The Hurst Phenomenon in Error Estimates Related to Atmospheric Turbulence

Abstract

The Hurst phenomenon is a well-known feature of long-range persistence first observed in hydrological and geophysical time series by E. Hurst in the 1950s. It has also been found in several cases in turbulence time series measured in the wind tunnel, the atmosphere, and in rivers. Here, we conduct a systematic investigation of the value of the Hurst coefficient H in atmospheric surface-layer data, and its impact on the estimation of random errors. We show that usually $H > 0.5$, which implies the non-existence (in the statistical sense) of the integral time scale. Since the integral time scale is present in the Lumley–Panofsky equation for the estimation of random errors, this has important practical consequences. We estimated H in two principal ways: (1) with an extension of the recently proposed filtering method to estimate the random error ($H_p$), and (2) with the classical rescaled range introduced by Hurst ($H_R$). Other estimators were tried but were found less able to capture the statistical behaviour of the large scales of turbulence. Using data from three micrometeorological campaigns we found that both first- and second-order turbulence statistics display the Hurst phenomenon. Usually, $H_R$ is larger than $H_p$ for the same dataset, raising the question that one, or even both, of these estimators, may be biased. For the relative error, we found that the errors estimated with the approach adopted by us, that we call the relaxed filtering method, and that takes into account the occurrence of the Hurst phenomenon, are larger than both the filtering method and the classical Lumley–Panofsky estimates. Finally, we found that there is no apparent relationship between H and the Obukhov stability parameter. The relative errors, however, do show stability dependence, particularly in the case of the error of the kinematic momentum flux in unstable conditions, and that of the kinematic sensible heat flux in stable conditions.

Appendices

Appendix 1 The Cases $q=1$ and $q=2$

When $q=1$, substitution of (14) into (12) yields

$$\begin{aligned} I&= \alpha _1{\mathscr {T}}_H + \int _{{\mathscr {T}}_H}^{{\Delta }} k \left( \frac{\eta }{{\Delta }}\right) ^{-1}\,\mathrm{d}\eta - \int _{{\mathscr {T}}_H}^{{\Delta }} \frac{k}{{\Delta }}\eta \left( \frac{\eta }{{\Delta }}\right) ^{-1}\,\mathrm{d}\eta \nonumber \\&= {\mathscr {T}}_H\left[ (\alpha _1 - k) + k\ln \left( \frac{{\Delta }}{{\mathscr {T}}_H}\right) + k\left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) \right] , \end{aligned}$$

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$$\begin{aligned} { MSE}({\widetilde{x}}_{\Delta })&= \frac{2{\mathscr {T}}_H}{{\Delta }}\left[ (\alpha _1 - k) + k\ln \left( \frac{{\Delta }}{{\mathscr {T}}_H}\right) + k\left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) \right] { Var}\{x\}\nonumber \\&\sim (2{\mathscr {T}}_H/{\Delta })\ln ({\Delta }/{\mathscr {T}}_H){ Var}\{x\} \end{aligned}$$

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for ${\Delta } \gg {\mathscr {T}}_H$. When $q=2$, substitution of (14) into (12) yields

$$\begin{aligned} I&= \alpha _1{\mathscr {T}}_H + \int _{{\mathscr {T}}_H}^{{\Delta }} k \left( \frac{\eta }{{\Delta }}\right) ^{-2}\,d\eta - \int _{{\mathscr {T}}_H}^{{\Delta }} \frac{k}{{\Delta }}\eta \left( \frac{\eta }{{\Delta }}\right) ^{-2}\,\mathrm{d}\eta \nonumber \\&= {\mathscr {T}}_H\left[ (\alpha _1 + k) - k\left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) + k\left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) \ln \left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) +\right] , \end{aligned}$$

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$$\begin{aligned} { MSE}({\widetilde{x}}_{\Delta })&= \frac{2{\mathscr {T}}_H}{{\Delta }}\left[ (\alpha _1 + k) - k\left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) + k\left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) \ln \left( \frac{{\mathscr {T}}_H}{{\Delta }}\right) \right] { Var}\{x\}\nonumber \\&\sim \frac{2{\mathscr {T}}_H}{{\Delta }}{ Var}\{ x\} \end{aligned}$$

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for ${\Delta } \gg {\mathscr {T}}_H$.

Appendix 2 A Test for Changing Variance

Consider a discretized dataset corresponding to $30\,\mathrm {min}$ at 20 Hz, with $N = 36{,}000$ data points $x_0, x_1, \ldots , x_{N-1}$. A running mean and running variance with window D are defined as

$$\begin{aligned} {\widetilde{x}}_{D}(k)&= \frac{1}{D}\sum _{i=k}^{k+D-1} x_i, \end{aligned}$$

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$$\begin{aligned} \widetilde{{ Var}(x)}_{D}(k)&= \frac{1}{D}\sum _{i=k}^{k+D-1} \left( x_i -{\widetilde{x}}_{D}(k)\right) ^2, \end{aligned}$$

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and the ratio of the largest to the smallest $\widetilde{{ Var}(x)}_{D}(k)$ values over all k is then calculated. The measurement period is rejected when this ratio is larger than r. We used $D=4800$ and $r=5$ for u and v, $D=2400$ and $r=3$ for w and $D=2400$ and $r=4$ for $\theta$.