Markov-random-field modeling for linear seismic tomography

We apply the Markov-random-field model to linear seismic tomography and propose a method to estimate the hyperparameters for the smoothness and the magnitude of the noise. Optimal hyperparameters can be determined analytically by minimizing the free energy function, which is defined by marginalizing the evaluation function. In synthetic inversion tests under various settings, the assumed velocity structures are successfully reconstructed, which shows the effectiveness and robustness of the proposed method. The proposed mathematical framework can be applied to inversion problems in various fields in the natural sciences.