Method for determining k-vector of a wave by a single spacecraft

A practical method\footnote{% P. M. Bellan, JGR-Space Phys. 121, 8589 (2016).} is described for determining the 3D wave-vector of quasi-neutral plasma waves using magnetic field and electric current density measurements made by a single spacecraft. This wave-vector knowledge can then be used to remove the space-time ambiguity produced by frequency Doppler shift associated with spacecraft motion so the actual plasma-frame wave dispersion relation is determined with no theoretical assumptions. The method involves applying the Wiener-Khinchin theorem to cross-correlations of the current and magnetic field oscillations and to auto-correlations of the magnetic field oscillations; the wave-vector is proportional to the ratio of the Fourier transforms of these cross- and auto-correlations. The method requires that each wave frequency component map to a unique wave-vector, a situation presumed true in many spacecraft measurement situations. Synthetic data examples that validate the method are presented. The method has recently been used successfully on data from the MMS spacecraft\footnote{% D. Gershman et al., Nature Comm. 8, 14719 (2017).}.