Accurate estimation of the diffuse attenuation coefficient is critical for our understanding and modelling of key physical, chemical, and biological processes in water bodies. For extremely turbid, shallow, Lake Taihu in China, we synchronously monitored the diffuse attenuation coefficient of photosynthetically active radiation (Kd(PAR)) and the remote sensing reflectance at 134 sites. Kd(PAR)) varied greatly among different sites from 1.62 to 14.68 m−1 with a mean value of 5.62 ± 2.99 m−1. A simple optical model from near-infrared remote sensing reflectance of MODIS channels 2 (859 nm) and 15 (748 nm) was calibrated, and validated, to estimate Kd(PAR). With the simple optical model, the root mean square error and mean relative error were 0.95 m−1 and 17.0% respectively at 748 nm, and 0.98 m−1 and 17.6% at 859 nm, based on an independent validation data set. Our results showed a good precision of estimation for Kd(PAR) using the new simple optical model, contrasting with the poor estimations derived from existing empirical and semi-analytical models developed in clear, open ocean waters or slightly turbid coastal waters. Although at 748 nm the model had slightly higher precision than at 859 nm, the spatial resolution at 859 nm was four times that at 748 nm. Therefore, we propose a new model based on the MODIS-derived normalized water-leaving radiances at a wavelength of 859 nm, for accurate retrieval of Kd(PAR) in extremely turbid, shallow lakes with Kd(PAR) larger than 1.5 m−1.
© 2012 OSA
The diffuse attenuation coefficient is defined in terms of the exponential decrease of the ambient irradiance with depth, and thus the degree of attenuation of the irradiance. The diffuse attenuation coefficient depends on both the composition of the medium, and the directional structure of the ambient light field, hence it is classified as an apparent optical property of water . Four optically significant substances control the diffuse attenuation coefficient, namely tripton (non-phytoplankton particles), phytoplankton, chromophoric dissolved organic matter (CDOM), and water itself.
Accurate estimation of the diffuse attenuation coefficient is critical for our understanding and modeling of physical processes such as sediment resuspension , chemical processes such as photobleaching , and biological processes such as phytoplankton photosynthesis in the euphotic zone . Furthermore, knowledge of the spatial-temporal variation of the diffuse attenuation coefficient is needed to estimate the primary production of a lake , and to take management actions to restore an underwater light climate that permits the growth of submerged aquatic vegetation .
Therefore, the diffuse attenuation coefficient has been an important parameter for ocean color satellite 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).
Three main types of models are used to estimate the diffuse attenuation coefficient, as summarized in Table 1 . (1) Empirical relationships between the diffuse attenuation coefficient and a single wavelength, or different wavelength ratios of the normalized water-leaving radiance (or remote sensing reflectance), such as the blue-green and blue-red (Eqs. (1)-(5) in Table 1) [7–13]. (2) The empirical relationship between the diffuse attenuation coefficient and the chlorophyll a concentration is derived through regression analyses (Eqs. (6), (7) in Table 1) . (3) Semi-analytical approaches based on radiative transfer models, as proposed for coastal waters; these models are generally applicable to clear, open ocean waters or slightly turbid coastal waters (Eqs. (8), (9) in Table 1) [10, 15, 16].
There are relatively few reports of remote sensing estimation of the diffuse attenuation coefficient in lakes, especially for extremely turbid lakes . The actual measured diffuse attenuation coefficients in extremely turbid, shallow lakes such as Lake Taihu (the third largest freshwater lake in China), are approximately 1-2 orders of magnitude higher than those in clear, open ocean waters or in slightly turbid coastal waters [2, 10, 15].
This large difference in the measured coefficients between the different types of water strongly suggests that the models developed for clear, open ocean waters, or slightly turbid coastal waters, might not be applicable to extremely turbid, shallow lakes. To address this concern, the objective of our present study, using Lake Taihu as an example of an extremely turbid and shallow lake, was to develop a simple model to estimate the diffuse attenuation coefficient using surface reflectance data, and compare the estimation precision with the previous models.
2 Materials and methods
2.1 Study sites and sampling schedule
Lake Taihu is a large, shallow, eutrophic lake with a surface area of 2,338 km2 and a mean depth of 1.9 m, with high spatial variability, including regions that are dominated by algae or macrophytes . The waters in most of Lake Taihu are consistently highly turbid, due to the shallow water and frequent sediment resuspension caused by wind waves . However, in East Lake Taihu and some of the East Lake regions (Fig. 1 ), the waters are often slightly turbid, with benthic submerged aquatic vegetation .
The optical measurements were made, and the water samples were collected, at 50 cm depth below the surface along 4 transects comprising a total of 50 sites covering different regions of the lake (Fig. 1). In order to get the high-quality in situ remote sensing reflectance and underwater PAR data, measurements were carried out under clear sky conditions with no wind or very low winds during the hours of 8:30–16:30. The mean wind speed for five cruises was 2.11 m/s during 7–9 January 2006, 0.45 m/s during 29 July–1 August 2006, 2.51 m/s during 12–15 October 2006, 2.91 m/s during 7–9 January 2007, and 2.90 m/s during 25–27 April 2007 .
2.2 Measurement of remote sensing reflectance and diffuse attenuation coefficient
Downwelling radiance and upwelling total radiance measurements were made with an ASD field spectrometer (Analytical Devices, Inc., Boulder, CO) with a spectral response of 350 to 1050 nm, a spectral resolution of 3 nm, and a sampling interval of 1 nm. The “above water method” was used to measure water surface spectra.
An optical fiber was positioned at nadir on a mount extending about 1 m away from the boat, to reduce the influence of reflectance from the vessel on the collected spectra. The radiance spectra from the reference panel (Lp(λ, 0+)), water (Lsw(λ, 0+)), and sky (Lsky(λ)) were measured approximately 0.3 m above the water surface under clear sky conditions. At each sampling site, the spectra were measured 10 times to optimize the signal-to-noise ratio, and thus reduce the error of in situ measurements. Each spectrum was sampled 90° azimuthally from the sun, and at a nadir viewing angle of 40°, to avoid the interference of the ship with the water surface and the influence of direct sunlight.
The water-leaving radiance Lw(λ, 0+) can be derived from the following equation:
The incident downwelling irradiance Ed(λ, 0+) was determined by measurement of the radiance of the Lambertian reference panel Lp(λ, 0+) as follows:
The remote sensing reflectance above the water surface Rrs(λ) was calculated as the ratio of water-leaving upwelling radiance Lw(λ, 0+) to incident downwelling irradiance Ed(λ, 0+). A total 250 Rrs(λ) (50 sites × 5 cruises) were collected. Some Rrs(λ) spectra were excluded from the data set if the sites had a thick algal bloom or macrophytes.
Underwater PAR measurements were taken at a subset of sites of Rrs(λ) measurement, and 134 PAR diffuse attenuation coefficient estimates was obtained in five cruises including 81 sites with spectral diffuse attenuation coefficients. Downwelling PAR was measured on the sunny side of the boat, just below the water surface (0- m) and at other six depths (0.20, 0.50, 0.75, 1.00, 1.50 and 2.00 m), using a Li-Cor 192SA cosine corrected underwater quantum sensor connected to a Li-Cor 1400 datalogger (http://www.licor.com). At each depth, 3 recordings were made, and their mean value was considered as the PAR intensity for that depth, to minimize the effect of wind and waves on underwater PAR measurement.
The spectra of underwater downward irradiance just below the water surface (0- m) and at different depths from 0 to 1 m were measured using an underwater spectroradiometer with a SAM 8180 sensor to obtain the spectral diffuse attenuation coefficients . The diffuse attenuation coefficients for the downward PAR and spectral irradiance were determined as the slope of log-transformed profile underwater irradiance data . Only diffuse attenuation coefficient values from regressions with r2≥0.99 were accepted. The number of depths used in these regressions was 4-11, depending on the penetration depth.
2.3 Data analysis
To calibrate and validate our Kd(PAR) estimation model, this Kd(PAR) data set was randomly divided into two groups across all seasons, namely the calibration data set (n = 90) and the validation data set (n = 44).
To assess the precision of using the MODIS data to estimate Kd(PAR), we used the central wavelength values of remote sensing reflectance measured in the MODIS spectral bands (channel 2: 859 nm with 250 m spatial resolution, channel 15: 748 nm with 1000 m spatial resolution) to calibrate and validate the Kd(PAR) estimation model, and the model values were compared with measured Kd(PAR).
Statistical analysis (mean value, linear and non-linear fitting) were performed with SPSS 17.0 software (Statistical Program for Social Sciences). Regression and correlation analyses were used to examine the relationships between variables using SPSS software. Significance levels are reported as not significant (p > 0.05), or significant (p < 0.05). The performance of the retrieval model was evaluated by the regression determination coefficient (r2), the root mean square error (RMSE), and the mean relative error (MRE). The calculations of RMSE and MRE were the same as the previous study .
3.1 Variation in Kd(PAR) and correlation between Kd(490) and Kd(PAR)
The depth profiles of the underwater PAR for three sites, and the underwater spectral irradiance for a site are shown in Fig. 2 . The high determination coefficient (>0.99) for the linear fitting between the natural logarithm values of PAR intensity, and spectral irradiance intensity vs depth, confirmed that PAR and spectral irradiance intensity exponentially attenuated with the increasing water depth. There were significant differences in PAR attenuation between sites. Kd(PAR) ranged from 1.62 to 14.68 m−1 with a mean value of 5.62 ± 2.99 m−1 (mean ± standard deviation). There was a difference in PAR attenuation of almost one order of magnitude between sites.
There was a significant positive correlation between Kd(490) and Kd(PAR) (Kd(PAR) = 0.896Kd(490)0.873, r2 = 0.98, n = 81, p<0.001) (Fig. 3 ), which was used to calculate Kd(PAR) from Kd(490) because the published empirical and semi-analytical models estimated Kd(490) using the reflectance data. The correlation express form was the same as the previous study but the empirical coefficients were different .
3.2 Estimation model of Kd(PAR): calibration
The calibration data set contained 90 water samples, with Kd(PAR) ranging from 1.62 to 14.68 m−1 with a mean value of 5.63 ± 3.05 m−1. To find the best wavelength band, or band ratio, by which to estimate Kd(PAR) in the extremely turbid Lake Taihu, the single bands and the ratios of any two wavelengths from 400 to 900 nm were tested for correlation with Kd(PAR) based on the linear and quadratic algorithms. Because the diffuse attenuation coefficient depends on the sun angle and the angular distribution of the light field , we used the ratio of Rrs(λ) to the mean cosine of the angles the photons make with the vertical just beneath the water surface (μ0) to correlate with Kd(PAR), in order to minimize the effect of the sun angle. The value of μ0 is dependent upon the solar elevation and the proportion of direct and diffuse radiation. μ0 is calculated according to the sampling time, latitude and solar declination.
Four algorithms showed a reasonable correlation with Kd(PAR) within wavelengths from 700 to 900 nm. The simple optical model showed that two peaks near 740 nm and 835 nm, and two valleys near 760 nm and 805 nm were found for the determination coefficient. Rrs(λ, 0+) at the two peaks could be used to estimate Kd(PAR) (Fig. 4 ). The trends of MRE and RMSE were in contrast to those of the determination coefficients. r2 was significantly higher, and MRE and RMSE were significantly lower, using Rrs(λ)/μ0 than Rrs(λ) (One-Way ANOVA, p<0.001), showingthat Kd(PAR) could be accurately estimated using Rrs(λ)/μ0 considering the effect of the sun angle.
The quadratic algorithm gave the best precision, with the highest determination coefficient and the lowest MRE and RMSE, using Rrs(λ)/μ0 and Kd(PAR) (Fig. 4). However, the quadratic algorithm was only slightly better than the linear algorithm, without significant differences in r2, MRE and RMSE (One-Way ANOVA, p>0.1). Therefore, the linear algorithm between Rrs(λ)/μ0 and Kd(PAR) is recommended, considering the simplicity of the model. In addition, the significant correlations between Kd(PAR) and Rrs(λ) at the red and near-infrared bands were similar to those between total suspended matter concentration and Rrs(λ) [20, 21] because Kd(PAR) was controlled by the tripton concentration in the extremely turbid lake waters.
To apply the new model presented in this study to the MODIS data, we chose channel 2 at 859 nm with 250 m spatial resolution, and channel 15 at 748 nm with 1000 m spatial resolution, to calibrate and validate the Kd(PAR) estimation model. Because Rrs(859) and Rrs(748) are highly linearly correlated to the integrated values of channels 2 and 15, with r2 higher than 0.999, respectively, the estimation precision using the central wavelength values at 859 nm and 748 nm is the same as using the integrated values of channels 2 and 15. The model Eqs. (10), (11) and error test are presented in Fig. 5 and Table 1. The simple optical model was highly significant, with r2, RME and RMSE of 0.866, 15.1%, and 1.11 m−1 at 859 nm (Fig. (5a)), and 0.874, 15.0%, and 1.08 m−1 at 748 (Fig. (5b)), respectively, showing that the MODIS data could be used to estimate Kd(PAR).
3.2 Estimation model of Kd(PAR): validation
To further understand the applicability of the simple optical model to estimate Kd(PAR), we evaluated its performance using a validation data set of 44 samples. Validation is fundamental in the development of an estimation procedure, to evaluate the overall reliability of the retrieval scheme, and characterize the uncertainty associated with the estimated values. The model performance must be evaluated using a validation data set which has not been used in the calibration phase. To validate our model, we used values for Kd(PAR) from 1.85 to 13.47 m−1, with a mean of 5.58 ± 2.91 m−1, which fell into the range of Kd(PAR) used to calibrate the model.
Comparisons of the measured and estimated Kd(PAR) using the calibrated simple optical model showed that these values were in close agreement, with a highly significant linear relationship, with an r2 of 0.884 at 859 nm and of 0.893 at 748 nm. The measured and estimated values for Kd(PAR) were distributed along the 1:1 line (Fig. (6a) , (6b)), indicating that the simple optical model could be used for the extremely turbid waters of Lake Taihu.
Overall, the precision of the model at 748 nm was slightly higher than that at 859 nm, for both the calibration and validation data sets (Figs. 5 and 6). However, at the 859 nm wavelength, the MODIS-measured high spatial remote sensing reflectance data (250 m) is particularly useful for estimation of Kd(PAR) in inland lake waters, due to the relatively small area, and high spatial variation in the optical and bio-optical properties of the water . Wang et al.  suggested that the MODIS-derived water-leaving radiance at the visible band of 859 nm compared reasonably well with the in situ measurements, based on the shortwave infrared iterative atmospheric correction algorithm for Lake Taihu water. The MODIS-derived water-leaving radiance at 859 nm in Lake Taihu was significant over most of the lake, and could be accurately accounted for in satellite remote sensing . Thus the application of the proposed model to the MODIS data in our study was feasible. We recommend the use of the MODIS data at 859 nm to estimate Kd(PAR) in extremely turbid, shallow lake waters.
4.1 Variation of Kd(PAR) in the extremely turbid waters
The high Kd(PAR) values ranging from 1.62 to 14.68 m−1 observed in the present study reflects the high turbidity of Lake Taihu, and are similar to the results found in other shallow turbid inland waters [22, 23]. In the present study, we measured the underwater PAR at only 4-7 depths because the lake was very shallow, and there were frequent waves and wind. These 4-7 depths are markedly fewer than are typically used in studies in deep lakes, estuaries and oceans . It could be argued that the numbers of measurement depth are limited because the lake was very shallow and the water was highly turbid, thus the PAR intensity was less than the detection limitation at the depth less than 1.5 or 2.0 m. Our fitting error of Kd(PAR) was probably very small, considering the high determination coefficients (r2> 0.99).
4.2 Assessment and application of the simple optical model
In order to compare the estimation precision of our simple optical model with that of more complex previous models, we firstly calibrated the coefficients for the same model expressions of Eqs. (2), (4), (6) using the calibration data set in Lake Taihu. There were no correlations between Kd(490) and chlorophyll a concentration, the ratio Rrs(490)/Rrs(555). Thus we only get the calibrated Eq. (12) using Rrs(670)/Rrs(490) with a determination coefficient of 0.54 in Table 1.
We assessed the two published empirical and semi-analytical models (Eqs. (8), (12) in Table 1) by using the same independent validation data set we used to validate our own model. In the published models, Kd(490) was used as an indicator of water turbidity, and could generally be used as a surrogate for Kd(PAR). Although some studies have shown that Kd(490) was robustly correlated to Kd(PAR), the relationship has quite wide regional variations [10, 14]. In addition, those relationships were usually derived from clear or slight turbid waters. In the present study, Kd(PAR) was calculated using Kd(490) based on the former correlation in Fig. 3 developed in the turbid waters, with a determination coefficient of 0.980 for the corresponding fitting.
Significant differences were found between the estimated and measured Kd(PAR) values, with RME and RMSE of 61.5%, 4.68 m−1, and 26.7%, 2.08 m−1 for Eqs. (8), (12). For Kd(PAR) model developed for the slight turbid waters , the model-derived Kd(PAR) values underestimated the in situ Kd(PAR) values by a factor of 1.4-6.3. In addition, Wang et al.  presented monthly climatology images of the MODIS-Aqua derived Kd(490) in Lake Taihu with Kd(490) less than 4.0 m−1, corresponding to Kd(PAR) less than 3.0 m−1 using the semi-analytical model developed in the slightly turbid Chesapeake Bay waters. The estimated Kd(PAR) by Wang et al.  was far lower than the actual measured Kd(PAR) in Lake Taihu.
The results from the semi-analytical model used in the slightly turbid ocean regions were better than results from the empirical models used in the clear ocean waters. However, the empirical and semi-analytical models developed in clear or slightly turbid ocean waters cannot be used in extremely turbid, shallow lake waters. For example, the mean value of the in situ Kd(PAR) of our independent data set was more than three times higher than the mean Kd(PAR) derived from the other published models. The apparent underestimations of the Kd(PAR) from models derived from clear or slightly turbid ocean waters, may have been due to the variations in the range of Kd(PAR). Kd(490) was in the range of 0.001-3.5 m−1 corresponding to Kd(PAR) in the range of 0.001-2.6 m−1 in clear or slightly turbid ocean waters [10, 12, 13]. However, in the present study we report that Kd(PAR) ranged from 1.62 to 14.68 m−1 in Lake Taihu, with only 12 Kd(PAR) values less than 2.6 m−1 (9.0%), which was significantly higher than those in clear or slightly turbid ocean waters.
The empirical and semi-analytical models developed in clear and slightly turbid ocean waters successfully used remote sensing reflectance less than 700 nm, using the ratios of blue-green (490/555) and blue-red (490/665, 490/709, 488/667, 488/645) [8, 10, 12, 13]. However, our calibration model showed that in the extremely turbid waters of Lake Taihu, remote sensing reflectance less than 700 nm could not be used to appropriately estimate Kd(PAR) (Fig. 4). The wavelength combinations developed in the clear ocean or lake waters cannot be used to appropriately estimate Kd(PAR) in the extremely turbid waters of Lake Taihu. The inconsistency of estimation model of Kd(PAR) are attributed to the differences of the optical properties (spectral absorption and scattering) and dominant attenuators in the clear Case 1 and turbid Case 2 waters. In clear ocean or lake waters, phytoplankton generally dominates the absorption, scattering, and attenuation of spectral irradiance and PAR. For example, chlorophyll a concentration can explain 91.7% variation in Kd(490) based on a large data set (n = 712) in open ocean (Case 1) waters . In contrast, in the extremely turbid, shallow lake waters, tripton (non-phytoplankton particles) dominated the absorption, scattering and attenuation of spectral irradiance and PAR [6, 22, 23]. In Lake Taihu, tripton concentration can explain 97.5% variation in Kd(PAR) (n = 67) . Result showed that tripton concentration was estimated from near-infrared remote sensing reflectance larger than 700 nm in the extremely turbid waters .
Significant differences in the bio-optical properties in the different waters determined that the new estimation model of Kd(PAR), based on remote sensing reflectance larger than 700 nm, needed to be calibrated and validated for extremely turbid lake waters when used in other turbid waters. Remote sensing algorithms for Kd(PAR) in extremely turbid waters are rare, contrasting with the many algorithms developed and used in clear or slightly turbid ocean waters [8, 12–16]. Our new algorithm thus has great potential in the study of physical, chemical and biological processes in the extremely turbid, shallow, lakes. It must be emphasized, however, that this algorithm may, at present, only be used for regional applications with Kd(PAR) larger than 1.5 m−1 because our data set with Kd(PAR) larger than this value In regions with different sediment types and bio-optical conditions, the coefficients of the simple optical model need to be calibrated. In addition, caution should be exercised when using the model if there are marked algal bloom and macrophytes signals on the MODIS image, because our model excluded the sites with thick algal bloom or macrophytes.
The successful application of this algorithm will allow the study of Kd(PAR) distributions in Lake Taihu over spatial and temporal scales not previously possible. In addition, the algorithm of Kd(PAR) will facilitate the estimation of euphotic depth, which is not only a quality index of an ecosystem , but also an important input parameter for primary production estimation  and heat transfer  in the upper water column. Since primary production has been correlated to fisheries yield [26, 27], future primary production estimation should provide useful information for fisheries management in relation to spatial and temporal changes in this important sub-tropical lake. Primary production estimation from remotely sensed estimates of optical properties Kd(PAR) and phytoplankton biomass (chlorophyll-a) will also present new insight for the regional lake carbon cycle and climate change.
The new simple optical model of Kd(PAR) using the MODIS channel 2 with 250 m spatial resolution was superior for application in a turbid lake when compared to the published empirical and semi-analytical models developed in clear or slightly turbid ocean waters. Thus, the simple optical model improved the Kd(PAR) estimation precision in the extremely turbid inland waters. The new simple optical model at 859 nm allowed estimation of Kd(PAR) with the MRE and RMSE of 17.6% and 0.98 m−1 respectively, whereas the published empirical and semi-analytical models had the RME and RMSE greater than 61.5% and 4.68 m−1 respectively, using an independent validation data set. Further study may include determining the temporal-spatial distribution of Kd(PAR) using the MODIS data at 859 nm based on the simple optical model calibrated and validated in this study, estimation of primary production, and investigating the interrelationship between Kd(PAR) and the spatial distribution of submerged aquatic vegetation in Lake Taihu. Caution should be taken in applying the new algorithm in other waters because this model was fitted for waters with Kd(PAR) lager than 1.5 m−1.
This study was jointly supported by the Provincial Nature Science Foundation of Jiangsu of China (BK2012050), the Major Projects on Control and Rectification of Water Body Pollution (2012ZX07101-010), and the National Natural Science Foundation of China (Grant No. 41271355, 40825004). We especially thank J. S. Li, S. Feng, X. Wang, Q. H. Zhao, and H. Zhang for their help with field sample collection.
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