Abstract

In this paper, we present a new spectral light transport model for sand. The model employs a novel approach to simulate light interaction with particulate materials which yields both the spectral and spatial (bidirectional reflectance distribution function, or BRDF) responses of sand. Furthermore, the parameters specifying the model are based on the physical and mineralogical properties of sand. The model is evaluated quantitatively, through comparisons with measured data. Good spectral reconstructions were achieved for the reflectances of several real sand samples. The model was also evaluated qualitatively, and compares well with descriptions found in the literature. Its potential applications include, but are not limited to, applied optics, remote sensing and image synthesis.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Prahl, "Light Transport in Tissue," Ph.D. thesis, University of Texas at Austin (1988).
  2. S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).
  3. Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996).
    [CrossRef] [PubMed]
  4. G. Baranoski, J. Rokne, and G. Xu, "Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering," The Visual Computer 17, 506-518 (2001).
    [CrossRef]
  5. A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools  9, 31-51 (2004).
  6. S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
    [CrossRef]
  7. R. Shuchman and D. Rea, "Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data," Remote Sensing of Environment 11, 295-310 (1981).
    [CrossRef]
  8. R. Morris and D. Golden, "Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars," Icarus 134, 1-10 (1998).
    [CrossRef]
  9. R. Singer, "Spectral Evidence for the Mineralogy of High-Albedo Soils and Dust on Mars," Journal of Geophysical Research 87, 10,159-10,168 (1982).
    [CrossRef]
  10. F. Pettijohn, P. Potter, and R. Siever, Sand and Sandstone, 2nd ed. (Springer-Verlag, New York, NY, 1987).
    [CrossRef]
  11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).
  12. J. Gerrard, Fundamentals of Soils (Routledge, New York, NY, 2000).
  13. Soil Science Division Staff, Soil Survey Manual (Soil Conservation Service, 1993). United States Department of Agriculture Handbook 18.
  14. K. Coulson and D. Reynolds, "The Spectral Reflectance of Natural Surfaces," Journal of Applied Meteorology 10, 1285-1295 (1971).
    [CrossRef]
  15. J. Norman, J. Welles, and E. Walter, "Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils," IEEE Transactions on Geoscience and Remote Sensing GE-23, 659-667 (1985).
    [CrossRef]
  16. K. Coulson, G. Bouricius, and E. Gray, "Optical Reflection Properties of Natural Surfaces," Journal of Geophysical Research 70, 4601-4611 (1965).
    [CrossRef]
  17. M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985).
    [CrossRef]
  18. D. Leu, "Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship," Remote Sensing of Environment 6, 169-182 (1977).
    [CrossRef]
  19. G. Hunt and J. Salisbury, "Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals," Modern Geology 1, 283-300 (1970).
  20. P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).
  21. A. Mottana, R. Crespi, and G. Liborio, Simon and Schuster’s Guide to Rocks and Minerals (Simon and Schuster, Inc., New York, NY, 1978).
  22. J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983).
    [CrossRef]
  23. R. Cornell and U. Schwertmann, The Iron Oxides, 2nd ed. (Wiley-VCH GmbH & Co. KGaA, Weinheim, Germany, 2003).
    [CrossRef]
  24. H. Wopfner and C. Twindale, "Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia," Geologische Rundschau 77, 815-834 (1988).
    [CrossRef]
  25. G. Hunt, J. Salisbury, and C. Lenhoff, "Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides," Modern Geology 2, 195-205 (1971).
  26. S. Twomey, C. Bohren, and J. Mergenthaler, "Reflectance and Albedo Differences Between Wet and Dry Surfaces," Applied Optics 25, 57-84 (1986).
    [CrossRef]
  27. M. Kühl and B. Jørgensen, "The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments," Limnology and Oceanography 39, 1368-1398 (1994).
    [CrossRef]
  28. R. Vincent and G. Hunt, "Infrared Reflectance from Mat Surfaces," Applied Optics 7, 53-59 (1968).
    [CrossRef] [PubMed]
  29. H. Wadell, "Volume, Shape, and Roundness of Rock Particles," Journal of Geology 40, 443-451 (1932).
    [CrossRef]
  30. N. Riley, "Projection Sphericity," Journal of Sedimentary Petrology 11, 94-95 (1941).
  31. W. Krumbein, "Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles," Journal of Sedimentary Petrology 11, 64-72 (1941).
  32. H. Zhang and K. Voss, "Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models," Applied Optics 44, 597-610 (2005).
    [CrossRef] [PubMed]
  33. Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
    [CrossRef]
  34. M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, Cambridge, 2006).
  35. M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
    [CrossRef] [PubMed]
  36. B. Hapke, "Bidirectional Reflectance Spectroscopy. 1. Theory," Journal of Geophysical Research 86, 3039-3054 (1981).
    [CrossRef]
  37. B. Hapke and E. Wells, "Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations," Journal of Geophysical Research 86, 3055-3054 (1981).
    [CrossRef]
  38. A. Emslie and J. Aronson, "Spectral Reflectance of Particulate Materials. 1: Theory," Applied Optics 12, 2563-2572 (1973).
    [CrossRef] [PubMed]
  39. W. Egan and T. Hilgeman, "Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering," Applied Optics 17, 245-252 (1978).
    [CrossRef] [PubMed]
  40. M. Oren and S. Nayar, "Generalization of Lambert’s Reflectance Model," in Computer Graphics Proceedings, Annual Conference Series, pp. 239-246 (1994).
  41. L. Wolff, "Diffuse-Reflectance Model for Smooth Dielectric Surfaces," Journal of the Optical Society of America A (Optics, Image Science, and Vision) 11, 2956-2968 (1994).
    [CrossRef]
  42. M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999).
    [CrossRef]
  43. D. Stankevich and Y. Shkuratov, "Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure," Journal of Quantitative Spectroscopy and Radiative Transfer 87, 289-296 (2004).
    [CrossRef]
  44. J. Peltoniemi, "Spectropolarised Ray-Tracing Simulations in Densely Packed Particulate Medium," Journal of Quantitative Spectroscopy and Radiative Transfer (2007). In Press, accepted, URL http://dx.doi.org/10.1016/j.jqsrt.2007.05.009.
    [CrossRef]
  45. H. Jensen, J. Legakis, and J. Dorsey, "Rendering ofWet Materials," in Proceedings of the Eurographics Workshop on Rendering, pp. 273-282 (1999).
  46. L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophysical Journal 93, 70-83 (1941).
    [CrossRef]
  47. Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
    [CrossRef]
  48. D. Lobell and G. Asner, "Moisture Effects on Soil Reflectance," Soil Science Society of America Journal 66, 722-727 (2002).
    [CrossRef]
  49. D. Neema, A. Shah, and A. Patel, "A Statistical Optical Model for Light Reflection and Penetration Through Sand," International Journal of Remote Sensing 8, 1209-1217 (1987).
    [CrossRef]
  50. V. Barron and L. Montealegre, "Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory," American Journal of Science 286, 792-802 (1986).
    [CrossRef]
  51. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche (An Article on Optics of Paint Layers)," Zeitschrift fur Technische Physik 12, 593-601 (1931).
  52. D. Nickerson, "History of the Munsell Color System and its Scientific Application," Journal of the Optical Society of America 30, 575-645 (1940).
    [CrossRef]
  53. G. Okin and T. Painter, "Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces," Remote Sensing of Environment 89, 272-280 (2004).
    [CrossRef]
  54. F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, United States Department of Commerce, 1977).
  55. R. Hunter and R. Harold, The Measurement of Appearance, 2nd ed. (JohnWiley and Sons, New York, NY, 1987).
  56. G. Ward, "Measuring and Modeling Anisotropic Reflection," Computer Graphics 26, 262-272 (1992).
    [CrossRef]
  57. V. Kuz’min and I. Meglinski, "Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering," Optics & Spectroscopy 97, 100-106 (2004).
    [CrossRef]
  58. F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).
  59. T. Nousiainen, K. Muinonen, and P. Räisänen, "Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation," Journal of Geophysical Research 108, AAC 12-1-17 (2003).
    [CrossRef]
  60. G. Hale and M. Querry, "Optical Constants of Water in the 200-nm to 200- m Wavelength Region," Applied Optics 12, 555-563 (1973).
    [CrossRef] [PubMed]
  61. W. Tropf, M. Thomas, and T. Harris, "Properties of Crystals and Glasses," in Handbook of Optics, M. Bass, E. Van Stryland, D. Williams, and W.Wolfe, eds., vol. 2, 2nd ed., chap. 33 (McGraw-Hill, 1995).
  62. I. Sokolik and O. Toon, "Incorporation of Mineralogical Composition into Models of the Radiative Properties of Mineral Aerosol from UV to IR Wavelengths," Journal of Geophysical Research 104, 9423-9444 (1999).
    [CrossRef]
  63. W. Egan and T. Hilgeman, Optical Properties of Inhomogeneous Materials: Applications to Geology, Astronomy, Chemistry, and Engineering (Academic Press, New York, NY, 1979).
    [PubMed]
  64. A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979).
    [CrossRef]
  65. M. Vepraskas and D. Cassel, "Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties," Soil Science Society of America Journal 51, 1108-1112 (1987).
    [CrossRef]
  66. B. Kimmel, "SPLITS: A Spectral Light Transport Model for Sand," Master’s thesis, School of Computer Science, University of Waterloo (2005).
  67. S. Ross, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).
  68. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, NY, 1983).
  69. J. Maxwell Garnett, "Colours in Metal Glasses and in Metallic Films," Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203, 385-420 (1904).
    [CrossRef]
  70. M. Shirazi, L. Boersma, and J. Hart, "A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale," Soil Science Society of America Journal 52, 181-190 (1988).
    [CrossRef]
  71. J. Snyder, Map Projections: A Working Manual (United States Government Printing Office, Washington, 1987). U.S. Geological Survey Professional Paper 1395.
  72. B. Hapke, "Bidirectional Reflectance Spectroscopy. 4. The Extinction Coefficient and the Opposition Effect," Icarus 67, 264-280 (1986).
    [CrossRef]
  73. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, Oxford, England, 1980).
  74. C. Gribble and A. Hall, Optical Mineralogy: Principles and Practice (Chapman & Hall, New York, NY, 1993).
  75. J. Rinker, C. Breed, J. McCauley, and P. Corl, "Remote Sensing Field Guide — Desert," Tech. rep., U.S. Army Topographic Engineering Center, Fort Belvoir, VA (1991).
  76. T. Cudahy and E. Ramanaidou, "Measurement of the Hematite:Goethite Ratio Using Field Visible and Near-Infrared Reflectance Spectrometry in Channel Iron Deposits, Western Australia," Australian Journal of Earth Sciences 44, 411-420 (1997).
    [CrossRef]
  77. J. Cierniewski, "A Model for Soil Surface Roughness Influence on the Spectral Response of Bare Soils in the Visible and Near-Infrared Range," Remote Sensing of Environment 23, 92-115 (1987).
    [CrossRef]

2007

M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
[CrossRef] [PubMed]

2006

Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
[CrossRef]

2005

H. Zhang and K. Voss, "Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models," Applied Optics 44, 597-610 (2005).
[CrossRef] [PubMed]

2004

A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools  9, 31-51 (2004).

D. Stankevich and Y. Shkuratov, "Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure," Journal of Quantitative Spectroscopy and Radiative Transfer 87, 289-296 (2004).
[CrossRef]

G. Okin and T. Painter, "Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces," Remote Sensing of Environment 89, 272-280 (2004).
[CrossRef]

V. Kuz’min and I. Meglinski, "Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering," Optics & Spectroscopy 97, 100-106 (2004).
[CrossRef]

2003

T. Nousiainen, K. Muinonen, and P. Räisänen, "Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation," Journal of Geophysical Research 108, AAC 12-1-17 (2003).
[CrossRef]

2002

D. Lobell and G. Asner, "Moisture Effects on Soil Reflectance," Soil Science Society of America Journal 66, 722-727 (2002).
[CrossRef]

2001

G. Baranoski, J. Rokne, and G. Xu, "Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering," The Visual Computer 17, 506-518 (2001).
[CrossRef]

1999

M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999).
[CrossRef]

I. Sokolik and O. Toon, "Incorporation of Mineralogical Composition into Models of the Radiative Properties of Mineral Aerosol from UV to IR Wavelengths," Journal of Geophysical Research 104, 9423-9444 (1999).
[CrossRef]

1998

R. Morris and D. Golden, "Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars," Icarus 134, 1-10 (1998).
[CrossRef]

1997

T. Cudahy and E. Ramanaidou, "Measurement of the Hematite:Goethite Ratio Using Field Visible and Near-Infrared Reflectance Spectrometry in Channel Iron Deposits, Western Australia," Australian Journal of Earth Sciences 44, 411-420 (1997).
[CrossRef]

1996

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996).
[CrossRef] [PubMed]

1994

M. Kühl and B. Jørgensen, "The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments," Limnology and Oceanography 39, 1368-1398 (1994).
[CrossRef]

L. Wolff, "Diffuse-Reflectance Model for Smooth Dielectric Surfaces," Journal of the Optical Society of America A (Optics, Image Science, and Vision) 11, 2956-2968 (1994).
[CrossRef]

1992

G. Ward, "Measuring and Modeling Anisotropic Reflection," Computer Graphics 26, 262-272 (1992).
[CrossRef]

1989

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).

1988

H. Wopfner and C. Twindale, "Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia," Geologische Rundschau 77, 815-834 (1988).
[CrossRef]

M. Shirazi, L. Boersma, and J. Hart, "A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale," Soil Science Society of America Journal 52, 181-190 (1988).
[CrossRef]

1987

M. Vepraskas and D. Cassel, "Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties," Soil Science Society of America Journal 51, 1108-1112 (1987).
[CrossRef]

J. Cierniewski, "A Model for Soil Surface Roughness Influence on the Spectral Response of Bare Soils in the Visible and Near-Infrared Range," Remote Sensing of Environment 23, 92-115 (1987).
[CrossRef]

D. Neema, A. Shah, and A. Patel, "A Statistical Optical Model for Light Reflection and Penetration Through Sand," International Journal of Remote Sensing 8, 1209-1217 (1987).
[CrossRef]

1986

V. Barron and L. Montealegre, "Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory," American Journal of Science 286, 792-802 (1986).
[CrossRef]

S. Twomey, C. Bohren, and J. Mergenthaler, "Reflectance and Albedo Differences Between Wet and Dry Surfaces," Applied Optics 25, 57-84 (1986).
[CrossRef]

B. Hapke, "Bidirectional Reflectance Spectroscopy. 4. The Extinction Coefficient and the Opposition Effect," Icarus 67, 264-280 (1986).
[CrossRef]

1985

M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985).
[CrossRef]

J. Norman, J. Welles, and E. Walter, "Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils," IEEE Transactions on Geoscience and Remote Sensing GE-23, 659-667 (1985).
[CrossRef]

1983

J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983).
[CrossRef]

1982

R. Singer, "Spectral Evidence for the Mineralogy of High-Albedo Soils and Dust on Mars," Journal of Geophysical Research 87, 10,159-10,168 (1982).
[CrossRef]

1981

R. Shuchman and D. Rea, "Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data," Remote Sensing of Environment 11, 295-310 (1981).
[CrossRef]

B. Hapke, "Bidirectional Reflectance Spectroscopy. 1. Theory," Journal of Geophysical Research 86, 3039-3054 (1981).
[CrossRef]

B. Hapke and E. Wells, "Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations," Journal of Geophysical Research 86, 3055-3054 (1981).
[CrossRef]

1979

A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979).
[CrossRef]

1978

W. Egan and T. Hilgeman, "Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering," Applied Optics 17, 245-252 (1978).
[CrossRef] [PubMed]

1977

D. Leu, "Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship," Remote Sensing of Environment 6, 169-182 (1977).
[CrossRef]

1973

A. Emslie and J. Aronson, "Spectral Reflectance of Particulate Materials. 1: Theory," Applied Optics 12, 2563-2572 (1973).
[CrossRef] [PubMed]

G. Hale and M. Querry, "Optical Constants of Water in the 200-nm to 200- m Wavelength Region," Applied Optics 12, 555-563 (1973).
[CrossRef] [PubMed]

1971

G. Hunt, J. Salisbury, and C. Lenhoff, "Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides," Modern Geology 2, 195-205 (1971).

K. Coulson and D. Reynolds, "The Spectral Reflectance of Natural Surfaces," Journal of Applied Meteorology 10, 1285-1295 (1971).
[CrossRef]

1970

G. Hunt and J. Salisbury, "Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals," Modern Geology 1, 283-300 (1970).

1968

R. Vincent and G. Hunt, "Infrared Reflectance from Mat Surfaces," Applied Optics 7, 53-59 (1968).
[CrossRef] [PubMed]

1965

K. Coulson, G. Bouricius, and E. Gray, "Optical Reflection Properties of Natural Surfaces," Journal of Geophysical Research 70, 4601-4611 (1965).
[CrossRef]

1941

N. Riley, "Projection Sphericity," Journal of Sedimentary Petrology 11, 94-95 (1941).

W. Krumbein, "Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles," Journal of Sedimentary Petrology 11, 64-72 (1941).

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophysical Journal 93, 70-83 (1941).
[CrossRef]

1940

D. Nickerson, "History of the Munsell Color System and its Scientific Application," Journal of the Optical Society of America 30, 575-645 (1940).
[CrossRef]

1932

H. Wadell, "Volume, Shape, and Roundness of Rock Particles," Journal of Geology 40, 443-451 (1932).
[CrossRef]

1904

J. Maxwell Garnett, "Colours in Metal Glasses and in Metallic Films," Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203, 385-420 (1904).
[CrossRef]

Alferez, F.

J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983).
[CrossRef]

Alvarado, S.

A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979).
[CrossRef]

Andreoli, G.

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

Aronson, J.

A. Emslie and J. Aronson, "Spectral Reflectance of Particulate Materials. 1: Theory," Applied Optics 12, 2563-2572 (1973).
[CrossRef] [PubMed]

Asner, G.

D. Lobell and G. Asner, "Moisture Effects on Soil Reflectance," Soil Science Society of America Journal 66, 722-727 (2002).
[CrossRef]

Baranoski, G.

A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools  9, 31-51 (2004).

G. Baranoski, J. Rokne, and G. Xu, "Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering," The Visual Computer 17, 506-518 (2001).
[CrossRef]

Barron, V.

V. Barron and L. Montealegre, "Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory," American Journal of Science 286, 792-802 (1986).
[CrossRef]

Baumgardner, M.

M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985).
[CrossRef]

Betty, C.

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

Biehl, L.

M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985).
[CrossRef]

Boersma, L.

M. Shirazi, L. Boersma, and J. Hart, "A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale," Soil Science Society of America Journal 52, 181-190 (1988).
[CrossRef]

Bohren, C.

S. Twomey, C. Bohren, and J. Mergenthaler, "Reflectance and Albedo Differences Between Wet and Dry Surfaces," Applied Optics 25, 57-84 (1986).
[CrossRef]

Bouricius, G.

K. Coulson, G. Bouricius, and E. Gray, "Optical Reflection Properties of Natural Surfaces," Journal of Geophysical Research 70, 4601-4611 (1965).
[CrossRef]

Cairns, B.

M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
[CrossRef] [PubMed]

Cassel, D.

M. Vepraskas and D. Cassel, "Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties," Soil Science Society of America Journal 51, 1108-1112 (1987).
[CrossRef]

Cierniewski, J.

J. Cierniewski, "A Model for Soil Surface Roughness Influence on the Spectral Response of Bare Soils in the Visible and Near-Infrared Range," Remote Sensing of Environment 23, 92-115 (1987).
[CrossRef]

Coulson, K.

K. Coulson and D. Reynolds, "The Spectral Reflectance of Natural Surfaces," Journal of Applied Meteorology 10, 1285-1295 (1971).
[CrossRef]

K. Coulson, G. Bouricius, and E. Gray, "Optical Reflection Properties of Natural Surfaces," Journal of Geophysical Research 70, 4601-4611 (1965).
[CrossRef]

Cudahy, T.

T. Cudahy and E. Ramanaidou, "Measurement of the Hematite:Goethite Ratio Using Field Visible and Near-Infrared Reflectance Spectrometry in Channel Iron Deposits, Western Australia," Australian Journal of Earth Sciences 44, 411-420 (1997).
[CrossRef]

Dlugach, J.

M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999).
[CrossRef]

Egan, W.

W. Egan and T. Hilgeman, "Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering," Applied Optics 17, 245-252 (1978).
[CrossRef] [PubMed]

Emslie, A.

A. Emslie and J. Aronson, "Spectral Reflectance of Particulate Materials. 1: Theory," Applied Optics 12, 2563-2572 (1973).
[CrossRef] [PubMed]

Fechter, H.

J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983).
[CrossRef]

Fung, A.

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

Gao, B.-C.

Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
[CrossRef]

Gibbs, D.

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

Golden, D.

R. Morris and D. Golden, "Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars," Icarus 134, 1-10 (1998).
[CrossRef]

Govaerts, Y.

Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996).
[CrossRef] [PubMed]

Gray, E.

K. Coulson, G. Bouricius, and E. Gray, "Optical Reflection Properties of Natural Surfaces," Journal of Geophysical Research 70, 4601-4611 (1965).
[CrossRef]

Greenstein, J.

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophysical Journal 93, 70-83 (1941).
[CrossRef]

Hale, G.

G. Hale and M. Querry, "Optical Constants of Water in the 200-nm to 200- m Wavelength Region," Applied Optics 12, 555-563 (1973).
[CrossRef] [PubMed]

Hapke, B.

B. Hapke, "Bidirectional Reflectance Spectroscopy. 4. The Extinction Coefficient and the Opposition Effect," Icarus 67, 264-280 (1986).
[CrossRef]

B. Hapke, "Bidirectional Reflectance Spectroscopy. 1. Theory," Journal of Geophysical Research 86, 3039-3054 (1981).
[CrossRef]

B. Hapke and E. Wells, "Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations," Journal of Geophysical Research 86, 3055-3054 (1981).
[CrossRef]

Hart, J.

M. Shirazi, L. Boersma, and J. Hart, "A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale," Soil Science Society of America Journal 52, 181-190 (1988).
[CrossRef]

Henyey, L.

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophysical Journal 93, 70-83 (1941).
[CrossRef]

Hilgeman, T.

W. Egan and T. Hilgeman, "Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering," Applied Optics 17, 245-252 (1978).
[CrossRef] [PubMed]

Hosgood, B.

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

Hunt, G.

G. Hunt, J. Salisbury, and C. Lenhoff, "Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides," Modern Geology 2, 195-205 (1971).

G. Hunt and J. Salisbury, "Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals," Modern Geology 1, 283-300 (1970).

R. Vincent and G. Hunt, "Infrared Reflectance from Mat Surfaces," Applied Optics 7, 53-59 (1968).
[CrossRef] [PubMed]

Irons, J.

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

Jacquemoud, S.

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996).
[CrossRef] [PubMed]

Jacques, S.

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).

Jørgensen, B.

M. Kühl and B. Jørgensen, "The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments," Limnology and Oceanography 39, 1368-1398 (1994).
[CrossRef]

Kattawar, G.

Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
[CrossRef]

Keijzer, M.

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).

Krishnaswamy, A.

A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools  9, 31-51 (2004).

Krumbein, W.

W. Krumbein, "Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles," Journal of Sedimentary Petrology 11, 64-72 (1941).

Kühl, M.

M. Kühl and B. Jørgensen, "The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments," Limnology and Oceanography 39, 1368-1398 (1994).
[CrossRef]

Kuz’min, V.

V. Kuz’min and I. Meglinski, "Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering," Optics & Spectroscopy 97, 100-106 (2004).
[CrossRef]

Lenhoff, C.

G. Hunt, J. Salisbury, and C. Lenhoff, "Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides," Modern Geology 2, 195-205 (1971).

Leu, D.

D. Leu, "Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship," Remote Sensing of Environment 6, 169-182 (1977).
[CrossRef]

Li, Z.

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

Liu, L.

M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
[CrossRef] [PubMed]

Lobell, D.

D. Lobell and G. Asner, "Moisture Effects on Soil Reflectance," Soil Science Society of America Journal 66, 722-727 (2002).
[CrossRef]

Mackowski, D.

M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
[CrossRef] [PubMed]

Maxwell Garnett, J.

J. Maxwell Garnett, "Colours in Metal Glasses and in Metallic Films," Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203, 385-420 (1904).
[CrossRef]

Meglinski, I.

V. Kuz’min and I. Meglinski, "Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering," Optics & Spectroscopy 97, 100-106 (2004).
[CrossRef]

Mergenthaler, J.

S. Twomey, C. Bohren, and J. Mergenthaler, "Reflectance and Albedo Differences Between Wet and Dry Surfaces," Applied Optics 25, 57-84 (1986).
[CrossRef]

Mishchenko, M.

M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
[CrossRef] [PubMed]

Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
[CrossRef]

M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999).
[CrossRef]

Montealegre, L.

V. Barron and L. Montealegre, "Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory," American Journal of Science 286, 792-802 (1986).
[CrossRef]

Morris, R.

R. Morris and D. Golden, "Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars," Icarus 134, 1-10 (1998).
[CrossRef]

Muinonen, K.

T. Nousiainen, K. Muinonen, and P. Räisänen, "Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation," Journal of Geophysical Research 108, AAC 12-1-17 (2003).
[CrossRef]

Neema, D.

D. Neema, A. Shah, and A. Patel, "A Statistical Optical Model for Light Reflection and Penetration Through Sand," International Journal of Remote Sensing 8, 1209-1217 (1987).
[CrossRef]

Nickerson, D.

D. Nickerson, "History of the Munsell Color System and its Scientific Application," Journal of the Optical Society of America 30, 575-645 (1940).
[CrossRef]

Norman, J.

J. Norman, J. Welles, and E. Walter, "Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils," IEEE Transactions on Geoscience and Remote Sensing GE-23, 659-667 (1985).
[CrossRef]

Nousiainen, T.

T. Nousiainen, K. Muinonen, and P. Räisänen, "Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation," Journal of Geophysical Research 108, AAC 12-1-17 (2003).
[CrossRef]

Okin, G.

G. Okin and T. Painter, "Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces," Remote Sensing of Environment 89, 272-280 (2004).
[CrossRef]

Painter, T.

G. Okin and T. Painter, "Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces," Remote Sensing of Environment 89, 272-280 (2004).
[CrossRef]

Patel, A.

D. Neema, A. Shah, and A. Patel, "A Statistical Optical Model for Light Reflection and Penetration Through Sand," International Journal of Remote Sensing 8, 1209-1217 (1987).
[CrossRef]

Prahl, S.

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).

Querry, M.

G. Hale and M. Querry, "Optical Constants of Water in the 200-nm to 200- m Wavelength Region," Applied Optics 12, 555-563 (1973).
[CrossRef] [PubMed]

Räisänen, P.

T. Nousiainen, K. Muinonen, and P. Räisänen, "Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation," Journal of Geophysical Research 108, AAC 12-1-17 (2003).
[CrossRef]

Ramanaidou, E.

T. Cudahy and E. Ramanaidou, "Measurement of the Hematite:Goethite Ratio Using Field Visible and Near-Infrared Reflectance Spectrometry in Channel Iron Deposits, Western Australia," Australian Journal of Earth Sciences 44, 411-420 (1997).
[CrossRef]

Rea, D.

R. Shuchman and D. Rea, "Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data," Remote Sensing of Environment 11, 295-310 (1981).
[CrossRef]

Reynolds, D.

K. Coulson and D. Reynolds, "The Spectral Reflectance of Natural Surfaces," Journal of Applied Meteorology 10, 1285-1295 (1971).
[CrossRef]

Riley, N.

N. Riley, "Projection Sphericity," Journal of Sedimentary Petrology 11, 94-95 (1941).

Rokne, J.

G. Baranoski, J. Rokne, and G. Xu, "Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering," The Visual Computer 17, 506-518 (2001).
[CrossRef]

Rokne, J. G.

A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools  9, 31-51 (2004).

Salisbury, J.

G. Hunt, J. Salisbury, and C. Lenhoff, "Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides," Modern Geology 2, 195-205 (1971).

G. Hunt and J. Salisbury, "Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals," Modern Geology 1, 283-300 (1970).

Schlegel, A.

A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979).
[CrossRef]

Schmuck, G.

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

Schwertmann, U.

J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983).
[CrossRef]

Shah, A.

D. Neema, A. Shah, and A. Patel, "A Statistical Optical Model for Light Reflection and Penetration Through Sand," International Journal of Remote Sensing 8, 1209-1217 (1987).
[CrossRef]

Shirazi, M.

M. Shirazi, L. Boersma, and J. Hart, "A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale," Soil Science Society of America Journal 52, 181-190 (1988).
[CrossRef]

Shkuratov, Y.

D. Stankevich and Y. Shkuratov, "Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure," Journal of Quantitative Spectroscopy and Radiative Transfer 87, 289-296 (2004).
[CrossRef]

Shuchman, R.

R. Shuchman and D. Rea, "Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data," Remote Sensing of Environment 11, 295-310 (1981).
[CrossRef]

Silva, L.

M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985).
[CrossRef]

Singer, R.

R. Singer, "Spectral Evidence for the Mineralogy of High-Albedo Soils and Dust on Mars," Journal of Geophysical Research 87, 10,159-10,168 (1982).
[CrossRef]

Sokolik, I.

I. Sokolik and O. Toon, "Incorporation of Mineralogical Composition into Models of the Radiative Properties of Mineral Aerosol from UV to IR Wavelengths," Journal of Geophysical Research 104, 9423-9444 (1999).
[CrossRef]

Stankevich, D.

D. Stankevich and Y. Shkuratov, "Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure," Journal of Quantitative Spectroscopy and Radiative Transfer 87, 289-296 (2004).
[CrossRef]

Stoner, E.

M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985).
[CrossRef]

Tjuatja, S.

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

Toon, O.

I. Sokolik and O. Toon, "Incorporation of Mineralogical Composition into Models of the Radiative Properties of Mineral Aerosol from UV to IR Wavelengths," Journal of Geophysical Research 104, 9423-9444 (1999).
[CrossRef]

Torrent, J.

J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983).
[CrossRef]

Twindale, C.

H. Wopfner and C. Twindale, "Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia," Geologische Rundschau 77, 815-834 (1988).
[CrossRef]

Twomey, S.

S. Twomey, C. Bohren, and J. Mergenthaler, "Reflectance and Albedo Differences Between Wet and Dry Surfaces," Applied Optics 25, 57-84 (1986).
[CrossRef]

Ustin, S.

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996).
[CrossRef] [PubMed]

Vepraskas, M.

M. Vepraskas and D. Cassel, "Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties," Soil Science Society of America Journal 51, 1108-1112 (1987).
[CrossRef]

Verdebout, J.

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

Verstraete, M.

Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996).
[CrossRef] [PubMed]

Videen, G.

M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
[CrossRef] [PubMed]

Vincent, R.

R. Vincent and G. Hunt, "Infrared Reflectance from Mat Surfaces," Applied Optics 7, 53-59 (1968).
[CrossRef] [PubMed]

Voss, K.

H. Zhang and K. Voss, "Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models," Applied Optics 44, 597-610 (2005).
[CrossRef] [PubMed]

Wachter, P.

A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979).
[CrossRef]

Wadell, H.

H. Wadell, "Volume, Shape, and Roundness of Rock Particles," Journal of Geology 40, 443-451 (1932).
[CrossRef]

Walter, E.

J. Norman, J. Welles, and E. Walter, "Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils," IEEE Transactions on Geoscience and Remote Sensing GE-23, 659-667 (1985).
[CrossRef]

Ward, G.

G. Ward, "Measuring and Modeling Anisotropic Reflection," Computer Graphics 26, 262-272 (1992).
[CrossRef]

Welch, A.

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).

Welles, J.

J. Norman, J. Welles, and E. Walter, "Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils," IEEE Transactions on Geoscience and Remote Sensing GE-23, 659-667 (1985).
[CrossRef]

Wells, E.

B. Hapke and E. Wells, "Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations," Journal of Geophysical Research 86, 3055-3054 (1981).
[CrossRef]

Wolff, L.

L. Wolff, "Diffuse-Reflectance Model for Smooth Dielectric Surfaces," Journal of the Optical Society of America A (Optics, Image Science, and Vision) 11, 2956-2968 (1994).
[CrossRef]

Wopfner, H.

H. Wopfner and C. Twindale, "Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia," Geologische Rundschau 77, 815-834 (1988).
[CrossRef]

Xie, Y.

Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
[CrossRef]

Xu, G.

G. Baranoski, J. Rokne, and G. Xu, "Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering," The Visual Computer 17, 506-518 (2001).
[CrossRef]

Yang, P.

Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
[CrossRef]

Yanovitskij, E.

M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999).
[CrossRef]

Zakharova, N.

M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999).
[CrossRef]

Zhang, H.

H. Zhang and K. Voss, "Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models," Applied Optics 44, 597-610 (2005).
[CrossRef] [PubMed]

Advances in Agronomy

M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985).
[CrossRef]

American Journal of Science

V. Barron and L. Montealegre, "Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory," American Journal of Science 286, 792-802 (1986).
[CrossRef]

Applied Optics

A. Emslie and J. Aronson, "Spectral Reflectance of Particulate Materials. 1: Theory," Applied Optics 12, 2563-2572 (1973).
[CrossRef] [PubMed]

W. Egan and T. Hilgeman, "Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering," Applied Optics 17, 245-252 (1978).
[CrossRef] [PubMed]

G. Hale and M. Querry, "Optical Constants of Water in the 200-nm to 200- m Wavelength Region," Applied Optics 12, 555-563 (1973).
[CrossRef] [PubMed]

Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996).
[CrossRef] [PubMed]

R. Vincent and G. Hunt, "Infrared Reflectance from Mat Surfaces," Applied Optics 7, 53-59 (1968).
[CrossRef] [PubMed]

S. Twomey, C. Bohren, and J. Mergenthaler, "Reflectance and Albedo Differences Between Wet and Dry Surfaces," Applied Optics 25, 57-84 (1986).
[CrossRef]

H. Zhang and K. Voss, "Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models," Applied Optics 44, 597-610 (2005).
[CrossRef] [PubMed]

Astrophysical Journal

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophysical Journal 93, 70-83 (1941).
[CrossRef]

Australian Journal of Earth Sciences

T. Cudahy and E. Ramanaidou, "Measurement of the Hematite:Goethite Ratio Using Field Visible and Near-Infrared Reflectance Spectrometry in Channel Iron Deposits, Western Australia," Australian Journal of Earth Sciences 44, 411-420 (1997).
[CrossRef]

Computer Graphics

G. Ward, "Measuring and Modeling Anisotropic Reflection," Computer Graphics 26, 262-272 (1992).
[CrossRef]

Containing Papers of a Mathematical or Physical Character

J. Maxwell Garnett, "Colours in Metal Glasses and in Metallic Films," Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203, 385-420 (1904).
[CrossRef]

Geologische Rundschau

H. Wopfner and C. Twindale, "Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia," Geologische Rundschau 77, 815-834 (1988).
[CrossRef]

Icarus

R. Morris and D. Golden, "Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars," Icarus 134, 1-10 (1998).
[CrossRef]

B. Hapke, "Bidirectional Reflectance Spectroscopy. 4. The Extinction Coefficient and the Opposition Effect," Icarus 67, 264-280 (1986).
[CrossRef]

IEEE Transactions on Geoscience and Remote Sensing

Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996).
[CrossRef]

J. Norman, J. Welles, and E. Walter, "Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils," IEEE Transactions on Geoscience and Remote Sensing GE-23, 659-667 (1985).
[CrossRef]

International Journal of Remote Sensing

D. Neema, A. Shah, and A. Patel, "A Statistical Optical Model for Light Reflection and Penetration Through Sand," International Journal of Remote Sensing 8, 1209-1217 (1987).
[CrossRef]

Journal of Applied Meteorology

K. Coulson and D. Reynolds, "The Spectral Reflectance of Natural Surfaces," Journal of Applied Meteorology 10, 1285-1295 (1971).
[CrossRef]

Journal of Geology

H. Wadell, "Volume, Shape, and Roundness of Rock Particles," Journal of Geology 40, 443-451 (1932).
[CrossRef]

Journal of Geophysical Research

B. Hapke, "Bidirectional Reflectance Spectroscopy. 1. Theory," Journal of Geophysical Research 86, 3039-3054 (1981).
[CrossRef]

B. Hapke and E. Wells, "Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations," Journal of Geophysical Research 86, 3055-3054 (1981).
[CrossRef]

K. Coulson, G. Bouricius, and E. Gray, "Optical Reflection Properties of Natural Surfaces," Journal of Geophysical Research 70, 4601-4611 (1965).
[CrossRef]

R. Singer, "Spectral Evidence for the Mineralogy of High-Albedo Soils and Dust on Mars," Journal of Geophysical Research 87, 10,159-10,168 (1982).
[CrossRef]

I. Sokolik and O. Toon, "Incorporation of Mineralogical Composition into Models of the Radiative Properties of Mineral Aerosol from UV to IR Wavelengths," Journal of Geophysical Research 104, 9423-9444 (1999).
[CrossRef]

T. Nousiainen, K. Muinonen, and P. Räisänen, "Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation," Journal of Geophysical Research 108, AAC 12-1-17 (2003).
[CrossRef]

Journal of Graphics Tools

A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools  9, 31-51 (2004).

Journal of Physics C: Solid State Physics

A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979).
[CrossRef]

Journal of Quantitative Spectroscopy and Radiative Transfer

M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999).
[CrossRef]

D. Stankevich and Y. Shkuratov, "Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure," Journal of Quantitative Spectroscopy and Radiative Transfer 87, 289-296 (2004).
[CrossRef]

Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006).
[CrossRef]

Journal of Sedimentary Petrology

N. Riley, "Projection Sphericity," Journal of Sedimentary Petrology 11, 94-95 (1941).

W. Krumbein, "Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles," Journal of Sedimentary Petrology 11, 64-72 (1941).

Journal of the Optical Society of America

D. Nickerson, "History of the Munsell Color System and its Scientific Application," Journal of the Optical Society of America 30, 575-645 (1940).
[CrossRef]

Journal of the Optical Society of America A (Optics, Image Science, and Vision)

L. Wolff, "Diffuse-Reflectance Model for Smooth Dielectric Surfaces," Journal of the Optical Society of America A (Optics, Image Science, and Vision) 11, 2956-2968 (1994).
[CrossRef]

Limnology and Oceanography

M. Kühl and B. Jørgensen, "The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments," Limnology and Oceanography 39, 1368-1398 (1994).
[CrossRef]

Modern Geology

G. Hunt, J. Salisbury, and C. Lenhoff, "Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides," Modern Geology 2, 195-205 (1971).

G. Hunt and J. Salisbury, "Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals," Modern Geology 1, 283-300 (1970).

Optics & Spectroscopy

V. Kuz’min and I. Meglinski, "Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering," Optics & Spectroscopy 97, 100-106 (2004).
[CrossRef]

Optics Express

M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007).
[CrossRef] [PubMed]

Remote Sensing of Environment

D. Leu, "Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship," Remote Sensing of Environment 6, 169-182 (1977).
[CrossRef]

S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996).
[CrossRef]

R. Shuchman and D. Rea, "Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data," Remote Sensing of Environment 11, 295-310 (1981).
[CrossRef]

G. Okin and T. Painter, "Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces," Remote Sensing of Environment 89, 272-280 (2004).
[CrossRef]

J. Cierniewski, "A Model for Soil Surface Roughness Influence on the Spectral Response of Bare Soils in the Visible and Near-Infrared Range," Remote Sensing of Environment 23, 92-115 (1987).
[CrossRef]

Soil Science

J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983).
[CrossRef]

Soil Science Society of America Journal

D. Lobell and G. Asner, "Moisture Effects on Soil Reflectance," Soil Science Society of America Journal 66, 722-727 (2002).
[CrossRef]

M. Vepraskas and D. Cassel, "Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties," Soil Science Society of America Journal 51, 1108-1112 (1987).
[CrossRef]

M. Shirazi, L. Boersma, and J. Hart, "A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale," Soil Science Society of America Journal 52, 181-190 (1988).
[CrossRef]

SPIE Institute Series

S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).

The Visual Computer

G. Baranoski, J. Rokne, and G. Xu, "Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering," The Visual Computer 17, 506-518 (2001).
[CrossRef]

Other

S. Prahl, "Light Transport in Tissue," Ph.D. thesis, University of Texas at Austin (1988).

F. Pettijohn, P. Potter, and R. Siever, Sand and Sandstone, 2nd ed. (Springer-Verlag, New York, NY, 1987).
[CrossRef]

N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

J. Gerrard, Fundamentals of Soils (Routledge, New York, NY, 2000).

Soil Science Division Staff, Soil Survey Manual (Soil Conservation Service, 1993). United States Department of Agriculture Handbook 18.

P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).

A. Mottana, R. Crespi, and G. Liborio, Simon and Schuster’s Guide to Rocks and Minerals (Simon and Schuster, Inc., New York, NY, 1978).

R. Cornell and U. Schwertmann, The Iron Oxides, 2nd ed. (Wiley-VCH GmbH & Co. KGaA, Weinheim, Germany, 2003).
[CrossRef]

M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, Cambridge, 2006).

J. Snyder, Map Projections: A Working Manual (United States Government Printing Office, Washington, 1987). U.S. Geological Survey Professional Paper 1395.

W. Egan and T. Hilgeman, Optical Properties of Inhomogeneous Materials: Applications to Geology, Astronomy, Chemistry, and Engineering (Academic Press, New York, NY, 1979).
[PubMed]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, Oxford, England, 1980).

C. Gribble and A. Hall, Optical Mineralogy: Principles and Practice (Chapman & Hall, New York, NY, 1993).

J. Rinker, C. Breed, J. McCauley, and P. Corl, "Remote Sensing Field Guide — Desert," Tech. rep., U.S. Army Topographic Engineering Center, Fort Belvoir, VA (1991).

B. Kimmel, "SPLITS: A Spectral Light Transport Model for Sand," Master’s thesis, School of Computer Science, University of Waterloo (2005).

S. Ross, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, NY, 1983).

F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).

W. Tropf, M. Thomas, and T. Harris, "Properties of Crystals and Glasses," in Handbook of Optics, M. Bass, E. Van Stryland, D. Williams, and W.Wolfe, eds., vol. 2, 2nd ed., chap. 33 (McGraw-Hill, 1995).

M. Oren and S. Nayar, "Generalization of Lambert’s Reflectance Model," in Computer Graphics Proceedings, Annual Conference Series, pp. 239-246 (1994).

J. Peltoniemi, "Spectropolarised Ray-Tracing Simulations in Densely Packed Particulate Medium," Journal of Quantitative Spectroscopy and Radiative Transfer (2007). In Press, accepted, URL http://dx.doi.org/10.1016/j.jqsrt.2007.05.009.
[CrossRef]

H. Jensen, J. Legakis, and J. Dorsey, "Rendering ofWet Materials," in Proceedings of the Eurographics Workshop on Rendering, pp. 273-282 (1999).

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, United States Department of Commerce, 1977).

R. Hunter and R. Harold, The Measurement of Appearance, 2nd ed. (JohnWiley and Sons, New York, NY, 1987).

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche (An Article on Optics of Paint Layers)," Zeitschrift fur Technische Physik 12, 593-601 (1931).

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 (11)

Fig. 1.
Fig. 1.

Projection of a sand grain onto a plane. The circles indicate measurements used in the calculation of the Riley sphericity [30].

Fig. 2.
Fig. 2.

Modeled sand medium.

Fig. 3.
Fig. 3.

An example of the light propagation through a modeled sand particle. Left: The particle consists of a core and an optional coating. The small box around the ray path at the coating interface corresponds to the layer interface diagram to the right. Right: A closer look at the modeled interface between the core, coating, and/or the surrounding medium.

Fig. 4.
Fig. 4.

Prolate spheroid.

Fig. 5.
Fig. 5.

Flowchart outlining the light transport algorithm used by the SPLITS model. The dashed line indicates the main loop. The term inner boundary refers to the inside of either extended boundary, whereas the outer boundary is the outside of the surface extended boundary.

Fig. 6.
Fig. 6.

Comparisons between real and simulated sand. The solid line indicates the reflectance of the sand sample. The dashed line indicates the modeled reflectance. The parameters used and corresponding RMS errors are provided in Table 5. Top Left: TEC #10019201, Top Right: TEC #10039240, Bottom Left: TEC #13j9823, Bottom Right: TEC #19au9815.

Fig. 7.
Fig. 7.

Computer generated images showing variation of sand colour with moisture as predicted by the model. The degree of saturation varies from S=0 at the top of each image to S=1 at the bottom. From left to right, the samples are TEC #10019201, TEC #10039240, TEC #13j9823, and TEC #19au9815.

Fig. 8.
Fig. 8.

Qualitative behaviour of simulated sand. The solid line in each plot indicates the spectral response from the SPLITS model with ϑ hg h g =0.01, rhg h /(ϑ h g )=0.00, ϑ m =0.00, ζ1=0.00, ζ2=0.10, ζ3=0.90, and µ′p =µ′m =µ′c =1/3. Top Left: Varying ϑ hg , Top Right: Varying rhg , Bottom Left: Varying ϑ m , Bottom Right: Varying the texture (ζ 123). See Table 2 for symbol definitions.

Fig. 9.
Fig. 9.

Computer generated images showing variation of sand colour as various parameters are changed. The image on the left (base image) corresponds to the solid lines in Fig. 8. The remaining images correspond to the spectral responses from the SPLITS model with the same parameters as in the base image except, from left to right, ϑ hg h g =0.05, rhg h /(ϑ h g )=0.90, ϑ m =0.30, ζ3=1.0. See Table 2 for symbol definitions.

Fig. 10.
Fig. 10.

Spectral reflectance curves predicted by the SPLITS model for two of the TEC sand samples, varying the water content, expressed as the degree of saturation, S. Left: TEC #13j9823, Right: TEC #19au9815.

Fig. 11.
Fig. 11.

BRDF of simulated sand at 600nm. Three dimensional plots of the BRDF are shown on the top. Profiles along the principal plane are shown on the bottom (the dashed line indicates the incident direction). Left: Normal incidence, Middle: For light incident 30° from the normal, Right: For light incident 60° from the normal.

Tables (5)

Tables Icon

Table 1: Soil separates (particle size classes) defined by the United States Department of Agriculture [13].

Tables Icon

Table 2. A summary of the data required in the simulations, along with typical ranges and values used.

Tables Icon

Table 3. Mass fractions, µj, of each of the different classes, j, of particles. The quantities βhg and βq are unknowns, given by Eqs. (8) and (9) respectively.

Tables Icon

Table 4. Mass concentration, ϑj,ℓ, of the constituent minerals, , in each of the different classes j of particles. The quantities ϑ′k and ϑ″k are unknowns given by Eqs. (10) and (11) respectively.

Tables Icon

Table 5. The parameters used in the SPLITS model for each of the four sand samples, and the root-mean-square error (RMSE) between the actual and simulated reflectances.

Equations (27)

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

Ψ = D i D c ,
Ψ = a c .
n · n = 1 X ,
ε avg = ε m ( 1 + 3 ν i ( ε ε m ε + 2 ε m ) 1 ν i ( ε ε m ε + 2 ε m ) ) ,
ϑ Fe = ϑ h + ϑ g + ϑ m .
ϑ hg = ϑ h + ϑ g ,
r hg = ϑ h ϑ h + ϑ g .
β hg = ϑ h μ p r hg = ϑ hg μ p .
β q = μ p ϑ m β hg .
ϑ k = γ k γ h ( γ h ν coat ( 1 ϑ hg ) γ q ( 1 ν coat ) ϑ hg γ q ( 1 ν coat ) + γ k ν coat ) .
ϑ k = γ k γ g ( γ g ν coat ( 1 ϑ hg ) γ q ( 1 ν coat ) ϑ hg γ q ( 1 ν coat ) + γ k ν coat ) .
ν coat A V ( s , Ψ ) h 1 + A V ( s , Ψ ) h ,
ν coat A V ( 1 , Ψ ) h 1 + A V ( 1 , Ψ ) h .
A V ( 1 , Ψ ) = 3 ( 1 + sin 1 1 Ψ 4 Ψ 2 1 Ψ 4 ) .
γ j = ( ϑ j , γ ) 1 .
γ = ( j μ j γ j ) 1 .
ν j , = γ j γ ϑ j , ,
ν j = γ γ j μ j .
f g ( s ) = 1 b g 2 π exp ( ( log s a g ) 2 2 b g 2 ) ,
𝕱 m ( s 1 , s 2 ) = s 1 s 2 s 1 f g ( s ) ds .
d j = 1 K j log ξ ,
K j = K 1 K 2 ,
K 1 = ν j s min s max s 2 f g ( s ) ds s min s max s 1 f g ( s ) ds ,
K 2 = Ψ min Ψ max A V ( 1 , Ψ ) Φ ( Ψ ¯ , σ Ψ 2 ) ( Ψ ) d Ψ 4 Ψ min Ψ max Φ ( Ψ ¯ , σ Ψ 2 ) ( Ψ ) d Ψ .
f s ( s ) = 1 C 1 f g ( s ) s 2 ,
f Ψ ( Ψ ) = 1 C 2 A V ( 1 , Ψ ) Φ ( Ψ ¯ , σ Ψ 2 ) ( Ψ ) ,
F ( u ) = 1 2 ( 1 + eu 1 e 2 u 2 + sin 1 eu e 1 e 2 + sin 1 e ) ,

Metrics