A regularization method for the extrapolation of the photospheric solar magnetic field - I. Linear force-free field

We present a method for reconstructing the magnetic field B above the photosphere {z = 0} as the solution of the boundary value problem (BVP) for a bounded regular force-free magnetic field in Omega = {z > 0} from its boundary values supposed to be given on {z = 0}. We propose a way for regularizing the class of standard extrapolation methods which turns out to diverge quickly with height, because of the ill posedness of the BVP that prevents extrapolation at a reasonable height. Our method, which is a Projection Method, allows us to improve considerably the possible height of extrapolation that can be reached by non regularized or even approximate filtering method. In this paper, because of its linear nature the method is applied to the case of linear force-free field.