Abstract

A three-component reflectance model (3C) is applied to above-water radiometric measurements to derive remote-sensing reflectance Rrs (λ). 3C provides a spectrally resolved offset Δ(λ) to correct for residual sun and sky radiance (Rayleigh- and aerosol-scattered) reflections on the water surface that were not represented by sky radiance measurements. 3C is validated with a data set of matching above- and below-water radiometric measurements collected in the Baltic Sea, and compared against a scalar offset correction Δ. Correction with Δ(λ) instead of Δ consistently reduced the (mean normalized root-mean-square) deviation between Rrs (λ) and reference reflectances to comparable levels for clear (Δ: 14.3 ± 2.5 %, Δ(λ): 8.2 ± 1.7 %), partly clouded (Δ: 15.4 ± 2.1 %, Δ(λ): 6.5 ± 1.4 %), and completely overcast (Δ: 10.8 ± 1.7 %, Δ(λ): 6.3 ± 1.8 %) sky conditions. The improvement was most pronounced under inhomogeneous sky conditions when measurements of sky radiance tend to be less representative of surface-reflected radiance. Accounting for both sun glint and sky reflections also relaxes constraints on measurement geometry, which was demonstrated based on a semi-continuous daytime data set recorded in a eutrophic freshwater lake in the Netherlands. Rrs (λ) that were derived throughout the day varied spectrally by less than 2 % relative standard deviation. Implications on measurement protocols are discussed. An open source software library for processing reflectance measurements was developed and is made publicly available.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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    [Crossref]
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2017 (1)

2016 (1)

K. Dörnhöfer, A. Göritz, P. Gege, B. Pflug, and N. Oppelt, “Water Constituents and Water Depth Retrieval from Sentinel-2A - A First Evaluation in an Oligotrophic Lake,” Rem. Sens. 8, 941 (2016).
[Crossref]

2015 (1)

2014 (2)

P. Gege, “WASI-2D: A software tool for regionally optimized analysis of imaging spectrometer data from deep and shallow waters,” Comput. Geosci. 62, 208–215 (2014).
[Crossref]

L. G. Sokoletsky and F. Shen, “Optical closure for remote-sensing reflectance based on accurate radiative transfer approximations: the case of the Changjiang (Yangtze) River Estuary and its adjacent coastal area, China,” Int. J. Rem. Sens. 35, 4193–4224 (2014).
[Crossref]

2013 (3)

T.-W. Cui, Q.-J. Song, J.-W. Tang, and J. Zhang, “Spectral variability of sea surface skylight reflectance and its effect on ocean color,” Opt. Express 21, 24929–24941 (2013).
[Crossref] [PubMed]

S. G. Simis and J. Olsson, “Unattended processing of shipborne hyperspectral reflectance measurements,” Rem. Sens. Environ. 135, 202–212 (2013).
[Crossref]

V. Martinez-Vicente, S. G. H. Simis, R. Alegre, P. E. Land, and S. B. Groom, “Above-water reflectance for the evaluation of adjacency effects in earth observation data: Initial results and methods comparison for near-coastal waters in the western channel, UK,” J. Euro. Opt. Soc. 8, 13060 (2013).
[Crossref]

2012 (1)

2010 (4)

Z. Lee, Y.-h. Ahn, C. Mobley, and R. Arnone, “Removal of surface-reflected light for the measurement of remote-sensing reflectance from an above-surface platform,” Opt. Express 18, 171–182 (2010).
[Crossref]

E. Aas, “Estimates of radiance reflected towards the zenith at the surface of the sea,” Ocean Sci. Discuss. 7, 1059–1102 (2010).
[Crossref]

Y. You, D. Stramski, M. Darecki, and G. W. Kattawar, “Modeling of wave-induced irradiance fluctuations at near-surface depths in the ocean: a comparison with measurements,” Appl. Opt. 49, 1041–1053 (2010).
[Crossref] [PubMed]

M. Hieronymi and A. Macke, “Spatiotemporal underwater light field fluctuations in the open ocean,” J. Euro. Opt. Soc. Rapid Publ. 5, 1–8 (2010).
[Crossref]

2006 (1)

K. G. Ruddick, V. De Cauwer, Y.-J. Park, and G. Moore, “Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters,” Limnol. Oceanog. 51, 1167–1179 (2006).
[Crossref]

2005 (1)

K. Ruddick, V. De Cauwer, and B. Van Mol, “Use of the near infrared similarity reflectance spectrum for the quality control of remote sensing data,” Proc. SPIE 5885, 1–12 (2005).

2004 (1)

P. Gege, “The water color simulator WASI: an integrating software tool for analysis and simulation of optical in situ spectra,” Comput. Geosci. 30, 523–532 (2004).
[Crossref]

2003 (1)

2002 (2)

G. Zibordi, S. B. Hooker, J. F. Berthon, and D. D’Alimonte, “Autonomous above-water radiance measurements from an offshore platform: A field assessment experiments,” J. Atmosph. Ocean. Technol. 19, 808–819 (2002).
[Crossref]

S. B. Hooker, G. Lazin, G. Zibordi, and S. Mclean, “An evaluation of above- and in-water methods for determining water-leaving radiances,” J. Atmosph. Ocean. Technol. 19, 486–515 (2002).
[Crossref]

2000 (2)

D. A. Toole, D. A. Siegel, D. W. Menzies, M. J. Neumann, and R. C. Smith, “Remote-sensing reflectance determinations in the coastal ocean environment: impact of instrumental characteristics and environmental variability,” Appl. Opt. 39, 456–469 (2000).
[Crossref]

C. Stedmon, S. Markager, and H. Kaas, “Optical Properties and Signatures of Chromophoric Dissolved Organic Matter (CDOM) in Danish Coastal Waters,” Estuar. Coastal Shelf Sci. 51, 267–278 (2000).
[Crossref]

1999 (2)

C. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999).
[Crossref]

H. J. Gons, “Optical teledetection of chlorophyll a in turbid inland waters,” Environ. Sci. Technol. 33, 1127–1132 (1999).
[Crossref]

1998 (2)

P. Gege, “Characterization of the phytoplankton in Lake Constance for classification by remote sensing,” Arch. Hydrobiol. Spec. Issues Advanc. Limnol. 0, 179–193 (1998).

Z. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. I. A semianalytical model,” Appl. Opt. 37, 6329–6338 (1998).
[Crossref]

1995 (1)

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[Crossref]

1994 (1)

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “Optical properties of pure water,” Proc. SPIE 174, 2258 (1994).

1990 (1)

W. Gregg and K. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limonol. Oceanog. 35, 1657–1675 (1990).
[Crossref]

1988 (2)

H. R. Gordon, J. W. Brown, O. B. Brown, R. H. Evans, and R. C. Smith, “A semianalytic radiance model of ocean color,” J. Coastal Res. 93, 10909–10924 (1988).

D. Stramski and J. Dera, “On the mechanism for producing flashing light under a wind-disturbed water surface,” Oceanologia 25, 5e21(1988).

1954 (1)

Aas, E.

E. Aas, “Estimates of radiance reflected towards the zenith at the surface of the sea,” Ocean Sci. Discuss. 7, 1059–1102 (2010).
[Crossref]

Ahn, Y.-h.

Z. Lee, Y.-h. Ahn, C. Mobley, and R. Arnone, “Removal of surface-reflected light for the measurement of remote-sensing reflectance from an above-surface platform,” Opt. Express 18, 171–182 (2010).
[Crossref]

Albert, A.

Alegre, R.

V. Martinez-Vicente, S. G. H. Simis, R. Alegre, P. E. Land, and S. B. Groom, “Above-water reflectance for the evaluation of adjacency effects in earth observation data: Initial results and methods comparison for near-coastal waters in the western channel, UK,” J. Euro. Opt. Soc. 8, 13060 (2013).
[Crossref]

Arnone, R.

Z. Lee, Y.-h. Ahn, C. Mobley, and R. Arnone, “Removal of surface-reflected light for the measurement of remote-sensing reflectance from an above-surface platform,” Opt. Express 18, 171–182 (2010).
[Crossref]

Berthon, J. F.

G. Zibordi, S. B. Hooker, J. F. Berthon, and D. D’Alimonte, “Autonomous above-water radiance measurements from an offshore platform: A field assessment experiments,” J. Atmosph. Ocean. Technol. 19, 808–819 (2002).
[Crossref]

Brown, J. W.

H. R. Gordon, J. W. Brown, O. B. Brown, R. H. Evans, and R. C. Smith, “A semianalytic radiance model of ocean color,” J. Coastal Res. 93, 10909–10924 (1988).

Brown, O. B.

H. R. Gordon, J. W. Brown, O. B. Brown, R. H. Evans, and R. C. Smith, “A semianalytic radiance model of ocean color,” J. Coastal Res. 93, 10909–10924 (1988).

Buiteveld, H.

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “Optical properties of pure water,” Proc. SPIE 174, 2258 (1994).

Byrd, R. H.

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[Crossref]

Carder, K.

W. Gregg and K. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limonol. Oceanog. 35, 1657–1675 (1990).
[Crossref]

Carder, K. L.

Cox, C.

Cui, T.-W.

D’Alimonte, D.

G. Zibordi, S. B. Hooker, J. F. Berthon, and D. D’Alimonte, “Autonomous above-water radiance measurements from an offshore platform: A field assessment experiments,” J. Atmosph. Ocean. Technol. 19, 808–819 (2002).
[Crossref]

Darecki, M.

De Cauwer, V.

K. G. Ruddick, V. De Cauwer, Y.-J. Park, and G. Moore, “Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters,” Limnol. Oceanog. 51, 1167–1179 (2006).
[Crossref]

K. Ruddick, V. De Cauwer, and B. Van Mol, “Use of the near infrared similarity reflectance spectrum for the quality control of remote sensing data,” Proc. SPIE 5885, 1–12 (2005).

Dera, J.

D. Stramski and J. Dera, “On the mechanism for producing flashing light under a wind-disturbed water surface,” Oceanologia 25, 5e21(1988).

Donze, M.

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “Optical properties of pure water,” Proc. SPIE 174, 2258 (1994).

Dörnhöfer, K.

K. Dörnhöfer, A. Göritz, P. Gege, B. Pflug, and N. Oppelt, “Water Constituents and Water Depth Retrieval from Sentinel-2A - A First Evaluation in an Oligotrophic Lake,” Rem. Sens. 8, 941 (2016).
[Crossref]

Du, K.

Evans, R. H.

H. R. Gordon, J. W. Brown, O. B. Brown, R. H. Evans, and R. C. Smith, “A semianalytic radiance model of ocean color,” J. Coastal Res. 93, 10909–10924 (1988).

Gege, P.

K. Dörnhöfer, A. Göritz, P. Gege, B. Pflug, and N. Oppelt, “Water Constituents and Water Depth Retrieval from Sentinel-2A - A First Evaluation in an Oligotrophic Lake,” Rem. Sens. 8, 941 (2016).
[Crossref]

P. Gege, “WASI-2D: A software tool for regionally optimized analysis of imaging spectrometer data from deep and shallow waters,” Comput. Geosci. 62, 208–215 (2014).
[Crossref]

P. Gege, “Analytic model for the direct and diffuse components of downwelling spectral irradiance in water,” Appl. Opt. 51, 1407–1419 (2012).
[Crossref] [PubMed]

P. Gege, “The water color simulator WASI: an integrating software tool for analysis and simulation of optical in situ spectra,” Comput. Geosci. 30, 523–532 (2004).
[Crossref]

P. Gege, “Characterization of the phytoplankton in Lake Constance for classification by remote sensing,” Arch. Hydrobiol. Spec. Issues Advanc. Limnol. 0, 179–193 (1998).

P. Gege, “A Case Study at Starnberger See for Hyperspectral Bathymetry Mapping Using Inverse Modeling,” in “WHISPERS 2014,” (2014), pp. 1–4.

P. Gege and P. Groetsch, “A spectral model for correcting sun glint and sky glint,” in “Proceedings of Ocean Optics XXIII” (2016).

Gons, H. J.

H. J. Gons, “Optical teledetection of chlorophyll a in turbid inland waters,” Environ. Sci. Technol. 33, 1127–1132 (1999).
[Crossref]

Gordon, H. R.

H. R. Gordon, J. W. Brown, O. B. Brown, R. H. Evans, and R. C. Smith, “A semianalytic radiance model of ocean color,” J. Coastal Res. 93, 10909–10924 (1988).

Göritz, A.

K. Dörnhöfer, A. Göritz, P. Gege, B. Pflug, and N. Oppelt, “Water Constituents and Water Depth Retrieval from Sentinel-2A - A First Evaluation in an Oligotrophic Lake,” Rem. Sens. 8, 941 (2016).
[Crossref]

Gregg, W.

W. Gregg and K. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limonol. Oceanog. 35, 1657–1675 (1990).
[Crossref]

Groetsch, P.

P. Gege and P. Groetsch, “A spectral model for correcting sun glint and sky glint,” in “Proceedings of Ocean Optics XXIII” (2016).

Groom, S. B.

V. Martinez-Vicente, S. G. H. Simis, R. Alegre, P. E. Land, and S. B. Groom, “Above-water reflectance for the evaluation of adjacency effects in earth observation data: Initial results and methods comparison for near-coastal waters in the western channel, UK,” J. Euro. Opt. Soc. 8, 13060 (2013).
[Crossref]

Hakvoort, J. H. M.

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “Optical properties of pure water,” Proc. SPIE 174, 2258 (1994).

He, S.

Heege, T.

T. Heege, “Flugzeuggestutzte Fernerkundung von Wasserinhaltsstoffen im Bodensee,” Ph.D. thesis, Freie UniversitatBerlin (2000).

Hieronymi, M.

M. Hieronymi and A. Macke, “Spatiotemporal underwater light field fluctuations in the open ocean,” J. Euro. Opt. Soc. Rapid Publ. 5, 1–8 (2010).
[Crossref]

Hooker, S. B.

G. Zibordi, S. B. Hooker, J. F. Berthon, and D. D’Alimonte, “Autonomous above-water radiance measurements from an offshore platform: A field assessment experiments,” J. Atmosph. Ocean. Technol. 19, 808–819 (2002).
[Crossref]

S. B. Hooker, G. Lazin, G. Zibordi, and S. Mclean, “An evaluation of above- and in-water methods for determining water-leaving radiances,” J. Atmosph. Ocean. Technol. 19, 486–515 (2002).
[Crossref]

Kaas, H.

C. Stedmon, S. Markager, and H. Kaas, “Optical Properties and Signatures of Chromophoric Dissolved Organic Matter (CDOM) in Danish Coastal Waters,” Estuar. Coastal Shelf Sci. 51, 267–278 (2000).
[Crossref]

Kattawar, G. W.

Kirk, J. T. O.

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems (Cambridge University, 1994).
[Crossref]

Land, P. E.

V. Martinez-Vicente, S. G. H. Simis, R. Alegre, P. E. Land, and S. B. Groom, “Above-water reflectance for the evaluation of adjacency effects in earth observation data: Initial results and methods comparison for near-coastal waters in the western channel, UK,” J. Euro. Opt. Soc. 8, 13060 (2013).
[Crossref]

Lazin, G.

S. B. Hooker, G. Lazin, G. Zibordi, and S. Mclean, “An evaluation of above- and in-water methods for determining water-leaving radiances,” J. Atmosph. Ocean. Technol. 19, 486–515 (2002).
[Crossref]

Lee, Z.

Z. Lee, Y.-h. Ahn, C. Mobley, and R. Arnone, “Removal of surface-reflected light for the measurement of remote-sensing reflectance from an above-surface platform,” Opt. Express 18, 171–182 (2010).
[Crossref]

Z. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. I. A semianalytical model,” Appl. Opt. 37, 6329–6338 (1998).
[Crossref]

Lu, P.

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. Sci. Comput. 16, 1190–1208 (1995).
[Crossref]

Macke, A.

M. Hieronymi and A. Macke, “Spatiotemporal underwater light field fluctuations in the open ocean,” J. Euro. Opt. Soc. Rapid Publ. 5, 1–8 (2010).
[Crossref]

Markager, S.

C. Stedmon, S. Markager, and H. Kaas, “Optical Properties and Signatures of Chromophoric Dissolved Organic Matter (CDOM) in Danish Coastal Waters,” Estuar. Coastal Shelf Sci. 51, 267–278 (2000).
[Crossref]

Martinez-Vicente, V.

V. Martinez-Vicente, S. G. H. Simis, R. Alegre, P. E. Land, and S. B. Groom, “Above-water reflectance for the evaluation of adjacency effects in earth observation data: Initial results and methods comparison for near-coastal waters in the western channel, UK,” J. Euro. Opt. Soc. 8, 13060 (2013).
[Crossref]

Mclean, S.

S. B. Hooker, G. Lazin, G. Zibordi, and S. Mclean, “An evaluation of above- and in-water methods for determining water-leaving radiances,” J. Atmosph. Ocean. Technol. 19, 486–515 (2002).
[Crossref]

Menzies, D. W.

Mobley,

Mobley, C.

Z. Lee, Y.-h. Ahn, C. Mobley, and R. Arnone, “Removal of surface-reflected light for the measurement of remote-sensing reflectance from an above-surface platform,” Opt. Express 18, 171–182 (2010).
[Crossref]

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

Mobley, C. D.

Moore, G.

K. G. Ruddick, V. De Cauwer, Y.-J. Park, and G. Moore, “Seaborne measurements of near infrared water-leaving reflectance: The similarity spectrum for turbid waters,” Limnol. Oceanog. 51, 1167–1179 (2006).
[Crossref]

Morel, A.

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Figures (6)

Fig. 1
Fig. 1 Measurement geometry for water leaving radiance Lw, upwelling radiance Lu, and sky radiance Ls (adapted from Simis and Olsson [10]). A minus sign (−) indicates sub-surface. Parameter dependencies were omitted for brevity and for further information on parameters and units, please refer to Table 1.
Fig. 2
Fig. 2 Stations that were sampled in the Baltic Sea between 2010 and 2012 and for which both above and below surface radiometric measurements were carried out. Four cases that represent a range of sky conditions are marked with an x, and are presented in detail in Fig. 4.
Fig. 3
Fig. 3 Spectrally resolved weighting factors that were used in 3C model optimization.
Fig. 4
Fig. 4 Each column depicts observations that belong to one of four cases representing clear, hazy, scattered cloud, and fully overcast sky conditions. Corresponding station locations are indicated in Fig. 2. In the first row, individual downwelling irradiance Ed (λ) spectra are plotted together with their mean spectrum. In the same fashion, the following rows depict downwelling radiance Ls (λ), upwelling radiance Lu (λ), the ratio L s ( λ ) E d ( λ ) , and the ratio L u ( λ ) E d ( λ ) . Remote-sensing reflectance R rs ref ( λ ) was derived from sub-surface reflectance (assuming a Q-factor of 5), which contains no surface reflections, and plotted in the last row for visual comparison with L u ( λ ) E d ( λ ) , which contains surface reflections. For further information on parameters and units, please refer to Table 1.
Fig. 5
Fig. 5 Processing results for the four cases depicted in Fig. 4, processed with 3C (left panels) and L10 (right panels). Modelled L u ( λ ) E d ( λ ) -ratios are depicted in red. Modelled L r ( λ ) E d ( λ ) (green) was subtracted from observations of L u ( λ ) E d ( λ ) (blue) to derive remote-sensing reflectances Rrs (λ) (thick grey lines, error bars indicate standard deviations). Reference remote-sensing reflectances R rs ref ( λ ) were derived from sub-surface reflectances R(λ) and are depicted as black lines (error bars indicate standard deviations). For further information on parameters and units, please refer to Table 1.
Fig. 6
Fig. 6 A daytime measurement cycle of radiometric observations that was recorded 11 October 2015 from 10:30 to 16:00 (local time, sun zenith angle lower than 70°) in Paterswoldsemeer (The Netherlands) by an Ecowatch instrument that features two remote-sensing reflectance Rrs (λ) channels, facing NNE (CH 1) and NNW (CH 2). Panel A depicts equalized Rrs (λ) that were derived with 3C for each channel over the course of the day, with standard deviations plotted as error bars. Panel B shows spectrally averaged L u ( λ ) E d ( λ ) -ratios (upper curves) and spectrally averaged Rrs (λ) (lower curves) as a function of azimuth angle difference between sun and sensor. Panels C to F depict 3C model parameters as a function of azimuth angle difference: diffuse and direct reflectance coefficients ρds and ρdd, respectively, Ångström exponent α, and turbidity coefficient β. Error bars in panel B to F indicate standard deviations over ten measurements that were recorded per cycle. The dashed blue vertical line indicates ±135° from the sun in the azimuthal plane, which is considered optimal for remote-sensing reflectance observations from above water. For further information on parameters and units, please refer to Table 1.

Tables (3)

Tables Icon

Table 1 Free model parameters (upper section) in 3C and L10 model optimization. Parameters not allowed to vary in the optimization were set to their start value (indicated by *). Parameters that were variable, but were determined outside of the optimisation, are marked with **. Parameters that were not used in a model run are marked with -. In the following, chlorophyll-a is abbreviated as chl, suspended particulate matter as SPM, and coloured dissolved organic matter as CDOM. The lower section lists further relevant parameter names and acronyms that are used in this manuscript. Units for radiances (mWm−2nm−1sr−1) and irradiances (mWm−2nm−1) are abbreviated with 1 and 2, respectively.

Tables Icon

Table 2 Average distribution statistics (mean and standard deviation (std)) of 3C model optimization results (upper section), and regression statistics between derived and reference remote-sensing reflectances (lower section), for Baltic Sea stations recorded under the following sky conditions: clear skies ( L s E d ( 750 nm ) < 0.1 , 17 stations), mixed sky conditions ( 0.1 > = L s E d ( 750 nm ) < 0.3 , 11 stations), and overcast skies ( L s E d ( 750 nm ) > = 0.3 , 9 stations). For further information on parameters and units, please refer to Table 1.

Tables Icon

Table 3 Average distribution statistics (mean and standard deviation (std)) of L10 model optimization results (upper section), and regression statistics between derived and reference remote-sensing reflectances (lower section), for Baltic Sea stations recorded under the following sky conditions: clear skies ( L s E d ( 750 nm ) < 0.1 , 15 stations), mixed sky conditions ( 0.1 > = L s E d ( 750 nm ) < 0.3 , 9 stations), and overcast skies ( L s E d ( 750 nm ) > = 0.3 , 9 stations). For further information on parameters and units, please refer to Table 1.

Equations (22)

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R rs ( λ ) L w ( λ ) E d ( λ ) ,
R rs ( λ ) = L u ( λ ) L r ( λ ) E d ( λ ) .
R rs ( λ ) = L u ( λ ) E d ( λ ) ρ s L s ( λ ) E d ( λ ) .
R rs ( λ ) = L u ( λ ) E d ( λ ) ρ s L s ( λ ) E d ( λ ) Δ .
max [ | z ( x ( λ ) ) ¯ z ( x ( λ ) ) | ] 0.3 ,
L u m ( λ ) E d m ( λ ) = R rs m ( λ ) + ρ f L s ( λ ) E d ( λ ) + Δ ( λ ) ,
a ( λ ) = a w ( λ ) + C chl a chl * ( λ ) + a CDOM 440 nm exp [ S ( λ 440 nm ) ]
b b ( λ ) = b b , w ( λ ) + C SPM b b , SPM * ( λ )
ω b ( λ ) = b b ( λ ) a ( λ ) + b b ( λ )
R ( λ ) = f ( λ , ω b , θ sun ) × ω b ( λ )
R rs ( λ ) = f rs ( λ , ω b , θ view , θ sun ) × ω b ( λ )
R rs m ( λ ) = ζ R rs ( λ ) 1 γ R ( λ ) ,
Δ ( λ ) = ρ dd E dd ( λ ) E d ( λ ) π + ρ ds [ E dsr ( λ ) E d ( λ ) π + E dsa ( λ ) E d ( λ ) π ] ,
T r ( λ ) = exp [ M ( 115.6406 λ 4 1.335 λ 2 ) 1 ]
T as ( λ ) = exp [ M ω a τ a ( λ ) ]
τ a ( λ ) = β ( λ λ a ) α
E dd ( λ ) E d ( λ ) = Tr ( λ ) T as ( λ ) [ Tr ( λ ) T as ( λ ) + 0.5 ( 1 T r ( λ ) 0.95 ) + T r ( λ ) 1.5 ( 1 T as ( λ ) ) F a ]
E dsa ( λ ) E d ( λ ) = 0.5 ( 1 T r ( λ ) 0.95 ) [ Tr ( λ ) T as ( λ ) + 0.5 ( 1 T r ( λ ) 0.95 ) + T r ( λ ) 1.5 ( 1 T as ( λ ) ) F a ]
E dsr ( λ ) E d ( λ ) = T r ( λ ) 1.5 ( 1 T as ( λ ) ) F a [ Tr ( λ ) T as ( λ ) + 0.5 ( 1 T r ( λ ) 0.95 ) + T r ( λ ) 1.5 ( 1 T as ( λ ) ) F a ] ,
RSS = i ( L u ( λ i ) E d ( λ i ) L u m ( λ i ) E d m ( λ i ) ) 2 W ( λ i ) .
R rs ( λ ) = L u ( λ ) E d ( λ ) ρ f L s ( λ ) E d ( λ ) Δ ( λ , α , β , ρ dd , ρ ds , θ sun ) .
R rs ref ( λ ) = ζ R ( λ ) Q [ 1 γ R ( λ ) ] + δ ,

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