## 2 Methods
### Development of hydro-optical models
#### 2.4.1 Lake Ladoga hydro-optical model  
In the second part of the 1980s, during the summer-time of three consecutive years, a large-scale ship campaign was conducted across the entire Lake Ladoga. In the course of this campaign, concurrently measured were spectral subsurface upwelling, _E\({}_{u}\)_(-0,_\(\lambda\)_) and downwelling, _E\({}_{d}\)_(-0,_\(\lambda\)_) irradiances to obtain spectral volume reflectance, _R\((\lambda)=E_{u}\)_(-0,_\(\lambda\)_) / _E\({}_{d}\)_(-0,_\(\lambda\)_), which is related to bulk coefficients of absorption, _a\((\lambda)\)_ and backscattering _b\({}_{b}\)*\((\lambda)\)_[21] and the concentrations of _chl_, _sm_ and _doc_. These data were employed for the development of a hydro-optical model of Lake Ladoga. To do this, the LM procedure (section II) was exploited, this time for determining specific spectral coefficients of absorption, _a*\((\lambda)\)_ and backscattering _b\({}_{b}\)*\((\lambda)\)_ (Fig. 2) of the water constituents addressed in this study (for details see [12]).  
It is important to underline that in the hydro-optical model of Lake Ladoga, _a*\({}_{doc}\)_ is calculated through the normalization to the _total doc_ concentration, i.e. not solely to the weight of the coloured fraction (called yellow substance or gelbstoff) - _cdom_. This has been done in order to facilitate the use of satellite data for validation of ecological models, which generally do not differentiate between sunlight-absorbing and non-absorbing _doc_ fractions. Such an approach opens door to the assimilation of satellite retrievals by aquatic ecological models.  
The efficiency of the developed algorithm performance was tested based on a fraction of the shipborne measurements that were not used for the hydro-optical model development. Table 3 lists WQP concentrations measured _in situ_ and retrieved from spectral volume reflectance concurrently measured at the same locations.  
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|} \hline \multicolumn{5}{|c|}{**Measured concentrations**} & \multicolumn{3}{|c|}{**Retrieved concentrations**} \\ \hline \(\mathbf{C_{chh}}\)**, mg/L** & \(\mathbf{C_{sm}}\)**, mg/L** & \(\mathbf{C_{dec}}\)**, mgC/L** & \(\mathbf{C_{chh}}\)**, mg/L** & \(\mathbf{C_{sm}}\)**, mg/L** & \(\mathbf{C_{dec}}\)**, mgC/L** \\ \hline
0.5 & 0.2 & 8 & 0.3 & 0.2 & 7 \\
6.6 & 1.0 & 9 & 5.5 & 1.1 & 10 \\
1.0 & 0.4 & 9 & 1.0 & 0.4 & 8 \\
0.6 & 0.3 & 9 & 0.8 & 0.2 & 7 \\
0.5 & 0.3 & 8 & 0.8 & 0.2 & 7 \\
1.0 & 0.2 & 8 & 1 & 0.05 & 9 \\
3.9 & 1.0 & 8 & 4 & 1.0 & 8 \\
7.1 & 0.8 & 8 & 9 & 1.0 & 7 \\
0.9 & 0.3 & 8 & 1 & 0.3 & 6 \\
2.1 & 0.3 & 8 & 1.5 & 0.3 & 7 \\ \hline \multicolumn{5}{|c|}{**Correlation (r)**} & \multicolumn{1}{c|}{**0.96**} & \multicolumn{1}{c|}{**0.98**} & \multicolumn{1}{c|}{**0.43**} \\ \hline \multicolumn{5}{|c|}{**Root mean square error (RMSE)**} & \multicolumn{1}{c|}{**0.73**} & \multicolumn{1}{c|}{**0.096**} & \multicolumn{1}{c|}{**1.2**} \\ \hline \end{tabular}
\end{table}
Table 3: Comparison of the _chl_, _sm_ and _doc_ concentrations measured _in situ_ and retrieved from shipborne optical measurements in Lake Ladoga  
Figure 2: Spectra of specific absorption (upper row of plates) and specific backscattering (second row of plates) for three hydro-optical models: 1 - Lake Ladoga; 2 – North Sea; 3 – Lake Erie. For chlorophyll (left column), the absorption and backscattering values are in m\({}^{2}\)/mg, for suspended minerals (central column), – in m\({}^{2}\)/g and for DOC (right column), – in m\({}^{2}\)/gC. Wavelengths on the X-axe are in nm.  
The developed algorithm restores the desired concentrations with an average error in the range about 6-12%. The attained accuracy is high enough, and indicates the intrinsic capacity of the developed LM algorithm to accurately retrieve the desired concentrations inasmuch as the adopted verification procedure excludes the intervening factor of imprecise atmospheric correction.