Abstract

An analytical expression is derived for the number density of sea surface specular points, λρ. Unlike previous results, the present result is not limited to small wave slopes and is valid for arbitrary source and receiver location as well as for arbitrary sea surface displacement. When the source is not in the horizon, it is shown that one immediate outcome from this generalization is that the standard method for obtaining the total number of specular points over the entire (flat) sea surface is invalid. Numerical results are also given for λρ as a function of wind speed and sun elevation.

© 1999 Optical Society of America

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References

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  2. O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurement of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13891–13901 (1988).
    [CrossRef]
  3. V. N. Nosov, S. Y. Pashin, “Statistical characteristics of wind-created sea waves in the gravity and capillary range,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 26, 851–856 (1990).
  4. T. Hara, E. J. Bock, D. Lyzenga, “In situ measurement of capillary-gravity wave spectra using a scanning laser slope gauge and microwave radar,” J. Geophys. Res. 99, 12593–12602 (1994).
    [CrossRef]
  5. J. A. Shaw, J. H. Churnside, “Scanning-laser glint measurements of sea-surface slope statistics,” Appl. Opt. 36, 4202–4213 (1997).
    [CrossRef] [PubMed]
  6. G. O. Marmorins, D. R. Lyzenga, J. A. C. Kaiser, “Comparison of airborne synthetic aperture radar imagery with in situ surface-slope measurements across Gulfstream slicks and a convergent front,” J. Geophys. Res. 104, 1405–1422 (1999).
    [CrossRef]
  7. C. Rufenach, C. Smith, “Observation of internal waves in LANDSAT and SEASAT satellite imagery,” Int. J. Remote Sens. 6, 1201–1207 (1985).
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  8. M. S. Longuet-Higgins, “Reflection and refraction at a random moving surface. I. Pattern and paths of specular points,” J. Opt. Soc. Am. 50, 838–844 (1960).
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  9. M. S. Longuet-Higgins, “Reflection and refraction at a random moving surface. II. Number of specular points in a Gaussian surface,” J. Opt. Soc. Am. 50, 845–850 (1960).
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  10. M. S. Longuet-Higgins, “Reflection and refraction at a random moving surface. III. Frequency of twinkling in a Gaussian surface,” J. Opt. Soc. Am. 50, 851–856 (1960).
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  11. R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), pp. 205–207, 315–321, and 341–343.
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    [CrossRef]
  13. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), pp. 345–353.
  14. The NAG Fortran Library Manual, Mark 17 (NAG Inc., Downers Grove, Ill., 1995).
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  16. C. R. Zeisse, “Radiance of the ocean horizon,” J. Opt. Soc. Am. A 12, 2022–2030 (1995).
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  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 307 (Integral 3.323.2), 433 (Integral 3.786.3), and 718 (Integral 6.633.4).

1999 (1)

G. O. Marmorins, D. R. Lyzenga, J. A. C. Kaiser, “Comparison of airborne synthetic aperture radar imagery with in situ surface-slope measurements across Gulfstream slicks and a convergent front,” J. Geophys. Res. 104, 1405–1422 (1999).
[CrossRef]

1997 (2)

1995 (1)

1994 (1)

T. Hara, E. J. Bock, D. Lyzenga, “In situ measurement of capillary-gravity wave spectra using a scanning laser slope gauge and microwave radar,” J. Geophys. Res. 99, 12593–12602 (1994).
[CrossRef]

1990 (1)

V. N. Nosov, S. Y. Pashin, “Statistical characteristics of wind-created sea waves in the gravity and capillary range,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 26, 851–856 (1990).

1988 (1)

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurement of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13891–13901 (1988).
[CrossRef]

1985 (1)

C. Rufenach, C. Smith, “Observation of internal waves in LANDSAT and SEASAT satellite imagery,” Int. J. Remote Sens. 6, 1201–1207 (1985).
[CrossRef]

1960 (3)

1954 (1)

Bock, E. J.

T. Hara, E. J. Bock, D. Lyzenga, “In situ measurement of capillary-gravity wave spectra using a scanning laser slope gauge and microwave radar,” J. Geophys. Res. 99, 12593–12602 (1994).
[CrossRef]

Churnside, J. H.

Constantikes, K. T.

Cox, C.

Donohue, D. J.

Freund, D. E.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 307 (Integral 3.323.2), 433 (Integral 3.786.3), and 718 (Integral 6.633.4).

Hara, T.

T. Hara, E. J. Bock, D. Lyzenga, “In situ measurement of capillary-gravity wave spectra using a scanning laser slope gauge and microwave radar,” J. Geophys. Res. 99, 12593–12602 (1994).
[CrossRef]

Joseph, R. I.

Kahaner, D.

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 153–157.

Kaiser, J. A. C.

G. O. Marmorins, D. R. Lyzenga, J. A. C. Kaiser, “Comparison of airborne synthetic aperture radar imagery with in situ surface-slope measurements across Gulfstream slicks and a convergent front,” J. Geophys. Res. 104, 1405–1422 (1999).
[CrossRef]

Longuet-Higgins, M. S.

Lyzenga, D.

T. Hara, E. J. Bock, D. Lyzenga, “In situ measurement of capillary-gravity wave spectra using a scanning laser slope gauge and microwave radar,” J. Geophys. Res. 99, 12593–12602 (1994).
[CrossRef]

Lyzenga, D. R.

G. O. Marmorins, D. R. Lyzenga, J. A. C. Kaiser, “Comparison of airborne synthetic aperture radar imagery with in situ surface-slope measurements across Gulfstream slicks and a convergent front,” J. Geophys. Res. 104, 1405–1422 (1999).
[CrossRef]

Marmorins, G. O.

G. O. Marmorins, D. R. Lyzenga, J. A. C. Kaiser, “Comparison of airborne synthetic aperture radar imagery with in situ surface-slope measurements across Gulfstream slicks and a convergent front,” J. Geophys. Res. 104, 1405–1422 (1999).
[CrossRef]

Moler, C.

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 153–157.

Munk, W.

Nash, S.

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 153–157.

Nosov, V. N.

V. N. Nosov, S. Y. Pashin, “Statistical characteristics of wind-created sea waves in the gravity and capillary range,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 26, 851–856 (1990).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), pp. 345–353.

Pashin, S. Y.

V. N. Nosov, S. Y. Pashin, “Statistical characteristics of wind-created sea waves in the gravity and capillary range,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 26, 851–856 (1990).

Rufenach, C.

C. Rufenach, C. Smith, “Observation of internal waves in LANDSAT and SEASAT satellite imagery,” Int. J. Remote Sens. 6, 1201–1207 (1985).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 307 (Integral 3.323.2), 433 (Integral 3.786.3), and 718 (Integral 6.633.4).

Shaw, J. A.

Shemdin, O. H.

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurement of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13891–13901 (1988).
[CrossRef]

Smith, C.

C. Rufenach, C. Smith, “Observation of internal waves in LANDSAT and SEASAT satellite imagery,” Int. J. Remote Sens. 6, 1201–1207 (1985).
[CrossRef]

Tran, H. M.

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurement of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13891–13901 (1988).
[CrossRef]

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), pp. 205–207, 315–321, and 341–343.

Wu, S. C.

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurement of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13891–13901 (1988).
[CrossRef]

Zeisse, C. R.

Appl. Opt. (1)

Int. J. Remote Sens. (1)

C. Rufenach, C. Smith, “Observation of internal waves in LANDSAT and SEASAT satellite imagery,” Int. J. Remote Sens. 6, 1201–1207 (1985).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

V. N. Nosov, S. Y. Pashin, “Statistical characteristics of wind-created sea waves in the gravity and capillary range,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 26, 851–856 (1990).

J. Geophys. Res. (3)

T. Hara, E. J. Bock, D. Lyzenga, “In situ measurement of capillary-gravity wave spectra using a scanning laser slope gauge and microwave radar,” J. Geophys. Res. 99, 12593–12602 (1994).
[CrossRef]

O. H. Shemdin, H. M. Tran, S. C. Wu, “Directional measurement of short ocean waves with stereophotography,” J. Geophys. Res. 93, 13891–13901 (1988).
[CrossRef]

G. O. Marmorins, D. R. Lyzenga, J. A. C. Kaiser, “Comparison of airborne synthetic aperture radar imagery with in situ surface-slope measurements across Gulfstream slicks and a convergent front,” J. Geophys. Res. 104, 1405–1422 (1999).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Other (5)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), pp. 345–353.

The NAG Fortran Library Manual, Mark 17 (NAG Inc., Downers Grove, Ill., 1995).

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 153–157.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 307 (Integral 3.323.2), 433 (Integral 3.786.3), and 718 (Integral 6.633.4).

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), pp. 205–207, 315–321, and 341–343.

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Figures (7)

Fig. 1
Fig. 1

Two-dimensional depiction of the coordinate system used for performing the analysis of the density of specular points. The wavy line represents the sea surface displacement given by z=h(x, y, t), with the thick portion of the wavy line representing an arbitrary facet that is displaced an amount h above the xy plane. The vectors r, rs, and rr locate, respectively, the positions of the facet, the source, and the receiver relative to the origin of the coordinate system. The vectors us and ur locate the source and the receiver relative to the facet. The unit vector u^n is the unit normal to the facet. The y axis (not shown) is coming straight out of the page.

Fig. 2
Fig. 2

Geometry of the rotated coordinate system. The solid lines are the x and y axes of the coordinate system shown in Fig. 1, and they define the mean sea level. The point (xr, yr) is the projection of the receiver’s location into the xy plane. The azimuthal orientation of the plane-wave source makes an angle ϕs with the x axis and is represented by the long-dashed line. The small-dashed lines represent the axes of the auxiliary urxury-coordinate system. The lines drawn with both long and short dashes are the x and y axes of the rotated xy-coordinate system. The origin of the xy-coordinate system is located at the point (xr, yr), and the x axis is along the azimuthal orientation of the plane-wave source.

Fig. 3
Fig. 3

Surface plot of specular point density λp as a function of (x, y) for the case when the source is in the zenith (θs=0°). The receiver is located 10 m above the origin. (a) Wind speed 2.5 m/s, ση2=0.0187 rad2, σκ2=66 m-2; (b) wind speed 10 m/s, ση2=0.0636 rad2, σκ2=1690 m-2. Note that the vertical scale in (b) is significantly different from that in (a).

Fig. 4
Fig. 4

Surface plot of specular point density λp as a function of (x, y) for the case when θs=37° and ϕs=180°. Recall that (θs, ϕs) are the polar and azimuthal angles of the unit vector u^s that points toward the plane-wave source. Rays emanating from the source propagate in the -u^s (=+sin θs xˆ-cos θs zˆ) direction. The receiver is located 10 m above the origin. (a) Wind speed 2.5 m/s, ση2=0.0187 rad2, σκ2=66 m-2; (b) wind speed 10 m/s, ση2=0.0636 rad2, σκ2=1690 m-2. Note that the vertical scale in (b) is significantly different from that in (a).

Fig. 5
Fig. 5

Same as Fig. 4, except that θs=85°.

Fig. 6
Fig. 6

Same as Fig. 4, except that θs=90°.

Fig. 7
Fig. 7

Surface plot of I(β, γ) as a function of (x, y) for the case when θs=85° and ϕs=180°. The wind speed is 10 m/s, and the receiver coordinates are (0, 0, 10) m.

Tables (1)

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Table 1 Theoretical and Experimental Values of the Maximum Specular Point Density

Equations (55)

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hx=x-xrur+x-xsuszr-hur+zs-hus,
hy=y-yrur+y-ysuszr-hur+zs-hus,
ur=[(xr-x)2+(yr-y)2+(zr-h)2]1/2,
us=[(xs-x)2+(ys-y)2+(zs-h)2]1/2.
hx=(x-xr)-ur sin θs cos ϕs(zr-h)+ur cos θsFx,
hy=(y-yr)-ur sin θs sin ϕs(zr-h)+ur cos θsFy.
u^ssin θs cos ϕs xˆ+sin θs sin ϕs yˆ+cos θs zˆ
λρ=|α3α5-α42|×Pα1  α5(α1=0, α2=0, α3, α4, α5)dα3dα4dα5,
α1hx-Fx,
α2hy-Fy,
α3α1x=2hx2-Fxx2hx2-Fxx,
α4α1y=2hyx-Fxy2hyx-Fxy,
α5α2y=2hy2-Fyy2hy2-Fyy.
ξ1hx,
ξ2hy,
ξ32hx2+2hy2,
ξ42hx2-2hy2,
ξ52hxy
Pα1  α5(α1, α2, α3, α4, α5)dα1  dα5=Pξ1  ξ5(ξ1, ξ2, ξ3, ξ4, ξ5)dξ1  dξ5=Pξ1(ξ1)Pξ2(ξ2)Pξ3(ξ3)Pξ4(ξ4)Pξ5(ξ5)dξ1  dξ5,
Pξi(ξi)=exp(-12ξi2/ξi2)2πξi2.
ξ12=ξ22=12ση2,
ξ32=σκ2,ξ42=12σκ2,ξ52=18σκ2.
λρ=exp[-(Fx2+Fy2)/ση2]πση2(2π)3/2σκ3×---|ξ˜32-ξ˜42-4ξ˜52|×exp[-(ξ32+2ξ42+8ξ52)/2σκ2]dξ3dξ4dξ5,
ξ˜3ξ3-(Fxx+Fyy),
ξ˜4ξ4-(Fxx-Fyy),
ξ˜5ξ5-Fxy.
λρ=exp[-(Fx2+Fy2)/ση2]πση2(2π)3/2σκ3×exp-14 4σκ2 [(Fxx-Fyy)2+4Fxy2]+2σκ2 (Fxx+Fyy)2×---|ξ˜32-ξ˜42-4ξ˜52|×exp(-{[ξ˜32+2(Fxx+Fyy)ξ˜3]+[2ξ˜42+4(Fxx-Fyy)ξ˜4]+(8ξ˜52+16Fxyξ˜5)}/2σκ2)dξ˜3dξ˜4dξ˜5.
λρ=σκ2(2π)3/2ση2 exp[-(tan2 θn)/ση2]I(β, γ),
I(β, γ)exp-14 (β2+2γ2)0du u exp(-u2)I0(βu)×-dv exp-12 v2exp(-γv)|u2-v2|.
tan2 θn=hx2+hy2Fx2+Fy2,
β2σκ [(Fxx-Fyy)2+4Fxy2]1/2,
γ1σκ (Fxx+Fyy).
x=(xr-x)cos ϕs+(yr-y)sin ϕs,
y=-(xr-x)sin ϕs+(yr-y)cos ϕs
x2+y2=(xr-x)2+(yr-y)2,
ur=[x2+y2+(zr-h)2]1/2.
tan2 θn=F2/G2,
β=4J sin θsσκG3 (x2+y2)1/2,
γ=2HJσκG3,
F2(x+ur sin θs)2+y2,
Gur cos θs+zr-h,
Hur+(zr-h)cos θs,
JH+x sin θs.
(λρ)max=12π3 σκ2ση2,
(x, y)maximum-densitypoint=(-zr tan θs, 0).
I(0, γ)=2π3 exp-13 γ2+γ2π2
=γ2π2 1+1γ2×23 exp-13 γ2.
I(β, γ)2π3 [1+(pβ2+qγ2)+],
p=8-3324,
q=33-26.
|a2-b2|=2π 0 dxx2 [(1-cos a2x)cos b2x+(1-cos b2x)cos a2x],
I(β, γ)=exp[-14(β2+2γ2)]π 0 dxx2×{[P(0)-P(x)]Q(x)+[Q(0)-Q(x)]P(x)},
P(x)=exp[β2/4(x2+1)]x2+1
×cosβ2x4(x2+1)-x sinβ2x4(x2+1),
Q(x)=exp[γ2/2(4x2+1)]4x2+1 (4x2+1+1)1/2×cosγ2x4x2+1-(4x2+1-1)1/2 sinγ2x4x2+1.

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