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This study addresses the impacts of a three-dimensional wind field on the weather hazard sensing and the engineering tradeoffs for the remote sensing of wind fields with a single-channel airborne Doppler radar. It is shown that the radial component of relative air speed in a hazardous weather cell can be retrieved through Doppler analysis. Under the forward-looking conditions, a micro-Doppler frequency component is observed from the cross-correlation of received signals with time separations, and can be used to estimate the tangential air speeds. It is also presented that multiple parameters have a significant impact on the wind-estimation performance, including the coherent observation time (COT), turbulence, and uncorrelated random noise. Monte-Carlo simulations for airborne wind hazard estimations are performed, and the results are discussed based on 1) a single radar resolution volume with arbitrary wind-vector, and 2) realistic weather model output generated from the advanced regional prediction system (ARPS). The relative air speed estimation techniques are also verified with flight test data obtained from an X-band airborne pulsed Doppler radar.