Uncertainty Estimation in Vector Wind Retrievals from Satellite-based Polarimetric Microwave Radiometer Measurements

The main characteristic of wind vector is circularity in any reference direction, which creates problems when analyzing the data statistically. Due to this periodicity, standard statistical techniques cannot be used to analyze circular data. For example, (1) in the regions including upwind/downwind ambiguity, the STD obtained from the linear statistics approach is about 60-80 degrees. However, we couldn't deduce that it is upwind/downwind ambiguity from the STD. (2) The Cramer-Rao bound (CRB) has been widely used as sensitivity estimation. The precondition of the CRB approach is the possibility distribution is normal one. Since the model of wind direction reveres is non-linear, the angel distribution is not normal. A problem appeared for wind-vector sensitivity estimation. The STD obtained from closest solution is lower than the sensitivity obtained from CRB. In this paper, we introduce the directional statistics approach for evaluating the uncertainty. The circular STD is always between zero and one inclusive, making it very different from its linear analog, which can have any positive value. It is indicative of the spread in a data set. If all samples point into the same direction, the circular STD will be small. If the samples are spread out evenly, the circular STD will be close to one. Importantly, however, a circular variance of 1 does not imply a uniform distribution around the circle. If all samples either point towards 0 degrees or 180 degrees, the circular variance is 1, yet the data is not distributed uniformly around the circle. The sensitivity is reassessed based on the directional statistics approach, and compared with CRB. The sensitivities of CRB are high where the derivatives of the brightness temperatures changing rapidly. The incorrect results come from the effect of nonlinearity. When the directional statistics approach is used, the peak disappeared where the derivatives of the brightness temperatures changing rapidly. In addition, the directional statistics approach can be used to estimate the characterization of uncertaintys in wind vector retrievals from Scatterometry.