Resampling hierarchical processes in the wavelet domain: A case study using atmospheric turbulence

There is a growing need for statistical methods that generate an ensemble of plausible realizations of a hierarchical process from a single run or experiment. The main challenge is how to construct such an ensemble in a manner that preserves the internal dynamics (e.g. intermittency) and temporal persistency of the hierarchical process. A popular hierarchical process often used as a case study in such problems is atmospheric turbulent flow. Analogies to turbulence are often called upon when information flow from large to small scales, non Gaussian statistics, and intermittency are inherent attributes of the process under consideration. These attributes are key defining syndromes of the turbulent cascade thereby making turbulence time series ideal for testing such ensemble generation schemes. In this study, we propose a wavelet based resampling scheme (WB) and compare it to the traditional Fourier based phase randomization bootstrap (FB) approach within the context of the turbulence energy cascade. The comparison between the two resampling methods and observed ensemble statistics constructed by clustering similar meteorological conditions demonstrate that the WB reproduces several features related to intermittency of the ensemble series when compared to $FB$. In particular, the WB exhibited an increase in wavelet energy activity and an increase in the wavelet flatness factor with increasing frequency consistent with the cluster of ensemble statistics. On the other hand, the $FB$ yielded no increase in such energy activity with scale and resulted in near Gaussian wavelet coefficients at all frequencies within the inertial subrange. The extension of WB to the multivariate case is also demonstrated via the conservation of co-spectra between longitudinal and vertical velocity time series. Because the resampling strategy proposed here is conducted in the wavelet domain, gap-infected and uneven sampled time series can be readily accommodated within the WB. Finally, recommendations about the filter and block sizes are discussed.