Should structure functions be used to estimate power laws in turbulence? A comparative study

Second-order structure functions are widely used to characterize turbulence in the inertial range because they are simple to estimate, particularly in comparison to spectral density functions and wavelet variances. Structure function estimators, however, are highly autocorrelated and, as a result, no suitable theory has been established to provide confidence intervals for turbulence parameters when determined via regression fits in log/log space. Monte Carlo simulations were performed to compare the performance of structure function estimators of turbulence parameters with corresponding multitaper spectral and wavelet variance estimators. The simulations indicate that these latter estimators have smaller variances than estimators based upon the structure function. In contrast to structure function estimators, the statistical properties of the multitaper spectral and wavelet variance estimators allow for the construction of confidence intervals for turbulence parameters. The Monte Carlo simulations also confirm the validity of the statistical theory behind the multitaper spectral and wavelet variance estimators. The strengths and weaknesses of the various estimators are further illustrated by analyzing an atmospheric temperature time series.