Abstract

Slope spectral density resolved in wave number and direction is an important statistical descriptor of water surface waves. Experimentalists have estimated this descriptor from optical wave imagery by assuming that light from the surface is modulated linearly by the component of wave slope aligned with the imaging azimuth. The level of error arising from this assumption of linearity depends on the optical conditions and can be severe. We have numerically explored this error when only reflected radiance is imaged by using a synthesized sea surface and a clear sky model to simulate sea surface imaging. Additionally, we have developed a method for identifying geometries which minimize nonlinearity. This paper describes our analytic models, our numerical techniques, and the character of our results.

© 1981 Optical Society of America

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References

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  1. N. F. Barber, Nature No. 4168, 485 (17Sept.1949).
    [CrossRef]
  2. D. Stilwell, J. Geophys. Res. 74, 1974 (1969).
    [CrossRef]
  3. D. Stilwell, R. O. Pilon, J. Geophys. Res. 79, 1277 (1974).
    [CrossRef]
  4. R. S. Kasevich, J. Geophys. Res. 80, 4535 (1975).
    [CrossRef]
  5. B. L. Gotwols, G. B. Irani, J. Geophys. Res. 85, 3964 (1980).
    [CrossRef]
  6. D. C. Epsley, IRE Proc. 21, 1439 (1933).
    [CrossRef]
  7. A. W. Bjerkaas, F. W. Riedel, “Proposed Model for the Elevation Spectrum of a Wind Roughened Sea,” JHU/APL TG-1328, NTIS AD-A08342617 (Apr.1979).
  8. G. Neumann, W. J. Pierson, Principles of Physical Oceanography (Prentice-Hall, Englewood Cliffs, N.J., 1966), pp. 336–346.
  9. G. I. Pokrowski, Z. Phys. 53, 67 (1929).
    [CrossRef]
  10. R. G. Hopkinson, J. Opt. Soc. Am. 44, 455 (1954).
    [CrossRef]
  11. K. L. Coulson, Solar and Terrestrial Radiation (Academic, New York, 1975), Chap. 7.

1980 (1)

B. L. Gotwols, G. B. Irani, J. Geophys. Res. 85, 3964 (1980).
[CrossRef]

1975 (1)

R. S. Kasevich, J. Geophys. Res. 80, 4535 (1975).
[CrossRef]

1974 (1)

D. Stilwell, R. O. Pilon, J. Geophys. Res. 79, 1277 (1974).
[CrossRef]

1969 (1)

D. Stilwell, J. Geophys. Res. 74, 1974 (1969).
[CrossRef]

1954 (1)

1949 (1)

N. F. Barber, Nature No. 4168, 485 (17Sept.1949).
[CrossRef]

1933 (1)

D. C. Epsley, IRE Proc. 21, 1439 (1933).
[CrossRef]

1929 (1)

G. I. Pokrowski, Z. Phys. 53, 67 (1929).
[CrossRef]

Barber, N. F.

N. F. Barber, Nature No. 4168, 485 (17Sept.1949).
[CrossRef]

Bjerkaas, A. W.

A. W. Bjerkaas, F. W. Riedel, “Proposed Model for the Elevation Spectrum of a Wind Roughened Sea,” JHU/APL TG-1328, NTIS AD-A08342617 (Apr.1979).

Coulson, K. L.

K. L. Coulson, Solar and Terrestrial Radiation (Academic, New York, 1975), Chap. 7.

Epsley, D. C.

D. C. Epsley, IRE Proc. 21, 1439 (1933).
[CrossRef]

Gotwols, B. L.

B. L. Gotwols, G. B. Irani, J. Geophys. Res. 85, 3964 (1980).
[CrossRef]

Hopkinson, R. G.

Irani, G. B.

B. L. Gotwols, G. B. Irani, J. Geophys. Res. 85, 3964 (1980).
[CrossRef]

Kasevich, R. S.

R. S. Kasevich, J. Geophys. Res. 80, 4535 (1975).
[CrossRef]

Neumann, G.

G. Neumann, W. J. Pierson, Principles of Physical Oceanography (Prentice-Hall, Englewood Cliffs, N.J., 1966), pp. 336–346.

Pierson, W. J.

G. Neumann, W. J. Pierson, Principles of Physical Oceanography (Prentice-Hall, Englewood Cliffs, N.J., 1966), pp. 336–346.

Pilon, R. O.

D. Stilwell, R. O. Pilon, J. Geophys. Res. 79, 1277 (1974).
[CrossRef]

Pokrowski, G. I.

G. I. Pokrowski, Z. Phys. 53, 67 (1929).
[CrossRef]

Riedel, F. W.

A. W. Bjerkaas, F. W. Riedel, “Proposed Model for the Elevation Spectrum of a Wind Roughened Sea,” JHU/APL TG-1328, NTIS AD-A08342617 (Apr.1979).

Stilwell, D.

D. Stilwell, R. O. Pilon, J. Geophys. Res. 79, 1277 (1974).
[CrossRef]

D. Stilwell, J. Geophys. Res. 74, 1974 (1969).
[CrossRef]

IRE Proc. (1)

D. C. Epsley, IRE Proc. 21, 1439 (1933).
[CrossRef]

J. Geophys. Res. (4)

D. Stilwell, J. Geophys. Res. 74, 1974 (1969).
[CrossRef]

D. Stilwell, R. O. Pilon, J. Geophys. Res. 79, 1277 (1974).
[CrossRef]

R. S. Kasevich, J. Geophys. Res. 80, 4535 (1975).
[CrossRef]

B. L. Gotwols, G. B. Irani, J. Geophys. Res. 85, 3964 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature No. 4168 (1)

N. F. Barber, Nature No. 4168, 485 (17Sept.1949).
[CrossRef]

Z. Phys. (1)

G. I. Pokrowski, Z. Phys. 53, 67 (1929).
[CrossRef]

Other (3)

A. W. Bjerkaas, F. W. Riedel, “Proposed Model for the Elevation Spectrum of a Wind Roughened Sea,” JHU/APL TG-1328, NTIS AD-A08342617 (Apr.1979).

G. Neumann, W. J. Pierson, Principles of Physical Oceanography (Prentice-Hall, Englewood Cliffs, N.J., 1966), pp. 336–346.

K. L. Coulson, Solar and Terrestrial Radiation (Academic, New York, 1975), Chap. 7.

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Figures (8)

Fig. 1
Fig. 1

Imaging geometry and nomenclature.

Fig. 2
Fig. 2

Contrast and THD for various imaging geometries. Solar zenith angle is 45°. Camera polarizer is horizontal.

Fig. 3
Fig. 3

Contrast and THD vs surface wave azimuth for γc = 60°, α = 135°, γs = 45°, and horizontal polarization.

Fig. 4
Fig. 4

Methodology of error estimation in two dimensions.

Fig. 5
Fig. 5

Variance preserving plot of the 1-D slope magnitude spectrum.

Fig. 6
Fig. 6

Ratio of the slope magnitude spectrum to the radiance image spectrum for a specific geometry. Ratio is logarithmically displayed with each gray level gradation equal to 2 dB (geometry: γc = 60°, αc = 135°, γs = 45°, αw = 135°, cos2αn = directionality).

Fig. 7
Fig. 7

Spectral density ratio averaged over vave number vs wave vector azimuth (geometry: γc = 60°, αc = 135°, γc = 45°, αw = 135°, cos2αn = directionality).

Fig. 8
Fig. 8

Extent of wave vector azimuths yielding a <3-dB error in the slope spectrum estimate for six geometries (for all geometries γc = 60°, γs = 45°, horizontal polarization).

Equations (14)

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N sea ( γ c , α c , γ n , α n , γ s ) = Γ ( ω ) N ( γ i , α i , γ s ) .
G ( k , α n ) = G ( k ) cos 2 α n π k ,
G ( k ) = 0 2 π G ( k , α n ) k d α n .
L ( γ i , α i ) = K ( 1 + cos 2 μ 1 - cos μ ) [ 1 - exp ( - 0.32 / cos γ i ) ] ,
cos μ = cos γ s cos γ i + sin γ s sin γ s sin γ i cos α i , K ~ 0.05 W / ( m · sr · nm ) .
β = 0.94 1 - cos 2 μ 1 + cos 2 μ .
cos θ = ( sin γ i cos γ s - sin γ s cos γ i cos α i ) / sin μ .
N H = K L [ ½ ( 1 - β ) + β cos 2 θ ] .
N V = K L [ ½ ( 1 - β ) + β ( 1 - cos 2 θ ) ] .
r ^ i = 2 r ^ n ( r ^ n · r ^ c ) - r ^ c .
cos ω = cos γ n cos γ c - sin γ n sin γ c cos ( α n - α c ) , cos γ i = 2 cos ω cos γ n - cos γ c , tan α i = 2 cos ω sin γ n sin α n + sin γ c sin α c 2 cos ω sin γ n cos α n + sin γ c cos α c .
Γ = sin 2 ( ω - r ) sin 2 ( ω + r ) , Γ = tan 2 ( ω - r ) tan 2 ( ω + r ) ,
Γ H = ½ ( Γ cos 2 α ω + Γ sin 2 α ω ) , Γ V = ½ ( Γ sin 2 α ω + Γ cos 2 α ω ) ,
cos α ω = sin γ i sin ( α c - α i ) / sin 2 ω .

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