How Long Is Long Enough When Measuring Fluxes and Other Turbulence Statistics?

TLDR

Systematic error in flux or moment estimates decreases with averaging time T and vanishes as T approaches infinity. The study determines the minimum time series length required to estimate covariances and fourth‑order moments with a specified statistical significance. By deriving expressions for systematic and random errors under Gaussian and skewed processes, the authors show that both errors decrease with T and that sampling intervals up to the integral scale increase errors by less than 8%. The results reveal that systematic error and error variance are inversely proportional to T, the random error exceeds the systematic error at large T, Gaussian assumptions underestimate random errors, and skewed models provide more accurate estimates.

Abstract

It is determined how long a time series must be to estimate covariances and moments up to fourth order with a specified statistical significance. For a given averaging time T there is a systematic difference between the true flux or moment and the ensemble average of the time means of the same quantities. This difference, referred to here as the systematic error, is a decreasing function of T tending to zero for T→∞. The variance of the time mean of the flux or moment, the so-called error variance, represents the random scatter of individual realizations, which, when T is much larger than the integral time scale T of the time series, is also a decreasing function of T. This makes it possible to assess the minimum value of T necessary to obtain systematic and random errors smaller than specified values. Assuming that the time series are either Gaussian processes with exponential correlation functions or a skewed process derived from a Gaussian, we obtain expressions for the systematic and random errors. These expressions show that the systematic error and the error variance in the limit of large T are both inversely proportional to T, which means that the random error, that is, the square root of the error variance, will in this limit be larger than the systematic error. It is demonstrated theoretically, as well as experimentally with aircraft data from the convective boundary layer over the ocean and over land, that the assumption that the time series are Gaussian leads to underestimation of the random errors, while derived processes with a more realistic skewness and kurtosis give better estimates. For fluxes, the systematic and random errors are estimated when the time series are sampled instantaneously, but the samples separated in time by an amount Δ. It is found that the random error variance and the systematic error increase by less than 8% over continuously sampled data if Δ is no larger than the integral scale obtained from the flux time series and the cospectrum, respectively.