## 3 Method
### Adaptation of WASI for Processing PRISMA and Sentinel-2 Imagery  
The level 2D products of PRISMA are atmospherically corrected and provided in units of reflectance. Therefore, we perform all WASI analyses in terms of reflectance. The image recorded reflectance \(R_{\text{rs}}(\lambda)\) can be decomposed into two components upwelling from water called remote sensing reflectance \(r_{\text{rs}}\) and another one reflected from the water surface \(R_{\text{rs}}^{\text{surf}}(\lambda)\):  
\[R_{\text{rs}}(\lambda)=\frac{\zeta\cdot r_{\text{rs}}(\lambda)}{1-\Gamma\cdot r _{\text{rs}}(\lambda)}+R_{\text{rs}}^{\text{surf}}(\lambda). \tag{1}\]  
where \(r_{\text{rs}}\) is defined as the ratio of upwelling radiance to downwelling irradiance just below the water surface [51,52]. The water-to-air radiance divergence is accounted for by the factor \(\zeta\approx 0.52\). The factor \(\Gamma\approx 1.6\) represents the reflection of upwelling radiance at the water surface. Given the turbid condition in the study area (-1.1 m Secchi depth), the bottom-reflected radiance is negligible over a large part of the lake. Therefore, the forward modeling, i.e., simulation, of \(r_{\text{rs}}(\lambda)\) is performed for optically deep waters [69,70]:  
\[\begin{split} r_{\text{rs}}^{\text{deep}}(\lambda)=\\ 0.0512\cdot& u(\lambda)\cdot\Big{(}1+4.6659\cdot u (\lambda)-7.8387\cdot u(\lambda)^{2}+5.4571\cdot u(\lambda)^{3}\Big{)}\\ &\quad\quad\quad\quad\quad  
to its insignificant effect on \(r_{\text{rs}}\). The wavelength-dependent function \(u(\lambda)\) characterizes the in-water absorption \(a(\lambda)\) and backscattering properties \(b_{b}(\lambda)\):  
\[u(\lambda)=\frac{b_{b}(\lambda)}{a(\lambda)+b_{b}(\lambda)}. \tag{3}\]  
Both \(a(\lambda)\) and \(b_{b}(\lambda)\) can be decomposed into the contributions of pure water and different in-water constituents which are optically active:  
\[ \begin{split} a(\lambda)=a_{w}(\lambda)&+a_{\text{ CDOM}}(\lambda_{0})\cdot\exp[-S_{\text{CDOM}}(\lambda-\lambda_{0})]+C_{\text{phy}} \cdot a_{\text{phy}}^{*}(\lambda)\\ &+C_{\text{NAP}}\cdot a_{\text{NAP}}^{*}(\lambda_{0})\cdot\exp[-S_ {\text{NAP}}\cdot(\lambda-\lambda_{0})],\end{split}  \tag{4}\]  
\[b_{b}(\lambda)=b_{b,w}(\lambda)+C_{\text{phy}}\cdot b_{b,\text{phy}}^{*}( \lambda)+C_{\text{NAP}}\cdot b_{b,\text{NAP}}^{*}(\lambda). \tag{5}\]  
The absorption [71; 72] and backscattering [73] coefficients of pure water are given by \(a_{w}(\lambda)\) and \(b_{b,w}(\lambda)\), respectively. Colored dissolved organic matter (CDOM), phytoplankton (phy), and non-algal particles (NAPs) are optically active constituents and their absorption and backscattering properties are considered in Equations (4) and (5). The equations are based on specific inherent optical properties (SIOPs) that indicate absorption and backscattering properties normalized by the concentration of the given constituent. Here, we use the SIOPs (denoted by asterisks) available for Lake Trasimeno [62]. Phytoplankton concentration \(C_{\text{phy}}\), NAP concentration \(C_{\text{NAP}}\), the absorption coefficient of CDOM at the wavelength \(\lambda_{0}\) = 440 nm, \(a_{\text{CDOM}}(440)\), are considered as fit parameters through the inverse modeling. The spectral slope of NAP in Lake Trasimeno is considered as \(S_{\text{NAP}}\) = 0.011 nm\({}^{-1}\)[62]. We also use the specific absorption spectrum of phytoplankton \(a_{\text{phy}}^{*}(\lambda)\) measured in Lake Trasimeno (Figure 3). We consider \(a_{\text{NAP}}^{*}(440)=0.0248\) m\({}^{2}\) g\({}^{-1}\)[62], which represents the specific absorption coefficient of NAP at \(\lambda_{0}\) = 440 nm. The spectral slope of CDOM is considered as \(S_{\text{CDOM}}=0.016\) nm\({}^{-1}\)[62]. The specific backscattering coefficient of NAP is considered as a wavelength-independent function for which \(b_{b,NAP}^{*}(\lambda)\) = 0.01271 m\({}^{2}\) g\({}^{-1}\)[62].