Abstract

It is possible to obtain useful maps of surface albedo from remotely sensed images by eliminating effects due to topography and the atmosphere, even when the atmospheric state is not known. A simple phenomenological model of earth radiance that depends on six empirically determined parameters is developed given certain simplifying assumptions. The model incorporates path radiance and illumination from sun and sky and their dependencies on surface altitude and orientation. It takes explicit account of surface shape, represented by a digital terrain model, and is therefore especially suited for use in mountainous terrain. A number of ways of determining the model parameters are discussed, including the use of shadows to obtain path radiance and to estimate local albedo and sky irradiance. The emphasis is on extracting as much information from the image as possible, given a digital terrain model of the imaged area and a minimum of site-specific atmospheric data. The albedo image, introduced as a representation of surface reflectance, provides a useful tool to evaluate the simple imaging model. Criteria for the subjective evaluation of albedo images are established and illustrated for Landsat multispectral data of a mountainous region of Switzerland. The method exposes some of the limitations found in computing reflectance information using only the image-forming equation.

© 1983 Optical Society of America

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References

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  1. S. I. Solomon, J. Cameron, J. Chadwick, A. S. Aggarwal, in Proceedings Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind.1975), p. 4B-1.
  2. R. M. Hoffer and staff, Natural Resource Mapping in Mountainous Terrain by Computer Analysis of ERTS-1 Satellite Data, LARS Info Note 061575, (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975); also available from NTIS as N76-14567.
  3. M. D. Fleming, J. S. Berkebile, R. M. Hoffer, in Proceedings, Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975), p. 1B-54.
  4. F. G. Sadowsky, W. A. Malila, “Investigation of Techniques for Inventorying Forested Regions. Vol. 1: Reflectance Modeling and Empirical Multispectral Analysis of Forest Canopy Components,” Final Report, NASA-CR-151561 (1977); also available from NTIS as N73-14462.
  5. B. N. Holben, C. O. Justice, Photogramm. Eng. Remote Sensing 46, 1191 (1980).
  6. J. N. Howard, J. S. Garing, Trans. Am. Geophys. Union 52, 371 (1971).
    [CrossRef]
  7. E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976).
  8. G. V. Rozenberg, Sov. Phys. Usp. 3, 346 (1960).
    [CrossRef]
  9. A. J. LaRocca, R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” IRIA State-of-the-Art Report, ERIM 107600-10-T (Environmental Research Institute of Michigan, Ann Arbor, 1975); also available from NTIS as AD-A017 459.
  10. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  11. S. Ueno, H. Kagiwada, R. Kalaba, J. Math. Phys. 12, 1279 (1971).
    [CrossRef]
  12. G. I. Marchuk et al., The Monte Carlo Methods in Atmospheric Optics (Springer, Berlin, 1980).
  13. F. E. Nicodemus et al., “Geometrical Considerations and Nomenclature for Reflectance,” Nat. Bur. Stand. U.S. Monogr. 160 (1977).
  14. P. Moon, Illum. Eng. 37, 707 (1942).
  15. B. K. P. Horn, R. W. Sjoberg, Appl. Opt. 18, 1770 (1979).
    [CrossRef] [PubMed]
  16. J. A. Smith, T. L. Lin, K. J. Ranson, Photogramm. Eng. Remote Sensing 46, 1183 (1980).
  17. S. L. Valley, Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).
  18. K. Coulson, J. Atmos. Sci. 25, 759 (1968). Sky irradiance data for μ0 = cosθ0 = 0.562 were interpolated from the published tables for a surface albedo of A (or ρ¯) = 0.25. Es(τ) = Fs + Fd of the published data, where the value for Fd for A = 0.25 was interpolated from the values for A = 0.20 and A = 0.30.
    [CrossRef]
  19. K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. California Press, Berkeley, 1960). Sky irradiance data were obtained by numerically integrating the tabulated values of downward radiance for μ0 = cosθ0 = 0.6 and A (or ρ¯) = 0.25. Path radiance data were obtained from the tabulated values of upward radiance at μ0 = cosθ0 = 0.6 and A = 0.25 by first subtracting the reflected target radiance. The reflected target radiance is Lt = (TdE0μ0 + Es)A/π.
  20. D. Deirmendjian, Z. Sekera, Tellus 6, 382 (1954). Sky irradiance data for μ0 = cosθ0 = 0.562 at A (or ρ¯) = 0.25 were interpolated from the published values of μ0 = 1.0,0.6,0.1 at A = 0.25.
    [CrossRef]
  21. J. Otterman, Appl. Opt. 17, 3431 (1978). Sky irradiance values for μ0 = 0.562 and ρ¯ = 0.25 were computed directly from the published formula for a Rayleigh atmosphere. Otterman fails to give an explicit formula for path radiance (which would be inappropriate for his approach) but does give one for radiant exitance. The plotted path radiance value is actually the radiant exitance at μ0 = 0.562 and ρ¯ = 0.25 divided by 2π.
    [CrossRef] [PubMed]
  22. R. E. Turner, M. M. Spencer, in Proceedings, Eighth International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1972), pp. 895–934.
  23. Sky irradiance and path radiance data were obtained directly from the published formula for a Rayleigh atmosphere with ρ¯ = 0.25 and μ0 = 0.562.
  24. B. K. P. Horn, B. L. Bachman, Commun. Assoc. Comput. Mach. 21, 914 (1978).
  25. R. J. Woodham, “Looking in the Shadows,” Working Paper 169 (Artificial Intelligence Laboratory, MIT, 1978).
  26. B. K. P. Horn, Proc. IEEE 69, 14 (1981).
    [CrossRef]
  27. B. K. P. Horn, R. J. Woodham, Comput. Graphics Image Process. 10, 69 (1979).
    [CrossRef]
  28. R. J. Woodham, “Two Simple Algorithms for Displaying Orthographic Projections of Surfaces,” Working Paper 126 (Artificial Intelligence Laboratory, MIT, 1976).
  29. R. J. Woodham, Proc. Soc. Photo-Opt. Instrum. Eng. 238, 361 (1980).
  30. F. J. Ahern, D. G. Goodenough, S. C. Jain, V. R. Rao, in Proceedings, Eleventh International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1977), pp. 731–755.
  31. K. R. Piech, J. R. Schott, Proc. Soc. Photo-Opt. Instrum. Eng. 51, 84 (1974).
  32. M. P. Thekaekara, J. Environ. Sci. 13, 6 (1970).
  33. J. Otterman, R. S. Fraser, Remote Sensing Environ.5, 247 (1976).
    [CrossRef]
  34. R. E. Turner, “Radiative Transfer in Real Atmospheres,” Technical Report ERIM 190100-24-T (Environmental Research Institute of Michigan, Ann Arbor, 1974); also available from NTIS as N74-31873.
  35. R. E. Turner, “Atmospheric Effects in Multispectral Remote Sensor Data,” Technical Report ERIM 109600-15-F (Environmental Research Institute of Michigan, Ann Arbor, 1975); also available from NTIS as N75-26474.
  36. W. S. Hering, “An Operational Technique for Estimating Visible Spectrum Contrast Transmittance,” Scientific Report 16 (Visibility Laboratory, Scripps Institution of Oceanography, La Jolla, Calif., 1981).
  37. R. W. Johnson, J. I. Gordon, “Airborne Measurements of Atmospheric Volume Scattering Coefficients in Northern Europe, Summer 1978,” Scientific Report 13 (Visibility Laboratory, Scripps Institution of Oceanography, La Jolla, Calif., 1980).
  38. B. W. Fitch, J. Atmos. Sci. 38, 2717 (1981).
    [CrossRef]
  39. B. Brennan, W. R. Bandeen, Appl. Opt. 9, 405 (1970).
    [CrossRef] [PubMed]

1981 (2)

B. K. P. Horn, Proc. IEEE 69, 14 (1981).
[CrossRef]

B. W. Fitch, J. Atmos. Sci. 38, 2717 (1981).
[CrossRef]

1980 (3)

R. J. Woodham, Proc. Soc. Photo-Opt. Instrum. Eng. 238, 361 (1980).

B. N. Holben, C. O. Justice, Photogramm. Eng. Remote Sensing 46, 1191 (1980).

J. A. Smith, T. L. Lin, K. J. Ranson, Photogramm. Eng. Remote Sensing 46, 1183 (1980).

1979 (2)

B. K. P. Horn, R. W. Sjoberg, Appl. Opt. 18, 1770 (1979).
[CrossRef] [PubMed]

B. K. P. Horn, R. J. Woodham, Comput. Graphics Image Process. 10, 69 (1979).
[CrossRef]

1978 (2)

1977 (1)

F. E. Nicodemus et al., “Geometrical Considerations and Nomenclature for Reflectance,” Nat. Bur. Stand. U.S. Monogr. 160 (1977).

1974 (1)

K. R. Piech, J. R. Schott, Proc. Soc. Photo-Opt. Instrum. Eng. 51, 84 (1974).

1971 (2)

J. N. Howard, J. S. Garing, Trans. Am. Geophys. Union 52, 371 (1971).
[CrossRef]

S. Ueno, H. Kagiwada, R. Kalaba, J. Math. Phys. 12, 1279 (1971).
[CrossRef]

1970 (2)

M. P. Thekaekara, J. Environ. Sci. 13, 6 (1970).

B. Brennan, W. R. Bandeen, Appl. Opt. 9, 405 (1970).
[CrossRef] [PubMed]

1968 (1)

K. Coulson, J. Atmos. Sci. 25, 759 (1968). Sky irradiance data for μ0 = cosθ0 = 0.562 were interpolated from the published tables for a surface albedo of A (or ρ¯) = 0.25. Es(τ) = Fs + Fd of the published data, where the value for Fd for A = 0.25 was interpolated from the values for A = 0.20 and A = 0.30.
[CrossRef]

1960 (1)

G. V. Rozenberg, Sov. Phys. Usp. 3, 346 (1960).
[CrossRef]

1954 (1)

D. Deirmendjian, Z. Sekera, Tellus 6, 382 (1954). Sky irradiance data for μ0 = cosθ0 = 0.562 at A (or ρ¯) = 0.25 were interpolated from the published values of μ0 = 1.0,0.6,0.1 at A = 0.25.
[CrossRef]

1942 (1)

P. Moon, Illum. Eng. 37, 707 (1942).

Aggarwal, A. S.

S. I. Solomon, J. Cameron, J. Chadwick, A. S. Aggarwal, in Proceedings Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind.1975), p. 4B-1.

Ahern, F. J.

F. J. Ahern, D. G. Goodenough, S. C. Jain, V. R. Rao, in Proceedings, Eleventh International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1977), pp. 731–755.

Bachman, B. L.

B. K. P. Horn, B. L. Bachman, Commun. Assoc. Comput. Mach. 21, 914 (1978).

Bandeen, W. R.

Berkebile, J. S.

M. D. Fleming, J. S. Berkebile, R. M. Hoffer, in Proceedings, Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975), p. 1B-54.

Brennan, B.

Cameron, J.

S. I. Solomon, J. Cameron, J. Chadwick, A. S. Aggarwal, in Proceedings Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind.1975), p. 4B-1.

Chadwick, J.

S. I. Solomon, J. Cameron, J. Chadwick, A. S. Aggarwal, in Proceedings Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind.1975), p. 4B-1.

Chandrasekhar, S.

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

Coulson, K.

K. Coulson, J. Atmos. Sci. 25, 759 (1968). Sky irradiance data for μ0 = cosθ0 = 0.562 were interpolated from the published tables for a surface albedo of A (or ρ¯) = 0.25. Es(τ) = Fs + Fd of the published data, where the value for Fd for A = 0.25 was interpolated from the values for A = 0.20 and A = 0.30.
[CrossRef]

Coulson, K. L.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. California Press, Berkeley, 1960). Sky irradiance data were obtained by numerically integrating the tabulated values of downward radiance for μ0 = cosθ0 = 0.6 and A (or ρ¯) = 0.25. Path radiance data were obtained from the tabulated values of upward radiance at μ0 = cosθ0 = 0.6 and A = 0.25 by first subtracting the reflected target radiance. The reflected target radiance is Lt = (TdE0μ0 + Es)A/π.

Dave, J. V.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. California Press, Berkeley, 1960). Sky irradiance data were obtained by numerically integrating the tabulated values of downward radiance for μ0 = cosθ0 = 0.6 and A (or ρ¯) = 0.25. Path radiance data were obtained from the tabulated values of upward radiance at μ0 = cosθ0 = 0.6 and A = 0.25 by first subtracting the reflected target radiance. The reflected target radiance is Lt = (TdE0μ0 + Es)A/π.

Deirmendjian, D.

D. Deirmendjian, Z. Sekera, Tellus 6, 382 (1954). Sky irradiance data for μ0 = cosθ0 = 0.562 at A (or ρ¯) = 0.25 were interpolated from the published values of μ0 = 1.0,0.6,0.1 at A = 0.25.
[CrossRef]

Fitch, B. W.

B. W. Fitch, J. Atmos. Sci. 38, 2717 (1981).
[CrossRef]

Fleming, M. D.

M. D. Fleming, J. S. Berkebile, R. M. Hoffer, in Proceedings, Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975), p. 1B-54.

Fraser, R. S.

J. Otterman, R. S. Fraser, Remote Sensing Environ.5, 247 (1976).
[CrossRef]

Garing, J. S.

J. N. Howard, J. S. Garing, Trans. Am. Geophys. Union 52, 371 (1971).
[CrossRef]

Goodenough, D. G.

F. J. Ahern, D. G. Goodenough, S. C. Jain, V. R. Rao, in Proceedings, Eleventh International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1977), pp. 731–755.

Gordon, J. I.

R. W. Johnson, J. I. Gordon, “Airborne Measurements of Atmospheric Volume Scattering Coefficients in Northern Europe, Summer 1978,” Scientific Report 13 (Visibility Laboratory, Scripps Institution of Oceanography, La Jolla, Calif., 1980).

Hering, W. S.

W. S. Hering, “An Operational Technique for Estimating Visible Spectrum Contrast Transmittance,” Scientific Report 16 (Visibility Laboratory, Scripps Institution of Oceanography, La Jolla, Calif., 1981).

Hoffer, R. M.

M. D. Fleming, J. S. Berkebile, R. M. Hoffer, in Proceedings, Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975), p. 1B-54.

R. M. Hoffer and staff, Natural Resource Mapping in Mountainous Terrain by Computer Analysis of ERTS-1 Satellite Data, LARS Info Note 061575, (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975); also available from NTIS as N76-14567.

Holben, B. N.

B. N. Holben, C. O. Justice, Photogramm. Eng. Remote Sensing 46, 1191 (1980).

Horn, B. K. P.

B. K. P. Horn, Proc. IEEE 69, 14 (1981).
[CrossRef]

B. K. P. Horn, R. J. Woodham, Comput. Graphics Image Process. 10, 69 (1979).
[CrossRef]

B. K. P. Horn, R. W. Sjoberg, Appl. Opt. 18, 1770 (1979).
[CrossRef] [PubMed]

B. K. P. Horn, B. L. Bachman, Commun. Assoc. Comput. Mach. 21, 914 (1978).

Howard, J. N.

J. N. Howard, J. S. Garing, Trans. Am. Geophys. Union 52, 371 (1971).
[CrossRef]

Jain, S. C.

F. J. Ahern, D. G. Goodenough, S. C. Jain, V. R. Rao, in Proceedings, Eleventh International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1977), pp. 731–755.

Johnson, R. W.

R. W. Johnson, J. I. Gordon, “Airborne Measurements of Atmospheric Volume Scattering Coefficients in Northern Europe, Summer 1978,” Scientific Report 13 (Visibility Laboratory, Scripps Institution of Oceanography, La Jolla, Calif., 1980).

Justice, C. O.

B. N. Holben, C. O. Justice, Photogramm. Eng. Remote Sensing 46, 1191 (1980).

Kagiwada, H.

S. Ueno, H. Kagiwada, R. Kalaba, J. Math. Phys. 12, 1279 (1971).
[CrossRef]

Kalaba, R.

S. Ueno, H. Kagiwada, R. Kalaba, J. Math. Phys. 12, 1279 (1971).
[CrossRef]

LaRocca, A. J.

A. J. LaRocca, R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” IRIA State-of-the-Art Report, ERIM 107600-10-T (Environmental Research Institute of Michigan, Ann Arbor, 1975); also available from NTIS as AD-A017 459.

Lin, T. L.

J. A. Smith, T. L. Lin, K. J. Ranson, Photogramm. Eng. Remote Sensing 46, 1183 (1980).

Malila, W. A.

F. G. Sadowsky, W. A. Malila, “Investigation of Techniques for Inventorying Forested Regions. Vol. 1: Reflectance Modeling and Empirical Multispectral Analysis of Forest Canopy Components,” Final Report, NASA-CR-151561 (1977); also available from NTIS as N73-14462.

Marchuk, G. I.

G. I. Marchuk et al., The Monte Carlo Methods in Atmospheric Optics (Springer, Berlin, 1980).

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976).

Moon, P.

P. Moon, Illum. Eng. 37, 707 (1942).

Nicodemus, F. E.

F. E. Nicodemus et al., “Geometrical Considerations and Nomenclature for Reflectance,” Nat. Bur. Stand. U.S. Monogr. 160 (1977).

Otterman, J.

Piech, K. R.

K. R. Piech, J. R. Schott, Proc. Soc. Photo-Opt. Instrum. Eng. 51, 84 (1974).

Ranson, K. J.

J. A. Smith, T. L. Lin, K. J. Ranson, Photogramm. Eng. Remote Sensing 46, 1183 (1980).

Rao, V. R.

F. J. Ahern, D. G. Goodenough, S. C. Jain, V. R. Rao, in Proceedings, Eleventh International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1977), pp. 731–755.

Rozenberg, G. V.

G. V. Rozenberg, Sov. Phys. Usp. 3, 346 (1960).
[CrossRef]

Sadowsky, F. G.

F. G. Sadowsky, W. A. Malila, “Investigation of Techniques for Inventorying Forested Regions. Vol. 1: Reflectance Modeling and Empirical Multispectral Analysis of Forest Canopy Components,” Final Report, NASA-CR-151561 (1977); also available from NTIS as N73-14462.

Schott, J. R.

K. R. Piech, J. R. Schott, Proc. Soc. Photo-Opt. Instrum. Eng. 51, 84 (1974).

Sekera, Z.

D. Deirmendjian, Z. Sekera, Tellus 6, 382 (1954). Sky irradiance data for μ0 = cosθ0 = 0.562 at A (or ρ¯) = 0.25 were interpolated from the published values of μ0 = 1.0,0.6,0.1 at A = 0.25.
[CrossRef]

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. California Press, Berkeley, 1960). Sky irradiance data were obtained by numerically integrating the tabulated values of downward radiance for μ0 = cosθ0 = 0.6 and A (or ρ¯) = 0.25. Path radiance data were obtained from the tabulated values of upward radiance at μ0 = cosθ0 = 0.6 and A = 0.25 by first subtracting the reflected target radiance. The reflected target radiance is Lt = (TdE0μ0 + Es)A/π.

Sjoberg, R. W.

Smith, J. A.

J. A. Smith, T. L. Lin, K. J. Ranson, Photogramm. Eng. Remote Sensing 46, 1183 (1980).

Solomon, S. I.

S. I. Solomon, J. Cameron, J. Chadwick, A. S. Aggarwal, in Proceedings Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind.1975), p. 4B-1.

Spencer, M. M.

R. E. Turner, M. M. Spencer, in Proceedings, Eighth International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1972), pp. 895–934.

Thekaekara, M. P.

M. P. Thekaekara, J. Environ. Sci. 13, 6 (1970).

Turner, R. E.

R. E. Turner, “Radiative Transfer in Real Atmospheres,” Technical Report ERIM 190100-24-T (Environmental Research Institute of Michigan, Ann Arbor, 1974); also available from NTIS as N74-31873.

R. E. Turner, “Atmospheric Effects in Multispectral Remote Sensor Data,” Technical Report ERIM 109600-15-F (Environmental Research Institute of Michigan, Ann Arbor, 1975); also available from NTIS as N75-26474.

R. E. Turner, M. M. Spencer, in Proceedings, Eighth International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1972), pp. 895–934.

A. J. LaRocca, R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” IRIA State-of-the-Art Report, ERIM 107600-10-T (Environmental Research Institute of Michigan, Ann Arbor, 1975); also available from NTIS as AD-A017 459.

Ueno, S.

S. Ueno, H. Kagiwada, R. Kalaba, J. Math. Phys. 12, 1279 (1971).
[CrossRef]

Valley, S. L.

S. L. Valley, Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).

Woodham, R. J.

R. J. Woodham, Proc. Soc. Photo-Opt. Instrum. Eng. 238, 361 (1980).

B. K. P. Horn, R. J. Woodham, Comput. Graphics Image Process. 10, 69 (1979).
[CrossRef]

R. J. Woodham, “Two Simple Algorithms for Displaying Orthographic Projections of Surfaces,” Working Paper 126 (Artificial Intelligence Laboratory, MIT, 1976).

R. J. Woodham, “Looking in the Shadows,” Working Paper 169 (Artificial Intelligence Laboratory, MIT, 1978).

Appl. Opt. (3)

Commun. Assoc. Comput. Mach. (1)

B. K. P. Horn, B. L. Bachman, Commun. Assoc. Comput. Mach. 21, 914 (1978).

Comput. Graphics Image Process. (1)

B. K. P. Horn, R. J. Woodham, Comput. Graphics Image Process. 10, 69 (1979).
[CrossRef]

Illum. Eng. (1)

P. Moon, Illum. Eng. 37, 707 (1942).

J. Atmos. Sci. (2)

K. Coulson, J. Atmos. Sci. 25, 759 (1968). Sky irradiance data for μ0 = cosθ0 = 0.562 were interpolated from the published tables for a surface albedo of A (or ρ¯) = 0.25. Es(τ) = Fs + Fd of the published data, where the value for Fd for A = 0.25 was interpolated from the values for A = 0.20 and A = 0.30.
[CrossRef]

B. W. Fitch, J. Atmos. Sci. 38, 2717 (1981).
[CrossRef]

J. Environ. Sci. (1)

M. P. Thekaekara, J. Environ. Sci. 13, 6 (1970).

J. Math. Phys. (1)

S. Ueno, H. Kagiwada, R. Kalaba, J. Math. Phys. 12, 1279 (1971).
[CrossRef]

Nat. Bur. Stand. U.S. Monogr. (1)

F. E. Nicodemus et al., “Geometrical Considerations and Nomenclature for Reflectance,” Nat. Bur. Stand. U.S. Monogr. 160 (1977).

Photogramm. Eng. Remote Sensing (2)

J. A. Smith, T. L. Lin, K. J. Ranson, Photogramm. Eng. Remote Sensing 46, 1183 (1980).

B. N. Holben, C. O. Justice, Photogramm. Eng. Remote Sensing 46, 1191 (1980).

Proc. IEEE (1)

B. K. P. Horn, Proc. IEEE 69, 14 (1981).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

K. R. Piech, J. R. Schott, Proc. Soc. Photo-Opt. Instrum. Eng. 51, 84 (1974).

R. J. Woodham, Proc. Soc. Photo-Opt. Instrum. Eng. 238, 361 (1980).

Sov. Phys. Usp. (1)

G. V. Rozenberg, Sov. Phys. Usp. 3, 346 (1960).
[CrossRef]

Tellus (1)

D. Deirmendjian, Z. Sekera, Tellus 6, 382 (1954). Sky irradiance data for μ0 = cosθ0 = 0.562 at A (or ρ¯) = 0.25 were interpolated from the published values of μ0 = 1.0,0.6,0.1 at A = 0.25.
[CrossRef]

Trans. Am. Geophys. Union (1)

J. N. Howard, J. S. Garing, Trans. Am. Geophys. Union 52, 371 (1971).
[CrossRef]

Other (20)

E. J. McCartney, Optics of the Atmosphere: Scattering by Molecules and Particles (Wiley, New York, 1976).

A. J. LaRocca, R. E. Turner, “Atmospheric Transmittance and Radiance: Methods of Calculation,” IRIA State-of-the-Art Report, ERIM 107600-10-T (Environmental Research Institute of Michigan, Ann Arbor, 1975); also available from NTIS as AD-A017 459.

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

S. I. Solomon, J. Cameron, J. Chadwick, A. S. Aggarwal, in Proceedings Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind.1975), p. 4B-1.

R. M. Hoffer and staff, Natural Resource Mapping in Mountainous Terrain by Computer Analysis of ERTS-1 Satellite Data, LARS Info Note 061575, (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975); also available from NTIS as N76-14567.

M. D. Fleming, J. S. Berkebile, R. M. Hoffer, in Proceedings, Second Symposium on Machine Processing of Remotely Sensed Data (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1975), p. 1B-54.

F. G. Sadowsky, W. A. Malila, “Investigation of Techniques for Inventorying Forested Regions. Vol. 1: Reflectance Modeling and Empirical Multispectral Analysis of Forest Canopy Components,” Final Report, NASA-CR-151561 (1977); also available from NTIS as N73-14462.

S. L. Valley, Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (U. California Press, Berkeley, 1960). Sky irradiance data were obtained by numerically integrating the tabulated values of downward radiance for μ0 = cosθ0 = 0.6 and A (or ρ¯) = 0.25. Path radiance data were obtained from the tabulated values of upward radiance at μ0 = cosθ0 = 0.6 and A = 0.25 by first subtracting the reflected target radiance. The reflected target radiance is Lt = (TdE0μ0 + Es)A/π.

G. I. Marchuk et al., The Monte Carlo Methods in Atmospheric Optics (Springer, Berlin, 1980).

R. J. Woodham, “Two Simple Algorithms for Displaying Orthographic Projections of Surfaces,” Working Paper 126 (Artificial Intelligence Laboratory, MIT, 1976).

R. J. Woodham, “Looking in the Shadows,” Working Paper 169 (Artificial Intelligence Laboratory, MIT, 1978).

R. E. Turner, M. M. Spencer, in Proceedings, Eighth International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1972), pp. 895–934.

Sky irradiance and path radiance data were obtained directly from the published formula for a Rayleigh atmosphere with ρ¯ = 0.25 and μ0 = 0.562.

F. J. Ahern, D. G. Goodenough, S. C. Jain, V. R. Rao, in Proceedings, Eleventh International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1977), pp. 731–755.

J. Otterman, R. S. Fraser, Remote Sensing Environ.5, 247 (1976).
[CrossRef]

R. E. Turner, “Radiative Transfer in Real Atmospheres,” Technical Report ERIM 190100-24-T (Environmental Research Institute of Michigan, Ann Arbor, 1974); also available from NTIS as N74-31873.

R. E. Turner, “Atmospheric Effects in Multispectral Remote Sensor Data,” Technical Report ERIM 109600-15-F (Environmental Research Institute of Michigan, Ann Arbor, 1975); also available from NTIS as N75-26474.

W. S. Hering, “An Operational Technique for Estimating Visible Spectrum Contrast Transmittance,” Scientific Report 16 (Visibility Laboratory, Scripps Institution of Oceanography, La Jolla, Calif., 1981).

R. W. Johnson, J. I. Gordon, “Airborne Measurements of Atmospheric Volume Scattering Coefficients in Northern Europe, Summer 1978,” Scientific Report 13 (Visibility Laboratory, Scripps Institution of Oceanography, La Jolla, Calif., 1980).

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Figures (14)

Fig. 1
Fig. 1

Comparison of the two scattering components of a reference atmosphere (labeled Ref. in the figure) and their sum to exponential functions (labeled Exp.) derived from least-squares fitting. The parameters of the exponentials are given in the text.

Fig. 2
Fig. 2

Variation of relative sky irradiance values with optical depth for a Rayleigh atmosphere. The approximations made in the text amount to assuming a linear form for this variation. See notes for derivations: Coulson 1968,18 Coulson et al. 1960,19 Deirmendjian and Sekera 1954,20 Otterman 197821 Turner and Spencer 1972.22,23

Fig. 3
Fig. 3

Comparing sky irradiance on a horizontal surface as a function of altitude for the Turner-Spencer model22 and the exponential approximation thereof at λ = 0.55 μm. The calculation for the Turner-Spencer curve used a forward-scattering coefficient η = 0.796, average background albedo ρ ¯ = 0.15, and a Rayleigh single-scattering phase function. The exponential curve Es(z) = Es0 exp(−z/Hs) was fit by least-squares to yield Es0 = 3.04 mW cm−2 and Hs = 2944.9 m.

Fig. 4
Fig. 4

Variation of relative path radiance values with optical depth in a Rayleigh atmosphere. The approximations made in the text amount to assuming a linear form for this variation. See notes for derivations: Coulson et al. 1960,19 Otterman 1978,21 Turner and Spencer 1972.22,23

Fig. 5
Fig. 5

Comparing path radiance as a function of altitude for the Turner-Spencer model22 and the exponential approximation thereof at λ = 0.55 μm. The calculation for the Turner-Spencer curve used a forward-scattering coefficient η = 0.796, average background albedo ρ ¯ = 0.15, and a Rayleigh single-scattering phase function. The exponential curve Lp(z) = Lp0 exp(−z/Hp) was fit by least-squares to yield Lp0 = 0.376 mW cm−2 sr−1 and Hp = 2732.56 m.

Fig. 6
Fig. 6

Digital elevation model of the Dent de Morcles test region is displayed here as an image. Elevation is encoded as brightness, the brighter the area, the higher it is. The figure to the right is a histogram of elevation values, 0 m on the far left, 5110 m on the far right. The lowest point in the scene is 410 m, the highest is 3210 m. The peculiar disparity of adjacent altitudes in the histogram is an artifact of the process used to interpolate elevations during digitization of the contour maps (even altitudes were favored over odd ones). There are two histograms here: the smaller shaded one represents shadowed targets and the larger solid one sunlit targets (see text).

Fig. 7
Fig. 7

Landsat 1 multispectral scanner image for the yellow-green channel 4, 500–600 nm. The original raw image was destriped and rectified to be commensurate with the digital terrain models as described in the text. Note the presence of clouds in the upper left corner of the image and the pronounced hazy appearance due to atmospheric path radiance. The histogram records digitized image brightness. A value of 0 corresponds to a sensor radiance of 0.0 mW cm−2 sr−1. A value of 511 corresponds to a sensor radiance of 2.48 mW cm−2 sr−1. (As provided on computer compatible tape, Landsat MSS channels 4, 5, and 6 are represented by 7-bit bytes, channel 7 by 6-bit bytes. These data were scaled to 9-bit bytes during destriping and rectification, hence the maximum value of 511.) The peak at the high end of the histogram is due to saturation on clouds and snow.

Fig. 8
Fig. 8

Figure on the left is a binary map of those targets in the Dent de Morcles region that were in cast shadow under the illumination conditions of the Landsat overflight. The figure on the right is a binary map of those targets determined to lie in self-shadow (oriented away from the sun) under the given illumination. The shadow computation was performed using a method derived from hidden-surface plotting.28,29

Fig. 9
Fig. 9

Synthetic image of the test region with the sun in the same position as during the Landsat overflight. The ground is assumed to be a Lambertian reflector of uniform albedo ρ = 1. No atmosphere is assumed. The peak in the center of the histogram is from radiances within the Rhone Valley (the nearly uniform gray area at bottom right), a relatively flat region of almost constant altitude. The peak at the extreme left of the histogram is zero radiance for all shadows.

Fig. 10
Fig. 10

Two possible exponential models of path radiance as inferred from minimum sensor data Lp(z) = Lp0 exp(−z/Hp), where each curve was fitted manually to the data. For the upper curve, Lp0 = 0.33 mW cm−2 sr−1; for the lower curve, Lp0 = 0.315 mW cm−2 sr−1. Hp = 4720 m for both curves.

Fig. 11
Fig. 11

Altitude profile of average albedo from sunlit targets in the Dent de Morcles test area. Albedo was calculated by assuming no intervening atmosphere and examining only sunlit targets at all altitudes. In such targets, reflected solar irradiance generally dominates, and computed albedo should approximate the true value. The data have been smoothed but show a distinct increase of albedo with altitude.

Fig. 12
Fig. 12

Albedo image, and its associated histogram, generated using model parameters for exponential approximations to functional forms of path radiance and sky irradiance computed from the Turner-Spencer model: τ 0 = 0.26185 , L p 0 = 0.376 mW cm - 2 sr - 1 , E s 0 = 3.04 mW cm - 2 , H = 2529 m , H p = 2734 m , H 2 = 2945 m . The average computed albedo for sunlit targets was ρ ¯ = 0.117 and for shadowed targets ρ ¯ = 0.276.

Fig. 13
Fig. 13

Improved albedo image and its associated histogram. Albedo was computed using the following atmospheric model parameters: τ 0 = 0.26185 , L p 0 = 0.315 mW cm - 2 sr - 1 , E s 0 = 3.0 mW cm - 2 , H = 2529 m , H p = 4720 m , H s = 4720 m . The average computed albedo for sunlit targets was ρ ¯ = 0.110 and for shadowed targets ρ ¯ = 0.200.

Fig. 14
Fig. 14

Third example albedo image and its associated histogram. Albedo was computed using the following atmospheric model parameters: τ 0 = 0.23 , L p 0 = 0.33 mW cm - 2 sr - 1 , E s 0 = 3.04 mW cm - 2 , H = 4000 m , H p = 2734 m , H s = 2945 m . The average computed albedo for sunlit targets was ρ ¯ = 0.108 and for shadowed targets ρ ¯ = 0.184.

Tables (4)

Tables Icon

Table I Surface Slope Angle for Which the Direct Solar Contribution to Target Irradiance Equals the Diffuse Sky Contribution; Values are Given at Various Altitudes and Two Azimuthal Angles for the Atmospheric Model of Fig. 14

Tables Icon

Table II Relative Sensitivities (∂ρ/∂X)/(ρ/X) of Computed Albedo to the Path Radiance Model Parameters X = Lp0 and X = Hp at Various Altitudes for MSS Band 4; Other Parameter Values are Those of Fig. 14; Model Sensor Radiance L is in mW cm−2 sr−1

Tables Icon

Table III Relative Sensitivities (∂ρ/∂X)/(ρ/X) of Computed Albedo to the Sky Irradiance X = Es0 and X = Hs and Optical Depth Model Parameters X = τ0 and X = H for Horizontal Sunlit Targets at Various Altitudes for MSS Band 4; Other Parameter Values are Those of Fig. 14

Tables Icon

Table IV Relative Sensitivities (∂ρ/∂X)/(ρ/X) of Computed Albedo to the Sky Irradiance X = Es0 and X = Hs and Optical Depth Model Parameters X = τ0 and X = H for Horizontal Shadowed Targets at Various Altitudes for MSS Band 4; Other Parameter Values are Those of Fig. 14

Equations (16)

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E i ( x i , y i ) = [ L t ( r t , r m ) T u ( r t , r m ) + L p ( r t , r m ) ] δ ω ,
L m ( r t , r m ) = E i ( x i , y i ) δ ω = L i ( r t , r m ) T u ( r t , r m ) + L p ( r t , r m ) .
T u ( r t , r m ) = exp [ - τ ( r t , r m ) ] ,
E sky ( r t ) = E s ( z ) h ( θ n , ϕ n ) = E s ( z ) ½ ( 1 + cos θ n ) ,
f r ( r t ; θ t , ϕ t ; θ m , ϕ m ) = ρ ( r t ) π .
L m ( x t y t ) = ρ ( x t , y t ) π T u ( z ) [ E 0 T d ( z ) R ( θ n , ϕ n ) + E s ( z ) h ( θ n , ϕ n ) ] + L p ( z ) ,
R ( θ n , ϕ n ) = { 0 , if traget is self - or cast - shadowed , cos θ 0 , if 0 θ 0 π / 2 ,
cos θ 0 = cos θ n cos θ n + sin θ n sin θ 0 cos ( ϕ n - ϕ 0 ) .
ρ ( x t , y t ) = π [ L m ( x t , y t ) - L p ( z ) ] T u ( z ) [ E 0 T d ( z ) R ( θ n , ϕ n ) + E s ( z ) h ( θ n , ϕ n ) ] .
τ ( z ) = τ 0 exp ( - z / H ) , L p ( z ) = L p 0 exp ( - z / H p ) , E s ( z ) = E s 0 exp ( - z / H s ) ,
Reyleigh : τ 0 R = 0.09917 H R = 8232 m , aerosol : τ 0 A = 0.19 H A = 1211 m , sum : τ 0 = 0.2619 H = 2529 m .
E s ( z ) π [ L - L p ( z ) ] ρ T u ( z ) h ( θ n , ϕ n ) ,
η = 0.5 τ R + 0.95 τ A τ R + τ A = 0.796 ,
τ 0 = 0.26185 , L p 0 = 0.376 mW cm - 2 sr - 1 , E s 0 = 3.04 mW cm - 2 , H = 2529 m , H p = 2734 m , H 2 = 2945 m .
τ 0 = 0.26185 , L p 0 = 0.315 mW cm - 2 sr - 1 , E s 0 = 3.0 mW cm - 2 , H = 2529 m , H p = 4720 m , H s = 4720 m .
τ 0 = 0.23 , L p 0 = 0.33 mW cm - 2 sr - 1 , E s 0 = 3.04 mW cm - 2 , H = 4000 m , H p = 2734 m , H s = 2945 m .

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