Abstract

A new analytical approach for retrieval of the vertically weighted chlorophyll a concentration (Chlrs) detected by remote sensors is presented. Model calculations were carried out for the turbid waters of Lake Kinneret, Israel, and showed that Chlrs may be replaced by the average chlorophyll a concentration (Chlp) within the upper “penetration layer” 0Zp. The study also showed a high correlation between Chlrs and Chl concentration averaged in the other depth layers, namely, the 01m layer, the euphotic layer (0Ze), and the production layer (0Zpr). Our findings are closely related to models developed for the world ocean, with the exception of periods when the dinoflagellate Peridinium gatunense blooms in the lake. We showed the effect of the pattern of vertical Chl distributions within the penetration layer on the difference between Chlrs and other Chl indices was conspicuous when the Chl maximum was in the uppermost 05m layer of the water column. We assume that the presented approaches are instrumental for further development of optimal, locally adapted algorithms for remote sensing of Chl in any type of natural waters.

© 2011 Optical Society of America

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    [CrossRef]

2009 (1)

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

2008 (2)

P. Xiu, Y. Liu, and J. Tang, “Variations of ocean colour parameters with nonuniform vertical profiles of chlorophyll concentration,” Int. J. Remote Sens. 29, 831–849 (2008).
[CrossRef]

J. Piscozub, T. Neumann, and L. Woźniak, “Ocean color remote sensing: choosing the correct depth weighting function,” Opt. Express 16, 14683–14688 (2008).
[CrossRef]

2006 (1)

Y. Z. Yacobi, “Temporal and vertical variation of chlorophyll a concentration, phytoplankton photosynthetic activity and light attenuation in Lake Kinneret: possibilities and limitations for simulation by remote sensing,” J. Plankton Res. 28, 725–736 (2006).
[CrossRef]

2005 (4)

2003 (2)

T. Hirata, “Irradiance inversion theory to retrieve volume scattering function of seawater,” Appl. Opt. 42, 1564–1573(2003).
[CrossRef] [PubMed]

L. Sokoletsky, Z. Dubinsky, M. Shoshany, and N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sens. 24, 5049–5073(2003).
[CrossRef]

2001 (1)

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106, 7163–7180(2001).
[CrossRef]

1999 (3)

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytical functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

D. Ballestero, “Remote sensing of vertically structured phytoplankton pigments,” Top. Meteor. Oceanogr. 6, 14–23 (1999).

N. Hoepffner, B. Sturm, Z. Finenko, and D. Larkin, “Depth-integrated primary production in the eastern tropical and subtropical North Atlantic basin from ocean colour imagery,” Int. J. Remote Sens. 20, 1435–1456 (1999).
[CrossRef]

1998 (3)

K. R. Arrigo, D. H. Robinson, D. L. Worthen, B. Schieber, and M. P. Lizotte, “Bio-optical properties of the southwestern Ross Sea,” J. Geophys. Res. 103, 21683–21695 (1998).
[CrossRef]

R. W. Gould, Jr., and R. A. Arnone, “Three-dimensional modeling of inherent optical properties in a coastal environment: coupling ocean color imagery and in situ measurements,” Int. J. Remote Sens. 19, 2141–2159 (1998).
[CrossRef]

A. Morel and H. Loisel, “Apparent optical properties of oceanic water, dependence on the molecular scattering contribution,” Appl. Opt. 37, 4765–4776 (1998).
[CrossRef]

1997 (3)

S. Sathyendranath and T. Platt, “Analytic model of ocean color,” Appl. Opt. 36, 2620–2629 (1997).
[CrossRef] [PubMed]

W. J. Rhea and C. O. Davis, “A comparison of the SeaWiFS chlorophyll and CZCS pigment algorithms using optical data from the 1992 JGOFS Equatorial Pacific Time Series,” Deep-Sea Res. Part II 44, 1907–1925 (1997).
[CrossRef]

H. Arst, S. Mäekivi, T. Lukk, and A. Herlevi, “Calculating irradiance penetration into water bodies from the measured beam attenuation coefficient,” Limnol. Oceanogr. 42, 379–385(1997).
[CrossRef]

1996 (1)

D. A. Antoine and A. Morel, “Oceanic primary production. 1. Adaptation of a spectral light-photosynthesis model in view of application to satellite chlorophyll observations,” Global Biogeochem. Cycles 10, 43–55 (1996).
[CrossRef]

1995 (2)

K. S. Prasad, S. E. Lohrenz, D. G. Redalje, and G. L. Fahnenstiel, “Primary production in the Gulf of Mexico coastal waters using ‘remotely-sensed’ trophic category approach,” Cont. Shelf Res. 15, 1355–1368 (1995).
[CrossRef]

Y. Z. Yacobi, A. Gitelson, and M. Mayo, “Remote sensing of chlorophyll in Lake Kinneret using high spectral resolution radiometer and Landsat Thematic Mapper: spectral features of reflectance and algorithm development,” J. Plankton Res. 17, 2155–2173 (1995).
[CrossRef]

1994 (2)

R. A. Arnone, R. W. Gould, Jr., R. A. Oriol, and G. E. Terrie, “Effects of vertical chlorophyll structure and solar irradiance on remote sensing ocean color spectrum,” Proc. SPIE 2258, 322–331 (1994).
[CrossRef]

S. Maritorena, A. Morel, and A. Gentili, “Diffuse reflectance of oceanic shallow waters: influence of water depth and bottom albedo,” Limnol. Oceanogr. 39, 1689–1703 (1994).
[CrossRef]

1992 (2)

J.-M. André, “Ocean color remote-sensing and the subsurface vertical structure of phytoplankton pigments,” Deep-Sea Res. Part A 39, 763–779 (1992).
[CrossRef]

H. R. Gordon, “Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile,” Appl. Opt. 31, 2116–2129 (1992).
[CrossRef] [PubMed]

1989 (3)

A. Morel and J. F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote-sensing applications,” Limnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

R. H. Stavn and A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441(1989).
[CrossRef]

1988 (3)

A. Morel, “Optical modeling of the upper ocean in relation to its biogeneous matter content (case I waters),” J. Geophys. Res. 93, 10749–10768 (1988).
[CrossRef]

T. Platt, S. Sathyendranath, C. M. Caverhill, and M. R. Lewis, “Ocean primary production and available light: further algorithms for remote sensing,” Deep-Sea Res. Part A 35, 855–879 (1988).
[CrossRef]

T. Platt and S. Sathyendranath, “Ocean primary production: estimation by remote sensing at local and regional scales,” Science 241, 1613–1620 (1988).
[CrossRef] [PubMed]

1987 (3)

1983 (2)

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36(1983).
[CrossRef] [PubMed]

M. R. Lewis, J. J. Cullen, and T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

1982 (2)

J. J. Cullen, “The deep chlorophyll maximum: comparing vertical profiles of chlorophyll a,” Can. J. Fish. Aquat. Sci. 39, 791–803 (1982).
[CrossRef]

J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982).
[CrossRef] [PubMed]

1981 (1)

R. C. Smith, “Remote sensing and depth distribution of ocean chlorophyll,” Mar. Ecol. Prog. Ser. 5, 359–361 (1981).
[CrossRef]

1980 (1)

1979 (1)

Z. Dubinsky and T. Berman, “Seasonal changes in the spectral composition of downwelling irradiance in Lake Kinneret (Israel),” Limnol. Oceanogr. 24, 652–663 (1979).
[CrossRef]

1975 (1)

Aas, E.

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytical functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
[CrossRef] [PubMed]

André, J.-M.

J.-M. André, “Ocean color remote-sensing and the subsurface vertical structure of phytoplankton pigments,” Deep-Sea Res. Part A 39, 763–779 (1992).
[CrossRef]

Antoine, D. A.

D. A. Antoine and A. Morel, “Oceanic primary production. 1. Adaptation of a spectral light-photosynthesis model in view of application to satellite chlorophyll observations,” Global Biogeochem. Cycles 10, 43–55 (1996).
[CrossRef]

Arnone, R. A.

R. W. Gould, Jr., and R. A. Arnone, “Three-dimensional modeling of inherent optical properties in a coastal environment: coupling ocean color imagery and in situ measurements,” Int. J. Remote Sens. 19, 2141–2159 (1998).
[CrossRef]

R. A. Arnone, R. W. Gould, Jr., R. A. Oriol, and G. E. Terrie, “Effects of vertical chlorophyll structure and solar irradiance on remote sensing ocean color spectrum,” Proc. SPIE 2258, 322–331 (1994).
[CrossRef]

Arrigo, K. R.

K. R. Arrigo, D. H. Robinson, D. L. Worthen, B. Schieber, and M. P. Lizotte, “Bio-optical properties of the southwestern Ross Sea,” J. Geophys. Res. 103, 21683–21695 (1998).
[CrossRef]

Arst, H.

H. Arst, S. Mäekivi, T. Lukk, and A. Herlevi, “Calculating irradiance penetration into water bodies from the measured beam attenuation coefficient,” Limnol. Oceanogr. 42, 379–385(1997).
[CrossRef]

Ballestero, D.

D. Ballestero, “Remote sensing of vertically structured phytoplankton pigments,” Top. Meteor. Oceanogr. 6, 14–23 (1999).

Barnard, A. H.

Bass, L. P.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Berman, T.

Z. Dubinsky and T. Berman, “Seasonal changes in the spectral composition of downwelling irradiance in Lake Kinneret (Israel),” Limnol. Oceanogr. 24, 652–663 (1979).
[CrossRef]

Berthon, J. F.

A. Morel and J. F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote-sensing applications,” Limnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

Boss, E.

Broenkow, W. W.

Brown, J. W.

Brown, O. B.

Budak, V. P.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Caverhill, C. M.

T. Platt, S. Sathyendranath, C. M. Caverhill, and M. R. Lewis, “Ocean primary production and available light: further algorithms for remote sensing,” Deep-Sea Res. Part A 35, 855–879 (1988).
[CrossRef]

Clark, D. K.

Clarke, D. C.

Cullen, J. J.

M. R. Lewis, J. J. Cullen, and T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

J. J. Cullen, “The deep chlorophyll maximum: comparing vertical profiles of chlorophyll a,” Can. J. Fish. Aquat. Sci. 39, 791–803 (1982).
[CrossRef]

Davis, C. O.

W. J. Rhea and C. O. Davis, “A comparison of the SeaWiFS chlorophyll and CZCS pigment algorithms using optical data from the 1992 JGOFS Equatorial Pacific Time Series,” Deep-Sea Res. Part II 44, 1907–1925 (1997).
[CrossRef]

Dickey, T. D.

D. A. Siegel and T. D. Dickey, “On the parameterization of irradiance for open ocean photoprocesses,” J. Geophys. Res. 92, 14648–14662 (1987).
[CrossRef]

Dubinsky, Z.

L. Sokoletsky, Z. Dubinsky, M. Shoshany, and N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sens. 24, 5049–5073(2003).
[CrossRef]

Z. Dubinsky and T. Berman, “Seasonal changes in the spectral composition of downwelling irradiance in Lake Kinneret (Israel),” Limnol. Oceanogr. 24, 652–663 (1979).
[CrossRef]

Evans, R. H.

Fahnenstiel, G. L.

K. S. Prasad, S. E. Lohrenz, D. G. Redalje, and G. L. Fahnenstiel, “Primary production in the Gulf of Mexico coastal waters using ‘remotely-sensed’ trophic category approach,” Cont. Shelf Res. 15, 1355–1368 (1995).
[CrossRef]

Finenko, Z.

N. Hoepffner, B. Sturm, Z. Finenko, and D. Larkin, “Depth-integrated primary production in the eastern tropical and subtropical North Atlantic basin from ocean colour imagery,” Int. J. Remote Sens. 20, 1435–1456 (1999).
[CrossRef]

Gentili, A.

S. Maritorena, A. Morel, and A. Gentili, “Diffuse reflectance of oceanic shallow waters: influence of water depth and bottom albedo,” Limnol. Oceanogr. 39, 1689–1703 (1994).
[CrossRef]

Gitelson, A.

Y. Z. Yacobi, A. Gitelson, and M. Mayo, “Remote sensing of chlorophyll in Lake Kinneret using high spectral resolution radiometer and Landsat Thematic Mapper: spectral features of reflectance and algorithm development,” J. Plankton Res. 17, 2155–2173 (1995).
[CrossRef]

Gordon, H. R.

Gould, R. W.

R. W. Gould, Jr., and R. A. Arnone, “Three-dimensional modeling of inherent optical properties in a coastal environment: coupling ocean color imagery and in situ measurements,” Int. J. Remote Sens. 19, 2141–2159 (1998).
[CrossRef]

R. A. Arnone, R. W. Gould, Jr., R. A. Oriol, and G. E. Terrie, “Effects of vertical chlorophyll structure and solar irradiance on remote sensing ocean color spectrum,” Proc. SPIE 2258, 322–331 (1994).
[CrossRef]

Herlevi, A.

H. Arst, S. Mäekivi, T. Lukk, and A. Herlevi, “Calculating irradiance penetration into water bodies from the measured beam attenuation coefficient,” Limnol. Oceanogr. 42, 379–385(1997).
[CrossRef]

Hirata, T.

Hoepffner, N.

N. Hoepffner, B. Sturm, Z. Finenko, and D. Larkin, “Depth-integrated primary production in the eastern tropical and subtropical North Atlantic basin from ocean colour imagery,” Int. J. Remote Sens. 20, 1435–1456 (1999).
[CrossRef]

Højerslev, N. K.

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytical functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

Iluz, D.

L. G. Sokoletsky, A. Oren, N. Stambler, and D. Iluz, “Practical algorithms for remote-sensing retrieval of the water column constituents in the Israeli waters,” in Proceedings of the V International Conference “Current Problems in Optics of Natural Waters, ONW-2009,” I.Levin and G.Gilbert, eds. (Rozhdestvensky Optical Society, 2009), pp. 287–292.

Kirk, J. T. O.

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

Kokhanovsky, A. A.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Kuznetsov, V. S.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Larkin, D.

N. Hoepffner, B. Sturm, Z. Finenko, and D. Larkin, “Depth-integrated primary production in the eastern tropical and subtropical North Atlantic basin from ocean colour imagery,” Int. J. Remote Sens. 20, 1435–1456 (1999).
[CrossRef]

Lewis, M. R.

T. Platt, S. Sathyendranath, C. M. Caverhill, and M. R. Lewis, “Ocean primary production and available light: further algorithms for remote sensing,” Deep-Sea Res. Part A 35, 855–879 (1988).
[CrossRef]

M. R. Lewis, J. J. Cullen, and T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

Liu, Y.

P. Xiu, Y. Liu, and J. Tang, “Variations of ocean colour parameters with nonuniform vertical profiles of chlorophyll concentration,” Int. J. Remote Sens. 29, 831–849 (2008).
[CrossRef]

Lizotte, M. P.

K. R. Arrigo, D. H. Robinson, D. L. Worthen, B. Schieber, and M. P. Lizotte, “Bio-optical properties of the southwestern Ross Sea,” J. Geophys. Res. 103, 21683–21695 (1998).
[CrossRef]

Lohrenz, S. E.

K. S. Prasad, S. E. Lohrenz, D. G. Redalje, and G. L. Fahnenstiel, “Primary production in the Gulf of Mexico coastal waters using ‘remotely-sensed’ trophic category approach,” Cont. Shelf Res. 15, 1355–1368 (1995).
[CrossRef]

Loisel, H.

Lukk, T.

H. Arst, S. Mäekivi, T. Lukk, and A. Herlevi, “Calculating irradiance penetration into water bodies from the measured beam attenuation coefficient,” Limnol. Oceanogr. 42, 379–385(1997).
[CrossRef]

Lunetta, R. S.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Mäekivi, S.

H. Arst, S. Mäekivi, T. Lukk, and A. Herlevi, “Calculating irradiance penetration into water bodies from the measured beam attenuation coefficient,” Limnol. Oceanogr. 42, 379–385(1997).
[CrossRef]

Maritorena, S.

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106, 7163–7180(2001).
[CrossRef]

S. Maritorena, A. Morel, and A. Gentili, “Diffuse reflectance of oceanic shallow waters: influence of water depth and bottom albedo,” Limnol. Oceanogr. 39, 1689–1703 (1994).
[CrossRef]

Mayo, M.

Y. Z. Yacobi, A. Gitelson, and M. Mayo, “Remote sensing of chlorophyll in Lake Kinneret using high spectral resolution radiometer and Landsat Thematic Mapper: spectral features of reflectance and algorithm development,” J. Plankton Res. 17, 2155–2173 (1995).
[CrossRef]

McCluney, W. R.

Morel, A.

K. J. Voss and A. Morel, “Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: measurements versus predictions,” Limnol. Oceanogr. 50, 698–705 (2005).
[CrossRef]

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106, 7163–7180(2001).
[CrossRef]

A. Morel and H. Loisel, “Apparent optical properties of oceanic water, dependence on the molecular scattering contribution,” Appl. Opt. 37, 4765–4776 (1998).
[CrossRef]

D. A. Antoine and A. Morel, “Oceanic primary production. 1. Adaptation of a spectral light-photosynthesis model in view of application to satellite chlorophyll observations,” Global Biogeochem. Cycles 10, 43–55 (1996).
[CrossRef]

S. Maritorena, A. Morel, and A. Gentili, “Diffuse reflectance of oceanic shallow waters: influence of water depth and bottom albedo,” Limnol. Oceanogr. 39, 1689–1703 (1994).
[CrossRef]

A. Morel and J. F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote-sensing applications,” Limnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

A. Morel, “Optical modeling of the upper ocean in relation to its biogeneous matter content (case I waters),” J. Geophys. Res. 93, 10749–10768 (1988).
[CrossRef]

Neumann, T.

Nikolaeva, O. V.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Oren, A.

L. G. Sokoletsky, A. Oren, N. Stambler, and D. Iluz, “Practical algorithms for remote-sensing retrieval of the water column constituents in the Israeli waters,” in Proceedings of the V International Conference “Current Problems in Optics of Natural Waters, ONW-2009,” I.Levin and G.Gilbert, eds. (Rozhdestvensky Optical Society, 2009), pp. 287–292.

Oriol, R. A.

R. A. Arnone, R. W. Gould, Jr., R. A. Oriol, and G. E. Terrie, “Effects of vertical chlorophyll structure and solar irradiance on remote sensing ocean color spectrum,” Proc. SPIE 2258, 322–331 (1994).
[CrossRef]

Philpot, W. D.

Piscozub, J.

Platt, T.

S. Sathyendranath and T. Platt, “Analytic model of ocean color,” Appl. Opt. 36, 2620–2629 (1997).
[CrossRef] [PubMed]

T. Platt and S. Sathyendranath, “Ocean primary production: estimation by remote sensing at local and regional scales,” Science 241, 1613–1620 (1988).
[CrossRef] [PubMed]

T. Platt, S. Sathyendranath, C. M. Caverhill, and M. R. Lewis, “Ocean primary production and available light: further algorithms for remote sensing,” Deep-Sea Res. Part A 35, 855–879 (1988).
[CrossRef]

M. R. Lewis, J. J. Cullen, and T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

Prasad, K. S.

K. S. Prasad, S. E. Lohrenz, D. G. Redalje, and G. L. Fahnenstiel, “Primary production in the Gulf of Mexico coastal waters using ‘remotely-sensed’ trophic category approach,” Cont. Shelf Res. 15, 1355–1368 (1995).
[CrossRef]

Redalje, D. G.

K. S. Prasad, S. E. Lohrenz, D. G. Redalje, and G. L. Fahnenstiel, “Primary production in the Gulf of Mexico coastal waters using ‘remotely-sensed’ trophic category approach,” Cont. Shelf Res. 15, 1355–1368 (1995).
[CrossRef]

Rhea, W. J.

W. J. Rhea and C. O. Davis, “A comparison of the SeaWiFS chlorophyll and CZCS pigment algorithms using optical data from the 1992 JGOFS Equatorial Pacific Time Series,” Deep-Sea Res. Part II 44, 1907–1925 (1997).
[CrossRef]

Robinson, D. H.

K. R. Arrigo, D. H. Robinson, D. L. Worthen, B. Schieber, and M. P. Lizotte, “Bio-optical properties of the southwestern Ross Sea,” J. Geophys. Res. 103, 21683–21695 (1998).
[CrossRef]

Sathyendranath, S.

S. Sathyendranath and T. Platt, “Analytic model of ocean color,” Appl. Opt. 36, 2620–2629 (1997).
[CrossRef] [PubMed]

T. Platt, S. Sathyendranath, C. M. Caverhill, and M. R. Lewis, “Ocean primary production and available light: further algorithms for remote sensing,” Deep-Sea Res. Part A 35, 855–879 (1988).
[CrossRef]

T. Platt and S. Sathyendranath, “Ocean primary production: estimation by remote sensing at local and regional scales,” Science 241, 1613–1620 (1988).
[CrossRef] [PubMed]

Schieber, B.

K. R. Arrigo, D. H. Robinson, D. L. Worthen, B. Schieber, and M. P. Lizotte, “Bio-optical properties of the southwestern Ross Sea,” J. Geophys. Res. 103, 21683–21695 (1998).
[CrossRef]

Shoshany, M.

L. Sokoletsky, Z. Dubinsky, M. Shoshany, and N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sens. 24, 5049–5073(2003).
[CrossRef]

Siegel, D. A.

D. A. Siegel and T. D. Dickey, “On the parameterization of irradiance for open ocean photoprocesses,” J. Geophys. Res. 92, 14648–14662 (1987).
[CrossRef]

Smith, R. C.

R. C. Smith, “Remote sensing and depth distribution of ocean chlorophyll,” Mar. Ecol. Prog. Ser. 5, 359–361 (1981).
[CrossRef]

Sokoletsky, L.

L. Sokoletsky, “Comparative analysis of selected radiative transfer approaches for aquatic environments,” Appl. Opt. 44, 136–148 (2005).
[CrossRef] [PubMed]

L. Sokoletsky, Z. Dubinsky, M. Shoshany, and N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sens. 24, 5049–5073(2003).
[CrossRef]

L. Sokoletsky, “In situ and remote sensing bio-optical methods for the estimation of phytoplankton concentration in the Gulf of Aqaba (Eilat),” Ph.D. dissertation (Bar-Ilan University, Israel, 2003).

Sokoletsky, L. G.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

L. G. Sokoletsky and Y. Z. Yacobi, “Use of radiative transfer approximations for estimating the optical properties and mean chlorophyll a concentration from reflectance spectra in highly turbid waters,” in Proceedings of the Third International Conference “Current Problems in Optics of Natural Waters, ONW-2005,” I.Levin and G.Gilbert, eds. (D. S. Rozhdestvensky Optical Society, 2005), pp. 47–55.

L. G. Sokoletsky, A. Oren, N. Stambler, and D. Iluz, “Practical algorithms for remote-sensing retrieval of the water column constituents in the Israeli waters,” in Proceedings of the V International Conference “Current Problems in Optics of Natural Waters, ONW-2009,” I.Levin and G.Gilbert, eds. (Rozhdestvensky Optical Society, 2009), pp. 287–292.

Stambler, N.

L. Sokoletsky, Z. Dubinsky, M. Shoshany, and N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sens. 24, 5049–5073(2003).
[CrossRef]

L. G. Sokoletsky, A. Oren, N. Stambler, and D. Iluz, “Practical algorithms for remote-sensing retrieval of the water column constituents in the Israeli waters,” in Proceedings of the V International Conference “Current Problems in Optics of Natural Waters, ONW-2009,” I.Levin and G.Gilbert, eds. (Rozhdestvensky Optical Society, 2009), pp. 287–292.

Stavn, R. H.

R. H. Stavn and A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441(1989).
[CrossRef]

Stramska, M.

Stramski, D.

Sturm, B.

N. Hoepffner, B. Sturm, Z. Finenko, and D. Larkin, “Depth-integrated primary production in the eastern tropical and subtropical North Atlantic basin from ocean colour imagery,” Int. J. Remote Sens. 20, 1435–1456 (1999).
[CrossRef]

Tang, J.

P. Xiu, Y. Liu, and J. Tang, “Variations of ocean colour parameters with nonuniform vertical profiles of chlorophyll concentration,” Int. J. Remote Sens. 29, 831–849 (2008).
[CrossRef]

Terrie, G. E.

R. A. Arnone, R. W. Gould, Jr., R. A. Oriol, and G. E. Terrie, “Effects of vertical chlorophyll structure and solar irradiance on remote sensing ocean color spectrum,” Proc. SPIE 2258, 322–331 (1994).
[CrossRef]

Voss, K. J.

K. J. Voss and A. Morel, “Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: measurements versus predictions,” Limnol. Oceanogr. 50, 698–705 (2005).
[CrossRef]

Weidemann, A. D.

R. H. Stavn and A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441(1989).
[CrossRef]

Wetzel, R. G.

R. G. Wetzel, Limnology: Lake and River Ecosystems, 3rd ed. (Academic, 2001).

Worthen, D. L.

K. R. Arrigo, D. H. Robinson, D. L. Worthen, B. Schieber, and M. P. Lizotte, “Bio-optical properties of the southwestern Ross Sea,” J. Geophys. Res. 103, 21683–21695 (1998).
[CrossRef]

Wozniak, L.

Xiu, P.

P. Xiu, Y. Liu, and J. Tang, “Variations of ocean colour parameters with nonuniform vertical profiles of chlorophyll concentration,” Int. J. Remote Sens. 29, 831–849 (2008).
[CrossRef]

Yacobi, Y. Z.

Y. Z. Yacobi, “Temporal and vertical variation of chlorophyll a concentration, phytoplankton photosynthetic activity and light attenuation in Lake Kinneret: possibilities and limitations for simulation by remote sensing,” J. Plankton Res. 28, 725–736 (2006).
[CrossRef]

Y. Z. Yacobi, A. Gitelson, and M. Mayo, “Remote sensing of chlorophyll in Lake Kinneret using high spectral resolution radiometer and Landsat Thematic Mapper: spectral features of reflectance and algorithm development,” J. Plankton Res. 17, 2155–2173 (1995).
[CrossRef]

L. G. Sokoletsky and Y. Z. Yacobi, “Use of radiative transfer approximations for estimating the optical properties and mean chlorophyll a concentration from reflectance spectra in highly turbid waters,” in Proceedings of the Third International Conference “Current Problems in Optics of Natural Waters, ONW-2005,” I.Levin and G.Gilbert, eds. (D. S. Rozhdestvensky Optical Society, 2005), pp. 47–55.

Zaneveld, J. R. V.

Appl. Opt. (12)

H. R. Gordon and W. R. McCluney, “Estimation of the depth of sunlight penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
[CrossRef] [PubMed]

J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982).
[CrossRef] [PubMed]

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36(1983).
[CrossRef] [PubMed]

E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
[CrossRef] [PubMed]

W. D. Philpot, “Radiative transfer in stratified waters: a single-scattering approximation for irradiance,” Appl. Opt. 26, 4123–4132 (1987).
[CrossRef] [PubMed]

H. R. Gordon, “Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile,” Appl. Opt. 31, 2116–2129 (1992).
[CrossRef] [PubMed]

S. Sathyendranath and T. Platt, “Analytic model of ocean color,” Appl. Opt. 36, 2620–2629 (1997).
[CrossRef] [PubMed]

A. Morel and H. Loisel, “Apparent optical properties of oceanic water, dependence on the molecular scattering contribution,” Appl. Opt. 37, 4765–4776 (1998).
[CrossRef]

T. Hirata, “Irradiance inversion theory to retrieve volume scattering function of seawater,” Appl. Opt. 42, 1564–1573(2003).
[CrossRef] [PubMed]

L. Sokoletsky, “Comparative analysis of selected radiative transfer approaches for aquatic environments,” Appl. Opt. 44, 136–148 (2005).
[CrossRef] [PubMed]

M. Stramska and D. Stramski, “Effects of nonuniform vertical profile of chlorophyll concentration on remote-sensing reflectance of the ocean,” Appl. Opt. 44, 1735–1747(2005).
[CrossRef] [PubMed]

H. R. Gordon and D. C. Clarke, “Remote sensing of optical properties of a stratified ocean: an improved interpretation,” Appl. Opt. 19, 3428–3430 (1980).
[CrossRef] [PubMed]

Can. J. Fish. Aquat. Sci. (1)

J. J. Cullen, “The deep chlorophyll maximum: comparing vertical profiles of chlorophyll a,” Can. J. Fish. Aquat. Sci. 39, 791–803 (1982).
[CrossRef]

Cont. Shelf Res. (1)

K. S. Prasad, S. E. Lohrenz, D. G. Redalje, and G. L. Fahnenstiel, “Primary production in the Gulf of Mexico coastal waters using ‘remotely-sensed’ trophic category approach,” Cont. Shelf Res. 15, 1355–1368 (1995).
[CrossRef]

Deep-Sea Res. Part A (2)

J.-M. André, “Ocean color remote-sensing and the subsurface vertical structure of phytoplankton pigments,” Deep-Sea Res. Part A 39, 763–779 (1992).
[CrossRef]

T. Platt, S. Sathyendranath, C. M. Caverhill, and M. R. Lewis, “Ocean primary production and available light: further algorithms for remote sensing,” Deep-Sea Res. Part A 35, 855–879 (1988).
[CrossRef]

Deep-Sea Res. Part II (1)

W. J. Rhea and C. O. Davis, “A comparison of the SeaWiFS chlorophyll and CZCS pigment algorithms using optical data from the 1992 JGOFS Equatorial Pacific Time Series,” Deep-Sea Res. Part II 44, 1907–1925 (1997).
[CrossRef]

Global Biogeochem. Cycles (1)

D. A. Antoine and A. Morel, “Oceanic primary production. 1. Adaptation of a spectral light-photosynthesis model in view of application to satellite chlorophyll observations,” Global Biogeochem. Cycles 10, 43–55 (1996).
[CrossRef]

Int. J. Remote Sens. (4)

R. W. Gould, Jr., and R. A. Arnone, “Three-dimensional modeling of inherent optical properties in a coastal environment: coupling ocean color imagery and in situ measurements,” Int. J. Remote Sens. 19, 2141–2159 (1998).
[CrossRef]

P. Xiu, Y. Liu, and J. Tang, “Variations of ocean colour parameters with nonuniform vertical profiles of chlorophyll concentration,” Int. J. Remote Sens. 29, 831–849 (2008).
[CrossRef]

N. Hoepffner, B. Sturm, Z. Finenko, and D. Larkin, “Depth-integrated primary production in the eastern tropical and subtropical North Atlantic basin from ocean colour imagery,” Int. J. Remote Sens. 20, 1435–1456 (1999).
[CrossRef]

L. Sokoletsky, Z. Dubinsky, M. Shoshany, and N. Stambler, “Estimation of phytoplankton pigment concentration in the Gulf of Aqaba (Eilat) by in situ and remote sensing single-wavelength algorithms,” Int. J. Remote Sens. 24, 5049–5073(2003).
[CrossRef]

J. Geophys. Res. (6)

K. R. Arrigo, D. H. Robinson, D. L. Worthen, B. Schieber, and M. P. Lizotte, “Bio-optical properties of the southwestern Ross Sea,” J. Geophys. Res. 103, 21683–21695 (1998).
[CrossRef]

M. R. Lewis, J. J. Cullen, and T. Platt, “Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

D. A. Siegel and T. D. Dickey, “On the parameterization of irradiance for open ocean photoprocesses,” J. Geophys. Res. 92, 14648–14662 (1987).
[CrossRef]

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: apparent optical properties and analytical functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015–8024 (1999).
[CrossRef]

A. Morel, “Optical modeling of the upper ocean in relation to its biogeneous matter content (case I waters),” J. Geophys. Res. 93, 10749–10768 (1988).
[CrossRef]

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106, 7163–7180(2001).
[CrossRef]

J. Plankton Res. (2)

Y. Z. Yacobi, A. Gitelson, and M. Mayo, “Remote sensing of chlorophyll in Lake Kinneret using high spectral resolution radiometer and Landsat Thematic Mapper: spectral features of reflectance and algorithm development,” J. Plankton Res. 17, 2155–2173 (1995).
[CrossRef]

Y. Z. Yacobi, “Temporal and vertical variation of chlorophyll a concentration, phytoplankton photosynthetic activity and light attenuation in Lake Kinneret: possibilities and limitations for simulation by remote sensing,” J. Plankton Res. 28, 725–736 (2006).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[CrossRef]

Limnol. Oceanogr. (7)

K. J. Voss and A. Morel, “Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: measurements versus predictions,” Limnol. Oceanogr. 50, 698–705 (2005).
[CrossRef]

R. H. Stavn and A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441(1989).
[CrossRef]

Z. Dubinsky and T. Berman, “Seasonal changes in the spectral composition of downwelling irradiance in Lake Kinneret (Israel),” Limnol. Oceanogr. 24, 652–663 (1979).
[CrossRef]

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

S. Maritorena, A. Morel, and A. Gentili, “Diffuse reflectance of oceanic shallow waters: influence of water depth and bottom albedo,” Limnol. Oceanogr. 39, 1689–1703 (1994).
[CrossRef]

H. Arst, S. Mäekivi, T. Lukk, and A. Herlevi, “Calculating irradiance penetration into water bodies from the measured beam attenuation coefficient,” Limnol. Oceanogr. 42, 379–385(1997).
[CrossRef]

A. Morel and J. F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote-sensing applications,” Limnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

Mar. Ecol. Prog. Ser. (1)

R. C. Smith, “Remote sensing and depth distribution of ocean chlorophyll,” Mar. Ecol. Prog. Ser. 5, 359–361 (1981).
[CrossRef]

Opt. Express (2)

Proc. SPIE (1)

R. A. Arnone, R. W. Gould, Jr., R. A. Oriol, and G. E. Terrie, “Effects of vertical chlorophyll structure and solar irradiance on remote sensing ocean color spectrum,” Proc. SPIE 2258, 322–331 (1994).
[CrossRef]

Science (1)

T. Platt and S. Sathyendranath, “Ocean primary production: estimation by remote sensing at local and regional scales,” Science 241, 1613–1620 (1988).
[CrossRef] [PubMed]

Top. Meteor. Oceanogr. (1)

D. Ballestero, “Remote sensing of vertically structured phytoplankton pigments,” Top. Meteor. Oceanogr. 6, 14–23 (1999).

Other (4)

L. G. Sokoletsky, A. Oren, N. Stambler, and D. Iluz, “Practical algorithms for remote-sensing retrieval of the water column constituents in the Israeli waters,” in Proceedings of the V International Conference “Current Problems in Optics of Natural Waters, ONW-2009,” I.Levin and G.Gilbert, eds. (Rozhdestvensky Optical Society, 2009), pp. 287–292.

R. G. Wetzel, Limnology: Lake and River Ecosystems, 3rd ed. (Academic, 2001).

L. Sokoletsky, “In situ and remote sensing bio-optical methods for the estimation of phytoplankton concentration in the Gulf of Aqaba (Eilat),” Ph.D. dissertation (Bar-Ilan University, Israel, 2003).

L. G. Sokoletsky and Y. Z. Yacobi, “Use of radiative transfer approximations for estimating the optical properties and mean chlorophyll a concentration from reflectance spectra in highly turbid waters,” in Proceedings of the Third International Conference “Current Problems in Optics of Natural Waters, ONW-2005,” I.Levin and G.Gilbert, eds. (D. S. Rozhdestvensky Optical Society, 2005), pp. 47–55.

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Figures (7)

Fig. 1
Fig. 1

Examples of PAR downwelling irradiance, E d ( PAR , z ) profiles observed at the Lake Kinneret from 1990 to 2010, central lake station A.

Fig. 2
Fig. 2

Examples of chlorophyll vertical profiles observed at the Lake Kinneret (a) when a Peridinium bloom was not dominant and (b) when a Peridinium bloom was dominant.

Fig. 3
Fig. 3

Comparison between Chl rs and other indices: (a)  Chl 0 1 m , (b)  Chl p , (c)  Chl e , and (d)  Chl pr for two datasets shown in Fig. 2. Note the same colors for each dataset, and corresponding colors for regression lines and text boxes. Relationships developed by Morel and Berthon (1989) and Antoine and Morel (1996) are also shown for some couples of Chl indices.

Fig. 4
Fig. 4

Relationships between the depth Z max of chlorophyll maximum and the ratios of Chl rs and the other Chl indices. The data for both datasets were used here.

Fig. 5
Fig. 5

The mean absolute percentage error, MAPE = 100 % | Chl rs / Chl ( 0 z ) 1 | ( z = 1 m or Z p ) versus coefficient of variation, CV = 100 % (StDev/Mean) in the first penetration layer. The data for both datasets were used here. Note the color correspondence between data’s symbols and regression lines.

Fig. 6
Fig. 6

Relationships between the ratio of the given layer’s thickness ( Z e or Z pr ) to the Z p and the Chl rs / Chl ( 0 z ) ratio. Only dataset (b) from Fig. 2 has been used here. The dashed vertical lines are shown for hypothetical homogeneous layers (yellow and orange lines for 0 Z e and 0 Z pr layers, respectively) while the dashed horizontal line means Chl rs / Chl ( 0 z ) . Note the color correspondence between the data’s symbols and regression lines.

Fig. 7
Fig. 7

Relationships between the Chl rs computed by Eq. (21) and that computed by Eq. (1) ( Chl rs , GC ), Eq. (4) ( Chl rs , ZBB ), and Eq. (22) ( Chl rs , bb = const ). Note the color correspondence between the data’s symbols and regression lines.

Tables (1)

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Table 1 Glossary of the Main Symbols Used in the Study

Equations (22)

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C rs = 0 Z p exp [ 2 K ¯ d ( 0 z ) z ] C ( z ) d z 0 Z p exp [ 2 K ¯ d ( 0 z ) z ] d z ,
K ¯ d ( 0 z ) = 1 z 0 z K d ( z ) d z ,
κ ¯ ( z 0 ) = 1 z 0 z κ ( z ) d z .
C rs = 0 [ K d ( z ) + K u ( z ) ] exp { [ K ¯ d ( 0 z ) + K ¯ u ( z 0 ) ] z } C ( z ) d z 2 0 K d ( z ) exp [ 2 K ¯ d ( 0 z ) z ] C ( z ) d z .
K d ( PAR , z ) = a 1 + 2 a 2 z + 3 a 3 z 2 ,
ln [ E d ( PAR , z ) ] = ( a 0 + a 1 z + a 2 z 2 + a 3 z 3 )
K ¯ d ( PAR , 0 z ) = a 1 + a 2 z + a 3 z 2 .
Chl rs = 0 Z b Chl ( z ) d E u ( z 0 ) 0 Z b d E u ( z , 0 ) .
d E u ( z 0 ) = d E u ( z ) exp [ κ ¯ ( z 0 ) z ] .
d E u z = b bd E d ( z ) d z ,
E d ( z ) = E d ( 0 ) exp [ K ¯ d ( 0 z ) z ] .
d E u ( z 0 ) = E d ( 0 ) b bd ( z ) exp { [ K ¯ d ( 0 z ) + κ ¯ ( z 0 ) ] z } d z ,
E u ( 0 ) = E d ( 0 ) 0 Z b b bd ( z ) exp { [ K ¯ d ( 0 z ) + κ ¯ ( z 0 ) ] z } d z .
Chl rs = 0 Z b b bd ( z ) exp { [ K ¯ d ( 0 z ) + κ ¯ ( z 0 ) ] z } Chl ( z ) d z 0 Z b b bd ( z ) exp { [ K ¯ d ( 0 z ) + κ ¯ ( z 0 ) ] z } d z .
b bd ( z ) = s ( z ) μ ¯ d ( z ) b b ( z ) ,
K d ( z ) = a ( z ) + b b ( z ) μ ¯ d ( z ) ,
κ ( z ) = a ( z ) + b b ( z ) μ ¯ u ( z ) ,
κ ¯ ( z 0 ) = μ ¯ d ( z ) μ ¯ u ( z ) K ¯ d ( 0 z ) .
κ ( z ) = K u ( z ) + K d ( z ) [ μ ¯ d ( z ) / μ ¯ u ( z ) 1 ] ,
Chl rs = 0 Z b b b ( z ) exp [ 3 K ¯ d ( 0 z ) z ] Chl ( z ) d z 0 Z b b b ( z ) exp [ 3 K ¯ d ( 0 z ) z ] d z .
Chl rs = 0 Z b K d ( z ) exp [ 3 K ¯ d ( 0 z ) z ] Chl ( z ) d z 0 Z b K d ( z ) exp [ 3 K ¯ d ( 0 z ) z ] d z ,
Chl rs = 0 Z b exp [ 3 K ¯ d ( 0 z ) z ] Chl ( z ) d z 0 Z b exp [ 3 K ¯ d ( 0 z ) z ] d z .

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