MODIS-Based Remote Estimation of Absorption Coe ﬃ cients of an Inland Turbid Lake in China

: Optical complexity and various properties of Case 2 waters make it essential to derive inherent optical properties (IOPs) through an appropriate method. Based on ﬁeld measured data of Lake Chaohu between 2009 and 2018, the quasi-analytical algorithm (QAA) was modiﬁed for the particular scenario of that lake to derive absorption coe ﬃ cients based on the moderate-resolution imaging spectroradiometer (MODIS) bands. By changing the reference wavelength to longer ones and building a relationship between the value of spectral power for particle backscattering coe ﬃ cient (Y), suspended particulate matter (SPM), and above-surface remote-sensing reﬂectance (R rs ), we improved the accuracy of the retrieval of total absorption coe ﬃ cients. The absorption coe ﬃ cients of gelbsto ﬀ and non-algal particulates (a dg ) and absorption coe ﬃ cients of phytoplankton (a ph ) in Lake Chaohu were also derived by changing important parameters according to Lake Chaohu. The derived a ph tend to be bigger than measured a ph in this study, while derived a dg tend to be smaller than measured data. We also used the corrected MODIS surface reﬂectance product (MOD09 / MYD09) to calculate the a ph (443), a ph (645), and a ph (678) by the model proposed in this study. It shows that in summer and autumn, a ph tended to be higher in the northwestern part of Lake Chaohu, and were relatively lower in the spring and winter, which is similar to previous studies. Overall, our study provides an algorithm that is e ﬀ ectively used in the case of Lake Chaohu and applicable to the data obtained by MODIS, which can be used for further study to investigate the change law of absorption coe ﬃ cients in long time series by applying MODIS data.


Introduction
The inversion of water color involves the derivation of inherent optical properties (IOPs) from apparent optical properties (AOPs). As a result, information about water constituents is retrieved from derived IOPs, such as concentrations of chlorophyll-a (Chl a), suspended sediments, and colored, dissolved organic matter (CDOM).
Recent studies have emphasized the importance of retrieving IOPs through remote sensing. Variations in IOPs can precisely indicate changes in water constituents and mass. AOPs refer to the parameters that vary with the change of illumination conditions, including water-leaving radiance (L w ), above-surface remote-sensing reflectance (R rs ) , and so on [1]. The IOPs and light field together determine the AOPs. The inversion of the IOPs can be achieved by using the related algorithms from

Measurements of Relevant Parameters
A handheld ASD (analytical spectral device) under the NASA Ocean Optics protocols was used to obtain Rrs [26]. Applying the method described by Mobley et al. [27], the viewing direction was 40 degree from the nadir and 135 degree from the Sun.
We filtered the water samples by Whatman GF/C glass-fiber filters with a pore size of approximately 1.2 μm and extracted pigments with a reference of 90% acetone. We used a Shimadzu UV-2600 (Kyoto, Japan) to measure the absorbance then achieved Chl a data [28]. We pre-combusted Whatman GF/F glass-fiber filters with a pore size of 0.7 μm at 450 °C for 6 h and pre-weighed them. We then filtered the water samples and dried them at high temperature (105 °C) for approximately 4-6 h for the measurement of SPM concentrations. Suspended particulate inorganic matter (SPIM) was similarly measured through weighing the filters before and after burning organic matter for 6 h [29].
The absorption coefficients of total particulate matter, absorption coefficients of non-algal particulate (ad), and absorption coefficients of phytoplankton (aph)at 350-800 nm were obtained by using the same machine with GF/F filters [26]. The baseline was obtained by a blank filter with distilled water [2]. The ad was measured after the pigments were bleached with sodium hypochlorite, and then aph were derived. The absorption coefficient of CDOM (ag) was measured using the same

Sampling and Data Collection
We conducted 11 cruise surveys, with 119 surface samples from October 2009 to July 2018 to measure optical properties in Lake Chaohu in this study ( Figure 1). From this, 94 samples were used to build models and 25 samples were used as test data. We measured R rs data at the sites. The water samples were kept at 4 • C in the dark before experiments of SPM, Chl a, dissolved organic carbon concentration (DOC), and absorption coefficients.

Measurements of Relevant Parameters
A handheld ASD (analytical spectral device) under the NASA Ocean Optics protocols was used to obtain R rs [26]. Applying the method described by Mobley et al. [27], the viewing direction was 40 degree from the nadir and 135 degree from the Sun.
We filtered the water samples by Whatman GF/C glass-fiber filters with a pore size of approximately 1.2 µm and extracted pigments with a reference of 90% acetone. We used a Shimadzu UV-2600 (Kyoto, Japan) to measure the absorbance then achieved Chl a data [28]. We pre-combusted Whatman GF/F glass-fiber filters with a pore size of 0.7 µm at 450 • C for 6 h and pre-weighed them. We then filtered the water samples and dried them at high temperature (105 • C) for approximately 4-6 h for the measurement of SPM concentrations. Suspended particulate inorganic matter (SPIM) was similarly measured through weighing the filters before and after burning organic matter for 6 h [29].
The absorption coefficients of total particulate matter, absorption coefficients of non-algal particulate (a d ), and absorption coefficients of phytoplankton (a ph ) at 350-800 nm were obtained by using the same machine with GF/F filters [26]. The baseline was obtained by a blank filter with distilled water [2]. The a d was measured after the pigments were bleached with sodium hypochlorite, and then a ph were derived. The absorption coefficient of CDOM (a g ) was measured using the same machine with Milli-Q water as the reference from water filtered by filters with a 0.22-µm pore size from 280 to 700 nm (1-nm interval) [2,30,31].

Satellite Image Data Preprocessing
In this study, MODIS data were selected as input data. MODIS has high spectral and time resolutions. It is set on Terra and Aqua and has five levels of data products. MODIS provides continuous global remote sensing data that have a wide range of applications in ecology and geography research. The MODIS data preprocessing in this study mainly refer to the geometric correction and atmospheric correction of the MODIS surface reflectance product (MOD09/MYD09) in Lake Chaohu.

The MOD09 Correction Method
The atmosphere is an important factor affecting the quantitative analysis and application of remote sensing. Therefore, removing the effects of atmospheric scattering and absorption from the radiance value received by the sensor has become the premise of remote sensing quantitative analysis. In some studies, MOD09 reflectance data are used directly [34][35][36]. Nevertheless, by comparing MOD09 data and measured data, it has been found that the MOD09 data generally show higher phenomena in some bands compared with the measured data [37].
In this study, we used a simple correction method based on near-infrared (NIR) and short-wave infrared (SWIR) bands [37]. The advantage of using this correction method is that it can eliminate the noise in MOD09 data by simple correction and then convert the surface reflectance to the scale of remote-sensing reflectance, so that a more accurate R rs (λ) is obtained and can be more easily applied to processing. The specific correction method is as follows: where min (R NIR : R SWIR ) refers to the minimum reflectance value of NIR and SWIR bands. Due to the strong absorption of water in the NIR and SWIR bands, reflectance will generally drop to 0 in the NIR band in the case of general water, while reflectance will generally decrease to 0 in the SWIR band in the Remote Sens. 2020, 12, 1940 5 of 21 case of turbid water [38]. Therefore, the reflectance value of the NIR and SWIR bands is considered as additive noise. The additive noise can be eliminated by subtracting the value from each band value. After the correction by this method, there are still some gaps between the corrected R rs and the measured values at 748 nm ( Figure 2a).Therefore, this study used the least-square method to calibrate the values at 748 nm, again based on the measured data. The equation between the measured R rs and the values derived from MOD09 is constructed to make the values closer to the measured data. The R rs after recalibration is shown in Figure 2b (N = 79). Similarly, the R rs values at 413 and 443 nm were also corrected again using this method.
Remote Sens. 2020, 12, x FOR PEER REVIEW 5 of 22 Rrs after recalibration is shown in Figure 2b (N = 79). Similarly, the Rrs values at 413 and 443 nm were also corrected again using this method.

Improving QAA
QAA is a semi-analytical method for calculating the absorption coefficient and backscattering coefficient of water proposed by Lee et al. [9]. The inversion process of QAA has two parts. The first one is deriving the total absorption coefficient and backscattering coefficient. The second, utilizing the coefficient of the total absorption obtained from the first part, is decomposing the total absorption coefficient into different elements. The QAA algorithm is proposed for open ocean and coastal waters; accordingly, some empirical formulas are used for specific research areas, which cannot be directly applied to the inversion of the absorption coefficients for inland Case 2 waters.

Inversion of Total Absorption Coefficients
In the first part of QAA, a relationship between Rrs and total absorption coefficients was established. Similarly in this study, a new relationship needs to be established according to Lake Chaohu by changing some parameters and formulas as follows.

Values of g0 and g1
The ratio of backscattering coefficient to the sum of absorption and backscattering coefficients (u(λ)) is calculated by rrs and g0, g1. Gordon et al. [39] found the values of g0 = 0.0949 and g1 = 0.0794 for oceanic Case 1 waters, whose optical properties are determined primarily by phytoplankton, CDOM, and detritus degradation products [40]. Lee et al. [9,41] advised that g0 = 0.084 and g1 = 0.17 is more accurate for higher-scattering coastal waters. In fact, the values of g0 and g1 are different because they depend on the particle phase function, which cannot be measured remotely. The values of these two parameters have to be estimated before being used in semi-analytical algorithms. Lee [9] used the averaged g0 and g1 values, which can be applied to coastal and open ocean waters. However, the values of g0 and g1 have a minimal influence on the inversion accuracy of the total absorption coefficient. Therefore, in this study, after trying to apply different values, we used the values of g0 of

Improving QAA
QAA is a semi-analytical method for calculating the absorption coefficient and backscattering coefficient of water proposed by Lee et al. [9]. The inversion process of QAA has two parts. The first one is deriving the total absorption coefficient and backscattering coefficient. The second, utilizing the coefficient of the total absorption obtained from the first part, is decomposing the total absorption coefficient into different elements. The QAA algorithm is proposed for open ocean and coastal waters; accordingly, some empirical formulas are used for specific research areas, which cannot be directly applied to the inversion of the absorption coefficients for inland Case 2 waters.

Inversion of Total Absorption Coefficients
In the first part of QAA, a relationship between R rs and total absorption coefficients was established. Similarly in this study, a new relationship needs to be established according to Lake Chaohu by changing some parameters and formulas as follows.
3.1.1. Values of g 0 and g 1 The ratio of backscattering coefficient to the sum of absorption and backscattering coefficients (u(λ)) is calculated by r rs and g 0 , g 1 . Gordon et al. [39] found the values of g 0 = 0.0949 and g 1 = 0.0794 for oceanic Case 1 waters, whose optical properties are determined primarily by phytoplankton, CDOM, and detritus degradation products [40]. Lee et al. [9,41] advised that g 0 = 0.084 and g 1 = 0.17 is more accurate for higher-scattering coastal waters. In fact, the values of g 0 and g 1 are different because they depend on the particle phase function, which cannot be measured remotely. The values of these two parameters have to be estimated before being used in semi-analytical algorithms. Lee [9] used the averaged g 0 and g 1 values, which can be applied to coastal and open ocean waters. However, the values of g 0 and g 1 have a minimal influence on the inversion accuracy of the total absorption coefficient. Therefore, in this study, after trying to apply different values, we used the values of g 0 of 0.08945 and g 1 of 0.1247 as in the QAA original algorithm, which is suitable for more types of waters.

Reference Wavelength
The measurements of absorption coefficients of water constituents include a ph (λ), a d (λ), and a g (λ). The total absorption coefficient a(λ) is the sum of a ph (λ), a d (λ), a g (λ), and the absorption coefficient of pure water a w (λ) [42]. a(λ) = a ph (λ) + a d (λ) + a g (λ) + a w (λ) The principle of selecting the reference wavelength in Step 2 (QAA) is that the absorption coefficient of pure water is dominant at the reference wavelength, and a w (λ 0 ) can basically replace a(λ 0 ). Table 1 provides examples of reference wavelengths in some relevant studies.

Reference Wavelength
Areas Reference 555 nm oligotrophic waters mesotrophic waters [9] 640 nm high-absorbing waters [9] 695 nm Lake Kuncheng [17] 715 nm Lake Taihu, Chaohu [15,43] According to the basic water quality in Lake Chaohu, a w increases and a g and a ph tend to decrease to 0 around 700 nm. Therefore, the reference wavelengths should be around 700 nm [43,44]. In our study, on the basis of the center wavelengths of MODIS bands, three reference wavelengths (645, 678, and 748 nm) were used to derive the total absorption coefficients of three different samples (Figure 3), which represent three kinds of water, namely, turbid, eutrophic, and general water. The turbid water has high SPM concentration (SPM concentration = 93 mg/L), the eutrophic water has high Chl a concentration (Chl a concentration = 183.39 µg/L) and the SPM and Chl a concentrations of general water are not high (SPM concentration = 27 mg/L, Chl a concentration = 17.04 µg/L). As the graphs suggest, in the case of using 748 nm as reference wavelength, the derived total absorption coefficients are much closer to the measured total absorption coefficients. We can conclude that, regardless of the type of water, setting 748 nm as the reference wavelength is most suitable for Lake Chaohu.
The principle of selecting the reference wavelength in Step 2 (QAA) is that the absorption coefficient of pure water is dominant at the reference wavelength, and aw(λ0) can basically replace a(λ0). Table 1 provides examples of reference wavelengths in some relevant studies.
According to the basic water quality in Lake Chaohu, aw increases and ag and aph tend to decrease to 0 around 700 nm. Therefore, the reference wavelengths should be around 700 nm [43,44]. In our study, on the basis of the center wavelengths of MODIS bands, three reference wavelengths (645, 678, and 748 nm) were used to derive the total absorption coefficients of three different samples ( Figure  3), which represent three kinds of water, namely, turbid, eutrophic, and general water. The turbid water has high SPM concentration (SPM concentration = 93mg/L), the eutrophic water has high Chl a concentration (Chl a concentration = 183.39 μg/L) and the SPM and Chl a concentrations of general water are not high (SPM concentration = 27mg/L, Chl a concentration = 17.04μg/L). As the graphs suggest, in the case of using 748 nm as reference wavelength, the derived total absorption coefficients are much closer to the measured total absorption coefficients. We can conclude that, regardless of the type of water, setting 748 nm as the reference wavelength is most suitable for Lake Chaohu.

Model to Estimate the Power Value Y
Power value Y is a parameter used to estimate the backscattering coefficients at different wavelengths. If a(λ0), u(λ0), and backscattering coefficients of pure water at wavelength λ0 (bbw(λ0)) are available, and the value of Y is estimated, then backscattering coefficients of particulate at wavelength λ0 (bbp(λ0)) can be efficiently obtained. The values of total backscattering coefficient at wavelength λ (bb(λ)) at all wavelengths are then derived. As shown in Table 2, different types of waters have different ranges of Y based on different reference wavelengths. A model to estimate the value of wavelength exponent Y in the case of Lake Chaohu should be established in this study. The initial value of Y is set to 0.1, and the step size is set to 0.1 for iteration, and then the data was used in calculation. [15] 0.61-1.99

Huanghai Sea East China Sea
The backscattering coefficient is calculated based on measured data. [45] 3.06 Lake Taihu Measured data are used to calculate the backscattering coefficient. [46]

1.3-3 Lake Kuncheng
The empirical model of the Y value is established by simulating the relationship between the reference Y value and reflectance ratio rrs(640)/rrs(715). [17] As no backscattering coefficient was measured in our research, the real Y value cannot be simulated by the original analytical model. Therefore, our research used the method of iteration [15]. We made Y iterate from 0.1 to 3 (step size of 0.1). When the minimum average absolute error (MAE) between the calculated total absorption coefficients and the measured total absorption coefficients at MODIS bands between 400 and 700 nm was less than 0.3, the best reference Y value of this sample was obtained. Figure 4 represents reference Y values of three different samples. The accuracy of the calculation results was greatly improved in the three cases mentioned.

Model to Estimate the Power Value Y
Power value Y is a parameter used to estimate the backscattering coefficients at different wavelengths. If a(λ 0 ), u(λ 0 ), and backscattering coefficients of pure water at wavelength λ 0 (b bw (λ 0 )) are available, and the value of Y is estimated, then backscattering coefficients of particulate at wavelength λ 0 (b bp (λ 0 )) can be efficiently obtained. The values of total backscattering coefficient at wavelength λ (b b (λ)) at all wavelengths are then derived. As shown in Table 2, different types of waters have different ranges of Y based on different reference wavelengths. A model to estimate the value of wavelength exponent Y in the case of Lake Chaohu should be established in this study. The initial value of Y is set to 0.1, and the step size is set to 0.1 for iteration, and then the data was used in calculation. [15] 0.61-1.99

Huanghai Sea East China Sea
The backscattering coefficient is calculated based on measured data. [45] 3.06 Lake Taihu Measured data are used to calculate the backscattering coefficient. [46] 1.3-3 Lake Kuncheng The empirical model of the Y value is established by simulating the relationship between the reference Y value and reflectance ratio r rs (640)/r rs (715). [17] As no backscattering coefficient was measured in our research, the real Y value cannot be simulated by the original analytical model. Therefore, our research used the method of iteration [15]. We made Y iterate from 0.1 to 3 (step size of 0.1). When the minimum average absolute error (MAE) between the calculated total absorption coefficients and the measured total absorption coefficients at MODIS bands between 400 and 700 nm was less than 0.3, the best reference Y value of this sample was obtained. Figure 4 represents reference Y values of three different samples. The accuracy of the calculation results was greatly improved in the three cases mentioned.
To obtain the reference Y value as accurately as possible, we had to establish a model for estimating Y. The Y value is related to parameters, such as R rs , and concentrations of water compositions. First, we found out that a very good correlation did not exist between the ratio of R rs and Y by applying measured data of Lake Chaohu. Thus, the relationships between the reference Y value and certain water quality parameters were established ( Figure 5). As the graphs show, the relationship between SPM To obtain the reference Y value as accurately as possible, we had to establish a model for estimating Y. The Y value is related to parameters, such as Rrs, and concentrations of water compositions. First, we found out that a very good correlation did not exist between the ratio of Rrs and Y by applying measured data of Lake Chaohu. Thus, the relationships between the reference Y value and certain water quality parameters were established ( Figure 5). As the graphs show, the relationship between SPM concentration and Y value had the best correlation. Therefore, this model (Y = 0.0103 * SPM + 1.6386) (N = 80) was used in our algorithm. Even though the relationship between the Y value and SPM is modeled, water quality parameters cannot be determined from previous calculations. Thus, the model of SPM concentration and R rs should be built. Based on several references, the general characteristics of SPM inversion algorithms are presented in Table 3. In this study, similar methods were used to construct the relationship between R rs and SPM concentration. We found that SPM concentration and R rs (555)/R rs (748) had the best correlation ( Figure 6).  Even though the relationship between the Y value and SPM is modeled, water quality parameters cannot be determined from previous calculations. Thus, the model of SPM concentration and Rrs should be built. Based on several references, the general characteristics of SPM inversion algorithms are presented in Table 3. In this study, similar methods were used to construct the relationship between Rrs and SPM concentration. We found that SPM concentration and Rrs(555)/Rrs(748) had the best correlation ( Figure 6).

Decomposition of Total Absorption Coefficient
Deriving aph(λ) and the absorption coefficients of gelbstoff and non-algal particulates (adg(λ)) from the total absorption coefficients (a(λ)) is a major challenge because the total absorption coefficient is the sum of aw, aph, ad, and ag. Lee has developed an empirical algorithm for the separation [9]. We should estimate two parameters first: ζ, which is equal to aph(410)/aph(440), and ξ, which amounts to adg(410)/adg(440). The value of ζ is obtained using the ratio of measured rrs(440)/rrs(555) data in the QAA algorithm.

Decomposition of Total Absorption Coefficient
Deriving a ph (λ) and the absorption coefficients of gelbstoff and non-algal particulates (a dg (λ)) from the total absorption coefficients (a(λ)) is a major challenge because the total absorption coefficient is the sum of a w , a ph , a d , and a g . Lee has developed an empirical algorithm for the separation [9].
We should estimate two parameters first: ζ, which is equal to a ph (410)/a ph (440), and ξ, which amounts to a dg (410)/a dg (440). The value of ζ is obtained using the ratio of measured r rs (440)/r rs (555) data in the QAA algorithm.
3.2.1. The Value of Spectral Slope of a dg Spectrum (S) The spectral slope of a dg spectrum (S) can depict the spectral shape of a dg (λ), which is the sum of a g (λ) and a d (λ). In the original QAA process, S was valued at 0.015 nm −1 . The measured data of a dg (410) and a dg (440) in Lake Chaohu were used to calculate S. We concluded that the average value of S is 0.01453, and the standard deviation is 0.001129, which means that all the values are basically distributed around the average. Therefore, we selected the mean, 0.01453, as the value of S in our study.

The Relationship of a ph and r rs
The value of a ph (410)/a ph (440) is calculated from the ratio of r rs (440)/r rs (555) through an empirical formula in the original QAA algorithm [9]. Considering the research area, we had to rebuild the relationship between a ph and r rs by simulating the relationship between the a ph (410)/a ph (440) and the spectral ratio r rs of the center wavelengths of MODIS bands between 400 and 800 nm. The model was constructed using r rs (645)/r rs (678) with a high correlation, as shown in Figure 7. The specific calculation process of this improved QAA is as follows (Table 4). Table 4.
Steps of the improved quasi-analytical algorithm (QAA).
Step Formula Approach The specific calculation process of this improved QAA is as follows (Table 4).
To evaluate the performance of this algorithm and the accuracy of the MOD09 data after correction, three parameters were calculated.
The accuracy evaluation indicators used in this study include average relative error (MRE) [51], average absolute error (MAE) and root mean square error (RMSE) [52]. The expression equations are as follows: where x represents the measured value, y represents the derived value, and N represents the number of samples. Coefficient of determination (R 2 ) was also used to assess the accuracy of the model. Table 4.
Steps of the improved quasi-analytical algorithm (QAA).
Step Formula Approach

MODIS Corrected Data Accuracy Evaluation
Compared with the measured R rs data, the errors of the R rs derived from MOD09 product were evaluated in this paper. Before correction, as shown in Figure 8a, most of the 79 points of R rs were above the 1:1 line, indicating that, on the same scale, the data at 413, 443, 469, 555, 645, 678, and 748 nm were generally larger than the measured data; after using this correction method, as shown in Figure 8b, the R rs of the points generally had a good linear correlation. The scatterplots were mostly distributed near the 1:1 line. At the wavelength 748 nm, the corrected values were still a little higher than the measured data. Tables 5 and 6 show the statistics of the errors before and after correction. The MRE at 413 nm was reduced from 76.46% to 30.02%, and the RMSE was from 0.012 sr −1 to 0.005 sr −1 . The MRE value before correction at the 555-nm band was 13.83%, while the MRE value after correction was only 9.23%. The MRE of 748 nm before correction was as high as 84.66%, while the MRE after correction was 31.23%, and the RMSE also decreased from 0.009 sr −1 to 0.006 sr −1 . Compared with the original MOD09, the RMSEs and MREs of the corrected MOD09 at all bands were significantly reduced. This shows that the correction method we used in this study can obtain more accurate R rs .
Remote Sens. 2020, 12, x FOR PEER REVIEW 12 of 22 where x represents the measured value, y represents the derived value, and N represents the number of samples. Coefficient of determination (R 2 ) was also used to assess the accuracy of the model.

MODIS Corrected Data Accuracy Evaluation
Compared with the measured Rrs data, the errors of the Rrs derived from MOD09 product were evaluated in this paper. Before correction, as shown in Figure 8 a, most of the 79 points of Rrs were above the 1:1 line, indicating that, on the same scale, the data at 413, 443, 469, 555, 645, 678, and 748 nm were generally larger than the measured data; after using this correction method, as shown in Figure 8b, the Rrs of the points generally had a good linear correlation. The scatterplots were mostly distributed near the 1:1 line. At the wavelength 748 nm, the corrected values were still a little higher than the measured data. Table 5 and Table 6 show the statistics of the errors before and after correction. The MRE at 413 nm was reduced from 76.46% to 30.02%, and the RMSE was from 0.012 sr -1 to 0.005 sr -1 . The MRE value before correction at the 555-nm band was 13.83%, while the MRE value after correction was only 9.23%. The MRE of 748 nm before correction was as high as 84.66%, while the MRE after correction was 31.23%, and the RMSE also decreased from 0.009 sr -1 to 0.006 sr -1 . Compared with the original MOD09, the RMSEs and MREs of the corrected MOD09 at all bands were significantly reduced. This shows that the correction method we used in this study can obtain more accurate Rrs.

Inversion of Absorption Coefficients in Different Water Types
The measured data of R rs of Lake Chaohu were applied to this model to obtain absorption coefficients. The wavelengths of input R rs were at center wavelengths of MODIS bands between 400 and 700 nm, including 413, 443, 469, 488, 531, 551, 555, 645, 667, and 678 nm. The retrieved IOPs included a ph (λ) and a dg (λ) of these wavelengths. Figure 9 shows the comparison of retrieved and measured a ph and a dg of the three different types of water. The concentrations of SPM and Chl a are shown in the graphs.

Inversion of Absorption Coefficients in Different Water Types
The measured data of Rrs of Lake Chaohu were applied to this model to obtain absorption coefficients. The wavelengths of input Rrs were at center wavelengths of MODIS bands between 400 and 700 nm, including 413, 443, 469, 488, 531, 551, 555, 645, 667, and 678 nm. The retrieved IOPs included aph(λ) and adg(λ) of these wavelengths. Figure 9 shows the comparison of retrieved and measured aph and adg of the three different types of water. The concentrations of SPM and Chl a are shown in the graphs. As shown in Figure 9, this model tends to be more suitable for general waters as it is more effective in obtaining aph when applied to general water than eutrophic water. The values of obtained adg are closer to those of measured adg than aph. Thus, the accuracy of derived aph in all water conditions needs to be improved especially in the case of turbid waters.

Derived Values at Typical Wavelengths
To present a general description of the performance of this algorithm, we used 25 field-measured test samples to derive a(λ), aph(λ), and adg(λ) at 410 nm, 440 nm, and center wavelengths of MODIS bands between 400 and 700 nm, then compared them with measured data. Table 7, Figure 10, and Figure 11 show the results of this analysis.  Figure 10 shows the total absorption coefficients of typical wavelengths such as 440, 488, and 555 nm.  As shown in Figure 9, this model tends to be more suitable for general waters as it is more effective in obtaining a ph when applied to general water than eutrophic water. The values of obtained a dg are closer to those of measured a dg than a ph . Thus, the accuracy of derived a ph in all water conditions needs to be improved especially in the case of turbid waters.

Derived Values at Typical Wavelengths
To present a general description of the performance of this algorithm, we used 25 field-measured test samples to derive a(λ), a ph (λ), and a dg (λ) at 410 nm, 440 nm, and center wavelengths of MODIS bands between 400 and 700 nm, then compared them with measured data. Table 7, Figures 10 and 11 show the results of this analysis.  Figure 10 shows the total absorption coefficients of typical wavelengths such as 440, 488, and 555 nm. As shown in Figure 9, this model tends to be more suitable for general waters as it is more effective in obtaining aph when applied to general water than eutrophic water. The values of obtained adg are closer to those of measured adg than aph. Thus, the accuracy of derived aph in all water conditions needs to be improved especially in the case of turbid waters.

Derived Values at Typical Wavelengths
To present a general description of the performance of this algorithm, we used 25 field-measured test samples to derive a(λ), aph(λ), and adg(λ) at 410 nm, 440 nm, and center wavelengths of MODIS bands between 400 and 700 nm, then compared them with measured data. Table 7, Figure 10, and Figure 11 show the results of this analysis.  Figure 10 shows the total absorption coefficients of typical wavelengths such as 440, 488, and 555 nm. Derived a (m -1 ) 440nm modeling 440nm test 488nm modeling 488nm test 555nm modeling 555nm test Figure 10. Comparison of improved QAA-derived total absorption coefficients a(λ) versus the measured total absorption coefficients a(λ) for wavelengths at 440, 488, and 555 nm. absorption coefficients were slightly higher than the measured total absorption coefficients. It was due to the errors between the estimated values of Y and the reference values of Y varying at different conditions. The selection of a reference wavelength may also have had some impact on this situation. Additionally, the errors in measurement of field data can also be responsible for the observed differences. As a result, the accuracy of this model to derive total absorption coefficients needs improvement in these aspects. Figure 11. Comparison of derived aph(λ) and adg(λ) versus measured aph(λ) and adg(λ) at 440, 488, and 555 nm.
The MRE values of derived aph at 440, 488, and 555 nm were 55.08%, 126.59%, and 184.72%, and the R 2 values were 0.76, 0.78, and 0.72, while the MRE values of adg were 24.64%, 27.52%, and 32.20% and the R 2 values were 0.38, 0.36, and 0.43 ( Figure 11). Comparison and analysis showed that the derived adg were moderately smaller than the measured values, but more errors were found at the retrieval of aph. When we compared the inversion of aph at three wavelengths, we easily concluded that the MREs of values at longer wavelengths were bigger than those at shorter wavelengths. The derived aph were mostly distributed below the 1:1 line, which means that the calculated values were larger than the measured values. Gelbstoff and detritus contribute substantially to the total absorption coefficients at 410 and 440 nm. In the direct decomposition of total a(λ) to aph(λ) and adg(λ), values of ζ and ξ were estimated. The errors in these sectors were transferred to the estimated values of adg(440), which then influenced the final inversion of this model. Thus, different wavelengths can be used in future studies to improve the accuracy of decomposition of total absorption coefficients. As this test included a limited range of water samples, it cannot represent the accuracy completely. Further detailed tests with measured data are desired and needed to improve this model to derive aph and adg optimally.

Error Propagation
Error propagation means errors that may occur at each step have varying effects on the analysis results. The errors in each step propagate to the next step in the step-by-step process. We analyzed the error propagation of some steps of this algorithm by using modeling data (Table 8). This part For these wavelengths, the MRE values of test data were 17.68%, 19.10%, and 18.71%, and R 2 values were 0.74, 0.75, and 0.88, which indicate a good correlation at the aforementioned wavelengths ( Figure 10). Some points were distributed below the 1:1 line, which shows that the derived total absorption coefficients were slightly higher than the measured total absorption coefficients. It was due to the errors between the estimated values of Y and the reference values of Y varying at different conditions. The selection of a reference wavelength may also have had some impact on this situation. Additionally, the errors in measurement of field data can also be responsible for the observed differences. As a result, the accuracy of this model to derive total absorption coefficients needs improvement in these aspects.
The MRE values of derived a ph at 440, 488, and 555 nm were 55.08%, 126.59%, and 184.72%, and the R 2 values were 0.76, 0.78, and 0.72, while the MRE values of a dg were 24.64%, 27.52%, and 32.20% and the R 2 values were 0.38, 0.36, and 0.43 ( Figure 11). Comparison and analysis showed that the derived a dg were moderately smaller than the measured values, but more errors were found at the retrieval of a ph . When we compared the inversion of a ph at three wavelengths, we easily concluded that the MREs of values at longer wavelengths were bigger than those at shorter wavelengths. The derived a ph were mostly distributed below the 1:1 line, which means that the calculated values were larger than the measured values. Gelbstoff and detritus contribute substantially to the total absorption coefficients at 410 and 440 nm. In the direct decomposition of total a(λ) to a ph (λ) and a dg (λ), values of ζ and ξ were estimated. The errors in these sectors were transferred to the estimated values of a dg (440), which then influenced the final inversion of this model. Thus, different wavelengths can be used in future studies to improve the accuracy of decomposition of total absorption coefficients. As this test included a limited range of water samples, it cannot represent the accuracy completely. Further detailed tests with measured data are desired and needed to improve this model to derive a ph and a dg optimally.

Error Propagation
Error propagation means errors that may occur at each step have varying effects on the analysis results. The errors in each step propagate to the next step in the step-by-step process. We analyzed the error propagation of some steps of this algorithm by using modeling data (Table 8). This part shows the performance assessment of absorption coefficients at wavelengths 410, 440 nm, and the center wavelengths of MODIS bands between 400 and 700 nm. As the table suggests, all the errors were from the empirical and semi-analytical algorithms. Through the steps to retrieve the total absorption coefficients, the MRE between calculated Y values and reference Y values was approximately 19.31%, which was due to the relationship of R rs , SPM concentration, and the value of Y. Therefore, this step consequently led to the errors of the calculated total absorption. The retrieval of a ph (410)/a ph (440) and a dg (410)/a dg (440) had errors of 12.62% and 2.74%, respectively, which showed that the main source of the errors of a dg was due to the calculation of a dg (440) and then extended to a dg in the full wavelength range. Also, the errors of derived total absorption coefficients affected the accuracy of a ph and a dg. At the same time, because the values of measured a ph at longer wavelengths and some samples were small, the MREs increased, thereby influencing the total MRE of a ph .

Comparison with QAA
As mentioned, QAA did not function well when applied to Lake Chaohu ( Figure 12). In the case below, especially at a wavelength longer than 600 nm, the values of a ph are almost less than zero, which was impossible for Lake Chaohu. Therefore, improving this algorithm is necessary.
center wavelengths of MODIS bands between 400 and 700 nm.
As the table suggests, all the errors were from the empirical and semi-analytical algorithms. Through the steps to retrieve the total absorption coefficients, the MRE between calculated Y values and reference Y values was approximately 19.31%, which was due to the relationship of Rrs, SPM concentration, and the value of Y. Therefore, this step consequently led to the errors of the calculated total absorption. The retrieval of aph(410)/aph(440) and adg(410)/adg(440) had errors of 12.62% and 2.74%, respectively, which showed that the main source of the errors of adg was due to the calculation of adg(440) and then extended to adg in the full wavelength range. Also, the errors of derived total absorption coefficients affected the accuracy of aph and adg. At the same time, because the values of measured aph at longer wavelengths and some samples were small, the MREs increased, thereby influencing the total MRE of aph.

Comparison with QAA
As mentioned, QAA did not function well when applied to Lake Chaohu ( Figure 12). In the case below, especially at a wavelength longer than 600 nm, the values of aph are almost less than zero, which was impossible for Lake Chaohu. Therefore, improving this algorithm is necessary. The algorithm in this article was further validated by the measured dataset collected in Lake Chaohu. The version, QAA_v6, tended to be more suitable for turbid coastal waters, so it was also applied to the same dataset ( Figure 13). The assessment results for retrieved absorption coefficients at some special wavelengths are summarized in Table 9. The RMSEs and MREs for the algorithm in this study were in the range of 0.21-1.06 sr −1 and 17.27-54.85%, while those of QAA_v6 were 0.99-4.17 sr −1 and 56.93-73.51%, respectively. Specifically, the RMSEs and MREs of the QAA_v6 were larger than those of the algorithm in this study. The values of total absorption coefficients derived from QAA_v6 were much smaller than measured data. The larger RMSEs' and MREs' values derived from QAA_v6 were mainly because of a short reference wavelength and inappropriate estimated formulas that do not work for turbid Case 2 waters. The a ph (443) was also retrieved based on QAA_v6 for comparison. Scatterplots of derived and measured a ph (443) are shown in Figure 13, and the evaluation indices are also demonstrated in Table 9. In general, we can conclude that the algorithm in this research had smaller errors of RMSE and MRE of 0.60 sr −1 and 54.85%, respectively, compared with those of 1.46 sr −1 and 56.93% from the a ph (443) estimated by QAA_v6. However, because the a ph derived from the algorithm in this study tended to be larger than measured data, at some points, a ph (443) derived from QAA-v6 were closer to measured data than those of this study. retrieved a (aph) versus measured data.
The algorithm in this article was further validated by the measured dataset collected in Lake Chaohu. The version, QAA_v6, tended to be more suitable for turbid coastal waters, so it was also applied to the same dataset ( Figure 13). The assessment results for retrieved absorption coefficients at some special wavelengths are summarized in Table 9. The RMSEs and MREs for the algorithm in this study were in the range of 0.21-1.06 sr -1 and 17.27%-54.85%, while those of QAA_v6 were 0.99-4.17 sr -1 and 56.93%-73.51%, respectively. Specifically, the RMSEs and MREs of the QAA_v6 were larger than those of the algorithm in this study. The values of total absorption coefficients derived from QAA_v6 were much smaller than measured data. The larger RMSEs' and MREs' values derived from QAA_v6 were mainly because of a short reference wavelength and inappropriate estimated formulas that do not work for turbid Case 2 waters. The aph(443) was also retrieved based on QAA_v6 for comparison. Scatterplots of derived and measured aph(443) are shown in Figure 13, and the evaluation indices are also demonstrated in Table 9. In general, we can conclude that the algorithm in this research had smaller errors of RMSE and MRE of 0.60 sr -1 and 54.85%, respectively, compared with those of 1.46 sr -1 and 56.93% from the aph(443) estimated by QAA_v6. However, because the aph derived from the algorithm in this study tended to be larger than measured data, at some points, aph(443) derived from QAA-v6 were closer to measured data than those of this study.

MODIS Data Inversion
We used the algorithm proposed in this study to calculate the aph at 443, 645, and 678 nm of Lake Chaohu in 2013 by applying corrected MOD09 data ( Figure 14). Because of the difference of bands, we used MODIS bands 8 and 9 data to be the input parameters at 410 and 440 nm. The missing parts of the image were due to saturation that usually occurs at long wavelengths of 1-km resolution of this product. Even though there were some problems with this product, we still could verify the general seasonal changes of Lake Chaohu. Generally, aph varied largely in Lake Chaohu. High aph was observed in the northwestern parts of Lake Chaohu in the summer and autumn, and aph were relatively lower in the spring and winter. This pattern is consistent with the known phenomenon. Cyanobacteria blooms occur mainly in the western area in the summer, and the large amount of cyanobacteria in the western part is due to the exogenous load of the lakes mainly from the northwest of the basin. The suitable temperature of 20-34 ℃ in summer, the higher N and P concentrations, the higher PH value, the appropriate light intensity, and other environmental conditions provide the perfect environment for the growth of cyanobacteria. Therefore, aph tends to be higher in the western part in summer.   Figure 13. Comparison of measured and derived total absorption coefficients at (a) 443 nm and (b) 555 nm, and a ph at (c) 443 nm for applying the algorithm in this study and QAA_v6, respectively.

MODIS Data Inversion
We used the algorithm proposed in this study to calculate the a ph at 443, 645, and 678 nm of Lake Chaohu in 2013 by applying corrected MOD09 data ( Figure 14). Because of the difference of bands, we used MODIS bands 8 and 9 data to be the input parameters at 410 and 440 nm. The missing parts of the image were due to saturation that usually occurs at long wavelengths of 1-km resolution of this product. Even though there were some problems with this product, we still could verify the general seasonal changes of Lake Chaohu. Generally, a ph varied largely in Lake Chaohu. High a ph was observed in the northwestern parts of Lake Chaohu in the summer and autumn, and a ph were relatively lower in the spring and winter. This pattern is consistent with the known phenomenon. Cyanobacteria blooms occur mainly in the western area in the summer, and the large amount of cyanobacteria in the western part is due to the exogenous load of the lakes mainly from the northwest of the basin. The suitable temperature of 20-34°C in summer, the higher N and P concentrations, the higher PH value, the appropriate light intensity, and other environmental conditions provide the perfect environment for the growth of cyanobacteria. Therefore, a ph tends to be higher in the western part in summer. Remote Sens. 2020, 12, x FOR PEER REVIEW 19 of 22

Conclusions
Based on the field-measured data of Lake Chaohu, this study improved the QAA algorithm to provide an effective inversion of the IOPs of Lake Chaohu according to MODIS bands. The appropriate reference wavelength was shifted to 748 nm according to the measured data from the lake, and the applicable empirical model of the Y value was established by building models with SPM concentration and Rrs. The adg and aph were also derived by changing important parameters according to Lake Chaohu. To test the accuracy of this model, we applied this algorithm to a test dataset. This algorithm tends to be more suitable for general waters. It works better in the retrieval of total absorption coefficients in the condition of Lake Chaohu than original QAA and QAA_v6 do. The derived adg of this algorithm tend to be smaller than measured data and the derived aph tend to be bigger than measured values at some points. We also used the corrected MOD09 data to calculate aph at 443, 645, and 678 nm by the model proposed in this study. It shows that, in summer and autumn, aph tend to be higher in the northwestern part of Lake Chaohu, which is similar to the previous studies.
More independent tests with field measurement are required for validating and improving the algorithm. This algorithm needs to be improved in several aspects. First, the accuracy of the empirical model for calculating the Y value can be developed because it is one of the errors of the derived total absorption coefficients. Secondly, basic wavelengths can be changed to derive accurate aph and adg.

Conclusions
Based on the field-measured data of Lake Chaohu, this study improved the QAA algorithm to provide an effective inversion of the IOPs of Lake Chaohu according to MODIS bands. The appropriate reference wavelength was shifted to 748 nm according to the measured data from the lake, and the applicable empirical model of the Y value was established by building models with SPM concentration and R rs . The a dg and a ph were also derived by changing important parameters according to Lake Chaohu. To test the accuracy of this model, we applied this algorithm to a test dataset. This algorithm tends to be more suitable for general waters. It works better in the retrieval of total absorption coefficients in the condition of Lake Chaohu than original QAA and QAA_v6 do. The derived a dg of this algorithm tend to be smaller than measured data and the derived a ph tend to be bigger than measured values at some points. We also used the corrected MOD09 data to calculate a ph at 443, 645, and 678 nm by the model proposed in this study. It shows that, in summer and autumn, a ph tend to be higher in the northwestern part of Lake Chaohu, which is similar to the previous studies.
More independent tests with field measurement are required for validating and improving the algorithm. This algorithm needs to be improved in several aspects. First, the accuracy of the empirical model for calculating the Y value can be developed because it is one of the errors of the derived total absorption coefficients. Secondly, basic wavelengths can be changed to derive accurate a ph and a dg .
Combining the measured backscattering coefficients with the derived backscattering coefficients in future research will help improve the accuracy of this algorithm.