## Theoretical Considerations  
### Backscattering Cross Section for Polydisperse Particle Assemblages  
Light scattering is produced by the presence of an object (such as a particle) with a refractive index different from that of the surrounding medium. The refractive index is expressed in complex form as \(n(\lambda)=n_{r}(\lambda)\ +\ i\ n_{i}(\lambda)\), where \(\lambda\) is the wavelength of the radiation in vacuum in units of nm. The real part determines the phase velocity of the propagating wave and the imaginary part accounts for the absorption. Please note that the refractive index is a relative value dependent upon the surrounding medium, i.e., relative to the refractive index of the medium. The single scattering process by a particle is described by the scattering cross section \(C_{sca}(D,\lambda)\) (units m\({}^{2}\)) and the scattering phase function \(\tilde{F}(D,\theta,\lambda)\) (dimensionless) as defined by [PERSON] et al. [26] (Equations (4.51)-(4.53), pp. 100-101):  
\[\frac{1}{2}\int_{0}^{\pi}\tilde{F}(D,\lambda,\theta)\,sin\theta\,d\theta=1 \tag{1}\]  
As particles are here assumed to be spherical, \(\tilde{F}\) depends only on the particle diameter \(D\), the scattering angle \(\theta\) within the arbitrary azimuthal plane of scattering, and the wavelength \(\lambda\). In the following, \(\lambda\) is omitted for brevity. To account for polydisperse particulate assemblages, the particle size distribution (PSD) is defined. For the present study, we adopt a power-law PSD (also named the Junge-like PSD) which is commonly used to represent the size distribution of marine particles in natural waters [5; 24; 27; 28]. The ensemble-average normalized phase function is:  
\[\tilde{F}(\theta)=\,\int_{D_{min}}^{D_{max}}\tilde{F}(D,\theta)\times A\,D^{- \xi}dD \tag{2}\]  
where \(D_{min}\) and \(D_{max}\) define the particle diameter range, \(\xi\) is the hyperbolic slope of PSD, and \(A\,D^{-\xi}\) (units, \(\upmu\)m\({}^{-1}\)) is the relative differential particle size distribution. As in many theoretical studies, the relative PSD is normalized such that the integral over the entire size range is unity. It follows that \(\tilde{F}(\theta)\) represents the average normalized phase function per particle. Equation (2) can be written for the scattering cross section replacing \(\tilde{F}(D,\theta)\) with \(C_{sca}(D)\) and \(\tilde{F}(\theta)\) with \(C_{sca}\). The backscattering cross section of the polydisperse assemblage is defined as:  
\[C_{sca}^{bb}=\frac{C_{sca}}{2}\,\int_{\pi/2}^{\pi}\tilde{F}(\theta)\,sin\theta \,d\theta \tag{3}\]  
It can be easily seen from Equations (1)-(3) that the integration of \(\tilde{F}(\theta)\) between 0 and \(\pi\) gives \(C_{sca}\), the scattering cross section of the polydisperse population. Many numerical codes (including those described in Section 3) use the normalized phase function \(\tilde{F}(\theta)\) to describe the angular distribution of the scattered radiation. However, in hydrologic optics, the volume scattering function (VSF), \(\tilde{\beta}(\theta)\) (m\({}^{-1}\) sr\({}^{-1}\)), is more commonly used instead of \(\tilde{F}(\theta)\)[29]. The relationship between \(\tilde{\beta}(\theta)\) and \(\tilde{F}(\theta)\) is:  
\[\tilde{\beta}(\theta)=\frac{N\,C_{sca}}{4\pi}\,\tilde{F}(\theta) \tag{4}\]  
with \(N\) the number of particles per cubic meter.