In the present communication we propose the application of unsupervised Artificial Neural Networks (ANN) to solve general ill-posed problems and particularly we apply them to the the estimation of the direction of arrival (DOA) of VLF/ELF radio waves. We use the wave distribution method which consists in the reconstruction of the energy distribution of magnetospheric VLF/ELF waves at the ionospheric base from observations of the wave's electromagnetic field on the ground. The present application is similar to a number of computerized tomography and image enhancement problems and the proposed algorithm can be straightforwardly extended to other applications in which observations are linearly related to unknowns. Then, we have proven the applicability and also we indicate the superiority of the ANN to the conventional methods to handle this kind of problems.

In this paper we consider the determination of direction of arrival of VLF/ELF radio waves and their energy distribution at the ionospheric base by means of the inversion of electromagnetic data observed on the ground. The observed data are too limited, leading us to deal with a severely ill-posed problem similar to those encountered in digital image enhancement and computerized tomography. To handle this situation, the a priori information if available, is supposed to bring as much weight as the observed data do. We used a regularization based on Bayesian information criterion to reconstruct the wave distribution function at the ionosphere, that is, to determine the wave arrival direction. Using computer-generated data, two main results were obtained: first, the electromagnetic field data observed on the ground are sufficient to give a good approximation to the exit region of VLF/ELF radio waves and to reconstruct the wave energy distribution nicely at the ionospheric base. Secondly, the Bayesian information criterion is shown efficient and very promising to handle the situations where the data number is too small compared to the number of unknowns which is the case of most reconstruction problems.

In this communication we propose to solve the problem of reconstruction from limited data using a statistical regularization method based on a Bayesian information criterion. The minimization of the Bayesian information criterion, which is used here as an objective index to measure the goodness of an estimate, gives the optimum value of the smoothing parameter. By doing so, we could reduce the inversion problem to a simple minimization of a one-variable nonlinear function. The application of such a technique overcomes the nonuniqueness of the solution of the ill-posed problem and all shortcomings of many iterative methods. In the light of simulation and application to real data, we propose a slight modification to the Bayesian information criterion to reconstruct the wave energy distribution at the ionospheric base from the observation of radio wave electromagnetic field on the ground. The achieved results in both the inversion problem and the wave direction finding are very promising and may support other works so far suggested the use of Bayesian methods in the inversion of ill-posed problems to benefit from the valuable information brought by the a priori knowledge.