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For a centered real scattering potential fluctuation $W$ and the imaginary part $b$ of the incident-field-weighted Green function, $\phi (\mathbf{r})=-\int b(\mathbf{r},{\mathbf{r}}^{\prime })W({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. Its mean vanishes, and its covariance function is the double integral of the two deterministic kernels against the covariance function of $W$. The correlation of the refractive index itself is insufficient when the scattering potential depends quadratically on that index.

The two Born approximation measurement operators $A_{\pm }$ map a supported scattering potential to fields on planes above and below it. The displayed Tikhonov regularization functional balances their residuals against a positive quadratic penalty. With an $L^2$ penalty, the unconstrained minimizer is $(A^{\ast }A+\alpha I{)}^{-1}{A}^{\ast }d$, where $A=(A_{+},A_{-})$. Regularization stabilizes estimation but cannot remove the nonuniqueness of Born non-scattering contrasts at one incident direction without extra information.