In this work, we present an efficient optic flow algorithm for the extraction of vertically resolved 3D atmospheric motion vector (AMV) fields from incomplete hyperspectral image data measures by infrared sounders. The model at the heart of the energy to be minimized is consistent with atmospheric dynamics, incorporating ingredients of thermodynamics, hydrostatic equilibrium and statistical turbulence. Modern optimization techniques are deployed to design a low-complexity solver for the energy minimization problem, which is non-convex, non-differentiable, high-dimensional and subject to physical constraints. In particular, taking advantage of the alternate direction of multipliers methods (ADMM), we show how to split the original high-dimensional problem into a recursion involving a set of standard and tractable optic-flow sub-problems. By comparing with the ground truth provided by the operational numerical simulation of the European Centre for Medium-Range Weather Forecasts (ECMWF), we show that the performance of the proposed method is superior to state-of-the-art optical flow algorithms in the context of real infrared atmospheric sounding interferometer (IASI) observations.