## Theoretical Considerations
### The Bulk Backscattering Ratio  
Marine particles are divided into five different categories: viruses (VIR), heterotrophic bacteria (BAC), phytoplankton (PHY), organic detritus (DET), and minerals (MIN). Table 1 displays the size ranges and the refractive indices of the different components as defined by previous studies [21; 24; 25]. The ensemble-average values of \(\tilde{F}_{j}(\theta)\), \(C_{sca,j}\), and \(C_{sca,j}^{bb}\) are computed from Equations (1)-(3) for each particulate component \(j\).  
The total normalized phase function and total scattering cross section of the water body are obtained as in [PERSON] et al., 2002 (Equations (4.74) p. 102 and (3.13) p. 71 ):  
\[F^{tot}(\theta)=\frac{\sum\limits_{j=1}^{5}N_{j}\,C_{sca,j}\,\tilde{F}_{j}( \theta)}{\sum\limits_{j=1}^{5}N_{j}\,C_{sca,j}} \tag{5}\]  
\[C_{sca}^{bb,\,tot}=\sum\limits_{j=1}^{5}N_{j}\,C_{sca,j}^{bb} \tag{6}\]  
where \(N_{j}\) is the relative concentration (i.e., the relative number of particles per unit volume of water) of the considered component. \(C_{sca}^{tot}\) is defined by replacing \(C_{sca,j}^{bb}\) with \(C_{sca,j}\) in Equation (6).  
The total (i.e., bulk) backscattering coefficient (b\({}_{bp}\)) (units m\({}^{-1}\)) of the water body is the sum of the relevant \(b_{bp,j}\) associated with each \(j\)th group. \(b_{bp,j}\) is equal to the polydisperse \(C_{sca,j}^{bb}\) weighted by the particle concentration of the \(j\)th group:  
\[b_{bp}=\sum\limits_{j=1}^{5}b_{bp,j}=N_{TOT}\times\,C_{sca}^{bb,\,tot} \tag{7}\]  
with \(N_{TOT}\) the total particle concentration (particles per m\({}^{3}\)) in the water body. Similarly, \(b_{p}\) is defined from Equation (7) by replacing \(b_{bp}\) with \(b_{p}\) and \(C_{sca}^{bb,\,tot}\) with \(C_{sca}^{tot}\). The bulk backscattering ratio \(\tilde{b_{bp}}\) is the dimensionless ratio:  
\[\tilde{b_{bp}}=\frac{b_{bp}}{b_{p}} \tag{8}\]  
In this study, we will use the bulk particulate real refractive index (\(\tilde{n_{r}}\)), which reproduces the bulk scattering properties of a water body. It represents the mean refractive index weighted by the scattering cross sections of all the particles:  
\[\tilde{n_{r}}=\frac{\sum\limits_{j=1}^{5}n_{r,j}\times N_{j}\,C_{sca,j}}{ \sum\limits_{j=1}^{5}N_{j}\,C_{sca,j}} \tag{9}\]  
Similarly, the bulk imaginary refractive index (\(\tilde{n_{i}}\)) is defined as follows:  
\[\tilde{n_{i}}=\frac{\sum\limits_{j=1}^{5}n_{i,j}\times N_{j}\,C_{abs,j}}{ \sum\limits_{j=1}^{5}N_{j}\,C_{abs,j}} \tag{10}\]  
where \(C_{abs,j}\) is the absorption cross section of particles.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Component (\(j\))** & **Sphere Model** & \(\mathbf{D_{min}}\)**-\(\mathbf{D_{max}}\) (\(\mathbf{\upmu}\))** & \(\mathbf{n_{r}}\) & \(\mathbf{n_{i}}\) \\ \hline Viruses & homogeneous & 0.03–0.2 & 1.05 & 0 \\ Heterotrophic bacteria & homogeneous & 0.2–2 & 1.05 & \(1.0\times 10^{-4}\) \\ Phytoplankton cells & two or three-layered & 0.3–40 & 1.044 * & \(1.5\times 10^{-3}\) * \\ Organic detritus & homogeneous & 0.05–500 & 1.04 & \(2.3\times 10^{-5}\) \\ Minerals & homogeneous & 0.05–500 & 1.18 & \(1.0\times 10^{-4}\) \\ \hline \end{tabular}
\end{table}
Table 1: Summary of the seawater constituents.