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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 6257–6273
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Impact of computational methods and spectral models on the retrieval of optical properties via spectral optimization

Shaohui Huang, Yonghong Li, Shaoping Shang, and Shaoling Shang  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 6257-6273 (2013)
http://dx.doi.org/10.1364/OE.21.006257


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Abstract

Spectral optimization algorithm (SOA) is a well-accepted scheme for the retrieval of water constituents from the measurement of ocean color radiometry. It defines an error function between the input and output remote sensing reflectance spectrum, with the latter modeled with a few variables that represent the optically active properties, while the variables are solved numerically by minimizing the error function. In this paper, with data from numerical simulations and field measurements as input, we evaluate four computational methods for minimization (optimization) for their efficiency and accuracy on solutions, and illustrate impact of bio-optical models on the retrievals. The four optimization routines are the Levenberg-Marquardt (LM), the Generalized Reduced Gradient (GRG), the Downhill Simplex Method (Amoeba), and the Simulated Annealing-Downhill Simplex (i.e. SA + Amoeba, hereafter abbreviated as SAA). The Garver-Siegel-Maritorena SOA model is used as a base to test these computational methods. It is observed that 1) LM is the fastest method, but SAA has the largest number of valid retrievals; 2) the quality of final solutions are strongly influenced by the forms of spectral models (or eigen functions); and 3) dynamically-varying eigen functions are necessary to obtain smaller errors for both reflectance spectrum and retrievals. Results of this study provide helpful guidance for the selection of a computational method and spectral models if an SOA scheme is to be used to process ocean color images.

© 2013 OSA

1. Introduction

Ocean color satellites have provided an unprecedented view of the global ocean owing to their ability to detect spatio-temporal patterns of bio-optical properties from space [1

1. C. R. McClain, “A decade of satellite ocean color observations,” Annu. Rev. Mar. Sci. 1(1), 19–42 (2009). [CrossRef] [PubMed]

]. However, a satellite sensor does not provide a direct measurement of biological or biogeochemical properties of the water, but top-of-atmosphere radiance (Lt, w/m2/nm/sr). Of this Lt, about 90% is a result of atmospheric scattering. Consequently, it is critical to accurately remove this atmospheric contribution before in-water properties could be adequately derived [2

2. IOCCG, Atmospheric Correction for Remotely-Sensed Ocean-Colour Products, M. Wang, ed., Reports of the International Ocean-Colour Coordinating Group, No.10, (IOCCG, Dartmouth, Canada, 2010).

]. After this correction, because the resulted water-leaving radiance (or remote-sensing reflectance) is still a complex function of in-water properties, ocean-color inversion algorithms for those properties play an important role. Over more than three decades of practice, a variety of algorithms have been developed, including simple empirical regression techniques [3

3. C. M. Hu, Z. Lee, and B. Franz, “Chlorophyll a algorithms for oligotrophic oceans: a novel approach based on three-band reflectance difference,” J. Geophys. Res., Oceans 117(C1), C01011 (2012). [CrossRef]

, 4

4. J. E. O'Reilly, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res., Oceans 103(C11), 24937–24953 (1998). [CrossRef]

], artificial neural networks (e.g. Doerffer & Schiller [5

5. R. Doerffer and H. Schiller, “The MERIS case 2 water algorithm,” Int. J. Remote Sens. 28(3-4), 517–535 (2007). [CrossRef]

]), algebraic solutions [6

6. F. E. Hoge and P. E. Lyon, “Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: an analysis of model and radiance measurement errors,” J. Geophys. Res., Oceans 101(C7), 16631–16648 (1996). [CrossRef]

, 7

7. Z. P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: a multiband quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41(27), 5755–5772 (2002). [CrossRef] [PubMed]

], and spectral optimization algorithms (SOA) [8

8. R. Doerffer and J. Fischer, “Concentrations of chlorophyll, suspended matter, and gelbstoff in case-ii waters derived from satellite coastal zone color scanner data with inverse modeling methods,” J. Geophys. Res., Oceans 99(C4), 7457–7466 (1994). [CrossRef]

15

15. Z. P. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. 2. deriving bottom depths and water properties by optimization,” Appl. Opt. 38(18), 3831–3843 (1999). [CrossRef] [PubMed]

].

The pioneering SOA was proposed in the 1990s [8

8. R. Doerffer and J. Fischer, “Concentrations of chlorophyll, suspended matter, and gelbstoff in case-ii waters derived from satellite coastal zone color scanner data with inverse modeling methods,” J. Geophys. Res., Oceans 99(C4), 7457–7466 (1994). [CrossRef]

]. Since then, various SOA schemes have been developed with different complexity in modeling the bio-optical properties that make up a reflectance spectrum [9

9. V. E. Brando and A. G. Dekker, “Satellite hyperspectral remote sensing for estimating estuarine and coastal water quality,” IEEE Trans. Geosci. Rem. Sens. 41(6), 1378–1387 (2003). [CrossRef]

14

14. C. S. Roesler and M. J. Perry, “In-situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res., Oceans 100(C7), 13279–13294 (1995). [CrossRef]

]. For example, Devred et al. [11

11. E. Devred, S. Sathyendranath, V. Stuart, H. Maass, O. Ulloa, and T. Platt, “A two-component model of phytoplankton absorption in the open ocean: theory and applications,” J. Geophys. Res., Oceans 111(C3), C03011 (2006). [CrossRef]

] incorporated more complicated components to model the spectral inherent optical properties (IOPs), while the Garver-Siegel-Maritorena (GSM) [13

13. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41(15), 2705–2714 (2002). [CrossRef] [PubMed]

] and the Hyperspectral Optimization Process Exemplar (HOPE) schemes [12

12. Z. P. Lee, Visible-Infrared Remote-Sensing Model and Applications for Ocean Waters (University of South Florida, 1994).

,15

15. Z. P. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. 2. deriving bottom depths and water properties by optimization,” Appl. Opt. 38(18), 3831–3843 (1999). [CrossRef] [PubMed]

] employed simpler bio-optical models with a smaller number of variables for the spectral IOPs. Because an SOA searches for the optimal set of numerical combinations among numerous candidates for each measured reflectance spectrum to get a solution, a large computational burden is encountered. This limited its application in processing satellite-measured ocean color images until the past decade when the rapid advance of computer technology reduced the computational stress. Its potential advantages over empirical approaches have renewed interest in using an SOA as an operational tool to process satellite ocean-color data collected by modern sensors, such as the Sea-viewing Wide Field-of-view Sensor (SeaWiFS), the Moderate Resolution Imaging Spectroradiometer (MODIS), and the Medium Resolution Imaging Spectrometer (MERIS) [16

16. S. Maritorena and D. A. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ. 94(4), 429–440 (2005). [CrossRef]

,17

17. J. Werdell, Global bio-optical algorithms for ocean color satelliteapplications: Inherent Optical Properties Algorithm Workshop at Ocean Optics XIX,” Eos Trans. AGU 90, 4 (2008).

]. A GIOP (stands for Generic Inherent Optical Properties) platform that offers freedom to specify various optimization approaches and parameterizations has been developed and implemented under SeaDAS [17

17. J. Werdell, Global bio-optical algorithms for ocean color satelliteapplications: Inherent Optical Properties Algorithm Workshop at Ocean Optics XIX,” Eos Trans. AGU 90, 4 (2008).

]. Because an SOA solves for multiple variables simultaneously in an implicit fashion, it is necessary and useful to illustrate the details of SOA computation, in particular the association and impact of mathematical methods and spectral models.

Here in this study, the GSM, of which the retrieved products have been widely used by the ocean color and oceanography community (e.g. Behrenfeld et al. [18

18. M. J. Behrenfeld, E. Boss, D. A. Siegel, and D. M. Shea, “Carbon-based ocean productivity and phytoplankton physiology from space,” Global Biogeochem. Cycles 19(1), 1006 (2005). [CrossRef]

], Siegel et al. [19

19. D. A. Siegel, S. Maritorena, N. B. Nelson, D. A. Hansell, and M. Lorenzi-Kayser, “Global ocean distribution and dynamics of colored dissolved and detrital organic materials,” J. Geophys. Res. 107(C12), 3228 (2002). [CrossRef]

]), is selected as an example of SOA to gain the insights of the SOA scheme. Four commonly used computational methods that can find local and global minimums are evaluated for their effectiveness in achieving optimization by applying them to a widely used data set simulated by the IOCCG Algorithm Working Group (http://www.ioccg.org/groups/OCAG_data.html). These methods are: the Levenberg-Marquardt (LM), the Generalized Reduced Gradient (GRG), the Downhill Simplex Method (Amoeba), and the Simulated Annealing-Downhill Simplex (SAA). The impact of forward spectral models on the closure between measured and modeled spectral remote-sensing reflectance, and on retrievals of spectral optical properties, is also evaluated using both the IOCCG simulated data set and an in situ data set.

2. Spectral optimization algorithm (SOA)

2.1 Spectral models

For optically deep waters, the remote sensing reflectance above the surface (Rrs, sr−1), which is defined as the ratio of water-leaving radiance to downwelling irradiance just above the surface [20

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

], can be modeled as a function of the total absorption (a(λ), m−1) and total backscattering coefficients (bb(λ), m−1) of the water column, i.e.
Rrs(λ)=Fun(a(λ),bb(λ))
(1)
And, more specifically, for a nadir-viewing angle, there is:
Rrs=0.52rrs11.7rrs
(2)
rrs=(0.0949+0.0794bba+bb)bba+bb
(3)
where rrs (sr−1) refers to the below-surface remote sensing reflectance [7

7. Z. P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: a multiband quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41(27), 5755–5772 (2002). [CrossRef] [PubMed]

,21

21. 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., Atmospheres 93(D9), 10909–10924 (1988). [CrossRef]

].

For an SOA, an error function is defined and minimized to derive the optimal model variables (a(λ) and bb(λ)), and it generally takes a form as
δRrs=Λλ1λ2(R˜rs(λ)Rrs(λ))2Λλ1λ2Rrs(λ)
(4)
with symbol Λλ1λ2representing the average of a spectrum between wavelengths of λ1 and λ2. Rrs(λ) is the spectrum from measurements, and R˜rs(λ)is the spectrum from modeling. δRrs thus provides a relative (percentage) measure of the closure between the measured and modeled Rrs(λ) spectra. Note that in the inversion process, it is always assumed that Eqs. (2)(3) are error free. Small errors in this modeling step will be propagated to the derived a(λ) and bb(λ) when these models are used in an inversion process. Discussions of such effects can be found in Brando et al. [22

22. V. E. Brando, A. G. Dekker, Y. J. Park, and T. Schroeder, “Adaptive semianalytical inversion of ocean color radiometry in optically complex waters,” Appl. Opt. 51(15), 2808–2833 (2012). [CrossRef] [PubMed]

].

The bio-optical models to calculate a(λ) and bb(λ) used in the GSM are detailed in Maritorena et al. [13

13. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41(15), 2705–2714 (2002). [CrossRef] [PubMed]

]. For easier understanding of the results in this study, they are briefly described as below,
a(λ)=aw(λ)+aph(λ)+adg(λ)
(5)
bb(λ)=bbw(λ)+bbp(λ)
(6)
aw(λ) and bbw(λ) are values of water molecules. Although they may change slightly with temperature and salinity [23

23. W. S. Pegau, D. Gray, and J. R. V. Zaneveld, “Absorption and attenuation of visible and near-infrared light in water: dependence on temperature and salinity,” Appl. Opt. 36(24), 6035–6046 (1997). [CrossRef] [PubMed]

], they are considered universal constants here. Thus there are three unknown spectra (aph(λ), adg(λ) and bbp(λ)), and they are further modeled as
aph(λ)=X1a˜ph(λ)
(7)
adg(λ)=X2a˜dg(λ)
(8)
bbp(λ)=X3b˜bp(λ)
(9)
X1-3 are three scalar variables to be derived from a measured Rrs spectrum via SOA, and they represent C, adg(λ0) and bbp(λ0). C is chlorophyll-a concentration (mg/m3). λ0 is a reference wavelength and is generally set as 440 nm.

a˜ph(λ), a˜dg(λ) and b˜bp(λ) are spectral shapes (eigen vectors or eigen functions) determined based on either field measurements [10

10. R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing Of Inland and Coastal Waters (CRC Press, 1995).

, 14

14. C. S. Roesler and M. J. Perry, “In-situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res., Oceans 100(C7), 13279–13294 (1995). [CrossRef]

, 24

24. A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption-coefficients of natural phytoplankton - analysis and parameterization,” J. Geophys. Res., Oceans 100(C7), 13321–13332 (1995). [CrossRef]

, 25

25. Z. P. 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(27), 6329–6338 (1998). [CrossRef] [PubMed]

] or “optimized” from a database [13

13. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41(15), 2705–2714 (2002). [CrossRef] [PubMed]

]. a˜dg(λ)and b˜bp(λ) are generally described as [26

26. 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(1), 43–53 (1981). [CrossRef]

, 27

27. H. R. Gordon and A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983).

]
a˜dg(λ)=eS(λλ0)
(10)
b˜bp(λ)=(λ0λ)Y
(11)
The spectral slope (S) of a˜dg(λ) and the power coefficient (Y) are generally not constants for global waters [28

28. M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Mar. Chem. 89(1-4), 69–88 (2004). [CrossRef]

, 29

29. S. Sathyendranath, L. Prieur, and A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sens. 10(8), 1373–1394 (1989). [CrossRef]

]. In GSM, to reduce variables involving in optimization calculations, S and Y are optimized as 0.0206 nm−1 and 1.03373 for global waters, respectively.

It is much more difficult to model a˜ph(λ) precisely (e.g. Devred et al. [11

11. E. Devred, S. Sathyendranath, V. Stuart, H. Maass, O. Ulloa, and T. Platt, “A two-component model of phytoplankton absorption in the open ocean: theory and applications,” J. Geophys. Res., Oceans 111(C3), C03011 (2006). [CrossRef]

], Bricaud et al. [24

24. A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption-coefficients of natural phytoplankton - analysis and parameterization,” J. Geophys. Res., Oceans 100(C7), 13321–13332 (1995). [CrossRef]

], Ciotti et al. [30

30. A. M. Ciotti, M. R. Lewis, and J. J. Cullen, “Assessment of the relationships between dominant cell size in natural phytoplankton communities and the spectral shape of the absorption coefficient,” Limnol. Oceanogr. 47(2), 404–417 (2002). [CrossRef]

]), although better fits could be achieved by increasing the number of free variables. In GSM, a˜ph(λ) is also optimized as a fixed spectrum for global waters, i.e.
a˜ph(λ)=aph*(λ)
(12)
which is the chlorophyll-specific absorption coefficient (m2/mg).

Therefore, the three eigen functions imbedded in GSM are all fixed globally. Although simple linear-matrix inversion schemes could be used to solve such a system of equations [31

31. P. Wang, E. S. Boss, and C. Roesler, “Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color,” Appl. Opt. 44(19), 4074–4085 (2005). [CrossRef] [PubMed]

], here and as in Maritorena et al. [13

13. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41(15), 2705–2714 (2002). [CrossRef] [PubMed]

], the SOA is used and the impact of other factors such as computational methods are investigated.

2.2 Initial guesses and constrained conditions

For all numerical solutions, a set of initial values (first guess) is required to start the process. In GSM, the initial values for the three variables are kept constant, i.e. X1 = 0.002, X2 = 0.01, and X3 = 0.0029.

To produce physically meaningful properties, boundaries of the three variables have to be defined. Retrievals from an Rrs(λ) spectrum via GSM are considered valid if all three variables satisfy:

0.01 ≤ X1 ≤ 64 mg m−3; 0.0001 ≤ X2 ≤ 2 m−1; 0.0001 ≤ X3 ≤ 0.1 m−1.

These constrains are either applied during the solution phase or used to filter out retrievals considered invalid after the completion of an optimization (see details in Section 3).

All these numbers (for initial guesses and boundaries) were obtained from the IOCCG website (http://www.ioccg.org/groups/software.html).

3. Computation methods for optimization

Under their default computational architectures, constrains can be applied during the solution phase for GRG (a unique feature of GRG [34

34. J. Abadie and J. Carpentier, “Generalization of the wolfe reduced gradient method to the case of nonlinear constraints,” in Optimization, R. Fletcher, ed. (Academic Press, 1969).

]), but not for the other three methods. In order to examine if this difference have any impacts on the solutions, in addition to the default GRG, we also implemented a modified GRG (MGRG) by widening the constraints to −500 < X < 500. After optimization is reached, the same validating constraints described in Section 2.2 are applied to results from LM, Amoeba, SAA and MGRG. Therefore, for these four methods, retrievals considered valid are those that each variable satisfies the same predefined boundaries. For GRG, the same validating constraints were applied during the solution phase.

Finally, the Interactive Data Language (ver.7.0) was used to implement the five optimization methods. Codes of the LM were downloaded from http://www.ioccg.org/groups/software.html, because the LM has been used by the GSM to solve the inversion equation numerically, likely due to its simplicity and effectiveness.

4. Data

Both numerically simulated and field-measured data are used for this study. The simulated data set came from the IOCCG Algorithm Working Group, and details of the data simulation can be found at http://www.ioccg.org/groups/OCAG_data.html. Basically the data set contains 500 spectral Rrs simulated by Hydrolight [38

38. C. D. Mobley, Hydrolight 3.0 Users’ Guide. SRI International (Menlo Park, California, 1995).

] along with their corresponding spectral IOP components, which cover optical properties from clear oceanic waters to turbid coastal waters, which generally follow the increase of the order of the data points. The wavelengths range from 400 nm to 800 nm with a 10 nm spectral resolution, which was sub-sampled to wavelengths at 410, 440, 490, 510, 550, and 670 nm, spectral bands closely matching the settings of SeaWiFS. Such a data set is considered measurement-error free, thus ideal for sensitivity analysis.

The NASA Bio-Optical Marine Algorithm Data set (NOMAD [39

39. P. J. Werdell and S. W. Bailey, “An improved bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98(1), 122–140 (2005). [CrossRef]

],), where all data came from field measurements, was downloaded from SeaBASS website (http://seabass.gsfc.nasa.gov). NOMAD provides optical properties that include IOPs and Rrs at bands of 411, 443, 489, 510, 555, 670 nm. Note that the slight band center difference, such as 410 nm vs 411 nm and 490 nm vs 489 nm, is considered negligible on the SOA solution so no fine adjustment of the eigen functions was made to accommodate the slight band shift [40

40. IOCCG, Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications, Z.-P. Lee, ed., Reports of the International Ocean Colour Coordinating Group, No. 5, (IOCCG, Dartmouth, Canada, 2006).

]. Between 550 and 555 nm, however, the absorption and backscattering coefficients of pure seawater were adjusted accordingly.

5. Result of computational methods

The results of computational methods on the SOA calculation and retrievals are presented in Fig. 1
Fig. 1 Distribution of δRrs obtained with the IOCCG simulated Rrs using GSM; (a) δRrs of each input (with X axis representing the sample number and lower for clearer waters and higher number for more turbid waters); (b) frequency distribution of δRrs.
, Fig. 2
Fig. 2 Distribution of δIOP obtained with the IOCCG simulated Rrs using GSM; the left column shows values for each input and the right column shows frequency distribution; (a) δaph; (b) δadg; (c) δbbp
, and Fig. 3
Fig. 3 Scatter plot of derived IOPs versus known values using the IOCCG simulated data set; (left) 410 nm; (right) 440 nm; (a) aph; (b) adg; (c) bbp.
. In addition to the total number of valid retrievals (N) from each method, a few more measures are included after optimization is achieved. They are: computing cost (T, seconds, with an Intel Core2 Duo E8400 CPU), δRrs(the value when optimization is achieved), δIOP, and root mean square error in log scale (RMSE). δIOP is the error between the derived and known spectral IOPs, which is calculated using Eq. (4), except that the Rrs(λ) spectrum is replaced by the corresponding spectral IOPs and δRrs is replaced by δIOP with the spectral IOPs that formed the optimized δRrs. Further, the calculation of δIOP is focused on the first five bands as the longer wavelength (670 nm here) has limited information on the active optical components for most oceanic waters. Thus δIOP provides a goodness of match between known and retrieved IOPs spectra, instead of products at a single wavelength. On the other hand, RMSE provides a goodness of match between known and retrieved IOPs at a single band. Range, average, median and standard deviation (std) of δRrs,δIOPand RMSE for valid retrievals are summarized in Table 1

Table 1. Results of Different Optimization Schemes for the IOCCG Simulated Data set (n = 500)

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.

A few findings can be drawn from these results:

  • (1) GRG appears to have the largest number of valid retrievals (fully retrieved), while LM and MGRG have the lowest number of valid retrievals. Note that the total valid number of retrieval by the LM in the IOCCG Report 5 [40

    40. IOCCG, Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications, Z.-P. Lee, ed., Reports of the International Ocean Colour Coordinating Group, No. 5, (IOCCG, Dartmouth, Canada, 2006).

    ] is 479. The different number of valid retrievals might be resulted from slightly different implementation of screening valid retrievals, as we could not generate 479 valid retrievals when using the first five bands of the subset. Nevertheless, this difference is not significant when we evaluating the five computational methods with the same screening conditions (see section 2.2).
  • (2) LM is the quickest method among the five, while SAA costs the most (about 800 times the cost of LM).
  • (3) The distribution of error, either in terms of δRrs or δIOP, is almost identical for all optimization methods. For example, δaph are centered around 0.5-0.7 and δadg around 0.2. δbbpmostly spans a range of 0.2 – 0.5 for the relatively clear waters (the first 200 of the data set). These results are consistent with that shown in the IOCCG Report 5 [40

    40. IOCCG, Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications, Z.-P. Lee, ed., Reports of the International Ocean Colour Coordinating Group, No. 5, (IOCCG, Dartmouth, Canada, 2006).

    ] and in Morel [41

    41. A. Morel, “Are the empirical relationships describing the bio-optical properties of case 1 waters consistent and internally compatible?” J. Geophys. Res. 114(C1), C01016 (2009). [CrossRef]

    ] (Fig. 8 of the article). Median values of error, either in terms of δRrs or δIOP, are also similar for all methods.
  • (4) When comparing derived IOP products with known values at each band (410, 440, 490, 510 and 550 nm), a larger RMSE for aph retrievals was found associated with the GRG, while the other four methods showed comparable RMSE numbers. For example, the RMSE of aph(440) using the GRG was roughly two times of that using the other four methods. The reason for this lower performance of the GRG was that it returned quite a few retrievals (~14%) sitting on the boundaries, which deviated from known values significantly.
  • (5) The RMSE for adg and bbp retrievals at each band (410, 440, 490, 510 and 550 nm) was similar for all five methods, and was much lower than that for aph (for example, ~0.1 versus ~0.3 at 440 nm).

For the NOMAD data set, a comparison of derived products with measured values at two bands is shown in Fig. 4
Fig. 4 Scatter plot of derived IOPs versus known values for the NOMAD data set; (left) 412 nm; (right) 443 nm; (a) aph; (b) adg; (c) bbp.
. Note that the N on the figure represents the number of valid data pairs (i.e. both retrieval and in situ data are valid), which was used for the calculation of RMSE. These results suggest consistent conclusions with that based on the IOCCG simulated data set.

In summary, three local optimization methods, i.e. LM, MGRG, and Amoeba, can achieve an optimized solution quickly and with similar precision (in terms of δRrs, δIOP, and RMSE), while LM is the quickest one. GRG can achieve 100% retrieval but those that landed on the boundary constraints are not reliable, as it is dependent on the definition of the boundary. The global optimization method, SAA, on the other hand, achieved the largest number of valid and not-hitting-boundary retrievals, but at a much higher computing cost. Applying the GSM optimization scheme to NOMAD, the NASA Ocean Biology Processing Group (http://oceancolor.gsfc.nasa.gov/MEETINGS/OOXIX/IOP/inversion_methods.html) also evaluated LM and Amoeba, and reached similar results as we shown here with both simulate and field-measured data, i.e. Amoeba is slower compared to LM. However, the results here suggest that SAA is a good candidate for retrieval performance through an SOA scheme, as long as the computational cost is tolerable.

6. Effects of spectral models on error functions and solutions

As shown in the IOCCG Report 5 [40

40. IOCCG, Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications, Z.-P. Lee, ed., Reports of the International Ocean Colour Coordinating Group, No. 5, (IOCCG, Dartmouth, Canada, 2006).

], different bio-optical models will impact the retrievals. The results there, however, show the cumulative effects of all the eigen functions. To highlight the impact of individual eigen function once a time, below we modify some of the eigen functions used in the GSM and to characterize the impacts on error functions and variable solutions. Both the IOCCG simulated data set and the NOMAD data set were used in the experiment.

For easier illustration, we simply change or adjust the three eigen functions in the GSM.

Modification 1 (m01_GSM): In this modification, the eigen function of a˜ph(λ)(used in Eq. (12) was changed to the following:
a˜ph(λ)=a0(λ)+a1(λ)ln(X1) hereX1=aph(λ0),
(13)
It is a model proposed by Lee et al. [25

25. Z. P. 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(27), 6329–6338 (1998). [CrossRef] [PubMed]

]. The other two eigen functions remain the same.

Modification 2 (m02_GSM): In this modification, the above change of a˜ph(λ)maintained, and the Y value used in Eq. (11) was changed from a constant (1.03373) to dynamic (Eq. (14),
Y=2(11.2e0.9Rrs(440)/Rrs(550))
(14)
while the eigen function for adg remains the same (e.g., S value kept as 0.0206 nm−1).

Modification 3 (m03_GSM): The above changes of a˜ph(λ)and b˜bp(λ) maintained, as in Modification 2, and further the adg spectral slope was changed from 0.0206 nm−1 to 0.015 nm−1, a forward bio-optical design matching that of HOPE [15

15. Z. P. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. 2. deriving bottom depths and water properties by optimization,” Appl. Opt. 38(18), 3831–3843 (1999). [CrossRef] [PubMed]

].

Here for easier implementation, MGRG was used to achieve optimization. Figure 5
Fig. 5 Distribution of δRrs and δIOP derived from the IOCCG simulated Rrs using GSM and modified models, computed by applying MGRG; the left column shows values for each input and the right column shows frequency distribution; (a) δRrs; (b) δaph; (c) δadg; (d) δbbp.
shows the δRrs and δIOP values from the GSM and the modified models for the IOCCG simulated data set, respectively, along with the distribution of the δRrs and δIOP values. It is found that δRrs from the m01_GSM, m02_GSM and m03_GSM is smaller than that from the GSM model (for m01_GSM, m02_GSM and m03_GSM, more than 70% of the data set has δRrs ≤ 0.02; δRrs from the original GSM spans a range between 0.04 and 0.10). And they also showed discernible improvement over the original GSM in terms of δIOP. Among the three variables, δaphdiffers the most between the original GSM and m01_GSM, m02_GSM and m03_GSM. δaphof the original GSM has ~25% of the data centered at 0.6, while δaphof the three modified models have ~20% data centered at 0.1 (see Fig. 5).

Figure 6
Fig. 6 Scatter plot of derived 440nm IOPs versus known values for the IOCCG simulated data set; GSM and three modified models were used, computed by applying MGRG; (a) aph, (b) adg, (c) bbp.
compares derived IOPs with known IOPs of the first five bands (except 667 nm) of the IOCCG data set (for convenience, Fig. 6 only shows the result at 440 nm as an example, see Table 2

Table 2. Compare Derived IOPs with Known IOPs of the First Five Bands (Except 667 nm) of the IOCCG Data set

table-icon
View This Table
| View All Tables
for more details). It was found that the modification of the eigen functions changed the retrievals of aph, adg and bbp spectra significantly, and this impact is not uniform among the properties. One modification may reduce the RMSE of one property, but may at the same time increase the RMSE of another property. For example, between the original GSM and m01_GSM, the RMSE for aph(440) reduced from 0.295 to 0.155, but the RMSE for adg(440) increased from 0.114 to 0.185. It is the application of m03_GSM reduced almost the RMSE of all three IOPs. In addition, although m01_GSM returned the lowest RMSE for aph, this may partially due to the much smaller number of valid retrievals (N = 228 with m01_GSM; but it is 420 with the original GSM). On the other hand, the valid number of retrievals with m03_GSM is a lot more than that of m01_GSM. All these results regarding the number of valid retrievals and their qualities suggest that because Rrs is a function of at least three active IOPs (aph, adg, bbp), simply modifying one component may not achieve optimal results of the three properties, although specific tuning may produce promising results of targeted property (such as chlorophyll-a concentration and diffused attenuation coefficient).

A further test of the NOMAD in situ data also provided consistent results (see Fig. 7
Fig. 7 Scatter plot of derived IOPs versus known values for the NOMAD data set; GSM and three modified models were used, computed by applying MGRG; while measured Rrs at six SeaWiFS bands were used as model input, output IOPs at five bands (without 670 nm) were put together and compared with measured data; (a) aph; (b) adg; (c) bbp
). Accuracy of retrievals in terms of absorption was much higher when all modifications were applied. For example, the RMSE for aph decreased from 0.504 using the original GSM to 0.281 using m03_GSM. Here the RMSE value is calculated for aph of the five bands to provide a measure of the quality of the retrieved aph spectrum, instead of the quality of aph at one specific band.

These results echo that spectral models have strong impact on the error function and final solutions of SOA [40

40. IOCCG, Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications, Z.-P. Lee, ed., Reports of the International Ocean Colour Coordinating Group, No. 5, (IOCCG, Dartmouth, Canada, 2006).

], and also suggest that simply changing one eigen function is probably insufficient to improve remote sensing retrievals through an SOA scheme. Of course, we did not consider all possibilities of forward model modification, and this study is just to demonstrate how significant the results will be when changing an eigen function. In addition, the a˜ph(λ) used in GSM is not likely to be the sole factor leading to relatively larger δRrs and δIOPvalue, although this eigen function does not necessarily represent the spectral shape observed from sample measurements [24

24. A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption-coefficients of natural phytoplankton - analysis and parameterization,” J. Geophys. Res., Oceans 100(C7), 13321–13332 (1995). [CrossRef]

], as indicated in Maritorena et al [13

13. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41(15), 2705–2714 (2002). [CrossRef] [PubMed]

].

7. Conclusions

Our analysis shows that the minimized value of the target function appears slightly different when different computational methods for optimization are used. For the five computational methods tested here, all of them can find numerical solutions for each Rrs spectrum, but the computing time and the number of valid retrievals are subject to the method used and the application of boundary constraints. Further, our results echo that the minimized values of the error function and the resulted retrievals depend on the details of the forward bio-optical models. It also suggested that for the implementation of an SOA scheme in ocean color remote sensing, it is desired to employee dynamic eigen functions to reduce errors in retrievals. These detailed illustrations regarding the impacts of computational methods and eigen functions on ocean color retrievals provide more insights associated with the spectral optimization scheme. These findings will lend helps to identify an optimal computational method and eigen functions if an SOA scheme be used to process ocean color images.

Acknowledgments

This study was supported jointly by the NSF-China (#40976068), the MOST-China through the National Basic Research Program (#2009CB421201), and the MOE-China through ITDU program (#B07034). We are grateful to Dr. E. Boss and an anonymous reviewer for their constructive comments that significantly improved this manuscript. And we are in debt to the NASA-OBPG, and data providers, for compiling and distributing NOMAD.

References and links

1.

C. R. McClain, “A decade of satellite ocean color observations,” Annu. Rev. Mar. Sci. 1(1), 19–42 (2009). [CrossRef] [PubMed]

2.

IOCCG, Atmospheric Correction for Remotely-Sensed Ocean-Colour Products, M. Wang, ed., Reports of the International Ocean-Colour Coordinating Group, No.10, (IOCCG, Dartmouth, Canada, 2010).

3.

C. M. Hu, Z. Lee, and B. Franz, “Chlorophyll a algorithms for oligotrophic oceans: a novel approach based on three-band reflectance difference,” J. Geophys. Res., Oceans 117(C1), C01011 (2012). [CrossRef]

4.

J. E. O'Reilly, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res., Oceans 103(C11), 24937–24953 (1998). [CrossRef]

5.

R. Doerffer and H. Schiller, “The MERIS case 2 water algorithm,” Int. J. Remote Sens. 28(3-4), 517–535 (2007). [CrossRef]

6.

F. E. Hoge and P. E. Lyon, “Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: an analysis of model and radiance measurement errors,” J. Geophys. Res., Oceans 101(C7), 16631–16648 (1996). [CrossRef]

7.

Z. P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: a multiband quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41(27), 5755–5772 (2002). [CrossRef] [PubMed]

8.

R. Doerffer and J. Fischer, “Concentrations of chlorophyll, suspended matter, and gelbstoff in case-ii waters derived from satellite coastal zone color scanner data with inverse modeling methods,” J. Geophys. Res., Oceans 99(C4), 7457–7466 (1994). [CrossRef]

9.

V. E. Brando and A. G. Dekker, “Satellite hyperspectral remote sensing for estimating estuarine and coastal water quality,” IEEE Trans. Geosci. Rem. Sens. 41(6), 1378–1387 (2003). [CrossRef]

10.

R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing Of Inland and Coastal Waters (CRC Press, 1995).

11.

E. Devred, S. Sathyendranath, V. Stuart, H. Maass, O. Ulloa, and T. Platt, “A two-component model of phytoplankton absorption in the open ocean: theory and applications,” J. Geophys. Res., Oceans 111(C3), C03011 (2006). [CrossRef]

12.

Z. P. Lee, Visible-Infrared Remote-Sensing Model and Applications for Ocean Waters (University of South Florida, 1994).

13.

S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41(15), 2705–2714 (2002). [CrossRef] [PubMed]

14.

C. S. Roesler and M. J. Perry, “In-situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res., Oceans 100(C7), 13279–13294 (1995). [CrossRef]

15.

Z. P. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. 2. deriving bottom depths and water properties by optimization,” Appl. Opt. 38(18), 3831–3843 (1999). [CrossRef] [PubMed]

16.

S. Maritorena and D. A. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ. 94(4), 429–440 (2005). [CrossRef]

17.

J. Werdell, Global bio-optical algorithms for ocean color satelliteapplications: Inherent Optical Properties Algorithm Workshop at Ocean Optics XIX,” Eos Trans. AGU 90, 4 (2008).

18.

M. J. Behrenfeld, E. Boss, D. A. Siegel, and D. M. Shea, “Carbon-based ocean productivity and phytoplankton physiology from space,” Global Biogeochem. Cycles 19(1), 1006 (2005). [CrossRef]

19.

D. A. Siegel, S. Maritorena, N. B. Nelson, D. A. Hansell, and M. Lorenzi-Kayser, “Global ocean distribution and dynamics of colored dissolved and detrital organic materials,” J. Geophys. Res. 107(C12), 3228 (2002). [CrossRef]

20.

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

21.

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., Atmospheres 93(D9), 10909–10924 (1988). [CrossRef]

22.

V. E. Brando, A. G. Dekker, Y. J. Park, and T. Schroeder, “Adaptive semianalytical inversion of ocean color radiometry in optically complex waters,” Appl. Opt. 51(15), 2808–2833 (2012). [CrossRef] [PubMed]

23.

W. S. Pegau, D. Gray, and J. R. V. Zaneveld, “Absorption and attenuation of visible and near-infrared light in water: dependence on temperature and salinity,” Appl. Opt. 36(24), 6035–6046 (1997). [CrossRef] [PubMed]

24.

A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption-coefficients of natural phytoplankton - analysis and parameterization,” J. Geophys. Res., Oceans 100(C7), 13321–13332 (1995). [CrossRef]

25.

Z. P. 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(27), 6329–6338 (1998). [CrossRef] [PubMed]

26.

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(1), 43–53 (1981). [CrossRef]

27.

H. R. Gordon and A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983).

28.

M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Mar. Chem. 89(1-4), 69–88 (2004). [CrossRef]

29.

S. Sathyendranath, L. Prieur, and A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sens. 10(8), 1373–1394 (1989). [CrossRef]

30.

A. M. Ciotti, M. R. Lewis, and J. J. Cullen, “Assessment of the relationships between dominant cell size in natural phytoplankton communities and the spectral shape of the absorption coefficient,” Limnol. Oceanogr. 47(2), 404–417 (2002). [CrossRef]

31.

P. Wang, E. S. Boss, and C. Roesler, “Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color,” Appl. Opt. 44(19), 4074–4085 (2005). [CrossRef] [PubMed]

32.

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

33.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963). [CrossRef]

34.

J. Abadie and J. Carpentier, “Generalization of the wolfe reduced gradient method to the case of nonlinear constraints,” in Optimization, R. Fletcher, ed. (Academic Press, 1969).

35.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965). [CrossRef]

36.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: the art of scientific computing, 2nd ed. (Cambridge University Press, 1992).

37.

S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983). [CrossRef] [PubMed]

38.

C. D. Mobley, Hydrolight 3.0 Users’ Guide. SRI International (Menlo Park, California, 1995).

39.

P. J. Werdell and S. W. Bailey, “An improved bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98(1), 122–140 (2005). [CrossRef]

40.

IOCCG, Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications, Z.-P. Lee, ed., Reports of the International Ocean Colour Coordinating Group, No. 5, (IOCCG, Dartmouth, Canada, 2006).

41.

A. Morel, “Are the empirical relationships describing the bio-optical properties of case 1 waters consistent and internally compatible?” J. Geophys. Res. 114(C1), C01016 (2009). [CrossRef]

OCIS Codes
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(010.0280) Atmospheric and oceanic optics : Remote sensing and sensors

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: February 8, 2013
Revised Manuscript: February 21, 2013
Manuscript Accepted: February 21, 2013
Published: March 5, 2013

Virtual Issues
Vol. 8, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Shaohui Huang, Yonghong Li, Shaoping Shang, and Shaoling Shang, "Impact of computational methods and spectral models on the retrieval of optical properties via spectral optimization," Opt. Express 21, 6257-6273 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6257


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References

  1. C. R. McClain, “A decade of satellite ocean color observations,” Annu. Rev. Mar. Sci.1(1), 19–42 (2009). [CrossRef] [PubMed]
  2. IOCCG, Atmospheric Correction for Remotely-Sensed Ocean-Colour Products, M. Wang, ed., Reports of the International Ocean-Colour Coordinating Group, No.10, (IOCCG, Dartmouth, Canada, 2010).
  3. C. M. Hu, Z. Lee, and B. Franz, “Chlorophyll a algorithms for oligotrophic oceans: a novel approach based on three-band reflectance difference,” J. Geophys. Res., Oceans117(C1), C01011 (2012). [CrossRef]
  4. J. E. O'Reilly, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res., Oceans103(C11), 24937–24953 (1998). [CrossRef]
  5. R. Doerffer and H. Schiller, “The MERIS case 2 water algorithm,” Int. J. Remote Sens.28(3-4), 517–535 (2007). [CrossRef]
  6. F. E. Hoge and P. E. Lyon, “Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: an analysis of model and radiance measurement errors,” J. Geophys. Res., Oceans101(C7), 16631–16648 (1996). [CrossRef]
  7. Z. P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: a multiband quasi-analytical algorithm for optically deep waters,” Appl. Opt.41(27), 5755–5772 (2002). [CrossRef] [PubMed]
  8. R. Doerffer and J. Fischer, “Concentrations of chlorophyll, suspended matter, and gelbstoff in case-ii waters derived from satellite coastal zone color scanner data with inverse modeling methods,” J. Geophys. Res., Oceans99(C4), 7457–7466 (1994). [CrossRef]
  9. V. E. Brando and A. G. Dekker, “Satellite hyperspectral remote sensing for estimating estuarine and coastal water quality,” IEEE Trans. Geosci. Rem. Sens.41(6), 1378–1387 (2003). [CrossRef]
  10. R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing Of Inland and Coastal Waters (CRC Press, 1995).
  11. E. Devred, S. Sathyendranath, V. Stuart, H. Maass, O. Ulloa, and T. Platt, “A two-component model of phytoplankton absorption in the open ocean: theory and applications,” J. Geophys. Res., Oceans111(C3), C03011 (2006). [CrossRef]
  12. Z. P. Lee, Visible-Infrared Remote-Sensing Model and Applications for Ocean Waters (University of South Florida, 1994).
  13. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt.41(15), 2705–2714 (2002). [CrossRef] [PubMed]
  14. C. S. Roesler and M. J. Perry, “In-situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res., Oceans100(C7), 13279–13294 (1995). [CrossRef]
  15. Z. P. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters. 2. deriving bottom depths and water properties by optimization,” Appl. Opt.38(18), 3831–3843 (1999). [CrossRef] [PubMed]
  16. S. Maritorena and D. A. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ.94(4), 429–440 (2005). [CrossRef]
  17. J. Werdell, Global bio-optical algorithms for ocean color satelliteapplications: Inherent Optical Properties Algorithm Workshop at Ocean Optics XIX,” Eos Trans. AGU90, 4 (2008).
  18. M. J. Behrenfeld, E. Boss, D. A. Siegel, and D. M. Shea, “Carbon-based ocean productivity and phytoplankton physiology from space,” Global Biogeochem. Cycles19(1), 1006 (2005). [CrossRef]
  19. D. A. Siegel, S. Maritorena, N. B. Nelson, D. A. Hansell, and M. Lorenzi-Kayser, “Global ocean distribution and dynamics of colored dissolved and detrital organic materials,” J. Geophys. Res.107(C12), 3228 (2002). [CrossRef]
  20. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press. 1994)
  21. 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., Atmospheres93(D9), 10909–10924 (1988). [CrossRef]
  22. V. E. Brando, A. G. Dekker, Y. J. Park, and T. Schroeder, “Adaptive semianalytical inversion of ocean color radiometry in optically complex waters,” Appl. Opt.51(15), 2808–2833 (2012). [CrossRef] [PubMed]
  23. W. S. Pegau, D. Gray, and J. R. V. Zaneveld, “Absorption and attenuation of visible and near-infrared light in water: dependence on temperature and salinity,” Appl. Opt.36(24), 6035–6046 (1997). [CrossRef] [PubMed]
  24. A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption-coefficients of natural phytoplankton - analysis and parameterization,” J. Geophys. Res., Oceans100(C7), 13321–13332 (1995). [CrossRef]
  25. Z. P. 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(27), 6329–6338 (1998). [CrossRef] [PubMed]
  26. 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(1), 43–53 (1981). [CrossRef]
  27. H. R. Gordon and A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983).
  28. M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Mar. Chem.89(1-4), 69–88 (2004). [CrossRef]
  29. S. Sathyendranath, L. Prieur, and A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sens.10(8), 1373–1394 (1989). [CrossRef]
  30. A. M. Ciotti, M. R. Lewis, and J. J. Cullen, “Assessment of the relationships between dominant cell size in natural phytoplankton communities and the spectral shape of the absorption coefficient,” Limnol. Oceanogr.47(2), 404–417 (2002). [CrossRef]
  31. P. Wang, E. S. Boss, and C. Roesler, “Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color,” Appl. Opt.44(19), 4074–4085 (2005). [CrossRef] [PubMed]
  32. K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math.2, 164–168 (1944).
  33. D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math.11(2), 431–441 (1963). [CrossRef]
  34. J. Abadie and J. Carpentier, “Generalization of the wolfe reduced gradient method to the case of nonlinear constraints,” in Optimization, R. Fletcher, ed. (Academic Press, 1969).
  35. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J.7(4), 308–313 (1965). [CrossRef]
  36. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: the art of scientific computing, 2nd ed. (Cambridge University Press, 1992).
  37. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science220(4598), 671–680 (1983). [CrossRef] [PubMed]
  38. C. D. Mobley, Hydrolight 3.0 Users’ Guide. SRI International (Menlo Park, California, 1995).
  39. P. J. Werdell and S. W. Bailey, “An improved bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ.98(1), 122–140 (2005). [CrossRef]
  40. IOCCG, Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Applications, Z.-P. Lee, ed., Reports of the International Ocean Colour Coordinating Group, No. 5, (IOCCG, Dartmouth, Canada, 2006).
  41. A. Morel, “Are the empirical relationships describing the bio-optical properties of case 1 waters consistent and internally compatible?” J. Geophys. Res.114(C1), C01016 (2009). [CrossRef]

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