Abstract

A model relating viewing geometry, sky conditions, and statistical sea surface parameters to the luminance of the sea surface is developed. The model is used to estimate the relative visibility of surface perturbations manifested by a variation in the rms surface slope. These estimates are presented for a variety of geometries in the solar plane using two sky conditions (clear and overcast) and two wind speeds (∼2 and 7 m/sec). The results of this analysis, applicable to a unidirectional radiometer with no temporal averaging, show comparable extrema in the visibility of surface perturbations for both the overcast and clear sky models. The visibility of surface perturbations is shown to be maximized by geometries with either large gradients in the slope-to-luminance transfer functions (within the glitter pattern for the clear sky and near the horizon for both sky models) or very small gradients in the slope-to-luminance transfer functions (90° away from the glitter pattern for the clear sky and nadir directed for the overcast sky). It is shown that improvements in the estimated values of the luminance SNR sensitivity to rms surface slope variations can be obtained through spatial and/or temporal averaging.

© 1981 Optical Society of America

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References

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  1. E. C. LaFond, in The Sea, Vol. 1, M. N. Hill, Ed. (Wiley, New York, 1962), p. 745.
  2. J. A. Shand, Trans. Am. Geophys. Union 34, 899 (1953).
  3. R. S. Dietz, Sea Front. 359 (Nov.–Dec. 1974).
  4. J. R. Apel et al., J. Geophys. Res. 80, No. 6, 865 (20Feb.1975).
    [CrossRef]
  5. B. A. Hughes, H. L. Grant, J. Geophys. Res. 83, No. C1, 443 (Jan. 1978).
    [CrossRef]
  6. B. A. Hughes, J. Geophys. Res. 83, No. C1, 455 (Jan.1978).
    [CrossRef]
  7. C. Cox, W. Munk, J. Opt. Soc. Am. 44, 838 (1954).
    [CrossRef]
  8. P. M. Saunders, J. Geophys. Res. 72, 4643 (1967).
    [CrossRef]
  9. R. G. Hopkinson, J. Opt. Soc. Am. 44, 455 (1954).
    [CrossRef]
  10. F. W. Reidel, “Analytical Investigation of a Simplified Model to Study Sun Glitter Temporal Fluctuations,” American Geophysical Union Spring Meeting, Washington, D.C., May 1979.
  11. G. N. Plass et al., Appl. Opt. 14, 1924 (1975).
    [CrossRef] [PubMed]
  12. G. N. Plass et al., Appl. Opt. 15, 3161 (1976).
    [CrossRef] [PubMed]

1978

B. A. Hughes, H. L. Grant, J. Geophys. Res. 83, No. C1, 443 (Jan. 1978).
[CrossRef]

B. A. Hughes, J. Geophys. Res. 83, No. C1, 455 (Jan.1978).
[CrossRef]

1976

1975

J. R. Apel et al., J. Geophys. Res. 80, No. 6, 865 (20Feb.1975).
[CrossRef]

G. N. Plass et al., Appl. Opt. 14, 1924 (1975).
[CrossRef] [PubMed]

1974

R. S. Dietz, Sea Front. 359 (Nov.–Dec. 1974).

1967

P. M. Saunders, J. Geophys. Res. 72, 4643 (1967).
[CrossRef]

1954

1953

J. A. Shand, Trans. Am. Geophys. Union 34, 899 (1953).

Apel, J. R.

J. R. Apel et al., J. Geophys. Res. 80, No. 6, 865 (20Feb.1975).
[CrossRef]

Cox, C.

Dietz, R. S.

R. S. Dietz, Sea Front. 359 (Nov.–Dec. 1974).

Grant, H. L.

B. A. Hughes, H. L. Grant, J. Geophys. Res. 83, No. C1, 443 (Jan. 1978).
[CrossRef]

Hopkinson, R. G.

Hughes, B. A.

B. A. Hughes, J. Geophys. Res. 83, No. C1, 455 (Jan.1978).
[CrossRef]

B. A. Hughes, H. L. Grant, J. Geophys. Res. 83, No. C1, 443 (Jan. 1978).
[CrossRef]

LaFond, E. C.

E. C. LaFond, in The Sea, Vol. 1, M. N. Hill, Ed. (Wiley, New York, 1962), p. 745.

Munk, W.

Plass, G. N.

Reidel, F. W.

F. W. Reidel, “Analytical Investigation of a Simplified Model to Study Sun Glitter Temporal Fluctuations,” American Geophysical Union Spring Meeting, Washington, D.C., May 1979.

Saunders, P. M.

P. M. Saunders, J. Geophys. Res. 72, 4643 (1967).
[CrossRef]

Shand, J. A.

J. A. Shand, Trans. Am. Geophys. Union 34, 899 (1953).

Appl. Opt.

J. Geophys. Res.

J. R. Apel et al., J. Geophys. Res. 80, No. 6, 865 (20Feb.1975).
[CrossRef]

B. A. Hughes, H. L. Grant, J. Geophys. Res. 83, No. C1, 443 (Jan. 1978).
[CrossRef]

B. A. Hughes, J. Geophys. Res. 83, No. C1, 455 (Jan.1978).
[CrossRef]

P. M. Saunders, J. Geophys. Res. 72, 4643 (1967).
[CrossRef]

J. Opt. Soc. Am.

Sea Front

R. S. Dietz, Sea Front. 359 (Nov.–Dec. 1974).

Trans. Am. Geophys. Union

J. A. Shand, Trans. Am. Geophys. Union 34, 899 (1953).

Other

E. C. LaFond, in The Sea, Vol. 1, M. N. Hill, Ed. (Wiley, New York, 1962), p. 745.

F. W. Reidel, “Analytical Investigation of a Simplified Model to Study Sun Glitter Temporal Fluctuations,” American Geophysical Union Spring Meeting, Washington, D.C., May 1979.

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Figures (13)

Fig. 1
Fig. 1

Surface imaging geometry. Surface normal is designated by rn. Unit vectors, r ̂ n, r ̂ i, and r ̂ d are coplanar.

Fig. 2
Fig. 2

(a) Logarthmic plot of the modified clear sky luminance model (γS = 45°). (b) Clear sky luminance vs incident zenith angle within the solar plane (γs = 45°).

Fig. 3
Fig. 3

Average sea surface luminance for a clear sky and a solar zenith angle of 45°. Dashed curves are for σ = 0.1; solid for σ = 0.2. Vertical lines represent the edges of the glitter pattern as explained in the text.

Fig. 4
Fig. 4

Standard deviation of the sea surface luminance for a clear sky.

Fig. 5
Fig. 5

Sea surface luminance sensitivity to rms surface slope variations for a clear sky.

Fig. 6
Fig. 6

Aerial photograph showing banded surface markings produced by internal waves: Strait of Georgia, British Columbia [taken by the British Columbia Government in 1950 and originally published by Shand (1953)].

Fig. 7
Fig. 7

Sea surface SNR sensitivity for a clear sky.

Fig. 8
Fig. 8

Sea surface SNR sensitivity for a clear sky.

Fig. 9
Fig. 9

Average sea surface luminance for an overcast sky. Units are relative.

Fig. 10
Fig. 10

Standard deviation of the sea surface luminance for an overcast sky.

Fig. 11
Fig. 11

Sea surface luminance sensitivity for an overcast sky.

Fig. 12
Fig. 12

Sea surface contrast sensitivity for an overcast sky.

Fig. 13
Fig. 13

Sea surface SNR sensitivity for an overcast sky.

Equations (21)

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L = S * ( γ d , σ ) R ( ω ) L s ( γ i , α i ) cos ω cos γ d cos γ n p ( s x , s y ) × Q ( s x , s y ) d s x d s y ,
S * ( γ d , σ ) = 2 [ 1 + erf ( υ ) + ( υ π ) 1 exp ( υ 2 ) ] 1 ,
R = ½ [ sin 2 ( ω r ) sin 2 ( ω + r ) + tan 2 ( ω r ) tan 2 ( ω + r ) ] ,
r = sin 1 ( sin ω 1.34 )
L s = L ( 1 + cos 2 μ 1 cos μ ) [ 1 exp ( 0.32 cos γ i ) ] ,
L 600 cd / m 2 , cos μ = cos γ s cos γ i + sin γ s sin γ i cos ( α s α i ) .
L 0 = 600 , L 1 = { L 0 if L 0 3400 , 3400 + ( L 0 3400 ) 0.95 if 3400 < L 0 , L 2 = { L 1 if L 1 6800 , 6800 + ( L 1 6800 ) 0.95 if 6800 < L 1 , L 3 = { L 2 if L 2 10 , 200 , 10 , 200 + ( L 2 10 , 200 ) 0.95 if 10 , 200 L 2 , L 4 = { L 3 if L 3 13 , 600 , 13 , 600 + ( L 3 13 , 600 ) 0.95 if 13 , 600 L 3 , L s = MIN ( L 4 , 2.7 × 10 6 ) .
p ( s x , s y ) = 1 π σ 2 exp ( s x 2 + s y 2 σ 2 ) .
Q = { 1 s cot γ d , 0 s > cot γ d ,
L 2 = S * ( γ d , σ ) R 2 ( ω ) L s 2 ( γ i , α i ) cos ω cos γ d cos γ n × p ( s x , s y ) Q ( s x , s y ) d s x d s y ,
SD ( L ) = L 2 L 2 .
L σ
L σ = S * σ I d s x d s y + S * I σ d s x d s y ,
I = R ( ω ) L s ( γ i , α i ) Q ( s x , s y ) cos ω cos γ d cos γ n p ( s x , s y ) , S * σ = ( S * ) 2 exp ( υ 2 ) 4 π γ 2 σ 2 tan γ d , I σ = R ( ω ) L s ( γ i , α i ) Q ( s x , s y ) cos ω cos γ n cos γ d × [ ( s x 2 + s y 2 ) σ 3 1 σ ] 2 p ( s x , s y ) .
Δ σ ( L σ ) .
( Δ σ L ) ( L σ ) .
1 L L σ .
SNRS = L σ SD ( L ) .
SNR a = t ( r ) L σ Δ σ n 2 N [ n w 2 t 2 ( r ) + n e 2 + n p 2 ] 1 / 2 ,
SNR = L σ Δ σ SD ( L ) ,
SNRS = SNR Δ σ = L σ SD ( L ) .

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