## 4 Results  
### Two-layered Semi-infinite Model  
The two-layered semi-infinite model as described in Equations (9) and (10), is employed to simulate the measurement data. The modelled measurement system consists of source-detector distances of 10 to 50 mm in steps of 10 mm, Figure 4, and wavelengths from 650 to 850 nm in steps of 10 nm, and a Gaussian random noise of 1% is added to the data. From previous studies, a noise level from 0.12 to 1.42% is generally added for source-detector separations varying from 13 to 48 mm [19], to represent a physical model. In this work, apart from multi-distance measurements varying from 10 to 50 mm, we have broadband wavelength data ranging from 650 to 850 nm, and therefore the distribution of noise levels is more complex than that. Therefore, an average of 1% random noise is applied for all source-detector separations and wavelengths. The tissue properties of the two layers are as shown in Table 1[19; 20], and 10 different cases of randomly varying scattering properties (i.e., both Sa and Sp) of both layers are considered with a standard deviation of 10% around the reference values, to include the inter-subject variability in scattering properties. To understand the response of the recovery method to different cerebral oxygenation levels, the TOI of layer-2 was also varied from 50% to 80% which are considered the practical extreme values [21; 22]. The sensitivity matrices corresponding to step-1 and step-2 of the recovery process as given in Equations (11) and (12), are calculated using perturbation method on the analytic expression of Equations (9) and (10). The whole simulation study is repeated for different layer-1 thickness values of L = 8, 10, 12 mm which are in the range of practical values for the extra-cerebral thickness of human head. The recovered TOI values of the two-layers for different thicknesses of layer-1 (8, 10, 12 mm) are as shown in Figure 5 in comparison to the homogenous parameter recovery. The homogenous parameter recovery is similar to step-1 in the proposed two-layered recovery process, but all the measured source-detector distances are considered. The retrieved values of layer-1 thickness is as shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline  & **HbT (mM)** & **TOI (\%)** & **Sa (mm\({}^{\text{-1}}\))** & **Sp** \\ \hline Layer-1 & 0.059 & 80 & \(0.64\pm 10\%\) & \(1.0685\pm 10\%\) \\ \hline Layer-2 & 0.076 & 50-80 & \(0.98\pm 10\%\) & \(0.611\pm 10\%\) \\ \hline \end{tabular}
\end{table} 
Table 1: Tissue parameters. Figure 4: Schematic showing semi-infinite model with source-detector locations on the boundary. Figure 5: Estimated tissue oxygenation index (TOI) values from the homogenous and two-layered parameter recovery methods corresponding to a two-layered semi-infinite medium of Layer-1 thickness as (**a**) 8 mm, (**b**) 10 mm, and (**c**) 12 mm, using multi-distance hyper-spectral continuous wave (CW) data. The ground-truth values of layer-1 and layer-2 is shown by dashed lines. The shaded region shows the standard deviation of recovery for different scattering parameters. The above results (Table 2) show that the proposed two-layered hyper-spectral parameter recovery method retrieves the tissue oxygenation index and tissue thickness within 4.4% and 4% error respectively. Recovery of other tissue parameters (hemoglobin concentrations and scattering amplitude of two layers) can be obtained within 16.6% error with this two-layered approach, as compared to 31.5% error with the homogenous parameter recovery. However, the scattering power parameter exhibits relatively higher recovery errors due to its low sensitivity to light-intensity.