## 5 Results  
### Accuracy of Numerical Computations  
A numerical integration over \(\theta\) is required to derive \(b_{p}^{\theta_{s}}\) and \(b_{bp}\) from the normalized phase function (Section 2). Due to the sharp increase of the normalized phase function in the forward scattering directions (Figure 3), the selection of the relevant angular step for the numerical integration is crucial. For that purpose, the impact of angular step (\(\Delta\theta\)) on the calculation of \(\widehat{b_{p}^{\theta_{s}}}\) is studied using Lorentz-Mie simulations in DS1 (Figure 1, N\({}^{\circ}\)M2, M3). The normalized phase function of polydisperse particles \(\tilde{F}(\theta)\) exhibits a maximum around \(\theta\) = 0\({}^{o}\)[26]. For small \(\xi\) value, that is when the proportion of large-sized particles compared to smaller particles increases, the forward peak is sharper. Indeed, for particles with a large diameter as compared to the wavelength, \(\tilde{F}(D,\theta)\) displays a sharp forward peak [26] due the concentration of light near \(\theta\) = 0\({}^{o}\) caused by diffraction. The presence of the peak in \(\tilde{F}(\theta)\) requires several integration points large enough to provide the desired numerical accuracy. The numerical integration over \(\theta\) (Figure 1, N\({}^{\circ}\)M2) is performed using the \"Trapz\" function from the Numpy package with Python. The \"Trapz\" function performs an integration along the given axis using the composite trapezoidal rule. To test the accuracy of the integration and to find the correct integration step, \(\Delta\theta\), we compare the result of the numerical integration of \(\tilde{F}(\theta)\) between 0 and \(\pi\) to its theoretical value (=2) (Figure 1, N\({}^{\circ}\)M3). When \(\Delta\theta\) = 0.05\({}^{o}\), corresponding to a total number of integration steps (N\({}_{\theta}\)) of 3600, the numerical integration value of \(\tilde{F}(\theta)\) is in the range [1.999-2.000] for small \(\xi\). For larger \(\xi\), it is in the range [1.800-1.999]. When the value of the numerical integration is in the range [1.800-1.999], a renormalization factor is applied to \(\tilde{F}(\theta)\) to ensure that the result of the numerical integration is exactly 2. We could also increase the number of integration points, but it would increase the computation time. Using a renormalization factor for large \(\xi\) is a good compromise to guarantee the accuracy and save computation time. For two-layered spheres (i.e., phytoplankton cells), the ScattnLay code provides only normalized phase functions for monodisperse particles (Figure 1, N\({}^{\circ}\)S1), so the numerical integration over the particle diameter range (Equation (2)) is realized as a separate calculation with the Python \"Trapz\" function (Figure 1, N\({}^{\circ}\)S2). For monodisperse particles, the normalized phase function displays a forward peak as explained above but can also display a sequence of maxima and minima due to interference and resonance features [26; 42]. The frequency of the maxima and minima over the \begin{table}
\begin{tabular}{c c c c} \hline  & **Oligotrophic-Like** & **Phytoplankton Bloom** & **Coastal-Like** \\ \cline{2-4} \(\xi\) & **[Chla]** & **[Chla]** & **[Chla]** \\ \hline
3 & 8.35 & 11.51 & 7.497 \\
3.5 & 0.773 & 1.580 & 0.6889 \\
4 & 0.102 & 0.341 & 0.0884 \\ \hline \end{tabular} The chlorophyll-a concentration (mg m\({}^{-3}\)) is estimated, as an indication, using the relative abundance of phytoplankton cells described in Tables 3–5 and considering N\({}_{TOT}\) is 1.1262 \(\times\) 10\({}^{4}\) particles per m\({}^{3}\).  
\end{table} 
Table 7: Total chlorophyll-a concentration for the three case studies. range of \(\theta\) increases with both increasing \(n_{r}\) and size parameter (=\(\pi\,D/\lambda\)).