# On the Adequacy of Representing Water Reflectance by Semi-Analytical Models in Ocean Color Remote Sensing

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Reflectance Models

^{−1}(λ is the wavelength). This allows applicability to Case 2 water situations. The model input also includes sun and sensor viewing geometry and wind speed. Specifically, the input geometry was set to match that of the specific dataset described later and the default value of wind speed was used, i.e., 5 m/s, as done in POLYMER. The output of this model is the spectral irradiance reflectance just below the surface, from which remote sensing reflectance, R

_{rs}, is deduced. In the transformation, the f/Q correction is performed according to Morel and Gentili [21]. The similarity spectrum of Ruddick et al. [22] is used to extend the model from 700 to 900 nm.

_{w}), defined as πL

_{w}/E

_{d}where L

_{w}is water-leaving radiance and E

_{d}the downward solar irradiance just above the surface, from which R

_{rs}= L

_{w}/E

_{d}is deduced by dividing by π.

#### 2.2. Inversion Scheme

_{rs}at a set of wavelengths in the 400–900 nm spectral range using, as in POLYMER, the Nelder–Mead optimization scheme [24]. A simplex method is used to minimize the cost function f, which is defined as:

_{rs}(λ

_{i}) − πR

_{rs}(λ

_{i}))

^{2}/norm, norm = 0.005 if πR

_{rs}(λ

_{i}) < 0.005, otherwise norm = πŘ

_{rs}(λ

_{i}),

_{rs}(λ

_{i}) is the modeled remote sensing reflectance at wavelength λ

_{i}, and R

_{rs}(λ

_{i}) is the Hydrolight simulated or field measured remote sensing reflectance at wavelength λ

_{i}. For the MM01 model, the first iteration simplex is defined by initial values log

_{10}([Chla]) = −1 and bbs = 0, and initial steps 0.05 and 0.0005 m

^{−1}. For the PR05 model, the first iteration simplex is defined by initial values log

_{10}([Chla]) = −1, log

_{10}(fb) = 0 and initial steps 0.2 and 0.2 when using two parameters, or by initial values log

_{10}([Chla]) = -1, log

_{10}(fb) = 0, log

_{10}(fa) = 0 and initial steps 0.2, 0.2, and 0.2 when using three parameters. The optimization scheme stops when f reaches a threshold value of 0.005 or the iteration reaches a maximum number.

#### 2.3. Data Source

- Phytoplankton absorption. The phytoplankton absorption, a
_{ph}, is expressed as a_{ph}= [Chla] a^{*}_{ph}, where a^{*}_{ph}is the chlorophyll-specific absorption coefficient, both provided in the IOCCG dataset. The values of a^{*}_{ph}at 350–400 nm were adopted from Morel [25] and extrapolated to 300 nm, which was then normalized to the IOCCG a^{*}_{ph}value at 400 nm to ensure continuity. The a^{*}_{ph}values in the 800-900 nm range were assumed constant and fixed at the value at 800 nm. - CDOM absorption. The CDOM absorption, a
_{g}, was modeled as a_{g}(λ) = a_{g}(440) exp(−S_{g}(λ − 440)), with S_{g}and a_{g}(440) provided by IOCCG [18]. - Detritus/mineral absorption. The absorption of detritus/mineral, a
_{dm}, was modeled as a_{dm}(λ) = a_{dm}(440) exp(−S_{dm}(λ − 440)), with S_{dm}and a_{dm}(440) provided by IOCCG [18]. - Backscattering of phytoplankton. The attenuation of phytoplankton, c
_{ph}_{,}was modeled as c_{ph}= c_{ph}(550) (550/λ)^{n1}in the IOCCG dataset. The value of n1 was determined using c_{ph}at 400–800 nm, which was then used to extend c_{ph}to 300-1000 nm. The backscattering of phytoplankton, bb_{ph}, was thus obtained using bb_{ph}= $\tilde{b}$_{ph}(c_{ph}− a_{ph}) with $\tilde{b}$_{ph}equal to 0.01. - Backscattering of detritus, mineral, and other particles. The backscattering of detritus, mineral, and others, bb
_{dm}, was modeled as bb_{dm}= bb_{dm}(550) (550/ λ)^{n2}. The value of n2 was determined using bb_{dm}at 400–800 nm, which was then used to extend bb_{dm}to 300–900 nm.

_{rs}, was further interpolated into MODIS wavelengths (412, 443, 488, 531, 547, 667, 678, 748, and 869 nm). Note that the IOCCG synthesized IOPs do not cover all possible natural waters. However, as the models and parameters of describing input IOPs are based on extensive field measurements of oligotrophic and eutrophic waters [8,29,30,31,32,33,34,35], the Hydrolight simulated data should be consistent with a wide range of field observations.

_{WN}(λ) at various wavelengths in the visible and near-infrared spectral regions is the primary ocean color radiometric product. Data at quality-assurance level 2.0, normalized to the geometry of the Sun at zenith and the view at nadir, was used. The level 2.0 data is fully quality controlled including pre- and post-field calibration with differences smaller than 5%, automatic cloud removal, and manual inspection. Conversion to R

_{rs}was accomplished by multiplying L

_{WN}(λ) by extraterrestrial solar irradiance. Note that uncertainties in the AERONET-OC in-situ measurements are about 5% in the blue to green bands, and about 8% in the red band [19]. The AERONET-OC measurements were collected using different radiometric instruments and thus the wavelengths used were not exactly the same. Since in this study R

_{rs}at MODIS wavelengths (see above) are considered, and since the simulated data set corresponds to those wavelengths, measurements at wavelengths located within the MODIS center wavelengths ± 3 nm were selected. At some of the AERONET-OC sites, in particular the Venise site (45°N,12°E), measurements at 547 ± 3 nm were missing. In the end measurements from a total of fourteen sites were used, including twelve coastal sites, i.e., COVE_SEAPRISM (36°N, 75°W), Gageocho_Station (33°N,124°E), Ieodo_Station (32°N,125°E), Socheongcho (37°N,124°E), Galata_Platform (43°N,28°E), GOT_SEAPRISM (9°N,101°E), LISCO (43°N,70°W), Lucinda (18°S,146°E), Thornton_C-Power (51°N,2°E), USC_SEAPRISM (33°N,118°W), WaveCIS_site_CSI_6 (28°N,90°W), and Zeebrugge-MOW1 (51°N,2°E), and two lake sites, i.e., Lake_Erie (41°N,83°W) and Palgrunden (58°N, 13°E). Additional criteria based on the similarity spectrum [22] were applied to check and remove outliers, i.e., only the data for which (i) R

_{rs}(869 nm) > −0.0001 sr

^{−1}and (ii) the ratio R

_{rs}(667 nm)/R

_{rs}(869 nm) > 3 if R

_{rs}(869 nm) >0.0008 sr

^{−1}were selected. This resulted in a total number of 9824 R

_{rs}measurements at wavelengths close to the MODIS wavelengths. For simplification, the AERONET-OC wavelengths will be referred to as 412, 443, 488, 531, 547, 667, and 869 nm, i.e., the MODIS wavelengths. The AERONET-OC measurements were made only under clear sky conditions. They were collected when wind speed is less than 15 m/s. Chlorophyll-a concentrations are provided in the AERONET-OC dataset, but not from field measurements; they are estimated through an iterative procedure making use of reflectance band ratios [19]. AERONET-OC sites for which band-ratio algorithms exist naturally benefit from those regional algorithms. For several AERONET-OC sites, the lack of regional algorithms imposes the application of the standard OC2V4 band-ratio. Because of this, the expected accuracy of AERONET-OC [Chla] varies from site to site.

## 3. Results and Discussion

#### 3.1. Model Performance with Hydrolight Simulations

_{p}(560), increases with increasing [Chla]. The upper limit of b

_{p}(560) can be expressed as

_{p}(560) = 0.69 [Chla]

^{0.766}

_{w}(560), as a function of [Chla]. The effect of solar zenith angle was taken into consideration. For a given [Chla], if the observed R

_{w}(560) exceeded the threshold R

_{w}(560), the water was determined as Case 2, otherwise as Case 1. Note that the threshold of Morel and Bélanger [36] only distinguishes Case 2 turbid water with excessive sediments and detritus and does not separate yellow substance-dominated Case 2 waters. Therefore, the determined Case 1 water situations still included yellow substance-dominated Case 2 water situations. An additional threshold of remote sensing reflectance at 670 nm, i.e., R

_{rs}(670) >0.0012 sr

^{-1}, now adopted as turbid water flag by the NASA Ocean Biology Processing Group (starting with the fourth processing), was also applied [37]. This second threshold may still not favor the identification of CDOM-dominated waters. With these two criteria, there are a total of 55 spectra classified as Case 1 and 445 spectra as Case 2 (Figure 1). Statistics including number of points (N), bias, root-mean-squared error (RMSE), and coefficient of determination (R

^{2}) were used to evaluate the model performance.

_{rs}spectra reconstructed using the MM01 and PR05 models. Only 2 variable parameters, i.e., [Chla] and fb, are used in PR05, with fa fixed to 1. The MM01 and PR05 spectra generally agree with the Hydrolight simulated spectra, especially for Case 1 water (Figure 2a–c). As [Chla] is increased (Figure 2b,c), the difference between reconstructed and simulated R

_{rs}around 680–700 nm becomes larger. This is expected since chlorophyll fluorescence effects are not considered in both models. The ability of the model to reproduce properly R

_{rs}is degraded for Case 2 water (Figure 2d–f). In the complex Case 2 situation of Figure 2f, one noticeable feature of the MM01 reconstructed R

_{rs}is the large discrepancy at 400–500 nm and 600–650 nm, which is attributed to neglecting CDOM absorption and crudely parameterizing the backscattering by non-algal particles. The disagreement around 550–650 nm between PR05 reconstructed and Hydrolight simulated R

_{rs}is due to the fact that the slope of CDOM absorption adopted in the PR05 model, modeled as a function of [Chla], does not reflect well the actual (i.e., prescribed) slope, which ultimately affects the red wavelengths since the spectral-matching scheme minimizes the overall difference.

_{rs}versus Hydrolight simulated R

_{rs}at selected wavelengths, i.e., 443, 488, 547, and 667 nm, for all cases. The MM01 model performs generally well in reproducing R

_{rs}at 443, 488, and 547 nm, with the performance for Case 1 water (Figure 3, blue) being better than that for Case 2 water (Figure 3, red), with overall bias and Root-Mean-Squared Error (RMSE) of 0.00038 (7.7%) and 0.00082 sr

^{−1}(16.5%) at 443 nm, −0.00016 (−2.7%) and 0.00070 sr

^{−1}(11.9%) at 488 nm, 0.00013 (2.1%) and 0.00075 sr

^{−1}(12.1%) at 547 nm, and −0.00039 (−19.3) and 0.00124 sr

^{−1}(61.4%) at 667 nm, respectively. The percent values in parentheses are biases or RMSEs normalized by the mean of the simulated data. The relatively high bias and RMSE for Case 2 water at 443 nm is due to the lack of CDOM absorption in the MM01 model and is consistent with Figure 2. The MM01 model performance is degraded at 667 nm. At this wavelength, large bias and RMSE are found for Case 2 water, although the reconstruction of R

_{rs}is still good for Case 1 water. The degradation at 667 nm is due primarily to the backscattering of non-algal particles, not being sufficiently well treated by introducing bbs, and secondarily to fluorescence and CDOM absorption not considered. Since the spectral-matching scheme utilizes information at all the input wavelengths, neglecting CDOM absorption, influential in the blue, may also impact results in the red. The PR05 model shows better performance than the MM01 model in reconstructing R

_{rs}at 443, 488, 547, and 667 nm, with bias and RMSE generally reduced, i.e., 0.00024 (4.8%) and 0.00038 sr

^{−1}(7.7%) at 443 nm, −0.00001 (−0.1%) and 0.00039 sr

^{−1}(6.8%) at 488 nm, −0.00030 (−4.9%) and 0.00061 sr

^{−1}(9.8%) at 547 nm, and 0.00015 (7.6%) and 0.00027 sr

^{−1}(13.4%) at 667 nm, respectively. The relatively large bias and RMSE at 547 nm, as compared to 443 and 488 nm, is due to the CDOM absorption slope, which is not adequately represented, as well as the spectral-matching scheme that minimizes the overall difference. The reconstructed R

_{rs}at 667 nm is in good agreement with the Hydrolight values, which is consistent with the results shown in Figure 2. The improvement is dramatic at this wavelength when using PR05 instead of MM01 (e.g., RMSE reduced from 61.4% to 13.4%).

_{rs}more accurately for Case 1 water than for Case 2 water from 412 nm to 547 nm as the bias and RMSE values are lower, which is consistent with Figure 2 and Figure 3. The bias and RMSE for Case 1 water are less than 4.0% and 6.0%, respectively, while the bias can increase to 17.0% and the RMSE to 26.1% for Case 2 water. As wavelength becomes longer, i.e., from 667 nm to 869 nm, the bias and RMSE error increase for both Case 1 and Case 2 water situations. The reconstruction of R

_{rs}at near infrared wavelengths, i.e., 748 and 869 nm, show large bias and RMSE (up to 45.6% and 78.3%, respectively). This is due mostly to the simplified treatment of non-algal particles via the addition of a backscattering coefficient with spectral dependence in λ

^{−1}(and no absorption). Extending R

_{rs}from 700 to 900 nm using the average similarity spectrum for turbid waters may also introduce errors. Similarly, the PR05 model generates more accurate R

_{rs}estimation for Case 1 water than for Case 2 water from 412 nm to 547 nm. At these wavelengths, the largest bias observed is 2.1% and the largest RMSE 4.8% for Case 1 water, and the bias and RMSE increase up to 5.2% and 9.7%, respectively, for Case 2 water. The bias and RMSE errors increase (up to 13.7% and 22.2%, respectively) at 667 and 678 nm, indicating significant degradation in R

_{rs}reconstruction at these wavelengths. Much larger relative errors are observed at the near infrared wavelengths, for both Case 1 and Case 2 situations, i.e., an RMSE of 47.3 and 21.7%, respectively, at 748 nm and 102.0 and 72.6%, respectively, at 869 nm, which reflects the generally lower R

_{rs}, especially at 869 nm. This could also be partly due to the fact that the input absorption and scattering coefficients at these wavelengths are extrapolated values based on the values at 700 nm instead of based on field measurements. Compared with the POLYMER performance against in situ data obtained for the Medium Resolution Imaging Spectroradiometer (MERIS) in Case 1 water (Müller et al. [14]), the error in reconstructing R

_{rs}from MM01 or PR05 (Table 1) is much smaller than the algorithm error, with RMSE values of 0.00008 (MM01) and 0.00012 (PR05) instead of 0.0011 sr

^{−1}at 443 nm, and 0.00008 (MM01) and 0.00008 (PR05) at 547 nm instead of 0.00032 sr

^{−1}at 560 nm. When data collected in both Case 1 and 2 waters are considered [14,38], the POLYMER RMSE values for MERIS are increased to 0.0037 sr

^{−1}and 0.0033 sr

^{−1}at 443 and 560 nm, respectively, which is much larger than the R

_{rs}reconstruction errors, i.e., 0.00082 and 0.00038 at 443 nm and 547, respectively (Table 1). For MODIS-Aqua, these authors reported similar POLYMER performance, i.e., RMSE of 0.0036 at 443 nm and 0.0031 sr

^{−1}at 547 nm. Thus, the error introduced by the semi-analytical modeling of water reflectance is unlikely to dominate the POLYMER performance, indicating that using MM01 and, especially, PR05 in the spectral-matching algorithm is adequate.

_{rs}ratios. Three ratios commonly used in ocean color algorithms [39,40] were selected, i.e., R

_{rs}(412)/R

_{rs}(547), R

_{rs}(443)/R

_{rs}(547), and R

_{rs}(488)/R

_{rs}(547). It is found that the MM01 and PR05 modeled band ratios R

_{rs}(412)/R

_{rs}(547), R

_{rs}(443)/R

_{rs}(547), and R

_{rs}(488)/R

_{rs}(547) are all in very good agreement with the Hydrolight simulations. The overall statistics indicate that the two models perform similarly in restituting R

_{rs}ratios: the bias is less than 6% and the RMSE less than 9% for the MM01 model, and the bias is less than 5% and the RMSE less than 10% for the PR05 model. Also, the performance of Case 1 water situations is similar to that of Case 2 water situations (Table 2). This indicates that both the MM01 and PR05 models are capable of provide accurate estimation of R

_{rs}ratios, which is important for ocean-color products using band ratio algorithms. If the MM01 and PR05 reconstructed R

_{rs}is used in the OCI algorithm [41], the resulted bias and RMSE for estimated [Chla] are −0.000 (−0.2%) and 0.006 mgm

^{−3}(4.3%), and −0.001 (−0.7%) and 0.008 mgm

^{−3}(6.6%), respectively, i.e., much less than the errors caused by the OCI algorithm itself. This is due to the fact that the errors on R

_{rs}at the various wavelengths considered in the CI equation, i.e., CI ≈ R

_{rs}(555) − 0.5(R

_{rs}(443) + R

_{rs}(667)) where R

_{rs}(555) is specified as 0.93R

_{rs}(547) [41], are small (on the order of 10

^{−4}sr

^{−1}) in the applicability domain of the algorithm (clear, Case 1 waters). Note for both models the Nelder–Mead retrieved [Chla] differs substantially from the prescribed [Chla] (Figure 4). The bias and RMSE of log

_{10}([Chla]) are 0.52 and 0.60, and 0.21 and 0.37, respectively, for MM01 and PR05 model, and the errors are similar for both Case 1 and Case 2 waters (Table 2). These values, especially biases, are much larger when compared with the MODIS-Aqua estimated [Chla] performance against in situ data which shows absolute bias and mean absolute error (MAE, always less or equal to RMSE) of 0.07 and 0.22 on log

_{10}scale, respectively [42]. This is probably due to the variations in the relation between the absorption/ backscattering coefficients and the model parameters, which are not accounted for in the models. It is thus concluded that the [Chla] parameter in the models does not represent well the actual [Chla] and should not be used as [Chla] estimate. Instead [Chla] should be derived from the reconstructed R

_{rs}, and using R

_{rs}in band-ratio or OCI algorithms is appropriate.

_{rs}than the MM01 model. Specifically, the large bias and RMSE of the MM01 model, noticeably at 412 nm, 667 nm and 678 nm, are mainly related to the simplified treatment of backscattering of non-algal particles, as well as lack of fluorescence and absorption of CDOM and non-algal particles in the model. The large bias and RMSE at 547 nm of the PR05 model is due to the non-representative CDOM absorption slope, as discussed previously. The results also suggest degraded performance for Case 2 waters since these waters have generally higher [Chla] than Case 1 waters (bias and RMSE are larger for productive waters).

_{rs}were also used as input to the QAA algorithm [20] to retrieve IOPs including absorption of detritus and CDOM at 443 nm (a

_{dg}(443)), phytoplankton absorption at 443 nm (a

_{ph}(443)), and backscattering of particles at 547 nm (b

_{bp}(547)). When using the Hydrolight simulated R

_{rs}as input to the QAA, the bias and RMSE for retrieved a

_{dg}(443), a

_{ph}(443), and b

_{bp}(547) are −3.3% and 29.3%, −3.4% and 34.4%, and −1.8% and 55.6%, respectively (Figure 6a–c). These values quantify algorithm errors. When using the reconstructed R

_{rs}as input, the retrieved a

_{dg}(443), a

_{ph}(443), and b

_{bp}(443) are largely biased, i.e., bias being −71.0%, 57.8%, and −23.7% for MM01, and −31.3%, −11.4%, and −14.6% for PR05, when compared with the prescribed values (Figure 6d–i). The errors of using the MM01 and PR05 reconstructed R

_{rs}are much higher than those of using Hydrolight simulated R

_{rs}. Specifically, the performance of using the PR05 reconstructed R

_{rs}is better than that of using the MM01 reconstructed R

_{rs}with relatively lower bias and RMSE. The QAA algorithm involves two important parameters: u and χ, the first one depending on R

_{rs}values and the second one depending on R

_{rs}ratios. The value of b

_{bp}(547) depends on u at 547 nm, u(547), and the total absorption coefficient at 547 nm, a(547). Since u(547) is related to R

_{rs}(547) only and a(547) is related to the ratio R

_{rs}(443)/R

_{rs}(547), R

_{rs}(488)/R

_{rs}(547), and R

_{rs}(667)/R

_{rs}(547), it is expected that the retrieved b

_{bp}(547) values show larger bias and RMSE compared with the retrieval using the Hydrolight simulated R

_{rs}, and the errors are larger for the MM01 model than for the PR05 model based on the model performance in reconstructing R

_{rs}(667) (Figure 3). The value of a

_{dg}(443) is expressed as a function of b

_{bp}(547), R

_{rs}(412), R

_{rs}(443), and R

_{rs}(443)/R

_{rs}(547). Therefore, the error in the b

_{bp}(547) propagates further and results in much larger error in the retrieved a

_{dg}(443), as shown in Figure 6d,g. Also, the retrieval error in a

_{dg}(443) is less when using the PR05 model reconstructed R

_{rs}due to the smaller error in b

_{bp}(547). The value of a

_{ph}(443) is calculated as the total absorption minus a

_{dg}and the absorption of pure sea water, in which way the error in retrieving a

_{ph}(443) is compensated to some extent as the total absorption is derived using b

_{bp}(547) and R

_{rs}(443). It is concluded that using successfully the QAA algorithm requires accurate input of R

_{rs}and R

_{rs}ratios. Other IOP algorithms such as GSM [43] and GIOP [44] also use R

_{rs}as input and may have similar problems. Therefore, the MM01 and PR05 reconstructed R

_{rs}should be used with caution to retrieve IOPs. One may envision, however, applying these inversion schemes to reflectance ratios, as long as sensitivity to IOPs remains adequate.

_{rs}reconstruction would be improved by introducing more parameters (Table 3). Results show that by introducing the third parameter fa (see Section 2.1), the PR05 model does improve the estimation of R

_{rs}at all wavelengths except 531 nm, wavelength for which the bias and RMSE are slightly higher than using a two-parameter model. However, the improvement is not significant for 412, 443, 488, 547, and 667 nm, with bias and RMSE reduced by about 1% or even less. Much more improvement is seen at longer wavelengths (i.e., 678, 748, and 869 nm), but the differences between Hydrolight simulated and PR05 modeled R

_{rs}are still large, for example the RMSE at 869 nm is 0.00010 sr

^{−1}(56.1%). Similar results of R

_{rs}are observed when separating Case 1 and Case 2 waters. Therefore, having a more intricate model may not be warranted. The trade-off is between efficiency/robustness and accuracy. It also should be noted that having more input parameters in the spectral-matching scheme might not guarantee convergence. It is concluded that the PR05 model with two parameters is preferred for R

_{rs}estimation than the three-parameter model (allows faster and easier convergence while keeping similar accuracy).

#### 3.2. Sensivity of R_{rs} Reconstruction to Atmospheric Transmittance

_{TOA}(λ) = t

_{oz}(λ)[ρ

_{mol}(λ) + T(λ)ρ

_{gli}(λ) + ρ

_{aer}(λ) + ρ

_{gam}(λ) + t(λ)ρ

_{w}(λ,0+)]

_{oz}(λ) is the ozone transmittance, ρ

_{mol}(λ) is the Rayleigh scattering reflectance, ρ

_{gli}(λ) is the (non-spectral) sun glint reflectance affected by the direct transmission factor T(λ), ρ

_{aer}(λ) is the aerosol reflectance, ρ

_{gam}(λ) accounts for the coupling between Fresnel reflectance and scattering by molecules and aerosols, t(λ) is the total (i.e., direct + diffuse) transmittance for atmospheric scattering, and ρ

_{w}(λ,0+) is the water reflectance above the water-air interface (ρ

_{w}= πR

_{rs}). In this decomposition, the atmosphere is assumed to be clear (i.e., no clouds). In the presence of clouds, Equation (2) should be modified to account for scattering by cloud droplets and its interaction with molecule/aerosol scattering and surface reflection, and t(λ) should include cloud transmittance, i.e., t(λ) = t

_{mol}(λ) t

_{aer}(λ) t

_{cloud}(λ), where subscripts mol, aer, and cloud refer to molecules, aerosols, and clouds, The transmittance t(λ) is pre-calculated with a successive orders of scattering (SOS) code [45] and stored in look-up tables, but only molecules are considered in the calculations, even though the algorithm is designed to work in the presence of aerosols and, at least, optically thin clouds. Neglecting the transmittance of aerosols and clouds is expected to affect the R

_{rs}reconstruction in the spectral-matching scheme. Therefore, it is important to evaluate the sensitivity of the R

_{rs}reconstruction using the MM01 and PR05 models to cloud and aerosol transmittance.

_{aer}was modeled following Tanre et al. [46]:

_{aer}= exp[−0.16τ

_{aer}(λ)(1/cos(θ

_{v}) +1/cos(θ

_{s}))],

_{s}and θ

_{v}are the Sun and view zenith angles, respectively, and τ

_{aer}is the aerosol optical thickness, τ

_{aer}(λ) = 0.1(λ/865)

^{−1}. The cloud transmittance t

_{cloud}was parameterized according to Fitzpatrick et al. [47]:

_{cloud}= [a(τ) + b(τ

_{cloud}) cosθ

_{v}][a(τ) + b(τ

_{cloud})cosθ

_{s}]/[1 + (c − dα)τ

_{cloud}]

^{2},

_{1}+ (1 − a

_{1})exp(−k

_{1}τ

_{cloud}),

_{1}[1 + b

_{2}exp(−k

_{2}τ

_{cloud}) + b

_{3}exp(−k

_{3}τ

_{cloud})],

_{cloud}is geometric-optics cloud optical thickness, α is surface albedo. Cloud optical depth τ

_{cloud}is set to 3 (thin cloud), and surface albedo α to 0.06. Sun and view zenith angles are those of the IOCCG dataset, i.e., 30° and 0°, respectively. The coefficients a

_{1}, b

_{1}, b

_{2}, b

_{3}, k

_{1}, k

_{2}, k

_{3}, c, and d are tabulated in Fitzpatrick et al. [47] as a function of wavelength. The resulting t

_{cloud}values were interpolated to the MODIS wavelengths. With the calculated t

_{aer}and t

_{cloud}, the Hydrolight simulated R

_{rs}were transformed to R

_{rs}t

_{aer}t

_{cloud}, and the MM01 and PR05 model parameters were retrieved using R

_{rs}t

_{aer}t

_{cloud}instead of R

_{rs}as input.

_{rs}, [Chla], and R

_{rs}ratios versus the Hydrolight simulations. The reconstructed R

_{rs}is systematically biased, i.e., by 20–40%, after taking into account the cloud and aerosol transmittance. However, the R

_{rs}ratios are still in good agreement with the Hydrolight simulated ratios, i.e., they exhibit bias and RMSE comparable to those obtained without considering the effects of transmittance. When using the MM01 and PR05 reconstructed R

_{rs}to estimate [Chla] with the OCI algorithm, the resulting bias and RMSE increase to 0.039 (32.5%) and 0.040 mgm

^{-3}(33.1%), and 0.036 (29.9%) and 0.036 mgm

^{-3}(30.5%), respectively. This is because the cloud and aerosol transmittances introduce a large bias, i.e., more than 20%, to R

_{rs}, which is further propagated to the CI index. The cloud and aerosol transmittances do not have much effect on the model-retrieved [Chla]; statistics are similar to those in Figure 4. However, again for both models, the Nelder–Mead [Chla] does not represent well the actual values. Note that most of the impact on the R

_{rs}reconstruction performance is due cloud transmittance (0.78 at 443 nm), as the aerosol transmittance (0.95 at 443 nm) is close to 1.

_{rs}as input to the QAA algorithm to retrieve a

_{dg}(443), a

_{ph}(443), and b

_{bp}(547), large errors are obtained compared with the results using the Hydrolight simulated reflectance spectra (Figure 9). As discussed in Section 3.1, the retrieved value of b

_{bp}(547) is dependent on R

_{rs}(547), R

_{rs}(443)/R

_{rs}(547), R

_{rs}(488)/R

_{rs}(547), and R

_{rs}(667)/R

_{rs}(547). Since the reconstructed R

_{rs}(547) is systematically biased when the effects of aerosol/cloud transmittance are taken into account (Figure 7), and even though the values of R

_{rs}(443)/R

_{rs}(547), R

_{rs}(488)/R

_{rs}(547), and R

_{rs}(667)/R

_{rs}(547) (Figure 8) are similar to those with no transmittance, the error in the retrieved b

_{bp}(547) is increased (compare Figure 9 with Figure 6). However, since a

_{dg}(443) and a

_{ph}(443) are not only determined by b

_{bp}(547) but also by R

_{rs}(412), R

_{rs}(443), and R

_{rs}(443)/R

_{rs}(547), in the end the errors somewhat compensate, yielding bias and RMSE for a

_{dg}(443) and a

_{ph}(443) that are similar to those before accounting for aerosol/cloud transmittance. Again, the errors in the retrieved IOPs are less when using the PR05 model reconstructed R

_{rs}, compared with the MM01 results.

#### 3.3. Model Performance Using AERONET-OC Dataset

_{rs}at all wavelengths, i.e., only R

_{rs}at 412, 443, 488, 531, 547, 667, and 869 nm are available. To determine if missing R

_{rs}at 678 and 748 nm affects the model performance, tests were made using the 500 Hydrolight simulated cases (Section 2.3) with full MODIS wavelengths and with partial MODIS wavelengths (excluding 678 and 748 nm). No significant difference (i.e., less than 1%) in bias and RMSE was observed between R

_{rs}, [Chla], and R

_{rs}ratios retrieved using the full or partial set of wavelengths. Therefore, the errors caused by missing wavelengths are regarded as negligible in the following study. Prior to evaluation, Case 1 and Case 2 water situations were separated by using the same criteria that were applied to the Hydrolight simulations, taking the solar zenith angle into account. This resulted in 6094 Case 1 water and 3730 Case 2 water situations (Figure 10).

_{rs}at 443, 488, and 547 nm. The bias and RMSE values are −0.00001 (−0.2%) and 0.00032 sr

^{−1}(9.2%), −0.00055 (−11.7%) and 0.00085 sr

^{−1}(18.0%), 0.00025 (5.5%) and 0.00056 sr

^{−1}(12.2%) respectively, i.e., higher than those obtained using Hydrolight simulations. Much larger bias and RMSE are found at 667 nm, which are −0.00015 (−12.1%) and 0.0010 sr

^{−1}(84.5%), respectively. Nevertheless, the errors in R

_{rs}are still smaller than typical POLYMER atmospheric correction errors. The PR05 model performs better in reconstructing R

_{rs}with generally lower bias and RMSE at each wavelength (Figure 11, bottom, and Table 4). Unlike for the MM01 model, the PR05 reconstructed R

_{rs}values at 667 nm are in good agreement with the AERONET-OC measurements, even though fluorescence is not considered in the PR05 model. Similar to the MM01 results, the bias and RMSE values of the PR05 reconstruction are higher than those obtained using Hydrolight simulations. As discussed in the previous section, the simplified representation of the backscattering of non-algal particles, together with the lack of consideration of fluorescence and CDOM and non-algal absorption in the MM01 model are the main contributors to the errors in reconstructed R

_{rs}. The PR05 model takes into account the absorption and backscattering of non-algal particles by using the [Chla] and fb parameters, but the slope used for CDOM absorption might not be sufficiently well representative, causing substantial errors at 547 nm for Case 2 water situations when applying the spectral-matching scheme for inversion. Note that the Hydrolight simulated R

_{rs}is not subjected to radiometric errors and other, e.g., environmental, perturbations and this may partly explain why the bias and RMSE in reconstructed R

_{rs}using the AERONET-OC dataset are larger than those obtained using Hydrolight simulations.

_{rs}ratios. Results show that the MM01 and PR05 band ratios R

_{rs}(412)/R

_{rs}(547), R

_{rs}(443)/R

_{rs}(547), and R

_{rs}(488)/ R

_{rs}(547) are all in good agreement with the AERONET-OC measurements. The overall bias is less than 15% and the overall RMSE less than 18% for the MM01 model. The PR05 reconstruction gives slightly better results, with the overall bias less than 11% and the overall RMSE less than 16%. As expected, the errors in the reconstructed R

_{rs}ratios are larger than those obtained using Hydrolight simulations. Table 5 shows that the model reconstructed R

_{rs}ratios agree well with the prescribed values for both Case 1 and Case 2 waters, with errors slightly lower for Case 1 situations than for Case 2 situations. When applying the MM01 and PR05 reconstructed R

_{rs}to estimate [Chla] using the OCI algorithm, the resulting bias and RMSE for [Chla] are 0.010 (4.8%) and 0.014 mgm

^{-3}(6.8%), and 0.007 (3.6%) and 0.011 mgm

^{-3}(5.4%), respectively. This confirms that both the MM01 and PR05 reconstructed R

_{rs}can be used to estimate [Chla] using band ratio and OCI algorithms. Figure 12 also shows that the model-retrieved [Chla] differs substantially from the AERONET-OC [Chla], both models tending to overestimate [Chla]. However, for both models the overall agreement is better than when using Hydrolight simulations: the bias and RMSE on log10 scale are 0.33 and 0.42 instead of 0.52 and 0.60, respectively, for the MM01 model, and 0.12 and 0.23 instead of 0.21 and 0.37, respectively, for the PR05 model. The statistics are similar when Case 1 and Case 2 waters are used separately (Table 5). The comparisons suggest that the bio-optical relationships used in both models, based on field measurements, are more representative of actual situations than those in the Hydrolight simulations. Despite the improvement in [Chla] estimate, the model-retrieved [Chla] should still not be considered as representative of the actual [Chla].

_{rs}reconstruction, relatively large bias and RMSE are observed at 488 nm (when [Chla] is within 2–4 mg m

^{−3}) and 667 nm (when [Chla] is larger than 2 mg m

^{−3}) (Figure 13). Such behavior could be due to the fact that the absorption and backscattering of non-algal particles, fluorescence, and CDOM absorption are not taken into account in the MM01 model. For other wavelengths the bias and RMSE are relatively small and do not quite change with [Chla], except when [Chla] is within 2–4 mg m

^{−3}(small increase). This is also the case when using PR05 model. This increased bias and RMS when [Chla] is in the range 2–4 mg m

^{−3}could be due to the fact that the AERONET-OC [Chla] are not measured but estimated from remote sensing reflectance [19], and that higher uncertainty exists in this [Chla] range. It is also possible the variability of chlorophyll-a absorption is not sufficiently well represented in the MM01 and PR05 model. The bias and RMSE are smaller over the entire [Chla] range when using the PR05 model, indicating the overall better performance of that model in reconstructing water reflectance, again confirming the results from simulations.

## 4. Summary and Conclusions

_{rs}at 412, 443, 488, 531, and 547 nm, even in optically complex waters. Neglecting CDOM absorption in the MM01 model is responsible for the relatively large bias and RMS found at 412 nm. The discrepancies found at 667 and 869 nm for Case 2 water are mainly due to the backscattering of inorganic particles not well represented in the MM01 model. Other factors, including compensation by the spectral-matching scheme of differences at shorter wavelengths caused by the lack of CDOM and non-algal absorption and R

_{rs}extrapolation above 700 nm using the similarity spectrum, may also contribute to degradation of performance at 667 and 869 nm. Despite the errors in R

_{rs}, the MM01 retrieved R

_{rs}ratios are in very good agreement with the Hydrolight R

_{rs}ratios. The PR05 model, on the other hand, shows better performance than the MM01 model in reconstructing R

_{rs}. The PR05 retrieved R

_{rs}values, obtained by fixing to unity the parameter allowing variable absorption by dissolved organic matter, agree with the Hydrolight values for both Case 1 and Case 2 situations at all visible wavelengths including 667 and 678 nm, even though fluorescence is not considered in the model. The relatively large bias and RMSE at 547 nm is due to the non-representative slope of CDOM absorption and the spectral-matching scheme that minimizes the overall difference. Large discrepancy with the Hydrolight R

_{rs}values is found at 748 and 869 nm, which reflects the generally lower R

_{rs}, especially at 869 nm, and could be partly due to the fact that the input absorption and scattering coefficients at these wavelengths are extrapolated from values at 700 nm instead of based on field measurements. Overall, the performance for Case 1 waters is better than that of for Case 2 waters. The PR05 reconstructed R

_{rs}ratios also agree well with the Hydrolight ratios. The reconstructed R

_{rs}from both models, when used in the OCI algorithm for oligotrophic waters, yields an accurate estimation of chlorophyll-a concentration. However, the MM01 and PR05 retrieved R

_{rs}, when used as input to the QAA algorithm, gives significant errors in the estimation of a

_{ph}(443), a

_{dg}(443), and b

_{bp}(547). This is due to the fact that the QAA algorithm uses R

_{rs}at all visible wavelengths and not only R

_{rs}ratios. Therefore, caution should be exercised when using the model-retrieved R

_{rs}in algorithms for IOP estimations such as QAA. Importantly, the model parameters that best fit the input data, in particular chlorophyll-a concentration, may not represent adequately actual values. The impact of cloud and aerosol transmittance (assumed to be 1 in POLYMER) on R

_{rs}reconstruction was also evaluated theoretically using Hydrolight simulations. While the cloud and aerosol transmittances do not affect significantly the reconstruction of R

_{rs}ratios, they do affect the accuracy of reconstructed R

_{rs}, the OCI estimated chlorophyll-a concentration, and the derived IOPs when using QAA and, presumably, other IOP algorithms.

_{rs}measurements at 412, 443, 488, 531, 547, 667, and 869 nm were available, but the missing data at 678 and 748 nm does not affect significantly the R

_{rs}retrievals as tested using the Hydrolight simulations. Similar conclusions to those obtained with simulated R

_{rs}are drawn for the model-retrieved R

_{rs}and R

_{rs}ratios, confirming the theoretical results, but with relatively higher bias and RMSE probably due to the uncertainties existing in field measurements, and the IOP variability not captured by both models. In any case, the uncertainty in reconstructing R

_{rs}is much smaller than typical POLYMER atmospheric correction errors obtained against in-situ measurements. Again, the model-retrieved chlorophyll-a concentration does not agree well with the AERONET-OC values, which may partly result from the application of non-site-specific bio-optical algorithms. Therefore, it is recommended that the reconstructed water reflectance, not the retrieved model parameters, should be used in bio-optical algorithms.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Spectra of remote sensing reflectance R

_{rs}for Case 1 (

**a**) and Case 2 (

**b**) water using Hydrolight simulations. To distinguish Case 1 and 2 water situations, two thresholds were used: the threshold of Morel and Bélanger [32], i.e., the limiting R

_{w}(560) values produced by the bio-optical model of Morel and Maritorena [13] with the upper limit of b

_{p}(560) = 0.69[Chla]

^{0.766}as input, and the threshold of Robinson et al. [33], i.e. R

_{rs}(670) ≤ 0.0012 sr

^{−1}for Case 1 water, otherwise Case 2 water.

**Figure 2.**Examples of MM01 and PR05 reconstructed R

_{rs}spectra versus Hydrolight simulated R

_{rs}spectra, for Case 1 (

**a**–

**c**) and Case 2 (

**d**–

**f**) water situations. The spectral matching procedure was performed at MODIS wavelengths (412, 443, 488, 531, 547, 667, 678, 748, and 869 nm), and the retrieved model parameters were then used to produce modeled R

_{rs}spectra from 400 nm to 900 nm at 10 nm interval. Black lines represent Hydrolight simulations, red lines represent MM01 reconstructed R

_{rs}spectra, and blue lines PR05 reconstructed R

_{rs}spectra. The dots show model reconstructed R

_{rs}at MODIS wavelengths.

**Figure 3.**Comparison of MM01 (

**a**–

**d**) and PR05 (

**e**–

**h**) reconstructed remote sensing reflectance R

_{rs}with Hydrolight simulated R

_{rs}at selected MODIS wavelengths: 443 nm, 488 nm, 547 nm, and 667 nm. Blue circles represent Case 1 water and red represent Case 2 water.

**Figure 4.**Comparison of MM01 (

**a**–

**d**) and PR05 (

**e**–

**h**) reconstructed and Hydrolight simulated band ratios R

_{rs}(412)/R

_{rs}(547), R

_{rs}(443)/R

_{rs}(547), and R

_{rs}(488)/R

_{rs}(547), as well as Nelder–Mead retrieved and prescribed [Chla]. Blue circles represent Case 1 water and red represent Case 2 water. The bias and RMSE of [Chla] were calculated on log10 scale.

**Figure 5.**Bias and RMSE as a function of [Chla], based on MM01 and PR05 reconstructed results using Hydrolight simulations. Eight [Chla] bins were used and the number of points was identical for each bin.

**Figure 6.**Comparison of QAA derived a

_{dg}(443), a

_{ph}(443), and b

_{bp}(547) using Hydrolight simulated R

_{rs}(

**a**–

**c**), MM01 reconstructed R

_{rs}(

**d**–

**f**), and PR05 reconstructed R

_{rs}(

**g**–

**i**) versus prescribed values. Blue circles represent Case 1 water and red represent Case 2 water.

**Figure 7.**Comparison of the MM01 (

**a**–

**d**) and PR05 (

**e**–

**h**) reconstructed remote sensing reflectance R

_{rs}with Hydrolight simulated R

_{rs}at selected MODIS wavelengths (443, 488, 547, and 667 nm), after applying cloud and aerosol transmittance. Blue circles represent Case 1 water and red represent Case 2 water.

**Figure 8.**Comparison of the MM01 (

**a**–

**d**) and PR05 (

**e**–

**h**) reconstructed versus Hydrolight simulated band ratios R

_{rs}(412)/R

_{rs}(547), R

_{rs}(443)/R

_{rs}(547), and R

_{rs}(488)/R

_{rs}(547), as well as Nelder–Mead retrieved versus prescribed [Chla], after applying the cloud and aerosol transmittance. Blue circles represent Case 1 water and red represent Case 2 water. The bias and RMSE of [Chla] were calculated on log10 scale.

**Figure 9.**Comparison of QAA derived a

_{dg}(443), a

_{ph}(443), and b

_{bp}(547) using MM01 reconstructed R

_{rs}(

**a**–

**c**) and PR05 reconstructed R

_{rs}(

**d**–

**f**) versus prescribed values, after applying cloud and aerosol transmittance. Blue circles represent Case 1 water and red represent Case 2 water.

**Figure 10.**AERONET-OC spectra of remote sensing reflectance R

_{rs}of Case 1 (

**a**) and Case 2 (

**b**) waters. To distinguish Case 1 and 2 water situations, two thresholds were used: the threshold of Morel and Bélanger [32], i.e., the limiting R

_{w}(560) values produced by the bio-optical model of Morel and Maritorena [13] with the upper limit of b

_{p}(560) = 0.69[Chla]

^{0.766}as input, and the threshold of Robinson et al. [33], i.e. R

_{rs}(670) ≤ 0.0012 sr

^{−1}for Case 1 water, otherwise Case 2 water.

**Figure 11.**Comparison of MM01 (

**a**–

**d**) and PR05 (

**e**–

**h**) reconstructed remote sensing reflectance R

_{rs}with AERNOET-OC R

_{rs}at selected MODIS wavelengths: 443 nm, 488 nm, 547 nm, and 667 nm. Blue circles represent Case 1 water and red represent Case 2 water.

**Figure 12.**Comparison of the MM01 (

**a**–

**d**) and PR05 (

**e**–

**h**) reconstructed versus AERONET-OC measured band ratios R

_{rs}(412)/R

_{rs}(547), R

_{rs}(443)/R

_{rs}(547), and R

_{rs}(488)/R

_{rs}(547), as well as Nelder–Mead retrieved versus prescribed [Chla]. Blue circles represent Case 1 water and red represent Case 2 water. The bias and RMSE of [Chla] were calculated on log10 scale.

**Figure 13.**Bias and RMSE as a function of [Chla], based on MM01 and PR05 reconstructed results using AERONET-OC dataset. Eight [Chla] bins were used and the number of points was identical for each bin.

**Table 1.**Performance statistics of MM01 and PR05 reconstructed R

_{rs}for Case 1 and Case 2 water using Hydrolight simulations.

Parameter | Case 1, N = 55 | Case 2, N = 445 | All Cases, N = 500 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3}sr^{−1} (RMSE%) | R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | ||

MM01 | R_{rs}(412) | 1.00 | 0.11 (3.7) | 0.18 (5.9) | 0.95 | 0.88 (17.0) | 1.35 (26.1) | 0.94 | 0.79 (16.0) | 1.27 (25.8) |

R_{rs}(443) | 1.00 | 0.01 (0.4) | 0.08 (2.6) | 0.95 | 0. 43 (8.2) | 0.87 (16.6) | 0.95 | 0.38 (7.7) | 0.82 (16.5) | |

R_{rs}(488) | 1.00 | −0.11 (−3.8) | 0.16 (5.4) | 0.93 | −0.16 (−2.6) | 0.74 (11.9) | 0.94 | −0.16 (−2.7) | 0.70 (11.9) | |

R_{rs}(531) | 1.00 | −0.04 (−1.5) | 0.06 (2.4) | 0.96 | −0.06 (−0.9) | 0.75 (11.8) | 0.96 | −0.06 (−1.0) | 0.70 (12.0) | |

R_{rs}(547) | 1.00 | 0.06 (2.5) | 0.08 (3.4) | 0.97 | 0.14 (2.1) | 0.79 (11.9) | 0.97 | 0.13 (2.1) | 0.75 (12.1) | |

R_{rs}(667) | 0.95 | 0.04 (8.1) | 0.07 (15.5) | 0.86 | −0.44 (−20.0) | 1.32 (59.5) | 0.87 | −0.39 (19.3) | 1.24 (61.4) | |

R_{rs}(678) | 0.92 | −0.09 (−15.4) | 0.45 (27.3) | 0.87 | −0.69 (−29.3) | 1.55 (65.8) | 0.88 | −0.62 (−28.9) | 1.46 (67.8) | |

R_{rs}(748) | 0.97 | 0.06 (80.2) | 0.07 (100.2) | 0.81 | 0.11 (25.1) | 0.28 (63.0) | 0.82 | 0.11 (26.1) | 0.27 (65.7) | |

R_{rs}(869) | 0.96 | 0.03 (119.3) | 0.04 (145.7) | 0.82 | 0.09 (44.2) | 0.15 (74.9) | 0.83 | 0.08 (45.6) | 0.14 (78.3) | |

PR05 | R_{rs}(412) | 1.00 | 0.05 (1.5) | 0.15 (4.8) | 1.00 | 0.27 (5.2) | 0.42 (8.1) | 1.00 | 0.25 (5.0) | 0.40 (8.0) |

R_{rs}(443) | 1.00 | 0.06 (2.1) | 0.12 (4.1) | 0.99 | 0.26 (4.9) | 0.40 (7.7) | 0.99 | 0.24 (4.8) | 0.38 (7.7) | |

R_{rs}(488) | 0.99 | −0.05 (−1.7) | 0.12 (4.0) | 0.98 | −0.00 (−0.0) | 0.42 (6.7) | 0.98 | −0.01 (−0.1) | 0.39 (6.8) | |

R_{rs}(531) | 0.99 | −0.03 (−1.3) | 0.07 (2.9) | 1.00 | −0.19 (−3.0) | 0.36 (5.6) | 1.00 | −0.17 (−2.9) | 0.34 (5.7) | |

R_{rs}(547) | 0.99 | −0.00 (−0.2) | 0.08 (3.3) | 0.99 | −0.34 (−5.1) | 0.64 (9.7) | 0.99 | −0.30 (−4.9) | 0.61 (9.8) | |

R_{rs}(667) | 0.99 | 0.06 (13.7) | 0.10 (22.2) | 0.99 | 0.17 (7.5) | 0.28 (12.8) | 0.99 | 0.15 (7.6) | 0.27 (13.4) | |

R_{rs}(678) | 0.97 | −0.07 (−12.5) | 0.09 (16.3) | 1.00 | −0.11 (−4.6) | 0.23 (9.8) | 1.00 | −0.10 (−4.8) | 0.22 (10.2) | |

R_{rs}(748) | 0.97 | 0.02 (30.7) | 0.03 (47.3) | 0.98 | 0.03 (7.6) | 0.10 (21.7) | 0.98 | 0.03 (8.0) | 0.09 (22.7) | |

R_{rs}(869) | 0.97 | 0.02 (67.8) | 0.03 (102.0) | 0.98 | 0.08 (39.6) | 0.14 (72.6) | 0.98 | 0.07 (40.1) | 0.14 (75.8) |

**Table 2.**Performance statistics of MM01 and PR05 reconstructed R

_{rs}ratios and [Chla] for Case 1 and Case 2 water using Hydrolight simulations. The bias and RMSE of log10([Chla]) are in the unit of mgm

^{-3}.

Parameter | Case 1, N =55 | Case 2, N = 445 | |||||
---|---|---|---|---|---|---|---|

R^{2} | Bias (bias%) | RMSE (RMSE%) | R^{2} | Bias (bias%) | RMSE (RMSE%) | ||

MM01 | R_{rs}(412)/R_{rs}(547) | 1.00 | 0.03 (2.1) | 0.05 (3.6) | 1.00 | 0.09 (6.5) | 0.13 (9.2) |

R_{rs}(443)/R_{rs}(547) | 1.00 | −0.04 (−0.6) | 0.08 (5.7) | 1.00 | 0.03 (2.4) | 0.09 (6.8) | |

R_{rs}(488)/R_{rs}(547) | 1.00 | −0.06 (−4.6) | 0.09 (6.9) | 1.00 | −0.05 (−3.9) | 0.08 (6.6) | |

log_{10}([Chla]) | 0.94 | 0.49 | 0.56 | 0.95 | 0.52 | 0.61 | |

PR05 | R_{rs}(412)/R_{rs}(547) | 1.00 | 0.03 (2.3) | 0.06 (4.1) | 1.00 | 0.06 (4.0) | 0.07 (5.4) |

R_{rs}(443)/R_{rs}(547) | 1.00 | 0.04 (3.1) | 0.14 (9.9) | 0.99 | 0.06 (4.4) | 0.13 (9.8) | |

R_{rs}(488)/R_{rs}(547) | 0.99 | −0.01 (−0.8) | 0.10 (7.5) | 0.99 | 0.01 (0.8) | 0.12 (9.1) | |

log_{10}([Chla]) | 0.93 | 0.37 | 0.41 | 0.95 | 0.19 | 0.37 |

**Table 3.**Performance statistics of three-parameter PR05 reconstructed R

_{rs}for Case 1 and Case 2 water using Hydrolight simulations.

Parameter | Case 1, N = 55 | Case 2, N = 445 | All Cases, N = 500 | ||||||
---|---|---|---|---|---|---|---|---|---|

R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | |

R_{rs}(412) | 1.00 | 0.05 (1.7) | 0.07 (2.4) | 1.00 | 0.25 (4.9) | 0.36 (6.9) | 1.00 | 0.23 (4.7) | 0.34 (6.8) |

R_{rs}(443) | 1.00 | 0.05 (1.8) | 0.07 (2.3) | 1.00 | 0.24 (4.6) | 0.35 (6.8) | 1.00 | 0.22 (4.5) | 0.33 (6.7) |

R_{rs}(488) | 1.00 | −0.05 (−1.7) | 0.08 (2.8) | 0.98 | −0.02 (−0.3) | 0.36 (5.8) | 0.99 | −0.02 (−0.3) | 0.34 (5.8) |

R_{rs}(531) | 0.99 | −0.03 (−1.2) | 0.07 (2.7) | 1.00 | −0.20 (−3.1) | 0.36 (5.7) | 1.00 | −0.18 (−3.1) | 0.34 (5.8) |

R_{rs}(547) | 0.99 | −0.01 (−0.3) | 0.07 (2.8) | 0.99 | −0.34 (−5.2) | 0.63 (9.5) | 0.99 | −0.31 (−4.9) | 0.59 (9.6) |

R_{rs}(667) | 0.99 | 0.07 (14.2) | 0.11 (22.9) | 1.00 | 0.20 (8.8) | 0.29 (13.2) | 1.00 | 0.18 (9.0) | 0.28 (13.7) |

R_{rs}(678) | 0.98 | −0.07 (−12.3) | 0.09 (15.4) | 1.00 | −0.08 (−3.2) | 0.14 (5.9) | 1.00 | −0.08 (−3.5) | 0.13 (6.3) |

R_{rs}(748) | 0.98 | 0.02 (25.9) | 0.03 (41.4) | 0.99 | 0.01 (2.2) | 0.07 (16.1) | 0.99 | 0.01 (2.6) | 0.07 (17.0) |

R_{rs}(869) | 0.96 | 0.02 (57.9) | 0.03 (92.0) | 0.99 | 0.07 (32.7) | 0.11 (53.6) | 0.99 | 0.06 (33.1) | 0.10 (56.1) |

**Table 4.**Performance statistics of MM01 and PR05 reconstructed R

_{rs}for Case 1 and Case 2 water using AERONET-OC dataset.

Parameter | Case 1, N = 6094 | Case 2, N = 3730 | All Cases, N = 9824 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | R^{2} | Bias, 10^{−3} sr^{−1} (bias%) | RMSE, 10^{−3} sr^{−1} (RMSE%) | ||

MM01 | R_{rs}(412) | 0.97 | 0.32 (12.2) | 0.43 (16.5) | 0.96 | 1.08 (30.8) | 1.23 (35.1) | 0.94 | 0.61 (20.7) | 0.83 (28.2) |

R_{rs}(443) | 0.99 | 0.01 (−0.2) | 0.11 (4.1) | 0.99 | −0.03 (−0.8) | 0.50 (10.7) | 0.99 | −0.01 (−0.2) | 0.32 (9.2) | |

R_{rs}(488) | 0.94 | −0.29 (−8.7) | 0.38 (11.5) | 0.98 | −0.97 (−13.9) | 1.28 (18.4) | 0.98 | −0.55 (−11.7) | 0.85 (18.0) | |

R_{rs}(531) | 0.98 | −0.08 (−2.7) | 0.15 (5.4) | 0.99 | −0.21 (−2.7) | 0.55 (7.1) | 0.99 | −0.13 (−2.7) | 0.36 (7.6) | |

R_{rs}(547) | 0.99 | 0.13 (5.0) | 0.16 (6.4) | 0.99 | 0.46 (5.8) | 0.89 (11.1) | 0.99 | 0.25 (5.5) | 0.56 (12.2) | |

R_{rs}(667) | 0.82 | 0.03 (6.9) | 0.13 (27.1) | 0.84 | −0.45 (−18.4) | 1.68 (69.1) | 0.84 | −0.15 (−12.1) | 1.05 (84.5) | |

R_{rs}(869) | 0.01 | −0.02 (−22.3) | 0.10 (28.1) | 0.44 | 0.01 (5.1) | 0.31 (124.4) | 0.42 | −0.01 (−4.2) | 0.21 (143.8) | |

PR05 | R_{rs}(412) | 0.98 | 0.25 (9.4) | 0.33 (12.8) | 0.95 | 0.42 (12.1) | 0.69 (19.6) | 0.96 | 0.31 (10.6) | 0.50 (16.9) |

R_{rs}(443) | 0.99 | 0.01 (0.5) | 0.12 (4.2) | 0.99 | −0.11 (−2.4) | 0.50 (10.9) | 0.99 | −0.03 (−1.0) | 0.32 (9.3) | |

R_{rs}(488) | 0.95 | −0.25 (−7.4) | 0.33 (10.1) | 0.99 | −0.46 (−6.6) | 0.71 (10.2) | 0.99 | −0.33 (−6.9) | 0.51 (10.8) | |

R_{rs}(531) | 0.98 | −0.05 (−1.6) | 0.14 (4.9) | 0.99 | 0.02 (0.2) | 0.63 (8.1) | 0.99 | −0.02 (−0.5) | 0.41 (8.5) | |

R_{rs}(547) | 0.97 | 0.08 (3.1) | 0.18 (7.2) | 0.99 | 0.33 (4.2) | 0.94 (11.8) | 0.99 | 0.18 (3.8) | 0.60 (12.9) | |

R_{rs}(667) | 0.85 | 0.04 (9.1) | 0.16 (32.3) | 0.98 | 0.03 (1.4) | 0.40 (16.6) | 0.98 | 0.04 (3.3) | 0.28 (22.6) | |

R_{rs}(869) | 0.01 | −0.03 (−42.7) | 0.10 (132.3) | 0.82 | −0.01 (−4.0) | 0.18 (71.1) | 0.76 | −0.02 (−17.1) | 0.14 (94.9) |

**Table 5.**Performance statistics of MM01 and PR05 reconstructed R

_{rs}ratios and log10([Chla]) for Case 1 and Case 2 water using AERONET-OC dataset. The bias and RMSE of log10([Chla]) are in the unit of mg m

^{−3}.

Parameter | Case 1, N = 6094 | Case 2, N = 3730 | |||||
---|---|---|---|---|---|---|---|

R^{2} | Bias (bias%) | RMSE (RMSE%) | R^{2} | Bias (bias%) | RMSE (RMSE%) | ||

MM01 | R_{rs}(412)/R_{rs}(547) | 0.99 | 0.03 (2.4) | 0.15 (11.8) | 0.87 | 0.12 (28.7) | 0.14 (32.6) |

R_{rs}(443)/R_{rs}(547) | 0.99 | −0.08 (−6.0) | 0.15 (11.4) | 0.94 | −0.02 (−3.5) | 0.06 (11.0) | |

R_{rs}(488)/R_{rs}(547) | 0.96 | −0.19 (−13.1) | 0.24 (16.4) | 0.89 | −0.16 (−18.2) | 0.18 (20.4) | |

log_{10}([Chla]) | 0.88 | 0.22 | 0.33 | 0.80 | 0.50 | 0.54 | |

PR05 | R_{rs}(412)/R_{rs}(547) | 0.99 | 0.04 (3.3) | 0.10 (8.0) | 0.85 | 0.06 (13.1) | 0.09 (21.1) |

R_{rs}(443)/R_{rs}(547) | 0.99 | −0.05 (−4.1) | 0.12 (9.1) | 0.91 | −0.02 (−3.1) | 0.08 (13.5) | |

R_{rs}(488)/R_{rs}(547) | 0.95 | −0.16 (−10.9) | 0.22 (15.2) | 0.84 | −0.08 (−9.8) | 0.14 (15.9) | |

log_{10}([Chla]) | 0.85 | 0.14 | 0.23 | 0.73 | 0.09 | 0.22 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Tan, J.; Frouin, R.; Ramon, D.; Steinmetz, F. On the Adequacy of Representing Water Reflectance by Semi-Analytical Models in Ocean Color Remote Sensing. *Remote Sens.* **2019**, *11*, 2820.
https://doi.org/10.3390/rs11232820

**AMA Style**

Tan J, Frouin R, Ramon D, Steinmetz F. On the Adequacy of Representing Water Reflectance by Semi-Analytical Models in Ocean Color Remote Sensing. *Remote Sensing*. 2019; 11(23):2820.
https://doi.org/10.3390/rs11232820

**Chicago/Turabian Style**

Tan, Jing, Robert Frouin, Didier Ramon, and François Steinmetz. 2019. "On the Adequacy of Representing Water Reflectance by Semi-Analytical Models in Ocean Color Remote Sensing" *Remote Sensing* 11, no. 23: 2820.
https://doi.org/10.3390/rs11232820