# Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette
## 3 Results
### Confounds to Fit Retrieved Absolute Optical Properties  
#### 3.3.1 Propagation of Noise to Optical Property Uncertainty  
To test how noise propagates through the recovery of \(\mu_{a}\) and \(\mu^{\prime}_{s}\), when using the fit developed in Section 3.2, we simulated \(\sigma_{|\widetilde{\mathrm{T}}|}/|\widetilde{\mathrm{T}}|=0.01\) and \(\sigma_{\angle\widetilde{\mathrm{T}}}=1.7\,\mathrm{mrad}=0.1^{\circ}\) as mentioned in Section 2.4. This was done by simulating measured \(\ln|\mathrm{DR}\left\{\widetilde{\mathrm{T}}\right\}|\) and \(\angle\mathrm{DR}\left\{\widetilde{\mathrm{T}}\right\}\) 101 times and each time adding Gaussian noise with the \(\sigma\)s stated above. For each of the 101, the fit was run to recover some \(\mu_{a}\) and \(\mu^{\prime}_{s}\). This was done for all 9 of the sets of _true_\(\mu_{a}\) and \(\mu^{\prime}_{s}\) shown in Figure 6.  
The results from this exercise are shown in Figure 8 and Table 1, from these three main observations can be drawn:  
* The fractional error in \(\mu_{a}\) is always larger compared to \(\mu^{\prime}_{s}\) suggesting the system can more precisely recover \(\mu^{\prime}_{s}\).
* Errors in \(\mu_{a}\) are much larger for small \(\mu^{\prime}_{s}\) and slightly larger for small \(\mu_{a}\) (with small \(\mu_{a}\) and \(\mu^{\prime}_{s}\) together being the worst case).
* That \(\mu_{a}\) and \(\mu^{\prime}_{s}\) are highly negatively correlated (as suggested by Figure 5b,c).  
Observation A again expounds upon what has been expected from the shape of \(\chi^{2}\) space presented in previous sections. Furthermore, observations B and C are typical for such diffusion theory based problems.  
To draw some quantitative values for this exercise, we can closely examine Table 1. It is helpful to extract the worst case (for the type of simulations we have done), and typical (considering typical being the case when \(\mu_{a,true}=0.010\,\mathrm{mm}^{-1}\) and \(\mu^{\prime}_{s,true}=1.0\,\mathrm{mm}^{-1}\)) fractional errors in \(\mu_{a}\) and \(\mu^{\prime}_{s}\). These are as follows:  
Figure 8: Result from 101 simulations of noise (\(\sigma_{\widetilde{|T|}}/|\widetilde{T}|=0.01\) and \(\sigma_{\widetilde{|T}}=1.7\,\mathrm{mrad}=0.1^{\circ}\)) for the 9 different _true_ sets of the \(\mu_{a}\) and the \(\mu^{\prime}_{s}\) shown in Figure 6. (**a**) Marginal histograms for recovered \(\mu^{\prime}_{s}\) values of the 9 sets of optical properties and 101 noise simulations. (**b**) Scatter plot of recovered \(\mu_{a}\) and \(\mu^{\prime}_{s}\) for the \(9\times 101\) noise simulations. (**c**) Marginal histograms for recovered \(\mu_{a}\) values of the 9 sets of optical properties and 101 noise simulations. Acronyms and Symbols: Green’s function for the complex Transmittance (\(\widetilde{T}\)), uncertainty (\(\sigma\)), absorption coefficient (\(\mu_{a}\)), and reduced scattering coefficient (\(\mu^{\prime}_{s}\)).  
For \(\mu_{a}\):  
* Typical error of 4 %.
* Worst case error of 20 % (for low \(\mu_{a}\) and \(\mu_{s}^{\prime}\)).
* For \(\mu_{s}^{\prime}\):
* Typical error of 1 %.
* Worst case error of 3 % (for high \(\mu_{a}\) and low \(\mu_{s}^{\prime}\)).  
Of course these values are dependent on the simulated measurement errors of \(\sigma_{|\widehat{T}|}/|\widehat{T}|=0.01\) and \(\sigma_{\angle\widehat{T}}=1.7\,\text{mrad}=0.1^{\circ}\) which may be different for different instruments.