$\Lambda_1$ represents the DN map corresponding to a constant conductivity of 1. The conductivity $\sigma(z)$ can be recovered from the CGOs $\mu(z,k)$ by: $$\label{eq:Nachcond}
\sigma(z)=\left(\lim_{|k|\to0}\mu(z,k)\right)^{1/2}.$$

A [*Born approximation*]{} $\operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{exp}}}}}$ to the fully nonlinear scattering data ${{\mathbf{t}}}(k)$ can be made by replacing $\psi_s$ in by its asymptotic condition $\psi_s\sim e^{ikz}$: $$\label{eq:Nach_texp}
\operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{exp}}}}}(k)=\int_{\operatorname{\partial\Omega}} e^{i\bar{k}\bar{z}}(\Lambda_\sigma-\Lambda_1)e^{ikz}\;dz.$$ This approximation circumvents computing the trace of $\psi_s(z,k)$ on $\partial\Omega$ which requires using another boundary integral equation. [[Approximating $\psi$ by its asymptotic behavior is equivalent to making a leading order approximation from the asymptotic series for $\psi$.]{}]{} This assumption is most accurate in the case of low contrast in conductivity/permittivity.

The reconstruction process is then:

1.  Given $[\Lambda_\sigma,\Lambda_1]$, evaluate for $\operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{exp}}}}}(k)$, $0<|k|\leq R$ for some chosen radius $R$.

2.  Solve the $\operatorname{{\bar{\partial}}_k}$ equation for each $z\in\Omega$ and recover the conductivity $\operatorname{\sigma^{\mbox{\tiny{\textbf{exp}}}}}(z)=\left(\operatorname{\mu^{\mbox{\tiny{\textbf{exp}}}}}(z,0)\right)^{1/2}$.

The method above produces [*absolute*]{}, also called [*static*]{}, EIT images. To obtain time-difference EIT images relative to a reference data set $\operatorname{\Lambda_{\mbox{\tiny{\textbf{ref}}}}}$, we use the [*differencing scattering transform*]{} $\operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{diff}}}}}$ introduced in \[[@Isaacson2006]\] $$\label{eq:Nach_tdiff}
\operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{diff}}}}}(k):=\operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{exp}}}}}^{\mbox{\tiny ,$\gamma$}}(k) - \operatorname{\mathbf{t}^{\mbox{\tiny{\textbf{exp}}}}}^{\mbox{\tiny ,$\gamma_{\mbox{\tiny ref}}$}}(k)=\int_{\operatorname{\partial\Omega}} e^{i\bar{k}\bar{z}}(\Lambda_\sigma-\operatorname{\Lambda_{\mbox{\tiny{\textbf{ref}}}}})e^{ikz}\;dz,$$ in equation and compute $\operatorname{\sigma^{\mbox{\tiny{\textbf{diff}}}}}\equiv \left[\operatorname{\mu^{\mbox{\tiny{\textbf{diff}}}}}(z,0)\right]^2-1$, since $\sigma=1$ near the boundary. If $\sigma=\sigma_0\neq 1$ near $\operatorname{\partial\Omega}$, the problem can be scaled as in \[[@Isaacson2004]\].

The D-bar Method 