in Section \[sec:results\], the method is remarkably stable against perturbations in electrodes locations as well as incorrect information about boundary shape. In fact, all of the D-bar methods presented here perform quite well (see Figures \[fig:ACT3\_HnL\_Nach\], \[fig:ACT3\_HnL\_FranSarah\] and \[fig:ACT3\_HnL\_FranNata\])[[. We remark that the action of the DN map $\Lambda_1$ on the complex exponential functions ($e^{ikz}$ for $\operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{exp}}}}}$, and $\frac{e^{ikz}}{ik}$ and $\frac{e^{-ik\bar{z}}}{-ik}$ for [*‘Approach 1’*]{}) can be approximated by using the definition of the DN map: $\Lambda_\gamma u=\gamma\nabla u\cdot \nu$. In \[[@Isaacson2004]\], this was done by analytically computing the action of $\Lambda_1$ on the basis of applied trigonometric current patterns. Alternatively, one could use a continuum type approximation directly computing, e.g., $\Lambda_1 e^{ikz}=1\nabla e^{ikz}\cdot\nu = ik\nu e^{ikz}$. However,]{}]{} due to the flexibility of not needing any simulated data, we focus this work on [*‘Approach 2’*]{}. To obtain time-difference EIT images, define the [*differencing scattering transform*]{}:

-   \[[@Hamilton2017_PhysMeas]\] $$\label{eq:Fran_scatPsiExpDiff}
    \begin{array}{lcl}
    S^{\mbox{\tiny{$\Psi$diff}}}_{12}(k)&=&\frac{i}{2\pi}\int_{\operatorname{\partial\Omega}}e^{-i\bar{k}{z}}\operatorname{\Psi_{12}^{\mbox{\tiny{\textbf{diff}}}}}(z,k){\nu(z)}ds(z)\\
    S^{\mbox{\tiny{$\Psi$diff}}}_{21}(k)&=&\frac{i}{2\pi}\int_{\operatorname{\partial\Omega}}e^{i\bar{k}\bar{z}}\operatorname{\Psi_{21}^{\mbox{\tiny{\textbf{diff}}}}}(z,k)\bar{\nu(z)}ds(z),
    \end{array}$$ where $$\label{eq:Fran_PsiDiff}
    \begin{array}{lcl}
    \operatorname{\Psi_{12}^{\mbox{\tiny{\textbf{diff}}}}}(z,k)&=& \int_{\operatorname{\partial\Omega}} \frac{e^{i(\bar{k}(k-\zeta))}}{4\pi(z-\zeta)}\left(\Lambda_\gamma-\operatorname{\Lambda_{\mbox{\tiny{\textbf{ref}}}}}\right)\left(\frac{e^{-ik\bar{\zeta}}}{-ik}\right)ds(\zeta)\\
    \operatorname{\Psi_{21}^{\mbox{\tiny{\textbf{diff}}}}}(z,k)&=& \int_{\operatorname{\partial\Omega}} \overline{\left[\frac{e^{i({k}(k-\zeta))}}{4\pi(z-\zeta)}\right]}\left(\Lambda_\gamma-\operatorname{\Lambda_{\mbox{\tiny{\textbf{ref}}}}}\right)\left(\frac{e^{ik\zeta}}{ik}\right)ds(\zeta).
    \end{array}$$

-   \[[@Herrera2015]\] $$\label{eq:Fran_scatdiff}
    \begin{array}{lcl}
    \operatorname{S_{12}^{\mbox{\tiny{\textbf{diff}}}}}(k) & =& \frac{i}{4\pi}\int_{\partial\Omega} e^{-i\bar{k}z}\left(\Lambda_{\gamma}-\operatorname{\Lambda_{\mbox{\tiny{\textbf{ref}}}}}\right)\left(\frac{e^{-ik\bar{z}}}{-ik}\right)ds(z)\\
    \operatorname{S_{21}^{\mbox{\tiny{\textbf{diff}}}}}(k) & =& -\frac{i}{4\pi}\int_{\partial\Omega} e^{i\bar{k}\bar{z}}\left(\Lambda_{\gamma}-\operatorname{\Lambda_{\mbox{\tiny{\textbf{ref}}}}}\right)\left(\frac{e^{ikz}}{ik}\right)ds(z),
    \end{array}$$

then, solve the $\operatorname{{\bar{\partial}}_k}$ equation using the differencing scattering data and recover $\operatorname{\gamma^{\mbox{\tiny{\textbf{diff}}}}}$ by subtracting $1$ from , as 1 is the constant admittivity 