## 2 Continuous Wave Data and Uniqueness in Parameter Recovery
Solving condition-1 inside region-1 and region-2 (i.e., \(\eta_{1}=\overline{\eta}_{1}\) and \(\eta_{2}=\overline{\eta}_{2}\) with \(\
abla^{2}\gamma=0\)), we have:  
\[\mu_{a_{1}}\cdot\mu_{s_{1}}^{{}^{\prime}}=\overline{\mu}_{a_{1}}\cdot \overline{\mu}_{s_{1}}, \tag{4}\]  
\[\mu_{a_{2}}\cdot\mu_{s_{2}}^{{}^{\prime}}=\overline{\mu}_{a_{2}}\cdot \overline{\mu}_{s_{2}}. \tag{5}\]  
At the boundary between region-1 and 2, knowing that \(\
abla^{2}\gamma\propto(\gamma_{2}-\gamma_{1})\), and from Equations (4) and (5):  
\[\frac{\gamma_{1}}{\gamma_{2}}=\frac{\overline{\gamma}_{1}}{\overline{\gamma} _{2}}\Rightarrow\frac{\mu_{s_{1}}^{{}^{\prime}}}{\overline{\mu}_{s_{2}}^{{}^ {\prime}}}=\frac{\overline{\mu}_{s_{1}}^{{}^{\prime}}}{\overline{\mu}_{s_{2}} ^{{}^{\prime}}} \tag{6}\]  
Similar analysis can be extended for more than two regions if needed and therefore the solution set \(\{\overline{\mu}_{a},\ \overline{\mu}_{s}^{{}^{\prime}}\}\) should be of the form \(\left\{\alpha\mu_{a},\ \alpha^{-1}\ \mu_{s}^{{}^{\prime}}\right\}\) everywhere in the medium, for an arbitrary constant \(\alpha\). An equivalent representation of this non-uniqueness expression in terms of spectral parameters chromophore concentrations (\(C_{\rm i}\)), scattering amplitude (\(Sa\)) and scattering power (\(Sp\)) can be shown as \(\left\{\overline{C_{\rm i}},\bar{Sa},\overline{Sp}\right\}=\left\{\alpha C_{ \rm i},\alpha^{-1}Sa,\mathsf{Sp}\right\}\), where these spectral parameters are related to the optical properties as follows:  
\[\mu_{a}(\lambda)=\sum_{\rm i=1}^{N}\varepsilon_{\rm i}(\lambda)\cdot C_{\rm i} \tag{7}\]  
\[\mu_{s}^{{}^{\prime}}(\lambda)=Sa\ \cdot\lambda^{-Sp} \tag{8}\]  
Here, \(C_{\rm i}\) denotes the concentration of the major chromophores contributing to light absorption, \(\varepsilon_{\rm i}\) denotes the extinction coefficient of the corresponding chromophore. Applying condition 2, \(\overline{\gamma}_{boundary}=\gamma_{boundary}\ \Rightarrow\ \overline{\gamma}_{1}=\gamma_{1}\ \Rightarrow\ \overline{\mu}_{s_{1}}^{{}^{\prime}}\), this gives the value of the arbitrary constant to be \(\alpha=1\). This shows that for measured boundary intensity, the solution set is unique with the assumed regional homogeneity. However, if the measured data is either a spatial or spectral derivative of the form \(\log\!\left(\Phi_{\rm boundary}({\bf r}_{1},\lambda_{1})/\Phi_{\rm boundary }({\bf r}_{2},\lambda_{2})\right)=\log\!\left(\Psi_{0}({\bf r}_{1},\lambda_{ 1})\cdot\gamma_{1}(\lambda_{2})/\Psi_{0}({\bf r}_{2},\lambda_{2})\cdot\gamma_ {1}(\lambda_{1})\right)\), any arbitrary value of the constant \(\alpha\) leads to the same measured data, thus allowing for a non-unique solution set. In such a scenario, this non-uniqueness leads to uncertain parameter recovery, and this is seen in terms of the cross-talk between chromophore concentrations and scattering amplitude. However, biomarkers such as tissue oxygenation index (TOI), which are based on the ratio of chromophore concentrations, will still be accurate, as all the chromophore concentrations in any of the non-unique solution sets differ only by a common constant factor from true values. To demonstrate the non-uniqueness in terms of the widely used spectral derivative of intensity data, let us consider an example of a two-layered medium with two major light-absorbing chromophores i.e., oxy-hemoglobin (its concentration denoted by HbO) and deoxy-hemoglobin (its concentration denoted by Hb) with their layer-specific values given as HbO\({}_{1}=0.047\) mM, HbO\({}_{2}=0.054\) mM, Hb\({}_{1}=0.011\) mM, Hb\({}_{2}=0.022\) mM, Sa\({}_{1}=0.64\) mm\({}^{-1}\), Sa\({}_{2}=0.98\) mm\({}^{-1}\), Sp\({}_{1}=1.06\), Sp\({}_{2}=0.611\), corresponding to layer-1 and layer-2 respectively, and L=10 mm.