Let's suppose the medium is not homogeneous, but as a Gaussian field, then reconstruction of coefficients is actually recovering the statistical mean, such problems are relevant to the submatrix detection which is supposed to equivalent to planted clique, which is hypothesized as an NP hard problem in certain regime. It can be shown that the signal-to-noise ratio will be critical.

Derive the nonlinear transport from nonlinear wave equation in random media. The Wignerization technique seems not simple to work anymore.  [nonlinear attenuation seems OK now, the Kerr effect is difficult to deal with. See the cross ref. in nonlinear diffusion and nonlinear optics. ]

The derivation of nonlinear attenuation requires an assumption on the regularity of the flux or angular dependence. Is it possible to lift the assumptions?