# Dual-Slope Diffuse Reflectance Instrument for Calibration-Free Broadband Spectroscopy
## 2 Materials and Methods
### Analysis
Thus, an iterative method is used to recover the \(\widetilde{\mu_{eff}}\):  
\[\widetilde{y}\left[\mu_{a,k},\mu_{s,k}^{{}^{\prime}}\right]=-\widetilde{\mu_{ eff}}f_{k+1}r_{1}\left[\mu_{s,k}^{{}^{\prime}}\right]+\widetilde{b}_{k+1}, \tag{5}\]  
where \(k\) is the iteration number, and \(\widetilde{b}\) is the complex intercept (dependent on instrumental factors not considered in Equation (4)). For the iterative method, an initial guess of \(\mu_{a}\) and \(\mu_{s}^{{}^{\prime}}\) is needed for \(k=0\). This guess was determined based on the linear slopes of complex reflectance amplitude and phase as previously described [19; 21]. At each iteration, the current values of \(\widetilde{y}\) and \(r_{1}\) were used to fit a complex linear slope (\(-\widetilde{\mu_{eff}}\)):  
\[\left[ \begin{matrix}-\widetilde{\mu_{eff}}\\ \widetilde{b}\end{matrix} \right]_{k+1}=\left[ \begin{matrix}r_{1,1}&1\\ \vdots&\vdots\\ r_{1,n}&1\end{matrix} \right]^{+}_{k}\left[ \begin{matrix}\widetilde{y}_{1}\\ \vdots\\ \widetilde{y}_{n}\end{matrix} \right]_{k}, \tag{6}\]  
where the \(+\) superscript is the Moore-Penrose inverse, \(\widetilde{b}\) is the complex intercept (ignored), and there are \(n\) measurements (distances). In the case of MD FD-NIRS, \(n=11\) since there were eleven measurements of complex reflectance at eleven different source-detector distances. To convert \(\widetilde{\mu_{eff}}\) to \(\mu_{a}\) and \(\mu_{s}^{{}^{\prime}}\) the following expressions were used:  
\[\mu_{t}^{{}^{\prime}}=\frac{-2c\Re\left[\widetilde{\mu_{eff}}\right]\Im \left[\widetilde{\mu_{eff}}\right]}{3n\omega}, \tag{7}\]  
\[\mu_{a}=\frac{\Re\left[\widetilde{\mu_{eff}}\right]^{2}-\Im\left[\widetilde{ \mu_{eff}}\right]^{2}}{3\mu_{t}^{{}^{\prime}}}, \tag{8}\]  
where \(\mu_{t}^{{}^{\prime}}\) is the total reduced attenuation coefficient, and \(\mu_{s}^{{}^{\prime}}=\mu_{t}^{{}^{\prime}}-\mu_{a}\). At each iteration, the new fitted \(-\widetilde{\mu_{eff}}\) yielded the \(\mu_{a}\) and \(\mu_{s}^{{}^{\prime}}\) used to calculate \(\widetilde{y}\) and \(r_{1}\) for the next iter ation (Equation (5)). The procedure terminated when the condition \(|\widetilde{\mu_{eff}}_{k}-\widetilde{\mu_{eff}}_{k-1}|<10^{-4}\) L/mm was met. The above expressions show how MD FD-NIRS data were used to find absolute \(\mu_{a}\) and \(\mu_{s}^{{}^{\prime}}\) at two wavelengths. To extend this to DS CW-bDRS, first, Equation (6) was modified to use DS data. DS yields two symmetric measurements of \(R\) versus \(\rho\) (where \(R\) is now real not complex since CW data is used). Thus, instead of Equation (6), the following was used for DS CW-bDRS:  
\[\mu_{eff}_{k+1}=\frac{-1}{2}\bigg{(}\frac{y_{2,1}-y_{1,1}}{r_{1,2,1}-r_{1,1,1} }+\frac{y_{2,2}-y_{1,2}}{r_{1,2,2}-r_{1,1,2}}\bigg{)}_{k}, \tag{9}\]  
where the first subscript of \(y\) and second subscript of \(r_{1}\) corresponds to source-detector distance, and the second and third subscript, respectively, corresponds to the symmetric slope in the DS set. This expression is simply the average of the slopes of \(y\) versus \(r_{1}\) between the two DS symmetric sets. In addition, note that the tildes were removed since \(y\) is now real (from real \(R\)). This results in a real effective attenuation coefficient (\(\mu_{eff}=\sqrt{3\mu_{a}(\mu_{s}^{{}^{\prime}}+\mu_{a})}\) ) instead of a complex one. Equations (1)-(5) may all be used as is, with removing the tilde hats to now represent real data. Since the DS CW-bDRS method yields a real \(\mu_{eff}\), \(\mu_{s}^{{}^{\prime}}\) must be known to find \(\mu_{a}\) using the following expression instead of Equation (8):  
\[\mu_{a}=\sqrt{\frac{\mu_{s}^{{}^{\prime}2}}{4}+\frac{\mu_{eff}^{2}}{3}-\frac{ \mu_{s}^{{}^{\prime}}}{2}}. \tag{10}\]  
To address the need to know \(\mu_{s}^{{}^{\prime}}\), the data from the MD FD-NIRS measurements were used. FD-NIRS yielded a measurement of \(\mu_{s}^{{}^{\prime}}\) at two wavelengths (\(\mu_{sFD}^{{}^{\prime}}(\lambda_{1})\) and \(\mu_{sFD}^{{}^{\prime}}(\lambda_{2})\); where \(\lambda\) is wavelength).