Abstract

The Cox–Munk–Plass interaction probability density is used for computing emissivity from a two-dimensional anisotropic model of a rough sea surface. The infrared wavelengths of interest are the 3–5-μm and 8–14-μm bands, and computations are performed in the co-wind and crosswind directions for receiver polar angles varying from the zenith to the horizon. Comparisons between co-wind and crosswind results are made as a function of windspeed, wavelength, and the polar angle of the receiver, with an emphasis on results at the horizon. A method for obtaining upper and lower bounds on the surface emissivity is also given which, in the context of reflection, provides a measure of the effects of multiple reflections.

© 1997 Optical Society of America

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References

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  1. M. R. Keller, B. L. Gotwols, W. J. Plant, W. C. Keller, “Comparison of optically-derived spectral densities and microwave cross sections in a wind-wave tank,” J. Geophys. Res. 100, 16163–16178 (1995).
    [CrossRef]
  2. R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), pp. 205–207, 228–230, 260, 297–310, 341–343, 366.
  3. R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
    [CrossRef]
  4. C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
    [CrossRef]
  5. C. R. Zeisse, “Radiance of the ocean horizon,” J. Opt. Soc. Am. A 12, 2022–2030 (1995).
    [CrossRef]
  6. G. N. Plass, G. W. Kattawar, J. A. Guinn, “Radiative transfer in the Earth’s atmosphere and ocean: influence of ocean waves” Appl. Opt. 14, 1924–1936 (1975).
    [CrossRef] [PubMed]
  7. K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. A 11, 1886–1893 (1994).
    [CrossRef]
  8. K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sensing Environ. 24, 313–329 (1988).
    [CrossRef]
  9. D. Lubin, “The role of the tropical super greenhouse effect in heating the ocean surface,” Science 265, 224–227 (1994).
    [CrossRef] [PubMed]
  10. A. R. Korb, P. Dybwad, W. Wadsworth, J. W. Salisbury, “Portable Fourier transform infrared spectroradiometer for field measurements of radiance and emissivity,” Appl. Opt. 35, 1679–1692 (1996).
    [CrossRef] [PubMed]
  11. K. T. Constantikes, E. D. Claussen, “Measuring marine infrared clutter,” Johns Hopkins APL Tech. Dig. 16, 207–210 (1995).
  12. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 33–34, 59–60.
  13. H. P. Baltes, “On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment,” in Progress in Optics XIII, E. Wolf ed. (Elsevier, New York, 1976), pp. 1–25.
  14. D. B. Burkhard, J. V. S. Lochhead, C. M. Penchina, “On the validity of Kirchhoff’s law in a nonequilibrium environment,” Am. J. Phys. 40, 1794–1798 (1972).
    [CrossRef]
  15. C. S. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 30–36, 123–126.
  16. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 381–388.
  17. M. A. Weinstein, “On the validity of Kirchhoff’s law for a freely radiating body,” Am. J. Phys. 28, 123–125 (1960).
    [CrossRef]
  18. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Philadelphia, Pa., 1992), pp. 55–56, 66, 75–76.
  19. T. L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956), pp. 1–17.
  20. The NAG Fortran Library Manual, Mark 16 (NAG Inc., Downers Grove, Ill., 1993).
  21. 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]

1996 (1)

1995 (3)

C. R. Zeisse, “Radiance of the ocean horizon,” J. Opt. Soc. Am. A 12, 2022–2030 (1995).
[CrossRef]

M. R. Keller, B. L. Gotwols, W. J. Plant, W. C. Keller, “Comparison of optically-derived spectral densities and microwave cross sections in a wind-wave tank,” J. Geophys. Res. 100, 16163–16178 (1995).
[CrossRef]

K. T. Constantikes, E. D. Claussen, “Measuring marine infrared clutter,” Johns Hopkins APL Tech. Dig. 16, 207–210 (1995).

1994 (2)

1988 (1)

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sensing Environ. 24, 313–329 (1988).
[CrossRef]

1986 (1)

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

1975 (1)

1973 (1)

1972 (1)

D. B. Burkhard, J. V. S. Lochhead, C. M. Penchina, “On the validity of Kirchhoff’s law in a nonequilibrium environment,” Am. J. Phys. 40, 1794–1798 (1972).
[CrossRef]

1960 (1)

M. A. Weinstein, “On the validity of Kirchhoff’s law for a freely radiating body,” Am. J. Phys. 28, 123–125 (1960).
[CrossRef]

1954 (1)

Baltes, H. P.

H. P. Baltes, “On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment,” in Progress in Optics XIII, E. Wolf ed. (Elsevier, New York, 1976), pp. 1–25.

Bohren, C. S.

C. S. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 30–36, 123–126.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 33–34, 59–60.

Burkhard, D. B.

D. B. Burkhard, J. V. S. Lochhead, C. M. Penchina, “On the validity of Kirchhoff’s law in a nonequilibrium environment,” Am. J. Phys. 40, 1794–1798 (1972).
[CrossRef]

Claussen, E. D.

K. T. Constantikes, E. D. Claussen, “Measuring marine infrared clutter,” Johns Hopkins APL Tech. Dig. 16, 207–210 (1995).

Constantikes, K. T.

K. T. Constantikes, E. D. Claussen, “Measuring marine infrared clutter,” Johns Hopkins APL Tech. Dig. 16, 207–210 (1995).

Cox, C.

Dybwad, P.

Gotwols, B. L.

M. R. Keller, B. L. Gotwols, W. J. Plant, W. C. Keller, “Comparison of optically-derived spectral densities and microwave cross sections in a wind-wave tank,” J. Geophys. Res. 100, 16163–16178 (1995).
[CrossRef]

Guinn, J. A.

Hale, G. M.

Hill, T. L.

T. L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956), pp. 1–17.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Philadelphia, Pa., 1992), pp. 55–56, 66, 75–76.

Huffman, D. R.

C. S. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 30–36, 123–126.

Ichioka, Y.

Itoh, K.

Kattawar, G. W.

Keller, M. R.

M. R. Keller, B. L. Gotwols, W. J. Plant, W. C. Keller, “Comparison of optically-derived spectral densities and microwave cross sections in a wind-wave tank,” J. Geophys. Res. 100, 16163–16178 (1995).
[CrossRef]

Keller, W. C.

M. R. Keller, B. L. Gotwols, W. J. Plant, W. C. Keller, “Comparison of optically-derived spectral densities and microwave cross sections in a wind-wave tank,” J. Geophys. Res. 100, 16163–16178 (1995).
[CrossRef]

Korb, A. R.

Lochhead, J. V. S.

D. B. Burkhard, J. V. S. Lochhead, C. M. Penchina, “On the validity of Kirchhoff’s law in a nonequilibrium environment,” Am. J. Phys. 40, 1794–1798 (1972).
[CrossRef]

Lubin, D.

D. Lubin, “The role of the tropical super greenhouse effect in heating the ocean surface,” Science 265, 224–227 (1994).
[CrossRef] [PubMed]

Masuda, K.

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sensing Environ. 24, 313–329 (1988).
[CrossRef]

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Munk, W.

Penchina, C. M.

D. B. Burkhard, J. V. S. Lochhead, C. M. Penchina, “On the validity of Kirchhoff’s law in a nonequilibrium environment,” Am. J. Phys. 40, 1794–1798 (1972).
[CrossRef]

Plant, W. J.

M. R. Keller, B. L. Gotwols, W. J. Plant, W. C. Keller, “Comparison of optically-derived spectral densities and microwave cross sections in a wind-wave tank,” J. Geophys. Res. 100, 16163–16178 (1995).
[CrossRef]

Plass, G. N.

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Querry, M. R.

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 381–388.

Salisbury, J. W.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Philadelphia, Pa., 1992), pp. 55–56, 66, 75–76.

Takashima, T.

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sensing Environ. 24, 313–329 (1988).
[CrossRef]

Takayama, Y.

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sensing Environ. 24, 313–329 (1988).
[CrossRef]

Wadsworth, W.

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), pp. 205–207, 228–230, 260, 297–310, 341–343, 366.

Weinstein, M. A.

M. A. Weinstein, “On the validity of Kirchhoff’s law for a freely radiating body,” Am. J. Phys. 28, 123–125 (1960).
[CrossRef]

Yoshimori, K.

Zeisse, C. R.

Am. J. Phys. (2)

M. A. Weinstein, “On the validity of Kirchhoff’s law for a freely radiating body,” Am. J. Phys. 28, 123–125 (1960).
[CrossRef]

D. B. Burkhard, J. V. S. Lochhead, C. M. Penchina, “On the validity of Kirchhoff’s law in a nonequilibrium environment,” Am. J. Phys. 40, 1794–1798 (1972).
[CrossRef]

Appl. Opt. (3)

J. Geophys. Res. (1)

M. R. Keller, B. L. Gotwols, W. J. Plant, W. C. Keller, “Comparison of optically-derived spectral densities and microwave cross sections in a wind-wave tank,” J. Geophys. Res. 100, 16163–16178 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. Oceanogr. (1)

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Johns Hopkins APL Tech. Dig. (1)

K. T. Constantikes, E. D. Claussen, “Measuring marine infrared clutter,” Johns Hopkins APL Tech. Dig. 16, 207–210 (1995).

Remote Sensing Environ. (1)

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sensing Environ. 24, 313–329 (1988).
[CrossRef]

Science (1)

D. Lubin, “The role of the tropical super greenhouse effect in heating the ocean surface,” Science 265, 224–227 (1994).
[CrossRef] [PubMed]

Other (8)

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), pp. 205–207, 228–230, 260, 297–310, 341–343, 366.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 33–34, 59–60.

H. P. Baltes, “On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment,” in Progress in Optics XIII, E. Wolf ed. (Elsevier, New York, 1976), pp. 1–25.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Philadelphia, Pa., 1992), pp. 55–56, 66, 75–76.

T. L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956), pp. 1–17.

The NAG Fortran Library Manual, Mark 16 (NAG Inc., Downers Grove, Ill., 1993).

C. S. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 30–36, 123–126.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 381–388.

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Figures (12)

Fig. 1
Fig. 1

Projection of a facet’s area onto a plane normal to a ray pointing in the zenith direction û.

Fig. 2
Fig. 2

Illustration of the meaning of a facet back. The facet in each part of the figure represents a planar element of the sea surface with u^n the unit vector normal to the facet front, u^nb the unit vector normal to the facet back, u^r the fixed unit vector in the direction of the receiver, and ωr the angle between u^n and u^r. The extension of the sea surface beyond the part represented by the facet is indicated by the thin curve. The facet front is the side of the facet that interfaces with the air, and the facet back is the side of the facet that interfaces with the water. For an opaque sea surface, no radiation is allowed to propagate within the water. (a) A facet tilted with respect to the receiver direction such that ωrπ/2 is included in the probability density q; (b) a facet tilted with respect to the receiver direction such that ωr>π/2 is excluded from q.

Fig. 3
Fig. 3

Illustration of the reflection processes that are included and excluded by the hemispherical ensemble average over coordinates [Eq. (16)] for the special case when the receiver is in the zenith and both the facet normal and the source direction are in the xz plane. (a) θn=30°, θs=60°, η=1/3; included: (b) θn=45°, θs=90°, η=1; included. (c) θn=60°, θs=120°, η=3; excluded.

Fig. 4
Fig. 4

Effect of sea surface roughness on emissivity is shown for three different windspeeds for a wavelength of 5 μm. The solid curves are co-wind results with the corresponding windspeed in meters per second placed above each curve. The dotted–dashed curve is the emissivity for a perfectly flat sea.

Fig. 5
Fig. 5

Comparison between sea surface emissivity computed in the co-wind and crosswind directions for a wavelength of 5 μm as a function of the polar angle of the receiver. When θr=0° the receiver is in the zenith, and when θr=90° it is at the horizon. The solid curves are the co-wind result and the dashed curves are the crosswind results. (a) Wind speed is 5 m/s, (b) wind speed is 10 m/s.

Fig. 6
Fig. 6

Comparison between the sea surface emissivity at the horizon computed in the co-wind and crosswind directions as a function of wavelength. The solid curves are co-wind results and the dashed curves are crosswind results. The windspeed W, in meters per second, is placed between the solid and dashed curves of each pair of co-wind and crosswind results. The dotted–dashed curve is for the case when the co-wind and crosswind values are equal (W=2.42 m/s). (a) Atmospheric window from 3 to 5 μm; (b) atmospheric window from 8 to 14 μm.

Fig. 7
Fig. 7

Plots of the sea surface emissivity computed for a wavelength of 5 μm in the co-wind and crosswind directions as functions of windspeed. The solid curves are co-wind results, and the dashed curves are crosswind results. The curves intersect at W2.42 m/s. (a) θr=90°, (b) θr=55°. Notice that the vertical scale in (b) is significantly different from that in (a).

Fig. 8
Fig. 8

Plots showing the lower bound, {1}, to sea surface emissivity (reflectivity) for the idealized case in which the sea is a blackbody (perfect reflector). The solid curves are co-wind lower bounds, and the dashed curves are crosswind lower bounds. (a) Windspeed is 1 m/s; (b) windspeed is 10 m/s.

Fig. 9
Fig. 9

Plot showing the upper and lower bounds to sea surface emissivity when the windspeed is 10 m/s and the wavelength of radiation is 5 μm. The solid curves are the emissivity [] and are computed with use of Eq. (12). The dashed curves are the upper and lower bounds to the emissivity and are computed with use of Eq. (16); also see Eq. (20). (a) Co-wind results, (b) crosswind results.

Fig. 10
Fig. 10

Demonstration of the use of the hemispherical ensemble average over coordinates in assessing the significance of multiply reflecting facets. The curve shows the value of θr where {1} first equals 0.99, as a function of windspeed. Symbols represent the Monte Carlo results of Preisendorfer and Mobley for the single-reflection theory in the small-slope approximation.

Fig. 11
Fig. 11

Plot of reflectivity as a function of the polar angle of the receiver for a windspeed of 5 m/s and a wavelength of 5 μm. Differences between {R} and [R] provide a measure of the contributions from multiple reflections. The reflectivity for a flat surface is shown for comparison.

Fig. 12
Fig. 12

Coordinate system used for computing surface emissivity. The z axis points along the zenith and the x axis points in the direction of the wind. The xy plane is the mean sea level. The unit vector u^s points in the direction of the source, u^r points in the direction of the receiver, and u^n points in the direction normal to a facet (not shown) used to represent a particular wave.

Equations (54)

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p(ηx, ηy; W)=exp-12 ηx2σx2+ηy2σy2/2πσxσy.
σx2=0.000+3.16×10-3W±0.004
σy2=0.003+1.92×10-3W±0.002,
q(ηx, ηy; uˆ, W)=q0(uˆ, W) cos ωucos θn p(ηx, ηy; W),
q0(uˆ, W)1ωuπ/2 uˆ=constantcos ωucos θn pdηxdηyωuπ/20ωu>π/2.
--dηxdηyq(ηx, ηy; uˆ, W)=1.
1=α+R+τ,
α=1-R.
=α,
=1-R.
R(ω, m)=12(R+R)ωπ/2,
[f]- dηx- dηyf(ηx, ηy)q(ηx, ηy; u^r, W).
L(T, λ; θr, ϕr; W)=LBB(T, λ)sea(T, λ; θr, ϕr; W),
sea(T, λ, θr, ϕr; W)[]=[1-R],
[f]=0π dθ sin θ02π dϕf(θ, ϕ; θr, ϕr)×q(θ, ϕ; u^r, W)J(η; θ, ϕ; θr, ϕr),
{f}0π/2 dθ sin θ02π dϕf(θ, ϕ; θr, ϕr)×q(θ, ϕ; u^r, W)J(η; θ, ϕ; θr, ϕr).
{f}[f].
{R}[R],
{1-R}[1-R]=[]sea.
{1-R}[]1-{R}.
{1}[1]=1.
{1}-[]{R}1-[].
u^s=sin θs cos ϕsxˆ+sin θs sin ϕsyˆ+cos θszˆ,
u^r=sin θr cos ϕrxˆ+sin θr sin ϕryˆ+cos θrzˆ,
u^n=sin θn cos ϕnxˆ+sin θn sin ϕnyˆ+cos θnzˆ,
cos ωs=u^su^n=cos θs cos θn+sin θs sin θn× cos(ϕs-ϕn)
cos ωr=u^ru^n=cos θr cos θn+sin θr sin θn× cos(ϕr-ϕn).
u^n=zˆ-η1+η2,
zˆu^n=cos θn=11+η2
η=zˆ-u^nzˆu^n.
ηx=-tan θn cos ϕn
ηy=-tan θn sin ϕn.
cos ωscos θn=cos θs-sin θs(ηx cos ϕs+ηy sin ϕs).
cos ωrcos θn=cos θr-sin θr(ηx cos ϕr+ηy sin ϕr).
u^s+u^r=(2 cos ω)u^n,
ω=ωs=ωr;
η=zˆ-u^s+u^rzˆ(u^s+u^r),
ηx=-(sin θs cos ϕs+sin θr cos ϕr)cos θs+cos θr
ηy=-(sin θs sin ϕs+sin θr sin ϕr)cos θs+cos θr.
cos θn=cos θs+cos θr2[1+cos θs cos θr+sin θs sin θr cos(ϕs-ϕr)]1/2,
cos ω=cos ωs=cos ωr=12 [1+cos θs cos θr+sin θs sin θr cos(ϕs-ϕr)]1/2=(cos θs+cos θr)/2 cos θn.
dηxdηy=J(η; θs, ϕs; θr, ϕr)sin θsdθsdϕs,
J(η; θs, ϕs; θr, ϕr)=14 cos ω cos3 θn.
R=|r˜|2,
R=|r˜|2,
r˜=cos Ω-m cos ωcos Ω+m cos ω,
r˜=cos ω-m cos Ωcos ω+m cos Ω.
sin Ω=sin ωm,
m=n+ik.
R={[b-(n2-k2)cos ω]2+[a-2nk cos ω]2}/{[b+(n2-k2)cos ω]2+[a+2nk cos ω]2},
R={[cos ω-b]2+a2}/{[cos ω+b]2+a2},
a[x2+(2nk)2-x]1/2/2,
b[x2+(2nk)2+x]1/2/2,
x(n2-k2)-sin2 ω.

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