Abstract

Remote-sensing reflectance (Rrs), which is defined as the ratio of water-leaving radiance (Lw) to downwelling irradiance just above the surface (Ed(0+)), varies with both water constituents (including bottom properties of optically-shallow waters) and angular geometry. Lw is commonly measured in the field or by satellite sensors at convenient angles, while Ed(0+) can be measured in the field or estimated based on atmospheric properties. To isolate the variations of Rrs (or Lw) resulting from a change of water constituents, the angular effects of Rrs (or Lw) need to be removed. This is also a necessity for the calibration and validation of satellite ocean color measurements. To reach this objective, for optically-deep waters where bottom contribution is negligible, we present a system centered on water’s inherent optical properties (IOPs). It can be used to derive IOPs from angular Rrs and offers an alternative to the system centered on the concentration of chlorophyll. This system is applicable to oceanic and coastal waters as well as to multiband and hyperspectral sensors. This IOP-centered system is applied to both numerically simulated data and in situ measurements to test and evaluate its performance. The good results obtained suggest that the system can be applied to angular Rrs to retrieve IOPs and to remove the angular variation of Rrs.

© 2011 Optical Society of America

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2010

2009

G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[CrossRef]

G. Zibordi, J.-F. Berthon, F. Mélin, D. D’Alimonte, and S. Kaitala, “Validation of satellite ocean color primary products at optically complex coastal sites: Northern Adriatic Sea, Northern Baltic Proper and Gulf of Finland,” Remote Sens. Environ. 113, 2574–2591 (2009).
[CrossRef]

A. Morel, “Are the empirical relationships describing the bio-optical properties of case 1 waters consistent and internally compatible?,” J. Geophys. Res. 114, C01016 (2009).
[CrossRef]

T. Hirata, N. Hardman-Mountford, J. Aiken, and J. Fishwick, “Relationship between the distribution function of ocean nadir radiance and inherent optical properties for oceanic waters,” Appl. Opt. 48, 3130–3139 (2009).
[CrossRef]

J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009).
[CrossRef] [PubMed]

2008

H. J. V. D. Woerd and R. Pasterkamp, “HYDROPT: a fast and flexible method to retrieve chlorophyll-a from multispectral satellite observations of optically complex coastal waters,” Remote Sens. Environ. 112, 1795–1807 (2008).
[CrossRef]

2007

B. Lubac and H. Loisel, “Variability and classification of remote sensing reflectance spectra in the eastern English Channel and southern North Sea,” Remote Sens. Environ. 110, 45–58 (2007).
[CrossRef]

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. 112, C03009 (2007).
[CrossRef]

K. J. Voss, A. Morel, and D. Antoine, “Detailed validation of the bidirectional effect in various case 1 waters for application to ocean color imagery,” Biogeosciences 4, 781–789(2007).
[CrossRef]

2006

Z. P. Lee and C. Hu, “Global distribution of case-1 waters: an analysis from SeaWiFS measurements,” Remote Sens. Environ. 101, 270–276 (2006).
[CrossRef]

G. Zibordi, B. Holben, S. B. Hooker, F. Mélin, J.-F. Berthon, and I. Slutsker, “A network for standardized ocean color validation measurements,” EOS Trans. AGU 87, 297–298 (2006).
[CrossRef]

2005

2004

Z. P. Lee, K. L. Carder, and K. P. Du, “Effects of molecular and particle scatterings on model parameters for remote-sensing reflectance,” Appl. Opt. 43, 4957–4964 (2004).
[CrossRef] [PubMed]

G. Zibordi, D. D’Alimonte, and J. F. Berthon, “An evaluation of depth resolution requirements for optical profiling in coastal waters,” J. Atmos. Ocean. Technol. 21, 1059–1073(2004).
[CrossRef]

G. Zibordi, F. Mélin, S. B. Hooker, D. D’Alimonte, and B. Holben, “An autonomous above-water system for the validation of ocean color radiance data,” IEEE Trans. Geosci. Remote Sens. 43, 401–415 (2004).
[CrossRef]

J.-F. Berthon and G. Zibordi, “Bio-optical relationships for the northern Adriatic Sea,” Int. J. Remote Sens. 25, 1527–1532(2004).
[CrossRef]

2003

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571 (2003).
[CrossRef]

A. Albert and C. D. Mobley, “An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters,” Opt. Express 11, 2873–2890 (2003).
[CrossRef] [PubMed]

C. D. Mobley, H. Zhang, and K. J. Voss, “Effects of optically shallow bottoms on upwelling radiances: bidirectional reflectance distribution function effects,” Limnol. Oceanog. 48, 337–345 (2003).
[CrossRef]

2002

2001

H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Remote Sens. 22, 275–295 (2001).
[CrossRef]

N. K. Hojerslev, “Analytic remote-sensing optical algorithms requiring simple and practical field parameter inputs,” Appl. Opt. 40, 4870–4874 (2001).
[CrossRef]

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

1998

1997

1996

1995

J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. 100, 13135–13142 (1995).
[CrossRef]

1994

1993

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters (2): bi-directional aspects,” Appl. Opt. 32, 6864–6879 (1993).
[CrossRef] [PubMed]

H. R. Gordon, K. J. Voss, and K. A. Kilpatrick, “Angular distribution of fluorescence from phytoplankton,” Limnol. Oceanog. 38, 1582–1586 (1993).
[CrossRef]

1992

H. R. Gordon and K. Ding, “Self-shading of in-water optical instruments,” Limnol. Oceanog. 37, 491–500 (1992).
[CrossRef]

1991

1989

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?,” Limnol. Oceanog. 34, 1389–1409 (1989).
[CrossRef]

1988

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

R. H. Stavn and A. D. Weidemann, “Optical modeling of clear ocean light fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
[CrossRef] [PubMed]

1987

1985

K. L. Carder and R. G. Steward, “A remote-sensing reflectance model of a red tide dinoflagellate off West Florida,” Limnol. Oceanogr. 30, 286–298 (1985).
[CrossRef]

1984

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanog. 29, 350–356 (1984).
[CrossRef]

1982

1981

1977

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanog. 22, 709–722 (1977).
[CrossRef]

1975

1972

T. J. Petzold, “Volume scattering functions for selected natural waters,” Scripps Inst. Oceanog. 72–78 (1972).

Åas, E.

Ahn, Y.-H.

Aiken, J.

T. Hirata, N. Hardman-Mountford, J. Aiken, and J. Fishwick, “Relationship between the distribution function of ocean nadir radiance and inherent optical properties for oceanic waters,” Appl. Opt. 48, 3130–3139 (2009).
[CrossRef]

J. E. O’ReillyS. Maritorena, D. Siegel, M. C. O’Brien, D. Toole, B. G. Mitchell, M. Kahru, F. P. Chavez, P. Strutton, G. Cota, S. B. Hooker, C. R. McClain, K. L. Carder, F. Muller-Karger, L. Harding, A. Magnuson, D. Phinney, G. F. Moore, J. Aiken, K. R. Arrigo, R. Letelier, andM. Culver, “SeaWiFS postlaunch calibration and validation analyses, part 3,” in SeaWiFS Postlaunch Technical Report Series, S.B.Hooker and E.R.Firestone, eds. (NASA, 2000), p. 58.

Albert, A.

Antoine, D.

K. J. Voss, A. Morel, and D. Antoine, “Detailed validation of the bidirectional effect in various case 1 waters for application to ocean color imagery,” Biogeosciences 4, 781–789(2007).
[CrossRef]

A. Morel, D. Antoine, and B. Gentili, “Bidirectional reflectance of oceanic waters: accounting for Raman emission and varying particle scattering phase function,” Appl. Opt. 41, 6289–6306 (2002).
[CrossRef] [PubMed]

Arnone, R.

Arrigo, K. R.

J. E. O’ReillyS. Maritorena, D. Siegel, M. C. O’Brien, D. Toole, B. G. Mitchell, M. Kahru, F. P. Chavez, P. Strutton, G. Cota, S. B. Hooker, C. R. McClain, K. L. Carder, F. Muller-Karger, L. Harding, A. Magnuson, D. Phinney, G. F. Moore, J. Aiken, K. R. Arrigo, R. Letelier, andM. Culver, “SeaWiFS postlaunch calibration and validation analyses, part 3,” in SeaWiFS Postlaunch Technical Report Series, S.B.Hooker and E.R.Firestone, eds. (NASA, 2000), p. 58.

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

Berthon, J. F.

G. Zibordi, D. D’Alimonte, and J. F. Berthon, “An evaluation of depth resolution requirements for optical profiling in coastal waters,” J. Atmos. Ocean. Technol. 21, 1059–1073(2004).
[CrossRef]

Berthon, J.-F.

G. Zibordi, J.-F. Berthon, F. Mélin, D. D’Alimonte, and S. Kaitala, “Validation of satellite ocean color primary products at optically complex coastal sites: Northern Adriatic Sea, Northern Baltic Proper and Gulf of Finland,” Remote Sens. Environ. 113, 2574–2591 (2009).
[CrossRef]

G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
[CrossRef]

G. Zibordi, B. Holben, S. B. Hooker, F. Mélin, J.-F. Berthon, and I. Slutsker, “A network for standardized ocean color validation measurements,” EOS Trans. AGU 87, 297–298 (2006).
[CrossRef]

J.-F. Berthon and G. Zibordi, “Bio-optical relationships for the northern Adriatic Sea,” Int. J. Remote Sens. 25, 1527–1532(2004).
[CrossRef]

Boss, E.

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationship between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427(1975).
[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. 112, C03009 (2007).
[CrossRef]

Z. P. Lee, K. L. Carder, and K. P. Du, “Effects of molecular and particle scatterings on model parameters for remote-sensing reflectance,” Appl. Opt. 43, 4957–4964 (2004).
[CrossRef] [PubMed]

Z. P. Lee, K. L. Carder, and R. Arnone, “Deriving inherent optical properties from water color: a multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002).
[CrossRef] [PubMed]

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H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
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J. E. O’ReillyS. Maritorena, D. Siegel, M. C. O’Brien, D. Toole, B. G. Mitchell, M. Kahru, F. P. Chavez, P. Strutton, G. Cota, S. B. Hooker, C. R. McClain, K. L. Carder, F. Muller-Karger, L. Harding, A. Magnuson, D. Phinney, G. F. Moore, J. Aiken, K. R. Arrigo, R. Letelier, andM. Culver, “SeaWiFS postlaunch calibration and validation analyses, part 3,” in SeaWiFS Postlaunch Technical Report Series, S.B.Hooker and E.R.Firestone, eds. (NASA, 2000), p. 58.

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G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
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Giles, D.

G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
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T. Hirata, N. Hardman-Mountford, J. Aiken, and J. Fishwick, “Relationship between the distribution function of ocean nadir radiance and inherent optical properties for oceanic waters,” Appl. Opt. 48, 3130–3139 (2009).
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Hooker, S. B.

G. Zibordi, B. Holben, S. B. Hooker, F. Mélin, J.-F. Berthon, and I. Slutsker, “A network for standardized ocean color validation measurements,” EOS Trans. AGU 87, 297–298 (2006).
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G. Zibordi, F. Mélin, S. B. Hooker, D. D’Alimonte, and B. Holben, “An autonomous above-water system for the validation of ocean color radiance data,” IEEE Trans. Geosci. Remote Sens. 43, 401–415 (2004).
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J. E. O’ReillyS. Maritorena, D. Siegel, M. C. O’Brien, D. Toole, B. G. Mitchell, M. Kahru, F. P. Chavez, P. Strutton, G. Cota, S. B. Hooker, C. R. McClain, K. L. Carder, F. Muller-Karger, L. Harding, A. Magnuson, D. Phinney, G. F. Moore, J. Aiken, K. R. Arrigo, R. Letelier, andM. Culver, “SeaWiFS postlaunch calibration and validation analyses, part 3,” in SeaWiFS Postlaunch Technical Report Series, S.B.Hooker and E.R.Firestone, eds. (NASA, 2000), p. 58.

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G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
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G. Zibordi, J.-F. Berthon, F. Mélin, D. D’Alimonte, and S. Kaitala, “Validation of satellite ocean color primary products at optically complex coastal sites: Northern Adriatic Sea, Northern Baltic Proper and Gulf of Finland,” Remote Sens. Environ. 113, 2574–2591 (2009).
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Lewis, M. R.

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571 (2003).
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McClain, C. R.

J. E. O’ReillyS. Maritorena, D. Siegel, M. C. O’Brien, D. Toole, B. G. Mitchell, M. Kahru, F. P. Chavez, P. Strutton, G. Cota, S. B. Hooker, C. R. McClain, K. L. Carder, F. Muller-Karger, L. Harding, A. Magnuson, D. Phinney, G. F. Moore, J. Aiken, K. R. Arrigo, R. Letelier, andM. Culver, “SeaWiFS postlaunch calibration and validation analyses, part 3,” in SeaWiFS Postlaunch Technical Report Series, S.B.Hooker and E.R.Firestone, eds. (NASA, 2000), p. 58.

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G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
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G. Zibordi, J.-F. Berthon, F. Mélin, D. D’Alimonte, and S. Kaitala, “Validation of satellite ocean color primary products at optically complex coastal sites: Northern Adriatic Sea, Northern Baltic Proper and Gulf of Finland,” Remote Sens. Environ. 113, 2574–2591 (2009).
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G. Zibordi, B. Holben, S. B. Hooker, F. Mélin, J.-F. Berthon, and I. Slutsker, “A network for standardized ocean color validation measurements,” EOS Trans. AGU 87, 297–298 (2006).
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J. E. O’ReillyS. Maritorena, D. Siegel, M. C. O’Brien, D. Toole, B. G. Mitchell, M. Kahru, F. P. Chavez, P. Strutton, G. Cota, S. B. Hooker, C. R. McClain, K. L. Carder, F. Muller-Karger, L. Harding, A. Magnuson, D. Phinney, G. F. Moore, J. Aiken, K. R. Arrigo, R. Letelier, andM. Culver, “SeaWiFS postlaunch calibration and validation analyses, part 3,” in SeaWiFS Postlaunch Technical Report Series, S.B.Hooker and E.R.Firestone, eds. (NASA, 2000), p. 58.

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Pasterkamp, R.

H. J. V. D. Woerd and R. Pasterkamp, “HYDROPT: a fast and flexible method to retrieve chlorophyll-a from multispectral satellite observations of optically complex coastal waters,” Remote Sens. Environ. 112, 1795–1807 (2008).
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A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanog. 22, 709–722 (1977).
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S. Sathyendranath and T. Platt, “Analytic model of ocean color,” Appl. Opt. 36, 2620–2629 (1997).
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E. Devred, S. Sathyendranath, and T. Platt, “Inversion based on a semi-analytical reflectance model,” in Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms and Applications, Z.-P.Lee, ed. (GKSS, 2006), p. 87–94.

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G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
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G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
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Slutsker, I.

G. Zibordi, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, F. Melin, J.-F. Berthon, D. Vandemark, H. Feng, G. Schuster, B. E. Fabbri, S. Kaitala, and J. Seppala, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009).
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Figures (9)

Fig. 1
Fig. 1

Angular coordinate used in this study and the IOP-centered system.

Fig. 2
Fig. 2

Particle phased function shapes (Petzold average and 1% Fournier and Forand) used for the numerical simulation. The open diamond represents one weighted average of the particle assemblage.

Fig. 3
Fig. 3

Schematic data flow chart of the IOP-centered BRDF correction scheme.

Fig. 4
Fig. 4

Distribution of absolute percentage difference (δ) of L w . Simulated data were used, and focused on the angular domain that θ S = 60 ° , θ v = 30 ° 70 ° , and φ = 0 ° 180 ° . Solid circle, no BRDF correction; open diamond, after IOP-based BRDF correction.

Fig. 5
Fig. 5

Remote sensing reflectance of the two sample stations: (left) measured in the Mediterranean Sea, (right) measured in the Monterey Bay.

Fig. 6
Fig. 6

(Top panel) Comparison between measured and derived L w ( λ , Ω ) / L w ( λ , 0 ° ) for selected wavelength and viewing angle of measurements made in the Mediterranean Sea. Red triangle represents measurements from the NuRADS; blue diamond represents values calculated from R rs (IOPs-centered approach). (Bottom panel) Comparison between measured and derived L w ( λ , Ω ) / L w ( λ , 0 ° ) for Ω in the remote sensing domain (blue circle, IOPs-centered approach; green square, Chl-centered approach).

Fig. 7
Fig. 7

As Fig. 5, for measurements made in the Monterey Bay.

Fig. 8
Fig. 8

Comparison between measured and derived L w ( λ , θ S , 0 ° , 0 ° ) for measurements made at the AAOT. The x axis represents L w ( λ , θ S , 0 ° , 0 ° ) measured by the WiSPER, while the y axis represents L w ( λ , θ S , 0 ° , 0 ° ) calculated from L w ( λ , θ S , 40 ° , 90 ° ) which was measured by a CE-318 radiometer (blue circle, IOPs-centered approach; green square, Chl-centered approach).

Fig. 9
Fig. 9

Scatter plot of L w ( λ , Ω ) / L w ( λ , 0 ° ) for different values of a ( λ 0 ) . The x axis represents the L w ( λ , Ω ) / L w ( λ , 0 ° ) ratio derived with a ( 550 ) in the current QAA scheme; the y-axis represents the L w ( λ , Ω ) / L w ( λ , 0 ° ) ratio derived after altering the estimated a ( 550 ) . Top panel, the blue water in the Mediterranean Sea, a ( 550 ) was altered by ± 10 % ; bottom panel, the green water in the Monterey Bay, a ( 550 ) was altered by ± 20 % .

Tables (2)

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Table 2 Sample Values of ( G 0 w ( Ω ) , G 1 w ( Ω ) , G 0 p ( Ω ) , G 1 p ( Ω ) ) for Angular R rs

Equations (21)

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[ L w ( λ ) ] N L w ( λ , Ω ) = F 0 ( λ ) E d ( 0 + , λ , θ S ) [ R ] N R ( θ S , θ v ) [ f ( λ ) ] N [ Q ( λ ) ] N Q ( λ , Ω ) f ( λ , θ S ) .
r rs ( λ , Ω ) = L u ( 0 , λ , Ω ) E d ( 0 , λ , θ S ) ,
L w ( λ , Ω ) = 1 ρ ( θ v , φ ) n 2 E d ( 0 , λ , θ S ) r rs ( λ , Ω ) ,
r rs ( λ , Ω ) D d ( λ , θ S ) c ( λ ) + k L ( λ , Ω ) f L ( λ , Ω ) b f ( λ ) 0 2 π 0 π / 2 β ( Ω , Ω ) L ( λ , Ω ) sin ( θ ) d θ d φ E o d ( 0 , λ , θ S ) .
r rs ( λ , 0 ° ) = D d ( λ , θ S ) β ( 180 θ S ) a ( λ ) + k L ( λ , 0 ° ) .
r rs ( λ , θ v = 0 ° ) = [ g 0 + g 1 ( b b ( λ ) a ( λ ) + b b ( λ ) ) ] b b ( λ ) a ( λ ) + b b ( λ ) .
r rs ( λ , Ω ) = q ( Ω , w ) i = 1 4 p i ( b b ( λ ) a ( λ ) + b b ( λ ) ) i .
r rs ( λ , Ω ) = g w ( Ω ) b b w ( λ ) a ( λ ) + b b ( λ ) + g p ( λ , Ω ) b b p ( λ ) a ( λ ) + b b ( λ ) .
r rs ( λ , Ω ) = i = 1 4 g i ( Ω , ν b ) ( b b ( λ ) a ( λ ) + b b ( λ ) ) i .
ln [ r rs ( λ , Ω ) ] = i = 1 4 j = 1 4 P i j ( Ω ) [ ln ( a ( λ ) ) ] i [ ln ( b ( λ ) ) ] j .
R rs ( λ , Ω ) = L w ( λ , Ω ) E d ( 0 + , λ , θ S ) .
R rs ( λ , Ω ) = G ( λ , Ω ) Fun ( IOP ( λ ) ) .
[ R rs ( λ ) ] N = [ G ( λ ) ] N Fun ( IOP ( λ ) ) ,
R rs ( λ , Ω ) = ( G 0 w ( Ω ) + G 1 w ( Ω ) b b w ( λ ) κ ( λ ) ) b b w ( λ ) κ ( λ ) + ( G 0 p ( Ω ) + G 1 p ( Ω ) b b p ( λ ) κ ( λ ) ) b b p ( λ ) κ ( λ ) ,
{ χ = log ( R rs ( 443 ) + R rs ( 490 ) R rs ( λ 0 ) + 5 R rs ( 667 ) R rs ( 490 ) R rs ( 667 ) ) , a ( λ 0 ) = a w ( λ 0 ) + 10 1.146 1.366 χ 0.469 χ 2 .
A x 2 + B x + C = 0 ,
{ A = G 0 p + G 1 p R rs , B = G 0 w b b w + G 0 p ( a + b b w ) 2 R rs ( a + b b w ) , C = G 0 w b b w ( a + b b w ) R rs ( a + b b w ) 2 + G 1 w ( b b w ) 2 .
R rs κ 2 X κ Y = 0 ,
{ X = G 0 w b b w + G 0 p b b p , Y = G 1 w ( b b w ) 2 + G 1 p ( b b p ) 2 .
a = κ b b w b b p ,
δ = | L L w ( λ , 0 ° ) | L w ( λ , 0 ° ) × 100 .

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