Abstract

The Markov chain formalism for polarized radiative transfer through a vertically inhomogeneous atmosphere is linearized comprehensively with respect to the aerosol and polarizing surface properties. For verification, numerical results are compared to those obtained by the finite difference method. We demonstrate the use of the linearized code as part of a retrieval of aerosol and surface properties for an atmosphere overlying a black and Fresnel-reflecting ocean surface.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. J. Diner, J. C. Beckert, T. H. Reilly, C. J. Bruegge, J. E. Conel, R. Kahn, J. V. Martonchik, T. P. Ackerman, R. Davies, S. A. W. Gerstl, H. R. Gordon, J. P. Muller, R. Myneni, R. J. Sellers, B. Pinty, and M. M. Verstraete, “Multi-angle Imaging SpectroRadiometer (MISR) instrument description and experiment overview,” IEEE Trans. Geosci. Remote Sens. 36, 1072–1087 (1998).
    [CrossRef]
  2. V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
    [CrossRef]
  3. P. Y. Deschamps, F. M. Bréon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Sèze, “The POLDER mission: Instrument characteristics and scientific objectives,” IEEE Trans. Geosci. Remote Sens. 32, 598–615 (1994).
    [CrossRef]
  4. R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
    [CrossRef]
  5. I. L. Katsev, A. S. Prikhach, E. P. Zege, A. P. Ivanov, and A. A. Kokhanovsky, “Iterative procedure for retrieval of spectral aerosol optical thickness and surface reflectance from satellite data using fast radiative transfer code and its application to MERIS measurements,” in Satellite Aerosol Remote Sensing Over Land, A. A. Kokhanovsky and G. de Leeuw, eds. (Springer, 2009).
  6. L. W. Esposito and L. L. House, “Radiative transfer calculated by a Markov chain formalism,” Astrophys. J. 219, 1058–1067 (1978).
    [CrossRef]
  7. F. Xu, A. B. Davis, R. A. West, and L. 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).
    [CrossRef]
  8. F. Xu, A. B. Davis, R. A. West, J. V. Martonchik, and D. J. Diner, “Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere overlying a polarizing surface,” Opt. Lett. 36, 2083–2085 (2011).
    [CrossRef]
  9. R. J. D. Spurr, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
    [CrossRef]
  10. O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
    [CrossRef]
  11. D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
    [CrossRef]
  12. D. W. Marquardt, “An algorithm for least squares estimation of nonlinear inequalities,” SIAM J. Appl. Math. 11, 431–441 (1963).
    [CrossRef]
  13. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  14. H. Wang, ed., Handbook of Modern Mathematics in Science and Engineering, Vol. 1 (Huazhong College of Technology, 1985).
  15. L. W. Esposito, “An ‘adding’ algorithm for the Markov chain formalism for radiation transfer,” Astrophys. J. 233, 661–663 (1979).
    [CrossRef]
  16. 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,” Atmos. Meas. Tech. 3, 909–932 (2010).
    [CrossRef]
  17. H. C. van de Hulst, Light Scattering by Small Particles, (Dover, 1981).
  18. H. Rahman, B. Pinty, and M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model 1. model description and inversion on synthetic data,” J. Geophys. Res. 98, 20779–20789 (1993).
    [CrossRef]
  19. J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
    [CrossRef]
  20. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
    [CrossRef]
  21. R. G. Grainger, J. Lucas, G. E. Thomas, and G. B. L. Ewen, “Calculation of Mie derivatives,” Appl. Opt. 43, 5386–5393 (2004).
    [CrossRef]
  22. P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
    [CrossRef]
  23. J. F. Potter, “The delta function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
    [CrossRef]
  24. 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, 2057–2079 (2009).
    [CrossRef]
  25. 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. Transfer 102, 316–342 (2006).
    [CrossRef]
  26. 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.”
  27. O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multiangle photo-polarimetry to vertical layering and mixing of absorbing aerosols: quantifying measurement uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 112, 2149–2163 (2011).
    [CrossRef]
  28. E. A. Ustinov, “Estimation of error in the solutions of inverse problems and application to the limb sounding of the emission of the upper planetary atmosphere,” Kosmicheskie issledovaniia 30, 120–126 (1992) (in Russian).
  29. S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).
  30. M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
    [CrossRef]
  31. M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
    [CrossRef]
  32. D. J. Diner, A. Davis, B. Hancock, S. Geier, B. Rheingans, V. Jovanovic, M. Bull, D. M. Rider, R. A. Chipman, A. Mahler, and S. C. McClain, “First results from a dual photoelastic modulator-based polarimetric camera,” Appl. Opt. 49, 2929–2946 (2010).
    [CrossRef]
  33. C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
  34. J. W. Hovenier, “Symmetry relations for scattering of polarized light in a slab of randomly oriented particles,” J. Atmos. Sci. 26, 488–499 (1969).
    [CrossRef]
  35. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41, 988–993 (2002).
    [CrossRef]
  36. C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley, 2006).
  37. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

2011

2010

D. J. Diner, A. Davis, B. Hancock, S. Geier, B. Rheingans, V. Jovanovic, M. Bull, D. M. Rider, R. A. Chipman, A. Mahler, and S. C. McClain, “First results from a dual photoelastic modulator-based polarimetric camera,” Appl. Opt. 49, 2929–2946 (2010).
[CrossRef]

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

2009

2007

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

2006

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. Transfer 102, 316–342 (2006).
[CrossRef]

2005

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

2004

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

R. G. Grainger, J. Lucas, G. E. Thomas, and G. B. L. Ewen, “Calculation of Mie derivatives,” Appl. Opt. 43, 5386–5393 (2004).
[CrossRef]

2002

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41, 988–993 (2002).
[CrossRef]

2001

R. J. D. Spurr, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

1998

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

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

1994

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

1993

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

H. Rahman, B. Pinty, and M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model 1. model description and inversion on synthetic data,” J. Geophys. Res. 98, 20779–20789 (1993).
[CrossRef]

1992

E. A. Ustinov, “Estimation of error in the solutions of inverse problems and application to the limb sounding of the emission of the upper planetary atmosphere,” Kosmicheskie issledovaniia 30, 120–126 (1992) (in Russian).

1989

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
[CrossRef]

1982

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

1979

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

1978

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

1970

J. F. Potter, “The delta function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

1969

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

1963

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear inequalities,” SIAM J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

1954

Ackerman, T. P.

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

Allen, D.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Averyt, K. B.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Barnes, W. L.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
[CrossRef]

Beckert, J. C.

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

Bohren, C. F.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley, 2006).

Box, M. A.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Bréon, F. M.

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

Bricaud, A.

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

Bruegge, C. J.

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

Bull, M.

Burg, R.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Buriez, J. C.

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

Cairns, B.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Chandrasekhar, S.

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

Chen, Z.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Chipman, R. A.

Clothiaux, E. E.

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley, 2006).

Conel, J. E.

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

Cox, C.

Davies, R.

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

Davis, A.

Davis, A. B.

F. Xu, A. B. Davis, R. A. West, and L. 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).
[CrossRef]

F. Xu, A. B. Davis, R. A. West, J. V. Martonchik, and D. J. Diner, “Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere overlying a polarizing surface,” Opt. Lett. 36, 2083–2085 (2011).
[CrossRef]

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multiangle photo-polarimetry to vertical layering and mixing of absorbing aerosols: quantifying measurement uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 112, 2149–2163 (2011).
[CrossRef]

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

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.”

DeFoor, T. E.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Deschamps, P. Y.

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

Deuzé, J. L.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Diner, D. J.

F. Xu, A. B. Davis, R. A. West, J. V. Martonchik, and D. J. Diner, “Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere overlying a polarizing surface,” Opt. Lett. 36, 2083–2085 (2011).
[CrossRef]

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multiangle photo-polarimetry to vertical layering and mixing of absorbing aerosols: quantifying measurement uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 112, 2149–2163 (2011).
[CrossRef]

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

D. J. Diner, A. Davis, B. Hancock, S. Geier, B. Rheingans, V. Jovanovic, M. Bull, D. M. Rider, R. A. Chipman, A. Mahler, and S. C. McClain, “First results from a dual photoelastic modulator-based polarimetric camera,” Appl. Opt. 49, 2929–2946 (2010).
[CrossRef]

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

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

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Dubovik, O.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Ducos, F.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Dutton, E. G.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Eck, T. F.

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

Emde, C.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Esposito, L. W.

F. Xu, A. B. Davis, R. A. West, and L. 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).
[CrossRef]

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

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

Ewen, G. B. L.

Fafaul, B. A.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

Gaitley, B. J.

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

Garay, M. J.

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multiangle photo-polarimetry to vertical layering and mixing of absorbing aerosols: quantifying measurement uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 112, 2149–2163 (2011).
[CrossRef]

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Geier, S.

Gerstl, S. A. W.

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

Gordon, H. R.

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

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

Grainger, R. G.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

R. G. Grainger, J. Lucas, G. E. Thomas, and G. B. L. Ewen, “Calculation of Mie derivatives,” Appl. Opt. 43, 5386–5393 (2004).
[CrossRef]

Hancock, B.

Hansen, J. E.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Hasekamp, O. P.

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

Heckel, A.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Herman, B. M.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Herman, M.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Hodos, R. A.

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Hofmann, D. J.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Holben, B. N.

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

Hooker, R. J.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

House, L. L.

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

Hovenier, J. W.

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

Hu, Y.

Itchkawich, T.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

Ivanov, A. P.

I. L. Katsev, A. S. Prikhach, E. P. Zege, A. P. Ivanov, and A. A. Kokhanovsky, “Iterative procedure for retrieval of spectral aerosol optical thickness and surface reflectance from satellite data using fast radiative transfer code and its application to MERIS measurements,” in Satellite Aerosol Remote Sensing Over Land, A. A. Kokhanovsky and G. de Leeuw, eds. (Springer, 2009).

Jovanovic, V.

Kahn, R.

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

Kahn, R. A.

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

Kalashnikova, O. V.

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multiangle photo-polarimetry to vertical layering and mixing of absorbing aerosols: quantifying measurement uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 112, 2149–2163 (2011).
[CrossRef]

Katsev, I. L.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

I. L. Katsev, A. S. Prikhach, E. P. Zege, A. P. Ivanov, and A. A. Kokhanovsky, “Iterative procedure for retrieval of spectral aerosol optical thickness and surface reflectance from satellite data using fast radiative transfer code and its application to MERIS measurements,” in Satellite Aerosol Remote Sensing Over Land, A. A. Kokhanovsky and G. de Leeuw, eds. (Springer, 2009).

Kaufman, Y. J.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Keller, J.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Kinne, S. A.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Knyazikhin, Y.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

Kokhanovsky, A. A.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

I. L. Katsev, A. S. Prikhach, E. P. Zege, A. P. Ivanov, and A. A. Kokhanovsky, “Iterative procedure for retrieval of spectral aerosol optical thickness and surface reflectance from satellite data using fast radiative transfer code and its application to MERIS measurements,” in Satellite Aerosol Remote Sensing Over Land, A. A. Kokhanovsky and G. de Leeuw, eds. (Springer, 2009).

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Kong, J. A.

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

Kopp, G.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

Landgraf, J.

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

Leroy, M.

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

Levy, R.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Livingston, J. M.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Lucas, J.

Lucker, P. L.

Mahler, A.

Manning, M.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Maring, H. B.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear inequalities,” SIAM J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Marquis, M.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Martins, J. V.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Martonchik, J. V.

F. Xu, A. B. Davis, R. A. West, J. V. Martonchik, and D. J. Diner, “Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere overlying a polarizing surface,” Opt. Lett. 36, 2083–2085 (2011).
[CrossRef]

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multiangle photo-polarimetry to vertical layering and mixing of absorbing aerosols: quantifying measurement uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 112, 2149–2163 (2011).
[CrossRef]

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

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Maymon, P. W.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
[CrossRef]

McClain, S. C.

Meier, S. R.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41, 988–993 (2002).
[CrossRef]

Miller, H. L.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Mishchenko, M. I.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Montgomery, H. E.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
[CrossRef]

Muller, J. P.

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

Munk, W.

Myneni, R.

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

Myneni, R. B.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

North, P. R. J.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Ostrow, H.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
[CrossRef]

Ota, Y.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Pilewskie, P.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Pinty, B.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

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

H. Rahman, B. Pinty, and M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model 1. model description and inversion on synthetic data,” J. Geophys. Res. 98, 20779–20789 (1993).
[CrossRef]

Podaire, A.

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

Potter, J. F.

J. F. Potter, “The delta function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

Priest, R. G.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41, 988–993 (2002).
[CrossRef]

Prikhach, A. S.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

I. L. Katsev, A. S. Prikhach, E. P. Zege, A. P. Ivanov, and A. A. Kokhanovsky, “Iterative procedure for retrieval of spectral aerosol optical thickness and surface reflectance from satellite data using fast radiative transfer code and its application to MERIS measurements,” in Satellite Aerosol Remote Sensing Over Land, A. A. Kokhanovsky and G. de Leeuw, eds. (Springer, 2009).

Pueschel, R. F.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Qin, D.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Rahman, H.

H. Rahman, B. Pinty, and M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model 1. model description and inversion on synthetic data,” J. Geophys. Res. 98, 20779–20789 (1993).
[CrossRef]

Reagan, J. A.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Reilly, T. H.

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

Rheingans, B.

Rider, D. M.

Rozanov, V. V.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Russell, P. B.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

Salomonson, V. V.

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
[CrossRef]

Sanghavi, S. V.

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Sayer, A. M.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Schueller, C. F.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

Sellers, R. J.

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

Sèze, G.

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

Shettle, E. P.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Shin, R. T.

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

Siewert, C. E.

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

Smirnov, A.

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

Solomon, S. D.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Spurr, R. J. D.

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. Transfer 102, 316–342 (2006).
[CrossRef]

R. J. D. Spurr, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

Tanré, D.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Thomas, G. E.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

R. G. Grainger, J. Lucas, G. E. Thomas, and G. B. L. Ewen, “Calculation of Mie derivatives,” Appl. Opt. 43, 5386–5393 (2004).
[CrossRef]

Tigor, M.

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Travis, L. D.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

Trepte, C. R.

Tsang, L.

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

Ustinov, E. A.

E. A. Ustinov, “Estimation of error in the solutions of inverse problems and application to the limb sounding of the emission of the upper planetary atmosphere,” Kosmicheskie issledovaniia 30, 120–126 (1992) (in Russian).

van de Hulst, H. C.

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

Verstraete, M. M.

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

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

H. Rahman, B. Pinty, and M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model 1. model description and inversion on synthetic data,” J. Geophys. Res. 98, 20779–20789 (1993).
[CrossRef]

von Allmen, P.

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

West, R. A.

Xu, F.

Zege, E. P.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

I. L. Katsev, A. S. Prikhach, E. P. Zege, A. P. Ivanov, and A. A. Kokhanovsky, “Iterative procedure for retrieval of spectral aerosol optical thickness and surface reflectance from satellite data using fast radiative transfer code and its application to MERIS measurements,” in Satellite Aerosol Remote Sensing Over Land, A. A. Kokhanovsky and G. de Leeuw, eds. (Springer, 2009).

Zhai, P. W.

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, 2057–2079 (2009).
[CrossRef]

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

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

Astrophys. J.

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

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

Atmos. Meas. Tech.

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,” Atmos. Meas. Tech. 3, 909–932 (2010).
[CrossRef]

Atmos. Res.

D. J. Diner, R. A. Hodos, A. B. Davis, M. J. Garay, J. V. Martonchik, S. V. Sanghavi, P. von Allmen, A. A. Kokhanovsky, and P. W. Zhai, “An optimization approach for aerosol retrievals using simulated MISR radiances,” Atmos. Res., doi:10.1016/j.atmosres.2011.05.020 (to be published).
[CrossRef]

Bull. Am. Meteorol. Soc.

M. I. Mishchenko, B. Cairns, G. Kopp, C. F. Schueller, 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, 677–691 (2007).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

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

V. V. Salomonson, W. L. Barnes, P. W. Maymon, H. E. Montgomery, and H. Ostrow, “MODIS: Advanced facility instrument for studies of the Earth as a system,” IEEE Trans. Geosci. Remote Sens. 27, 145–153 (1989).
[CrossRef]

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

J. V. Martonchik, D. J. Diner, B. Pinty, M. M. Verstraete, R. B. Myneni, Y. Knyazikhin, and H. R. Gordon, “Determination of land and ocean reflective, radiative, and biophysical properties using multiangle imaging,” IEEE Trans. Geosci. Remote Sens. 36, 1266–1281 (1998).
[CrossRef]

J. Atmos. Sci.

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

J. F. Potter, “The delta function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

J. Geophys. Res.

P. B. Russell, J. M. Livingston, E. G. Dutton, R. F. Pueschel, J. A. Reagan, T. E. DeFoor, M. A. Box, D. Allen, P. Pilewskie, B. M. Herman, S. A. Kinne, and D. J. Hofmann, “Pinatubo and pre-Pinatubo optical-depth spectra: Mauna Loa measurements, comparisons, inferred particle size distributions, radiative effects, and relationship to lidar data,” J. Geophys. Res. 98, 22969–22985 (1993).
[CrossRef]

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

H. Rahman, B. Pinty, and M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model 1. model description and inversion on synthetic data,” J. Geophys. Res. 98, 20779–20789 (1993).
[CrossRef]

R. A. Kahn, B. J. Gaitley, M. J. Garay, D. J. Diner, T. F. Eck, A. Smirnov, and B. N. Holben, “MISR global aerosol product assessment by comparison with AERONET,” J. Geophys. Res. 115, D23209, doi:10.1029/2010JD014601 (2010).
[CrossRef]

J. Opt. Soc. Am.

J. Quant. Spectrosc. Radiat. Transfer

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. V. Martins, and E. P. Shettle, “Monitoring of aerosol forcing of climate from space: Analysis of measurement requirements,” J. Quant. Spectrosc. Radiat. Transfer 88, 149–161 (2004).
[CrossRef]

O. V. Kalashnikova, M. J. Garay, A. B. Davis, D. J. Diner, and J. V. Martonchik, “Sensitivity of multiangle photo-polarimetry to vertical layering and mixing of absorbing aerosols: quantifying measurement uncertainties,” J. Quant. Spectrosc. Radiat. Transfer 112, 2149–2163 (2011).
[CrossRef]

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. Transfer 102, 316–342 (2006).
[CrossRef]

R. J. D. Spurr, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

Kosmicheskie issledovaniia

E. A. Ustinov, “Estimation of error in the solutions of inverse problems and application to the limb sounding of the emission of the upper planetary atmosphere,” Kosmicheskie issledovaniia 30, 120–126 (1992) (in Russian).

Opt. Eng.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41, 988–993 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

SIAM J. Appl. Math.

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear inequalities,” SIAM J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Other

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

H. Wang, ed., Handbook of Modern Mathematics in Science and Engineering, Vol. 1 (Huazhong College of Technology, 1985).

I. L. Katsev, A. S. Prikhach, E. P. Zege, A. P. Ivanov, and A. A. Kokhanovsky, “Iterative procedure for retrieval of spectral aerosol optical thickness and surface reflectance from satellite data using fast radiative transfer code and its application to MERIS measurements,” in Satellite Aerosol Remote Sensing Over Land, A. A. Kokhanovsky and G. de Leeuw, eds. (Springer, 2009).

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

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.”

C. F. Bohren and E. E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley, 2006).

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

S. D. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tigor, and H. L. Miller, eds., Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University, 2007).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1.
Fig. 1.

The percentage retrieval error bar estimated by use of Eq. (13) when both Iobs and Qobs from 10 pixels are used to retrieve aerosol properties mr, rm, S, and τa for Cases 6–10 [16]. For clarity of view the results of mr are multiplied by a factor 100.

Tables (5)

Tables Icon

Table 1. Mean Percentage Error of the Derivatives of Intensity Calculated for the Viewing Angles Varying between 0° and 90° in the Principal Plane ϕϕ0=0° by FD Approach as a Function of Δxq/xqa

Tables Icon

Table 2. Aerosol Retrieval from Synthetic Data for Cases 6–8 and 10 [16], Using Both BRF and Polarizationa

Tables Icon

Table 3. Same as Table 2, but Retrieval by Using Only BRF

Tables Icon

Table 4. Aerosol and Surface Retrieval from Synthetic Data for the Atmospheres in Cases 6–8 and 10 [16] Overlying an Ocean Surface of Wind Speed v=5m/s, Using Both BRF and Polarization

Tables Icon

Table 5. Same as Table 4, but Retrieval by Using Only BRF

Equations (147)

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

(2δ0m)IM(m)=R(m)(EQ(m))1Π0(m),
I=IS+m=0(2δ0m)[IM,c(m)cosm(ϕϕ0)+IM,s(m)sinm(ϕϕ0)].
(A+B)xq=Axq+Bxq,
(AB)xq=AxqB+ABxq,
(A1)xq=A1AxqA1.
IM(m)xq=1(2δ0m)[X(m)xqΠ0(m)+X(m)Π0(m)xq],
X(m)=R(m)(EQ(m))1
X(m)xq=R(m)xq(EQ(m))1+R(m)[(EQ(m))1Q(m)xq(EQ(m))1].
Kxq=p=1NKypypxq.
τa=Cext,a(mr,mi,rm/λ,S)NaHa,
[JkTWJk+λkdiag(JkTWJk)]Δxk=JkTW(ykyobs),
χ2(xk+1)=12(yk+1yobs)TW(yk+1yobs),
σj=[(JTWJ)1/2]jj,
yobs=yobs+ασs,y,
p(α)=12πexp(12α2).
IS=n=1NIS(n),
IS(n)=ω0(n)4μ0μe+μ0{exp[τn(1μe+1μ0)]exp[τn+1(1μe+1μ0)]}Pr(μe,ϕe;μ0,ϕ0,n)F0,
IS(n)τn=ω0(n)41μeexp[τn(1μe+1μ0)]Pr(n)F0,
IS(n)τn+1=ω0(n)41μeexp[τn+1(1μe+1μ0)]Pr(n)F0,
IS(n)ω0(n)=IS(n)ω0(n),
IS(n)Pr(n)=[Pr(n)]1IS(n).
IS(N)=exp[τN(1μe+1μ0)]ρMμ0F0,
IS(N)τN=(1μe+1μ0)IS(N).
(Δτn)x=(Cext,axNa(n)+Cext,aNa(n)x)ΔHn.
τnx=i=1n1(Δτi)x
ω0(n)=Δτn,R+Csca,aNa(n)ΔHnΔτn,R+Cext,aNa(n)ΔHn,
Pr(n)=Δτn,RPr,R+Csca,aNa(n)ΔHnPr,aΔτn,R+Csca,aNa(n)ΔHn,
ω0(n)x=ΔHnNa(n)ΔτnCsca,axΔτn,R+Csca,aNa(n)ΔHn(Δτn)2(Δτn)x,
Pr(n)x=Na(n)ΔHnCsca,aΔτn,R+Csca,aNa(n)ΔHnPr,ax+Δτn,RΔHnNa(n)(Pr,aPr,R)(Δτn,R+Csca,aNa(n)ΔHn)2Csca,ax,
ω0(n)Na(n)=Cabs,aΔHnΔτn,R(Δτn,R+Cext,aNa(n)ΔHn)2,
Pr(n)Na(n)=Δτn,RCsca,aΔHn(Pr,aPr,R)(Δτn,R+Csca,aNa(n)ΔHn)2.
Π(n,i),0,r(m)=wiexp(τnμ0)μ0μiμi+μ0ω0(n)Pr(m)(μi,μ0;n)2F0{1exp[Δτn(1μ0+1μi)]}{μiΔτn[1exp(Δτnμi)]}1.
Π(n,i),0,r(m)(Δτn)=wiexp(τnμ0)μ0μiμi+μ0ω0(n)Pr(m)(n)2F0(μ0+μiμ0μiexp[Δτn(1μ0+1μi)]{μiΔτn[1exp(Δτnμi)]}1+{1exp[Δτn(1μ0+1μi)]}[μiexp(Δτnμi)(μi+Δτn)]{μi[1exp(Δτnμi)]}2).
Π(n,i),0,t(m)=wiexp(τnμ0)μ0μiμiμ0ω0(n)Pt(m)(μi,μ0;n)2F0{exp(Δτnμi)exp(Δτnμ0)}{μiΔτn[1exp(Δτnμi)]}1
Π(n,i),0,t(m)(Δτn)=wiexp(τnμ0)μ0μiμiμ0ω0(n)Pt(m)(n)2F0([1μiexp(Δτnμi)+1μ0exp(Δτnμ0)]{μiΔτn[1exp(Δτnμi)]}1+[exp(Δτnμi)exp(Δτnμ0)][μiexp(Δτnμi)(μi+Δτn)]{μi[1exp(Δτnμi)]}2).
Π(n,i),0,t(m)=wiexp(τnμ0)ω0(n)Pt(m)(μi,μ0;n)2F0{Δτnexp(Δτnμi)}{μiΔτn[1exp(Δτnμi)]}1,
Π(n,i),0,t(m)(Δτn)=wiexp(τnμ0)ω0(n)Pt(m)(n)2F0([exp(Δτnμi)Δτnμiexp(Δτnμi)]{μiΔτn[1exp(Δτnμi)]}1+[Δτnexp(Δτnμi)][μiexp(Δτnμi)(μi+Δτn)]{μi[1exp(Δτnμi)]}2).
Π(n,i),0,r/t(m)τn=1μ0Π(n,i),0,r/t(m)
Π(n,i),0,r/t(m)ω0(n)=Π(n,i),0,r/t(m)ω0(n)
Π(n,i),0,r/t(m)Pr/t(m)(n)=[Pr/t(m)(n)]1Π(n,i),0,r/t(m)
Q(n,j),(n,i)(m)=wjω0(n)P(m)(μj,μi;n)2W(n,i),(n,i),
W(n,i),(n,i)=μiΔτn[1exp(Δτnμi)]exp(τnτn+1μi)[1exp(Δτnμi)],
W(n,i),(n,i)=μiΔτn[1exp(Δτnμi)]exp(τnτn+1μi)[1exp(Δτnμi)],
W(n,i),(n,i)=1μiΔτn[1exp(Δτnμi)].
Q(n,j),(n,i)(Δτn)=wjω0(n)P(m)(n)2{W(n,i),(n,i)Δτn+1Δτnexp(Δτnμi)exp(τnτn+1μi)[1exp(Δτnμi)]},
Q(n,j),(n,i)(Δτn)=wjω0(n)P(m)(n)2{1Δτn[1exp(Δτnμi)]exp(τnτn+1μi)exp(Δτnμi)},
Q(n,j),(n,i)τn=Q(n,j),(n,i)μi,
Q(n,j),(n,i)τn+1=Q(n,j),(n,i)μi,
Q(n,j),(n,i)(Δτn)=wjω0(n)P(m)(n)2{W(n,i),(n,i)Δτn+1Δτnexp(Δτnμi)exp(τnτn+1μi)[1exp(Δτnμi)]},
Q(n,j),(n,i)(Δτn)=wjω0(n)P(m)(n)2{1Δτn[1exp(Δτnμi)]exp(τnτn+1μi)exp(Δτnμi)},
Q(n,j),(n,i)τn=Q(n,j),(n,i)μi,
Q(n,j),(n,i)τn+1=Q(n,j),(n,i)μi,
Q(n,j),(n,i)(Δτn)=wjω0(n)P(m)(n)2{μiΔτn2[1exp(Δτnμi)]1Δτnexp(Δτnμi)},
Q(n,j),(n,i)[ω0(n)]=Q(n,j),(n,i)ω0(n),
Q(n,j),(n,i)P(m)(n)=[P(m)(n)]1Q(n,j),(n,i).
Re,(n,j)(m)=n=nNR(n,e),(n,j)(m),
R(n,e),(n,j)(m)=121Δτnμjμe+μjω0(n)Pr(m)(μe,μj;n)2{[exp(τn+1τnμjτnμe)exp(τnτnμjτnμe)][exp(τn+1τn+1μjτn+1μe)exp(τnτn+1μjτn+1μe)]},
R(n,e),(n,j)(m)(Δτn)=R(n,e),(n,j)(m)Δτn,
R(n,e),(n,j)(m)τn=121Δτnω0(n)Pr(m)(n)21μe[exp(τn+1τnμjτnμe)exp(τnτnμjτnμe)],
R(n,e),(n,j)(m)τn+1=121Δτnω0(n)Pr(m)(n)21μe[exp(τn+1τn+1μjτn+1μe)exp(τnτn+1μjτn+1μe)],
R(n,e),(n,j)(m)τn=121Δτn1μe+μjω0(n)Pr(m)(n)2[exp(τnτnμjτnμe)exp(τnτn+1μjτn+1μe)],
R(n,e),(n,j)(m)τn+1=121Δτn1μe+μjω0(n)Pr(m)(n)2[exp(τn+1τnμjτnμe)exp(τn+1τn+1μjτn+1μe)],
R(n,e),(n,j)(m)=121Δτn1μj+μeω0(n)Pr(m)(μe,μj;n)2{μe[exp(τnμe)exp(τn+1μe)]μjexp(τn+1μe)[1exp(Δτnμj)]},
R(n,e),(n,j)(m)(Δτn)=R(n,e),(n,j)(m)Δτn121Δτn1μj+μeω0(n)Pr(m)(n)2exp(τn+1μe)exp(Δτnμj),
R(n,e),(n,j)(m)τn=121Δτn1μj+μeω0(n)Pr(m)(n)2exp(τnμe),
R(n,e),(n,j)(m)τn+1=121Δτnμjμj+μeω0(n)Pr(m)(n)2{1μjexp(τn+1μe)+1μeexp(τn+1μe)[1exp(Δτnμj)]}.
Re,(n,j)(m)=n=1nR(n,e),(n,j)(m),
R(n,e),(n,j)(m)=121Δτnμjμeμjω0(n)Pt(m),*(μe,μj;n)2{[exp(τn+1τnμjτn+1μe)exp(τn+1τn+1μjτn+1μe)][exp(τnτnμjτnμe)exp(τnτn+1μjτnμe)]},
R(n,e),(n,j)(m)Δτn=R(n,e),(n,j)(m)Δτn,
R(n,e),(n,j)(m)τn=121Δτn1μeω0(n)Pt(m),*(n)2[exp(τnτnμjτnμe)exp(τnτn+1μjτnμe)],
R(n,e),(n,j)(m)τn+1=121Δτn1μeω0(n)Pt(m),*(n)2[exp(τn+1τnμjτn+1μe)exp(τn+1τn+1μjτn+1μe)],
R(n,e),(n,j)(m)τn=121Δτn1μeμjω0(n)Pt(m),*(n)2[exp(τn+1τnμjτn+1μe)exp(τnτnμjτnμe)],
R(n,e),(n,j)(m)τn+1=121Δτn1μeμjω0(n)Pt(m),*(n)2[exp(τn+1τn+1μjτn+1μe)exp(τnτn+1μjτnμe)],
R(n,e),(n,j)(m)=121Δτn1μjω0(n)Pt(m),*(μe,μj;n)2[exp(τnμj)exp(τn+1μj)],
R(n,e),(n,j)(m)(Δτn)=R(n,e),(n,j)(m)Δτn,
R(n,e),(n,j)(m)(Δτn)=R(n,e),(n,j)(m)Δτn,
R(n,e),(n,j)(m)τn=121Δτn1μjω0(n)Pt(m),*(n)2Δτnμjexp(τnμj),
R(n,e),(n,j)(m)τn+1=121Δτn1μjω0(n)Pt(m),*(n)2Δτnμjexp(τn+1μj),
R(n,e),(n,j)(m)τn=R(n,e),(n,j)(m)τn+1=0,
R(n,e),(n,j)(m)=121Δτn1μjμeω0(n)Pt(m),*(μe,μj;n)2{μjexp(τnμe)[1exp(Δτnμj)]μe[exp(τnμe)exp(τn+1μe)]},
R(n,e),(n,j)(m)(Δτn)=R(n,e),(n,j)(m)Δτn+121Δτn1μjμeω0(n)Pt(m),*(n)2exp(τnμe)exp(Δτnμj),
R(n,e),(n,j)(m)τn=121Δτn1μjμeω0(n)Pt(m),*(n)2exp(τnμe){μjμe[1exp(Δτnμj)]+1},
R(n,e),(n,j)(m)τn+1=121Δτn1μjμeω0(n)Pt(m),*(n)2exp(τn+1μe),
R(n,e),(n,j)(m)=121Δτnω0(n)Pt(m),*(μe,μj;n)2exp(τnμj)[1exp(Δτnμj)Δτnμjexp(Δτnμj)],
R(n,e),(n,j)(m)(Δτn)=R(n,e),(n,j)(m)Δτn+121μj2ω0(n)Pt(m),*(n)2exp(τnμj)exp(Δτnμj),
R(n,e),(n,j)(m)τn=1μjR(n,e),(n,j)(m).
R(n,e),(n,j)(m)ω0(n)=R(n,e),(n,j)(m)ω0(n),
R(n,e),(n,j)(m)P(m)(n)=[P(m)(n)]1R(n,e),(n,j)(m).
ρM(μ,μ0,φφ0)=L(πi2)ρ(μ,φ;μ0,φ0)L(i1),
ρM(μ,μ0,ϕϕ0)=ρM,c(0)(μ,μ0)+2m=1[ρM,c(m)(μ,μ0)cosm(ϕϕ0)+ρM,s(m)(μ,μ0)sinm(ϕϕ0)].
Π(N,i),0,c/s(m)=2wiμ0μiexp(τNμ0)ρM,c/s(m)(μi,μ0)F0,
Π(N,i),0,c/s(m)τN=1μ0Π(N,i),0,c/s(m),
Π(N,i),0,c/s(m)ρM,c/s(m)=[ρM,c/s(m)]1Π(N,i),0,c/s(m).
Q(n,j),(n,i)(m)=wjω0(n)P(m)(μj,μi;n)2exp(τNτn+1μi)[1exp(Δτnμi)],
Q(n,j),(n,i)(m)τN=1μiQ(n,j),(n,i)(m),
Q(n,j),(n,i)(m)τn+1=1μiQ(n,j),(n,i)(m),
Q(n,j),(n,i)(m)(Δτn)=wjω0(n)P(m)(n)2exp(τNτn+1μi)1μiexp(Δτnμi),
Q(n,j),(n,i)[ω0(n)]=Q(n,j),(n,i)ω0(n),
Q(n,j),(n,i)P(m)(n)=[P(m)(n)]1Q(n,j),(n,i).
Q(N,j),(n,i)(m)=2wjμjρM,c/s(m)(μj,μi)μiΔτnexp(τNτn+1μi)[1exp(Δτnμi)],
Q(N,j),(n,i)(m)(Δτn)=Q(N,j),(n,i)(m)Δτn+2wjμjρM,c/s(m)(μj,μi)1Δτnexp(τNτn+1μi)exp(Δτnμi),
Q(N,j),(n,i)(m)τN=1μiQ(N,j),(n,i)(m),
Q(N,j),(n,i)(m)τn+1=1μiQ(N,j),(n,i)(m),
Q(N,j),(n,i)ρM,c/s(m)=[ρM,c/s(m)]1Q(N,j),(n,i).
Re,(n,j)(m)={n=nNR(n,e),(n,j)(m),nN0,n=N
R(N,e),(n,j)(m)=1ΔτnμjρM,c/s(m)(μe,μj)exp(τNμe)[exp(τNτn+1μj)exp(τNτnμj)],
R(N,e),(n,j)(m)(Δτn)=RΔτn,
R(N,e),(n,j)(m)τN=(1μe+1μj)R(N,e),(n,j)(m),
R(N,e),(n,j)(m)τn=1ΔτnρM,c/s(m)exp(τNμe)exp(τNτnμj),
R(N,e),(n,j)(m)τn+1=1ΔτnρM,c/s(m)exp(τNμe)exp(τNτn+1μj),
R(N,e),(n,j)(m)ρM,c/s(m)=[ρM,c/s(m)]1R(N,e),(n,j)(m).
Re,(n,j)(m)={n=1nR(n,e),(n,j)(m),nNn=1N1R(n,e),(n,j)(m),n=N,
R(n,e),(N,j)(m)=121μeμjω0(n)P(m)(μe,μj;n)2[exp(τn+1τNμjτn+1μe)exp(τnτNμjτnμe)],
R(n,e),(N,j)(m)τn=121μjμeω0(n)P(m)(n)2[exp(τnτNμjτnμe)],
R(n,e),(N,j)(m)τn+1=121μjμeω0(n)P(m)(n)2[exp(τn+1τNμjτn+1μe)],
R(n,e),(N,j)(m)τN=1μjR(n,e),(N,j)(m),
R(n,e),(N,j)(m)=121μjμeω0(n)P(m)(μe,μj;n)2exp(τNμj)(τn+1τn),
R(n,e),(N,j)(m)τn=121μjμeω0(n)P(m)(n)2exp(τNμj),
R(n,e),(N,j)(m)τn+1=R(n,e),(N,j)(m)τn,
R(n,e),(N,j)(m)τN=1μjR(n,e),(N,j)(m).
R(n,e),(n,j)(m)ω0(n)=R(n,e),(n,j)(m)ω0(n),
R(n,e),(n,j)(m)P(m)(n)=[P(m)(n)]1R(n,e),(n,j)(m).
ρ=ρd+ρp,
ρd=a[μμ0(μ+μ0)]k1exp(bcosΘ)[1000000000000000],
ρp=ξπp(β)sin(Θ/2)2μμ0(μ+μ0)[F11F1200F12F110000F33F3400F34F33]fsh(μ,μ0),
F11=12(rprp*+rsrs*),
F12=12(rprp*rsrs*),
F33=12(rprs*+rp*rs),
F34=i2(rp*rsrprs*),
rp=mScosγcosγmScosγ+cosγ,
rs=cosγmScosγcosγ+mScosγ,
p(β)=12πσ2cos3βexp(tan2β2σ2),
fsh(μ,μ0)=[1+Λ(μ)+Λ(μ0)]1,
Λ(μ)=12{σμ2(1μ2)πexp[μ22σ2(1μ2)]erfc[μσ12(1μ2)]}.
ρa=ρda,
ρb=cosΘρd,
ρk=ln[μμ0(μ+μ0)]ρd,
ρξ=ρpξ.
ρijmS=ρp,ijmS=ξπp(β)sin(Θ/2)2μμ0(μ+μ0)FijmSfsh(μ,μ0).
F11mS=12(rpmSrp*+rp*mSrp+rsmSrs*+rs*mSrs),
rpmS=2[mS3cosγmS2sin2γ2mScosγmS2sin2γ](mS2cosγ+mS2sin2γ)2,
rsmS=2mScosγmS2sin2γ(cosγ+mS2sin2γ)2.
ρijσ=ρp,ijσ=ρp,ijp(β)p(β)σ+ρp,ijfshfshσ,
p(β)σ=1π(σcosβ)3exp(tan2β2σ2)(1tan2β2σ2),
fshσ=[Λ(μ)σ+Λ(μ0)σ][1+Λ(μ)+Λ(μ0)]2,
Λ(μ)σ=12μ[2(1μ2)π]1/2exp[μ22σ2(1μ2)].
ρp,ijv=ρp,ijσσv.

Metrics