## 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]. 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/m^{2}/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]. 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, 4], artificial neural networks (e.g. Doerffer & Schiller [5]), algebraic solutions [6, 7], and spectral optimization algorithms (SOA) [8–15].

The pioneering SOA was proposed in the 1990s [8]. Since then, various SOA schemes have been developed with different complexity in modeling the bio-optical properties that make up a reflectance spectrum [9–14]. For example, Devred et al. [11] incorporated more complicated components to model the spectral inherent optical properties (IOPs), while the Garver-Siegel-Maritorena (GSM) [13] and the Hyperspectral Optimization Process Exemplar (HOPE) schemes [12,15] 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) [16,17]. 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]. 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], Siegel et al. [19]), 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 (*R _{rs}*, sr

^{−1}), which is defined as the ratio of water-leaving radiance to downwelling irradiance just above the surface [20], can be modeled as a function of the total absorption (

*a*(

*λ*), m

^{−1}) and total backscattering coefficients (

*b*(

_{b}*λ*), m

^{−1}) of the water column, i.e.

*r*(sr

_{rs}^{−1}) refers to the below-surface remote sensing reflectance [7,21].

For an SOA, an error function is defined and minimized to derive the optimal model variables (*a*(*λ*) and *b _{b}*(

*λ*)), and it generally takes a form as

*λ*

_{1}and

*λ*

_{2}.

*R*(

_{rs}*λ*) is the spectrum from measurements, and ${\tilde{R}}_{rs}(\lambda )$is the spectrum from modeling.

*δ*thus provides a relative (percentage) measure of the closure between the measured and modeled

_{Rrs}*R*(

_{rs}*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

*b*(

_{b}*λ*) when these models are used in an inversion process. Discussions of such effects can be found in Brando et al. [22].

The bio-optical models to calculate *a*(*λ*) and *b _{b}*(

*λ*) used in the GSM are detailed in Maritorena et al. [13]. For easier understanding of the results in this study, they are briefly described as below,

*a*(

_{w}*λ*) and

*b*(

_{bw}*λ*) are values of water molecules. Although they may change slightly with temperature and salinity [23], they are considered universal constants here. Thus there are three unknown spectra (

*a*(

_{ph}*λ*),

*a*(

_{dg}*λ*) and

*b*(

_{bp}*λ*)), and they are further modeled as

*X*

_{1-3}are three scalar variables to be derived from a measured

*R*spectrum via SOA, and they represent

_{rs}*C*,

*a*(

_{dg}*λ*

_{0}) and

*b*(

_{bp}*λ*

_{0}).

*C*is chlorophyll-a concentration (mg/m

^{3}).

*λ*

_{0}is a reference wavelength and is generally set as 440 nm.

_{${\tilde{a}}_{ph}(\lambda )$}, ${\tilde{a}}_{dg}(\lambda )$ and ${\tilde{b}}_{bp}(\lambda )$ are spectral shapes (eigen vectors or eigen functions) determined based on either field measurements [10, 14, 24, 25] or “optimized” from a database [13]. ${\tilde{a}}_{dg}(\lambda )$and ${\tilde{b}}_{bp}(\lambda )$ are generally described as [26, 27]

*S*) of ${\tilde{a}}_{dg}(\lambda )$ and the power coefficient (

*Y*) are generally not constants for global waters [28, 29]. 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 ${\tilde{a}}_{ph}(\lambda )$ precisely (e.g. Devred et al. [11], Bricaud et al. [24], Ciotti et al. [30]), although better fits could be achieved by increasing the number of free variables. In GSM, ${\tilde{a}}_{ph}(\lambda )$ is also optimized as a fixed spectrum for global waters, i.e.

which is the chlorophyll-specific absorption coefficient (m^{2}/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], here and as in Maritorena et al. [13], 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. *X*_{1} = 0.002, *X*_{2} = 0.01, and *X*_{3} = 0.0029.

To produce physically meaningful properties, boundaries of the three variables have to be defined. Retrievals from an *R _{rs}*(

*λ*) spectrum via GSM are considered valid if all three variables satisfy:

0.01 ≤ *X*_{1} ≤ 64 mg m^{−3}; 0.0001 ≤ *X*_{2} ≤ 2 m^{−1}; 0.0001 ≤ *X*_{3} ≤ 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

There are various computational methods to reach, numerically, an optimization (minimum) status of the error function. Widely used methods include the Levenberg-Marquardt (LM), Generalized Reduced Gradient (GRG), Downhill Simplex Method (Amoeba), and Simulated Annealing-Downhill Simplex (SAA). The LM employs the partial derivative of the objective function to search for the set of variables where the objective function reaches a minimum. This is based on the theorem that the partial derivative of a function is zero when this function reaches an extreme (either maximum or minimum) [32,33]. The GRG is similar as LM in searching for minimization, but uses difference to calculate the gradient. What differs GRG from LM the most is the handling of boundary constraints. While LM assumes open boundary (or no boundary) in the solution phase, GRG uses both the gradient and the boundary conditions to determine the direction of the search for minimization and thus constrains the search within pre-determined boundaries [34]. The Amoeba, also known as Nelder-Mead algorithm, uses the downhill simplex concept to search for optimization, and thus evaluates the error function only, not its derivatives [35]. It operates by constructing a simplex between searching points and manipulating this simplex [35]. The SAA, on the other hand, is a combination of the Amoeba and simulated annealing approaches for optimization [36]. In particular, it uses stochastic transition to avoid local minimums [37]. While the LM, GRG and Amoeba belong to the category of local optimization schemes (locate a local minimum), SAA is a global optimization scheme, thus SAA is capable to overcome local minimums and achieve global optimization of an objective function.

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

Note that there are many computational details, such as the use of partial derivatives or differences, etc., are associated with these optimization methodologies. Discussions of these computational details are beyond the scope of the study here, as our focus is to evaluate the performance of the four search schemes on the SOA, and eventually identify an optimal method for processing satellite ocean-color images.

## 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 *R _{rs}* simulated by Hydrolight [38] 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],), 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 *R _{rs}* 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]. 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. 2
, and Fig. 3
. 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), ${\delta}_{{R}_{rs}}$(the value when optimization is achieved), ${\delta}_{IOP}$, and root mean square error in log scale (RMSE). ${\delta}_{IOP}$ is the error between the derived and known spectral IOPs, which is calculated using Eq. (4), except that the *R _{rs}*(

*λ*) spectrum is replaced by the corresponding spectral IOPs and ${\delta}_{{R}_{rs}}$ is replaced by ${\delta}_{IOP}$ with the spectral IOPs that formed the optimized ${\delta}_{{R}_{rs}}$. Further, the calculation of ${\delta}_{IOP}$ 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 ${\delta}_{IOP}$ 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 ${\delta}_{{R}_{rs}}$,${\delta}_{IOP}$and RMSE for valid retrievals are summarized in Table 1 .

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] 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 ${\delta}_{{R}_{rs}}$ or ${\delta}_{IOP}$, is almost identical for all optimization methods. For example, ${\delta}_{{a}_{ph}}$ are centered around 0.5-0.7 and ${\delta}_{{a}_{dg}}$ around 0.2. ${\delta}_{{b}_{bp}}$mostly 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] and in Morel [41] (Fig. 8 of the article). Median values of error, either in terms of ${\delta}_{{R}_{rs}}$ or ${\delta}_{IOP}$, 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
*a*retrievals was found associated with the GRG, while the other four methods showed comparable RMSE numbers. For example, the RMSE of_{ph}*a*(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._{ph} - (5) The RMSE for
*a*and_{dg}*b*retrievals at each band (410, 440, 490, 510 and 550 nm) was similar for all five methods, and was much lower than that for_{bp}*a*(for example, ~0.1 versus ~0.3 at 440 nm)._{ph}

For the NOMAD data set, a comparison of derived products with measured values at two bands is shown in Fig. 4
. 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 ${\delta}_{{R}_{rs}}$, ${\delta}_{IOP}$, 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], 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 ${\tilde{a}}_{ph}(\lambda )$(used in Eq. (12) was changed to the following:

Modification 2 (m02_GSM): In this modification, the above change of ${\tilde{a}}_{ph}(\lambda )$maintained, and the *Y* value used in Eq. (11) was changed from a constant (1.03373) to dynamic (Eq. (14),

*a*remains the same (e.g.,

_{dg}*S*value kept as 0.0206 nm

^{−1}).

Modification 3 (m03_GSM): The above changes of ${\tilde{a}}_{ph}(\lambda )$and ${\tilde{b}}_{bp}(\lambda )$ maintained, as in Modification 2, and further the *a _{dg}* spectral slope was changed from 0.0206 nm

^{−1}to 0.015 nm

^{−1}, a forward bio-optical design matching that of HOPE [15].

Here for easier implementation, MGRG was used to achieve optimization. Figure 5 shows the ${\delta}_{{R}_{rs}}$ and ${\delta}_{IOP}$ values from the GSM and the modified models for the IOCCG simulated data set, respectively, along with the distribution of the ${\delta}_{{R}_{rs}}$ and ${\delta}_{IOP}$ values. It is found that ${\delta}_{{R}_{rs}}$ 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 ${\delta}_{{R}_{rs}}$ ≤ 0.02; ${\delta}_{{R}_{rs}}$ 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 ${\delta}_{IOP}$. Among the three variables, ${\delta}_{{a}_{ph}}$differs the most between the original GSM and m01_GSM, m02_GSM and m03_GSM. ${\delta}_{{a}_{ph}}$of the original GSM has ~25% of the data centered at 0.6, while ${\delta}_{{a}_{ph}}$of the three modified models have ~20% data centered at 0.1 (see Fig. 5).

Figure 6
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
for more details). It was found that the modification of the eigen functions changed the retrievals of *a _{ph}*,

*a*and

_{dg}*b*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

_{bp}*a*(440) reduced from 0.295 to 0.155, but the RMSE for

_{ph}*a*(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

_{dg}*a*, 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

_{ph}*R*is a function of at least three active IOPs (

_{rs}*a*,

_{ph}*a*,

_{dg}*b*), 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).

_{bp}A further test of the NOMAD *in situ* data also provided consistent results (see Fig. 7
). Accuracy of retrievals in terms of absorption was much higher when all modifications were applied. For example, the RMSE for *a _{ph}* decreased from 0.504 using the original GSM to 0.281 using m03_GSM. Here the RMSE value is calculated for

*a*of the five bands to provide a measure of the quality of the retrieved

_{ph}*a*spectrum, instead of the quality of

_{ph}*a*at one specific band.

_{ph}These results echo that spectral models have strong impact on the error function and final solutions of SOA [40], 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 ${\tilde{a}}_{ph}(\lambda )$ used in GSM is not likely to be the sole factor leading to relatively larger ${\delta}_{{R}_{rs}}$ and ${\delta}_{IOP}$value, although this eigen function does not necessarily represent the spectral shape observed from sample measurements [24], as indicated in Maritorena et al [13].

## 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 *R _{rs}* 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.

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