Abstract

We describe a methodology to quantify and separate the errors of inherent optical properties (IOPs) derived from ocean-color model inversion. Their total error is decomposed into three different sources, namely, model approximations and inversion, sensor noise, and atmospheric correction. Prior information on plausible ranges of observation, sensor noise, and inversion goodness-of-fit are employed to derive the posterior probability distribution of the IOPs. The relative contribution of each error component to the total error budget of the IOPs, all being of stochastic nature, is then quantified. The method is validated with the International Ocean Colour Coordinating Group (IOCCG) data set and the NASA bio-Optical Marine Algorithm Data set (NOMAD). The derived errors are close to the known values with correlation coefficients of 60–90% and 67–90% for IOCCG and NOMAD data sets, respectively. Model-induced errors inherent to the derived IOPs are between 10% and 57% of the total error, whereas atmospheric-induced errors are in general above 43% and up to 90% for both data sets. The proposed method is applied to synthesized and in situ measured populations of IOPs. The mean relative errors of the derived values are between 2% and 20%. A specific error table to the Medium Resolution Imaging Spectrometer (MERIS) sensor is constructed. It serves as a benchmark to evaluate the performance of the atmospheric correction method and to compute atmospheric-induced errors. Our method has a better performance and is more appropriate to estimate actual errors of ocean-color derived products than the previously suggested methods. Moreover, it is generic and can be applied to quantify the error of any derived biogeophysical parameter regardless of the used derivation.

© 2009 Optical Society of America

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2009 (1)

M. S. Salama, A. G. Dekker, Z. Su, C. M. Mannaerts, and W. Verhoef, “Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes,” Hydrol. Earth Syst. Sci. 13, 1113-1121 (2009).
[CrossRef]

2008 (1)

R. van der Velde, Z. Su, and Y. Ma, “Impact of soil moisture dynamics on ASAR σo signatures and its spatial variability observed over the Tibetan Plateau,” Sensors 8, 5479-5491(2008).
[CrossRef]

2007 (1)

J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
[CrossRef]

2006 (3)

A. Morel and S. Bélanger, “Improved detection of turbid waters from ocean color sensors information,” Remote Sens. Environ. 102, 237-249 (2006).
[CrossRef]

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
[CrossRef]

G. Zibordi, “A network for standardized ocean color validation measurements,” EOS Trans. Am. Geophys. Union 87, 293-297 (2006).
[CrossRef]

2005 (3)

S. Maritorena and D. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ. 94, 429-440 (2005).
[CrossRef]

J. Werdell and S. Bailey, “An improved in-situ bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98, 122-140(2005).
[CrossRef]

P. Wang, E. Boss, and C. Roesler, “Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color,” Appl. Opt. 44, 4074-4084 (2005).
[CrossRef] [PubMed]

2004 (3)

M. Sydor, R. Gould, R. Arnone, V. Haltrin, and W. Goode, “Uniqueness in remote sensing of the inherent optical properties of ocean water,” Appl. Opt. 43, 2156-2162(2004).
[CrossRef] [PubMed]

C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
[CrossRef]

S. Salama, J. Monbaliu, and P. Coppin, “Atmospheric correction of advanced very high resolution radiometer imagery,” Int. J. Remote Sens. 25, 1349-1355 (2004).
[CrossRef]

2003 (1)

J. Duarte, M. Vélez-Reyes, S. Tarantola, F. Gilbes, and R. Armstrong, “A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters,” Proc. SPIE 5155, 162-173 (2003).
[CrossRef]

2002 (1)

1999 (1)

1998 (2)

Z. Lee, K. Carder, C. Mobley, R. Steward, and J. Patch, “Hyperspectral remote sensing for shallow waters: 1. A semianalytical model,” Appl. Opt. 37, 6329-6338 (1998).
[CrossRef]

C. Goutis and C. Robert, “Model choice in generalised linear models: a Bayesian approach via Kullback-Leibler projections,” Biometrika 85, 29-37 (1998).
[CrossRef]

1997 (3)

E. Vermote, D. Tanre, J. Deuze, M. Herman, and J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

R. Pope and E. Fry, “Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710-8723 (1997).
[CrossRef]

H. Gordon, “Atmospheric correction of ocean color imagery in the earth observing system era,” J. Geophys. Res. 102, 17081-17106 (1997).
[CrossRef]

1995 (1)

J. Campbell, “The log-normal distribution as a model for bio-optical variability in the sea,” J. Geophys. Res. 100, 13237-13254 (1995).
[CrossRef]

1994 (1)

1991 (1)

B. Carlin and N. Polson, “An expected utility approach to influence diagnostics,” J. Am. Stat. Assoc. 86, 1013-1021(1991).
[CrossRef]

1990 (1)

G. Christakos, “A Bayesian/maximum-entropy view to the spatial estimation problem,” Math. Geol. 22, 763-777 (1990).
[CrossRef]

1988 (2)

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

H. Gordon, J. Brown, and R. Evans, “Exact Rayleigh scattering calculation for the use with the Nimbus-7 Coastal Zone Color Scanner,” Appl. Opt. 27, 862-871 (1988).
[CrossRef] [PubMed]

1985 (1)

W. Johnson and S. Geisser, “Estimative influence measures of the multivariate general linear model,” J. Stat. Planning Inference 11, 33-56 (1985).
[CrossRef]

1983 (1)

1981 (1)

A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic-matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43-53 (1981).
[CrossRef]

1979 (1)

J. Bernardo, “Expected information as expected utility,” Ann. Stat. 7, 686-690 (1979).
[CrossRef]

1968 (1)

E. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. 4, 227-241 (1968).
[CrossRef]

1967 (1)

1957 (2)

E. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620-630 (1957).
[CrossRef]

E. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 108, 171-190 (1957).
[CrossRef]

1954 (1)

1951 (1)

S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

1948 (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Armstrong, R.

J. Duarte, M. Vélez-Reyes, S. Tarantola, F. Gilbes, and R. Armstrong, “A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters,” Proc. SPIE 5155, 162-173 (2003).
[CrossRef]

Arnone, R.

Austin, R.

R. Austin and T. Petzold, “The determination of the diffuse attenuation coefficient of seawater using the Coastal Zone Color Scanner,” in Oceanography from Space, J. Gower, ed. (Plenum, 1981), pp. 239-256.
[CrossRef]

Bailey, S.

J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
[CrossRef]

J. Werdell and S. Bailey, “An improved in-situ bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98, 122-140(2005).
[CrossRef]

Baker, K.

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

Bates, D.

D. Bates and D. Watts, “Nonlinear regression: iterative estimation and linear approximations,” in Nonlinear Regression Analysis and Its Applications, D. Bates and D. Watts, eds. (Wiley, 1988), pp. 32-65.

Bélanger, S.

A. Morel and S. Bélanger, “Improved detection of turbid waters from ocean color sensors information,” Remote Sens. Environ. 102, 237-249 (2006).
[CrossRef]

Bernardo, J.

J. Bernardo, “Expected information as expected utility,” Ann. Stat. 7, 686-690 (1979).
[CrossRef]

Bernardo, J. M.

J. M.Bernardo, “Reference analysis,” in Handbook of Statistics, D. K. Dey and C. R. Rao, eds. (North-Holland, 2005), Vol. 25, pp. 17-90.
[CrossRef]

Blaas, M.

M. Eleveld, H. van der Woerd, G. El Serafy, M. Blaas, T. van Kessel, and G. de Boer, “Assimilation of remotely sensed observations in a sediment transport model,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Boss, E.

Brad, Y.

Y. Brad, Nonlinear Parameter Estimation (Academic, 1974).

Bricaud, A.

A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic-matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43-53 (1981).
[CrossRef]

Brock, J.

C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
[CrossRef]

Broenkow, W.

Brown, J.

Brown, O.

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

H. Gordon, D. Clark, J. Brown, O. Brown, R. Evans, and 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]

Campbell, J.

J. Campbell, “The log-normal distribution as a model for bio-optical variability in the sea,” J. Geophys. Res. 100, 13237-13254 (1995).
[CrossRef]

Carder, K.

Carlin, B.

B. Carlin and N. Polson, “An expected utility approach to influence diagnostics,” J. Am. Stat. Assoc. 86, 1013-1021(1991).
[CrossRef]

Caselles, V.

B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

Chen, Z.

C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
[CrossRef]

Christakos, G.

G. Christakos, “A Bayesian/maximum-entropy view to the spatial estimation problem,” Math. Geol. 22, 763-777 (1990).
[CrossRef]

Clark, D.

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

H. Gordon, D. Clark, J. Brown, O. Brown, R. Evans, and 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]

Clayton, T.

C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
[CrossRef]

Coll, C.

B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

Coppin, P.

S. Salama, J. Monbaliu, and P. Coppin, “Atmospheric correction of advanced very high resolution radiometer imagery,” Int. J. Remote Sens. 25, 1349-1355 (2004).
[CrossRef]

Cox, C.

de Boer, G.

M. Eleveld, H. van der Woerd, G. El Serafy, M. Blaas, T. van Kessel, and G. de Boer, “Assimilation of remotely sensed observations in a sediment transport model,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Dekker, A.

H. Hoogenboom and A. Dekker, “The sensitivity of medium resolution imaging spectrometer MERIS for detecting chlorophyll and seton dry weight in coastal and inland waters,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing (IEEE, 1998), p. 183.

Dekker, A. G.

M. S. Salama, A. G. Dekker, Z. Su, C. M. Mannaerts, and W. Verhoef, “Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes,” Hydrol. Earth Syst. Sci. 13, 1113-1121 (2009).
[CrossRef]

Deuze, J.

E. Vermote, D. Tanre, J. Deuze, M. Herman, and J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

Doerffer, R.

R. Doerffer, “Analysis of the signal/noise and the water leaving radiance Finnish lakes,” Tech. rep. (Brockmann Consult, 2008).

Duarte, J.

J. Duarte, M. Vélez-Reyes, S. Tarantola, F. Gilbes, and R. Armstrong, “A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters,” Proc. SPIE 5155, 162-173 (2003).
[CrossRef]

Eckdagger, T.

B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

Eleveld, M.

M. Eleveld, H. van der Woerd, G. El Serafy, M. Blaas, T. van Kessel, and G. de Boer, “Assimilation of remotely sensed observations in a sediment transport model,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Evans, R.

Feldman, G.

J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
[CrossRef]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2002).

Franz,

J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
[CrossRef]

Franza, B.

J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
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Fry, E.

Garon, V.

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
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Geisser, S.

W. Johnson and S. Geisser, “Estimative influence measures of the multivariate general linear model,” J. Stat. Planning Inference 11, 33-56 (1985).
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Gilbes, F.

J. Duarte, M. Vélez-Reyes, S. Tarantola, F. Gilbes, and R. Armstrong, “A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters,” Proc. SPIE 5155, 162-173 (2003).
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Goody, R.

R. Goody, Atmospheric Radiation 1, Theoretical Basis (Oxford University, 1964).

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Gould, R.

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C. Goutis and C. Robert, “Model choice in generalised linear models: a Bayesian approach via Kullback-Leibler projections,” Biometrika 85, 29-37 (1998).
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Haltrin, V.

Harding, L.

J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
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Herman, M.

E. Vermote, D. Tanre, J. Deuze, M. Herman, and J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
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Holben, B.

B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

Hoogenboom, H.

H. Hoogenboom and A. Dekker, “The sensitivity of medium resolution imaging spectrometer MERIS for detecting chlorophyll and seton dry weight in coastal and inland waters,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing (IEEE, 1998), p. 183.

Hu, C.

C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
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E. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. 4, 227-241 (1968).
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E. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 108, 171-190 (1957).
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E. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620-630 (1957).
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Johnson, W.

W. Johnson and S. Geisser, “Estimative influence measures of the multivariate general linear model,” J. Stat. Planning Inference 11, 33-56 (1985).
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M. D. Kendall and A. Stuart, The Advanced Theory of Statistics: Distribution Theory, 5th ed. (Griffin, 1987), Vol. 1.

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O. Kopelevich, “Small-parameter model of optical properties of sea waters,” in Ocean Optics, A. Monin, ed. (Nauka, 1983), pp. 208-234.

Kroese, D. P.

R. Y. Rubinstein and D. P. Kroese, “Information science and statistics,” in The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning (Springer, 2004).
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S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
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Larnicol, G.

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
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Le Traon, P.-Y.

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
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Lee, Z.

Leibler, R.

S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
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R. van der Velde, Z. Su, and Y. Ma, “Impact of soil moisture dynamics on ASAR σo signatures and its spatial variability observed over the Tibetan Plateau,” Sensors 8, 5479-5491(2008).
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Malkmus, W.

Mannaerts, C. M.

M. S. Salama, A. G. Dekker, Z. Su, C. M. Mannaerts, and W. Verhoef, “Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes,” Hydrol. Earth Syst. Sci. 13, 1113-1121 (2009).
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S. Maritorena and D. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ. 94, 429-440 (2005).
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S. Maritorena, D. Siegel, and A. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41, 2705-2714 (2002).
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Monbaliu, J.

S. Salama, J. Monbaliu, and P. Coppin, “Atmospheric correction of advanced very high resolution radiometer imagery,” Int. J. Remote Sens. 25, 1349-1355 (2004).
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Morcrette, J.

E. Vermote, D. Tanre, J. Deuze, M. Herman, and J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
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A. Morel and S. Bélanger, “Improved detection of turbid waters from ocean color sensors information,” Remote Sens. Environ. 102, 237-249 (2006).
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Muller-Karger, F.

C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
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Nechad, B.

G. Neukermans, B. Nechad, and K. Ruddick, “Optical remote sensing of coastal waters from geostationary platforms: a feasibility study--mapping total suspended matter with SEVIRI,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Neukermans, G.

G. Neukermans, B. Nechad, and K. Ruddick, “Optical remote sensing of coastal waters from geostationary platforms: a feasibility study--mapping total suspended matter with SEVIRI,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Patch, J.

Perry, M.

R. Zaneveld, “Optical closure: from theory to measurement,” in Ocean Optics, Vol. 25 of Oxford Monographs on Geology and Geophysics, R. Spinrad, K. Carder, and M. Perry, eds. (Oxford University, 1994).

Peterson, A.

Petzold, T.

R. Austin and T. Petzold, “The determination of the diffuse attenuation coefficient of seawater using the Coastal Zone Color Scanner,” in Oceanography from Space, J. Gower, ed. (Plenum, 1981), pp. 239-256.
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Pottier, C.

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
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W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2002).

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A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic-matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43-53 (1981).
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C. Goutis and C. Robert, “Model choice in generalised linear models: a Bayesian approach via Kullback-Leibler projections,” Biometrika 85, 29-37 (1998).
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Rubinstein, R. Y.

R. Y. Rubinstein and D. P. Kroese, “Information science and statistics,” in The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning (Springer, 2004).
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Rubio, E.

B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

Ruddick, K.

G. Neukermans, B. Nechad, and K. Ruddick, “Optical remote sensing of coastal waters from geostationary platforms: a feasibility study--mapping total suspended matter with SEVIRI,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Salama, M. S.

M. S. Salama, A. G. Dekker, Z. Su, C. M. Mannaerts, and W. Verhoef, “Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes,” Hydrol. Earth Syst. Sci. 13, 1113-1121 (2009).
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S. Salama, J. Monbaliu, and P. Coppin, “Atmospheric correction of advanced very high resolution radiometer imagery,” Int. J. Remote Sens. 25, 1349-1355 (2004).
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Schaeffer, P.

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
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M. Eleveld, H. van der Woerd, G. El Serafy, M. Blaas, T. van Kessel, and G. de Boer, “Assimilation of remotely sensed observations in a sediment transport model,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Shannon, C.

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

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S. Maritorena and D. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ. 94, 429-440 (2005).
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S. Maritorena, D. Siegel, and A. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41, 2705-2714 (2002).
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V. Singh, “Entropy-based parameter estimation in hydrology,” in Water Science and Technology Library (Kluwer Academic, 1998), Vol. 30.

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B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

Smith, R.

H. Gordon, O. Brown, R. Evans, J. Brown, R. Smith, K. Baker, and D. Clark, “A semianalytical radiance model of ocean color,” J. Geophys. Res. 93, 10909-10924 (1988).
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B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

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Stuart, A.

M. D. Kendall and A. Stuart, The Advanced Theory of Statistics: Distribution Theory, 5th ed. (Griffin, 1987), Vol. 1.

Su, Z.

M. S. Salama, A. G. Dekker, Z. Su, C. M. Mannaerts, and W. Verhoef, “Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes,” Hydrol. Earth Syst. Sci. 13, 1113-1121 (2009).
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R. van der Velde, Z. Su, and Y. Ma, “Impact of soil moisture dynamics on ASAR σo signatures and its spatial variability observed over the Tibetan Plateau,” Sensors 8, 5479-5491(2008).
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S. Salama and Z. Su, “Bayesian approach to resolve the sub-scale spatial variability of ocean color match-ups: methodology and performance,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2009).

Sudre, J.

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
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C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
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Sydor, M.

Tanre, D.

E. Vermote, D. Tanre, J. Deuze, M. Herman, and J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
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J. Duarte, M. Vélez-Reyes, S. Tarantola, F. Gilbes, and R. Armstrong, “A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters,” Proc. SPIE 5155, 162-173 (2003).
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W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C++, The Art of Scientific Computing (Cambridge U. Press, 2002).

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B. Holben, T. Eckdagger, I. Slutsker, E. Sospedra, V. Caselles, C. Coll, E. Valor, and E. Rubio, “Validation of cloud detection algorithms,” EARSeL Symposium on Remote Sensing in the 21st century: Economic and Environmental Applications (Balkema, 2000), pp. 119-123.

van der Velde, R.

R. van der Velde, Z. Su, and Y. Ma, “Impact of soil moisture dynamics on ASAR σo signatures and its spatial variability observed over the Tibetan Plateau,” Sensors 8, 5479-5491(2008).
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van der Woerd, H.

M. Eleveld, H. van der Woerd, G. El Serafy, M. Blaas, T. van Kessel, and G. de Boer, “Assimilation of remotely sensed observations in a sediment transport model,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

van Kessel, T.

M. Eleveld, H. van der Woerd, G. El Serafy, M. Blaas, T. van Kessel, and G. de Boer, “Assimilation of remotely sensed observations in a sediment transport model,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

Vélez-Reyes, M.

J. Duarte, M. Vélez-Reyes, S. Tarantola, F. Gilbes, and R. Armstrong, “A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters,” Proc. SPIE 5155, 162-173 (2003).
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Verhoef, W.

M. S. Salama, A. G. Dekker, Z. Su, C. M. Mannaerts, and W. Verhoef, “Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes,” Hydrol. Earth Syst. Sci. 13, 1113-1121 (2009).
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E. Vermote, D. Tanre, J. Deuze, M. Herman, and J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
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J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
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J. Werdell and S. Bailey, “An improved in-situ bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98, 122-140(2005).
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R. Zaneveld, “Optical closure: from theory to measurement,” in Ocean Optics, Vol. 25 of Oxford Monographs on Geology and Geophysics, R. Spinrad, K. Carder, and M. Perry, eds. (Oxford University, 1994).

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G. Zibordi, “A network for standardized ocean color validation measurements,” EOS Trans. Am. Geophys. Union 87, 293-297 (2006).
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S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
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J. Bernardo, “Expected information as expected utility,” Ann. Stat. 7, 686-690 (1979).
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Appl. Opt. (9)

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R. Pope and E. Fry, “Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710-8723 (1997).
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P. Wang, E. Boss, and C. Roesler, “Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color,” Appl. Opt. 44, 4074-4084 (2005).
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C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Biometrika (1)

C. Goutis and C. Robert, “Model choice in generalised linear models: a Bayesian approach via Kullback-Leibler projections,” Biometrika 85, 29-37 (1998).
[CrossRef]

EOS Trans. Am. Geophys. Union (1)

G. Zibordi, “A network for standardized ocean color validation measurements,” EOS Trans. Am. Geophys. Union 87, 293-297 (2006).
[CrossRef]

Hydrol. Earth Syst. Sci. (1)

M. S. Salama, A. G. Dekker, Z. Su, C. M. Mannaerts, and W. Verhoef, “Deriving inherent optical properties and associated inversion-uncertainties in the Dutch Lakes,” Hydrol. Earth Syst. Sci. 13, 1113-1121 (2009).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

E. Vermote, D. Tanre, J. Deuze, M. Herman, and J. Morcrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

C. Pottier, V. Garon, G. Larnicol, J. Sudre, P. Schaeffer, and P.-Y. Le Traon, “Merging SeaWiFS and MODIS/Aqua ocean color data in North and Equatorial Atlantic using weighted averaging and objective analysis,” IEEE Trans. Geosci. Remote Sens. 44, 3436-3451 (2006).
[CrossRef]

IEEE Trans. Syst. Sci. Cybern. (1)

E. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. 4, 227-241 (1968).
[CrossRef]

Int. J. Remote Sens. (1)

S. Salama, J. Monbaliu, and P. Coppin, “Atmospheric correction of advanced very high resolution radiometer imagery,” Int. J. Remote Sens. 25, 1349-1355 (2004).
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J. Am. Stat. Assoc. (1)

B. Carlin and N. Polson, “An expected utility approach to influence diagnostics,” J. Am. Stat. Assoc. 86, 1013-1021(1991).
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J. Geophys. Res. (3)

H. Gordon, O. Brown, R. Evans, J. Brown, R. Smith, K. Baker, and D. Clark, “A semianalytical radiance model of ocean color,” J. Geophys. Res. 93, 10909-10924 (1988).
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J. Stat. Planning Inference (1)

W. Johnson and S. Geisser, “Estimative influence measures of the multivariate general linear model,” J. Stat. Planning Inference 11, 33-56 (1985).
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A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic-matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43-53 (1981).
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E. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 108, 171-190 (1957).
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Proc. SPIE (2)

J. Werdell, Franz, B. Franza, S. Bailey, L. Harding, and G. Feldman, “Approach for the long-term spatial and temporal evaluation of ocean color satellite data products in a coastal environment,” Proc. SPIE 6680, 66800G (2007).
[CrossRef]

J. Duarte, M. Vélez-Reyes, S. Tarantola, F. Gilbes, and R. Armstrong, “A probabilistic sensitivity analysis of water-leaving radiance to water constituents in coastal shallow waters,” Proc. SPIE 5155, 162-173 (2003).
[CrossRef]

Remote Sens. Environ. (4)

S. Maritorena and D. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ. 94, 429-440 (2005).
[CrossRef]

J. Werdell and S. Bailey, “An improved in-situ bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98, 122-140(2005).
[CrossRef]

C. Hu, Z. Chen, T. Clayton, P. Swarzenski, J. Brock, and F. Muller-Karger, “Assessment of estuarine water-quality indicators using MODIS medium-resolution bands: initial results from Tampa Bay, FL,” Remote Sens. Environ. 93, 423-441 (2004).
[CrossRef]

A. Morel and S. Bélanger, “Improved detection of turbid waters from ocean color sensors information,” Remote Sens. Environ. 102, 237-249 (2006).
[CrossRef]

Sensors (1)

R. van der Velde, Z. Su, and Y. Ma, “Impact of soil moisture dynamics on ASAR σo signatures and its spatial variability observed over the Tibetan Plateau,” Sensors 8, 5479-5491(2008).
[CrossRef]

Other (21)

G. Neukermans, B. Nechad, and K. Ruddick, “Optical remote sensing of coastal waters from geostationary platforms: a feasibility study--mapping total suspended matter with SEVIRI,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

M. Eleveld, H. van der Woerd, G. El Serafy, M. Blaas, T. van Kessel, and G. de Boer, “Assimilation of remotely sensed observations in a sediment transport model,” in Proceedings of the XIX Ocean Optics Conference (Oceanography Society, 2008).

R. Austin and T. Petzold, “The determination of the diffuse attenuation coefficient of seawater using the Coastal Zone Color Scanner,” in Oceanography from Space, J. Gower, ed. (Plenum, 1981), pp. 239-256.
[CrossRef]

R. Goody, Atmospheric Radiation 1, Theoretical Basis (Oxford University, 1964).

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Figures (5)

Fig. 1
Fig. 1

Generated N ( 0 , 1 ) values under the condition that Eq. (13) defines a unique ordered pair of N ( 0 , 1 ) values. A Gaussian distribution (solid curve) is fitted to the generated values (gray area).

Fig. 2
Fig. 2

Schematic summary of the proposed method and related concepts and their implementations.

Fig. 3
Fig. 3

Derived versus known errors (dot symbols) of the IOPs estimated from the IOCCG data set: [(a),(b),(c)] Chl-a absorption at 440 nm ; [(d),(e),(f)] CDOM and detritus absorption at 440 nm ; [(g),(h),(i)] SPM scattering at 550 nm . Vertical panels are for the different error sources: [(a),(d),(g)] model; [(b),(e),(h)] noise; and [(c),(f),(i)] aerosol. Nonlinear regression errors are also superimposed on derived model errors as plus symbols (panels a,d,g). r 2 and r reg 2 are the correlation coefficients for dot and plus symbols, respectively; f is the fraction of successfully derived error values.

Fig. 4
Fig. 4

Derived versus known errors (dots) of IOPs estimated from SeaWiFS spectra of the NOMAD data set for: (a) Chl-a absorption at 440 nm ; (b) detritus and CDOM absorption at 440 nm ; (c) SPM scattering at 550 nm ; and (d) total absorption at 440 nm . Nonlinear regression results are also superimposed as plus symbols. r 2 and r reg 2 are the correlation coefficients for the dot and plus symbols, respectively; f is the fraction of successful retrievals.

Fig. 5
Fig. 5

Sum of variances versus the total variance of the IOCCG data set for (a) Chl-a absorption at 440 nm , (b) absorption of detritus and CDOM at 440 nm , (c) SPM scattering at 550 nm , and (d) total absorption coefficient at 440 nm .

Tables (4)

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Table 1 Average Relative Contribution (%) of Error Components on IOCCG Data Set

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Table 2 Average Relative Contribution (%) of Error Components on SeaWiFS Observations in the NOMAD Data Set

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Table 3 Mean and Standard Deviation (stdv) of Relative Error % on the Derived Values of Posterior Standard Deviation

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Table 4 Average Relative Contribution (%) of Error Components on Derived IOPs Using the Ocean-Color Model [Eq. (1)] and Simulated MERIS Bands from IOCCG Data Set

Equations (34)

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Rs w ( λ ) = t n w 2 i = 1 2 g i ( b b ( λ ) b b ( λ ) + a ( λ ) ) i ,
a ( λ ) = a w ( λ ) + a ph ( λ ) + a dg ( λ ) ,
b b ( λ ) = 0.5 b w ( λ ) + η b spm ( λ ) ,
a ph ( λ ) a 0 ( λ ) a ph ( 440 ) + a 1 ( λ ) a ph ( 440 ) ln a ph ( 440 ) ,
a dg ( λ ) = a dg ( 440 ) exp [ s ( λ 440 ) ] ,
b spm ( λ ) = b spm ( 550 ) ( 550 λ ) y .
iop = [ a ph ( 440 ) , a dg ( 440 ) , s , b spm ( 550 ) , y ] ,
iop = [ a ph ( 440 ) , a dg ( 440 ) , b spm ( 550 ) ] .
Rs t ( λ ) = Rs path ( λ ) + T ( λ ) Rs sfc ( λ ) + T ( λ ) Rs w ( λ ) .
σ t 2 σ t 0 2 + σ t 2 σ inv 2 σ inv 2 + σ t 2 σ ner 2 σ ner 2 + σ t 2 σ a 2 σ a 2 ,
σ t 2 w inv σ inv 2 + w ner σ ner 2 + w a σ a 2 ,
α u = log iop u log iop obs σ ,
r u , l = α u α l = log iop u log iop obs log iop l log iop obs ,
H { P ( iop ) } = 1 N P ( iop ) · log P ( iop ) ,
D KL { P ( iop | ω ) | P ( iop ) } = 1 N P ( iop | ω ) · log P ( iop | ω ) P ( iop ) ,
D KL { P ( iop | ω ) | P ( iop ) } = H { P ( iop | ω ) , P ( iop ) } H { P ( iop | ω ) } ,
H { P ( iop | ω ) , P ( iop ) } = 1 N P ( iop | ω ) · log P ( iop ) .
log iop obs > log iop u > log iop 1 , log iop obs < log iop 1 < log iop u , log iop 1 < log iop obs < log iop u .
m = e μ e 0.5 σ 2 ,
v = e 2 μ e σ 2 ( e σ 2 1 ) .
σ t 2 σ inv 2 + σ ner 2 + σ a 2 .
Δ b spm ( λ ) = b spm ( λ ) b spm ( 550 ) Δ b spm ( 550 ) + b spm ( λ ) y Δ y + b spm ( λ ) λ Δ λ .
Δ b spm ( λ ) = ( 550 λ ) y Δ b spm ( 550 ) + b spm ( 550 ) ( 550 λ ) y ln 550 λ Δ y y λ b spm ( 550 ) ( 550 λ ) y Δ λ .
Δ b spm ( 400 ) = 1.718 Δ b spm ( 550 ) + 0.547 b spm ( 550 ) Δ y ,
Δ b spm ( 680 ) = 0.697 Δ b spm ( 550 ) 0.148 b spm ( 550 ) Δ y .
Δ b spm ( 550 ) b spm ( 550 ) > 0.681 Δ y .
Δ a dg ( λ ) = exp [ s ( λ 440 ) ] Δ a dg ( 440 ) a dg ( 440 ) ( λ 440 ) exp [ s ( λ 440 ) ] Δ s s × a dg ( 440 ) exp [ s ( λ 440 ) ] Δ λ .
Δ a dg ( 400 ) = 2.316 Δ a dg ( 440 ) + 92.654 × 10 9 a dg ( 440 ) Δ S ,
Δ a dg ( 680 ) = 6.47 × 10 3 Δ a dg ( 440 ) 1.554 × 10 9 a dg ( 440 ) Δ S .
Δ a dg ( 440 ) a dg ( 440 ) > 4.08 × 10 8 Δ s .
Δ a ph ( λ ) = a 1 ( λ ) + a 0 ( λ ) + a 1 ( λ ) log a ph ( 440 ) Δ a ph ( 440 ) .
Δ a ph ( 400 ) = 0.731 + 0.012 log a ph ( 440 ) Δ a ph ( 440 ) ,
Δ a ph ( 680 ) = 0.945 + 0.149 log a ph ( 440 ) Δ a ph ( 440 ) .
log a ph ( 440 ) Δ a ph ( 440 ) < 1.562.

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