Abstract

An analytical approach of the two-dimensional emissivity of a rough sea surface in the infrared band is presented. The emissivity characterizes the intrinsic radiation of the sea surface. Because the temperature measured by the infrared camera depends on the emissivity, the emissivity is a relevant parameter for retrieving the sea-surface temperature from remotely sensed radiometric measurements, such as from satellites. This theory is developed from the first-order geometrical-optics approximation and is based on recent research. The typical approach assumes that the slope in the upwind direction is greater than the slope in the crosswind direction, involving the use of a one-dimensional shadowing function with the observed surface assumed to be infinite. We introduce the two-dimensional shadowing function and the surface observation length parameters that are included in the modeling of the two-dimensional emissivity.

© 2000 Optical Society of America

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References

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  1. K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sens. Environ. 24, 313–329 (1988).
    [CrossRef]
  2. P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
    [CrossRef]
  3. X. Wu, W. L. Smith, “Emissivity of rough sea surface for 8–13 mm: modeling and verification,” Appl. Opt. 36, 2609–2619 (1997).
    [CrossRef] [PubMed]
  4. K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
    [CrossRef]
  5. K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
    [CrossRef] [PubMed]
  6. R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).
  7. B. G. Smith, “Lunar surface roughness, shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
    [CrossRef]
  8. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-5, 668–671 (1967).
    [CrossRef]
  9. R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–627 (1967).
  10. C. Bourlier, J. Saillard, G. Berginc, “Spatial autocorrelation function of the heights of an even sea spectrum,” presented at Poster Sessions (Phenomenology), 5th International Conference on Radar Systems, La Société des Electriciens et des Electroniciens, 17–21 May 1999, Brest, France.
  11. C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–226 (1954).
  12. 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]
  13. M. Kunt, Traitement de l’information VI: techniques modernes de traitement numériques des signaux (Presses Polytechniques Romandes, France, 1991).
  14. F. Daout, “Etude de la dépolarisation des ondes centimétiques par une surface rugueuse—Application au domaine maritime,” Thèse de Doctorat (Institut de Recherche de l’Enseignement Supérieur aux Techniques de l’Electronique, Nantes, France, 1996).
  15. H. L. Chan, A. K. Fung, “A theory of sea scatter at large incident angles,” J. Geophys. Res. 82, 3439–3444 (1977).
    [CrossRef]
  16. C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
    [CrossRef]
  17. L. M. Ricciardi, S. Sato, “On the evaluation of first passage time densities for Gaussian processes,” Signal Process. 11, 339–357 (1986).
    [CrossRef]
  18. L. M. Ricciardi, S. Sato, “A note on first passage time problems for Gaussian processes and varying boundaries,” IEEE Trans. Inf. Theory IT-29, 454–457 (1983).
    [CrossRef]

2000 (1)

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

1997 (1)

1995 (1)

1994 (1)

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

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 Sens. Environ. 24, 313–329 (1988).
[CrossRef]

1986 (1)

L. M. Ricciardi, S. Sato, “On the evaluation of first passage time densities for Gaussian processes,” Signal Process. 11, 339–357 (1986).
[CrossRef]

1983 (1)

L. M. Ricciardi, S. Sato, “A note on first passage time problems for Gaussian processes and varying boundaries,” IEEE Trans. Inf. Theory IT-29, 454–457 (1983).
[CrossRef]

1977 (1)

H. L. Chan, A. K. Fung, “A theory of sea scatter at large incident angles,” J. Geophys. Res. 82, 3439–3444 (1977).
[CrossRef]

1973 (1)

1967 (4)

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
[CrossRef]

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-5, 668–671 (1967).
[CrossRef]

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–627 (1967).

1966 (1)

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

1954 (1)

C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–226 (1954).

Berginc, G.

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

C. Bourlier, J. Saillard, G. Berginc, “Spatial autocorrelation function of the heights of an even sea spectrum,” presented at Poster Sessions (Phenomenology), 5th International Conference on Radar Systems, La Société des Electriciens et des Electroniciens, 17–21 May 1999, Brest, France.

Bourlier, C.

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

C. Bourlier, J. Saillard, G. Berginc, “Spatial autocorrelation function of the heights of an even sea spectrum,” presented at Poster Sessions (Phenomenology), 5th International Conference on Radar Systems, La Société des Electriciens et des Electroniciens, 17–21 May 1999, Brest, France.

Brokelman, R. A.

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–627 (1967).

Chan, H. L.

H. L. Chan, A. K. Fung, “A theory of sea scatter at large incident angles,” J. Geophys. Res. 82, 3439–3444 (1977).
[CrossRef]

Cox, C.

C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–226 (1954).

Daout, F.

F. Daout, “Etude de la dépolarisation des ondes centimétiques par une surface rugueuse—Application au domaine maritime,” Thèse de Doctorat (Institut de Recherche de l’Enseignement Supérieur aux Techniques de l’Electronique, Nantes, France, 1996).

Fung, A. K.

H. L. Chan, A. K. Fung, “A theory of sea scatter at large incident angles,” J. Geophys. Res. 82, 3439–3444 (1977).
[CrossRef]

Hagfors, T.

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–627 (1967).

Hale, G. M.

Ichioka, Y.

K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
[CrossRef] [PubMed]

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

Itoh, K.

K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
[CrossRef] [PubMed]

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

Kunt, M.

M. Kunt, Traitement de l’information VI: techniques modernes de traitement numériques des signaux (Presses Polytechniques Romandes, France, 1991).

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 Sens. Environ. 24, 313–329 (1988).
[CrossRef]

Munk, W.

C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–226 (1954).

Querry, M. R.

Ricciardi, L. M.

L. M. Ricciardi, S. Sato, “On the evaluation of first passage time densities for Gaussian processes,” Signal Process. 11, 339–357 (1986).
[CrossRef]

L. M. Ricciardi, S. Sato, “A note on first passage time problems for Gaussian processes and varying boundaries,” IEEE Trans. Inf. Theory IT-29, 454–457 (1983).
[CrossRef]

Saillard, J.

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

C. Bourlier, J. Saillard, G. Berginc, “Spatial autocorrelation function of the heights of an even sea spectrum,” presented at Poster Sessions (Phenomenology), 5th International Conference on Radar Systems, La Société des Electriciens et des Electroniciens, 17–21 May 1999, Brest, France.

Sato, S.

L. M. Ricciardi, S. Sato, “On the evaluation of first passage time densities for Gaussian processes,” Signal Process. 11, 339–357 (1986).
[CrossRef]

L. M. Ricciardi, S. Sato, “A note on first passage time problems for Gaussian processes and varying boundaries,” IEEE Trans. Inf. Theory IT-29, 454–457 (1983).
[CrossRef]

Saunders, P. M.

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
[CrossRef]

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-5, 668–671 (1967).
[CrossRef]

Smith, W. L.

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 Sens. 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 Sens. Environ. 24, 313–329 (1988).
[CrossRef]

Wagner, R. J.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

Wu, X.

Yoshimori, K.

K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
[CrossRef] [PubMed]

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (3)

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-5, 668–671 (1967).
[CrossRef]

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–627 (1967).

IEEE Trans. Inf. Theory (1)

L. M. Ricciardi, S. Sato, “A note on first passage time problems for Gaussian processes and varying boundaries,” IEEE Trans. Inf. Theory IT-29, 454–457 (1983).
[CrossRef]

J. Geophys. Res. (3)

H. L. Chan, A. K. Fung, “A theory of sea scatter at large incident angles,” J. Geophys. Res. 82, 3439–3444 (1977).
[CrossRef]

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

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
[CrossRef]

J. Mar. Res. (1)

C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–226 (1954).

J. Opt. Soc. Am. (2)

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

Remote Sens. 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 Sens. Environ. 24, 313–329 (1988).
[CrossRef]

Signal Process. (1)

L. M. Ricciardi, S. Sato, “On the evaluation of first passage time densities for Gaussian processes,” Signal Process. 11, 339–357 (1986).
[CrossRef]

Other (3)

M. Kunt, Traitement de l’information VI: techniques modernes de traitement numériques des signaux (Presses Polytechniques Romandes, France, 1991).

F. Daout, “Etude de la dépolarisation des ondes centimétiques par une surface rugueuse—Application au domaine maritime,” Thèse de Doctorat (Institut de Recherche de l’Enseignement Supérieur aux Techniques de l’Electronique, Nantes, France, 1996).

C. Bourlier, J. Saillard, G. Berginc, “Spatial autocorrelation function of the heights of an even sea spectrum,” presented at Poster Sessions (Phenomenology), 5th International Conference on Radar Systems, La Société des Electriciens et des Electroniciens, 17–21 May 1999, Brest, France.

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Figures (13)

Fig. 1
Fig. 1

Illustration of shadowing function.

Fig. 2
Fig. 2

One-dimensional shadowing function for an infinite surface as a function of the parameter v.

Fig. 3
Fig. 3

Difference between one-dimensional shadowing functions for an infinite surface and that obtained from the infinite Gaussian surface.

Fig. 4
Fig. 4

Smith one-dimensional shadowing function as a function of parameter v and normalized correlation length.

Fig. 5
Fig. 5

Relative error between the Smith one-dimensional shadowing function and that obtained from an infinite surface.

Fig. 6
Fig. 6

Incidence limit angle as a function of wind friction speed and observation length.

Fig. 7
Fig. 7

Two-dimensional configuration of shadowing function.

Fig. 8
Fig. 8

Smith two-dimensional shadowing function for infinite observation length with u f = 20 cm/s, function of direction ϕ, and incidence angle θ.

Fig. 9
Fig. 9

Smith two-dimensional shadowing function for infinite observation length with u f = 40 cm/s, function of the direction ϕ, and incidence angle θ.

Fig. 10
Fig. 10

Two-dimensional configuration of the facet.

Fig. 11
Fig. 11

Comparison of authors’ results with those obtained in Ref. 5.

Fig. 12
Fig. 12

Emissivity as a function of the direction ϕ and the incidence angle, for finite observation length of 60 m. {λ = 4 µm, u f = 40 cm/s}.

Fig. 13
Fig. 13

Emissivity as a function of direction ϕ and incidence angle, for infinite observation length. {λ = 4 µm, u f = 40 cm/s}.

Tables (1)

Tables Icon

Table 1 Definition of the Autocorrelation Functions

Equations (54)

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Sθ, F, L0=Υμ-γ0exp-0L0 gθ|F; ldl,
Υμ-γ0=0if γ0μ1if γ0<μ,
gWθ|F; l=μγ-μpξ; γ|ξ0, γ0dγ,  ξ=ξ0+μl, gSθ|F; l=μγ-μpξ; γ|ξ0, γ0dγ--ξ0+μl pξ; γ|ξ0, γ0dξdγ=gWθ|F; l--ξ0+μl pξ; γ|ξ0, γ0dξdγ,
pξpγ=12πσωexp-ξ22ω2-γ22σ2,
gWl|F, θ=μΛv1ω2πexp-ξ0+μl2ω2, gSl|F, θ=gWl|F, θ21+erfξ0+μl2ωgWl|F, θ,
Λv=exp-v2-vπ erfcv2vπ, v=μ2σ=cot θ2σ,
SWθ, F, L0=Υμ-γ0exp-Λ2erfcξ02ω-erfcξ0+μL02ω, SSθ, F, L0=Υμ-γ01-12erfcξ02ω1-12erfcξ0+μL02ωΛ.
Sθ, L0=-- Sθ, Fξ0, γ0, L0pξ0, γ0dξ0dγ0.
SWv, L0=1π1-12erfcv-exp-h02-Λ2erfch0-erfch0+y0vdh0, SSv, L0=1π1-12erfcv-exp-h02×1-12erfch01-12erfch0+y0vΛdh0,
y0=L0l0,  l0=ωσ,  v=cot θ2σ,
Φf=|Gf|2Φxf.
Gf=Φf/Φxf1/2.
yi=gi * xi,
yi=1/ωbwi * xi,
wi=TF-1Φf.
ES=100×SSv, y0-SSvSSv, y0.
R0l=ω2cosl/Lc1+l/Lc2,
ω2=3.953×10-5u104.04, Lc=0.154u102.04, Lc=0.244u101.91;
l0=ωσ=ω-d2R0dl2l=01/2=Lc2+Lc/Lc21/2,
σ2=0.003+5.08×10-3u12,
uz=uf0.4lnzz0,
z0=0.684uf+4.28×10-5uf2-4.43×10-2.
pξ, γX=1ωσX2πexp-γX22σX2-ξ22ω2,
σX2=α+β cos2ϕ, α=σx2+σy22, β=σx2-σy22,
γ  with γX, σ  with σX.
SSθ, ϕ, L0=1π1-12erfcv-exp-h02×1-1/2 erfch01-1/2 erfch0+y0vΛvdh0,
y0ϕ=L0l0ϕ,  v=cot θ2σXϕ.
n=001, n=11+γx2+γy21/2-γx-γy1, s=sin θ cos ϕsin θ sin ϕcos θ,
cos φ=n·s=cos θ-γx cos ϕ+γy sin ϕsin θ1+γx2+γy21/2.
=1-|r|φ||2,
rVφ=n cos φ-cos φn cos φ+cos φ, rHφ=cos φ-n cos φcos φ+n cos φ,  with sin φ=sin φ/n,
pγx, γy=12πσx2σy2-σxy41/2exp-12yx γy×σx2σxy2σxy2σy2-1γxγy,
pγx, γy=12πσxσyexp-γx22σx2-γy22σy2.
¯θ, ϕ=12πσxσy--1-|r|ϕ||2exp-γx22σx2-γy22σy2×g×Sdγxdγy,
g=n·sn·nn·s=1-γx cos ϕ+γy sin ϕtan θ.
γx=γX cos ϕ-γY sin ϕ, γy=γX sin ϕ+γY cos ϕ.
¯θ, ϕ=12πα2-β21/2--1-|r|φ||2exp-aγY2-2bγYγX-cγX2×g×SdγXdγY,
cos φ=cos θ-γX sin θ1+γX2+γY21/2  g=1-γX tan θ, a=α+β cos2ϕ2α2-β2,  b=β sin2ϕ2α2-β2, c=α-β cos2ϕ2α2-β2, α=σx2+σy2/2, β=σx2-σy2/2.
SSθ, ϕ, γX=Υcot θ-γXπ-exp-h02×1-1/2 erfch01-1/2 erfch0+y0vΛvdh0,
¯θ, ϕ=12ππα2-β2--cotθ×1-|r|φ||2exp-aγY2-2bγYγX-cγX2×1-γX tan θdγXdγY-exp-h02×1-1/2 erfch01-1/2 erfch0+y0vΛvdh0.
¯θ, ϕ=12πΛv+1α2-β21/2×-cotθexp-cγX2×1-γX tan θ-1-|r|φ||2exp-aγY2-2bγYγXdγYdγY.
¯90°, ϕ=1σx2πα2-β21/20 γX exp-cγX2-×1-|r|φ||2exp-aγY2-2bγYγXdγγdγX.
R2d1, ϕ=ω2R0lω2-A cos2ϕJ2lL21+lL22,
A=3.439  L2=0.157u101.95  L2=0.138u102.05;
l0ϕ=ωσ=ω2l2-R2dl, ϕl=01/2=Lc2+Lc/Lc2+A/4cos2ϕLc/L221/2,
σx2=3.16×10-3u12, σy2=0.003+1.92×10-3u12.
pξ, γx, γy=12π3 |Cxy|exp-1/2VxyTCxy-1Vxy,
Cxy=ω2000σx2000σy2,  Vxy=ξγxγy,
Vxy=ξγxγy=1000cos ϕ-sin ϕ0sin ϕcos ϕξγXγY=O3VXY.
VxyTCxy-1Vxy=VXYTO3-1CxyO3-1VXY,
pξ, γX, γY=12π3ωσXσY1-ρ21/2exp-ξ22ω2-121-ρ2γX2σX2+γY2σY2-2ργXγYσXσY,
σX2=α+β cos2ϕ, σY2=α-β cos2ϕ, ρ=-β sin2ϕα2-β cos2ϕ21/2,
pξ, γX=- pξ, γX, γYdγY,
pξ, γX=1ωσX2πexp-γX22σX2-ξ22ω2.

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