Abstract

The Zaneveld–Wells algorithm for calculating N inherent optical expansion coefficients from N + 1 measured angle-integrated moments of the radiant light field is investigated. Because the algorithm is well conditioned but sensitive to errors in the spatial derivatives, different approximations for the spatial derivatives are considered. The effects of noise and sensor error on the performance of the algorithm have been evaluated analytically, and testing with randomly sampled simulated noise was performed to assess the stability and sensitivity of the algorithm. Results show that the algorithm is fairly insensitive to sensor noise, but neither using a higher-order finite-difference approximation for the derivatives nor reformulating the algorithm into an integral form was successful in overcoming the large errors observed.

© 1995 Optical Society of America

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References

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  1. J. R. V. Zaneveld, “New development of the theory of radiative transfer in the oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 121–134.
  2. J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
    [CrossRef]
  3. W. H. Wells, “Techniques for measuring radiance in sea and air,” Appl. Opt. 22, 2313–2321 (1983).
    [CrossRef] [PubMed]
  4. W. Doss, W. H. Wells, “Undersea compound radiometer,” Appl. Opt. 31, 4268–4274 (1992).
    [CrossRef] [PubMed]
  5. W. Doss, W. H. Wells, “Radiometer for light in the sea,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 363–372 (1990).
  6. L. J. Holl, N. J. McCormick, “Inherent optical property estimation in ocean water using the Zaneveld–Wells algorithm,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2258, 2–11 (1994).
  7. C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).
  8. S. Chandrasehkar, Radiative Transfer(Dover, NewYork, 1960), pp. 54–61 and 153.
  9. K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic “fisheye” camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
    [CrossRef]
  10. K. J. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).
  11. R. D. O'Dell, “Revised user's manual for onedant: a code package for one-dimensional, diffusion-accelerated, neutral-particle transport,” Rep. LA-9184-M (Los Alamos National Laboratory, Los Alamos, New Mexico, 1989).
  12. R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
    [CrossRef] [PubMed]
  13. A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.
  14. H. R. Gordon, “Bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
    [CrossRef] [PubMed]
  15. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical methods for solving the radiative transfer equation,” Appl. Opt. 32, 7484–7504 (1993).
    [CrossRef] [PubMed]
  16. P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
    [CrossRef]
  17. L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
    [CrossRef]
  18. C. D. Mobley, “The optical properties of water,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995), pp. 43.1–43.56.
  19. H. Gordon, A. Morel, Remote Assesment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.
  20. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]

1994

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

1993

1992

W. Doss, W. H. Wells, “Undersea compound radiometer,” Appl. Opt. 31, 4268–4274 (1992).
[CrossRef] [PubMed]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

1989

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic “fisheye” camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

K. J. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).

1987

1983

1981

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

1978

C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).

1972

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[CrossRef]

Baker, K. S.

Chandrasehkar, S.

S. Chandrasehkar, Radiative Transfer(Dover, NewYork, 1960), pp. 54–61 and 153.

Doss, W.

W. Doss, W. H. Wells, “Undersea compound radiometer,” Appl. Opt. 31, 4268–4274 (1992).
[CrossRef] [PubMed]

W. Doss, W. H. Wells, “Radiometer for light in the sea,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 363–372 (1990).

Francisco, P. W.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Gentili, B.

Gordon, H.

H. Gordon, A. Morel, Remote Assesment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.

Gordon, H. R.

Holl, L. J.

L. J. Holl, N. J. McCormick, “Inherent optical property estimation in ocean water using the Zaneveld–Wells algorithm,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2258, 2–11 (1994).

Jin, Z.

Kattawar, G. W.

McCormick, N. J.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

L. J. Holl, N. J. McCormick, “Inherent optical property estimation in ocean water using the Zaneveld–Wells algorithm,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2258, 2–11 (1994).

Mobley, C. D.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical methods for solving the radiative transfer equation,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

C. D. Mobley, “The optical properties of water,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995), pp. 43.1–43.56.

Morel, A.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical methods for solving the radiative transfer equation,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

H. Gordon, A. Morel, Remote Assesment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.

O'Dell, R. D.

R. D. O'Dell, “Revised user's manual for onedant: a code package for one-dimensional, diffusion-accelerated, neutral-particle transport,” Rep. LA-9184-M (Los Alamos National Laboratory, Los Alamos, New Mexico, 1989).

Pak, H.

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[CrossRef]

Prieur, L.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Reinersman, P.

Sathyendranath, S.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Siewert, C. E.

C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).

Smith, R. C.

Stamnes, K.

Stavn, R. H.

Voss, K. J.

K. J. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic “fisheye” camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

Wells, W. H.

W. Doss, W. H. Wells, “Undersea compound radiometer,” Appl. Opt. 31, 4268–4274 (1992).
[CrossRef] [PubMed]

W. H. Wells, “Techniques for measuring radiance in sea and air,” Appl. Opt. 22, 2313–2321 (1983).
[CrossRef] [PubMed]

W. Doss, W. H. Wells, “Radiometer for light in the sea,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 363–372 (1990).

Zaneveld, J. R. V.

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[CrossRef]

J. R. V. Zaneveld, “New development of the theory of radiative transfer in the oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 121–134.

Zibordi, G.

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic “fisheye” camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

Appl. Opt.

J. Atmos. Oceanic Technol.

K. J. Voss, G. Zibordi, “Radiometric and geometric calibration of a visible spectral electro-optic “fisheye” camera radiance distribution system,” J. Atmos. Oceanic Technol. 6, 652–662 (1989).
[CrossRef]

J. Geophys. Res.

J. R. V. Zaneveld, H. Pak, “Some aspects of the axially symmetric submarine daylight field,” J. Geophys. Res. 77, 2677–2680 (1972).
[CrossRef]

Limnol. Oceanogr.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

Nucl. Sci. Eng.

C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).

Opt. Eng.

K. J. Voss, “Electro-optic camera system for measurement of the underwater radiance distribution,” Opt. Eng. 28, 241–247 (1989).

Other

R. D. O'Dell, “Revised user's manual for onedant: a code package for one-dimensional, diffusion-accelerated, neutral-particle transport,” Rep. LA-9184-M (Los Alamos National Laboratory, Los Alamos, New Mexico, 1989).

J. R. V. Zaneveld, “New development of the theory of radiative transfer in the oceans,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 121–134.

S. Chandrasehkar, Radiative Transfer(Dover, NewYork, 1960), pp. 54–61 and 153.

W. Doss, W. H. Wells, “Radiometer for light in the sea,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 363–372 (1990).

L. J. Holl, N. J. McCormick, “Inherent optical property estimation in ocean water using the Zaneveld–Wells algorithm,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2258, 2–11 (1994).

C. D. Mobley, “The optical properties of water,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995), pp. 43.1–43.56.

H. Gordon, A. Morel, Remote Assesment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

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Figures (3)

Fig. 1
Fig. 1

Plot of the incident surface illumination versus the cosine of the zenith angle.

Fig. 2
Fig. 2

Effect of the use of a higher-order spatial derivative on the percent error in the calculated values of An as a function of n for C = 0 and C = 0.1 mg/m3 at 570 nm and an optical depth of 10. Open symbols denote results obtained with a first-order backward-difference scheme, and filled symbols indicate that a fourth-order backward-difference was used.

Fig. 3
Fig. 3

Increase in the percent error in the calculated An coefficients as a result of the addition of noise to the simulated data at a wavelength of 570 nm and an optical depth of 10. Results were averaged over 1000 runs with the noise randomly sampled from a normal distribution with a variance of 1% and no values lying outside the range of ±4%.

Tables (5)

Tables Icon

Table 1 Theoretical Optical Properties for the Numerical Tests at 570 nm

Tables Icon

Table 2 Percent Error in the Calculated Values of An for Various Nondimensional Distances between Measurements with a First-Order Finite-Difference Scheme at 570 nm in the Asymptotic Regimea

Tables Icon

Table 3 Percent Error in the Calculated Values of An for Various Nondimensional Distances between Measurements with a First-Order Finite-Difference Scheme at 570 nm in the Asymptotic Regimea

Tables Icon

Table 4 Percent Error in Calculated An Coefficients at 570 nm as a Function of the Derivative Approximation Order and the Chlorophyll Concentration (mg/m3) at an Optical Depth of 10

Tables Icon

Table 5 Percent Error in Calculated AnCoefficients at 570 nm with the Integral Form of the Algorithm as a Function of the Number of Measurements in Depth Used and the Chlorophyll Concentration (mg/m3) at an Optical Depth of 10

Equations (28)

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β ( η ) = n = 0 2 n + 1 2 ( c A n ) P n ( η ) ,
μ z L ( μ , z ) + c L ( μ , z ) = 1 1 β ( μ μ ) L ( μ , z ) d μ ,
( n + 1 ) L n + 1 ( z ) + ( 2 n + 1 ) A n L n ( z ) + n L n 1 ( z ) = 0 , n = 0 , 1 , 2 , ,
L n ( z ) = 1 1 L ( μ , z ) P n ( μ ) d μ .
A n = ( n + 1 ) L n + 1 ( z ) + n L n 1 ( z ) ( 2 n + 1 ) L n ( z ) , n = 0 , 1 , 2 , .
A n = ( n + 1 ) L n + 1 ( z ) | z 1 z 2 + n L n 1 ( z ) | z 1 z 2 ( 2 n + 1 ) z 1 z 2 L n ( z ) d z , n = 0 , 1 , 2 , .
Λ j ( z ) = 1 1 L ( μ , z ) R j ( μ ) d μ , for j = 1 to N + 1 ,
Λ j ( z ) = n = 0 N R j n L n ( z ) + X j ( z ) ,
R j n = 2 n + 1 2 1 1 R j ( μ ) P n ( μ ) d μ .
Λ = RL ( z ) + X .
L n ( z ) = j = 1 N + 1 [ R 1 ] n j [ Λ j ( z ) X j ( z ) ] ,
δ A n A n = δ L n L n + ( n + 1 ) δ L n + 1 + n δ L n 1 ( n + 1 ) L n + 1 + n L n 1 .
δ Λ = R δ L ( z ) + δ RL ( z ) + δ X .
δ L ( z ) R 1 [ δ ( Λ X ) + δ R L ( z ) ] .
δ L ( z ) L ( z ) R 1 R { δ [ Λ ( z ) X ] Λ ( z ) X + δ R R } .
R j ( μ ) = i j ( μ μ i ) ( μ j μ i ) , μ j 1 μ μ j + 1 , = 0 , otherwise ,
a mix ( λ ) = a water ( λ ) + 0.06 C 0.602 a C ( λ ) a C ( 440 nm ) ,
b mix ( λ ) = b water ( λ ) + C 0.62 b C ( 550 nm ) ( 550 λ ) n ,
L n ( z ) L n ( z ) L n ( z Δ z ) Δ z .
L n ( z ) = D g n ( c / K ) exp ( K z ) ,
δ L n ( z ) = Dg n ( c / K ) K exp ( K z ) × { 1 + [ 1 exp ( K Δ z ) ] K Δ z } .
δ A n A n = δ L n L n + { 1 + [ 1 exp ( K Δ z ) ] K Δ z } .
err ( A n ) = ( p n k q n + 1 k ) ( n + 1 ) r n + ( p n k q n 1 k ) n ( 1 + p n k ) [ ( n + 1 ) r n + n ] ,
| p n k | = | δ L n ( z k ) L n ( z k ) | R 1 R ξ k , for all n ,
q n k = δ L n ( z k ) L n ( z k ) = p n k L n k p n k 1 L n k 1 L n k L n k 1 .
U = R 1 R S ( N + 1 ) 1 / 2 .
3 U ( 1 U ) err ( A n ) 3 U ( 1 + U ) for U < 1 / 2 ,
3 U ( 1 U ) err ( A n ) 2 U for U > 1 / 2 .

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