Estimating higher-order structure functions from geophysical turbulence time series: Confronting the curse of the limited sample size

Utilizing synthetically generated random variates and laboratory measurements, we document the inherent limitations of the conventional structure function approach in limited sample size settings. We demonstrate that an alternative approach, based on the principle of maximum likelihood, can provide nearly unbiased structure function estimates with far less uncertainty under such unfavorable conditions. The superiority of this approach over the conventional approach does not diminish even in the case of strongly correlated samples. Two entirely different types of probability distributions, which have been reported in the turbulence literature, are found to be compatible with the proposed approach.

Figure 1

NIG distributed variates with three different parameter combinations: (a) $\alpha =0.1,\delta =1$, (b) $\alpha =1,\delta =1$, and (c) $\alpha =2,\delta =2$. For all these cases, the parameters $\mu$ and $\beta$ are assumed to be equal to zero. Please refer to Appendix pp1 for a detailed description of the NIG parameters. For each case, ${10}^7$ samples were generated using Rydberg's algorithm [60]. The distributions were normalized by the standard deviation ${\sigma }_{\delta u}$. A Gaussian pdf is overlaid (dashed line) as a reference.

Figure 2

“Empirical” $S_6$ box plots for three different NIG distributions with parameter combinations: (a) $\alpha =0.1,\delta =1$, (b) $\alpha =1,\delta =1$, and (c) $\alpha =2,\delta =2$. The parameters $\mu$ and $\beta$ are assumed to be equal to zero. The sample sizes ($N$) are varied from ${10}^3$ to ${10}^7$. One hundred realizations are used for the construction of the box plots. The dashed magenta lines represent the true $S_6$ values based on Eq. (A4).

Figure 3

Rank-order (a.k.a. Zipf) plots for NIG distributed variates with three different parameter combinations: (a) $\alpha =0.1,\delta =1$, (b) $\alpha =1,\delta =1$, and (c) $\alpha =2,\delta =2$. The parameters $\mu$ and $\beta$ are assumed to be equal to zero. The sample sizes ($N$) are varied from ${10}^4$ to ${10}^7$. The tail indices (${\gamma }^{*}$) are estimated for $N={10}^4$ (dot-dashed) and $N={10}^7$ (dashed) and reported in the bottom-left corner of the plots.

Figure 4

Hill plots for NIG distributed variates with three different parameter combinations: (a) $\alpha =0.1,\delta =1$, (b) $\alpha =1,\delta =1$, and (c) $\alpha =2,\delta =2$. The parameters $\mu$ and $\beta$ are assumed to be equal to zero. The sample sizes ($N$) are varied from ${10}^4$ to ${10}^7$. The dashed black lines represent $p_{max}=6$.

Figure 5

Box plots of K-S test statistic ($D$) comparing MME (top panel) versus MLE (bottom panel) results. The following parameters are utilized to generate the random variates: $\alpha =0.1,\beta =0,\mu =0$, and $\delta =1$ (heavy-tailed). The sample sizes ($N$) are varied from ${10}^3$ to ${10}^6$. As before, 100 realizations are used for the construction of the box plots.

Figure 6

MLE-based $S_6$ box plots for three NIG distributions with parameter combinations: (a) $\alpha =0.1,\delta =1$, (b) $\alpha =1,\delta =1$, and (c) $\alpha =2,\delta =2$. The parameters $\mu$ and $\beta$ are assumed to be equal to zero. The sample sizes ($N$) are varied from ${10}^3$ to ${10}^6$. One hundred realizations are used for the construction of the box plots. The dashed magenta lines represent the true $S_6$ values based on Eq. (A4).

Figure 7

Realizations of i.i.d (top panel) and correlated (bottom panel) NIG variates. Both realizations follow the same NIG distribution with prescribed parameters: $\alpha =1, \beta =0,\mu =0$, and $\delta =1$.

Figure 8

“Empirical” (top panel) and MLE-based (bottom panel) $S_6$ box plots for correlated NIG variates. The following parameters are utilized to generate the random variates: $\alpha =1,\beta =0,\mu =0$, and $\delta =1$. The sample sizes ($N$) are varied from ${10}^3$ to ${10}^6$. One hundred realizations are used for the construction of the box plots. The dashed magenta lines represent the true $S_6$ value based on Eq. (A4).

Figure 9

Analyses of ONERA S1 wind tunnel data. Selected increment: $r=8.2$ mm or $\mathrm{\Delta }t=4\times {10}^{-4}$ sec, (a) corresponding PDF with NIG-MLE fit, (b) “Empirical” $S_6$ box plots, (c) NIG-MLE $S_6$ box plots. One hundred realizations are used for the construction of the box plots. The dashed magenta line represent the $S_6$ values based on Eq. (1) using the entire data set. The dot-dashed lines represent an uncertainty of $\pm 10\%$ around the magenta line.

Figure 10

Same a Fig. 9, except for $r=277.8$ mm or $\mathrm{\Delta }t=1.35\times {10}^{-2}$ sec.

Figure 11

Top panel: comparison of fitted NIG and LNSS pdfs for i.i.d NIG variates. The sample size is ${10}^6$. Middle panel: magnifies the right-tail estimation by multiplying the pdf by $\delta u^6$. Bottom panel: MLE-based $S_6$ box plot for i.i.d. NIG variates with the (incorrect) assumption of LNSS as the underlying pdf. Due to high computational costs associated with numerical integration of the LNSS pdf, the sample sizes ($N$) are only limited to ${10}^4$ in this example. The following parameters are utilized to generate the random variates: $\alpha =1,\beta =0,\mu =0$, and $\delta =1$. One hundred realizations are used for the construction of the box plot. The dashed magenta line represents the true $S_6$ value based on Eq. (A4).