Current and next generation of telescopes resort to adaptive optics in order to compensate for atmospheric turbulence and correct the incident wavefront, which is affected mainly as far as the phase is concerned, in order to get clearer images of the observed objects. Starting from the statistical description of the atmospheric turbulence, the reconstruction of the turbulent phase is therefore a key problem in astronomical seeing and is central to designing control systems to command the adaptive optic deformable mirror. Moreover, the introduction of a dynamical model able to predict the turbulence helps improving the performance of the control system and its description through a principal component analysis (PCA) approach yields compactness to the representation while preserving the physical insight of a modal decomposition.