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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 9 — Oct. 3, 2011
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Ensemble uncertainty of inherent optical properties

Mhd. Suhyb Salama, Frederic Mélin, and Rogier Van der Velde  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 16772-16783 (2011)
http://dx.doi.org/10.1364/OE.19.016772


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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 R2 > 0.95 and mean absolute difference (MADi) <0.25 m−1. Due to the influence of observation errors, these values deteriorate to 0.45 < R2 < 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−1decade−1. This value is translated to an error of 0.04 m−1decade−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

Various methods have been developed to evaluate the uncertainty of derived IOPs. 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]

] estimated the uncertainty embedded within chlorophyll-a (chla) 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]

] quantified the uncertainty of Chla 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

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]

] have employed gradient-based methods to evaluate IOP uncertainties via application of a specific ocean color model, utilizing model derivatives and the difference between model best-fit and observed radiances. Alternatively, Salama and Stein [8

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

] developed a non-gradient based method to quantify and separate uncertainty sources within derived IOPs based on a prior estimate of the radiometric uncertainties and their propagation through the retrievals. However, they also showed that the gradient based method has two inherent limitations: it produces uncertainty values that are proportional to the magnitude of derived IOPs, and it cannot handle radiometric residuals resulting from sensor noise and atmospheric correction.

The uncertainties in derived IOPs are determined by radiometric uncertainties, imperfection in the forward ocean color model and the adopted parametrization. The effect of the used ocean color model and set of parameterizations, or model-parametrization setup on the accuracy of retrieved IOPs has, so far, not been quantified in a single measure. In this paper, we present a method that quantifies the uncertainty of IOPs that follows from a model-parametrization setup. As such, the method provides a single measure of ensemble (collective) uncertainty of IOPs and avoids computing the radiometric uncertainty.

Our method is validated using three data sets. The first is a set of radiative transfer simulations of synthetic IOPs obtained from the International Ocean Color Coordination Group (IOCCG), report-5 [9, IOCCG data set]. The second is 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]

]. From hereon, we assume the difference between known and derived IOPs per unit of radiometric error is equivalent to the uncertainty.

2. Method

2.1. Ensemble Uncertainty of IOPs

Equation (1) facilitates estimating the uncertainties of all IOPs if ΔRs 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 ΔRs w(λ). It is, therefore, more convenient to evaluate the uncertainty of IOPs with respect to ΔRs w(λ). We divide both sides of Eq. (1) by ΔRs w(λ), and denote the ratio Δiopi(λ 0)/ΔRs w(λ) as ϕi(λ), we have: Σ wi(λ)ϕi(λ) = 1. Dividing both sides by Σwi2, yields the collective uncertainty of derived IOPs per unit of radiometric error:
Φ(λ)=i=1i=nwi(λ)φi(λ)/i=1i=nwi(λ)=1/i=1i=nwi(λ).
(2)
The term Φ(λ) in Eq. (2) is referred to as the ensemble uncertainty of IOPs and can be expressed as,
Φ(λ)=Δiop(λ)ΔRsw(λ),
(3)
where 〈Δiop(λ)〉 is the weighted sum of IOPs errors, with the ith weight being the ratio wi(λ)/Σ wi(λ).

Assuming that the IOPs are mutually independent, we have from the Taylor series approximation of the second moment:
σr2(λ)=i=1i=nwi2(λ)σi2(λ0)+δ2+,
(4)
where σr2(λ) is the radiometric variance and σi2(λ0) is the variance of the ith IOP at the reference wavelength λ 0; δ 2 is an error component analogous to δ in Eq. (1); represents the covariance terms in the Taylor series expansion. Assuming that the water observed-radiance is governed by independently varying IOPs, gives ≈ 0. For now, the term δ 2 is embedded in σr2(λ) for the following steps, but is further elaborated on in the analysis and discussion. Dividing both sides of Eq. (4) by the radiometric variance term yields,
i=1i=nwi2(λ)ψi2(λ)=1,
(5)
with ψi2(λ)=σi2(λ0)/σr2(λ). The ensemble uncertainty of IOPs per radiometric error, Ψ(λ), is derived from Eq. (5) by normalizing both sides by the squared sum of partial derivatives and taking its square-root:
Ψ(λ)=(i=1i=nwi2(λ)ψi2(λ)/i=1i=nwi2(λ))0.5=(i=1i=nwi2(λ))0.5.
(6)
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

From the definition of Ψ(λ), Eq. (6) can be rewritten as,
Ψ(λ)=σiop(λ)/σr(λ),
(7)
where 〈σ iop(λ 0)〉 is the sum of weighted IOPs uncertainties:
σiop(λ)=(Σwi2(λ)σi2(λ0)Σwi2(λ))0.5=σr(λ)(i=1i=nwi2(λ))0,5.
(8)
Dividing both sides of Eq. (7) by the sum of derived IOPs, cb d(λ) = Σ iopi(λ), we have
ΨN(λ)=CV(λ)σr(λ)=Ψ(λ)cbd(λ),
(9)
where the parameter CV(λ) is the ratio, CV(λ) = 〈σ iop(λ)〉/cb d(λ). The ratio, ΨN(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),
σrN(λ)=σr(λ)CV(λ)=cbd(λ)Ψ(λ).
(10)
σrN(λ) in Eq. (10) is called the normalized radiometric uncertainty, and quantifies the radiometric uncertainty per unit of relative error of IOPs with the sr−1 unit.

3. Used Data Sets and Ocean Color Model

Four data sets are used in this study, three to validate the proposed method and one to demonstrate its potential for assessing global ocean color products. The validation data include simulated, 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]

] and covered the period between 1997 and 2007. This set includes SeaWiFS monthly products of IOPs derived from the Garver-Siegel-Maritorena (GSM) [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]

] model-parametrization setup.

The GSM model is used to relate the IOPs to the radiometric quantities of ocean color data as [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

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]

],
Rsw(λ)=t2/n2i=1i=2giui(λ),
(11)
where 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; gi are model expansion parameters, for which 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

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]

]; and u(λ) is the ratio bb(λ)/(a(λ) + bb(λ)), with a(λ) and bb(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 (Chla), a chl a(440), the lumped absorption effect of detritus and gelbstoff, a dg(440) and the backscattering of suspended particulate matter (SPM), bb 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

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]

] are used for simulated, in-situ measured and SeaWiFS matchup data. For global ocean color products of IOPs, however, the original parameterizations of the GSM model [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]

] are applied. The difference between these two parameterizations is only in the expression of a chl a(λ) at the reference wavelength λ 0.

We present an application of our method to the existing global products (SeaWiFS) to demonstrate its potential for evaluating the uncertainty of existing model-parametrization setups. In this context, slight differences in the model-parameterizations setup will not affect the conclusions of this application. Parameterizations of IOPs with respect to the reference wavelength 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) loga 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

17. 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.

]: bb spm(λ) = bb spm(440)ζ spm, in which ζ spm = (440 · λ −1)y describes the spectral dependency with y = 1.1.

4. Validation

The cross-entropy method of Salama [14

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]

] is used to derive the IOPs from radiometric measurements. The partial derivatives, wi, of Eq. (11) are computed with respect to the derived IOPs. The ensemble uncertainty is derived from Eq. (6) as derived error = Ψ(λ)σ r(λ). The radiometric uncertainty, σr2(λ), is computed as (x dx 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,
known_error2(λ)=(Σwi*2(λ)Δiopi2(λ0)+δ2)/Σwi*2(λ),
(12)
where Δiopi is the difference between the derived and measured ith IOPs, and wi* is the ith partial derivatives computed using the measured values of IOPs. The error parameter, δ 2, is estimated as δ 2 = (x mx 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).

The NOMAD data subset is selected such that each site has radiometric observations and three IOPs measurements, 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

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

, 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]

]. In total, there are 90 NOMAD and 123 SeaWiFS-matchup sites satisfying this criterion.

Derived and known values for the wavelength 440 nm are log-transformed and compared in Fig. 1 for the three data sets.

Fig. 1 Known versus estimated ensemble-uncertainties at 440 nm expressed in m−1, of derived IOPs from the three data sets of (a) : IOCCG, gray circles ; (b): NOMAD, red triangles and; (c) : SeaWiFS-matchup in blue squares. The values are log transformed.

The fits between derived and known uncertainties in Fig. 1 are assessed using three goodness-of-fit parameters: i - R2, coefficient of determination;

ii - MADi, mean absolute differences between derived values and the regression line;

iii - RMSE, root mean of squared error.

The regression line was computed using type-II model [19

19. E. Laws, Mathematical Methods for Oceanographers: An Introduction (John Wiley and Sons, 1997).

] and following the practice of the IOCCG report [9

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

]. The goodness-of-fit parameters are computed for each data set separately as well as for all data points and are given in Table 1.

Table 1. Goodness-of-Fit Parameters between Known and Derived Ensemble Uncertainties at 440 nm Using the Three Data Sets

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Figure 1 and Table 1 show that derived and known uncertainty values have a good linear relationship, with MADi and RMSE values ranging from minima for IOCCG to maxima for SeaWiFS-matchup. Correspondingly, the values of R2 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

The error term, δ 2, was included in Eq. (4) to account for the uncertainty of the forward model. Figure 2 shows the comparison of σr2 on the X-axis, computed as (x dx 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.

Fig. 2 Comparison between known and estimated radiometric uncertainty for different wavelengths: (a) 400 nm, (b) 440 nm, (c) 550 nm, and (d) 670 nm. The values are log transformed.

The best-fit parameters, R2, 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 R2 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]

]. Willmot and Matsuura [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]

] recommend using mean absolute error (MADi in this study) instead of RMSE.

Table 2. Goodness-of-Fit Parameters Between Known and Derived Radiometric Uncertainty as Retrieved from IOCCG Data Set

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Recall that the following approximations were used: δ 2 = (x mx k)2 and σr2=(xdxk)2, from which the following relation can be derived δ2=σr2+xm2xd2+2xk(xdxm). Substituting δ in Eq. (4) gives Σwi2σi2=xd2xm2+2xk(xmxd). For this to hold, i.e. Σwi2σi20, the following condition should be satisfied x k ⩽ 0.5 (x d + x m). So assuming δ 2 = (x mx k)2 and σr2=(xdxk)2 implies that the forward and inversion models should on average produce radiance that is equal to or larger than the actual one. This implicit assumption does not reflect the practice of model inversion. Equation (11) may under- or overestimate the observed radiance based on noise level and water contents of dissolved and suspended matters. We avoid, however, this implication by using absolute values, such as δ = |x mx k| and σ r = |x dx k|, which prohibits expanding δ 2 and σr2, and yet guarantees positive value.

5.2. Validation Results

The ensemble uncertainty, Ψ in Eq. (6), is a function of the weights, wi, which are obtained from the partial derivatives of Rs w(λ) = f(iop) with respect to IOPs. The weight wi is only a function of IOPs and Ψ (uncertainty per radiometric error) is nearly independent of the error on 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.

Fig. 3 The relationship between the normalized radiometric uncertainty, σrN, and the sum of known IOPs values, cb k. The values are log transformed.

The term Δiopi2 in Eq. (12) includes the uncertainties associated with 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

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]

]. Thus, the effect of the model-parametrization setup contributes for 30% of the total uncertainty in the derived IOPs. On the other hand, the sensor noise signal in the NOMAD data set contributes by an average of 20% to the total error on derived IOP [8, their Table 4], Moreover, the NOMAD data are affected by surface reflectance, which can be in the same order of magnitude of water signal in case-1 waters [25

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

]. This leaves about 40% of the total error to model-parametrization setup. Two additional error components will be added due to approximating σi2 by Δiopi2, and truncating the covariance terms in the Taylor expansion, 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

Application of a gradient-based method for estimating the uncertainties in derived IOPs produces values proportional to the magnitude of IOPs and radiometric uncertainty [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]

,8

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

]. This proportionality is compensated for by normalizing Ψ to the sum of derived IOPs, ΨN(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 abb, the sum of IOPs (cb) is equivalent to the total absorption coefficient, which makes cb hard to interpret statistically. The reciprocal of ΨN(λ) is the radiometric uncertainty normalized to the relative ensemble error, σrN(λ). The normalized radiometric uncertainty, σrN, is expected to increase with water load of dissolved and suspended matters (turbidity), and approaches a constant value for waters with a high turbidity. Figure 3 shows σrN, computed from derived values, versus the sum of known IOPs, cb k.

For the three data sets, the normalized radiometric uncertainty approaches a constant value for cb k(440) ≥ 0.4 m−1, with σrN of about 0.0783 sr−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 bba). In this case, SoR = limu→1 Rsw = 0.0922 sr−1. Conversely, the reciprocal of σrN(λ), ΨλN, decreases with increasing water turbidity reaching a constant value of 1/0.0922 = 10.85 sr. It is clear from the above discussion that higher radiometric uncertainty ( σrN(λ)) is expected for turbid waters, and the relative ensemble uncertainty is expected to be higher for clear water. In addition, Fig. 3 allows us also to provide an estimate of the optimum accuracy that is achievable with a model-parametrization setup. The maximum value of normalized radiometric uncertainty (at 440 nm) is computed as 0.0783 sr. This value can be converted to ensemble uncertainty using 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.

Nominal values for radiometric error at 440 nm are in the range of 0.0024 sr−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 Δcb(440) = 0.0024 × 0.33 = 0.0008 m−1 for IOCCG and Δcb(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

Fig. 4 Trends of ensemble uncertainty at 440 nm between the period 1997–2007.

Analyzing the spatial distribution of ΨN(440) annual values (not shown here) show persistent patterns of high values of ΨN(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]

] for the subtropical gyres, whereas, the global distribution of Ψ(440), see Eq.(9), corresponds to the results of 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]

] (see their Fig. 5).

6. Conclusions

A method is presented for the quantification of the combined accuracy of ocean color models and the parameterizations of inherent optical proprieties (IOPs), or model-parametrization setup. Its application produces a single (or ensemble) uncertainty measure for the collective errors in the derived IOPs relative to the radiometric uncertainty without the need for model inversion or prior information on the radiometric errors. As such, the method is directly applicable to existing satellite based IOP products. It can be used to compare different satellite data products and in a context of data merging where the respective uncertainties associated with the input data streams are required.

A thorough validation of the method is presented using simulated, in-situ measured and ocean color matchup data. For the synthetic data a very good fit is obtained between the derived and known values (R2 > 0.95 and mean absolute difference (MADi) < 0.25 m−1). A reduced performance (0.45 < R2 < 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.

Quantifying the ensemble uncertainty of IOPs provides the optimum accuracy which can be achieved by a model-parametrization setup. Moreover, it has the potential of detecting areas for which the performance of a given model-parametrization setup is suboptimal or deteriorating. This provides the information needed for updating the global model-parametrization setup, which contributes to the overall uncertainty reduction within IOP products.

Acknowledgments

The authors would like to thank NASA Ocean Biology Processing Group and individual data contributors for maintaining and updating the SeaBASS database and the NASA Goddard Earth Sciences Data and Information Services Center for providing Giovanni online data system. SeaWiFS and MODIS missions scientists and associated NASA personnel are acknowledged for the production of the data used in this paper. Three anonymous referees are acknowledged for reviewing the manuscript.

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 Treatise on Water Science, P. Wilderer, ed. (Academic Press, 2011). [CrossRef]

2.

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

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]

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]

5.

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]

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]

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]

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. 98, 122–140 (2005). [CrossRef]

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]

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]

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]

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]

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]

17.

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.

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]

19.

E. Laws, Mathematical Methods for Oceanographers: An Introduction (John Wiley and Sons, 1997).

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]

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]

23.

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]

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. 86, 499–511 (2010). [CrossRef]

25.

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

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

  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 Treatise on Water Science , P. Wilderer, ed. (Academic Press, 2011). [CrossRef]
  2. F. Mélin, “Global distribution of the random uncertainty associated with satellite-derived chla,” IEEE Geosci. Remote Sens. Lett. 7, 220–224 (2010). [CrossRef]
  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]
  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]
  5. 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]
  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]
  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]
  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. 98, 122–140 (2005). [CrossRef]
  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]
  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]
  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]
  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]
  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]
  17. 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.
  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]
  19. E. Laws, Mathematical Methods for Oceanographers: An Introduction (John Wiley and Sons, 1997).
  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]
  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]
  23. 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]
  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. 86, 499–511 (2010). [CrossRef]
  25. 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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