Measuring second- through fourth-order moments in noisy data

The authors derive expressions for correcting second- through fourth-order moments of measured variables that are contaminated by random uncorrelated noise. These expressions are then tested by applying them to an artificiall y produced time series as well as measurements from two upward-pointing gro und-based lidar systems: a differential absorption lidar that measures wate r vapor density and a high-resolution Doppler lidar that measures Vertical wind velocity. Both sets of measurements were obtained in a convective boun dary layer, and contain sufficient noise to significantly affect measuremen ts of second- and fourth-order moments (as well as integral scale and skewn ess) throughout the boundary layer. It is shown that the corrections derive d here can be used to obtain useful measurements of these moments from inst ruments such as lidars, which are inherently noisy. The authors also obtain information on higher-order moments of the noise as well as the correlatio n between noise and atmospheric measurements.