Abstract

The widely-used Secchi disk method is re-examined from the modulation transfer aspect. Namely, by assuming a volume scattering function and applying small angle scattering approximation, we show that the Secchi depth and horizontal visibility can be determined using the water modulation transfer function and the corresponding spatial frequencies associated with the disk. A basic equation of Secchi disk is reached that is comparable to the radiative transfer approach, in that the Secchi depth is inversely proportional to the attenuation coefficient (c). With typical values for parameters applied, we demonstrate that the modulation transfer technique produces a horizontal visibility range of about 4.8/c, which is inline with previous studies. The improvement lies in the fact that the current approach correctly addresses the response of all spatial frequencies according to the modeled optical transfer function of the water. In terms of Secchi disk theory, the current approach helps to understand the effect of disk size as well as the role of scattering on the Secchi disk depth. The approach presented provides an understanding of Secchi disk disappearance by showing that as the disk is moved away from the observer, the spatial frequencies corresponding to the disk size increase, while the modulation transfer dampens contrast at an increased rate.

© 2007 Optical Society of America

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References

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  1. R. W. Preisendorfer, and Environmental Research Laboratories (U.S.), Eyeball optics of natural waters : Secchi disk science (U.S. Dept. of Commerce, National Oceanic and Atmospheric Administration, Environmental Research Laboratories, [Washington, D.C.], 1986).
  2. R. W. Preisendorfer, "Secchi disk science: Visual optics of natural waters," Limnol. Oceanogr. 31, 909-926 (1986).
    [CrossRef]
  3. Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
    [PubMed]
  4. J. R. Zaneveld, and W. S. Pegau, "Robust underwater visibility parameter," Opt. Express 11, 2997-3009 (2004).
    [CrossRef]
  5. S. Q. Duntley, "Light in the sea," J.Opt. Soc. Am. 53, 214-233 (1963).
    [CrossRef]
  6. H. R. Blackwell, "Contrast thresholds of the human eye," J.Opt. Soc. Am. 36, 624-643 (1946).
    [CrossRef] [PubMed]
  7. R. J. Davies-Colley, "Measuring water clarity with a black disk," Limnol. Oceanogr. 33, 616-623 (1988).
    [CrossRef]
  8. E. A. Steel, and S. Neuhausser, "Comparison of methods for measuring visual water clarity," J. of the North American Benthological Society 21, 326-335 (2002).
    [CrossRef]
  9. H. H. Barrett, and K. J. Myers, Foundations of image science (Wiley-Interscience, Hoboken, NJ, 2004).
  10. J. W. Coltman, "The specification of imaging properties by response to a Sine wave input," J. Opt. Soc. Am. 44, 468-471 (1954).
    [CrossRef]
  11. W. H. Wells, "Theory of small angle scattering," in AGARD Lec. Series No. 61 (NATO, 1973).
  12. C. D. Mobley, Light and Water: radiative transfer in natural waters (Academic Press, New York, 1994).
  13. J. W. McLean and K. J. Voss, "Point spread function in ocean water: comparison between theory and experiment," Appl. Opt. 34, 2027-2030 (1991).
  14. C. F. Bohren, "Optics atmospheric," in Encyclopedia of Applied Physics (New York, 1995), pp. 405-434.
  15. J. Jaffe, "Monte Carlo modeling of underwater image formation: validity of the linear and small-angle approximations," Appl. Opt. 34, 5413 (1995).
    [CrossRef] [PubMed]
  16. R. E. Carlson, "The Secchi disk and the volunteer monitor," LakeLine 15, 28-29 (1995).
  17. N. K. Hojerslev, "Visibility of the sea with special reference to the Secchi disc," in Proc. Soc. Photo-Opt. Inst. Eng. 294-305 (1986).

2006 (1)

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

2004 (1)

2002 (1)

E. A. Steel, and S. Neuhausser, "Comparison of methods for measuring visual water clarity," J. of the North American Benthological Society 21, 326-335 (2002).
[CrossRef]

1995 (2)

1991 (1)

J. W. McLean and K. J. Voss, "Point spread function in ocean water: comparison between theory and experiment," Appl. Opt. 34, 2027-2030 (1991).

1988 (1)

R. J. Davies-Colley, "Measuring water clarity with a black disk," Limnol. Oceanogr. 33, 616-623 (1988).
[CrossRef]

1986 (1)

R. W. Preisendorfer, "Secchi disk science: Visual optics of natural waters," Limnol. Oceanogr. 31, 909-926 (1986).
[CrossRef]

1963 (1)

S. Q. Duntley, "Light in the sea," J.Opt. Soc. Am. 53, 214-233 (1963).
[CrossRef]

1954 (1)

1946 (1)

H. R. Blackwell, "Contrast thresholds of the human eye," J.Opt. Soc. Am. 36, 624-643 (1946).
[CrossRef] [PubMed]

Arnone, R.

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

Blackwell, H. R.

H. R. Blackwell, "Contrast thresholds of the human eye," J.Opt. Soc. Am. 36, 624-643 (1946).
[CrossRef] [PubMed]

Carder, K. L.

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

Carlson, R. E.

R. E. Carlson, "The Secchi disk and the volunteer monitor," LakeLine 15, 28-29 (1995).

Coltman, J. W.

Davies-Colley, R. J.

R. J. Davies-Colley, "Measuring water clarity with a black disk," Limnol. Oceanogr. 33, 616-623 (1988).
[CrossRef]

Davis, C.

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

Duntley, S. Q.

S. Q. Duntley, "Light in the sea," J.Opt. Soc. Am. 53, 214-233 (1963).
[CrossRef]

Jaffe, J.

Kindle, J.

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

Lee, Z. P.

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

McLean, J. W.

J. W. McLean and K. J. Voss, "Point spread function in ocean water: comparison between theory and experiment," Appl. Opt. 34, 2027-2030 (1991).

Neuhausser, S.

E. A. Steel, and S. Neuhausser, "Comparison of methods for measuring visual water clarity," J. of the North American Benthological Society 21, 326-335 (2002).
[CrossRef]

Pegau, W. S.

Preisendorfer, R. W.

R. W. Preisendorfer, "Secchi disk science: Visual optics of natural waters," Limnol. Oceanogr. 31, 909-926 (1986).
[CrossRef]

Steel, E. A.

E. A. Steel, and S. Neuhausser, "Comparison of methods for measuring visual water clarity," J. of the North American Benthological Society 21, 326-335 (2002).
[CrossRef]

Voss, K. J.

J. W. McLean and K. J. Voss, "Point spread function in ocean water: comparison between theory and experiment," Appl. Opt. 34, 2027-2030 (1991).

Weidemann, A.

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

Zaneveld, J. R.

Appl. Opt. (2)

J. W. McLean and K. J. Voss, "Point spread function in ocean water: comparison between theory and experiment," Appl. Opt. 34, 2027-2030 (1991).

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

J. Geophys. Res. (1)

Z. P. Lee, A. Weidemann, J. Kindle, R. Arnone, K. L. Carder, and C. Davis, "Euphotic zone depth: Its derivation and implication to ocean-color remote sensing," J. Geophys. Res., Accepted (2006).
[PubMed]

J. of the North American Benthological Society (1)

E. A. Steel, and S. Neuhausser, "Comparison of methods for measuring visual water clarity," J. of the North American Benthological Society 21, 326-335 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J.Opt. Soc. Am. (2)

S. Q. Duntley, "Light in the sea," J.Opt. Soc. Am. 53, 214-233 (1963).
[CrossRef]

H. R. Blackwell, "Contrast thresholds of the human eye," J.Opt. Soc. Am. 36, 624-643 (1946).
[CrossRef] [PubMed]

LakeLine (1)

R. E. Carlson, "The Secchi disk and the volunteer monitor," LakeLine 15, 28-29 (1995).

Limnol. Oceanogr. (2)

R. W. Preisendorfer, "Secchi disk science: Visual optics of natural waters," Limnol. Oceanogr. 31, 909-926 (1986).
[CrossRef]

R. J. Davies-Colley, "Measuring water clarity with a black disk," Limnol. Oceanogr. 33, 616-623 (1988).
[CrossRef]

Opt. Express (1)

Other (6)

N. K. Hojerslev, "Visibility of the sea with special reference to the Secchi disc," in Proc. Soc. Photo-Opt. Inst. Eng. 294-305 (1986).

R. W. Preisendorfer, and Environmental Research Laboratories (U.S.), Eyeball optics of natural waters : Secchi disk science (U.S. Dept. of Commerce, National Oceanic and Atmospheric Administration, Environmental Research Laboratories, [Washington, D.C.], 1986).

C. F. Bohren, "Optics atmospheric," in Encyclopedia of Applied Physics (New York, 1995), pp. 405-434.

H. H. Barrett, and K. J. Myers, Foundations of image science (Wiley-Interscience, Hoboken, NJ, 2004).

W. H. Wells, "Theory of small angle scattering," in AGARD Lec. Series No. 61 (NATO, 1973).

C. D. Mobley, Light and Water: radiative transfer in natural waters (Academic Press, New York, 1994).

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Figures (3)

Fig. 1.
Fig. 1.

Well’s phase function Eq. (18) with different MSA parameters compared to measured coastal water phase function by Petzold (reproduced from Table 3.10 of [12]), up to 20o. All curves are normalized at 1o for comparison.

Fig. 2.
Fig. 2.

MTF of water at different mean square scattering settings Eq. (19), with optical length DZ=4.8. The curve with circles demonstrate the MTF without e-2πθ0ψ term in Eq. (19). See text for details.

Fig. 3.
Fig. 3.

Visibility range comparisons between previous radiative transfer approach and current MTF approach. c(532) ranges from 0.24 to 0.45. See text for details.

Equations (33)

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C V = L T θ ϕ z L B θ ϕ z L B θ ϕ z ,
C VZ = C V 0 e cz ,
C VZ = C V 0 e ( c + K ) z ,
c + K = ln ( C L ) Z SD ,                    C L = C VZ C V 0 ,
g ( x , y ) = Ξ f ( x i , y i ) h ( x x i , y y i ) d x i d y i ,
g ( x , y ) = f ( x , y ) h ( x , y ) ,
G ( u , v ) = F ( u , v ) H ( u , v ) ,
H ( u , v ) = h ( x , y ) e j 2 π ( u x + v y ) d x d y ,
H ( u , v ) = H system ( u , v ) H medium ( u , v ) .
M ( ψ , z ) = S max ( ψ , z ) S min ( ψ , z ) S max ( ψ , z ) + S min ( ψ , z ) ,
M ( ψ , 0 ) = S max ( ψ , 0 ) S min ( ψ , 0 ) S max ( ψ , 0 ) + S min ( ψ , 0 ) = M 0 ,
H ( ψ , z ) = M ( ψ , z ) M 0 .
H ( ψ , z ) = 2 π θ = 0 θ max J 0 ( 2 πθψ ) h ( θ , z ) θdθ .
H ( ψ , z ) = e D ( ψ ) Z ,
D ( ψ ) = c S ( ψ ) ,
S ( ψ ) = 2 π θ = 0 θ max J 0 ( 2 πθψ ) β ( θ ) θdθ ,
b = 2 π 0 π β ( θ ) sin θdθ .
β ( θ ) = 0 2 π ( θ 0 2 + θ 2 ) 3 2 ,
D ( ψ ) = c S ( ψ )
= c b ( 1 e 2 πθ 0 ψ ) 2 π θ 0 ψ .
ψ SD = Z SD ( d 2 ) = 2 Z SD d .
L T L B or C L 1 .
H ( ψ SD ) = [ S max ( ψ SD ) S min ( ψ SD ) S max ( ψ SD ) + S min ( ψ SD ) ] M 0
= ( L T L B L T + L B ) M 0 = C L ( C L + 2 ) M 0 C L 2 M 0 .
D ( ψ ) = ln [ H ( ψ ) ] R .
c b ( 1 e 2 π θ 0 ψ SD ) 2 π θ 0 ψ SD = 1 Z SD ln ( C L 2 M 0 ) .
c b 2 π θ 0 ψ SD = Ϛ Z SD .
c + K b 2 π θ 0 ψ SD = Ϛ Z SD .
c Z SD = Γ for the horizontal case , and
( c + K ) Z SD = Γ for the vertical case .
Z SD = Γ c .
visibility 4.8 c ,
Z SD d = b 4 π θ 0 ( c + K ) .

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