  We develop a method to infer log-normal random fields from measurement data
affected by Gaussian noise. The log-normal model is well suited to describe
strictly positive signals with fluctuations whose amplitude varies over several
orders of magnitude. We use the formalism of minimum Gibbs free energy to
derive an algorithm that uses the signal's correlation structure to regularize
the reconstruction. The correlation structure, described by the signal's power
spectrum, is thereby reconstructed from the same data set. We show that the
minimization of the Gibbs free energy, corresponding to a Gaussian
approximation to the posterior marginalized over the power spectrum, is
equivalent to the empirical Bayes ansatz, in which the power spectrum is fixed
to its maximum a posteriori value. We further introduce a prior for the power
spectrum that enforces spectral smoothness. The appropriateness of this prior
in different scenarios is discussed and its effects on the reconstruction's
results are demonstrated. We validate the performance of our reconstruction
algorithm in a series of one- and two-dimensional test cases with varying
degrees of non-linearity and different noise levels.
