Abstract

The sea surface turbulent trailing wake of a ship, which can be rather easily observed in the infrared by airborne surveillance systems, is a consequence of the difference in roughness and temperature between the wake and the sea background. We have developed a phenomenological model for the infrared radiance of the turbulent wake by assuming that the sea surface roughness is dependent upon the turbulent intensity near the sea surface. Describing the sea surface roughness with a Cox and Munk probability distribution function of slopes, we distinguish on the sea surface between the sea background and the turbulent wake by the variance of sea surface slopes, σCM2=constant and σTW2(x,y)constant. The latter dependence is assumed to be inversely proportional to the turbulent intensity of the wake, Urms(x,y). Given the incident solar, atmospheric, and sky infrared radiances, we calculate the reflected and emitted sea surface radiance from both the wake and the background. We compare the infrared contrast of the wake with infrared image data obtained in an airborne trial. Our predictions and the measurements agree very well in trend over a significant range of observer zenith angles. Our calculations reveal the strong dependence of the wake radiance on the observer zenith angle, allowing for positive and negative contrasts with the background.

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References

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  1. C. R. Zeisse, C. P. McGrath, K. M. Littfin, and H. G. Hughes, “Infrared radiance of the wind-ruffled sea,” J. Opt. Soc. Am. A 16, 1439–1452 (1999).
    [CrossRef]
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    [CrossRef]
  3. X. Wu and W. L. Smith, “Emissivity of rough sea surface for 8–13  νm: modeling and verification,” Appl. Opt. 36, 2609–2619 (1997).
    [CrossRef]
  4. C. Monzon, D. W. Forester, R. Burkhart, and J. Bellemare, “Rough ocean surface and sunglint region characteristics,” Appl. Opt. 45, 7089–7096 (2006).
    [CrossRef]
  5. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
    [CrossRef]
  6. G. Zilman, A. Zapolski, and M. Marom, “The speed and beam of a ship from its wake’s SAR images,” IEEE Trans. Geosci. Remote Sens. 42, 2335–2343 (2004).
    [CrossRef]
  7. ShipIR/NTCS (Naval Threat & Countermeasures Simulator) software, W. R. Davis Engineering Limited, Canada.
  8. V. Issa and Z. A. Daya, “Sea surface infrared radiance simulator part 1: roughness and temperature models of the sea surface radiance,” (Defence R&D Canada-Atlantic, 2010).
  9. I. B. Schwartz and R. G. Priest, “Reflection driven ship wake contrasts in the infrared,” , 1988.
  10. Z. A. Daya and C. Galloway, “Infrared intensity of the trailing turbulent surface wake of cfav quest in trial q300,” (Defence R&D Canada-Atlantic, 2010).
  11. G. Birkhoff and E. Zarantonello, Jets, Wakes, and Cavities (Academic, 1957).
  12. H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT, 1974).
  13. A. M. Reed and J. H. Milgram, “Ship wakes and their radar images,” Annu. Rev. Fluid Mech. 34, 469–502 (2002).
    [CrossRef]
  14. MODTRAN (MODerate resolution atmospheric TRANsmission) software, Air Force Research Laboratory, US, 1999.

2006

2004

G. Zilman, A. Zapolski, and M. Marom, “The speed and beam of a ship from its wake’s SAR images,” IEEE Trans. Geosci. Remote Sens. 42, 2335–2343 (2004).
[CrossRef]

2002

A. M. Reed and J. H. Milgram, “Ship wakes and their radar images,” Annu. Rev. Fluid Mech. 34, 469–502 (2002).
[CrossRef]

1999

1997

1995

1954

Bellemare, J.

Birkhoff, G.

G. Birkhoff and E. Zarantonello, Jets, Wakes, and Cavities (Academic, 1957).

Burkhart, R.

Cox, C.

Daya, Z. A.

Z. A. Daya and C. Galloway, “Infrared intensity of the trailing turbulent surface wake of cfav quest in trial q300,” (Defence R&D Canada-Atlantic, 2010).

V. Issa and Z. A. Daya, “Sea surface infrared radiance simulator part 1: roughness and temperature models of the sea surface radiance,” (Defence R&D Canada-Atlantic, 2010).

Forester, D. W.

Galloway, C.

Z. A. Daya and C. Galloway, “Infrared intensity of the trailing turbulent surface wake of cfav quest in trial q300,” (Defence R&D Canada-Atlantic, 2010).

Hughes, H. G.

Issa, V.

V. Issa and Z. A. Daya, “Sea surface infrared radiance simulator part 1: roughness and temperature models of the sea surface radiance,” (Defence R&D Canada-Atlantic, 2010).

Littfin, K. M.

Lumley, J. L.

H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT, 1974).

Marom, M.

G. Zilman, A. Zapolski, and M. Marom, “The speed and beam of a ship from its wake’s SAR images,” IEEE Trans. Geosci. Remote Sens. 42, 2335–2343 (2004).
[CrossRef]

McGrath, C. P.

Milgram, J. H.

A. M. Reed and J. H. Milgram, “Ship wakes and their radar images,” Annu. Rev. Fluid Mech. 34, 469–502 (2002).
[CrossRef]

Monzon, C.

Munk, W.

Priest, R. G.

I. B. Schwartz and R. G. Priest, “Reflection driven ship wake contrasts in the infrared,” , 1988.

Reed, A. M.

A. M. Reed and J. H. Milgram, “Ship wakes and their radar images,” Annu. Rev. Fluid Mech. 34, 469–502 (2002).
[CrossRef]

Schwartz, I. B.

I. B. Schwartz and R. G. Priest, “Reflection driven ship wake contrasts in the infrared,” , 1988.

Smith, W. L.

Tennekes, H.

H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT, 1974).

Wu, X.

Zapolski, A.

G. Zilman, A. Zapolski, and M. Marom, “The speed and beam of a ship from its wake’s SAR images,” IEEE Trans. Geosci. Remote Sens. 42, 2335–2343 (2004).
[CrossRef]

Zarantonello, E.

G. Birkhoff and E. Zarantonello, Jets, Wakes, and Cavities (Academic, 1957).

Zeisse, C. R.

Zilman, G.

G. Zilman, A. Zapolski, and M. Marom, “The speed and beam of a ship from its wake’s SAR images,” IEEE Trans. Geosci. Remote Sens. 42, 2335–2343 (2004).
[CrossRef]

Annu. Rev. Fluid Mech.

A. M. Reed and J. H. Milgram, “Ship wakes and their radar images,” Annu. Rev. Fluid Mech. 34, 469–502 (2002).
[CrossRef]

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

G. Zilman, A. Zapolski, and M. Marom, “The speed and beam of a ship from its wake’s SAR images,” IEEE Trans. Geosci. Remote Sens. 42, 2335–2343 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

ShipIR/NTCS (Naval Threat & Countermeasures Simulator) software, W. R. Davis Engineering Limited, Canada.

V. Issa and Z. A. Daya, “Sea surface infrared radiance simulator part 1: roughness and temperature models of the sea surface radiance,” (Defence R&D Canada-Atlantic, 2010).

I. B. Schwartz and R. G. Priest, “Reflection driven ship wake contrasts in the infrared,” , 1988.

Z. A. Daya and C. Galloway, “Infrared intensity of the trailing turbulent surface wake of cfav quest in trial q300,” (Defence R&D Canada-Atlantic, 2010).

G. Birkhoff and E. Zarantonello, Jets, Wakes, and Cavities (Academic, 1957).

H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT, 1974).

MODTRAN (MODerate resolution atmospheric TRANsmission) software, Air Force Research Laboratory, US, 1999.

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Figures (24)

Fig. 1.
Fig. 1.

Flowchart of our approach and overall model.

Fig. 2.
Fig. 2.

Sea surface radiance IR simulator high level architecture.

Fig. 3.
Fig. 3.

Coordinate system and geometry of a sea surface facet.

Fig. 4.
Fig. 4.

Geometric parameters for the geographic area seen by one pixel.

Fig. 5.
Fig. 5.

Turbulence intensity, Urms (top), and sea surface roughness (bottom) of the trailing wake for a wind speed of 4m/s (σcm=0.0117).

Fig. 6.
Fig. 6.

Cox and Munk PDF as a function of the slopes, zx and zy, for two different wind speeds, (top) 4m/s and (bottom) 6m/s.

Fig. 7.
Fig. 7.

Trailing wake PDF as a function of the slopes, zx and zy, at a wind speed of 4m/s. for two different positions in the TW (top) 30 m and (bottom) 500 m.

Fig. 8.
Fig. 8.

Contrast radiance of the central line of the trailing wake as a function of downstream distance, (top) in a linear scale and (bottom) in a log–log scale.

Fig. 9.
Fig. 9.

Parameter, C, of the radiance contrast of the trailing wake toward a receiver at 300 m, (top) for various approach range, and (bottom) for different zenith angle.

Fig. 10.
Fig. 10.

Dependence of emissivity and reflectivity of the sea surface on the angle of incidence.

Fig. 11.
Fig. 11.

Examples of the geometry for a receiver at a high zenith angle; (left) with no roughness, (middle) example of a high roughness: ω increases but θs decreases; (right) example of a high roughness: hidden facet.

Fig. 12.
Fig. 12.

Total radiance (top), the emitted radiance (middle), and the reflected radiance (bottom) of the background (blue line), the trailing wake with a variable roughness and constant temperature (°), the trailing wake with a variable temperature and constant roughness (+), the trailing wake with a variable temperature and roughness (red line) toward a low altitude broadside receiver (high zenith angle) located at (0, 1150, 100).

Fig. 13.
Fig. 13.

Examples of the geometry for a receiver at a low zenith angle: (left) with no roughness; (middle) example of a high roughness: ω decreases but θs increases; (right) example of a high roughness: ω and θs increase.

Fig. 14.
Fig. 14.

Total radiance (top), the emitted radiance (middle), and the reflected radiance (bottom) of the background (blue line), the trailing wake with a variable roughness and constant temperature (°), the trailing wake with a variable temperature and constant roughness (+), the trailing wake with a variable temperature and roughness (red line) toward a high altitude broadside receiver (low zenith angle) located at (0 1000 1000).

Fig. 15.
Fig. 15.

Typical image of the data showing the Quest, her trailing wake and the coordinate system used in [10].

Fig. 16.
Fig. 16.

Example of the cross stream intensity of the trailing wake. The blue line is a polynomial fit to the background variation outside the wake, i.e., beyond the red dashed lines. Figure from Ref. [10].

Fig. 17.
Fig. 17.

Peak wake contrast intensity for various approach ranges for the MPA flights at (top) 300 m and (bottom) 30 m. Squares with dotted lines denote measured peak wake intensity and the solid lines show the corresponding fits to constantx(4/5).

Fig. 18.
Fig. 18.

Peak wake contrast intensity from SSRS for various approach ranges for the MPA flights at (top) 300 m and (bottom) 30 m. Squares with dotted lines denote exact contrast radiance from the SSRS calculation [Eq. (22)] and the solid lines refer to the approximated results [Eq. (31)] with Cx4/5.

Fig. 19.
Fig. 19.

Different C values corresponding to each range for the flight at 300 m, from the Q300 data and SSRS.

Fig. 20.
Fig. 20.

Different C values corresponding to each range for the flight at 30 m from, the Q300 data and SSRS.

Fig. 21.
Fig. 21.

Different C values corresponding to each range for the flight at 30 and 300 m from the Q300 data and from SSRS, with and without atmospheric propagation as a function of the zenith angle.

Fig. 22.
Fig. 22.

Orientation of the normal vector n^=(θn,ϕn) for a sea surface facet as a function of the slopes zx and zy. (Top)θn and (bottom) ϕn.

Fig. 23.
Fig. 23.

Zenith orientation of the sky reflected on the sea surface facet as a function of the slopes zx and zy.

Fig. 24.
Fig. 24.

Angle, ωr, (top) between the receiver and the normal to the facet as a function of the slopes zx and zy and (bottom) the function where the condition ω(π/2) applies for a receiver at (θr=85, ϕr=90).

Tables (1)

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Table 1. Background Parameters for the Trial Q300

Equations (40)

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L(λ,T,θr,ϕr,zx,zy)=Le(λ,T,θr,ϕr,zx,zy)+Lr(λ,θr,ϕr,zx,zy)=[1ρ(ω,λ)]×P(T,λ)+ρ(ω)×Ls(λ,θs,ϕs),
L(λ,T,θr,ϕr)=ωπ2dAr(θr,ϕr,zx,zy)Ar(θr,ϕr)L(λ,T,θr,ϕr,zx,zy)dzxdzy
dAr(θr,ϕr,zx,zy)=cosω(θr,ϕr,zx,zy)cosθn(θr,ϕr,zx,zy)×dAh(θr,ϕr,zx,zy).
dAh(θr,ϕr,zx,zy)Ah(θr,ϕr)=pcm(zx,zy,w)dzxdzy,
pcm(zx,zy,w)12πσuσcexp[12(zx2σu2+zy2σc2)].
σu2=3.16×103w,σc2=0.003+1.92×103w.
dAr(θr,ϕr,zx,zy)Ar(θr,ϕr)=dAr(θr,ϕr,zx,zy)ωπ2dAr(θr,ϕr,zx,zy)dzxdzy,
dAr(θr,ϕr,zx,zy)Ar(θr,ϕr)=cosω(θr,ϕr,zx,zy)cosθn(θr,ϕr,zx,zy)pcm(zx,zy,w)dzxdzyωπ2cosω(θr,ϕr,zx,zy)cosθn(θr,ϕr,zx,zy)pcm(zx,zy,w)dzxdzy.
L(λ,θr,ϕr,x,y)=ωπ2ξpcm(zx,zy,w,x,y)dzxdzyωπ2αpcm(zx,zy,w,x,y)dzxdzy
L(T,θr,ϕr,x,y)=λ1λ2L(λ,T,θr,ϕr,x,y)dλ.
Urmsxx45,
Urmsy(112ζ2)e12ζ2,
ζ=yl(x)
lx15.
σTW2=[1UA*Urmsy*Urmsx+UB]*Γ+(1Γ)*1UB
Γ={0ify>l,1ifyl,
σcm2=0.003+0.00512w2.
ptw(zx,zy,w,x,y)[Urmsyσcm2σA2x4/5+1]pcm(zx,zy,w)*e12zx2+zy2σA2Urmsyx4/5,
pcm(zx,zy,w)12πσcm2e12zx2+zy2σcm2.
Ltw(λ,θr,ϕr,x,y)=ωπ2ξtwptw(zx,zy,w,x,y)dzxdzyωπ2αptw(zx,zy,w,x,y)dzxdzy
Ltw(θr,ϕr,x,y)=λ1λ2Ltw(λ,θr,ϕr,x,y)dλ.
Contrast(λ,x,y)=Ltw(λ,Ttw,θr,ϕr,x,y)Lbck(λ,Tbckθr,ϕr,x,y)=ωπ2ξtwptw(zx,zy,w,x,y)dzxdzyωπ2αptw(zx,zy,w,x,y)dzxdzyωπ2ξbckpcm(zx,zy,w,x,y)dzxdzyωπ2αpcm(zx,zy,w,x,y)dzxdzy,
ptw(zx,zy,w,x,y)[σcm2σA2x4/5+1]pcm(zx,zy,w)e12zx2+zy2σA2x4/5.
Contrast(λ,x,y)=ωπ2ξtwe12zx2+zy2σA2x4/5pcm(zx,zy,w,x,y)dzxdzyωπ2αe12zx2+zy2σA2x4/5pcm(zx,zy,w,x,y)dzxdzyωπ2ξbckpcm(zx,zy,w,x,y)dzxdzyωπ2αpcm(zx,zy,w,x,y)dzxdzy,
Contrast(λ,x,y)=n=0(x4/52σA2)n1n!γξtwnm=0(x4/52σA2)m1m!γαmγξbck0γα0,
γξk=ωπ2ξ(zx2+zy2)kpcm(zx,zy,w,x,y)dzxdzy
γαk=ωπ2α(zx2+zy2)kpcm(zx,zy,w,x,y)dzxdzy.
Contrast(λ,x,y)2σA2γα0[γξtw0γξbck0]2σA2(γα0)2+[γξbck0γα1γξbck1γα0]x4/52σA2(γα0)2.
Contrast(λ,x,y)[γξ0γα1γξ1γα02σA2(γα0)2]x4/5.
Contrast(x,y)=λ1λ2Contrast(λ,x,y)dλ.
Contrast(x,y)Cx4/5,
C=λ1λ2[γξ0γα1γξ1γα02σA2(γα0)2]dλ.
Contrast(λ,x,y)=τ(λ)Ltw(λ,Ttw,θr,ϕr,x,y)τ(λ)Lbck(λ,Tbckθr,ϕr,x,y).
Contrast(x,y)Cx4/5,
C=λ1λ2τ(λ)[γξ0γα1γξ1γα02σA2(γα0)2]dλ.
θn=atan(zx2+zy2)
ϕn={atan(zyzx)zx<0andzy<0,atan(zyzx)+2πzx<0andzy>0,atan(zyzx)+πzx>0.
cosθs=2ζcosθncosθr
cosϕs=2ζsinθncosϕnsinθrcosϕrsinθs,
ζ=sinθncosϕnsinθrcosϕr+sinθnsinϕnsinθrsinϕr+cosθncosθr=n^·r^=cosω.

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