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 artificially produced time series as well as measurements from two upward-pointing ground-based lidar systems:a differential absorption lidar that measures water vapor density and a high-resolution Doppler lidar that measures vertical wind velocity. Both sets of measurements were obtained in a convective boundary layer, and contain sufficient noise to significantly affect measurements of second- and fourth-order moments (as well as integral scale and skewness) throughout the boundary layer. It is shown that the corrections derived here can be used to obtain useful measurements of these moments from instruments such as lidars, which are inherently noisy. The authors also obtain information on higher-order moments of the noise as well as the correlation between noise and atmospheric measurements.