Abstract

By means of a specially constructed portable goniophotometer the directional reflectance of numerous surfaces of snow was measured during the winter of 1951–1952. While some samples reflected more nearly diffusely than others, all showed much specular reflection at high angles of incidence. An approximate theory of the specular reflection is given and its results compared with experiment. The experimental fact that the angle of maximum reflectance is greater than the angle of incidence is explained by the theory.

© 1952 Optical Society of America

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References

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  1. R. W. Gerdel, Trans. Am. Geophys. Union 29, 366 (1948).
    [Crossref]
  2. R. W. Gerdel, Trans. Am. Geophys. Union (N.V.) Part 1, 118 (1944).
    [Crossref]
  3. A. Gershun, J. Opt. Soc. Am. 35, 162 (1945).
    [Crossref]
  4. E. O. Hulbert, J. Opt. Soc. Am. 17, 23 (1928).
    [Crossref]
  5. I. F. Hand and R. E. Lundquist, Monthly Weather Rev. 70, 22 (1942).
  6. N. N. Kalitin, Monthly Weather Rev. 58, 59 (1930).
    [Crossref]
  7. U. Nakaya and K. Hasikura, J. Faculty of Sci., Hokkaido Imperial University 1, 63 (1934). This paper has fourteen beautiful plates of snowflakes.
  8. Klein, Pearce, and Gold, Method of Measuring the Significant Characteristics of a Snow Cover, , November, 1950 (N.R.C. No. 2269).
  9. C. Seligman, Snow Structure and Ski Fields (MacMillan and Company, Ltd., London, 1936).
  10. A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 27.
  11. See Rayleigh, Phil. Mag. 33, 1–18 (1892).
    [Crossref]

1948 (1)

R. W. Gerdel, Trans. Am. Geophys. Union 29, 366 (1948).
[Crossref]

1945 (1)

1944 (1)

R. W. Gerdel, Trans. Am. Geophys. Union (N.V.) Part 1, 118 (1944).
[Crossref]

1942 (1)

I. F. Hand and R. E. Lundquist, Monthly Weather Rev. 70, 22 (1942).

1934 (1)

U. Nakaya and K. Hasikura, J. Faculty of Sci., Hokkaido Imperial University 1, 63 (1934). This paper has fourteen beautiful plates of snowflakes.

1930 (1)

N. N. Kalitin, Monthly Weather Rev. 58, 59 (1930).
[Crossref]

1928 (1)

1892 (1)

See Rayleigh, Phil. Mag. 33, 1–18 (1892).
[Crossref]

Gerdel, R. W.

R. W. Gerdel, Trans. Am. Geophys. Union 29, 366 (1948).
[Crossref]

R. W. Gerdel, Trans. Am. Geophys. Union (N.V.) Part 1, 118 (1944).
[Crossref]

Gershun, A.

Gold,

Klein, Pearce, and Gold, Method of Measuring the Significant Characteristics of a Snow Cover, , November, 1950 (N.R.C. No. 2269).

Hand, I. F.

I. F. Hand and R. E. Lundquist, Monthly Weather Rev. 70, 22 (1942).

Hardy, A. C.

A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 27.

Hasikura, K.

U. Nakaya and K. Hasikura, J. Faculty of Sci., Hokkaido Imperial University 1, 63 (1934). This paper has fourteen beautiful plates of snowflakes.

Hulbert, E. O.

Kalitin, N. N.

N. N. Kalitin, Monthly Weather Rev. 58, 59 (1930).
[Crossref]

Klein,

Klein, Pearce, and Gold, Method of Measuring the Significant Characteristics of a Snow Cover, , November, 1950 (N.R.C. No. 2269).

Lundquist, R. E.

I. F. Hand and R. E. Lundquist, Monthly Weather Rev. 70, 22 (1942).

Nakaya, U.

U. Nakaya and K. Hasikura, J. Faculty of Sci., Hokkaido Imperial University 1, 63 (1934). This paper has fourteen beautiful plates of snowflakes.

Pearce,

Klein, Pearce, and Gold, Method of Measuring the Significant Characteristics of a Snow Cover, , November, 1950 (N.R.C. No. 2269).

Perrin, F. H.

A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 27.

Rayleigh,

See Rayleigh, Phil. Mag. 33, 1–18 (1892).
[Crossref]

Seligman, C.

C. Seligman, Snow Structure and Ski Fields (MacMillan and Company, Ltd., London, 1936).

J. Faculty of Sci., Hokkaido Imperial University (1)

U. Nakaya and K. Hasikura, J. Faculty of Sci., Hokkaido Imperial University 1, 63 (1934). This paper has fourteen beautiful plates of snowflakes.

J. Opt. Soc. Am. (2)

Monthly Weather Rev. (2)

I. F. Hand and R. E. Lundquist, Monthly Weather Rev. 70, 22 (1942).

N. N. Kalitin, Monthly Weather Rev. 58, 59 (1930).
[Crossref]

Phil. Mag. (1)

See Rayleigh, Phil. Mag. 33, 1–18 (1892).
[Crossref]

Trans. Am. Geophys. Union (1)

R. W. Gerdel, Trans. Am. Geophys. Union 29, 366 (1948).
[Crossref]

Trans. Am. Geophys. Union (N.V.) (1)

R. W. Gerdel, Trans. Am. Geophys. Union (N.V.) Part 1, 118 (1944).
[Crossref]

Other (3)

Klein, Pearce, and Gold, Method of Measuring the Significant Characteristics of a Snow Cover, , November, 1950 (N.R.C. No. 2269).

C. Seligman, Snow Structure and Ski Fields (MacMillan and Company, Ltd., London, 1936).

A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 27.

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Figures (6)

Fig. 1
Fig. 1

Cross section of goniophotometer.

Fig. 2
Fig. 2

Goniophotometer in use in the snow, showing tripod and toboggans.

Fig. 3
Fig. 3

Measured luminance of a snow surface compared with that of a perfect diffuser, both illuminated by a source placed at angles of incidence 0°, 30°, 45°, 60°, 75°. The angle of incidence corresponding to each curve is shown by an arrow. Snow surface ——. Perfect diffuser – – – –. (a) Wind-packed snow. (b) New snow fallen in calm. (c) Rain crust. (d) Settling snow. (e) Surface hoar. (f) Glazed rain crust.

Fig. 4
Fig. 4

Geometry of reflection of a ray from a facet.

Fig. 5
Fig. 5

Masking of a facet by adjacent facets.

Fig. 6
Fig. 6

Calculated luminance of a snow surface compared with that of a perfect diffuser, both illuminated by a source placed at angles of incidence 0°, 30°, 45°, 60°, 75°. The angle of incidence corresponding to each curve is shown by an arrow. Snow surface ——. Perfect diffuser – – – –. (a) h=0.01. (b) h=0.02. (c) h=0.03. (d) h=0.30.

Equations (23)

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B = B 0 cos θ ,
β = ( π / 2 ) - θ 1 - tan - 1 [ cotan θ 1 ( cos 2 δ - sin 2 δ ) ] , β = sin - 1 [ cos θ 1 sin 2 δ ] .
α = cos - 1 ( cos α cos δ ) .
d N = C exp ( - h 2 α 2 ) d α .
C exp ( - h 2 α 2 ) a cos α d α = 1 ,
I 0 F ( n , α + θ ) a cos ( α + θ ) r 1 2 ,
F = sin 2 ( α + θ - r ) 2 sin 2 ( α + θ + r ) + tan 2 ( α + θ - r ) 2 tan 2 ( α + θ + r ) , r 1 = sin - [ sin ( α + θ ) n ]             ( n = 1.31 ) .
I 0 F ( α + θ ) a cos ( α + θ ) r 1 2 · Δ N d ω ,
Δ N d ω = 1 cos r [ α α + γ / 2 C exp ( - h 2 α 2 ) d α ] · p ,
0 γ / 2 C a exp ( - h 2 α 2 ) cos α d α ,
m 2 = l 2 sin 2 α + n 2 - 2 n l sin α cos r , n 2 = ( l - m ) 2 + l 2 cos 2 α - 2 ( l - m ) l cos 2 α .
m = l sin 2 α sin 2 r ± [ l 2 sin 4 α sin 4 r - ( sin 2 α - cos 2 r ) l 2 sin 2 α sin 2 r ] 1 2 sin 2 α - cos 2 r ,
M ( α , r ) = l - m l = 1 - 1 - ( 1 - A ) 1 2 A ,
A = sin 2 α - cos 2 r sin 2 α sin 2 r             for             sin 2 α cos 2 r .
F p cos r [ α α + γ / 2 C exp ( - h 2 α 2 ) d α ] · I 0 F ( α + θ ) a cos ( α + θ ) r 1 2 · M ( α , r ) ,
p = c 0 γ / 2 C a cos α exp ( - n 2 α 2 ) d α             [ c constant ] .
F c 2 cos r · γ C a exp [ - h 2 ( r - θ ) 2 / 4 ] · α = 0 α = γ / 2 C a cos α exp ( - h 2 α 2 ) Δ α · I 0 F ( α + θ ) cos ( α + θ ) r 1 2 · M ( α , r ) .
B = c 2 cos r · C a exp [ - h 2 ( r + θ ) 2 / 4 ] · [ 1 γ α = 0 α = γ / 2 C a exp ( - h 2 α 2 ) Δ α ] · I 0 F [ 1 2 ( r - θ ) ] cos [ 1 2 ( r - θ ) ] r 1 2 · M [ 1 2 ( r + θ ) , r ] + B 0 cos θ .
lim r π / 2 M ( α 1 r ) cos r .
M ( α 1 r ) cos r = q l cos r = q [ ( l - q ) 2 cos 2 α + q 2 sin 2 α ] 1 2 l q sin α = [ ( l - q ) 2 cos 2 α + q 2 sin 2 α ] 1 2 l sin α ;
lim r π / 2 M ( α 1 r ) cos r = ( l 2 cos 2 α ) 1 2 l sin α = cot α .
B B = F ( n , α - θ ) F ( n , α - θ ) ,
F ( n , α - θ ) = sin 2 ( α - θ - r ) sin 2 ( α - θ + r ) ,             r = sin - 1 ( sin ( α - θ ) n ) F ( n , α - θ ) = tan 2 ( α - θ - r ) tan 2 ( α - θ + r ) ,             n = 1.31 for ice .