## 2 Methods
### Laboratory Experiments and Mie Scattering Calculations to Evaluate LISST-VSF
#### 2.1.1 Instrumentation  
A LISST-VSF instrument (S/N 1475) was equipped with a custom designed 2 L sample chamber for benchtop laboratory use. This chamber effectively rejects ambient light and promotes good mixing conditions to maintain particles in suspension. For sample illumination the LISST-VSF uses a frequency-doubled YAG laser to produce a beam of light at a wavelength of 532 nm with a Gaussian beam profile of 3 mm in diameter. A single measurement takes approximately 4 s and consists of two scans of a 15-cm path within the sample, each with a different linear polarization state of the incident beam, i.e., parallel and perpendicular to the scattering plane. Scattered intensity is measured at multiple scattering angles \(\psi\) from 0.09\({}^{\circ}\) to 15.17\({}^{\circ}\) with 32 logarithmically-spaced ring detectors and from 14\({}^{\circ}\) to 155\({}^{\circ}\) with 1\({}^{\circ}\) interval using a fixed axis Roving Eyeball sensor equipped with photomultiplier tubes (PMTs). For the Roving Eyeball, scattered light is split between two PMTs with a polarizing prism allowing for only parallel or perpendicularly polarized light to be detected by each PMT. To enable measurements of large dynamic range of scattered intensity with a single PMT, the laser power is dimmed by a factor of 8 for the angular range 14-63\({}^{\circ}\) and returned to full power for 64-155\({}^{\circ}\). The beam attenuation coefficient, \(c\), is also measured at light wavelength of 532 nm for the 15-cm path length of the sample. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Nominal \(D\)** & **Catalog** & \(\overline{D}\) & \(SD\) & \(c_{p}^{SPEC}\) & **Dilution Factor** & **Dilution Factor** \\
**[nm]** & **No.** & **[nm]** & **[nm]** & **[m\({}^{-1}\)]** & **(PMT 500)** & **(PMT 550)** \\ \hline
100 & 3100A & 100 \(\pm\) 3 & 7.8 & 58.63 & \
\begin{tabular}{c} DF1: 96, DF2: 48.5, \\ _DF3: 32.7_ \\ \end{tabular} &
\begin{tabular}{c} DF1: 96, _DF2: 48.5,_ \\ _DF3: 32.7_ \\ \end{tabular} \\
400 & 3400A & 400 \(\pm\) 9 & 7.3 & 51.44 & \
\begin{tabular}{c} DF1: 87.4, _DF2: 44.2,_ \\ _DF3: 29.8_ \\ \end{tabular} &
\begin{tabular}{c} DF1: 87.4 \\ _DF2: 20_ \\ \end{tabular} \\
500 & 3500A & 508 \(\pm\) 8 & 8.5 & 20.64 & **DF2: 20** \\
700 & 3700A & 707 \(\pm\) 9 & 8.3 & 50.93 &
\begin{tabular}{c} DF1: 96, _DF2: 48.5_ \\ \end{tabular} \\
2000 & 4202A & 2020 \(\pm\) 15 & 21 & 18.21 & **DF2: 20** \\ \hline \end{tabular}
\end{table} 
Table 1: Information on the polystyrene bead size standards used to create laboratory sample suspensions for experiments. The nominal bead diameter (\(D\)), catalog number, and actual mean diameter \(\overline{D}\) (\(\pm\) standard error of estimate) and standard deviation of the mean (\(SD\)) provided by the manufacturer (Thermo Fisher Scientific, Inc.) is listed. The particulate beam attenuation coefficient at light wavelength 532 nm of the master sample as determined with a spectrophotometer, \(c_{p}^{SPEC}\), is listed in addition to specific dilution names and factors (e.g., DF1, DF2, etc.) of the master suspension used for LISST-VSF measurements at different PMT gain settings. The dilution factors in italic font denote the experimental data used for generation of the final correction functions \(CF_{f}\) and \(BF_{f}\), and those in boldface font denote the six examples used for statistical evaluation in Table 2.  
For incoherent elastic scattering of light at a given wavelength \(\lambda\) by a collection of particles suspended in water, the Stokes vector of incident light beam, \(S_{i}=\left[I_{i}\ Q_{i}\ U_{i}\ V_{i}\right]^{T}\), where T represents the transpose operation, is transformed into the Stokes vector of scattered beam, \(S_{s}(\psi)\), by a scattering matrix, \(P(\psi)\). For an ensemble of randomly-oriented particles exhibiting certain symmetry properties and no optical activity, the scattering matrix simplifies to 6 independent non-zero elements [20; 36; 51]  
\[S_{s}(\psi)=\left[ \begin{array}{c}I_{s}(\psi)\\ Q_{s}(\psi)\\ U_{s}(\psi)\\ V_{s}(\psi)\end{array} \right]=P(\psi)S_{i}=C\left[