We present a mathematical and numerical analysis of the stability and accuracy of the NMLA (Numerical MicroLocal Analysis) method [J. Comput. Phys. 199(2) (2004), 717-741] and its discretization.We restrict to homogeneous space and focus on the two simplest cases: (1) Noisy plane wave packets, (2) Noisy point source solutions. A stability result is obtained through the introduction of a new "impedance" observable. The analysis of the point source case leads to a modified second order (curvature dependent) correction of the algorithm. Since NMLA is local, this second order improved version can be applied to general data (heterogeneous media). See [J. Comput. Phys. 231(14) (2012), 4643-4661] for a an application to a source discovery inverse problem.