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This work presents a novel GA-SA approach to retrieve the constituents of water bodies from remote sensing of ocean color. This approach is validated and compared to the existing algorithms using the same synthetic and in-situ datasets compiled by the International Ocean Color Coordinate Group. Comparing to the other methods, the GA-SA approach provides better retrievals for both the inherent optical properties and various water constituents. This novel approach is successfully applied in processing the images taken by MODerate resolution Imaging Spectroradiometer (MODIS) and generates regional maps of chlorophyll-a concentration, total suspended matter, and the absorption coefficient of color dissolved organic matter at 443nm.
©2007 Optical Society of America
1. Introduction
Retrieving information of water constituents is an extremely important application of ocean color analysis. For instance, retrieved chlorophyll concentration is essential for estimating the primary productivity of an ocean [1]. Over the past two decades, most studies relied on regression analysis to derive the empirical relationship between chlorophyll (Chl) concentration and the above-surface remote sensing reflectance Rrs(λ). For application on a global scale, however, the applicability of regression algorithms is limited to the local region for which empirical relationships are derived. Semianalytical (SA) models [2] can simulate sea surface reflectance (SSR) based on the fundamental physical principles and can account for measurement uncertainties [3]. Therefore, numerous retrieval algorithms shifted toward coupling SA models to simulate apparent optical properties (AOP), and then inversely retrieving the constituents of water bodies via optimization techniques [3, 4] such as the Levenberg-Marquart approach [5], and nonlinear least-square schemes [6, 7]. Alternatively, SA paradigms can also be modified to a simplified expression for linear inversion, such as the simplex method [8] and the matrix inversion based on the least-square approach [9].
Consensus has been reached in recent years that retrieval of inherent optical properties (IOP) is theoretically straightforward and more accurate than retrieving water constituents on a global scale [10]. For practical applications that require not only IOPs but also the constituents of water bodies, however, it remains necessary to further decompose contributions to total IOPs from individual water constituents based on biooptical models (BOM). The BOMs are comprised of various formulations and parameters that are generally nonlinear and vary globally [3]. Retrieving water constituents from complex BOMs is a challenging task as conventional optimization approaches might reduce their accuracy without constraining initial values [11] or limiting the number of decision variables [6]. Consequently, soft computing and global optimization methods, such as simulated annealing [3], neural networks [12] and genetic algorithms [11] (GA), were introduced in numerous works to resolve this inverse problem. The advantage of GA over conventional optimization methods were demonstrated by Zhan et al. [11]. For example, GA is able to retrieve IOPs with limited amount of a priori information. It is particularly suitable for solving problems of complex model for which the interaction of parameters is highly non-linear [13].
This work presents a novel GA-SA approach for retrieving water constituents from remote sensing of ocean color under the assumption that other IOPs covary with phytoplankton absorption coefficient at 440nm aph(440). Application of this novel GA-SA approach is validated against a synthetic dataset (N=500) and an in-situ dataset (N=656) compiled by the International Ocean Color Coordinate Group (IOCCG) [14]. Note that the synthetic dataset comprises a wide range of parameters characterizing the global ocean (Case 1 waters), and most of the in-situ data come from locations that are relatively close to the coast (some are Case 2 waters) [15]. Nevertheless, the GA-SA approach can retrieve aph(440) (linear percentage error, 40%) for synthetic data and aph(443) (linear percentage error, 86%) for in-situ data. Compared with existing retrieval algorithms published in the recent IOCCG report [14], the proposed GA-SA approach provides better retrievals of water constituents and IOPs, and can be implemented to process images taken by MODerate resolution Imaging Spectroradiometer (MODIS) on a regional scale. This work encourages application of the proposed GA-SA approach for deriving other products of ocean color at a regional scale, such as the color dissolved organic matter (CDOM) and non-algal particle/detritus/mineral (NAP).
2. Semianalytical model
Lee et al. [16] presents a full description of the SA model. Briefly, the SA model simplifies the radiative transfer process by relating Rrs(λ) to the dimensionless number urs(λ) as
where rrs(λ) is the below-surface remote-sensing reflectance spectra; g 0=0.0895 and g 1=0.1247 are taken from Lee et al. [17]. Note that u(λ) derived from rrs(λ) is denoted as urs(λ). In addition, u(λ) can also be derived by a function of the absorption coefficient a(λ) and the backscattering coefficient bb(λ) [18], both of which can be derived from in-water constituents
The distinction of urs(λ) and uiwc(λ) is just based on how they are derived, for the purpose of forming an explicit objective function of optimization in the subsequent section.
Although the retrievals of a(λ) and bb(λ) from Rrs(λ) are robust and accurate, to further decompose the contributions to total a(λ) and bb(λ) from individual water constituents still requires a set of BOMs that are sufficiently flexible to characterize the wide variety of biooptical relationships among various water constituents [19]. The four-component BOM [20] utilized in this research is summarized as follows. The BOM is typically a function of Chl [21] or aph(λ). This research follows the work by Lee et al. [16] to parameterize aph(λ) as
where the empirical coefficients for a0(λ) and a1(λ) are in Table 2 of Lee et al. [16]. The optical properties of additional water constituents are generally also co-varied with aph [21]. Therefore, this work uses the similar co-variation formulations for an ocean color model for global applications [3] to yield a general expression of optical properties contributed of other constituents
In this formulation, Ri is the covariance between aph(440) and other IOPs (Xi). Thus, there would be four combinations of Xi and Ri, i.e., Rg is the ratio between the absorption coefficient of CDOM at 440nm ag(440) and aph(440); Rd is the ratio between the absorption coefficient of NAP at 440nm ad(440) and aph(440); Rbph is the ratio between the backscattering coefficient of phytoplankton at 550nm bbph(550) and aph(440), and Rbd is the ratio between the backscattering coefficient of NAP at 550nm bbd(550) and aph(440). Except for the contribution of aph(λ) and the absorption of pure water [22] aw(λ), the absorption coefficients for other water constituents can be modeled as an exponential function of wavelength [23]. Thus, five parameters are used to express a(λ) as
where Sg and Sd are the spectral slope of absorption coefficients for CDOM and NAP, respectively. To compare with the IOP derived by other algorithms in subsequent sections, the absorption coefficient due to NAP and CDOM adg(λ) is calculated by the summation of ag(λ) and ad(λ). Similarly, the backscattering coefficients of other water constituents can be modeled as a power law function [16]. Therefore, four parameters are used to express bb(λ) as
where Yph and Yd are the spectral shapes of backscattering coefficients for phytoplankton and NAP, respectively, and bw(λ) is the scattering coefficient of pure seawater [24]. To compare with the IOP derived by other algorithms in subsequent sections, the backscattering coefficient due to particles bbp(λ) is calculated by the summation of bbph(λ) and bbd(λ).
3. GA-SA approach
The theoretical foundations of GAs and details of the searching approach for optimization are found in Goldberg [13]. Briefly, the GAs are stochastic algorithms with searching approaches that mimic phenomena in nature, such as inheritance and Darwinian struggle for survival. To solve inverse problems, the concept of GA can be utilized to optimize a quantifiable objective, namely, fitness. The fitness value is an evaluation of the solutions purposed for the problem. Solutions with high fitness values would have high probabilities to survive in the revolution process. Evaluation of fitness depends on “decision variables” of the forward model, which are usually nonlinear and must subject to “constraints.”
Therefore, this work has three key points in combining GAs and SA to solve the inverse problem in ocean optics. Because urs(λ) can be obtained semi-analytically from SSR with few errors [17] [Eqs. (1) and (2)], the first key point is to calculate uiwc(λ) in Eq. (3). Nine unknown variables in Eq. (4) to Eq. (7) need to be decided before Eq. (3) can be calculated, which are so-called decision variables, i.e., aph(440), Rg, Rd, Sg, Sd, Rbph, Rbd, Yph, and Yd. Since the minimal divergence between uiwc(λ) and urs(λ) represents a good agreement between the actual amount of water constituents and the retrievals from remote sensing. Therefore, the second point is to minimize obj and with an intention to obtain high fitness values, as
where N is the number of available satellite bands and i is the band index. GA searches the best set of decision variables that falls within the range specified by users, which means that GA would yield a near-best solution if the best set of decision variables does not exist in the searchable range. For example, GA would generate a solution with aph(440) equals to 0.5 if the actual solution of aph(440) is 0.6 and the upper bound for searching is 0.5. Additionally, extra computing time is required for GA to find the best solution in a broader range. To achieve a better performance and efficiency, the third point is to constrain all parameters in a dynamic range when uiwc(λ) is examined. In this work, the constraints on the range of all decision variables are determined by Lee’s model [15] that was specifically designed to generate a comprehensive synthetic data set based on observation and theory (Table 1). Note that some parameters are specified in GA to control the efficiency of GA searching (see Table 2). The initial ranges of these parameter values are referred to practical rules [13], and their final ranges are determined by trial and error tests. The more accurate solution can be obtained with large number of generations at the cost of more computing time. Fig. 1 presents a flowchart and some examples of the GA operators to illustrate the procedures of GA-SA; these procedures are summarized as follows.
Table 1. The dynamic ranges of nine decision variables in biooptical model
Table 2. Values of control parameters used in GA
- 1. Each GA run comprises a number of generations in a population of individuals. Each individual, which represents one possible solution to the SA inverse problem, can be any combinations of parameters to be determined.
- 2. Since GA attempts to identify the best individual in a genetic manner, the GA-SA process begins by randomly generating a population of individuals (chromosomes, strings). In the decoding example of Fig. 1, three alphabets (0 or 1) are used to represent one real value of each decision variable, therefore, there are 8 (23) possible values for each of them. The chromosome is a combination of nine decision variables, which comprises 27 alphabets (genes) to represent one possible solution of the inverse problem. A binary decoding skill is utilized to translate an individual from a chromosome form to a real form, as
where xi is the real value of the ith parameter; ximax is the upper bound of the ith parameter; ximin is the lower bound of the ith parameter; li is the string length of the ith parameter; j is the index number of genes; bitij is the alphabet in binary coding (0 or 1) of the jth gene of the ith parameter. The fitness value is then assessed by the “real form” of the individual using BOMs, a given urs(λ), and Eq. (8).
- 3. The second and subsequent generations are created via a reproduction process. In this process, an individual with a high fitness value has a likely opportunity to be selected and pass its genetic characteristics to its offspring by duplicating or processing a “crossover operation” with other selected individuals (Fig. 1). The crossover operator is applied to create offspring by randomly selecting the corresponding genes of two parent chromosomes for exchanging genetic features. Following the crossover operation, offspring have various genetic features from their parents and in some cases, “good” features are destroyed, resulting in an ineffective search. A GA parameter called crossover probability (Pc) is introduced to obtain a balance between diversity and search efficiency. When Pc=0.5, approximately 50% of selected individuals in the previous generation participate the crossover process and the remaining individuals pass on their genetic features to the next generation.
- 4. Retaining the diversity of genetic features is the most important goal that prevents the process of numerical evolution ending at a local solution. To achieve this aim, a “mutation operator” is utilized for the reproduction process by randomly varying genetic features (Fig. 1). The mutation operator dramatically changes the genetic features; thus, the probability of mutation is commonly considerably lower than Pc, as is the case for humans.
- 5. In the final generation, the optimum set of all decision variables for given urs(λ) can be obtained. With this optimum set, all required information for IOPs can be derived and the inverse problem is solved.
4. Results and discussion
To evaluate and compare the performance of various retrieval algorithms, the IOCCG compiled a synthetic dataset [25] and an in-situ dataset containing IOPs and corresponding AOPs [26]. Uncertainties in IOP measurements were integrated into the synthetic dataset by mapping one Chl to 25 different sets of IOPs based on theory and field data [15], resulting in 500 IOPs that are generated from 20 levels of Chl in the range of 0.03–30 mg/m3, and the corresponding SSRs simulated by Hydrolight for the sun at 30° from zenith. The in-situ dataset comprises 656 cases originating from NASA’s SeaWiFS Bio-optical Archive and Storage System (SeaBASS), including Chl-a, a(λ), adg(λ), aph(λ) and Rrs(λ) at the first five SeaWiFS bands (λi=412, 443, 490, 510 and 555nm). Utilizing these two datasets, Lee et al. [14] assessed and compared the performance of nine retrieval algorithms by calculating the Root-Mean-Square-Error in log phase (RMSE). The RMSE value for these retrieval algorithms were taken to calculate the linear percentage errors ε in log scale [17], as
Tables 3 and 4 present summaries of the ε for retrieving a(λ), adg(λ), bbp(λ) and aph(λ). Note that bbp is not included in the in-situ data. This work employs the same datasets and applies the same procedures as described by Lee et al. [14] to validate the proposed GA-SA model.
4.1 Test using the synthetic dataset
Figure 2 and Table 3 present the performance of the proposed GA-SA by comparing the retrievals of 500 synthetic cases to the known IOPs. For the purpose of deriving water constituents, a distinction between individual and combinational IOPs is required. The combinational IOP contains the optical information attributed to more than one water constituent, e.g., a, bb, adg, and bbp. The individual IOP, e.g., aph and bbd, is needed for accurately deriving the concentration of the specified water constituent. For the category of combinational IOPs, most of the existing algorithms are able to attain a good retrieval. Notably, the Quasi Analytical Algorithm [17] (QAA) obtains the best retrievals for adg in this category, whereas the proposed GA-SA approach ranks second among all existing algorithms and provides the best retrieval of bbp(550). As illustrated in Table 3, aph(λ) retrievals is approximately 1.6–6.7 times worse than the a(λ) retrievals when using other algorithms. The results of Figs. 2(a) and 2(e)-2(g) also show that it is difficult for GA-SA to divide the contributions between aph(λ), ag(λ) and ad(λ), from a(λ) to obtain the retrievals of individual IOPs. However, the proposed GA-SA still yields a good result for aph(440) with ε as low as 40%, which is only 1.5 times the error rate for a(λ) retrievals. This result demonstrates that the GA-SA approach is ideal for further decomposing the contributions of individual water constituents to total a(λ) and bb(λ).
Table 3. Linear percentage errors ε (%) between derived and known values of the synthetic dataset
Fig. 2. Comparisons between GA-SA derived and known IOPs, for synthetic data compiled by IOCCG [14]. (a) a(440), (b) bb(550), (c) adg(440), (d) bbp(550), (e) aph(440), (f) ag(440), (g) ad(440), (h) bbph(550), and (i) bbd(550).
Another significant advantage of the proposed GA-SA approach is that ag(440), ad(440), bbph(550) and bbd(550) can be retrieved simultaneously. As the spectral shape of one IOP might be very similar to the spectral shape of the other IOPs, e.g., ad and ag in Eq. (5) ,and bbph and bbd in Eq. (6), a relatively high contribution comes from one IOP would overlap the contribution from the other IOPs. Therefore, the overlapping absorption and backscattering makes it difficult to differentiate individual IOPs from the total optical properties using the general SA models. In the synthetic dataset, for example, this is particularly true for ad(440) that accounts for 12% of a(440), and bbph(550) that accounts for 35% of bb(550). By contrast, ag(440) accounts for 64% of a(440), and bbd(550) accounts for 61% of bb(550). As a result, the values of R2 for retrieving ag(440) and bbd(550) are as high as 0.965 and 0.870, respectively. Despite of the relative low contribution of aph(440) (23%) to a(440), the GA-SA approach is able to retrieve this key component with the lowest error among all IOPs. This is because aph(440) has a particular BOM [Eq. (4)] and most variables are directly or indirectly associated with aph(440).
4.2 Test using the in-situ dataset
Figure 3 and Table 4 show the performance of the GA-SA approach by comparing the retrievals of GA-SA approach to the 656 in-situ cases of measured IOPs. Due to the high uncertainty in measurement errors in the in-situ dataset, the retrieving errors (Table 4) of the QAA is increased from 2.4 (for aph) to 3.3 (for a) times to the results of the 500 synthetic cases (Table 3). Conversely, the GA-SA approach is relatively robust with only 2.0 (for adg) to 2.2 (for a) times to the results of the 500 synthetic cases. To retrieve the combinational IOPs, the GA-SA approach is ranked second and is only slightly less accurate than the QAA. In retrieving the key component aph(443), however, the GA-SA scheme performs best. Compared with algorithms that disregard several cases due to an error checking mechanism, GA-SA performance was evaluated for all 1156 cases—synthetic and in-situ datasets combined. No unreasonable retrieval (i.e., negative IOPs) was found in any case.
Fig. 3. Comparisons between GA-SA derived and measured IOPs, for in-situ data compiled by IOCCG [14]. (a) a(443), (b) adg(443), and (c) aph(443).
Table 4. Linear percentage errors ε (%) between derived and measured values of the in-situ dataset.
4.3 Applications to processing satellite imagery
The proposed GA-SA approach is accurate and robust, and easy to use and sufficiently fast at processing satellite imagery on a regional scale. Figure 4 presents an example of Taiwan’s coastal region taken by MODIS-Aqua on May 9, 2006. Although the GA-SA approach is applied in a pixel-by-pixel manner, the processed image is fairly smooth without apparent discontinuity. All processing is conducted on a personal computer equipped with a Pentium 2.4GHz CPU. It took 3.6 hours to process 15,000 pixels. Processing can be accelerated by parallelizing the computation on a cluster machine.
Compared to the standard product of MODIS, the proposed GA-SA approach yields comparable values of Chl [Figs. 4(a) and 4(b)], except for coastal waters where the NAP or CDOM dominates the spectral shape of SSR [29]. This is exactly the problem in existing global algorithms that typically overestimate Chl in coastal waters. In order to clarify the contributions to SSR from Chl, NAP and CDOM, the IOPs of individual water constituents need to be retrieved as accurate as possible. As shown in Fig. 2, the retrievals of aph(440) and bbd(550) derived from the GA-SA approach are more representative and accurate, comparing to other retrievals, such as bbph(550) and ad(440). Therefore, the Chl (mg/m3) is calculated by [30]
and the NAP (g/m3) is calculated by [31]
where Bd depends on the selected phase function, and is 0.0183 when Petzold average particle phase function [32] is used. Finally, the concentration of CDOM (m−1) is expressed by ag(443) that is calculated by the retrievals of aph(440), Rg and Eq. (6).
One remarkable feature shown in Fig. 4 is the river plumes along Taiwan’s west coast [referring to the two red frames in Fig. 4(a)]. The standard product of Chl [Fig. 4(a)] does show the distribution and dispersion of the plumes; however, the high values of Chl seem to be not realistic and no information for NAP and CDOM exists. Conversely, the coastal GASA value [Fig. 4(b)] is roughly of the same order as the retrieval of Chl in offshore waters, except for two major estuaries for which the concentrations of Chl seem to be generally high [referring to the two arrows in Fig. 4(b)]. Furthermore, the maps for NAP [Fig. 4(c)] and CDOM [Fig. 4(d)] derived using the GA-SA approach clearly delineate the distributions and dispersions of several major river plumes. These two new products facilitate the study of river plume dynamics and the estimation of the annual amount of sediment discharged through the river-sea system.
5. Summary
This work presents a novel GA-SA approach for retrieving water constituents from remote sensing of ocean color. This approach is validated and evaluated against a synthetic data set and an in-situ data set compiled by IOCCG [14]. The result of comparison demonstrates that the GA-SA approach is as almost as accurate as the QAA method in retrieving combinational IOPs. Furthermore, the proposed approach is particularly ideal for further decomposing the contributions from individual water constituents to total IOPs. The GA-SA approach is accurate and robust, and is easily utilized and sufficiently fast at processing the satellite imagery on a regional scale. After processing by the GA-SA approach, the MODIS-Aqua image of Taiwan’s coastal region (2006/5/9) yields a realistic map of Chl, both for coastal and offshore waters. Furthermore, the maps of NAP and CDOM clearly delineate the distributions and dispersions of several major river plumes. This work suggests that the constituents of water bodies, such as Chl, NAP and CDOM, can be acquired routinely from remote sensing of ocean color.
Fig. 4. Applications of the GA-SA approach in processing MODIS-Aqua imagery of the coastal region of Taiwan (2006/05/09). (a) Chl (standard MODIS product), (b) Chl (GA-SA approach), (c) NAP (GA-SA approach), and (d) CDOM (ag(443), GA-SA approach). Note that no data is given by MODIS in those white areas.
6. Acknowledgment
The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract Nos. NSC-95-2625-Z-006-004-MY3, NSC-95-2211-E006-283 and NSC-95-2611-M-006-002.
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