Errors in Turbulence Statistics: The Hurst Phenomenon Approach

The Hurst phenomenon was first observed by Hurst (1951) in hydrological and geopysical time series. Some works also show that turbulence displays this phenomenon. The Hurst phenomenon was characterized by H (0 < H < 1), called the Hurst exponent. We investigated the value of H in atmospheric surface layer data, measured in three micrometeorological campaigns, and its impact on the estimation of random errors. Lumley and Panofsky (1964) proposed a method to estimate random errors that relate the integral scale, the variance of the statistics, and the length of the data record. When H > 1/2, the time series exhibits long-range persistence, and the integral time scale does not exist; this is known as Hurst's phenomenon. An alternative to estimate the random error avoiding the use of the integral scale is the Filtering Method recently proposed by Salesky et al (2012). The method consists in calculating the mean square error (MSE) of a process x(t), tilde{x}_DELTA, for several values of a time scale DELTA. The behavior of MSE(tilde{x}_DELTA) decresses with DELTA according to a power law with exponent -p. In Salesky et al.'s method, p = 1, forcing the existence of an integral scale. This is equivalent to H = 1/2. The error at the time scale T of the sample size is obtained by extrapolation of the power law to T. In this work we show that the Lumley-Panofsky equation can be adapted to dispense with the need of the existence of the integral time scale, and that, by allowing p to vary freely, the Filtering Method can be adapted and still provide reliable error estimates even when the integral scale doesn't exist. We call it the Relaxed Filtering Method. We estimated the Hurst exponent using two approaches: (1) with the Relaxed Filtering Method (H_p), the Hurst exponent can be estimated by H = 1 - p/2; and (2) with the classical rescaled range (R/S) proposed by Hurst (H_R). The data analyzed in this study were measured at a grass farm in Tijucas do Sul, Paraná, Southern Brazil at 1.85 m over grass; a very small island over Itaipu Lake, Parana, Southern Brazil at 3.76 m over the water surface; and at a site of the AHATS study near Kettlemand City, at 8 m above the ground. From all campaigns we used wind velocity (u, v and w) and air temperature (theta) measured with a CSAT-3 sonic anemometer and a FWTC-3 fine-wire thermocuple. For the AHATS data only, theta is the sonic temperature measured by the CSAT-3. A quality control procedure was applied to eliminate all periods with strong evidence of nonstationarity in the first- and second-order moments. After that, a 2-D coordinate rotation was applied and the fluctuations were obtained around a 30-min straight line used for detrending. The following statistics were analyzed: u', v', w' and theta' (first order) and u'w', w'theta', u'u', v'v', w'w', and theta'theta' (second order). The two estimates of H (H_p and H_R) for all campaigns show that both first- and second-order turbulence statistics display the Hurst phenomenon. Usually, H_R is larger than H_p for the same variable. This was also found in hidrology, rasing the question that one, or even both of these estimators, may be biased. To analyze the effect of the Hurst phenomenon on the estimation of random errors we compare the estimatives of relative errors using the following approaches: (1) the Relaxed Filtering Method, (2) the Filtering Method, and (3) the classical Lumley-Panofsky estimate. We found that the error estimates by the Relaxed Filtering Method, which takes account the Hurst phenomenon, are larger than the other two. We also analyzed the relationship between the random error and Obukhov’s stability variable zeta. We found that in unstable conditions the kinematic momentum flux depends on zeta, as well as the kinematic sensible heat flux in stable conditions.