## 2 Materials and Methods  
### Theory  
#### 2.1.1 Particle Mass Modelling from Apparent Density of Hydrated Matter  
Calculations for modelled values of SPM, OSM, and ISM were made following the technique presented by [PERSON] et al. [6]. Building on the approach presented by [PERSON] [7] and [PERSON] et al. [8] the technique estimates a density value for the particulate matter which is dependent on the real part of the refractive index of the particles and is designed to account for their water content. Since this value is neither the value of the dry matter fraction of the particle nor that of water, but rather a combination of the two, this global density is also known as \"apparent\" density. The equation takes the form  
\[\rho=\rho_{o}V_{o}=\rho_{o}\frac{n_{r}-1}{n_{o}-1}, \tag{1}\]  
where \(n_{r}\) is the real refractive index of the whole particle and \(n_{o}\), \(\rho_{o}\), and \(V_{o}\) are respectively the real refractive index, density, and fractional volume of the dry matter fraction of the particle. All refractive index values are given relative to water. Values for the \(\rho_{o}/(n_{o}-1)\) ratio were defined following those employed by [PERSON] et al. [6]. For organic particles (defined as the fraction of the particle population with \(n_{r}<1.1\)) the mean value of the ratio was set at (8.56 \(\pm\) 1.1) \(\times\) 10\({}^{6}\) g/m\({}^{3}\). These are particles with high water content, as high as -80 \(\pm\) 10% for algal cells [9]. For mineral particles (defined as the fraction of the particle population with \(n_{r}\geq 1.1\)) the mean value of the ratio was instead set at (15.52 \(\pm\) 1.84) \(\times\) 10\({}^{6}\) g/m\({}^{3}\). These particles have low water content; when the fractional volume of dry matter reaches unity (i.e., water content within the particles is zero) the apparent density of the particle becomes equal to the density of the dry mineral matter and can be calculated accordingly. [PERSON] et al. [6] find \(n_{r}=1.16\) as the threshold above which \(V_{o}=1\), and give \(\rho=[(6.42\pm 0.85)n_{r}-(4.86\pm 0.99)]\times 10^{6}\) g/m\({}^{3}\) as the corresponding \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(N(D)\) & Number concentration of particles within particle size bin corresponding to particle diameter D, mL\({}^{-1}\) \\ \(N^{\prime}(D)\) & Density function of the particle size distribution, mL\({}^{-1}\)\(\mu\)m\({}^{-1}\) \\ \(N_{r}\) & Number of particles within particle size bin corresponding to particle radius \(r\), dimensionless \\ \(N_{D,nr}\) & Number of particles within particle bin corresponding to particle diameter \(D\) and real refractive index \(n_{r}\), dimensionless \\ \(n_{o}\) & Real refractive index of the dry matter fraction of the particle, dimensionless \\ \(n_{r}\) & Real refractive index of the particle, dimensionless \\ \(n_{i}\) & Imaginary refractive index of the particle, dimensionless \\ \(\overline{n}_{r,1},\overline{n}_{r,2}\) & Average real refractive indices at the upper and lower extremes of the particle size distribution respectively, dimensionless \\ \(r\), \(D\) & Particle radius and particle diameter, \(\mu\)m \\ \(r_{o}\) & Primary particle radius, \(\mu\)m \\ \(V_{D}\) & Particle volume, \(\mu\)m\({}^{3}\) \\ \(V_{o}\) & Volume of the dry matter fraction of the particle, \(\mu\)m\({}^{3}\) \\ \(y(r)\) & Volume scaling function \\ \(B\) & Fractal dimension exponent, dimensionless \\ \(\gamma\) & Power law slope, dimensionless \\ \(P\) & Particle density, g/m\({}^{3}\) \\ \(\rho_{nr}\) & Density of a particle with real refractive index \(n_{r}\), g/m\({}^{3}\) \\ \(\rho_{o}\) & Density of the dry matter fraction of the particle, g/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._density based on a linear regression of literature values of density and refractive index for a number of mineral species. Overall, the final expression of Equation (1) used in practice was  
\[\rho=\left\{