# Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette
## 3 Results
### Confounds to Fit Retrieved Absolute Optical Properties
#### 3.3.2 Assumption of Index of Refraction  
Finally, we examine how the assumption of \(n\) (and by extension the model boundary conditions) affects the recovered \(\mu_{a}\) and \(\mu_{s}^{\prime}\). We have done this by running the fit _assuming_ sets of \(n_{i}\) and \(n_{o}\), but generating forward data with different \(true\)\(n\)s in the range 1 to 2 (we do not co-vary \(n_{i}\) and \(n_{o}\) for simplicity). 2 sets of \(n\)s _assumed_ in the fit were invested:  
* \(n_{i,assumed}=1.3\) & \(n_{o,assumed}=1.0\) (Figure 9 solid lines). * \(n_{i,assumed}=1.3\) & \(n_{o,assumed}=2.0\) (Figure 9 dashed lines). This exercise was done for all 9 sets of \(\mu_{a}\) and \(\mu_{s}^{\prime}\) shown in Figure 6.  
Figure 9 shows these recovered \(\mu_{a}\) and \(\mu_{s}^{\prime}\) for the 2 _assumed_ cases while varying \(n_{i,true}\) and \(n_{o,true}\). First, we note that \(n_{i}\) has a larger effect on the recovered \(\mu_{a}\) and \(\mu_{s}^{\prime}\) compared to \(n_{o}\), with \(\mu_{a}\) having a negative, and \(\mu_{s}^{\prime}\) a positive, correlation with \(n_{i,true}\) (Figure 9a,c). Furthermore, \(\mu_{a}\) is much more strongly affected by \(n_{i,true}\) than \(\mu_{s}^{\prime}\), with recovered values being up to about 7 times greater than the \(true\) value when there is a low \(n_{i,true}\) value. For high \(n_{i,true}\) the recovered \(\mu_{a}\) often approaches 0 mm\({}^{-1}\) (hitting the fmincon constraint). All of this suggests that the method's ability to accurately recover \(\mu_{s}^{\prime}\) and particularly \(\mu_{a}\) is dependent on knowledge of \(n_{i}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mu_{a,true}\) & \(\mu_{s,true}^{\prime}\) & \(\sigma_{\mu_{a}}\) & \(\sigma_{\mu_{a}}\mu_{\overline{\mu}_{a}}\) & \(\sigma_{\mu_{s}^{\prime}}\) & \(\sigma_{\mu_{s}^{\prime}}\mu_{\overline{\mu}_{s}^{\prime}}\) & \(\tau_{\mu_{a},\mu_{s}^{\prime}}\) \\ (mm\({}^{-1}\)) & (mm\({}^{-1}\)) & (mm\({}^{-1}\)) & & (mm\({}^{-1}\)) & & (mm\({}^{-1}\)) \\ \hline
0.005 & 0.5 & 0.0008 & 0.2 & 0.01 & 0.02 & \(-0.9986\) \\
0.005 & 1.0 & 0.0003 & 0.06 & 0.01 & 0.01 & \(-0.9970\) \\
0.005 & 2.0 & 0.0001 & 0.02 & 0.02 & 0.008 & \(-0.9931\) \\
0.010 & 0.5 & 0.001 & 0.1 & 0.01 & 0.03 & \(-0.9986\) \\
0.010 & 1.0 & 0.0004 & 0.04 & 0.01 & 0.01 & \(-0.9983\) \\
0.010 & 2.0 & 0.0002 & 0.02 & 0.02 & 0.009 & \(-0.9971\) \\
0.020 & 0.5 & 0.001 & 0.07 & 0.02 & 0.03 & \(-0.9990\) \\
0.020 & 1.0 & 0.0007 & 0.04 & 0.02 & 0.02 & \(-0.9988\) \\
0.020 & 2.0 & 0.0003 & 0.01 & 0.02 & 0.009 & \(-0.9981\) \\ \hline \end{tabular} Symbols: Absorption coefficient (\(\mu_{a}\)), reduced scattering coefficient (\(\mu_{s}^{\prime}\)), Green’s function for the complex Transmittance (\(\widehat{T}\)), uncertainty (\(\sigma\)), and correlation coefficient (\(r\)).  
\end{table} 
Table 1: Errors for 9 sets of \(true\) optical properties and 101 noise simulations using \(\sigma_{|\widehat{T}|}/|\widehat{T}|=0.01\) and \(\sigma_{\angle\widehat{T}}=1.7\,\text{mrad}=0.1^{\circ}\). Now, focusing on Figure 9b,d, we see the effect of \(n_{o}\). In this case \(\mu_{s}^{\prime}\) is almost not affected at all by \(n_{o,true}\) and \(\mu_{a}\) is much more significantly affected when \(n_{o,true}<n_{i,true}\) or \(n_{r,true}>1\). In this case the correlation between \(\mu_{a}\) and \(n_{o}\) is positive (opposite to that for \(n_{i}\)), suggesting a connection to the dependence on \(n_{r}\). These results further re-enforce the idea that recovered \(\mu_{a}\) would be highly effected by the true \(n\)s, thus control or knowledge of the \(n\)s for this method is critical. Lastly, by comparing the two \(n_{i,assumed}\) and \(n_{o,assumed}\) sets (solid versus dashed lines) we notice that the recovered \(\mu_{a}\) and \(\mu_{s}^{\prime}\) vary less for the dashed lines (Figure 9).