## 2 Materials and Methods
### Diffusion Theory
#### 2.1.2 Diffuse Light in Two Dimensions
Moreover, we obtain  
\[u_{0}(\mathbf{\rho},t;\mathbf{\rho}_{s}^{i})=\frac{e^{-\mu_{a0}ct}}{2\pi Dt}e^{-\frac{ (x-y_{s}^{i})^{2}+y^{2}}{4 Dct}}\left[1-\frac{\sqrt{\pi Dt}}{\ell}e^{\left( \frac{y+2 Dct/\ell}{\sqrt{4 Dct}}\right)^{2}}\text{erfc}\left(\frac{y+2 Dct/\ell }{\sqrt{4 Dct}}\right)\right]. \tag{15}\]  
The above expression of \(u_{0}\) is similar to the formula in Equation (6) but does not have a time integral because in this case we assumed the delta function \(\delta(t)\) for the incident beam. To obtain the expression of the Rytov approximation, we introduce \(\psi_{0},\psi_{1}\) as [31]  
\[u_{0}=e^{\psi_{0}},\qquad u=e^{\psi_{0}+\psi_{1}}. \tag{16}\]By plugging the above expressions of \(u,u_{0}\) into the Born approximation and neglecting terms of \(O((\delta\mu_{a})^{2})\), we obtain  
\[ \begin{split} e^{\psi_{1}}&=1-e^{-\psi_{0}}\int_{0}^{t }\int_{0}^{\infty}\int_{-\infty}^{\infty}G(\mathbf{\rho},t;\mathbf{\rho}^{\prime},s) \delta\mu_{a}(\mathbf{\rho}^{\prime})u_{0}(\mathbf{\rho}^{\prime},s;\mathbf{\rho}^{i}_{s}) \,dx^{\prime}dy^{\prime}ds\\ &=\exp\left[-e^{-\psi_{0}}\int_{0}^{t}\int_{0}^{\infty}\int_{- \infty}^{\infty}G(\mathbf{\rho},t;\mathbf{\rho}^{\prime},s)\delta\mu_{a}(\mathbf{\rho}^{ \prime})u_{0}(\mathbf{\rho}^{\prime},s;\mathbf{\rho}^{i}_{s})\,dx^{\prime}dy^{\prime} ds\right].\end{split}  \tag{17}\]  
The (first) Rytov approximation is thus derived as  
\[u(\mathbf{\rho},t;\mathbf{\rho}^{i}_{s})=u_{0}(\mathbf{\rho},t;\mathbf{\rho}^{i}_{s})\exp\left[ -\frac{1}{u_{0}(\mathbf{\rho},t;\mathbf{\rho}^{i}_{s})}\int_{0}^{t}\int_{0}^{\infty} \int_{-\infty}^{\infty}G(\mathbf{\rho},t;\mathbf{\rho}^{\prime},s)\delta\mu_{a}(\mathbf{ \rho}^{\prime})u_{0}(\mathbf{\rho}^{\prime},s;\mathbf{\rho}^{i}_{s})\,dx^{\prime}dy^ {\prime}ds\right]. \tag{18}\]  
Let us define  
\[ \begin{split} g(y,t;y^{\prime},s)&=\frac{1}{4\pi D (t-s)}e^{-\frac{(y+y^{\prime})^{2}}{4 Dc(t-s)}}\left[1+e^{\frac{(y+y^{\prime})^ {2}-(y-y^{\prime})^{2}}{4 Dc(t-s)}}-\frac{\sqrt{4\pi Dc(t-s)}}{\ell}\right.\\ &\times e^{\left(\frac{y+y^{\prime}}{2\sqrt{Dc(t-s)}}+\frac{ \sqrt{Dc(t-s)}}{\ell}\right)^{2}}\operatorname{erfc}\left(\frac{y+y^{\prime}}{ 2\sqrt{Dc(t-s)}}+\frac{\sqrt{Dc(t-s)}}{\ell}\right)\right].\end{split}  \tag{19}\]  
Then we have  
\[\int_{0}^{t}G(\mathbf{\rho},t;\mathbf{\rho}^{\prime},s)u_{0}(\mathbf{\rho}^{\prime},s)\,ds =e^{-\mu_{0}ct}\int_{0}^{t}e^{-\frac{(x-y^{\prime})^{2}}{4 Dc(t-s)}}e^{- \frac{(x^{\prime}-x^{\prime})^{2}}{4 DcS}}g(y,t;y^{\prime},s)g(y^{\prime},s;0, 0)\,ds. \tag{20}\]  
Therefore, Equation (18) can be rewritten as  
\[ \begin{split} u(\mathbf{\rho},t;\mathbf{\rho}^{i}_{s})&=u_{ 0}(\mathbf{\rho},t;\mathbf{\rho}^{i}_{s})\exp\Bigg{[}-\frac{e^{-\mu_{0}ct}}{u_{0}(\mathbf{ \rho},t;\mathbf{\rho}^{i}_{s})}\int_{0}^{\infty}\int_{-\infty}^{\infty}\delta\mu_{ a}(\mathbf{\rho}^{\prime})\\ &\times\left(\int_{0}^{t}e^{-\frac{(x-y^{\prime})^{2}}{4 Dc(t-s)}} e^{-\frac{(x^{\prime}-x^{\prime})^{2}}{4 DcS}}g(y,t;y^{\prime},s)g(y^{\prime},s;0,0) \,ds\right)\,dx^{\prime}dy^{\prime}\Bigg{]}.\end{split}  \tag{21}\]  
This expression (21) is used to compute the forward data in Section 3.2.