the corresponding voltage measurement. We form $\mathbf{R}_\gamma$ using the normalized currents $\phi^j=\frac{\Phi^j}{\|\Phi^j\|_2}$ and voltages $v^j_\ell$ satisfying $\sum_{\ell=1}^L v^j_\ell=0$ and $v^j=\frac{v^j}{\|\Phi^j\|_2}$: $$\label{eq:NDcomp}
\mathbf{R}_\gamma(m,n):=\sum_{\ell=1}^L \frac{\phi^m_\ell v^n_\ell}{|e_\ell|},\quad 1\leq m,n\leq L-1,\;\; 1\leq \ell\leq L,$$ where $|e_\ell |$ is the area of the $\ell$-th electrode. The discrete approximation $\mathbf{L}_\gamma$ to the DN map $\Lambda_\gamma$ is then $\mathbf{L}_\gamma = \left(\mathbf{R}_\gamma\right)^{-1}$. While the method as described assumes [[$\gamma=\gamma_0 =1$]{}]{} on $\operatorname{\partial\Omega}$ and the domain has maximal radius 1, the DN matrix $\mathbf{L}_\gamma$ can be scaled by $1/{\gamma_0}$ and to a max radius 1 by multiplying the DN matrix $\mathbf{L}_\gamma$ by $r$, the radius of the smallest circle containing the domain, as in \[[@Isaacson2004]\]. Note that an estimate to $\gamma_0$ is the best constant-admittivity approximation to measured data, as in \[[@Cheney1990a]\]. The resulting conductivity or admittivity at the end of the algorithm is then rescaled by multiplying by $\sigma_0$ or $\gamma_0$ respectively.

The evaluation of the scattering data requires computing the action of the DN map on the asymptotic behaviors of $u_1$ and $u_2$, namely $\mathbf{L}_\gamma \left(\frac{e^{ikz}}{ik}\right)$ and $\mathbf{L}_\gamma \left(\frac{e^{-ik\bar{z}}}{-ik}\right)$. To accomplish this, we follow the method outlined in \[[@DeAngelo2010]\] and expand the functions $\frac{e^{ikz}}{ik}$ and $\frac{e^{-ik\bar{z}}}{-ik}$ in the basis of normalized current patterns $\{\phi^j\}_{j=1}^{L-1}$ as $$\frac{e^{ikz_\ell}}{ik}\approx \sum_{j=1}^{L-1} 