## 3 Results
### Remote Sensing Reflectance Model Considering Vertical Distribution Of Phytoplankton  
The results of numerical simulations indicated that the traditional remote sensing reflectance model is insufficient for describing the relationship between \(r_{\mathrm{rs}}\)(0\({}^{-}\)) and \(IOP^{\prime}\)(z) in vertically inhomogeneous waters. In general, multiple \(g\) values (Equation (1)) will exist for the same \(IOP^{\prime}\)(z = 0) value on the water surface with a vertical variation of phytoplankton. Based on weighting average theory and  
Figure 10: Contribution of each layer to \(r_{\mathrm{rs}}\)(\(\lambda\), 0\({}^{-}\)) at different values of SPIM (0–60 mg/L with an interval of 5 mg/L, increasing from top to bottom in each large grid) and \(a_{g}\)(440) (0–4 m\({}^{-1}\) with an interval of 0.5 m\({}^{-1}\), increasing from top to bottom in each small grid) in the wavelength region from 400 to 750 nm. Layer 1 is the water depth between 0.0 and 0.1 m; Layer 2 is the water depth between 0.1 and 0.2 m; Layer 3 is the water depth between 0.2 and 0.3 m. The grey dashed lines represent the cases with average SPIM and \(a_{g}\)(440) values. Equation (8), \(r_{rs}(0^{-})\) in vertically inhomogeneous waters (\(r_{rs-v}(0^{-})\)) can be related to \(IOP^{\prime}(z)\), \(Fr(z)\), and \(g_{x}(z)\) of different water depths. \[ \begin{array}{l}r_{rs-v}(0^{-})=\int_{0}^{\infty}IOP^{\prime}(z)\frac{d}{dz}[ \exp\{-\int_{0}^{z}[g_{x}(z^{\prime})]dz^{\prime}]dz\\ =\sum\limits_{i=1}^{n-1}Fr(z_{i},z_{i+1})\times\int_{z_{i}}^{z_{i+1}}IOP^{ \prime}(z)\frac{d}{dz}[\exp\{-\int_{0}^{z}[g_{x}(z^{\prime})]dz^{\prime}]\}dz\\ =\sum\limits_{i=1}^{n-1}IOP^{\prime}(z_{i}\to z_{i+1})\times Fr(z_{i},z_{i+1}) \times\int_{z_{i}}^{z_{i+1}}\frac{d}{dz}[\exp\{-\int_{0}^{z}[g_{x}(z^{\prime}) ]dz^{\prime}]\}dz\end{array} , \tag{14}\]  
where  
\[IOP^{\prime}(z_{i}\to z_{i+1})=\frac{b_{b}(\lambda,z_{i}\to z_{i+1})}{a( \lambda,z_{i}\to z_{i+1})+b_{b}(\lambda,z_{i}\to z_{i+1})}. \tag{15}\]  
By assuming  
\[S_{i}=Fr(z_{i},z_{i+1})\times\int_{z_{i}}^{z_{i+1}}\frac{d}{dz}[\exp\{-\int_{ 0}^{z}[g_{x}(z^{\prime})]dz^{\prime}]\}dz, \tag{16}\]  
\(r_{rs-v}(\lambda,0^{-})\) can be written as the relationship between \(IOP^{\prime}(\lambda,z)\) and parameter \(S_{i}(\lambda,z)\):  
\[ \begin{array}{l}r_{rs-v}(\lambda,0^{-})=\sum\limits_{i=1}^{n}\left[S_{i}( \lambda)\times IOP^{\prime}(\lambda,z_{i})\right]\\ =S_{1}(\lambda)\times IOP^{\prime}(\lambda,z_{1})+S_{2}(\lambda)\times IOP^{ \prime}(\lambda,z_{2})+\ldots+S_{n-1}(\lambda)\times IOP^{\prime}(\lambda,z_ {n})\end{array} . \tag{17}\]  
As waters of the top 0.3 m contributed a large part of \(r_{rs}(0^{-})\) in the simulated data in Lake Chaohu; hence, a simple, stratified remote sensing model for nadir-viewed \(r_{rs}(0^{-})\) can be built based on Equation (17). Three simple models (Model A, Model B, and Model C) with polynomial expression were built using the absorption and backscattering coefficients of the first two or three layers. The model parameters (\(S_{Ai}\), \(S_{Bi}\), and \(S_{Ci}\)) of each model can be determined using the curve fitting toolbox in MATLAB R2015b. Model A:  
\[r_{rs-v1}(\lambda,0^{-})=S_{A1}(\lambda,z(1))\frac{b_{b}(\lambda,z(1))}{a+b_{ b}(\lambda,z(1))}+S_{A2}(\lambda,z(2))\frac{b_{b}(\lambda,z(2))}{a+b_{b}( \lambda,z(2))}+S_{A3}(\lambda,z(3))\frac{b_{b}(\lambda,z(3))}{a+b_{b}(\lambda,z(3))} \tag{18}\]  
Model B:  
\[r_{rs-v2}(\lambda,0^{-})=S_{\text{BI}}(\lambda,z(1))\frac{b_{b}(\lambda,z(1))} {a+b_{b}(\lambda,z(1))}+S_{\text{B2}}(\lambda,z(2))\frac{b_{b}(\lambda,z(2))}{ a+b_{b}(\lambda,z(2))} \tag{19}\]  
Model C:  
\[r_{rs-v3}(\lambda,0^{-})=S_{\text{C1}}(\lambda,z(1))\frac{b_{b}(\lambda,z(1))} {a+b_{b}(\lambda,z(1))}+S_{\text{C2}}(\lambda,z(2))\frac{b_{b}(\lambda,z(2))}{ a+b_{b}(\lambda,z(2))}+S_{\text{C3}}(\lambda) \tag{20}\]  
A similar relative difference between modeled and simulated \(r_{rs}(0^{-})\) occurred in Models A and B, and approximately 20% of the calibration data had an average error of \(r_{rs}(0^{-})>10\%\) (Figure 11).