Abstract

Two new sets of analytical equations are derived with which the albedo of single scattering and the coefficients of a Legendre polynomial expansion of the scattering phase function can be determined for a source-free, homogeneous plane-parallel medium uniformly illuminated over the surfaces. The equations, essentially linear in the unknowns, require measurements of the radiance in the interior of the medium, but no iterative forward-problem calculations are needed. Sets of equations for both unpolarized and polarized radiation applications are given, as well as a side-by-side comparison with previously known sets of analytic inversion equations. Applications of the equations are suggested.

© 2004 Optical Society of America

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  1. H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).
  2. S. Stephany, H. F. de Campos Velho, F. M. Ramos, C. D. Mobley, “Identification of inherent optical properties and bioluminescence source term in a hydrologic optics problem,” J. Quant. Spectrosc. Radiat. Transf. 67, 113–123 (2000).
    [CrossRef]
  3. E. S. Chalhoub, H. F. Campos Velho, “Estimation of the optical properties of seawater from measurements of exit radiance,” J. Quant. Spectrosc. Radiat. Transf. 72, 551–565 (2002).
    [CrossRef]
  4. K. M. Case, “Inverse problem in transport theory,” Phys. Fluids 16, 1607–1611 (1973).
    [CrossRef]
  5. N. J. McCormick, I. Kuščer, “On the inverse problem in radiative transfer,” J. Math. Phys. 15, 926–927 (1974).
    [CrossRef]
  6. C. E. Siewert, “On a possible experiment to evaluate the validity of the one-speed or constant cross section model of the neutron-transport equation,” J. Math. Phys. 19, 1587–1588 (1978).
    [CrossRef]
  7. C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transf. 22, 441–446 (1979).
    [CrossRef]
  8. N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
    [CrossRef]
  9. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 149–154.
  10. R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
    [CrossRef]
  11. N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
    [CrossRef]
  12. R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transf. 28, 169–184 (1982).
    [CrossRef]
  13. J. C. Oelund, N. J. McCormick, “Sensitivity of multiple-scattering inverse transport methods to measurement errors,” J. Opt. Soc. Am. A 2, 1972–1978 (1985).
    [CrossRef]
  14. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  15. C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey–Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transf. 72, 827–835 (2002).
    [CrossRef]
  16. A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” Appl. Opt. 42, 931–938 (2003).
    [CrossRef] [PubMed]
  17. N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
    [CrossRef]
  18. T. Viik, N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235–241 (2003).
    [CrossRef]
  19. K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
    [CrossRef]
  20. M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
    [CrossRef]
  21. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91.
  22. E. W. Larsen, “Solution of the inverse problem in multigroup transport theory,” J. Math. Phys. 22, 158–160 (1981).
    [CrossRef]
  23. G. B. Rybicki, “Integrals of the transfer equation. I. Quadratic integrals for monochromatic, isotropic scattering,” Astrophys. J. 213, 165–176 (1977).
    [CrossRef]
  24. N. J. McCormick, “Methods for solving inverse problems for radiation transport—an update,” Transp. Theory Stat. Phys. 15, 759–772 (1986).
    [CrossRef]
  25. C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
    [CrossRef]
  26. C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
  27. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  28. J. W. Hovenier, 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).
  29. The definition of Uin Ref. 17should have been stated in this way.
  30. C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523–528 (1983).
    [CrossRef]
  31. K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fish-eye’ camera radiance distribution system,” J. Atmos. Ocean. Phys. 6, 652–662 (1989).
  32. E. Aas, N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
    [CrossRef]
  33. A. H. Hakim, B. D. Piening, N. J. McCormick, “Near-asymptotic angle dependence of ocean optical radiance,” manuscript available from N. J. McCormick, mccor@u.washington.edu.
  34. C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 437–439.
  35. C. E. Siewert, “Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection,” J. Quant. Spectrosc. Radiat. Transf. 72, 299–313 (2002).
    [CrossRef]
  36. N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
    [CrossRef]
  37. N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1995).
    [CrossRef]

2003 (2)

T. Viik, N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235–241 (2003).
[CrossRef]

A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” Appl. Opt. 42, 931–938 (2003).
[CrossRef] [PubMed]

2002 (4)

C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey–Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transf. 72, 827–835 (2002).
[CrossRef]

E. S. Chalhoub, H. F. Campos Velho, “Estimation of the optical properties of seawater from measurements of exit radiance,” J. Quant. Spectrosc. Radiat. Transf. 72, 551–565 (2002).
[CrossRef]

C. E. Siewert, “Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection,” J. Quant. Spectrosc. Radiat. Transf. 72, 299–313 (2002).
[CrossRef]

H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).

2000 (1)

S. Stephany, H. F. de Campos Velho, F. M. Ramos, C. D. Mobley, “Identification of inherent optical properties and bioluminescence source term in a hydrologic optics problem,” J. Quant. Spectrosc. Radiat. Transf. 67, 113–123 (2000).
[CrossRef]

1999 (1)

E. Aas, N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

1995 (2)

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1995).
[CrossRef]

1990 (1)

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

1989 (2)

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fish-eye’ camera radiance distribution system,” J. Atmos. Ocean. Phys. 6, 652–662 (1989).

1986 (1)

N. J. McCormick, “Methods for solving inverse problems for radiation transport—an update,” Transp. Theory Stat. Phys. 15, 759–772 (1986).
[CrossRef]

1985 (1)

1983 (3)

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

J. W. Hovenier, 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).

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523–528 (1983).
[CrossRef]

1982 (2)

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

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transf. 28, 169–184 (1982).
[CrossRef]

1981 (4)

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

E. W. Larsen, “Solution of the inverse problem in multigroup transport theory,” J. Math. Phys. 22, 158–160 (1981).
[CrossRef]

1979 (2)

C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transf. 22, 441–446 (1979).
[CrossRef]

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

1978 (1)

C. E. Siewert, “On a possible experiment to evaluate the validity of the one-speed or constant cross section model of the neutron-transport equation,” J. Math. Phys. 19, 1587–1588 (1978).
[CrossRef]

1977 (1)

G. B. Rybicki, “Integrals of the transfer equation. I. Quadratic integrals for monochromatic, isotropic scattering,” Astrophys. J. 213, 165–176 (1977).
[CrossRef]

1974 (2)

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

N. J. McCormick, I. Kuščer, “On the inverse problem in radiative transfer,” J. Math. Phys. 15, 926–927 (1974).
[CrossRef]

1973 (1)

K. M. Case, “Inverse problem in transport theory,” Phys. Fluids 16, 1607–1611 (1973).
[CrossRef]

1941 (1)

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aas, E.

E. Aas, N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

Campos Velho, H. F.

E. S. Chalhoub, H. F. Campos Velho, “Estimation of the optical properties of seawater from measurements of exit radiance,” J. Quant. Spectrosc. Radiat. Transf. 72, 551–565 (2002).
[CrossRef]

Case, K. M.

K. M. Case, “Inverse problem in transport theory,” Phys. Fluids 16, 1607–1611 (1973).
[CrossRef]

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91.

Chalhoub, E. S.

E. S. Chalhoub, H. F. Campos Velho, “Estimation of the optical properties of seawater from measurements of exit radiance,” J. Quant. Spectrosc. Radiat. Transf. 72, 551–565 (2002).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 149–154.

de Campos Velho, H. F.

S. Stephany, H. F. de Campos Velho, F. M. Ramos, C. D. Mobley, “Identification of inherent optical properties and bioluminescence source term in a hydrologic optics problem,” J. Quant. Spectrosc. Radiat. Transf. 67, 113–123 (2000).
[CrossRef]

Gordon, H. R.

H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hakim, A. H.

A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” Appl. Opt. 42, 931–938 (2003).
[CrossRef] [PubMed]

A. H. Hakim, B. D. Piening, N. J. McCormick, “Near-asymptotic angle dependence of ocean optical radiance,” manuscript available from N. J. McCormick, mccor@u.washington.edu.

Hansen, J. E.

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

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hobbs, P. V.

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

Højerslev, N. K.

E. Aas, N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier, 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).

King, M. D.

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

Kušcer, I.

N. J. McCormick, I. Kuščer, “On the inverse problem in radiative transfer,” J. Math. Phys. 15, 926–927 (1974).
[CrossRef]

Larsen, E. W.

E. W. Larsen, “Solution of the inverse problem in multigroup transport theory,” J. Math. Phys. 22, 158–160 (1981).
[CrossRef]

McCormick, N. J.

T. Viik, N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235–241 (2003).
[CrossRef]

A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” Appl. Opt. 42, 931–938 (2003).
[CrossRef] [PubMed]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1995).
[CrossRef]

N. J. McCormick, “Methods for solving inverse problems for radiation transport—an update,” Transp. Theory Stat. Phys. 15, 759–772 (1986).
[CrossRef]

J. C. Oelund, N. J. McCormick, “Sensitivity of multiple-scattering inverse transport methods to measurement errors,” J. Opt. Soc. Am. A 2, 1972–1978 (1985).
[CrossRef]

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transf. 28, 169–184 (1982).
[CrossRef]

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

N. J. McCormick, I. Kuščer, “On the inverse problem in radiative transfer,” J. Math. Phys. 15, 926–927 (1974).
[CrossRef]

A. H. Hakim, B. D. Piening, N. J. McCormick, “Near-asymptotic angle dependence of ocean optical radiance,” manuscript available from N. J. McCormick, mccor@u.washington.edu.

Mobley, C. D.

S. Stephany, H. F. de Campos Velho, F. M. Ramos, C. D. Mobley, “Identification of inherent optical properties and bioluminescence source term in a hydrologic optics problem,” J. Quant. Spectrosc. Radiat. Transf. 67, 113–123 (2000).
[CrossRef]

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 437–439.

Oelund, J. C.

Piening, B. D.

A. H. Hakim, B. D. Piening, N. J. McCormick, “Near-asymptotic angle dependence of ocean optical radiance,” manuscript available from N. J. McCormick, mccor@u.washington.edu.

Radke, L. F.

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

Ramos, F. M.

S. Stephany, H. F. de Campos Velho, F. M. Ramos, C. D. Mobley, “Identification of inherent optical properties and bioluminescence source term in a hydrologic optics problem,” J. Quant. Spectrosc. Radiat. Transf. 67, 113–123 (2000).
[CrossRef]

Rybicki, G. B.

G. B. Rybicki, “Integrals of the transfer equation. I. Quadratic integrals for monochromatic, isotropic scattering,” Astrophys. J. 213, 165–176 (1977).
[CrossRef]

Sanchez, R.

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transf. 28, 169–184 (1982).
[CrossRef]

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

Siewert, C. E.

C. E. Siewert, “Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection,” J. Quant. Spectrosc. Radiat. Transf. 72, 299–313 (2002).
[CrossRef]

C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey–Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transf. 72, 827–835 (2002).
[CrossRef]

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523–528 (1983).
[CrossRef]

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

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transf. 22, 441–446 (1979).
[CrossRef]

C. E. Siewert, “On a possible experiment to evaluate the validity of the one-speed or constant cross section model of the neutron-transport equation,” J. Math. Phys. 19, 1587–1588 (1978).
[CrossRef]

Stephany, S.

S. Stephany, H. F. de Campos Velho, F. M. Ramos, C. D. Mobley, “Identification of inherent optical properties and bioluminescence source term in a hydrologic optics problem,” J. Quant. Spectrosc. Radiat. Transf. 67, 113–123 (2000).
[CrossRef]

Travis, L. D.

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

van der Mee, C. V. M.

J. W. Hovenier, 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).

Viik, T.

T. Viik, N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235–241 (2003).
[CrossRef]

Voss, K. J.

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fish-eye’ camera radiance distribution system,” J. Atmos. Ocean. Phys. 6, 652–662 (1989).

Zibordi, G.

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fish-eye’ camera radiance distribution system,” J. Atmos. Ocean. Phys. 6, 652–662 (1989).

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91.

Ann. Nucl. Energy (1)

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1995).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (2)

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

J. W. Hovenier, 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).

Astrophys. J. (3)

G. B. Rybicki, “Integrals of the transfer equation. I. Quadratic integrals for monochromatic, isotropic scattering,” Astrophys. J. 213, 165–176 (1977).
[CrossRef]

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[CrossRef]

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

J. Atmos. Ocean. Phys. (1)

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic ‘fish-eye’ camera radiance distribution system,” J. Atmos. Ocean. Phys. 6, 652–662 (1989).

J. Atmos. Sci. (1)

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

J. Geophys. Res. (1)

E. Aas, N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

J. Math. Phys. (6)

E. W. Larsen, “Solution of the inverse problem in multigroup transport theory,” J. Math. Phys. 22, 158–160 (1981).
[CrossRef]

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

N. J. McCormick, I. Kuščer, “On the inverse problem in radiative transfer,” J. Math. Phys. 15, 926–927 (1974).
[CrossRef]

C. E. Siewert, “On a possible experiment to evaluate the validity of the one-speed or constant cross section model of the neutron-transport equation,” J. Math. Phys. 19, 1587–1588 (1978).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (9)

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523–528 (1983).
[CrossRef]

C. E. Siewert, “Inverse solutions to radiative-transfer problems based on the binomial or the Henyey–Greenstein scattering law,” J. Quant. Spectrosc. Radiat. Transf. 72, 827–835 (2002).
[CrossRef]

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transf. 28, 169–184 (1982).
[CrossRef]

C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transf. 22, 441–446 (1979).
[CrossRef]

S. Stephany, H. F. de Campos Velho, F. M. Ramos, C. D. Mobley, “Identification of inherent optical properties and bioluminescence source term in a hydrologic optics problem,” J. Quant. Spectrosc. Radiat. Transf. 67, 113–123 (2000).
[CrossRef]

E. S. Chalhoub, H. F. Campos Velho, “Estimation of the optical properties of seawater from measurements of exit radiance,” J. Quant. Spectrosc. Radiat. Transf. 72, 551–565 (2002).
[CrossRef]

C. E. Siewert, “Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection,” J. Quant. Spectrosc. Radiat. Transf. 72, 299–313 (2002).
[CrossRef]

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527–535 (1983).
[CrossRef]

T. Viik, N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235–241 (2003).
[CrossRef]

Limnol. Oceanogr. (2)

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

Oceanologia (1)

H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).

Phys. Fluids (1)

K. M. Case, “Inverse problem in transport theory,” Phys. Fluids 16, 1607–1611 (1973).
[CrossRef]

Space Sci. Rev. (1)

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

Transp. Theory Stat. Phys. (1)

N. J. McCormick, “Methods for solving inverse problems for radiation transport—an update,” Transp. Theory Stat. Phys. 15, 759–772 (1986).
[CrossRef]

Other (5)

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91.

The definition of Uin Ref. 17should have been stated in this way.

A. H. Hakim, B. D. Piening, N. J. McCormick, “Near-asymptotic angle dependence of ocean optical radiance,” manuscript available from N. J. McCormick, mccor@u.washington.edu.

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 437–439.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 149–154.

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Equations (124)

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μ I(τ, μ)τ+I(τ, μ)=ϖ-11p(μμ)I(τ, μ)dμ=ϖ2n=0N(2n+1)fnPn(μ)×-11Pn(μ)I(τ, μ)dμ,
0ττ*,
-μ I(τr-τ, μ)τ+I(τr-τ, μ)=ϖ2n=0N(2n+1)fnPn(μ)×-11Pn(μ)I(τr-τ, μ)dμ,0ττr,
ddτ-11I(τ, μ)I(τr-τ, μ)dμ
=ϖ2ddτn=0N(2n+1)fnIn(τ)In(τr-τ),
In(τ)=-11Pn(μ)I(τ, μ)dμ.
2-11I(τ, μ)I(τr-τ, μ)dμτ1τ2
=ϖn=0N(2n+1)fnIn(τ)In(τr-τ)τ1τ2,
F(τ)|τ1τ2=F(τ2)-F(τ1).
-μ I(τ, -μ)τ+I(τ, -μ)
=ϖ2n=0N(-1)n(2n+1)fnPn(μ)×-11Pn(μ)I(τ, μ)dμ,
0ττ*.
ddτ-11I(τ, μ)I(τ, -μ)dμ
=ϖ2ddτn=0N(-1)n(2n+1)fnIn2(τ).
2-11I(τ, μ)I(τ, -μ)dμτ1τ2
=ϖn=0N(-1)n(2n+1)fnIn2(τ)τ1τ2.
ddτ-11μ2I(τ, μ)I(τr-τ, μ)dμ
=ϖ2n=0N(2n+1)fn[In(τ)I˜n(τr-τ)
-I˜n(τ)In(τr-τ)],
I˜n(τ)=-11μPn(μ)I(τ, μ)dμ.
In(τ)=-11-ϖfndI˜n(τ)dτ.
In(τr-τ)=11-ϖfndI˜n(τr-τ)dτ.
ddτ-11μ2I(τ, μ)I(τr-τ, μ)dμ
=-ϖ2ddτn=0N(2n+1)f˜nI˜n(τ)I˜n(τr-τ),
f˜n=fn/(1-ϖfn),
2-11μ2I(τ, μ)I(τr-τ, μ)dμτ1τ2
=-ϖn=0N(2n+1)f˜nI˜n(τ)I˜n(τr-τ)τ1τ2.
ddτ-11μ2I(τ, μ)I(τ, -μ)dμ
=-ϖn=0N(-1)n(2n+1)fnIn(τ)I˜n(τ).
ddτ-11μ2I(τ, μ)I(τ, -μ)dμ
=ϖ2ddτn=0N(-1)n(2n+1)f˜nI˜n2(τ).
2-11μ2I(τ, μ)I(τ, -μ)dμτ1τ2
=ϖn=0N(-1)n(2n+1)f˜nI˜n2(τ)τ1τ2.
μ I(τ, μ, ϕ)τ+I(τ, μ, ϕ)
=ϖ02π-11p(μμ, ϕϕ)I(τ, μ, ϕ)dμdϕ,
0ττ*,
I(τ, μ, ϕ)=m=0N[Φ1m(ϕ)I1m(τ, μ)+Φ2m(ϕ)I2m(τ, μ)]+I>N(τ, μ, ϕ)
Ijm(τ, μ)=[π(1+δm0)]-102πΦjm(ϕ)I(τ, μ, ϕ)dϕ,
j=1,2,
02πI>N(τ, μ, ϕ)cos m(ϕ-ϕr)dϕ
=02πI>N(τ, μ, ϕ)sin m(ϕ-ϕr)dϕ=0,
m=0toN.
I(0, τ, μ)=S1δ(μ-μ1)δ(ϕ-ϕ1)+S2δ(μ-μ2)δ(ϕ-ϕ2),
0μ1,
μ Ijm(τ, μ)τ+Ijm(τ, μ)=ϖ2n=mN(2n+1)fnκnmPnm(μ)×-11Pnm(μ)Ijm(τ, μ)dμ,
0ττ*,
κnm=(n-m)!(n+m)!.
2-11Ijm(τ, μ)Ikm(τr-τ, μ)dμτ1τ2
=ϖn=mN(2n+1)fnκnmIj,nm(τ)Ik,nm(τr-τ)τ1τ2,
2-11μ2Ijm(τ, μ)Ikm(τr-τ, μ)dμτ1τ2
=-ϖn=mN(2n+1)f˜nκnmI˜j,nm(τ)I˜k,nm(τr-τ)τ1τ2,
Ij,nm(τ)=-11Pnm(μ)Ijm(τ, μ)dμ,
I˜j,nm(τ)=-11μPnm(μ)Ijm(τ, μ)dμ.
2-11Ijm(τ, μ)Ikm(τ, -μ)dμτ1τ2
=ϖn=mN(-1)n-m(2n+1)fnκnmIj,nm(τ)Ik,nm(τ)τ1τ2,
2-11μ2Ijm(τ, μ)Ikm(τ, -μ)dμτ1τ2
=ϖn=mN(-1)n-m(2n+1)f˜nκnmI˜j,nm(τ)I˜k,nm(τ)τ1τ2.
μ I(τ, μ, ϕ)τ+I(τ, μ, ϕ)
=ϖ02π-11p(μμ, ϕϕ)I(τ, μ, ϕ)dμdϕ,
0ττ*,
I=m=0N[Φ1m(ϕ)I1m(τ, μ)+Φ2m(ϕ)I2m(τ, μ)]+I>N(τ, μ, ϕ),
Φ1m(ϕ)=diag{cos m(ϕ-ϕr), cos m(ϕ-ϕr),sin m(ϕ-ϕr), sin m(ϕ-ϕr)},
Φ2m(ϕ)=diag{-sin m(ϕ-ϕr), -sin m(ϕ-ϕr),cos m(ϕ-ϕr), cos m(ϕ-ϕr)}.
Ijm(τ, μ)=[π(1+δm0)]-102πΦjm(ϕ)I(τ, μ, ϕ)dϕ,
j=1,2,
02πI>N(τ, μ, ϕ)cos m(ϕ-ϕr)dϕ
=02πI>N(τ, μ, ϕ)sin m(ϕ-ϕr)dϕ=0,
m=0toN.
μ Ijm(τ, μ)τ+Ijm(τ, μ)
=ϖ2n=mNκnmΠnm(μ)Bn-11Πnm(μ)Ijm(τ, μ)dμ,
0ττ*.
Πnm(μ)=Pnm(μ)0000Rnm(μ)-Tnm(μ)00-Tnm(μ)Rnm(μ)0000Pnm(μ),
Bn=βnγn00γnαn0000ρn-n00nδn,n=0toN,
Πnm(μ)=FΠnm(μ)F,
(hnT)-1FBn=FBnhn-1,
BnT=FBnF,F=diag{1, 1, 1, -1},
hn=U-(2n+1)-1ϖBn
Ij,n(τ)=-(hn)-1dI˜j,n(τ)/dτ,
2-11[Ijm(τ, μ)]TFIkm(τr-τ, μ)dμτ1τ2
=ϖn=mNκnm[Ij,nm(τ)]TFBnIk,nm(τr-τ)τ1τ2,
2-11μ2[Ijm(τ, μ)]TFIkm(τr-τ, μ)dμτ1τ2
=-ϖn=mNκnm[I˜j,nm(τ)]TFB˜nI˜k,nm(τr-τ)τ1τ2,
Ij,nm(τ)=-11Πnm(μ)Ijm(τ, μ)dμ,
I˜j,nm(τ)=-11μΠnm(μ)Ijm(τ, μ)dμ.
B˜n=β˜nγ˜n00γ˜nα˜n0000ρ˜n-˜n00˜nδ˜n,n=0toN,
ϖBn=(2n+1){U-[U+ϖBn/(2n+1)]-1}.
Πnm(-μ)=(-1)n-mEΠnm(μ)E,
(hnT)-1EBn=EBnhn-1,
BnT=EBnE,
E=diag{1, 1, -1, 1}.
2-11[Ijm(τ, μ)]TEIkm(τ, -μ)dμτ1τ2
=ϖn=mN(-1)n-mκnm[Ij,nm(τ)]TEBnIk,nm(τ)τ1τ2,
2-11μ2[Ijm(τ, μ)]TEIkm(τ, -μ)dμτ1τ2
=ϖn=mN(-1)n-mκnm[I˜j,nm(τ)]TEB˜nI˜k,nm(τ)τ1τ2.
2-11L-1l=1LwlI(τl, μ)I(τl, -μ)-I(0, μ)I(0, -μ)dμ=ϖn=0N(-1)n(2n+1)fnL-1l=1LwlIn2(τl)-In2(0),
I(τ, μ, ϕ)I(τ, μ, ϕ=180°) 1+(τ, μ)1-(τ, μ)cos ϕ,
(τ, μ)=I(τ, μ, ϕ=0°)-I(τ, μ, ϕ=180°)I(τ, μ, ϕ=0°)+I(τ, μ, ϕ=180°)
Ijm(τ, μ)=I(τ, μ, ϕ=180°) 2(1+δm0)1+(τ, μ)1-(τ, μ)×1-1-2(τ, μ)(τ, μ)m.
fn/fn-1=(α+1-n)/(α+1+n),n1.
I(τ*, -μ)=ρ[(1-ξ)I(τ*, μ)+2ξ01I(τ*, μ)μdμ],μ>0,
I(τ, μ)=j=1J[A(νj)ϕ(νj, μ)exp(-τ/νj)+A(-νj)ϕ(-νj, μ)exp(τ/νj)]+-11A(ν)ϕ(ν, μ)exp(-τ/ν)dνσA(ν)ϕ(ν, μ)exp(-τ/ν)dν,
ϕ(±νj, μ)=ϖνj2g(±νj, μ)νjμ,
gn(ν)=-11Pn(μ)ϕ(ν, μ)dμ,
(n+1)gn+1(ν)-(2n+1)(1-ϖfn)νgn(ν)+ngn-1(ν)
=0.
-11ϕ(ν, μ)ϕ(ν, μ)μdμ=0,νν,
=N(±νj),ν=ν=±νj,
=N(ν)δ(ν-ν),
-1ν,ν1,
σσA(ν)A(ν)expτ-τrν-τν
×2-11ϕ(ν, μ)ϕ(ν, μ)dμ
-ϖn=0N(2n+1)fngn(ν)gn(ν)dνdν|τ1τ2=0.
2-11ϕ(ν, μ)ϕ(ν, μ)dμ-ϖn=0N(2n+1)fngn(ν)gn(ν)
=0,νν,
=2N(±νj)±νj,ν=ν=νj,
=2N(ν)ν δ(ν-ν),-1ν,ν1.
σσA(ν)A(ν)expτ-τrν-τν
2-11μ2ϕ(ν, μ)ϕ(ν, μ)dμ+ϖn=0N(2n+1)×fn(1-ϖfn)ννgn(ν)gn(ν)dνdντ1τ2=0.
I˜n(τ)=-11μPn(μ)ϕ(ν, μ)dμ=(1-ϖfn)νgn(ν).
2-11μ2ϕ(ν, μ)ϕ(ν, μ)dμ
+ϖn=0N(2n+1)fn(1-ϖfn)ννgn(ν)gn(ν)
=0,νν,
=±2νjN(±νj),ν=ν=νj,
=2νN(ν)δ(ν-ν),-1ν,ν1,

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