  We develop an ultrawideband (UWB) inverse scattering technique for
reconstructing continuous random media based on Bayesian compressive sensing.
In addition to providing maximum a posteriori estimates of the unknown weights,
Bayesian inversion provides estimate of the confidence level of the solution,
as well as a systematic approach for optimizing subsequent measurement(s) to
maximize information gain. We impose sparsity priors directly on spatial
harmonics to exploit the spatial correlation exhibited by continuous media, and
solve for their posterior probability density functions efficiently using a
fast relevance vector machine. We linearize the problem using the first-order
Born approximation which enables us to combine, in a single inversion,
measurements from multiple transmitters and ultrawideband frequencies. We
extend the method to high-contrast media using the distorted-Born iterative
method. We apply time-reversal strategies to adaptively focus the inversion
effort onto subdomains of interest, and hence reduce the overall inversion
cost. The proposed techniques are illustrated in a number of canonical
scenarios including crosshole and borehole sensing.
