## 2 Methods
### Laboratory Experiments and Mie Scattering Calculations to Evaluate LISST-VSF
#### 2.1.4 Determination of Correction Functions  
In addition to \(\beta_{p}^{LISST*}(\psi)\), Figure 1 shows results for the 200 nm and 2000 nm polystyrene beads based on Mie scattering calculations (more details about these calculations are provided below). These results are significantly higher (nearly a factor of 2) than the measured values of \(\beta_{p}^{LISST*}(\psi)\). We assume that the Mie scattering calculations for samples of spherical polystyrene beads are sufficiently accurate to provide reference values for such samples. In order to correct for the mismatch between the measured and reference values, a calibration correction function \(CF(\psi)\) is defined as  
\[CF(\psi)=\frac{\beta_{p}^{REF}(\psi)}{\beta_{p}^{LISST*}(\psi)}, \tag{5}\]  
where \(\beta_{p}^{REF}(\psi)\) is a reference volume scattering function determined according to  
\[\beta_{p}^{REF}(\psi)=\widetilde{\beta}_{p}^{ Mie}(\psi)b_{p}^{REF}, \tag{6}\]  
where \(\widetilde{\beta}_{p}^{ Mie}(\psi)\) is the scattering phase function [sr\({}^{-1}\)] obtained from Mie scattering computations and \(b_{p}^{REF}\) is the reference particulate scattering coefficient [m\({}^{-1}\)]. Note that all quantities in Equations (5) and (6) are for the LISST-VSF light wavelength of 532 nm. For each examined suspension of standard polystyrene beads, \(\widetilde{\beta}_{p}^{ Mie}(\psi)\) was determined from Mie scattering computations for homogeneous spherical particles. We used the Mie scattering code for homogeneous spheres of Bohren and Huffman [20], which included our modifications to account for polydispersity of the sample, i.e., to use particle size distribution as input to the code rather than just a single particle diameter as in the original code. The computations were performed assuming a relative particle size distribution (PSD) of Gaussian shape, with 300 evenly spaced size bins about the nominal mean diameter \(\pm\) 3 standard deviations, as provided by the manufacturer for each bead size (Table 1). The use of such PSDs allows us to account for the realistic, small degree of polydispersity of each sample. The Mie computations also require input of the refractive index of particles. Based on the study of [PERSON] et al. [68] we assumed that the complex refractive index of polystyrene relative to water at 532 nm is \(m=1.193+0.0003i\), where the first component is the real part and the second component is the imaginary part of refractive index. Note that the imaginary part is very small because polystyrene is a weakly absorbing material in the examined spectral region. Equation (6) also requires \(b_{p}^{REF}\), which was determined from the combination of beam attenuation measurements and Mie scattering calculations as  
\[b_{p}^{REF}=c_{p}^{LISST*}\frac{Q_{b}^{Mie}}{Q_{c}^{Mie}}, \tag{7}\]  
where \(Q_{b}^{Mie}\) and \(Q_{c}^{Mie}\) are the single-particle scattering and attenuation efficiency factors, respectively, obtained from Mie computations. Because the populations of examined beads exhibit a slight degree of polydispersity, the calculated \(Q_{b}^{Mie}\) and \(Q_{c}^{Mie}\) represent the average values of efficiency factors for a given particle population [69]. Given very weak light absorption of polystyrene beads at 532 nm, the ratio \(\frac{Q_{b}^{Mie}}{Q_{b}^{Mie}}\) was found to be \(>\)95%. We also note that in addition to \(c_{p}^{LISST*}\), we have another potential measurement of beam attenuation coefficient from the spectrophotometer (\(c_{p}^{SPEC}\)). Figure 2 shows that the measurements of \(c_{p}^{LISST*}\) and \(c_{p}^{SPEC}\) are consistent, and generally agree very well. By combining Equations (6) and (7), \(\beta_{p}^{REF}(\psi)\) can be determined for each LISST-VSF measurement as  
\[\beta_{p}^{REF}(\psi)=\widetilde{\beta}_{p}^{Mie}(\psi)\;c_{p}^{LISST*}\;\frac {Q_{b}^{Mie}}{Q_{c}^{Mie}}. \tag{8}\]  
Note that the estimates of \(\beta_{p}^{REF}(\psi)\) can vary between individual measurements because of variations in \(c_{p}^{LISST*}\).