Abstract

The development of a simple model of the seawater inherent optical properties (IOPs) associated with bubbles and sediments would represent a great advance in surf zone optics. We present one solution for this problem using a combination of geometrical optics and Fraunhofer diffraction. An analytic model of the IOPs of bubbles and sediments (the extinction and absorption coefficients, and phase function) is developed in terms of the moments of the particle size distribution and the complex refractive index of particles.

© 2006 Optical Society of America

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References

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  1. O. V. Kopelevich, "Factors determining the optical characteristics of seawaters," in Ocean Optics, A. S. Monin ed. (Nauka, Moscow, 1983), Vol. 1, pp. 150-163.
  2. K. S. Shifrin, Scattering of Light in a Turbid Media (Gostekhteorizdat, Moscow, 1951) [English translation: NASA Report TT F-447 (NASA, Washington DC, 1968).
  3. E. P. Zege, I. L. Katsev, and A. A. Kokhanovsky, "Phenomenological optical model of dense packed scattering media and its application to the foam optics," Opt. Spektrosk. 71, 835-841 (1991).
  4. E. Terrill and M. R. Lewis, "Tiny bubbles: an overlooked optical constituent," Oceanogr. 17, 11-18 (2004).
  5. X. Zhang, M. Lewis, B. Johnson, and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
    [CrossRef]
  6. X. Zhang, M. Lewis, and B. Johnson, "Influence of bubbles on scattering of light in the ocean," Appl. Opt. 37, 6525-6536 (1998).
    [CrossRef]
  7. E. J. Terrill, W. K. Melvlle, and D. Stramski, "Bubble entrainment by breaking waves and their effect on the inherent optical properties of the upper ocean," Ocean Optics OOXIV, Kona, Hl (Office of Naval Research, November 10-13, 1998), Vol. 1231, on CD.
  8. A. A. Kokhanovsky and E. P. Zege, "Scattering optics of snow," Appl. Opt. 43, 1589-1602 (2004).
    [CrossRef] [PubMed]
  9. A. A. Kokhanovsky, "Optical properties of irregularly shaped particles," J. Appl. Phys. 36, 915-923 (2003).
  10. E. P. Zege and A. A. Kokhanovsky, "Integral characteristics of light scattering by large spherical particles," Izv. Akad. Nauk SSSR , Fiz. Atmos. Okeana 24, 695-701 (1987).
  11. A. A. Kokhanovsky and E. P. Zege, "Parameterization of the local optical characteristics of clouds," Izv. Akad. Nauk , Fiz. Atmos. Okeana 33, 209-218 (1997).
  12. W. E. McBride III, A. D. Weidemann, and J. T. Shoemaker, "Meeting Navy needs with the generic lidar model," in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE 3761, 71-82 (1999).
  13. E. P. Zege and A. A. Kokhanovsky, "Analytical solution to the optical transfer function of a scattering medium with large particles," Appl. Opt. 33, 6547-6554 (1994).
    [CrossRef] [PubMed]

2004

E. Terrill and M. R. Lewis, "Tiny bubbles: an overlooked optical constituent," Oceanogr. 17, 11-18 (2004).

A. A. Kokhanovsky and E. P. Zege, "Scattering optics of snow," Appl. Opt. 43, 1589-1602 (2004).
[CrossRef] [PubMed]

2003

A. A. Kokhanovsky, "Optical properties of irregularly shaped particles," J. Appl. Phys. 36, 915-923 (2003).

2002

X. Zhang, M. Lewis, B. Johnson, and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

1999

W. E. McBride III, A. D. Weidemann, and J. T. Shoemaker, "Meeting Navy needs with the generic lidar model," in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE 3761, 71-82 (1999).

1998

1997

A. A. Kokhanovsky and E. P. Zege, "Parameterization of the local optical characteristics of clouds," Izv. Akad. Nauk , Fiz. Atmos. Okeana 33, 209-218 (1997).

1994

1991

E. P. Zege, I. L. Katsev, and A. A. Kokhanovsky, "Phenomenological optical model of dense packed scattering media and its application to the foam optics," Opt. Spektrosk. 71, 835-841 (1991).

1987

E. P. Zege and A. A. Kokhanovsky, "Integral characteristics of light scattering by large spherical particles," Izv. Akad. Nauk SSSR , Fiz. Atmos. Okeana 24, 695-701 (1987).

Johnson, B.

X. Zhang, M. Lewis, B. Johnson, and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

X. Zhang, M. Lewis, and B. Johnson, "Influence of bubbles on scattering of light in the ocean," Appl. Opt. 37, 6525-6536 (1998).
[CrossRef]

Katsev, I. L.

E. P. Zege, I. L. Katsev, and A. A. Kokhanovsky, "Phenomenological optical model of dense packed scattering media and its application to the foam optics," Opt. Spektrosk. 71, 835-841 (1991).

Kokhanovsky, A. A.

A. A. Kokhanovsky and E. P. Zege, "Scattering optics of snow," Appl. Opt. 43, 1589-1602 (2004).
[CrossRef] [PubMed]

A. A. Kokhanovsky, "Optical properties of irregularly shaped particles," J. Appl. Phys. 36, 915-923 (2003).

A. A. Kokhanovsky and E. P. Zege, "Parameterization of the local optical characteristics of clouds," Izv. Akad. Nauk , Fiz. Atmos. Okeana 33, 209-218 (1997).

E. P. Zege and A. A. Kokhanovsky, "Analytical solution to the optical transfer function of a scattering medium with large particles," Appl. Opt. 33, 6547-6554 (1994).
[CrossRef] [PubMed]

E. P. Zege, I. L. Katsev, and A. A. Kokhanovsky, "Phenomenological optical model of dense packed scattering media and its application to the foam optics," Opt. Spektrosk. 71, 835-841 (1991).

E. P. Zege and A. A. Kokhanovsky, "Integral characteristics of light scattering by large spherical particles," Izv. Akad. Nauk SSSR , Fiz. Atmos. Okeana 24, 695-701 (1987).

Kopelevich, O. V.

O. V. Kopelevich, "Factors determining the optical characteristics of seawaters," in Ocean Optics, A. S. Monin ed. (Nauka, Moscow, 1983), Vol. 1, pp. 150-163.

Korotaev, G.

X. Zhang, M. Lewis, B. Johnson, and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

Lewis, M.

X. Zhang, M. Lewis, B. Johnson, and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

X. Zhang, M. Lewis, and B. Johnson, "Influence of bubbles on scattering of light in the ocean," Appl. Opt. 37, 6525-6536 (1998).
[CrossRef]

Lewis, M. R.

E. Terrill and M. R. Lewis, "Tiny bubbles: an overlooked optical constituent," Oceanogr. 17, 11-18 (2004).

McBride, W. E.

W. E. McBride III, A. D. Weidemann, and J. T. Shoemaker, "Meeting Navy needs with the generic lidar model," in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE 3761, 71-82 (1999).

Melvlle, W. K.

E. J. Terrill, W. K. Melvlle, and D. Stramski, "Bubble entrainment by breaking waves and their effect on the inherent optical properties of the upper ocean," Ocean Optics OOXIV, Kona, Hl (Office of Naval Research, November 10-13, 1998), Vol. 1231, on CD.

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Media (Gostekhteorizdat, Moscow, 1951) [English translation: NASA Report TT F-447 (NASA, Washington DC, 1968).

Shoemaker, J. T.

W. E. McBride III, A. D. Weidemann, and J. T. Shoemaker, "Meeting Navy needs with the generic lidar model," in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE 3761, 71-82 (1999).

Stramski, D.

E. J. Terrill, W. K. Melvlle, and D. Stramski, "Bubble entrainment by breaking waves and their effect on the inherent optical properties of the upper ocean," Ocean Optics OOXIV, Kona, Hl (Office of Naval Research, November 10-13, 1998), Vol. 1231, on CD.

Terrill, E.

E. Terrill and M. R. Lewis, "Tiny bubbles: an overlooked optical constituent," Oceanogr. 17, 11-18 (2004).

Terrill, E. J.

E. J. Terrill, W. K. Melvlle, and D. Stramski, "Bubble entrainment by breaking waves and their effect on the inherent optical properties of the upper ocean," Ocean Optics OOXIV, Kona, Hl (Office of Naval Research, November 10-13, 1998), Vol. 1231, on CD.

Weidemann, A. D.

W. E. McBride III, A. D. Weidemann, and J. T. Shoemaker, "Meeting Navy needs with the generic lidar model," in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE 3761, 71-82 (1999).

Zege, E. P.

A. A. Kokhanovsky and E. P. Zege, "Scattering optics of snow," Appl. Opt. 43, 1589-1602 (2004).
[CrossRef] [PubMed]

A. A. Kokhanovsky and E. P. Zege, "Parameterization of the local optical characteristics of clouds," Izv. Akad. Nauk , Fiz. Atmos. Okeana 33, 209-218 (1997).

E. P. Zege and A. A. Kokhanovsky, "Analytical solution to the optical transfer function of a scattering medium with large particles," Appl. Opt. 33, 6547-6554 (1994).
[CrossRef] [PubMed]

E. P. Zege, I. L. Katsev, and A. A. Kokhanovsky, "Phenomenological optical model of dense packed scattering media and its application to the foam optics," Opt. Spektrosk. 71, 835-841 (1991).

E. P. Zege and A. A. Kokhanovsky, "Integral characteristics of light scattering by large spherical particles," Izv. Akad. Nauk SSSR , Fiz. Atmos. Okeana 24, 695-701 (1987).

Zhang, X.

X. Zhang, M. Lewis, B. Johnson, and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

X. Zhang, M. Lewis, and B. Johnson, "Influence of bubbles on scattering of light in the ocean," Appl. Opt. 37, 6525-6536 (1998).
[CrossRef]

Appl. Opt.

Izv. Akad. Nauk

A. A. Kokhanovsky and E. P. Zege, "Parameterization of the local optical characteristics of clouds," Izv. Akad. Nauk , Fiz. Atmos. Okeana 33, 209-218 (1997).

Izv. Akad. Nauk SSSR

E. P. Zege and A. A. Kokhanovsky, "Integral characteristics of light scattering by large spherical particles," Izv. Akad. Nauk SSSR , Fiz. Atmos. Okeana 24, 695-701 (1987).

J. Appl. Phys.

A. A. Kokhanovsky, "Optical properties of irregularly shaped particles," J. Appl. Phys. 36, 915-923 (2003).

Limnol. Oceanogr.

X. Zhang, M. Lewis, B. Johnson, and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

Oceanogr.

E. Terrill and M. R. Lewis, "Tiny bubbles: an overlooked optical constituent," Oceanogr. 17, 11-18 (2004).

Opt. Spektrosk.

E. P. Zege, I. L. Katsev, and A. A. Kokhanovsky, "Phenomenological optical model of dense packed scattering media and its application to the foam optics," Opt. Spektrosk. 71, 835-841 (1991).

Other

O. V. Kopelevich, "Factors determining the optical characteristics of seawaters," in Ocean Optics, A. S. Monin ed. (Nauka, Moscow, 1983), Vol. 1, pp. 150-163.

K. S. Shifrin, Scattering of Light in a Turbid Media (Gostekhteorizdat, Moscow, 1951) [English translation: NASA Report TT F-447 (NASA, Washington DC, 1968).

W. E. McBride III, A. D. Weidemann, and J. T. Shoemaker, "Meeting Navy needs with the generic lidar model," in Airborne and In-Water Underwater Imaging, G. D. Gilbert, ed., Proc. SPIE 3761, 71-82 (1999).

E. J. Terrill, W. K. Melvlle, and D. Stramski, "Bubble entrainment by breaking waves and their effect on the inherent optical properties of the upper ocean," Ocean Optics OOXIV, Kona, Hl (Office of Naval Research, November 10-13, 1998), Vol. 1231, on CD.

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Figures (4)

Fig. 1
Fig. 1

Examples of approximations of experimental data (Ref. 6) (solid curves) for 1, r = 52   μm and 2, r = 110   μm compared to the lognormal distributions (dashed curves).

Fig. 2
Fig. 2

Small-angle phase function: Mie calculation (solid curves) and the simple approximation obtained with Eq. (51) substituted into Eq. (49) (dashed curves).

Fig. 3
Fig. 3

GO phase function (dashed curve) for the bubbles in comparison with the Mie calculations (solid curve).

Fig. 4
Fig. 4

Phase functions of bubbles with r = 3 , 50, and 180   μm , sediments sand and clay with r = 15 , 50, and 180   μm . Solid curves, Mie calculation; dashed curves, our approximation.

Tables (3)

Tables Icon

Table 1 Some Constants for Bubbles and Sediments at a Wavelength of λ = 0.524 μm

Tables Icon

Table 2 Values of S ( n ), α, and η

Tables Icon

Table 3 Parameters of the Phase Function (72) for the Bubbles, Sand, and Clay Particles

Equations (75)

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n ( r ) = N 1 ( 2 π ) 1 / 2 σ r exp [ ln 2 ( r / r 0 ) 2 σ 2 ] .
M k = ( 1 / N ) 0 r k n ( r ) d r
M k = r k = r 0 k exp ( σ 2 k 2 / 2 ) .
r = r 0 exp ( σ 2 / 2 ) ,
r 2 = r 0 2 exp ( 2 σ 2 ) ,
r 2 = r 2 exp ( σ 2 ) .
r 0 = 0.59 r 1.09 ,
σ = ln ( 3.06 r 0.13 ) .
Q ext = 2 π r 2 .
c = 0 Q ext ( r ) n ( r ) d r .
c = 2 π N r 2 = 3 C v 2 r ef .
r ef = r 3 / r 2 ,
C v = 4 3 π N r 3 .
c = 2 π N r 2 exp ( σ 2 ) .
Q abs = π r 2 { 1 φ ( l , n ) S ( n ) [ 1 exp ( b l ) ] 2 } .
l = 2 r ( 4 π χ / λ ) = 4 χ ρ ,
φ ( l , n ) = ( 2 n 2 / l 2 ) [ ( 1 + b l ) exp ( b l ) ( 1 + l ) exp ( l ) ] ,
b ( n ) = ( 1 n 2 ) 1 / 2 .
ρ = 2 π r / λ .
S ( n ) = 0 π / 2 R ( β ) sin ( 2 β ) d β ,
S ( n ) = 0.0841 n 2 + 0.3816 n 0.2927.
a = 0 Q abs ( r ) n ( r ) d r .
Q abs ( r ) = π r 2 [ 1 S ( n ) ] ,
a = π N r 2 [ 1 S ( n ) ] .
a = π N r 2 exp ( σ 2 ) [ 1 S ( n ) ] .
ω 0 = 0.5 [ 1 + S ( n ) ] .
ω 0 = { 1 for   bubbles , 0 .523 for   clay, 0 .518 for   sand .
a = ( c / 2 ) ( 1 φ ( l , n ) S ( n ) { 1 exp [ b ( n ) l ] } 2 ) ,
l = 8 π χ r / λ ,
φ ( l , n ) = 2 n 2 l 2 { ( 1 + b ( n ) l ) exp [ b ( n ) l ] ( 1 + l ) exp ( l ) } .
P ( β ) = σ dif P dif ( β ) + σ GO P GO ( β ) σ dif + σ GO ,
σ dif ( r , β ) = ρ 2 r 2 4 [ 2 J 1 ( ρ β ) ρ β ] 2 ,
P dif ( r , β ) = 4 π σ dif ( r , β ) σ dif ( r ) ,
σ dif ( r ) = 2 π 0 σ dif ( r , β ) β d β = π r 2 .
1 2 0 P dif ( r , β ) β d β = 1 .
P dif ( r , β ) = ρ 2 [ 2 J 1 ( ρ β ) ρ β ] 2 .
σ dif ( β ) = 0 σ dif ( r , β ) n ( r ) d r = 0 ρ 2 r 2 4 [ 2 J 1 ( ρ β ) ρ β ] 2 n ( r ) d r ,
σ dif = 2 π 0 σ dif ( β ) β d β = N π r 2 ,
P dif ( β ) = 1 r 2 0 ρ 2 r 2 [ 2 J 1 ( ρ β ) ρ β ] 2 n ( r ) N d r .
J 1 ( z ) = z 2 ( 1 z 2 8 ) .
r m n = r m / r n , ρ m n = ρ m / ρ n ,
P dif ( β ) = ρ 42 ( 1 ρ 64 β 2 / 4 ) .
r n = a 0 n μ n Γ ( μ + n + 1 ) Γ ( μ + 1 ) ,
J 1 ( z ) = 2 / π z cos 2 ( z 3 π / 4 ) .
P dif ( β ) = 4 λ π 2 β 3 r 2 0 r cos 4 ( 2 π r λ β 3 π 4 ) n ( r ) N d r .
cos 4 ( 2 π r λ β 3 π 4 ) cos 4 ( 2 π r λ β 3 π 4 ) = 3 8
P dif ( β ) = 3 λ r 2 π 2 β 3 r 2 .
P dif ( β ) C β 3 ,
P dif ( β ) = 2 a ( a 2 + β 2 ) 3 / 2 .
P dif ( β ) = 2 a 2 ( 1 3 β 2 2 a 2 ) .
a 2 = 2 / ρ 42 ,
P dif ( β ) = ρ 42 ( 1 3 ρ 42 β 2 4 ) .
a 2 = 6 / ρ 64 ,
P dif ( β ) = ρ 64 3 ( 1 ρ 64 β 2 12 ) .
ρ 64 = 3 ρ 42 .
ρ 64 / ρ 42 = exp ( 4 σ 2 ) .
σ = ln 3.06 0.14 ln r = 1.12 0.14 ln r ,
ρ 64 ρ 42 = ( μ + 6 ) ( μ + 5 ) ( μ + 4 ) ( μ + 3 ) .
σ GO ( β ) = 0 I ( r , β ) n ( r ) d r ,
I ( r , β ) = j = 1 I j ( r , β ) .
I 1 ( r , β ) = r 2 R ( β ) 4 ,
I 2 ( r , β ) = r 2 T ( β ) 4 exp [ l Δ ( β ) ] .
R ( β ) = 1 2 j = 1 2 [ N j     2 ( 1 q 2 ) 1 / 2 ( n 2 q 2 ) 1 / 2 N j     2 ( 1 q 2 ) 1 / 2 + ( n 2 q 2 ) 1 / 2 ] 2 ,
T ( β ) = ( 2 n n 2 1 ) 4 ( n q 1 ) 3 ( n q ) 3 ( 1 + q 4 ) 2 q 5 ( 1 + n 2 2 n q ) 2 ,
Δ ( β ) = ( n q ) ( 1 + n 2 2 n q ) 1 / 2 ,
q = cos β 2 , N 1 = 1 , N 2 = n 2 .
R ( β ) = exp ( α β ) ,
T ( β ) = T ( 0 ) exp ( η β 2 l ) ,
Δ ( β ) = 1 2 n 1 4 ( n 1 ) β 2 ,
T ( 0 ) = 1 ( n 1 ) 2 ( 2 n n + 1 ) 4 .
P ( β ) = 4 π σ dif ( β ) + σ GO ( β ) σ dif + σ GO .
P ( β ) = 2 F [ α P dif ( β ) + ( 1 α ) c 22           2 exp ( c 22 β ) ] + P min + k = 1 3 d k exp [ ( π β ) 2 / 2 V d k ] ,
F = 1 ( 2 P min + k = 1 3 d k V d k ) / 2
α = 0.5 / ω 0 F .
P ( π ) = 0.002 r 32 + 0.0118 , r 32 = r 0 exp ( 2.5 σ 2 ) .

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