Abstract

The bidirectional reflectance distribution function (BRDF) model developed by Torrance and Sparrow [J. Opt. Soc. Am. 57, 1105–1114 (1967)] is used to describe the specular reflection of rough surfaces. We compare this model with the BRDF measurements of four manmade surfaces with different roughnesses. The model can be used to describe the basic features of the measured BRDFs. We found that the width of the specular peak perpendicular to the principal plane decreases strongly with an increasing illumination zenith angle in the data as well as in the model. A model analysis shows that the width is approximately proportional to the cosine of the illumination angle θi, and the deviations are determined by the roughness of the surface. This relationship is accompanied by an increase in reflectance in the specular direction in the principal plane that is 1/cos θi stronger than the increase for a perfectly smooth surface.

© 2001 Optical Society of America

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References

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    [CrossRef]
  23. A. Rothkirch, G. Meister, H. Spitzer, J. Bienlein, “BRDF measurements of urban surface materials at the EGO facility using a laser source,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing, (2000), pp. 777–784.
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    [CrossRef]

2000 (3)

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “BRDF field studies for remote sensing of urban areas,” Remote Sens. Rev. 19, 37–57 (2000).
[CrossRef]

A. Rothkirch, G. Meister, B. Hosgood, H. Spitzer, J. Bienlein, “BRDF measurements on urban materials using laser light,” Remote Sens. Rev. 19, 21–35 (2000).
[CrossRef]

G. Meister, A. Rothkirch, B. Hosgood, H. Spitzer, J. Bienlein, “Error analysis for BRDF measurements at the European Goniometric Facility,” Remote Sens. Rev. 19, 111–131 (2000).
[CrossRef]

1999 (1)

K. J. Dana, B. v. Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real world surfaces,” ACM (Assoc. Comput. Mach.) Trans. Graphics 18, 1–34 (1999).
[CrossRef]

1998 (1)

1997 (1)

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

1995 (1)

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

1994 (1)

1993 (1)

1991 (1)

1990 (1)

1989 (1)

1987 (1)

1975 (1)

B. Phong, “Illumination for computer generated pictures,” Commun. ACM (Association for Computing Machinery) 18, 311–317 (1975).
[CrossRef]

1967 (1)

1954 (1)

Andreoli, G.

S. Solheim, B. Hosgood, G. Andreoli, J. Piironen, “Calibration and characterization of data from the European Goniometric Facility (EGO),” (Space Applications Institute, Joint Research Centre, Ispra, Italy, 1996), pp. 1–31.

Barnsley, M. J.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 81.

Bennett, J. M.

J. M. Bennett, L. Mattson, Introduction to Rough Surface Scattering (Optical Society of America, Washington, D.C., 1989).

Bienlein, J.

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “BRDF field studies for remote sensing of urban areas,” Remote Sens. Rev. 19, 37–57 (2000).
[CrossRef]

A. Rothkirch, G. Meister, B. Hosgood, H. Spitzer, J. Bienlein, “BRDF measurements on urban materials using laser light,” Remote Sens. Rev. 19, 21–35 (2000).
[CrossRef]

G. Meister, A. Rothkirch, B. Hosgood, H. Spitzer, J. Bienlein, “Error analysis for BRDF measurements at the European Goniometric Facility,” Remote Sens. Rev. 19, 111–131 (2000).
[CrossRef]

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “The shape of the specular peak of rough surfaces,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing (2000), pp. 852–860.

A. Rothkirch, G. Meister, H. Spitzer, J. Bienlein, “BRDF measurements of urban surface materials at the EGO facility using a laser source,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing, (2000), pp. 777–784.

Brandt, S.

S. Brandt, Data Analysis (Springer-Verlag, Berlin, 1999), pp. 1–652.

Cox, C.

Dana, K. J.

K. J. Dana, B. v. Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real world surfaces,” ACM (Assoc. Comput. Mach.) Trans. Graphics 18, 1–34 (1999).
[CrossRef]

Donnel, K. A.

Ginneken, B. v.

K. J. Dana, B. v. Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real world surfaces,” ACM (Assoc. Comput. Mach.) Trans. Graphics 18, 1–34 (1999).
[CrossRef]

B. V. Ginneken, M. Stavridi, J. J. Koenderink, “Diffuse and specular reflectance from rough surfaces,” Appl. Opt. 37, 130–139 (1998).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Lamperis, “Geometric considerations and nomenclature for reflectance,” (U. S. Department of Commerce, National Bureau of Standards, 1977), pp. 1–33.

Hosgood, B.

G. Meister, A. Rothkirch, B. Hosgood, H. Spitzer, J. Bienlein, “Error analysis for BRDF measurements at the European Goniometric Facility,” Remote Sens. Rev. 19, 111–131 (2000).
[CrossRef]

A. Rothkirch, G. Meister, B. Hosgood, H. Spitzer, J. Bienlein, “BRDF measurements on urban materials using laser light,” Remote Sens. Rev. 19, 21–35 (2000).
[CrossRef]

S. Solheim, B. Hosgood, G. Andreoli, J. Piironen, “Calibration and characterization of data from the European Goniometric Facility (EGO),” (Space Applications Institute, Joint Research Centre, Ispra, Italy, 1996), pp. 1–31.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Lamperis, “Geometric considerations and nomenclature for reflectance,” (U. S. Department of Commerce, National Bureau of Standards, 1977), pp. 1–33.

Hu, B.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Knotts, M. E.

Koenderink, J. J.

K. J. Dana, B. v. Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real world surfaces,” ACM (Assoc. Comput. Mach.) Trans. Graphics 18, 1–34 (1999).
[CrossRef]

B. V. Ginneken, M. Stavridi, J. J. Koenderink, “Diffuse and specular reflectance from rough surfaces,” Appl. Opt. 37, 130–139 (1998).
[CrossRef]

Lamperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Lamperis, “Geometric considerations and nomenclature for reflectance,” (U. S. Department of Commerce, National Bureau of Standards, 1977), pp. 1–33.

Lewis, P.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Li, X.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Mattson, L.

J. M. Bennett, L. Mattson, Introduction to Rough Surface Scattering (Optical Society of America, Washington, D.C., 1989).

Maystre, D.

Meister, G.

A. Rothkirch, G. Meister, B. Hosgood, H. Spitzer, J. Bienlein, “BRDF measurements on urban materials using laser light,” Remote Sens. Rev. 19, 21–35 (2000).
[CrossRef]

G. Meister, A. Rothkirch, B. Hosgood, H. Spitzer, J. Bienlein, “Error analysis for BRDF measurements at the European Goniometric Facility,” Remote Sens. Rev. 19, 111–131 (2000).
[CrossRef]

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “BRDF field studies for remote sensing of urban areas,” Remote Sens. Rev. 19, 37–57 (2000).
[CrossRef]

G. Meister, “Bidirectional reflectance of urban surfaces,” Ph.D. dissertation, (Universität Hamburg, Hamburg, Germany, 2000), pp. 15–78, http://www.sub.uni-hamburg.de/disse/253/diss.pdf .

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “The shape of the specular peak of rough surfaces,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing (2000), pp. 852–860.

A. Rothkirch, G. Meister, H. Spitzer, J. Bienlein, “BRDF measurements of urban surface materials at the EGO facility using a laser source,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing, (2000), pp. 777–784.

Mendez, E. R.

Michel, T. R.

Muller, J.-P.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Munk, W.

Nayar, S. K.

K. J. Dana, B. v. Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real world surfaces,” ACM (Assoc. Comput. Mach.) Trans. Graphics 18, 1–34 (1999).
[CrossRef]

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Lamperis, “Geometric considerations and nomenclature for reflectance,” (U. S. Department of Commerce, National Bureau of Standards, 1977), pp. 1–33.

Nieto-Vesperinas, M.

O’Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institut of Physics Publishing, Bristol, UK, 1992), pp. 1–250.

Oren, M.

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

Phong, B.

B. Phong, “Illumination for computer generated pictures,” Commun. ACM (Association for Computing Machinery) 18, 311–317 (1975).
[CrossRef]

Piironen, J.

S. Solheim, B. Hosgood, G. Andreoli, J. Piironen, “Calibration and characterization of data from the European Goniometric Facility (EGO),” (Space Applications Institute, Joint Research Centre, Ispra, Italy, 1996), pp. 1–31.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Lamperis, “Geometric considerations and nomenclature for reflectance,” (U. S. Department of Commerce, National Bureau of Standards, 1977), pp. 1–33.

Rothkirch, A.

G. Meister, A. Rothkirch, B. Hosgood, H. Spitzer, J. Bienlein, “Error analysis for BRDF measurements at the European Goniometric Facility,” Remote Sens. Rev. 19, 111–131 (2000).
[CrossRef]

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “BRDF field studies for remote sensing of urban areas,” Remote Sens. Rev. 19, 37–57 (2000).
[CrossRef]

A. Rothkirch, G. Meister, B. Hosgood, H. Spitzer, J. Bienlein, “BRDF measurements on urban materials using laser light,” Remote Sens. Rev. 19, 21–35 (2000).
[CrossRef]

A. Rothkirch, G. Meister, H. Spitzer, J. Bienlein, “BRDF measurements of urban surface materials at the EGO facility using a laser source,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing, (2000), pp. 777–784.

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “The shape of the specular peak of rough surfaces,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing (2000), pp. 852–860.

Saillard, M.

Sanchez-Gil, J. A.

Schaaf, C.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Solheim, S.

S. Solheim, B. Hosgood, G. Andreoli, J. Piironen, “Calibration and characterization of data from the European Goniometric Facility (EGO),” (Space Applications Institute, Joint Research Centre, Ispra, Italy, 1996), pp. 1–31.

Soto-Crespo, J. M.

Sparrow, E.

Spitzer, H.

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “BRDF field studies for remote sensing of urban areas,” Remote Sens. Rev. 19, 37–57 (2000).
[CrossRef]

A. Rothkirch, G. Meister, B. Hosgood, H. Spitzer, J. Bienlein, “BRDF measurements on urban materials using laser light,” Remote Sens. Rev. 19, 21–35 (2000).
[CrossRef]

G. Meister, A. Rothkirch, B. Hosgood, H. Spitzer, J. Bienlein, “Error analysis for BRDF measurements at the European Goniometric Facility,” Remote Sens. Rev. 19, 111–131 (2000).
[CrossRef]

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “The shape of the specular peak of rough surfaces,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing (2000), pp. 852–860.

A. Rothkirch, G. Meister, H. Spitzer, J. Bienlein, “BRDF measurements of urban surface materials at the EGO facility using a laser source,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing, (2000), pp. 777–784.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 81.

Stavridi, M.

Stover, J. C.

J. C. Stover, Optical Scattering—Measurement and Analysis (The International Society for Optical Engineering, Bellingham, Wash., 1995), 2nd ed. pp. 1–321.

Strahler, A. H.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Torrance, K.

Wanner, W.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

Wolff, L. B.

L. B. Wolff, “On the relative brightness of specular and diffuse reflection,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Seattle, (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 369–376.

ACM (Assoc. Comput. Mach.) Trans. Graphics (1)

K. J. Dana, B. v. Ginneken, S. K. Nayar, J. J. Koenderink, “Reflectance and texture of real world surfaces,” ACM (Assoc. Comput. Mach.) Trans. Graphics 18, 1–34 (1999).
[CrossRef]

Appl. Opt. (1)

Commun. ACM (Association for Computing Machinery) (1)

B. Phong, “Illumination for computer generated pictures,” Commun. ACM (Association for Computing Machinery) 18, 311–317 (1975).
[CrossRef]

Int. J. Comput. Vision (1)

M. Oren, S. K. Nayar, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vision 14, 227–251 (1995).
[CrossRef]

J. Geophys. Res. (1)

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS modis and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17143–17161 (1997).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Remote Sens. Rev. (3)

G. Meister, A. Rothkirch, B. Hosgood, H. Spitzer, J. Bienlein, “Error analysis for BRDF measurements at the European Goniometric Facility,” Remote Sens. Rev. 19, 111–131 (2000).
[CrossRef]

A. Rothkirch, G. Meister, B. Hosgood, H. Spitzer, J. Bienlein, “BRDF measurements on urban materials using laser light,” Remote Sens. Rev. 19, 21–35 (2000).
[CrossRef]

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “BRDF field studies for remote sensing of urban areas,” Remote Sens. Rev. 19, 37–57 (2000).
[CrossRef]

Other (11)

S. Solheim, B. Hosgood, G. Andreoli, J. Piironen, “Calibration and characterization of data from the European Goniometric Facility (EGO),” (Space Applications Institute, Joint Research Centre, Ispra, Italy, 1996), pp. 1–31.

G. Meister, “Bidirectional reflectance of urban surfaces,” Ph.D. dissertation, (Universität Hamburg, Hamburg, Germany, 2000), pp. 15–78, http://www.sub.uni-hamburg.de/disse/253/diss.pdf .

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Institut of Physics Publishing, Bristol, UK, 1992), pp. 1–250.

J. C. Stover, Optical Scattering—Measurement and Analysis (The International Society for Optical Engineering, Bellingham, Wash., 1995), 2nd ed. pp. 1–321.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 81.

J. M. Bennett, L. Mattson, Introduction to Rough Surface Scattering (Optical Society of America, Washington, D.C., 1989).

A. Rothkirch, G. Meister, H. Spitzer, J. Bienlein, “BRDF measurements of urban surface materials at the EGO facility using a laser source,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing, (2000), pp. 777–784.

S. Brandt, Data Analysis (Springer-Verlag, Berlin, 1999), pp. 1–652.

L. B. Wolff, “On the relative brightness of specular and diffuse reflection,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Seattle, (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 369–376.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Lamperis, “Geometric considerations and nomenclature for reflectance,” (U. S. Department of Commerce, National Bureau of Standards, 1977), pp. 1–33.

G. Meister, A. Rothkirch, H. Spitzer, J. Bienlein, “The shape of the specular peak of rough surfaces,” in Proceedings of the Nineteenth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), 16–23 July 2000, Amsterdam, Vol. 33 of International Archives of Photogrammetry and Remote Sensing (2000), pp. 852–860.

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Fig. 1
Fig. 1

Fresnel reflectance for unpolarized illumination as a function of the illumination angle for different parameters n, k: solid curve, Fresnel reflectance for k = 0 and n = 1.5; crosses, Fresnel reflectance for k = 0.4 and n = 1.35, normalized to the value θ i = 0° of the solid curve; dotted curve, Fresnel reflectance for k = 0.2, n = 1.6; stars, Fresnel reflectance for k = 0.55, n = 1.4, normalized to the value of the dotted curve at θ i = 0°. It can be seen that different parameters k produce a very similar Fresnel reflectance if the index of refraction n and the specular intensity parameter t 1 are adjusted.

Fig. 2
Fig. 2

BRDF of the samples at different illumination angles θ i as a function of viewing zenith angle θ r in the forward-scattering direction (φ = 180°): stars, measured values; solid curves, TS model predictions from the parameters in Table 1. The samples have specular peaks of different intensity and width (widest for roof tile, narrowest for aluminum). The vertical bars within the stars show the measurement error. The shift of the maximum of the specular peak toward higher zenith angles can be seen especially well at illumination angle θ i = 50°.

Fig. 3
Fig. 3

BRDF of the samples at constant zenith angles θ i , θ r as a function of azimuth angle φ: stars, measured values; solid curve, TS model predictions with the parameters in Table 1; vertical bars within the stars, measurement error. The fourth column shows the measured specular peak only (i.e., the measured values minus the coefficient t 0) normalized to its maximum value: solid curve, θ i = θ r = 30°; dashed curve, θ i = θ r = 50°; dotted curve, θ i = 65°, θ r = 70°. The azimuthal width of the specular peak decreases strongly with increasing zenith angles.

Fig. 4
Fig. 4

Derivative of α [Eq. (3)] with respect to φ as a function of the illumination zenith angle θ i with θ r = θ i and φ increasing from 180°.

Fig. 5
Fig. 5

Solid curve, BRF in the specular direction (θ r = θ i , φ = 180°) for a perfectly smooth surface; dashed curve, BRF in the specular direction for a rough surface, normalized to the BRF value of the smooth surface at θ i = 0°. The Fresnel reflectance is determined by n = 1.5 and k = 0.25 in both cases.

Fig. 6
Fig. 6

Thin lines, unit sphere and the light path for perfect specular direction (θ i = θ r , φ = 180°); thick curve, angles perpendicular to the principal plane for calculating the FWHM (=2β) (see text). The thick curve runs on the unit sphere, connecting the specular direction and (θ = 90°, φ = 90°).

Fig. 7
Fig. 7

FWHM of the specular peak perpendicular to the principal plane as a function of the illumination angle θ i for the four samples derived from the TS model.

Fig. 8
Fig. 8

FWHM perpendicular to the principal plane (Fig. 7) divided by the cosine of the illumination angle θ i normalized to the value at θ i = 0° as a function of θ i for the four samples derived from the TS model. The FWHM decreases proportionally to the cosine of θ i for the concrete tiles and the aluminum sample, with deviations as great as 5% for the concrete tiles and as great as only 2.5% for the aluminum sample. The very rough sample roof tile shows much stronger deviations.

Fig. 9
Fig. 9

FWHM perpendicular to the principal plane (Fig. 7) divided by the cosine of the illumination angle θ i normalized to the value at θ i = 0° (Fig. 8) as a function of θ i for the Cox–Munk model. The FWHM decreases in proportional to the cosine of θ i with deviations of only 2%, which are due to numerical uncertainties.

Tables (1)

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Table 1 Parameter Obtained from Fitting the TS Model [Eq. (1)] to the BRDF Data of the Respective Sample at a Wavelength of 660 nm

Equations (8)

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frTS=t0+t1frspec, frspec=Fθi, θr, φ, n, kcos θi cos θr Gθi, θr, φexp-w2α2,
Pα  exp-w2α2,
cos α=cos θi+cos θrcos φ sin θr+sin θi2+sin2 φ sin2 θr+cos θi+cos θr2-1/2.
χ2=iNfr,imeasured-fr,imodeled2σi2,
Fk+σk=Fk+δFδk σk,
α=θr-θi2.
BRFωi, ωr=πΩiΩrωiωr frθi, ϕi, θr, ϕrdΩrdΩi.
BRFπBRDF=π LrE0 cos θi=π L0Fn, k, θiπL0 cos θi=Fn, k, θicos θi,

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