## 2 Methods
### Development of hydro-optical models
#### 2.4.3 Lake Erie hydro-optical model  
A rather extensive field campaign (over 60 stations) was being conducted by the Michigan State University (MTU) across Lake Erie in 2005 during June - September. The _in situ_ measurements encompassed spectrometric measurements to retrieve subsurface remote sensing reflectance, \(R_{rsw}(\lambda)\) at 6 MODIS wavelengths in the visible (412, 443, 488, 531, 551, 667 nm) as well as the concentrations of _chl_, _sm_ and _doc_. The Lake Ontario hydro-optical model [21] was selected as a starting option for elaborating the Lake Erie hydro-optical model (Fig. 2). An iterative algorithm was used to find the desired five variables (i.e. \(a^{*}_{~{}chh}\), \(b^{*}_{~{}chh}\), \(a^{*}_{~{}sm}\), \(b^{*}_{~{}sm}\), \(a^{*}_{~{}doc}\):). In each iteration step:  
* the \(a^{*}_{~{}chh}\), \(b^{*}_{~{}chh}\), \(a^{*}_{~{}sm}\), \(b^{*}_{~{}sm}\), \(a^{*}_{~{}doc}\) cross-sections for Lake Ontario are shifted up or down using the following equation (example for \(a^{*}_{~{}chl}\)): \[a^{*}_{~{}chl}=a^{*}_{~{}chl}\,{}_{0}+O_{chl}\] (8) where \(a_{chl~{}i}\) is the value of \(a^{*}_{chl}\) in the \(i\)th iteration step, \(a^{*}_{~{}chl}\) 0 is the value of \(a^{*}_{~{}chl}\) for Lake Ontario, \(O_{chl}\) is the random offset for \(a^{*}_{~{}chl}\);
* the shifted hydro-optical model is further used for the retrieval of CPA concentrations from the \(R_{rsw}(\lambda)\) values measured _in situ_ employing our LM- based algorithm;
* the difference between the retrieved and measured CPA concentration vector \(C\) is then calculated. In performing this step, only the difference between the concentrations of _chl_ and _sm_ was tested;
* the obtained difference is further used for the selection of the best vector \(O\). This step yields five offsets (\(O_{achl},O_{bbchl}\), \(O_{asm}\), \(O_{bbsm}\), \(O_{docl}\)) that are further used for calculating a _rough_ hydro-optical model for Lake Erie. Thus, due to application of the above procedure, the cross-sections of the established rough model only differ from the Ontario cross-sections by offsets \(O\). However, in reality, the spectral distributions of the desired CPA cross-sections can also differ from the \"benchmark\" model in terms of their spectral envelope. In order to accommodate this option and reveal possible spectral deviations of the sought for hydro-optical model from the \"benchmark\" model, applied were both the LM optimization technique and the Genetic algorithm [11] to find the best set of \(a^{*}\) and \(b^{*}_{b}\) in each channel. The results (i.e. desired spectral specific absorption and backscattering coefficients) proved to be very similar. The LM optimization was run six times (for six channels), and it was found that \(a^{*}\) and \(b_{b}{}^{*}\) values in different MODIS channels were independent. In each step of the LM optimization:  
* \(R_{rsw}\) values in a certain channel were calculated using the established five spectral values of \(a^{*}\) and \(b^{*}_{b}\) (\(a^{*}_{ch}\), \(b^{*}_{b}\)\({}^{*}_{ch}\), \(a^{*}_{sm}\), \(b^{*}_{b}\)\({}^{*}_{sm}\), \(a^{*}_{doc}\)) and the CPA concentrations. All three concentrations measured _in situ_ were used to calculate \(R_{rsw}(\lambda)\);
* \(\bullet\) the calculated spectral values of \(R_{rsw}\) were further compared with the measured ones and the respective cost function \(f\) was determined;
* \(\bullet\) the Jacobian of \(f\) was calculated;
* \(\bullet\) each optimization iteration was run using the estimated cost function and calculated Jacobian;
* \(\bullet\) vectors \(a^{*}\) and \(b_{b}{}^{*}\)(i.e. \(a^{*}_{ch}\), \(b^{*}_{b}\)\({}^{*}_{ch}\), \(a^{*}_{sm}\), \(b^{*}_{b}\)\({}^{*}_{sm}\), \(a^{*}_{doc}\)) were modified according to the yield of the previous iteration step;  
In each iteration step, each of the above 5 variables (i.e. the above specific absorption and backscattering coefficients) were operated. The model established in the previous step was used to retrieve CPA concentrations from \(R_{rsw}\) spectra measured _in situ_.