## 3 Results
### Particulate Organic Carbon and Chlorophyll-a Concentration Modelling Optimization  
A simple inversion of the procedure used to calculate POC and ChIA values allows for the empirical optimization of the parameter pairs used in the cell volume scaling models. Maintaining the general form  
\[y(r)=kV(r)^{h}, \tag{12}\]  
Figure 9: Comparison of (**a**) modelled vs. measured POC and (**b**) modelled vs. measured ChIA when a spherical model of particle volume is employed, and comparison of (**c**) modelled vs. measured POC and (**d**) modelled vs. measured ChIA when a fractal model of particle volume is employed instead. POC and ChIA values calculated using the [PERSON] [19] and [PERSON] et al. [20] models are represented by dark grey squares and light grey diamonds respectively. The dashed grey lines indicate the 1:1 relationship. arrays of values for the parameters \(k\) and \(h\) can be generated, combined, and substituted in Equation (12), and the results compared and fitted against available measurements to identify the best parameter combinations, respectively (\(h_{1},k_{1}\)) and (\(h_{2},k_{2}\)) for POC and chlorophyll-a intraparticle concentrations. The optimization was initially applied to the whole UKCW dataset. Three parameter arrays were generated, one shared by exponents \(h_{1}\) and \(h_{2}\) plus one each for factors \(k_{1}\) and \(k_{2}\). The ranges were designed to encompass the parameter values of the models used thus far: specifically, 201 linearly spaced values for exponents \(h_{1}\) and \(h_{2}\) in a 0.6-1 range, 301 linearly spaced values for factor \(k_{1}\) in a 0.05-0.65 range and 461 linearly spaced values for factor \(k_{2}\) in a 0.004-0.05 range. Each (\(h_{1}\), \(k_{1}\)) and (\(h_{2}\), \(k_{2}\)) combination was then applied dataset-wide, compared against measured POC and ChlA and evaluated using the goodness-of-fit of a forced linear fit of the 1:1 line. The best parameter combinations were selected as those that minimized the RMSE values of the forced fit. The best POC model for the UKCW dataset was found as  
\[y(r)=0.442V(r)^{0.720} \tag{13}\]  
while the best chlorophyll-a model was found as  
\[y(r)=0.029V(r)^{0.736}. \tag{14}\]  
Comparisons of the optimized model results with the measured POC and ChlA values of the UKCW dataset are shown in Figure 10. RMS%E values for the comparisons were found to be 45.6% and 51.8% for POC and ChlA respectively. An analogous optimization procedure can be applied to the refractive index-based \(C_{i}\) and \(Chl_{i}\) estimation models. Maintaining the use of fractal particle volumes and the general form  
\[C_{x}=an_{x}-b \tag{15}\]  
common to both models as described by Equations (4)-(7), where \(C_{x}\) is either \(C_{i}\) or \(Chl_{i}\) and \(n_{x}\) is respectively either \(n_{r}\) or \(n_{i}\), arrays of values for the parameters \(a\) and \(b\) can be generated, combined and substituted in Equation (15), and the results compared and fitted against available measurements to identify the best parameter combinations. Interestingly, for both POC and ChlA the procedure could not identify single parameter combinations that minimized the RMSE; indeed, in both cases what was found was instead a continuum of parameter pairs which all minimized the RMSE to very similar values across the entire parameter ranges. We interpret this to be indicative of an excess of degrees of freedom in the relationship described by Equation (15) for the models to be properly constrained. Returning to the physical basis of these relationships given in [PERSON] [19], the empirical linear relationship between \(C_{i}\) and \(n_{r}\) reflects a physical realization that increasing carbon content will  
Figure 10: Comparison of (**a**) POC and (**b**) ChlA values as determined by the optimized models of Equations (13) and (14) vs. their respective measured values. The dashed grey lines indicate the 1:1 relationship. generally increase real refractive index.