## 1 Introduction  
The frequent occurrence of algal blooms in eutrophic lakes is a worldwide problem that threatens ecological and environmental safety, drinking water resources, and human activities. Under the regulation of self-buoyancy and environmental conditions (e.g., light intensity, wind, waves), algae can move up and down in the water column, and exhibit a vertically inhomogeneous distribution of chlorophyll-a (Chla(z), z refers to water depth, Table 1). The Gaussian and shifted Gaussian models have often been used to illustrate the vertical non-uniform profiles of phytoplankton in oceanic waters [1; 2; 3; 4; 5; 6]. Meanwhile, four vertical distribution classes (vertically uniform, Gaussian, exponential, and hyperbolic) of Chla(z), with the maximum value on the water surface, were observed in a shallow eutrophic lake (i.e., Lake Chaohu, China) with a large content of cyanobacteria [7]. models of coupled ocean-atmosphere system (e.g., studies in References [30; 31; 32; 33; 34; 35; 36]) were developed to predict the radiance and degree of polarization of the light using various numerical methods. Through theoretical analyses and numerical modeling, sub-surface remote sensing reflectance (\(r_{rs}(0^{-})\)) was determined to be generally proportional to the ratio of \(b_{b}\)/(\(a+b_{b}\)), where \(a\) is the absorption coefficient and \(b_{b}\) is the backscattering coefficient [37; 38]. The most extensively used forward remote sensing model was reported by [PERSON] et al. [39] and is expressed as follows:  
\[r_{rs}(0^{-})=g\frac{b_{b}}{a+b_{b}}, \tag{1}\]  
\[g=g_{0}+g_{1}\frac{b_{b}}{a+b_{b}}, \tag{2}\]  
where, \(g_{0}\) = 0.0949, and \(g_{1}\) = 0.0794 for nadir-viewed \(r_{rs}(0^{-})\). Parameter \(g\) is related to the sun's zenith angle, viewing angle, bottom reflection, and radiance distribution [40]. However, this model was built based on the assumption of vertical homogeneity, and thus, the use of Equation (1) to model \(r_{rs}(0^{-})\) would lead to limitations in stratified waters. For example, two waters with the same surface \(a\) and \(b_{b}\) but different vertical profiles of Chl\(a\)(z) may have varying values and spectrum shapes of \(r_{rs}(0^{-})\) and \(R_{rs}(\lambda)\). In such case, \(r_{rs}(0^{-})\) contains the signal from vertical variation of phytoplankton, and is not related well to the \(b_{b}\)/(\(a+b_{b}\)) near the water surface in stratified waters. [PERSON] and [PERSON] [19] emphasized the necessity of understanding the relationship between the vertical distribution of IOPs(z) and \(R_{rs}(\lambda)\). The relationship between \(b_{b}\)(z)/(\(a\)(z) + \(b_{b}\)(z)) in each water depth and \(r_{rs}(0^{-})\) could explain the effects of vertical distribution of phytoplankton on underwater light field. Several approaches have been proposed to reduce the effects of vertical distribution of phytoplankton on \(R_{rs}(\lambda)\)[9; 10; 17]. The \(R_{rs}(\lambda)\) of stratified Case I waters was interpreted and found to be identical to that of hypothetical homogeneous waters with phytoplankton pigment concentration (C\({}_{rs}\)), which is a depth-weighted average of the actual profiles of pigment concentration [8]. This hypothesis was accepted and used in several studies on stratified waters by changing the average weighting function [2; 3; 10]. [PERSON] et al. [10] proposed that \(R_{rs}(\lambda)\) is a function of the derivative of the round trip attenuation of the downwelling and backscattered light, while it was usually modeled as being dependent on the round trip attenuation [8]. [PERSON] et al. [41] added a degree of freedom between the aforementioned two functions [8; 10] to retrieve vertical profiles of the total suspended matter concentration. Besides, the average Chl\(a\)(z) within the penetration layer or certain water depth was used to replace the vertically weighted Chl\(a\) in Lake Kinneret [9].