Abstract

The point spread and modulation transfer functions for natural waters are derived using the small-angle approximation to radiative transfer theory. The functional forms are expanded into a summation of terms that represent each order-of-scattering contribution to the total. The beam spread function is shown to be a product of an angle function that depends only on the phase function of the medium and a weighting factor that depends only on the optical properties and depth. The modulation transfer function is similarly shown as a product of a function depending only on the spatial frequency and a weighting function. These results are compared with Monte Carlo calculations using two different phase functions, with excellent agreement. The results suggest the small-angle approximation to be valid over a much larger angular range than previously thought.

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    [CrossRef]
  2. C. J. Funk, S. B. Bryant, and P. J. Heckman, Handbook of Underwater Imaging System Design (Ocean Technology Department, Naval Undersea Center, 1972).
  3. W. H. Wells, “Theory of small-angle scattering,” AGARD Lect. Ser. 61, 1–19 (1973).
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    [CrossRef]
  5. E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
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    [CrossRef]
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    [CrossRef]
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  9. D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. Zaneveld, “Comparison of near-forward light scattering on oceanic turbulence and particle,” Appl. Opt. 37, 4669–4677 (1998).
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  10. W. T. Scott, “The theory of small-angle multiple-scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
    [CrossRef]
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    [CrossRef]
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2008 (1)

2003 (1)

2001 (2)

1998 (1)

1995 (1)

1994 (1)

1990 (1)

J. S. Jaffe, “Computer modeling and the design of optimal underwater imaging systems,” IEEE J. Ocean Eng. 15, 101–111 (1990).
[CrossRef]

1979 (1)

1977 (1)

1973 (1)

W. H. Wells, “Theory of small-angle scattering,” AGARD Lect. Ser. 61, 1–19 (1973).

1969 (2)

D. M. Bravo-Zhivotovskiy, L. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, “Structure of a narrow light beam in sea water,” Izv. Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) 5, 160–167 (1969) (translated by P. A. Kaehn).

W. H. Wells, “Loss of resolution in water as a result of multiple small-angle scattering,” J. Opt. Soc. Am. 59, 686–691 (1969).
[CrossRef]

1963 (1)

W. T. Scott, “The theory of small-angle multiple-scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
[CrossRef]

Arnone, R. A.

Billard, B. D.

Bogucki, D.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).

Bravo-Zhivotovskiy, D. M.

D. M. Bravo-Zhivotovskiy, L. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, “Structure of a narrow light beam in sea water,” Izv. Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) 5, 160–167 (1969) (translated by P. A. Kaehn).

Bryant, S. B.

C. J. Funk, S. B. Bryant, and P. J. Heckman, Handbook of Underwater Imaging System Design (Ocean Technology Department, Naval Undersea Center, 1972).

Conchello, J.-A.

Dolin, L. S.

D. M. Bravo-Zhivotovskiy, L. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, “Structure of a narrow light beam in sea water,” Izv. Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) 5, 160–167 (1969) (translated by P. A. Kaehn).

Domaradzki, J. A.

Funk, C. J.

C. J. Funk, S. B. Bryant, and P. J. Heckman, Handbook of Underwater Imaging System Design (Ocean Technology Department, Naval Undersea Center, 1972).

Gehman, V. M.

Gennaro, T. L.

Gordon, A.

A. Gordon and M. R. Kittel, “Underwater multiple scattering of light for system designers,” Naval Undersea Center Technical Report 371 (National Technical Information Service, 1973).

Gordon, H. R.

Gray, D. J.

Haltrin, V. I.

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, 1997).

Heckman, P. J.

C. J. Funk, S. B. Bryant, and P. J. Heckman, Handbook of Underwater Imaging System Design (Ocean Technology Department, Naval Undersea Center, 1972).

Hou, W.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).

Jaffe, J. S.

J. S. Jaffe, “Monte Carlo modeling of underwater-image formation: validity of the linear and small-angle approximations,” Appl. Opt. 34, 5413–5421 (1995).
[CrossRef]

J. S. Jaffe, “Computer modeling and the design of optimal underwater imaging systems,” IEEE J. Ocean Eng. 15, 101–111 (1990).
[CrossRef]

Katsev, I. L.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).

Kittel, M. R.

A. Gordon and M. R. Kittel, “Underwater multiple scattering of light for system designers,” Naval Undersea Center Technical Report 371 (National Technical Information Service, 1973).

Luchinin, A. G.

D. M. Bravo-Zhivotovskiy, L. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, “Structure of a narrow light beam in sea water,” Izv. Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) 5, 160–167 (1969) (translated by P. A. Kaehn).

Markham, J.

Mertens, L. E.

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Reference 72–78 (Scripps Institute of Oceanography, University of California, San Diego,1972).

Replogle, F. S.

Savel’yev, V. A.

D. M. Bravo-Zhivotovskiy, L. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, “Structure of a narrow light beam in sea water,” Izv. Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) 5, 160–167 (1969) (translated by P. A. Kaehn).

Scott, W. T.

W. T. Scott, “The theory of small-angle multiple-scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
[CrossRef]

Stotts, L. B.

Stramski, D.

Swanson, N. L.

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley, 1994).

Weidemann, A. D.

Wells, W. H.

Zaneveld, J. R.

Zege, E. P.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).

AGARD Lect. Ser. (1)

W. H. Wells, “Theory of small-angle scattering,” AGARD Lect. Ser. 61, 1–19 (1973).

Appl. Opt. (4)

IEEE J. Ocean Eng. (1)

J. S. Jaffe, “Computer modeling and the design of optimal underwater imaging systems,” IEEE J. Ocean Eng. 15, 101–111 (1990).
[CrossRef]

Izv. Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) (1)

D. M. Bravo-Zhivotovskiy, L. S. Dolin, A. G. Luchinin, and V. A. Savel’yev, “Structure of a narrow light beam in sea water,” Izv. Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) 5, 160–167 (1969) (translated by P. A. Kaehn).

J. Opt. Soc. Am. (3)

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

Opt. Express (1)

Rev. Mod. Phys. (1)

W. T. Scott, “The theory of small-angle multiple-scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
[CrossRef]

Other (9)

C. J. Funk, S. B. Bryant, and P. J. Heckman, Handbook of Underwater Imaging System Design (Ocean Technology Department, Naval Undersea Center, 1972).

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).

R. E. Walker, Marine Light Field Statistics (Wiley, 1994).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Reference 72–78 (Scripps Institute of Oceanography, University of California, San Diego,1972).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, 1997).

A. Gordon and M. R. Kittel, “Underwater multiple scattering of light for system designers,” Naval Undersea Center Technical Report 371 (National Technical Information Service, 1973).

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).

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Figures (11)

Fig. 1.
Fig. 1.

Geometry of the Monte Carlo calculation of the beam spread function.

Fig. 2.
Fig. 2.

Phase functions. The vertical line at 15° demarcates the change in scale from logarithmic on the left to linear on the right.

Fig. 3.
Fig. 3.

Comparison of numerical (solid) to analytical (dashed) calculations of the order-of-scattering modulation transfer functions for the Wells phase function. The optical depth is 1, and the single-scatter albedo is 0.75.

Fig. 4.
Fig. 4.

Order-of-scattering beam spread functions calculated from Fig. 3.

Fig. 5.
Fig. 5.

Order-of-scattering modulation transfer functions using the Wells phase function at an optical depth of 6.

Fig. 6.
Fig. 6.

Order-of-scattering beam spread functions using the Wells phase functions at an optical depth of 6. The solid curves were calculated from the MTFs of Fig. 5 by numerical Hankel transform, and the dashed cuves are the results of the Monte Carlo calculation.

Fig. 7.
Fig. 7.

Comparison of the total beam spread functions from Fig. 6 for the two single-scatter albedos.

Fig. 8.
Fig. 8.

Angular shapes of the order-of-scattering beam spread functions for different optical depths and single-scatter albedos for the Wells phase function. Each beam spread function is normalized to its integral over a solid angle.

Fig. 9.
Fig. 9.

Values of the integrated beam spread function over the detector sphere for each scattering order, along with the SAA prediction. Results are shown for two single-scatter albedos. The optical depth is 6, and the phase function is Wells’.

Fig. 10.
Fig. 10.

Order-of-scattering modulation transfer and beam spread functions using the Petzold phase function for an optical depth of 6 and single-scatter albedo of 0.5.

Fig. 11.
Fig. 11.

Comparison of total modulation transfer and beam spread functions using the Wells (solid) and Petzold phase functions. The optical depth is 6, and single-scatter albedo is 0.5.

Equations (40)

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s ^ · L ( r⃗ , s ^ ) = c ( r⃗ ) L ( r⃗ , s ^ ) + 4 π β ( r⃗ , s ^ s ^ ) L ( r⃗ , s ^ ) d Ω .
r⃗ = ρ⃗ + z z ^ , s ^ = s⃗ + w z ^ , = x x ^ + y y ^ + z z ^ = t + z z ^ ,
z L ( z , ρ⃗ , s⃗ ) + s⃗ · t L ( z , ρ⃗ , s⃗ ) = c L ( z , ρ⃗ , s⃗ ) + β ( s⃗ s⃗ ) L ( z , ρ⃗ , s⃗ ) d s⃗ .
L 1 ( z , κ⃗ , s⃗ ) = L ( z , ρ , s⃗ ) e ι κ⃗ · ρ⃗ d ρ⃗ ,
( z ι κ⃗ · s⃗ + c ) L 1 ( z , κ⃗ , s⃗ ) = β ( s⃗ s⃗ ) L 1 ( z , κ⃗ , s⃗ ) d s⃗ .
L 1 ( z , κ⃗ , s⃗ ) = L ( z , κ⃗ , s⃗ ) exp ( 0 z ( ι s⃗ · κ⃗ c ) d z )
z L ( z , κ⃗ , s⃗ ) = β ( s⃗ s⃗ ) L ( z , κ⃗ , s⃗ ) e ι κ⃗ · ( s⃗ s⃗ ) d s⃗ .
d d z F ( z , κ⃗ , q⃗ ) = B ( q⃗ κ⃗ z ) F ( z , κ⃗ , q⃗ ) ,
F ( z , κ⃗ , q⃗ ) = L ( z , κ⃗ , s⃗ ) e ι s⃗ · q⃗ d s⃗ ,
B ( q⃗ ) = β ( s⃗ ) e ι s⃗ · q⃗ d s⃗
F ( z , κ⃗ , q⃗ ) = F o ( κ⃗ , q⃗ ) exp ( 0 z B ( q⃗ κ⃗ z ) d z ) ,
F o ( κ⃗ , q⃗ ) = L o ( ρ⃗ , s⃗ ) e ι ( κ⃗ · ρ⃗ + q⃗ · s⃗ ) d ρ⃗ d s⃗
L ( z , ρ⃗ , q⃗ ) = 1 ( 2 π ) 4 F o ( κ⃗ , q⃗ ) exp [ 0 z ( B ( q⃗ κ⃗ z ) c ) d z ] e ι ( κ⃗ · ρ + s⃗ · ( q⃗ κ⃗ z ) ) d κ⃗ d q⃗ .
L ( z , ρ⃗ , s⃗ ) = 1 ( 2 π ) 4 F o ( κ⃗ , q⃗ + κ⃗ z ) K ( z , κ⃗ , q⃗ ) e ι ( κ⃗ · ρ⃗ + q⃗ · s⃗ ) d κ⃗ d q⃗ ,
K ( z , κ⃗ , q⃗ ) = exp { 0 z [ c ( z ζ ) B ( z ζ , q⃗ + κ⃗ ζ ) ] d ζ }
L o ( ρ⃗ , s⃗ ) = P o δ ( ρ⃗ ) δ ( s⃗ ) .
L ( z , ρ⃗ , q⃗ ) = P o ( 2 π ) 4 K ( z , κ⃗ , q⃗ ) e ι ( κ⃗ · ρ⃗ + q⃗ · s⃗ ) d κ⃗ d q⃗ .
E ( z , ρ⃗ ) = 2 π L ( z , ρ⃗ , s⃗ ) cos θ d Ω = 2 π L ( z , ρ⃗ , s⃗ ) w d Ω L ( z , ρ⃗ , s⃗ ) d s⃗
E ( z , ρ⃗ ) = P o ( 2 π ) 4 ( K ( z , κ⃗ , q⃗ ) e ι ( κ⃗ · ρ⃗ + q⃗ · s⃗ ) d κ⃗ d q⃗ ) d s⃗ = P o ( 2 π ) 4 K ( z , κ⃗ , q⃗ ) e ι κ⃗ · ρ⃗ ( ( 2 π ) 2 δ ( q⃗ ) ) d κ⃗ d q⃗ = P o ( 2 π ) 2 K ( z , κ⃗ , q⃗ = 0 ) e ι κ⃗ · ρ⃗ d κ⃗ .
BSF ( z , ρ⃗ ) E ( z , ρ⃗ ) P o = 1 ( 2 π ) 2 K ( z , κ⃗ , q⃗ = 0 ) e ι κ⃗ · ρ⃗ d κ⃗ .
OTF ( z , κ⃗ ) = K ( z , κ⃗ , q⃗ = 0 ) = exp [ c z + 0 z B ( κ⃗ ζ ) d ζ ]
B ( | κ⃗ ζ | ) = β ( | s⃗ | ) e ι s⃗ · q⃗ d s⃗ ,
B ( ψ ζ / z ) = 2 π 0 β ( θ ) J o ( 2 π θ ψ ζ / z ) θ d θ ,
MTF ( z , ψ ) = exp [ c z + 0 z B ( ψ ζ / z ) d ζ ] .
MTF ( z , ψ ) = exp [ c z + z 0 1 B ( ψ t ) d t ] .
BSF ( z , θ ) = 2 π z 2 0 MTF ( z , ψ ) J o ( 2 π θ ψ ) ψ d ψ .
MTF ( z , ψ ) = 2 π 0 [ z 2 BSF ( z , θ ) ] J o ( 2 π θ ψ ) θ d θ .
MTF ( z , ψ ) = exp [ c z + b z 0 1 B ˜ ( ψ t ) d t ] .
MTF ( z , ψ ) = e τ exp [ ϖ τ B ¯ ( ψ ) ] .
lim ψ 0 B ¯ ( ψ ) = lim ψ 0 2 π 0 β ˜ ( θ ) J o ( 2 π θ ψ ) θ d θ = 1 , lim ψ B ¯ ( ψ ) = lim ψ 2 π 0 β ˜ ( θ ) J o ( 2 π θ ψ ) θ d θ = 0 .
lim ψ 0 MTF ( z , ψ ) = e a z , lim ψ MTF ( z , ψ ) = e c z ,
MTF ( z , ψ ) = e τ n = 0 ( ϖ τ ) n n ! [ B ¯ ( ψ ) ] n .
BSF ( z , θ ) = e τ z 2 n = 0 ( ϖ τ ) n n ! ( 2 π 0 [ B ¯ ( ψ ) ] n J o ( 2 π θ ψ ) ψ d ψ ) .
BSF ( z , θ ) = e τ z 2 n = 0 ( ϖ τ ) n n ! H n ( θ ) ,
BSF ( z , θ ) = e ( 1 ϖ ) τ z 2 n = 0 ( ϖ τ ) n n ! e ϖ τ H n ( θ ) = e a z z 2 n = 0 ( b z ) n n ! e b z H n ( θ ) .
BSF o ( z ) = e τ z 2 δ ( θ⃗ ) ,
β ˜ ( θ ) = N θ o 2 π ( θ 2 + θ o 2 ) 3 / 2 ,
MTF ( z , ψ ) = exp [ c z + b z 1 exp ( 2 π θ o ψ ) 2 π θ o ψ ] = e τ exp [ ϖ τ 1 exp ( 2 π θ o ψ ) 2 π θ o ψ ] .
MTF n ( z , ψ ) = e τ ( ω τ ) n n ! [ 1 exp ( 2 π θ o ψ ) 2 π θ o ψ ] n .
I n = 2 π z 2 0 π BSF n ( z , θ ) sin θ d θ .

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