part II. Sunlight reflected by terrestrial water clouds," J. Atmos. Sci. 28, 1400-1426 (1971).
[CrossRef]

Heidinger, A.

T. Greenwald, R. Bennartz, C. O’Dell, and A. Heidinger, "Fast computation of microwave radiances for data assimilation using the Successive Order of Scattering method," J. Appl. Meteorol. 44, 960-966 (2005).
[CrossRef]

Heidinger, A. K.

A. K. Heidinger, C. O’Dell, R. Bennartz, and T. Greenwald, "The successive-order-of-interaction radiative transfer model. part I: model development," J. Appl. Meteorol. Climatol. 45, 1388-1402 (2005).
[CrossRef]

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Herman, M.

J. Lenoble, M. Herman, J.L. Deuzé, B. Lafrance, R. Santer, D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107, 479-507 (2007).
[CrossRef]

M. Herman, J.L. Deuzé, A. Marchand, B. Roger, and P. Lallart, "Aerosol remote sensing from POLDER/ADEOS over the ocean: improved retrieval using a nonspherical particle model," J. Geophys. Res. 110, D10S02 (2005).
[CrossRef]

J. L. Deuzé, M. Herman, R. Santer, "Fourier series expansion of the transfer equation in the oceanatmosphere system," J. Quant. Spectrosc. Radiat. Transfer 41, 483-494 (1989).
[CrossRef]

Hooker, R. J.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

Hovenier, J. W.

W. M. F. Wauben and J. W. Hovenier, "Polarized radiation of an atmosphere containing randomly-oriented spheroids," J. Quant. Spectrosc. Radiat. Transfer 47, 491-504 (1992).
[CrossRef]

J. W. Hovenier and C. V. M. van der Mee, "Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere," Astron. Astrophys. 128, 1-16 (1983).

Hovenier, J.W.

C. V. M. Van der Mee, J.W. Hovenier, "Expansion coefficients in polarized light transfer," Astron. Astrophys. 228, 559-568 (1990).

Hu, Y.

Y. Hu, D. Winker, P. Yang, B. Baum, L. Poole, and L. Vann, "Identification of cloud phase from PICASSO-CENA lidar depolarization: a multiple scattering sensitivity study," J. Quant. Spectrosc. Radiat. Transfer 70, 569-579 (2001).
[CrossRef]

Hu, Y. -X.

Y. -X. Hu, B. Wielicki, B. Lin, G. Gibson, S. -C. Tsay, K. Stamnes, and T. Wong, "™ -Fit: A fast and accurate treatment of particle scattering phase functions with weighted singular-value decomposition least-squares fitting," J. Quant. Spectrosc. Radiat. Transfer 65, 681-690 (2000).
[CrossRef]

Itchkawich, T.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

Jayaweera, K.

Jin, Z.

Kanemasu, E. T.

R.B. Myneni, G. Asrar, and E. T. Kanemasu, "Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA)," Agric. For. Meteorol. 39, 1-12 (1987).
[CrossRef]

Katsev, I. L.

Kattawar, G. W.

P. Zhai, G. W. Kattawar, and P. Yang, "Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method," Appl. Opt. 47, 1037-1047 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=ao-47-8-1037.
[CrossRef] [PubMed]

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, "Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations," Appl. Opt. 40, 400-412 (2001).
[CrossRef]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, "Comparison of numerical models for computing underwater light fields," Appl. Opt. 32, 7484-7504 (1993).
[CrossRef] [PubMed]

G. W. Kattawar and C. N. Adams, "Stokes vector calculations of the submarine light field in an atmosphereocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization," Limnol. Oceanogr. 34, 1453-1472 (1989).
[CrossRef]

G. N. Plass, G. W. Kattawar, and F. E. Catchings, "Matrix operator theory of radiative transfer. 1: Rayleighscattering," Appl. Opt. 12, 314-329 (1973).
[CrossRef] [PubMed]

C. N. Adams and G. W. Kattawar, "Solutions of the equation of radiative transfer by an invariant imbedding approach," J. Quant. Spectrosc. Radiat. Transfer 10, 341-366 (1970).
[CrossRef]

G. N. Plass and G. W. Kattawar, "Monte Carlo calculations of light scattering from clouds," Appl. Opt. 7, 415-419 (1968).
[CrossRef] [PubMed]

G. W. Kattawar and G. N. Plass, "Radiance and polarization of multiple scattered light from haze and clouds," Appl. Opt. 7, 1519-1527 (1968).
[CrossRef] [PubMed]

Kattawar, G.W.

Kisselev, V. B.

Klemm, F. J.

Kopp, G.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

Kotchenova, S. Y.

Krajewski, W. F.

J. L. Haferman, T. F. Smith, and W. F. Krajewski, "A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles," J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

A. Sánchez, T. F. Smith, and W. F. Krajewski, "A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method," Atmos. Research 33, 283-308 (1994).
[CrossRef]

Ku?s?cer, I.

I. Kuˇsˇcer and M. Ribariˇc, "Matrix formalism in the theory of diffusion of light," Opt. Acta 6, 42-51 (1959).
[CrossRef]

Kummerow, C.

L. Roberti and C. Kummerow, "Monte Carlo calculations of polarized microwave radiation emerging from cloud structures," J. Geophy. Res. 104, 2093-2104 (1999).
[CrossRef]

Kylling, A.

A. Kylling and K. Stamnes, "Efficient yet accurate solution of the linear transport equation in the presence of internal sources: the exponential-linear-in depth approximation," J. Comput. Phys. 102, 265-276 (1992).
[CrossRef]

Lafrance, B.

J. Lenoble, M. Herman, J.L. Deuzé, B. Lafrance, R. Santer, D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107, 479-507 (2007).
[CrossRef]

Lallart, P.

M. Herman, J.L. Deuzé, A. Marchand, B. Roger, and P. Lallart, "Aerosol remote sensing from POLDER/ADEOS over the ocean: improved retrieval using a nonspherical particle model," J. Geophys. Res. 110, D10S02 (2005).
[CrossRef]

LaVeigne, J. D.

B. Cairns, E. E. Russell, J. D. LaVeigne, and P. M. W. Tennant. "Research scanning polarimeter and airborne usage for remote sensing of aerosols," Proc SPIE 5158, 33-44 (2003).
[CrossRef]

Lenoble, J.

J. Lenoble, M. Herman, J.L. Deuzé, B. Lafrance, R. Santer, D. Tanré, "A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols," J. Quant. Spectrosc. Radiat. Transfer 107, 479-507 (2007).
[CrossRef]

Lin, B.

Y. -X. Hu, B. Wielicki, B. Lin, G. Gibson, S. -C. Tsay, K. Stamnes, and T. Wong, "™ -Fit: A fast and accurate treatment of particle scattering phase functions with weighted singular-value decomposition least-squares fitting," J. Quant. Spectrosc. Radiat. Transfer 65, 681-690 (2000).
[CrossRef]

Liou, K. N.

K. N. Liou, "A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer: Application to cloudy and hazy atmospheres," J. Atmos. Sci. 30, 1303-1326 (1973).
[CrossRef]

Liu, Q.

Marchand, A.

M. Herman, J.L. Deuzé, A. Marchand, B. Roger, and P. Lallart, "Aerosol remote sensing from POLDER/ADEOS over the ocean: improved retrieval using a nonspherical particle model," J. Geophys. Res. 110, D10S02 (2005).
[CrossRef]

Maring, H. B.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

Matarrese, R.

Min, Q.

M. Duan and Q. Min, "A semi-analytic technique to speed up successive order of scattering model for optically thick media," J. Quant. Spectrosc. Radiat. Transfer 95, 21-32 (2005).
[CrossRef]

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

Mishchenko, M. I.

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. Amer. Meteorol. Soc. 88, 677-691 (2007).
[CrossRef]

M. I. Mishchenko and L. D. Travis, "Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight," J. Geophys. Res. 102, 16989-17013 (1997).
[CrossRef]

M. I. Mishchenko, "Reflection of polarized light by plane-parallel slabs containing randomly-oriented, nonspherical particles," J. Quant. Spectrosc. Radiat. Transfer 46, 171-181 (1991).
[CrossRef]

Mobley, C. D.

Morel, A.

Munk, W.

C. Cox and W. Munk, "Statistics of sea surface derived from sun glitter," J. Mar. Res. 13, 198-227 (1954).

Myneni, R.B.

R.B. Myneni, G. Asrar, and E. T. Kanemasu, "Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA)," Agric. For. Meteorol. 39, 1-12 (1987).
[CrossRef]

Nakajima, T.

T. Suzuki, T. Nakajima, M. Tanaka, "Scaling algorithms for the calculation of solar radiative fluxes," J. Quant. Spectrosc. Radiat. Transfer 107, 458-469 (2007).
[CrossRef]

T. Nakajima and M. Tanaka, "Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation," J. Quant. Spectrosc. Radiat. Transfer 40, 51-69 (1988).
[CrossRef]

T. Nakajima, M. Tanaka, "Effect of wind-generated waves on the transfer of solar radiation in the atmosphereocean system," J. Quant. Spectrosc. Radiat. Transfer 29, 521-537 (1983).
[CrossRef]

O’Brien, D. M.

D. M. O’Brien, "Accelerated quasi Monte Carlo integration of the radiative transfer equation," J. Quant. Spectrosc. Radiat. Transfer 48, 41-59 (1992).
[CrossRef]

O’Dell, C.

A. K. Heidinger, C. O’Dell, R. Bennartz, and T. Greenwald, "The successive-order-of-interaction radiative transfer model. part I: model development," J. Appl. Meteorol. Climatol. 45, 1388-1402 (2005).
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Figures (4)

Fig. 1.
Fig. 1.

The Stokes parameters at the top of the atmosphere. The solid lines are calculated by the Monte Carlo (MC) method. The circle and cross symbols are computed by the SOS method with the single Gaussian (SG) quadrature and double Gaussian quadrature (DG) schemes, respectively. The Gaussian quadrature order of Pa = 36 is used in the atmosphere. The ocean Gaussian quadrature order of Po = 72 is used for the DG scheme. The Fourier series order used is M = 8. The solar incident angle is θ 0 = 120°. The azimuthal angle of the observation is 90° from the solar principle plane. The atmosphere is a conservative Rayleigh medium with an optical depth of 0.1. The ocean scattering function is a Henyey- Greenstein function with an asymmetry factor of 0.9185. The optical depth of the ocean is 10.0. The ocean bottom is a Lambertian surface. The single scattering albedos of both the ocean medium and bottom are 0.5. The total order of scattering is N = 10.

Fig. 2.
Fig. 2.

The Stokes parameters at the bottom of the atmosphere (just above the ocean surface) for the same system as in Fig. 1.

Fig. 3.
Fig. 3.

The water leaving Stokes parameters at the bottom of the atmosphere (just above the ocean surface) for the same system as in Fig. 1.

Fig. 4.
Fig. 4.

The effects of the orders of the Gaussian quadrature and Fourier series orders

Tables (2)

Tables Icon

Table 1. Average CPU time for the SG and DG schemes.

Tables Icon

Table 2. Scattering orders needed for convergence.

Equations (104)

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μ d L ( τ , μ , ϕ ) = L ( τ , μ , ϕ ) S ( τ , μ , ϕ ) ,
S ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π 1 1 P ( τ , μ , ϕ , μ , ϕ ) L ( τ , μ , ϕ ) ,
P ( τ , μ , ϕ , μ , ϕ ) = R ( π i 2 ) F ( τ , Θ ) R ( i 1 ) .
L ( τ , μ < 0 , ϕ ) = L ( τ l , μ , ϕ ) e ( τ l τ ) / μ τ l τ e ( τ t ) / μ S ( τ , μ , ϕ ) / μ ,
L ( τ , μ > 0 , ϕ ) = L ( τ u , μ , ϕ ) e ( τ u τ ) / μ + τ τ u e ( τ t ) / μ S ( τ , μ , ϕ ) / μ ,
L ( τ , μ , ϕ ) = n = 1 N L n ( τ , μ , ϕ ) .
L n ( τ , μ < 0 , ϕ ) = L n ( τ l , μ , ϕ ) e ( τ l τ ) / μ τ l τ e ( τ t ) / μ S n ( τ , μ , ϕ ) / μ ,
L n ( τ , μ > 0 , ϕ ) = L n ( τ u , μ , ϕ ) e ( τ u τ ) / μ + τ τ u e ( τ t ) / μ S n ( τ , μ , ϕ ) / μ ,
S 1 ( τ < τ a * , μ , ϕ ) = ω ( τ ) 4 π { e τ / μ 0 P ( τ , μ , ϕ , μ 0 , ϕ 0 )
+ e ( 2 τ a * τ ) / μ 0 P ( τ , μ , ϕ , μ 0 , ϕ 0 ) r ( π θ 0 ) } E 0 ,
S 1 ( τ > τ o , μ , ϕ ) = ω ( τ ) 4 π μ 0 μ 0 e ( τ τ o ) / μ 0 P ( τ , μ , ϕ , μ 0 , ϕ 0 ) t ( π θ 0 ) E 0 e τ a * / μ 0 ,
S n > 1 ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π 1 1 P ( τ , μ , ϕ , μ , ϕ ) L n 1 ( τ , μ , ϕ ) d μ d ϕ .
L n ( τ = 0 , μ < 0 , ϕ ) = 0 ,
L 1 ( τ a * , μ > 0 , ϕ ) = 0 ,
L n > 1 ( τ o * , μ > 0 , ϕ ) = r ( θ ) L n 1 ( τ o * , μ , ϕ ) + t ( θ ) n w 2 L n 1 ( τ a , μ , ϕ ) ,
L 1 ( τ o , μ < 0 , ϕ ) = 0 ,
L n > 1 ( τ o , μ < 0 , ϕ ) = r ( π θ ) L n 1 ( τ o , μ , ϕ ) + n w 2 t ( π θ ) L n 1 ( τ a * , μ , ϕ ) ,
L 1 ( τ o * , μ > 0 , ϕ ) = μ 0 π r * ( μ , ϕ , μ 0 , ϕ 0 ) e ( τ o * τ o ) / μ 0 t ( π θ 0 ) μ 0 μ 0 E 0 e τ a * / μ 0 ,
L n > 1 ( τ o * , μ > 0 , ϕ ) = 0 2 π 1 0 μ π r * ( μ , ϕ , μ , ϕ ) L n 1 ( τ o * , μ , ϕ ) d μ d ϕ
L n ( τ , μ , ϕ ) = m = 0 M ( 2 δ 0 m ) { cos [ m ( ϕ ϕ 0 ) ] L n , cos m ( τ , μ ) + sin [ m ( ϕ ϕ 0 ) ] L n , sin m ( τ , μ ) } ,
L n , cos m ( τ , μ ) = ( I n m , Q n m , 0 , 0 ) T ,
L n , sin m ( τ , μ ) = ( 0,0 , U n m , V n m ) T .
P ( τ , μ , ϕ , μ , ϕ ) = P 1 ( τ , μ , ϕ , μ , ϕ ) + P 2 ( τ , μ , ϕ , μ , ϕ ) ,
P 1 ( τ , μ , ϕ , μ , ϕ ) = ( P 11 P 12 0 0 P 21 P 22 0 0 0 0 P 33 P 34 0 0 P 43 P 44 )
= m = 0 M ( 2 δ 0 m ) cos [ m ( ϕ ϕ ) ] P cos m ( τ , μ , μ ) ,
P 2 ( τ , μ , ϕ , μ , ϕ ) = ( 0 0 P 13 P 14 0 0 P 23 P 24 P 31 P 32 0 0 P 41 P 42 0 0 )
= m = 0 M ( 2 δ 0 m ) sin [ m ( ϕ ϕ ) ] P sin m ( τ , μ , μ )
L n m ( τ , μ < 0 ) = L n m ( τ l , μ ) e ( τ l τ ) / μ τ l τ e ( τ τ ) / μ S n m ( τ , μ ) / μ ,
L n m ( τ , μ > 0 ) = L n m ( τ u , μ ) e ( τ u τ ) / μ + τ τ u e ( τ τ ) / μ S n m ( τ , μ ) d τ / μ ,
S 1 m ( τ < τ a * , μ ) = ω ( τ ) 4 π { e τ / μ 0 P m ( τ , μ , μ 0 )
+ e ( 2 τ a * τ ) / μ 0 P m ( τ , μ , μ 0 ) r ( π θ 0 ) } E 0 ,
S 1 m ( τ > τ o , μ ) = ω ( τ ) 4 π μ 0 μ 0 e ( τ τ o ) / μ 0 P m ( τ , μ , μ 0 ) t ( π θ 0 ) E 0 e τ a * / μ 0 ,
S n > 1 m ( τ , μ ) = ω ( τ ) 2 1 1 P m ( τ , μ , μ " ) L n 1 m ( τ , μ " ) d μ " ,
L n m ( τ = 0 , μ < 0 ) = 0 ,
L 1 m ( τ a * , μ > 0 ) = 0 ,
L n > 1 m ( τ a * , μ > 0 ) = r ( θ ) L n 1 m ( τ a * , μ ) + t ( θ ) n w 2 L n 1 m ( τ o , μ ) ,
L 1 m ( τ o , μ < 0 ) = 0 ,
L n > 1 m ( τ o , μ < 0 ) = r ( π θ ) L n 1 m ( τ o , μ ) + n w 2 t ( π θ ) L n 1 m ( τ a * , μ ) ,
L 1 m ( τ a * , μ > 0 ) = μ 0 π r * m ( μ , μ 0 ) e ( τ o * τ o ) / μ 0 t ( π θ 0 ) μ 0 μ 0 E 0 e τ a * / μ 0 ,
L n > 1 m ( τ a * , μ > 0 ) = 2 1 0 ( μ ) r * m ( μ , μ ) L n 1 m ( τ a * , μ ) d μ .
L n m ( τ , μ < 0 ) = n w 2 t ( π θ ) L n 1 m ( τ a * , μ ) e ( τ l τ ) / μ + L n , diff m ( τ , μ ) ,
L n , diff m ( τ , μ ) = r ( π θ ) L n 1 m ( τ o , μ ) e ( τ l τ ) / μ τ l τ e ( τ ' τ ) / μ S n m ( τ , μ ) d τ / μ ,
n w 2 d μ " = n w 2 sin ( θ " ) d θ "
= n w sin ( θ ) cos ( θ ) d θ n w cos ( θ " )
= μ μ " d μ .
S n > 1 , drct m ( τ , μ p ) = ω ( τ ) 2 1 μ c P m ( τ , μ p , μ " ) n w 2 t ( π θ ) L n 1 m ( τ a * , μ ) e ( τ l τ ) / μ " d μ "
= ω ( τ ) 2 1 0 P m ( τ , μ p , μ " ) t ( π θ ) L n 1 m ( τ a * , μ ) e ( τ l τ ) / μ " μ μ " d μ '
= ω ( τ ) 2 q = 1 P / 2 w q P m ( τ , μ p , μ q " ) t ( π θ q ) L n 1 m ( τ a * , μ q ) e ( τ l τ ) / μ q " μ q μ q " ,
S n > 1 , diff m ( τ , μ p ) = ω ( τ ) 2 1 1 P m ( τ , μ p , μ " ) L n 1 , diff m ( τ , μ " ) d μ "
= ω ( τ ) 2 q = 1 p w q P m ( τ , μ p , μ q ) L n 1 , diff m ( τ , μ q ) .
S n > 1 m ( τ , μ p ) = S n > 1 , drct m ( τ , μ p ) + S n > 1 , diff m ( τ , μ p )
L n , wlr m ( τ , μ > 0 ) = t ( θ ) n w 2 L n 1 m ( τ o , μ ) e ( τ u τ ) / μ ,
L n m ( τ k , μ p < 0 ) = τ l τ k e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p ,
= τ l τ k 1 e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p + τ k 1 τ k e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p ,
= e δτ / μ p L n m ( τ k 1 , μ p < 0 ) - τ k - 1 τ k e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p .
L n m ( τ k , μ p > 0 ) = τ k τ u e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p ,
= τ k + 1 τ u e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p + τ k τ k + 1 e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p ,
= e −δτ / μ p L n m ( τ k + 1 , μ p > 0 ) + τ k τ k + 1 e ( τ τ k ) / μ p S n m ( τ , μ p ) / μ p .
L n m ( τ k > τ o , μ p > 0 ) L n m ( τ k > τ o , μ p > 0 ) + L n m ( τ o * , μ p > 0 ) e ( τ k τ o * ) / μ p ,
L n m ( τ k > τ o , μ p < 0 ) L n m ( τ k > τ o , μ p < 0 ) + r ( π θ p ) L n m ( τ o , μ p ) e ( τ k τ o ) / μ p ,
L n m ( τ k < τ o * , μ p > 0 ) L n m ( τ k < τ o * , μ p > 0 ) + r ( θ p ) L n m ( τ o * , μ p ) e ( τ k τ o ) / μ p ,
L n , det m ( τ k > τ o , μ p < 0 ) L n , det m ( τ k , μ p ) + L n m ( τ k , μ p )
+ e ( τ o τ k ) / μ p n w 2 t ( π θ p ) L n 1 m ( τ a * , μ p ) ,
L n , det m ( τ k > τ o , μ p > 0 ) L n , det m ( τ k , μ p ) + L n m ( τ k , μ p ) ,
L n , det m ( τ k < τ a * , μ p < 0 ) L n , det m ( τ k , μ p ) + L n m ( τ k , μ p ) ,
L n , det m ( τ k < τ a * , μ p > 0 ) L n , det m ( τ k , μ p ) + L n m ( τ k , μ p )
+ e ( τ k τ a * ) / μ p n w 2 t ( π θ p ) L n 1 m ( τ o , μ p ) .
q n > N m ( τ k , μ p ) = L n m ( τ k , μ p ) L n 1 m ( τ k , μ p ) = Cons tan t .
L m ( τ , μ ) = n = 1 N L n m ( τ , μ ) + L n m ( τ , μ ) q N m ( τ k , μ p ) 1 q N m ( τ k , μ p ) .
F ( Θ ) = 3 4 ( cos 2 ( Θ ) + 1 cos 2 ( Θ ) 1 0 0 cos 2 ( Θ ) 1 cos 2 ( Θ ) + 1 0 0 0 0 2 cos ( Θ ) 0 0 0 0 2 cos ( Θ ) ) .
F 11 ( μ , g ) = 1 g 2 ( 1 2 g μ + g 2 ) 3 / 2 ,
g = 1 P ( θ ) cos ( θ ) d Ω
F i j H G F 11 H G = F i j RAY F 11 RAY ,
P mn l ( ξ ) = i m n d mn l ( θ ) ,
P m , ± 2 l ( ξ ) = i m 2 d m , ± 2 l ( θ ) = i m d m , ± 2 l ( θ ) .
R l m ( ξ ) = ( 1 ) m 2 ( l + m ) ! ( l m ) ! ( d m , 2 l + d m , 2 l )
= ( l + m ) ! ( l m ) ! d m , 2 l , + ,
T l m ( ξ ) = ( 1 ) m 2 ( l + m ) ! ( l m ) ! ( d m , 2 l d m , 2 l )
= ( l + m ) ! ( l m ) ! d m , 2 l , ,
d m , 2 l , + = ( 1 ) m d m , 2 l + d m , 2 l 2 ,
d m , 2 l , = ( 1 ) m d m , 2 l d m , 2 l 2 .
d 00 l ( θ ) = P i ( ξ ) .
d m 0 l ( θ ) = ( l m ) ! l + m ) ! P l m ( ξ ) .
P l 2 ( ξ ) = ( l + 2 ) ! l 2 ) ! d 20 l ( θ )
F ( Θ ) = ( a 1 ( ξ ) b 1 ( ξ ) 0 0 b 1 ( ξ ) a 2 ( ξ ) 0 0 0 0 a 3 ( ξ ) b 2 ( ξ ) 0 0 b 2 ( ξ ) a 4 ( ξ ) ) .
β l = 2 l + 1 2 1 1 a 1 ( ξ ) d 00 l ( θ ) ,
δ l = 2 l + 1 2 1 1 a 4 ( ξ ) d 00 l ( θ ) .
γ l = 2 l + 1 2 1 1 b 1 ( ξ ) d 20 l ( θ ) ,
ε l = 2 l + 1 2 1 1 b 2 ( ξ ) d 20 l ( θ ) .
ζ 1 = 2 l + 1 2 1 1 [ a 2 ( ξ ) d 2,2 l , ( θ ) + a 3 ( ξ ) d 2,2 l , + ( θ ) ] ,
α 1 = 2 l + 1 2 1 1 [ a 2 ( ξ ) d 2,2 l , + ( θ ) + a 3 ( ξ ) d 2,2 l , ( θ ) ] .
β l = { α 1 l } M , α l = { α 2 l } M ,
ζ l = { α 3 l } M , δ l = { α 4 l } M ,
γ l = { β 1 l } M , ε l = { β 2 l } M ,
P m = l = m L ( β l d m 0 l d m 0 l γ l d m 0 l d m 2 l ,+ γ l d m 0 l d m 2 l , 0 γ l d m 2 l , + d m 0 l α l d m 2 l , + d m 2 l ,+ + ζ l d m 2 l , d m 2 l , α l d m 2 l , + d m 2 l , ζ l d m 2 l , d m 2 l , + ε l d m 2 l , d m 0 l γ l d m 2 l , d m 0 l α l d m 2 l , d m 2 l ,+ ζ l d m 2 l , + d m 2 l , α l d m 2 l , d m 2 l , + ζ l d m 2 l , + d m 2 l , + ε l d m 2 l , + d m 0 l 0 ε l d m 0 l d m 2 l , ε l d m 0 l d m 2 l , + δ l d m 0 l d m 0 l ) ,
a 1 * ( θ ) = 2 ( 1 cos θ ) + ( 1 f ) l = 0 L β l * d 00 l ( θ )
β l * = β l f 1 f , 0 <= l <= L .
ε = k w k ( a 1 * ( θ k ) a 1 ( θ k ) 1 ) 2 ,
a 1 * ( θ k ) = l = 0 L β l * d 00 l ( θ k ) .
F ij * ( θ k ) = a 1 * ( θ k ) . F ij ( θ k ) a 1 ( θ k ) ,
ε ij = k [ l = 0 L c ij l * d mn l ( θ k ) F ij * ( θ k ) ] 2 ,
ε ij c ij l = 0.0 .
τ = ( 1 ωf ) τ ,
ω = ω ( 1 f ) 1 ωf .

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