It is well known that observational data is generally insufficient to provide a unique ,and stable solution when tackling an inverse problem. The recommended approach in this case is the the use of any regularization technique, in order to assure that parameter variations are bounded to such a degree that the final solution looks physically reasonable. Generally, this rather vague notion of reasonable means in fact smootlmess, Indeed, classical regularization techniques yield the smoothest reconstructions which are consistent with the available data.
In this work, two new entropic regularization techniques are introduced. The first technique is based on the minimization of the entropy, in Shannon's sense, of the vector of first-differences of the unknown parameters. Adopting the standard terminology, it is called the minimum first-order entropy method (MinEnt-1). Unlike the classical maximum entropy formalism, this method constrains the class of possible solutions into a restricted set of low entropy models, constituted by locally smooth regions separated by sharp discontinuities. To illustrate the essential feature of the method, MinEnt-1 is applied to the reconstruction of two-dimensional geoelectric conductivity distributions from magnetotelluric data. Numerical simulations, using synthetic data corrupted with gaussian noise, show that the MinEnt-1 algorithm converges to excellent 2D earth models, yielding, in many cases, results that are superior to those obtained by the maximum entropy method.
The second technique is based on the maxhnization of the entropy of the vector of second-differences of the unknown parameters, and is denoted as the MaxEnt-2 method. The MaxEnt-2 method is applied to the retrieval of vertical profiles of temperature in the atmosphere from remote sensing data. Extensive simulations, using both synthetic and satellite data, and comparison with in situ baloon measurements demonstrate the accuracy ,and the practical applicability of the MaxEnt-2 method.