## Ensemble uncertainty of inherent optical properties |

Optics Express, Vol. 19, Issue 18, pp. 16772-16783 (2011)

http://dx.doi.org/10.1364/OE.19.016772

Acrobat PDF (1288 KB)

### Abstract

We present a method to evaluate the combined accuracy of ocean color models and the parameterizations of inherent optical proprieties (IOPs), or model-parametrization setup. The method estimates the ensemble (collective) uncertainty of derived IOPs relative to the radiometric error and is directly applicable to ocean color products without the need for inversion. Validation shows a very good fit between derived and known values for synthetic data, with R^{2} > 0.95 and mean absolute difference (MADi) <0.25 m^{−1}. Due to the influence of observation errors, these values deteriorate to 0.45 < R^{2} < 0.5 and 0.65 < MADi < 0.9 for *in-situ* and ocean color matchup data. The method is also used to estimate the maximum accuracy that could be achieved by a specific model-parametrization setup, which represents the optimum accuracy that should be targeted when deriving IOPs. Application to time series of ocean color global products collected between 1997–2007 shows few areas with increasing annual trends of ensemble uncertainty, up to 8 sr m^{−1}decade^{−1}. This value is translated to an error of 0.04 m^{−1}decade^{−1} in the sum of derived absorption and backscattering coefficients at the blue wavelength 440 nm. As such, the developed method can be used as a tool for assessing the reliability of model-parametrization setups for specific biophysical conditions and for identifying hot-spots for which the model-parametrization setup should be reconsidered.

© 2011 OSA

## 1. Introduction

2. F. Mélin, “Global distribution of the random uncertainty associated with satellite-derived chla,” IEEE Geosci. Remote Sens. Lett. **7**, 220–224 (2010). [CrossRef]

*a*) concentrations derived from Sea viewing Wide Field-of-view Sensor (SeaWiFS) and Moderate Resolution Imaging Spectroradiometer (MODIS) measurements. Moore

*et al.*[3

3. T. S. Moore, J. W. Campbell, and M. D. Dowell, “A class-based approach to characterizing and mapping the uncertainty of the MODIS ocean chlorophyll product,” Remote Sens. Environ. **113**, 2424–2430 (2009). [CrossRef]

*a*concentrations derived from MODIS based on a fuzzy-logic approach to define memberships to specific optical water types. Other studies [4

4. 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]

7. Z. Lee, R. Arnone, C. Hu, J. Werdell, and B. Lubac, “Uncertainties of optical parameters and their propagations in an analytical ocean color inversion algorithm,” Appl. Opt. **49**, 369–381 (2010). [CrossRef] [PubMed]

8. M. S. Salama and A. Stein, “Error decomposition and estimation of inherent optical properties,” Appl. Opt. **48**, 4926–4962 (2009). [CrossRef]

*in-situ*measured data of water-leaving radiance and IOPs obtained from the NASA bio-Optical Marine Algorithm Data set (NOMAD), Version 2.a [10, NOMAD data set]. The third consists of concurrent SeaWiFS observations and NOMAD measured inherent and apparent optical properties, Version 1.3 [10, SeaWiFS matchup data set]. Finally, the operational application of the method is demonstrated using time series of IOPs derived from SeaWiFS monthly acquisitions from 1997 to 2007 [11

11. J. G. Acker and G. Leptoukh, “Online analysis enhances use of NASA earth science data,” Eos, Trans. AGU **88**, 14–17 (2005). [CrossRef]

## 2. Method

### 2.1. Ensemble Uncertainty of IOPs

*Rs*

_{w}(

*λ*), to the IOPs (generally absorption and backscattering coefficients); (ii) parameterizations of the IOPs as functions of their values at a reference wavelength

*λ*

_{0}. So we have:

*Rs*

_{w}(

*λ*) =

*f*(

**iop**), with

*λ*being the wavelength and

**iop**being the set of derived IOPs at the reference wavelength

*λ*

_{0}:

**iop**= [

*iop*

_{i}_{=1},...,

*iop*

_{i}_{=}

*]. As such, the radiometric uncertainty is propagated towards the derived IOPs as follows, where Δ*

_{n}*Rs*

_{w}(

*λ*) and Δ

*iop*(

_{i}*λ*

_{0}) represent the infinitesimal-change in

*Rs*

_{w}(

*λ*) and i

*th*IOP at the reference wavelength

*λ*

_{0},

*iop*(

_{i}*λ*

_{0}), respectively;

*w*is the partial derivative of remote sensing reflectance with respect to i

_{i}*th*IOP; i.e.

*w*=

_{i}*∂Rs*

_{w}(

*λ*)/

*∂iop*(

_{i}*λ*

_{0}). The term

*δ*(

*λ*) is an error component that represents the accuracy of the used forward ocean color model in describing the relationship between apparent and inherent optical properties. To simplify the mathematical derivations, the term

*δ*(

*λ*) will be imbedded in Δ

*Rs*

_{w}(

*λ*) and its spectral dependence will be dropped.

*Rs*

_{w}(

*λ*) is provided for at least

*n*wavelengths, with

*n*being the number of derived IOPs which requires prior knowledge on the magnitude of Δ

*Rs*

_{w}(

*λ*). It is, therefore, more convenient to evaluate the uncertainty of IOPs with respect to Δ

*Rs*

_{w}(

*λ*). We divide both sides of Eq. (1) by Δ

*Rs*

_{w}(

*λ*), and denote the ratio Δ

*iop*(

_{i}*λ*

_{0})/Δ

*Rs*

_{w}(

*λ*) as

*ϕ*(

_{i}*λ*), we have: Σ

*w*(

_{i}*λ*)

*ϕ*(

_{i}*λ*) = 1. Dividing both sides by

*λ*) in Eq. (2) is referred to as the ensemble uncertainty of IOPs and can be expressed as, where 〈Δ

**iop**(

*λ*)〉 is the weighted sum of IOPs errors, with the i

*th*weight being the ratio

*w*(

_{i}*λ*)/Σ

*w*(

_{i}*λ*).

*th*IOP at the reference wavelength

*λ*

_{0};

*δ*

^{2}is an error component analogous to

*δ*in Eq. (1);

*ℓ*represents the covariance terms in the Taylor series expansion. Assuming that the water observed-radiance is governed by independently varying IOPs, gives

*ℓ*≈ 0. For now, the term

*δ*

^{2}is embedded in

*λ*), is derived from Eq. (5) by normalizing both sides by the squared sum of partial derivatives and taking its square-root: Both, Φ(

*λ*) and Ψ(

*λ*) represent the ensemble uncertainty of IOPs per unit error of remote sensing reflectance and have the unit of sr m

^{−1}. Since underestimation of the absorption coefficient is generally associated with overestimation of the backscattering and vice versa, Φ(

*λ*) is expected to be smaller than the individual uncertainties of absorption and backscattering coefficients. In other words, the under/overestimations cancel each other out. Conversely, Ψ(

*λ*) is additive; errors always add up. From hereon, Ψ(

*λ*) will be used as the measure of uncertainty instead of Φ(

*λ*).

### 2.2. Relative Measure of Uncertainty

*λ*), Eq. (6) can be rewritten as, where 〈

*σ*

_{iop}(

*λ*

_{0})〉 is the sum of weighted IOPs uncertainties: Dividing both sides of Eq. (7) by the sum of derived IOPs,

*cb*

_{d}(

*λ*) = Σ

*iop*(

_{i}*λ*), we have where the parameter CV(

*λ*) is the ratio, CV(

*λ*) = 〈

*σ*

_{iop}(

*λ*)〉/

*cb*

_{d}(

*λ*). The ratio, Ψ

^{N}(

*λ*), in Eq. (9) is a measure of the relative ensemble uncertainty per radiometric error and has units of sr. The reciprocal of Eq. (9) is a measure of the radiometric uncertainty with respect to CV(

*λ*),

^{−1}unit.

## 3. Used Data Sets and Ocean Color Model

*in-situ*measured and ocean color matchup data. Simulated data are radiative transfer simulations with the synthetic IOPs [9, IOCCG data set] as input, performed for a 30° sun zenith over the 400 nm to 720 nm spectral range with 10 nm interval. Inelastic scattering, such as Raman scattering, chlorophyll fluorescence etc, were excluded from the simulations.

*In-situ*measured data of water-leaving radiance and IOPs are taken from the NOMAD data set, Version 2.a [10, NOMAD data set]. Ocean color matchup data are concurrent SeaWiFS observations and NOMAD measured inherent and apparent optical properties, Version 1.3 [10, SeaWiFS matchup data set]. Information on the different versions of NOMAD data sets can be found on SeaWiFS Bio-optical Archive and Storage System (SeaBASS): http://seabass.gsfc.nasa.gov/seabasscgi/nomad.cgi. Global ocean color products of monthly IOPs data are downloaded from the Goddard Earth Sciences Data and Information Services Center, Interactive Online Visualization and Analysis Infrastructure (Giovanni) [11

11. J. G. Acker and G. Leptoukh, “Online analysis enhances use of NASA earth science data,” Eos, Trans. AGU **88**, 14–17 (2005). [CrossRef]

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

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

13. 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]

*t*is the transmission function from water to air and taken equal to

*t*= 0.95 for the nadir viewing angle;

*n*is the water index of refraction and is taken equal to 1.334;

*g*are model expansion parameters, for which

_{i}*g*

_{1}= 0.0949 and

*g*

_{2}= 0.0794 are adopted [12

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

13. 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]

*u*(

*λ*) is the ratio

*b*(

_{b}*λ*)/(

*a*(

*λ*) +

*b*(

_{b}*λ*)), with

*a*(

*λ*) and

*b*(

_{b}*λ*) as the bulk absorption and the backscattering coefficients of the water upper layer, respectively. Three IOPs will be considered at the reference wavelength

*λ*

_{0}= 440 nm: the absorption of chlorophyll-a (Chl

*a*),

*a*

_{chl}

*(440), the lumped absorption effect of detritus and gelbstoff,*

_{a}*a*

_{dg}(440) and the backscattering of suspended particulate matter (SPM),

*b*

_{b}_{p}(440). The parameterizations of Salama

*et al.*[6

6. 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]

14. M. S. Salama and F. Shen, “Stochastic inversion of ocean color data using the cross-entropy method,” Opt. Express **18**, 479–499 (2010). [CrossRef] [PubMed]

*in-situ*measured and SeaWiFS matchup data. For global ocean color products of IOPs, however, the original parameterizations of the GSM model [12

**41**, 2705–2714 (2002). [CrossRef] [PubMed]

*a*

_{chl}

*(*

_{a}*λ*) at the reference wavelength

*λ*

_{0}.

*λ*

_{0}= 440 nm are as follows: the coefficient

*a*

_{chl}

*(*

_{a}*λ*) is obtained from [15

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

*a*

_{ph}(

*λ*) =

*a*

_{ph}(440)[

*a*

_{0}(

*λ*) +

*a*

_{1}(

*λ*) log

*a*

_{ph}(440)] with

*a*

_{0}and

*a*

_{1}tabulated; the coefficient

*a*

_{dg}(

*λ*) is defined as in [16

16. 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]

*a*

_{dg}(

*λ*) =

*a*

_{dg}(440)

*ζ*

_{dg}, where

*ζ*

_{dg}= exp [−

*s*(

*λ*− 440)] describes the spectral shape via

*s*= 0.021 nm

^{−1}; the SPM backscattering coefficient is parameterized as in [17]:

*b*

_{b}_{spm}(

*λ*) =

*b*

_{b}_{spm}(440)

*ζ*

_{spm}, in which

*ζ*

_{spm}= (440 ·

*λ*

^{−1})

*describes the spectral dependency with*

^{y}*y*= 1.1.

## 4. Validation

14. M. S. Salama and F. Shen, “Stochastic inversion of ocean color data using the cross-entropy method,” Opt. Express **18**, 479–499 (2010). [CrossRef] [PubMed]

*w*, of Eq. (11) are computed with respect to the derived IOPs. The ensemble uncertainty is derived from Eq. (6) as derived error = Ψ(

_{i}*λ*)

*σ*

_{r}(

*λ*). The radiometric uncertainty,

*x*

_{d}–

*x*

_{k})

^{2}with

*x*being the reflectance. The quantity,

*x*

_{d}, is the best-fit spectrum derived from inverting Eq. (11), whereas

*x*

_{k}refers to the known reflectance, in this case from the IOCCG-simulated or NOMAD-measured spectra. For the SeaWiFS-matchup set, these two quantities (

*x*

_{d}and

*x*

_{k}) are replaced by

*in-situ*measured and satellite observed spectra, respectively. The known values of ensemble-uncertainties are estimated as, where Δ

*iop*is the difference between the derived and measured i

_{i}*th*IOPs, and

*th*partial derivatives computed using the measured values of IOPs. The error parameter,

*δ*

^{2}, is estimated as

*δ*

^{2}= (

*x*

_{m}–

*x*

_{k})

^{2}, in which

*x*

_{m}is the output of Eq. (11) using the measured IOPs and

*x*

_{k}is the observed (or known) spectra (e.g. IOCCG-simulated, NOMAD-measured or SeaWiFS-observed).

*a*

_{chla}(440),

*a*

_{dg}(440) and

*b*

_{bp}(440). On the other hand, the SeaWiFS-matchup subset is composed such that each site has radiometric observations and at least two measured absorption coefficients,

*a*

_{chla}(440) and

*a*

_{dg}(440). Missing measurements of

*b*

_{bp}(440) in the SeaWiFS-matchup subset are substituted by their estimates as derived from the measured spectra. This is justified by studies showing that the uncertainties associated with the derivation of the backscattering coefficient,

*b*

_{bp}(440), are much lower than those found for absorption [9, 18

18. F. Mélin, G. Zibordi, and JF. Berthon, “Assessment of satellite ocean color products at a coastal site,” Remote Sens. Environ. **110**, 192–215 (2007). [CrossRef]

^{2}, coefficient of determination;

^{2}decrease from about 0.96 for IOCCG to 0.50 for NOMAD and reaches 0.45 for SeaWiFS-matchup. Table 1 and Fig. 1 confirm that the proposed method produces acceptable estimates of uncertainty for the three data sets. Overall, the simulated, measured and ocean color matchup yield a

*R*

^{2}∼ 0.61, MADi < 0.7 and RMSE < 1.2.

## 5. Discussions

### 5.1. Formulation

*δ*

^{2}, was included in Eq. (4) to account for the uncertainty of the forward model. Figure 2 shows the comparison of

*x*

_{d}–

*x*

_{k})

^{2}, against Eq. (4) on the Y-axis, with

*δ*included (red dots) and without

*δ*(grey circles). This figure is produced using the IOCCG data set and serves as a reference.

^{2}, MADi and RMSE, between known and derived radiometric uncertainties and the effect of including

*δ*are shown in Table 2. It is obvious from Fig. 2 and Table 2 that adding

*δ*improves the results for all wavelengths, with 13–35 % increase in R

^{2}and 13–44% decrease in MADi. The same improvement in RMSE is, however, more difficult to note. This can be attributed to the nature of RMSE which depends, apart from accuracy, also on the distribution of errors [20

20. C. J. Willmott and K. Matsuura, “Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance,” Climate Res. **30**, 79–82 (2005). [CrossRef]

20. C. J. Willmott and K. Matsuura, “Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance,” Climate Res. **30**, 79–82 (2005). [CrossRef]

*δ*

^{2}= (

*x*

_{m}–

*x*

_{k})

^{2}and

*δ*in Eq. (4) gives

*x*

_{k}⩽ 0.5 (

*x*

_{d}+

*x*

_{m}). So assuming

*δ*

^{2}= (

*x*

_{m}–

*x*

_{k})

^{2}and

*δ*= |

*x*

_{m}–

*x*

_{k}| and

*σ*

_{r}= |

*x*

_{d}–

*x*

_{k}|, which prohibits expanding

*δ*

^{2}and

### 5.2. Validation Results

*w*, which are obtained from the partial derivatives of

_{i}*Rs*

_{w}(

*λ*) =

*f*(

**iop**) with respect to IOPs. The weight

*w*is only a function of IOPs and Ψ (uncertainty per radiometric error) is nearly independent of the error on

_{i}*Rs*

_{w}. As such, Ψ computed for the three data sets should be comparable for similar IOPs, which is confirmed in Fig. 3. For validation, we express the uncertainty in terms of IOPs, which is done by multiplying Ψ by

*σ*

_{r}for the derived error, and by expressing the known error as a sum of weighted IOPs’s errors, Eq. (12). The reason behind this weighting is that the ensemble uncertainty of IOPs is also a weighted sum.

*Rs*

_{w}(

*λ*) (for NOMAD and SeaWiFS-matchup), so that the error will increase as the input spectra might have their spectral shapes affected by errors from various sources (sensor noise, atmospheric correction and spatial scale differences). For the SeaWiFS sensor, and the same model-parametrization setup as used in this paper, residuals from atmospheric correction and sensor noise are on average 50% of the total error [8, their Table 2]. The discrepancy in the spatial scale accounts for about 20% of the total error [21

21. M. Salama and Z. Su, “Bayesian model for matching the radiometric measurements of aerospace and field ocean color sensors,” Sensors **10**, 7561–7575 (2010). [CrossRef] [PubMed]

22. M. S. Salama and Z. Su, “Resolving the subscale spatial variability of apparent and inherent optical properties in ocean color matchup sites,” IEEE Trans. Geosci. Remote Sens. **49**, 2612–2622 (2011). [CrossRef]

25. E. Aas, “Estimates of radiance reflected towards the zenith at the surface of the sea,” Ocean Sci. **6**, 861–876, (2010) [CrossRef]

*ℓ*in Eq. (4). These reasons could explain the large discrepancy shown in Fig. 1 between known and derived values for the NOMAD and SeaWiFS-matchup data sets.

### 5.3. Optimum Accuracy of Model-Parametrization Setup

6. 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]

8. M. S. Salama and A. Stein, “Error decomposition and estimation of inherent optical properties,” Appl. Opt. **48**, 4926–4962 (2009). [CrossRef]

^{N}(

*λ*). Doing so will result in a measure of relative ensemble uncertainty per unit error of radiance. However, for most of the ocean waters, where

*a*≫

*b*, the sum of IOPs (

_{b}*cb*) is equivalent to the total absorption coefficient, which makes

*cb*hard to interpret statistically. The reciprocal of Ψ

^{N}(

*λ*) is the radiometric uncertainty normalized to the relative ensemble error,

*cb*

_{k}.

*cb*

_{k}(440) ≥ 0.4 m

^{−1}, with

^{−1}(−2.547 on the log scale). This value, 0.0783, is very high and close to the saturation-of-reflectance (SoR) of the employed ocean color model, i.e. Eq. (11), SoR is defined here as being the highest radiometric value that can be produced by the ocean color model. This situation occurs in waters loaded with non-absorbing particles such that the fraction

*u*in Eq. (11) approaches unity (i.e. absorption by constituents is small in comparison to backscattering

*b*≫

_{b}*a*). In this case, SoR = lim

_{u→1}Rs

_{w}= 0.0922 sr

^{−1}. Conversely, the reciprocal of

*cb*

_{k}(440) ≈ 0.4 m

^{−1}as: 0.4/0.0783 = 5.11 sr m

^{−1}. The minimum value of normalized radiometric uncertainty (also at 440 nm, see Fig. 3) is 0.0369 sr which corresponds to

*cb*

_{k}(440) ≈ 0.0123 m

^{−1}. In the same way we compute the minimum value of ensemble uncertainty as: 0.0123/0.0369 = 0.33 sr m

^{−1}. The lower limit of the ensemble uncertainty shows that the ocean color model and IOPs parameterizations have an inherent error at 440 nm of at least 0.333 sr m

^{−1}. Therefore, each error of 1 sr

^{−1}results in 0.33 m

^{−1}collective error of IOPs. This lower limit of error represents the optimum (maximum) accuracy that can be achieved by a model-parametrization setup.

^{−1}(for IOCCG) and 0.0050 sr

^{−1}(for NOMAD and SeaWiFS-matchup). These values are derived as the difference between model best-fit and observed remote reflectance, and averaged for each data set. In consequence, the highest (optimum) accuracy of the used model-parametrization setup is Δ

*cb*(440) = 0.0024 × 0.33 = 0.0008 m

^{−1}for IOCCG and Δ

*cb*(440) = 0.005 × 0.33 = 0.0017 m

^{−1}for NOMAD and SeaWiFS-matchup. These values increase to their maxima for SoR (using the value 5.11 sr m

^{−1}) of 0.0123 m

^{−1}and 0.0256 m

^{−1}, respectively.

### 5.4. Global product of the ensemble uncertainty

^{−1}decade

^{−1}. Using an average radiometric error of 0.005 sr

^{−1}(obtained from the SeaWiFS-matchup data), this value is equivalent to a Δ

*cb*(440) of 0.04 m

^{−1}decade

^{−1}. Via this application of our method, it is shown that the GSM model-parametrization setup produces IOPs subject to an increasing level of uncertainty in specific regions. As such, the method has the potential of detecting areas for which the model-parametrization setup should be considered. Moreover it provides also a means to identify changes in the biophysical characteristics of waters associated to, for example, changes in climate or anthropogenic influences.

^{N}(440) annual values (not shown here) show persistent patterns of high values of Ψ

^{N}(440) throughout the last decade in the subtropical gyres, whereas lower values are observed in most coastal areas. Moreover, the spatial distribution of the relative ensemble uncertainty largely resembles the observed values of remote sensing reflectance at 443 nm. These results can also be deduced from Fig. 3 and are in accordance with the global uncertainty maps for Chlorophyll-a by Mélin [2

2. F. Mélin, “Global distribution of the random uncertainty associated with satellite-derived chla,” IEEE Geosci. Remote Sens. Lett. **7**, 220–224 (2010). [CrossRef]

*et al.*[3

3. T. S. Moore, J. W. Campbell, and M. D. Dowell, “A class-based approach to characterizing and mapping the uncertainty of the MODIS ocean chlorophyll product,” Remote Sens. Environ. **113**, 2424–2430 (2009). [CrossRef]

## 6. Conclusions

*in-situ*measured and ocean color matchup data. For the synthetic data a very good fit is obtained between the derived and known values (R

^{2}> 0.95 and mean absolute difference (MADi) < 0.25 m

^{−1}). A reduced performance (0.45 < R

^{2}< 0.5 and 0.65 < MADi < 0.9) is, however, found for the

*in-situ*and ocean color matchup data, which is attributed to additional error sources such as sensor noise, atmospheric correction and spatial scale differences. Further, we employ the method also for estimating optimum accuracy that could be achieved with the three data sets for a specific model-parametrization setup, which could be seen as the target accuracy in retrieving IOPs.

## Acknowledgments

## References and links

1. | Z. Su, R. A. Roebeling, J. Schulz, I. Holleman, V. Levizzani, W. J. Timmermans, H. Rott, N. Mognard-Campbell, R. de Jeu, W. Wagner, M. Rodell, M. S. Salama, G. Parodi, and L. Wang, “Observation of Hydrological Processes Using Remote Sensing,” in |

2. | F. Mélin, “Global distribution of the random uncertainty associated with satellite-derived chla,” IEEE Geosci. Remote Sens. Lett. |

3. | T. S. Moore, J. W. Campbell, and M. D. Dowell, “A class-based approach to characterizing and mapping the uncertainty of the MODIS ocean chlorophyll product,” Remote Sens. Environ. |

4. | P. Wang, E. Boss, and C. Roesler, “Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color,” Appl. Opt. |

5. | S. Maritorena and D. Siegel, “Consistent merging of satellite ocean color data sets using a bio-optical model,” Remote Sens. Environ. |

6. | 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. |

7. | Z. Lee, R. Arnone, C. Hu, J. Werdell, and B. Lubac, “Uncertainties of optical parameters and their propagations in an analytical ocean color inversion algorithm,” Appl. Opt. |

8. | M. S. Salama and A. Stein, “Error decomposition and estimation of inherent optical properties,” Appl. Opt. |

9. | Z. Lee, “Remote sensing of inherent optical properties: Fundamentals, tests of algorithms, and applications,” Tech. Rep. 5, International Ocean-Colour Coordinating Group (2006). |

10. | 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. |

11. | J. G. Acker and G. Leptoukh, “Online analysis enhances use of NASA earth science data,” Eos, Trans. AGU |

12. | S. Maritorena, D. Siegel, and A. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. |

13. | 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. |

14. | M. S. Salama and F. Shen, “Stochastic inversion of ocean color data using the cross-entropy method,” Opt. Express |

15. | Z. Lee, K. Carder, C. Mobley, R. Steward, and J. Patch, “Hyperspectral remote sensing for shallow waters: 2. deriving bottom depths and water properties by optimization,” Appl. Opt. |

16. | 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. |

17. | O. Kopelevich, “Small-parameter model of optical properties of sea waters,” in “ |

18. | F. Mélin, G. Zibordi, and JF. Berthon, “Assessment of satellite ocean color products at a coastal site,” Remote Sens. Environ. |

19. | E. Laws, |

20. | C. J. Willmott and K. Matsuura, “Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance,” Climate Res. |

21. | M. Salama and Z. Su, “Bayesian model for matching the radiometric measurements of aerospace and field ocean color sensors,” Sensors |

22. | M. S. Salama and Z. Su, “Resolving the subscale spatial variability of apparent and inherent optical properties in ocean color matchup sites,” IEEE Trans. Geosci. Remote Sens. |

23. | M. S. Salama, J. Monbaliu, and P. Coppin, “Atmospheric correction of advanced very high resolution radiometer imagery,” Int. J. Remote Sens. |

24. | M. S. Salama and F. Shen, “Simultaneous atmospheric correction and quantification of suspended particulate matters from orbital and geostationary earth observation sensors,” Estuarine Coastal Shelf Sci. |

25. | E. Aas, “Estimates of radiance reflected towards the zenith at the surface of the sea,” Ocean Sci. |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(010.7340) Atmospheric and oceanic optics : Water

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: April 25, 2011

Revised Manuscript: July 4, 2011

Manuscript Accepted: August 2, 2011

Published: August 15, 2011

**Virtual Issues**

Vol. 6, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Mhd. Suhyb Salama, Frederic Mélin, and Rogier Van der Velde, "Ensemble uncertainty of inherent optical properties," Opt. Express **19**, 16772-16783 (2011)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-18-16772

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### References

- Z. Su, R. A. Roebeling, J. Schulz, I. Holleman, V. Levizzani, W. J. Timmermans, H. Rott, N. Mognard-Campbell, R. de Jeu, W. Wagner, M. Rodell, M. S. Salama, G. Parodi, and L. Wang, “Observation of Hydrological Processes Using Remote Sensing,” in Treatise on Water Science , P. Wilderer, ed. (Academic Press, 2011). [CrossRef]
- F. Mélin, “Global distribution of the random uncertainty associated with satellite-derived chla,” IEEE Geosci. Remote Sens. Lett. 7, 220–224 (2010). [CrossRef]
- T. S. Moore, J. W. Campbell, and M. D. Dowell, “A class-based approach to characterizing and mapping the uncertainty of the MODIS ocean chlorophyll product,” Remote Sens. Environ. 113, 2424–2430 (2009). [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]
- 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]
- 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]
- Z. Lee, R. Arnone, C. Hu, J. Werdell, and B. Lubac, “Uncertainties of optical parameters and their propagations in an analytical ocean color inversion algorithm,” Appl. Opt. 49, 369–381 (2010). [CrossRef] [PubMed]
- M. S. Salama and A. Stein, “Error decomposition and estimation of inherent optical properties,” Appl. Opt. 48, 4926–4962 (2009). [CrossRef]
- Z. Lee, “Remote sensing of inherent optical properties: Fundamentals, tests of algorithms, and applications,” Tech. Rep. 5, International Ocean-Colour Coordinating Group (2006).
- 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]
- J. G. Acker and G. Leptoukh, “Online analysis enhances use of NASA earth science data,” Eos, Trans. AGU 88, 14–17 (2005). [CrossRef]
- S. Maritorena, D. Siegel, and A. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt. 41, 2705–2714 (2002). [CrossRef] [PubMed]
- 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]
- M. S. Salama and F. Shen, “Stochastic inversion of ocean color data using the cross-entropy method,” Opt. Express 18, 479–499 (2010). [CrossRef] [PubMed]
- Z. Lee, K. Carder, C. Mobley, R. Steward, and J. Patch, “Hyperspectral remote sensing for shallow waters: 2. deriving bottom depths and water properties by optimization,” Appl. Opt. 38, 3831–3843 (1999). [CrossRef]
- 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]
- O. Kopelevich, “Small-parameter model of optical properties of sea waters,” in “Ocean Optics ,”, vol. 1 Physical Ocean Optics, A. Monin, ed. (Nauka, 1983), pp. 208–234.
- F. Mélin, G. Zibordi, and JF. Berthon, “Assessment of satellite ocean color products at a coastal site,” Remote Sens. Environ. 110, 192–215 (2007). [CrossRef]
- E. Laws, Mathematical Methods for Oceanographers: An Introduction (John Wiley and Sons, 1997).
- C. J. Willmott and K. Matsuura, “Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance,” Climate Res. 30, 79–82 (2005). [CrossRef]
- M. Salama and Z. Su, “Bayesian model for matching the radiometric measurements of aerospace and field ocean color sensors,” Sensors 10, 7561–7575 (2010). [CrossRef] [PubMed]
- M. S. Salama and Z. Su, “Resolving the subscale spatial variability of apparent and inherent optical properties in ocean color matchup sites,” IEEE Trans. Geosci. Remote Sens. 49, 2612–2622 (2011). [CrossRef]
- M. 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]
- M. S. Salama and F. Shen, “Simultaneous atmospheric correction and quantification of suspended particulate matters from orbital and geostationary earth observation sensors,” Estuarine Coastal Shelf Sci. 86, 499–511 (2010). [CrossRef]
- E. Aas, “Estimates of radiance reflected towards the zenith at the surface of the sea,” Ocean Sci. 6, 861–876, (2010) [CrossRef]

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