Abstract

The directional characteristics of the thermal emissivity and reflectivity of a model water surface roughened by wind are analytically studied. The study of emissivity and reflectivity is of importance for accurate measurement of the temperature distribution of a wind-roughened water surface by infrared thermal imaging. Our statistical model of a water surface assumes that the surface displacements are a two-dimensional Gaussian random process whose spectrum is specified by the Joint North Sea Wave Project (JONSWAP) wave spectral model. The effective emissivity and the effective bistatic reflectivity of the water surface in the presence of the shadowing effect are derived. They can be determined by three external parameters: wind velocity, wind direction, and wind fetch. Numerical results show that, as the surface wave grows, the effective emissivity tends to increase at distances far from the detector. This fact indicates that the measurable distance for a wind-roughened water surface may be extended somewhat farther than that for a flat water surface.

© 1994 Optical Society of America

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References

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  1. K. Yoshimori, K. Itoh, Y. Ichioka, “Statistical formulation for an inhomogeneous random water surface: a basis for optical remote sensing of oceans,” J. Opt. Soc. Am. A 11, 723–730 (1994).
    [CrossRef]
  2. D. E. Hasselmann, M. Dunckel, J. A. Ewing, “Directional wave spectra observed during JONSWAP 1973,” J. Phys. Oceanogr. 10, 1264–1280 (1980).
    [CrossRef]
  3. N. H. S. Long, C. T. Y. Yuen, L. F. Bliven, “A unified two-parameter wave spectral model for a general sea state,” J. Fluid Mech. 112, 203–224 (1981).
    [CrossRef]
  4. W. J. Pierson, L. Moskowitz, “A proposed spectral form for fully developed wind sea based on the similarity theory of S. A. Kitaigorodskii,” J. Geophys. Res. 69, 5181–5190 (1956).
    [CrossRef]
  5. H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
    [CrossRef]
  6. B. G. Smith, “Lunar surface roughness: shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
    [CrossRef]
  7. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15668–671 (1967).
    [CrossRef]
  8. See, for example, A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surface,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), p. 1.See also A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
    [CrossRef]
  9. See, for example, R. Siegel, J. R. Howell, Thermal Radiation and Heat Transfer (Hemisphere, New York, 1981), p. 57.
  10. B. Kindsman, Wind Waves (Prentice-Hall, Englewood Cliffs, N.J., 1965), p. 344.
  11. Ref. 10, p. 399.
  12. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surface (Pergamon, Oxford, 1979), p. 297.
  13. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  14. Ref. 6, p. 44.

1994 (1)

1981 (1)

N. H. S. Long, C. T. Y. Yuen, L. F. Bliven, “A unified two-parameter wave spectral model for a general sea state,” J. Fluid Mech. 112, 203–224 (1981).
[CrossRef]

1980 (2)

D. E. Hasselmann, M. Dunckel, J. A. Ewing, “Directional wave spectra observed during JONSWAP 1973,” J. Phys. Oceanogr. 10, 1264–1280 (1980).
[CrossRef]

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

1973 (1)

1967 (2)

B. G. Smith, “Lunar surface roughness: shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15668–671 (1967).
[CrossRef]

1956 (1)

W. J. Pierson, L. Moskowitz, “A proposed spectral form for fully developed wind sea based on the similarity theory of S. A. Kitaigorodskii,” J. Geophys. Res. 69, 5181–5190 (1956).
[CrossRef]

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surface (Pergamon, Oxford, 1979), p. 297.

Bliven, L. F.

N. H. S. Long, C. T. Y. Yuen, L. F. Bliven, “A unified two-parameter wave spectral model for a general sea state,” J. Fluid Mech. 112, 203–224 (1981).
[CrossRef]

Dunckel, M.

D. E. Hasselmann, M. Dunckel, J. A. Ewing, “Directional wave spectra observed during JONSWAP 1973,” J. Phys. Oceanogr. 10, 1264–1280 (1980).
[CrossRef]

Ewing, J. A.

D. E. Hasselmann, M. Dunckel, J. A. Ewing, “Directional wave spectra observed during JONSWAP 1973,” J. Phys. Oceanogr. 10, 1264–1280 (1980).
[CrossRef]

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surface (Pergamon, Oxford, 1979), p. 297.

Hale, G. M.

Hasselmann, D. E.

D. E. Hasselmann, M. Dunckel, J. A. Ewing, “Directional wave spectra observed during JONSWAP 1973,” J. Phys. Oceanogr. 10, 1264–1280 (1980).
[CrossRef]

Honda, T.

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Howell, J. R.

See, for example, R. Siegel, J. R. Howell, Thermal Radiation and Heat Transfer (Hemisphere, New York, 1981), p. 57.

Ichioka, Y.

Ishimaru, A.

See, for example, A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surface,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), p. 1.See also A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

Itoh, K.

Kindsman, B.

B. Kindsman, Wind Waves (Prentice-Hall, Englewood Cliffs, N.J., 1965), p. 344.

Long, N. H. S.

N. H. S. Long, C. T. Y. Yuen, L. F. Bliven, “A unified two-parameter wave spectral model for a general sea state,” J. Fluid Mech. 112, 203–224 (1981).
[CrossRef]

Mitsuyasu, H.

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Mizuno, S.

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Moskowitz, L.

W. J. Pierson, L. Moskowitz, “A proposed spectral form for fully developed wind sea based on the similarity theory of S. A. Kitaigorodskii,” J. Geophys. Res. 69, 5181–5190 (1956).
[CrossRef]

Ohkutsu, M.

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Pierson, W. J.

W. J. Pierson, L. Moskowitz, “A proposed spectral form for fully developed wind sea based on the similarity theory of S. A. Kitaigorodskii,” J. Geophys. Res. 69, 5181–5190 (1956).
[CrossRef]

Querry, M. R.

Rikiishi, K.

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Siegel, R.

See, for example, R. Siegel, J. R. Howell, Thermal Radiation and Heat Transfer (Hemisphere, New York, 1981), p. 57.

Smith, B. G.

B. G. Smith, “Lunar surface roughness: shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15668–671 (1967).
[CrossRef]

Suhara, T.

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Tasi, F.

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Yoshimori, K.

Yuen, C. T. Y.

N. H. S. Long, C. T. Y. Yuen, L. F. Bliven, “A unified two-parameter wave spectral model for a general sea state,” J. Fluid Mech. 112, 203–224 (1981).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15668–671 (1967).
[CrossRef]

J. Fluid Mech. (1)

N. H. S. Long, C. T. Y. Yuen, L. F. Bliven, “A unified two-parameter wave spectral model for a general sea state,” J. Fluid Mech. 112, 203–224 (1981).
[CrossRef]

J. Geophys. Res. (2)

W. J. Pierson, L. Moskowitz, “A proposed spectral form for fully developed wind sea based on the similarity theory of S. A. Kitaigorodskii,” J. Geophys. Res. 69, 5181–5190 (1956).
[CrossRef]

B. G. Smith, “Lunar surface roughness: shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. Oceanogr. (2)

D. E. Hasselmann, M. Dunckel, J. A. Ewing, “Directional wave spectra observed during JONSWAP 1973,” J. Phys. Oceanogr. 10, 1264–1280 (1980).
[CrossRef]

H. Mitsuyasu, F. Tasi, T. Suhara, S. Mizuno, M. Ohkutsu, T. Honda, K. Rikiishi, “Observation of the power spectrum of ocean wave using a cloverleaf buoy,” J. Phys. Oceanogr. 10, 286–296 (1980).
[CrossRef]

Other (6)

See, for example, A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surface,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), p. 1.See also A. Ishimaru, J. S. Chen, “Scattering from very rough surfaces based on the modified second order Kirchhoff approximation with angular and propagation shadowing,” J. Acoust. Soc. Am. 88, 1877–1883 (1990).
[CrossRef]

See, for example, R. Siegel, J. R. Howell, Thermal Radiation and Heat Transfer (Hemisphere, New York, 1981), p. 57.

B. Kindsman, Wind Waves (Prentice-Hall, Englewood Cliffs, N.J., 1965), p. 344.

Ref. 10, p. 399.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surface (Pergamon, Oxford, 1979), p. 297.

Ref. 6, p. 44.

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Figures (10)

Fig. 1
Fig. 1

Cartesian coordinate system.

Fig. 2
Fig. 2

Illustration of the shadowing that occurs when we observe the surface with viewing angle ϕ.

Fig. 3
Fig. 3

Effective-distribution functions p0(γ; ϕ) for several values of the viewing angle: curve (a) ϕ = 0, curve (b) ϕ = 80, curve (c) ϕ = 85, and curve (d) ϕ = 89 deg. The value of the rms surface slope obtained by the JONSWAP model is γ0 = tan 8.20 deg, under the wind condition U10 = 5 m/s, X = 105 m, and Ψ = 0 deg.

Fig. 4
Fig. 4

Normalization condition for p0(γ; ϕ).

Fig. 5
Fig. 5

Probability density functions of observable surface slope p0(γ; ϕ)g(γ; ϕ) under the same wind conditions as in Fig. 3: curve (a) ϕ = 0, curve (b) ϕ = 80, curve (c) ϕ = 85, and curve (d) ϕ = 89 deg.

Fig. 6
Fig. 6

Effective emissivities for two sets of external parameters as a function of the normalized observation range tan ϕ: (a) U10 = 5 m/s, X = 105 m, and Ψ = 0 deg; (b) U10 = 15 m/s, X = 105 m, and Ψ = 0 deg (γ0 = tan 10.88 deg); (c) emissivity for a flat water surface is shown for comparison. The refractive index of water is assumed to be n = 1.19.

Fig. 7
Fig. 7

Illustration of the shadowing in the first-order reflection process.

Fig. 8
Fig. 8

Effective bistatic distribution functions for mirror reflection process p1(γ; ϕ) for several values of the viewing angles under the same wind condition as in Fig. 3: curve (a) ϕ = 0, curve (b) ϕ = 80, curve (c) ϕ = 85, and curve (d) ϕ = 89 deg.

Fig. 9
Fig. 9

Dependence of the effective bistatic reflectivity on the incident angle χ for several values of the viewing angles under the same wind condition as in Fig. 3: curve (a) ϕ = 45, curve (b) ϕ = 60, curve (c) ϕ = 70, curve (d) ϕ = 80, and curve (e) ϕ = 85 deg. The refractive index of water is assumed to be n = 1.19.

Fig. 10
Fig. 10

Sum of the effective emissivity and the directional-hemispherical reflectivity (energy loss that is due to the second shadowing effect) as a function of the viewing angle, under the same wind condition as in Fig. 3. The refractive index of water is assumed to be n = 1.19.

Equations (52)

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s = ( sin ϕ , 0 , cos ϕ ) ,
n · s = cos θ .
B ( x , s ; ω ) = ε ( θ ; ω ) B ( ω ; T ) ,
ε ( θ ; ω ) = 1 ρ ( θ ; ω ) .
ρ ( θ , ω ) = 1 2 { [ tan ( θ θ ) tan ( θ + θ ) ] 2 + [ sin ( θ θ ) sin ( θ + θ ) ] 2 } ,
sin θ = 1 n ( ω ) sin θ ,
γ = η ( x , t ) x = tan μ ,
θ = ϕ + μ .
P ( γ ) = 1 2 π γ 0 exp ( γ 2 / 2 γ 0 2 ) ,
γ 0 2 = 0 + d ω π π d φ S ( x , φ ; ω ) [ k ( ω ) cos φ ] 2 .
ω ( k ) = [ ( g / k + Γ k / ρ ) tanh ( h k ) ] 1 / 2 k ,
S ( x , φ ; ω ) = Ψ ( U 10 , X , ω ) G ( φ , ω ) ,
Ψ ( U 10 , X , ω ) = α g 2 ω 5 exp [ 5 4 ( ω m ω ) 4 ] κ β ,
β = exp [ ( ω ω m ) 2 2 σ 2 ω m 2 ] ,
σ = { σ a ω ω m σ b ω > ω m ,
ω m = 2 π × 3.5 g U 10 X ¯ 0.33 ,
X ¯ = X g U 10 2 ,
α = 0.076 X ¯ 0.22 ,
κ = 3.3 ,
σ a = 0.07 ,
σ b = 0.09 .
π + π d φ G ( φ , ω ) = 1 .
G ( φ , ω ) = { 2 / π cos 2 ( φ ψ ) for π / 2 + ψ < φ < π / 2 + ψ , 0 otherwise
γ 0 2 = 2 + cos 2 ψ 4 0 + d ω Ψ ( U 10 , X , ω ) [ k ( ω ) ] 2 .
p 0 ( γ ; ϕ ) = P ( γ ) ϑ ( cot | ϕ | γ sgn ϕ ) Q ( a ) ,
ϑ ( γ ) = { 1 γ 0 0 γ < 0 ,
Q ( a ) = 1 1 + Λ ( a ) ,
Λ ( a ) = 1 2 π a exp ( a 2 / 2 ) 1 2 erfc ( a 2 ) ,
a = cot | ϕ | γ 0 ,
L - observed d σ ( n · s ) = L 0 ( n · s ) .
L d σ + d γ p 0 ( γ ; ϕ ) ( n · s ) = L 0 ( n · s ) .
d x = ( n · n ) d σ
L 0 + d γ p 0 ( γ ; ϕ ) ( n · s ) ( n · n ) = L 0 ( n · s ) .
+ d γ p 0 ( γ ; ϕ ) g ( γ ; ϕ ) = 1 ,
g ( γ ; ϕ ) = ( n · s ) ( n · n ) ( n · s ) = 1 γ tan φ .
A = A g ϕ 0 = + d γ p 0 ( γ ; ϕ ) g ( γ ; ϕ ) A ( γ ) ,
ε eff ( ϕ , ω ) = π / 2 π / 2 d μ p 0 ( tan μ ; ϕ ) cos 2 μ × ε ( | ϕ + μ | , ω ) g ( tan μ ; ϕ ) .
p 1 ( γ ; ϕ , χ ) = { p 0 [ γ ; max ( | ϕ | , χ ) ] for ϕ χ 0 , P ( γ ) ϑ ( cot | ϕ | γ sgn ϕ ) × ϑ ( cot | χ | + γ sgn χ ) Q ( a , b ) for ϕ χ > 0
Q ( a , b ) = 1 [ 1 + Λ ( a ) + Λ ( b ) ] ,
b = cot | χ | γ 0 ·
χ = ϕ + 2 μ = ϕ + 2 arctan γ .
p 1 ( γ ; ϕ ) = p 1 ( γ ; ϕ , ϕ + 2 tan 1 γ ) ,
Q ( χ ) = Q i δ ( χ χ ) ,
L observed from both sides d σ Q ( ϕ + 2 tan 1 γ ) ( n · s ) ρ ( | θ | ; ω ) = L 0 Q i + d γ p 1 ( γ ; ϕ ) δ ( ϕ + 2 tan 1 γ χ ) × ρ ( | θ | ; ω ) ( n · s ) ( n · n ) ,
w 1 ( χ , ϕ ; ω ) = Q r / Q i = + d γ p 1 ( γ ; ϕ ) δ ( ϕ + 2 arctan γ χ ) × ρ ( | θ | ; ω ) g ( γ ; ϕ ) = δ ( ϕ + 2 arctan γ χ ) ρ ( | ϕ + arctan γ | ; ω ) × g ( γ ; ϕ ) ϕ 1 ,
w 1 ( χ , ϕ ; ω ) = π / 2 π / 2 d μ p 1 ( tan μ ; ϕ ) cos 2 μ × δ ( ϕ + 2 μ χ ) ρ ( | ϕ + μ | ; ω ) g ( tan μ ; ϕ ) = [ p 1 ( tan χ ϕ 2 ; ϕ ) ] / [ 2 cos 2 χ ϕ 2 ] × ρ ( | χ + ϕ | 2 ; ω ) g ( tan χ ϕ 2 ; ϕ ) .
ρ 1 ( ϕ , ω ) = π / 2 + π / 2 d χ w 1 ( χ , ϕ ; ω ) .
ε eff ( ϕ , ω ) + ρ 1 ( ϕ , ω ) < 1 .
I = 0 ω max d ω Ψ ( ω ) [ k ( ω ) ] 2 ,
Φ ( ω ) ω + α g 2 ω 5 .
k ( ω ) ω + { ω 2 / g for Γ = 0 ( ρ / Γ ) 1 / 3 ω 2 / 3 for Γ is finite .
Ψ ( ω ) [ k ( ω ) ] 2 ω + { α / ω for Γ = 0 α g 2 ( ρ / Γ ) 2 / 3 1 / ω 11 / 3 for Γ is finite .

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