## 2 Methods
### Assessment of Robustness of the LM Procedures against Simulated Measurement Errors  
The LM procedure in the Matlab environment has been realized using the recipes given by [PERSON] [11]. The code for the L-M procedure used in the present study was developed by one of us ([PERSON]) using the C++ language. Calculations of the resultant bulk absorption and backscattering coefficients for the simulated aquatic medium were performed by using the hydro-optical model suggested by [PERSON] [12, 10]. The model is a collection of tabulated spectral values of _a*_ and _b\({}_{b}\)*_ for the major CPAs, i.e. _chl, sm_ and _doc_ within the spectral region 400-700 nm, as obtained for case 2 waters with the trophic status varying from oligotrophic to eutrophic. In order to test the algorithm performance, the following numerical experiment was carried out. 1000 concentration vectors (_C\(=\) \(\langle C_{chb}\)_C\({}_{sm}\)_C\({}_{doc}\rangle\)_) were randomly selected in rather wide ranges (0-70 \(\mu\)g/L; 0-30 mg/L and 0-30 mgC/L for _chl, sm_ and _doc_, respectively). One thousand spectra of \(T\) were generated from concentration vectors using equations 1, 2 and 3. Another 13,000 spectra of \(S\) were simulated by contaminating the calculated \(T\) with 'noise' for the following options:  
* four levels of noise in input data: 0, 5, 10 and 15%;
* two types of noise distribution: uniform and normal;
* two types of \(\lambda\)-dependency: noise is independent and dependent on \(\lambda\) (the spectral dependence was assumed to be linearly decreasing with \(\lambda\)) [13, 14]. Our comparison of simulated and retrieved results showed a very high retrieval accuracy for the case with a zero input noise (correlation, r equals 0.999 for all three parameters, root mean square error, RMSE equals 1.8 \(\mu\)g/L for _chl_, 1.0 mg/L for _sm_ and 1.5 mgC/L for _doc_). Moreover, our simulations indicate that the L-M procedure remains stable for measurement errors up to 15 %. It was also found that the error level and its distribution as well as its dependence on \(\lambda\) strongly influence the attainable retrieval accuracy. The latter is also a function of the CPA concentration vector _per se_. For oceanographic and immunological research we assume that, at least, for case 2 waters, the _chl_ retrieval errors specified in Table 1 for a number of concentration ranges are admissible [15]. The simulations indicate that these accuracy requirements can only be fulfilled for specific hydro-optical conditions, i.e. the concentrations of _chl_, _sm_ and _doc_ have to fall into specific concentration ranges (Table 2). Our simulations also indicate (Table 2) that the _sm_ and _doc_ concentration ranges (within which the LM procedure yields _chl_ retrieval results with the permissible error as in table 1) shrink with increasing measurement error. The most stringent limitations arise in the case of normally distributed and \(\lambda\)-dependent measurement errors. Under favourable conditions (e.g. when _doc_ and _sm_ concentrations are in the range 0-2 mgC/L and 0-0.5 mg/L respectively, and the noise is \(\lambda\)-independent) _chl_ concentrations can be retrieved with acceptable accuracy in the entire range 0-30 \(\mu\)g/L (the corresponding cell is marked with asterisk in Table 2). In turbid waters (_C\({}_{sm}\)_ varies within 5-10 mg/L, _C\({}_{doc}\)_ - within 5-10 mgC/L) and with \(\lambda\) \begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline _chl_ **range, \(\mu\)g/L** & 0 - 5 & 5 - 10 & 10 - 20 & 20 - 30 \\ \hline \hline
**admissible error, \%** & 50 & 40 & 30 & 20 \\ \hline \end{tabular}
\end{table} 
Table 1: Acceptable accuracy of _chl_ retrieval for different _chl_ concentration rangesdependent noise (the corresponding cell is marked with double asterisk) chlorophyll can be retrieved accurately only in the range 0-5 \(\mu\)g/L.