Abstract

Reflection measurements were performed on dry and moistened sand grains and glass spheres, respectively. A simple model for determining the water content is proposed from looking at the reflection distribution in the plane of incidence. The model is a combination of two sheared cosine-functions and consists of only two parameters. One parameter controls whether the reflection is mainly in the forward or backward direction. The former is true when the water content is high and the latter is true when the material is dry. The other parameter gives an idea of the homogeneity of the material.

© 2013 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, 1960), pp. 1–287.
  2. B. Hapke, “Bidirectional reflectance spectroscopy 1. Theory,” J. Geophys. Res. 86, 3039–3054 (1981).
    [CrossRef]
  3. B. Hapke, “Bidirectional reflectance spectroscopy: 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
    [CrossRef]
  4. B. Hapke, “Bidirectional reflectance spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
    [CrossRef]
  5. B. Hapke, “Bidirectional reflectance spectroscopy 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
    [CrossRef]
  6. B. Hapke, “Bidirectional reflectance spectroscopy 6. Effects of porosity,” Icarus 195, 918–926 (2008).
    [CrossRef]
  7. J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
    [CrossRef]
  8. M. K. Shepard and P. Helfenstein, “A test of the Hapke photometric model,” J. Geophys. Res. 112, E03001 (2007).
    [CrossRef]
  9. Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
    [CrossRef]
  10. B. Hapke, “Comment on a critical assessment of the Hapke photometric model by Y. Shkuratov et al.” J. Quant. Spectrosc. Radiat. Transfer 116, 184–190 (2013).
    [CrossRef]
  11. Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
    [CrossRef]
  12. S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).
  13. S. Jacquemoud, F. Baret, and J. F. Hanocq, “Modeling spectral and bidirectional soil reflectance,” Remote Sens. Environ. 41, 123–132 (1992).
    [CrossRef]
  14. G.-J. Yang, C.-J. Zhao, W.-J. Huang, and J.-H. Wang, “Extension of the Hapke bidirectional reflectance model to retrieve soil water content,” Hydrol. Earth Syst. Sci. 15, 2317–2326 (2011).
    [CrossRef]
  15. J. Casselgren, M. Sjödahl, and J. LeBlanc, “Angular spectral response from covered asphalt,” Appl. Opt. 46, 4277–4288 (2007).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), p. 195.
  17. S. A. Twomey, C. F. Bohren, and J. L. Mergenthaler, “Reflectance and albedo differences between wet and dry surfaces,” Appl. Opt. 25, 431–437 (1986).
    [CrossRef]

2013 (2)

B. Hapke, “Comment on a critical assessment of the Hapke photometric model by Y. Shkuratov et al.” J. Quant. Spectrosc. Radiat. Transfer 116, 184–190 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

2012 (1)

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

2011 (1)

G.-J. Yang, C.-J. Zhao, W.-J. Huang, and J.-H. Wang, “Extension of the Hapke bidirectional reflectance model to retrieve soil water content,” Hydrol. Earth Syst. Sci. 15, 2317–2326 (2011).
[CrossRef]

2008 (1)

B. Hapke, “Bidirectional reflectance spectroscopy 6. Effects of porosity,” Icarus 195, 918–926 (2008).
[CrossRef]

2007 (2)

M. K. Shepard and P. Helfenstein, “A test of the Hapke photometric model,” J. Geophys. Res. 112, E03001 (2007).
[CrossRef]

J. Casselgren, M. Sjödahl, and J. LeBlanc, “Angular spectral response from covered asphalt,” Appl. Opt. 46, 4277–4288 (2007).
[CrossRef]

2004 (1)

J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
[CrossRef]

2002 (1)

B. Hapke, “Bidirectional reflectance spectroscopy 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[CrossRef]

1992 (1)

S. Jacquemoud, F. Baret, and J. F. Hanocq, “Modeling spectral and bidirectional soil reflectance,” Remote Sens. Environ. 41, 123–132 (1992).
[CrossRef]

1986 (2)

S. A. Twomey, C. F. Bohren, and J. L. Mergenthaler, “Reflectance and albedo differences between wet and dry surfaces,” Appl. Opt. 25, 431–437 (1986).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

1984 (1)

B. Hapke, “Bidirectional reflectance spectroscopy: 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

1981 (1)

B. Hapke, “Bidirectional reflectance spectroscopy 1. Theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

1975 (1)

S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).

Baret, F.

S. Jacquemoud, F. Baret, and J. F. Hanocq, “Modeling spectral and bidirectional soil reflectance,” Remote Sens. Environ. 41, 123–132 (1992).
[CrossRef]

Bohren, C. F.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), p. 195.

Casselgren, J.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960), pp. 1–287.

Hale, A. S.

J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
[CrossRef]

Hanocq, J. F.

S. Jacquemoud, F. Baret, and J. F. Hanocq, “Modeling spectral and bidirectional soil reflectance,” Remote Sens. Environ. 41, 123–132 (1992).
[CrossRef]

Hapke, B.

B. Hapke, “Comment on a critical assessment of the Hapke photometric model by Y. Shkuratov et al.” J. Quant. Spectrosc. Radiat. Transfer 116, 184–190 (2013).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 6. Effects of porosity,” Icarus 195, 918–926 (2008).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy: 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 1. Theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

Hapke, B. W.

J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
[CrossRef]

Helfenstein, P.

M. K. Shepard and P. Helfenstein, “A test of the Hapke photometric model,” J. Geophys. Res. 112, E03001 (2007).
[CrossRef]

Huang, W.-J.

G.-J. Yang, C.-J. Zhao, W.-J. Huang, and J.-H. Wang, “Extension of the Hapke bidirectional reflectance model to retrieve soil water content,” Hydrol. Earth Syst. Sci. 15, 2317–2326 (2011).
[CrossRef]

Idso, S. B.

S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).

Jackson, R. D.

S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).

Jacquemoud, S.

S. Jacquemoud, F. Baret, and J. F. Hanocq, “Modeling spectral and bidirectional soil reflectance,” Remote Sens. Environ. 41, 123–132 (1992).
[CrossRef]

Kaydash, V.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

Kimball, B. A.

S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).

Korokhin, V.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

LeBlanc, J.

Mergenthaler, J. L.

Nakayama, F. S.

S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).

Nelson, R. M.

J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
[CrossRef]

Petrov, D.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

Piatek, J. L.

J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
[CrossRef]

Reginato, R. J.

S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).

Shepard, M. K.

M. K. Shepard and P. Helfenstein, “A test of the Hapke photometric model,” J. Geophys. Res. 112, E03001 (2007).
[CrossRef]

Shkuratov, Y.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

Sjödahl, M.

Smythe, W. D.

J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
[CrossRef]

Stankevich, D.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

Twomey, S. A.

Velikodsky, Y.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

Videen, G.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

Wang, J.-H.

G.-J. Yang, C.-J. Zhao, W.-J. Huang, and J.-H. Wang, “Extension of the Hapke bidirectional reflectance model to retrieve soil water content,” Hydrol. Earth Syst. Sci. 15, 2317–2326 (2011).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), p. 195.

Yang, G.-J.

G.-J. Yang, C.-J. Zhao, W.-J. Huang, and J.-H. Wang, “Extension of the Hapke bidirectional reflectance model to retrieve soil water content,” Hydrol. Earth Syst. Sci. 15, 2317–2326 (2011).
[CrossRef]

Zhao, C.-J.

G.-J. Yang, C.-J. Zhao, W.-J. Huang, and J.-H. Wang, “Extension of the Hapke bidirectional reflectance model to retrieve soil water content,” Hydrol. Earth Syst. Sci. 15, 2317–2326 (2011).
[CrossRef]

Zubko, E.

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

Appl. Opt. (2)

Hydrol. Earth Syst. Sci. (1)

G.-J. Yang, C.-J. Zhao, W.-J. Huang, and J.-H. Wang, “Extension of the Hapke bidirectional reflectance model to retrieve soil water content,” Hydrol. Earth Syst. Sci. 15, 2317–2326 (2011).
[CrossRef]

Icarus (5)

B. Hapke, “Bidirectional reflectance spectroscopy: 3. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy 6. Effects of porosity,” Icarus 195, 918–926 (2008).
[CrossRef]

J. L. Piatek, B. W. Hapke, R. M. Nelson, W. D. Smythe, and A. S. Hale, “Scattering properties of planetary regolith analogs,” Icarus 171, 531–545 (2004).
[CrossRef]

J. Appl. Meteorol. (1)

S. B. Idso, R. D. Jackson, R. J. Reginato, B. A. Kimball, and F. S. Nakayama, “The dependence of bare soil albedo on soil water content,” J. Appl. Meteorol. 14, 109–113 (1975).

J. Geophys. Res. (2)

B. Hapke, “Bidirectional reflectance spectroscopy 1. Theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

M. K. Shepard and P. Helfenstein, “A test of the Hapke photometric model,” J. Geophys. Res. 112, E03001 (2007).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (3)

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “A critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 113, 2431–2456 (2012).
[CrossRef]

B. Hapke, “Comment on a critical assessment of the Hapke photometric model by Y. Shkuratov et al.” J. Quant. Spectrosc. Radiat. Transfer 116, 184–190 (2013).
[CrossRef]

Y. Shkuratov, V. Kaydash, V. Korokhin, Y. Velikodsky, D. Petrov, E. Zubko, D. Stankevich, and G. Videen, “Response to the comment by B. Hapke on a critical assessment of the Hapke photometric model,” J. Quant. Spectrosc. Radiat. Transfer 116, 191–195 (2013).
[CrossRef]

Remote Sens. Environ. (1)

S. Jacquemoud, F. Baret, and J. F. Hanocq, “Modeling spectral and bidirectional soil reflectance,” Remote Sens. Environ. 41, 123–132 (1992).
[CrossRef]

Other (2)

S. Chandrasekhar, Radiative Transfer (Dover, 1960), pp. 1–287.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), p. 195.

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Figures (4)

Fig. 1.
Fig. 1.

Experimental setup. The light source is fiber coupled and the light passes through an aperture (ap) and a collimating lens (L). θ is the illumination angle and α is the measurement angle, given as positive on the side of the illumination and negative on the opposite side of the surface normal.

Fig. 2.
Fig. 2.

Four experimentally achieved reflection distributions compared to a cosine function (solid line). The circles and the diamonds are measurements on dry sand, the circles using an illumination angle of 40° and the diamonds using an illumination angle of 60°. The square measurements are done on moistened sand using an illumination angle of 40°; the closed squares have lower water content than the open ones.

Fig. 3.
Fig. 3.

Example of two sheared cosine functions and the combination of them. In the figure the shape function denoted by squares is C and the one denoted by circles is C+; in this case the illumination angle was θ=40° and γ=1.0. The function denoted by “x” is the combination of C and C+ according to Eq. (1) using n=0.7.

Fig. 4.
Fig. 4.

Measurement result from (a) sand 40°, (b) sand 60°, (c) glass 40°, and (d) glass 60°. In all figures the measurements at the top are the results from the measurements on the dry surface, i.e. surfaces with water content 0%. The other results are from measurements done on surfaces with higher water content according to the figures.

Tables (1)

Tables Icon

Table 1. Model Parameters

Equations (3)

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

r(α,θ)=n·C+(1n)·C+,
C±=S±[αcosα],
S±=[1±γθ01],

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