Cross-correlation based time delay estimation for turbulent flow velocity measurements: Statistical considerations

Abstract

Time delay estimation methods (TDE) are well-known techniques to investigate poloidal flows in hot magnetized plasmas through the propagation properties of turbulent structures in the medium. One of these methods is based on the estimation of the time lag at which the cross-correlation function (CCF) estimation reaches its maximum value. The uncertainty of the peak location refers to the smallest determinable flow velocity modulation, and therefore the standard deviation of the time delay imposes important limitation to the measurements. In this article, the relative standard deviation of the CCF estimation and the standard deviation of its peak location are calculated analytically using a simple model of turbulent signals. This model assumes independent (non interacting) overlapping events (coherent structures) with randomly distributed spatio-temporal origins moving with background flow. The result of our calculations is the derivation of a general formula for the CCF variance, which is valid not exclusively in the high event density limit, but also for arbitrary event densities. Our formula reproduces the well known expression for high event densities previously published in the literature. In this paper we also present a derivation of the variance of time delay estimation that turns out to be inversely proportional to the applied time window. The derived formulas were tested in real plasma measurements. The calculations are an extension of the earlier work of Bencze and Zoletnik [Phys. Plasmas 12, 052323 (2005)] where the autocorrelation-width technique was developed. Additionally, we show that velocities calculated by a TDE method possess a broadband noise which originates from this variance, its power spectral density cannot be decreased by worsening the time resolution and can be coherent with noises of other velocity measurements where the same turbulent structures are used. This noise should not be confused with the impact of zero mean frequency zonal flow modulations and can be the explanation for the TEXTOR velocity spectra measured by beam emission spectroscopy.

Comparison of edge turbulence imaging at two different poloidal locations in the scrape-off layer of Alcator C-Mod

“…V.C. Some other limitations of the cross-correlation method for velocity analysis have been described recently [20,56,57]. There are fundamental ambiguities in velocity estimates made using 1-D cross-correlations in a 2-D velocity field; however, the method of Sec.…”

Experimental determination of the correlation properties of plasma turbulence using 2D BES systems

“…This is shown analytically in Appendix B, where the details of our model field are presented. We stress that this model is purely phenomenological and contains no physical prescription for the formation of turbulent structures, and is thus similar to the models used in other tests of measurement techniques [1,35,36]. A time series of density fluctuations is generated by constructing a signal from the sum of N localised perturbations.…”

“…We start from the laboratory-frame covariance function (B.30), apply the PSFs using the integral (33) and define the BES correlation function analogously to (36). This expression is too cumbersome to reproduce here: it is similar in form to (B.39), but includes a cross-term between ∆r and ∆Z similar to that in (38), arising due to the PSF integral.…”

Determination of structure tilting in magnetized plasmas—Time delay estimation in two dimensions

“…The results in this section can be considered twodimensional generalizations of the model presented in [12], but derived without further approximations.…”