Robust estimation of the integral scale for quantifying uncertainty of the sample mean from non-independent velocity data

Using large data sets, we evaluate statistical bootstrapping schemes for approximating uncertainty in the sample mean of highly correlated velocity field measurements. Interest in time-resolved velocity field data has led to sampling rates high enough that non-independent samples are commonplace. Uncertainty of the sample mean collected from stationary but correlated data is given by $s/ \sqrt{N_{eff}}$, where $s$ is the standard deviation of the samples, and $N_{eff}$ is the "effective" number of samples, that is, the number of samples $N$ divided by twice the integral time scale $T_u$. We can approximate $T_u$ using of a sum of auto-correlation coefficients, but it is necessary to truncate the sum at a prescribed lag $K$. This lag parameter $K$ is equivalent to a bootstrapping parameter in statistics, and we can optimize selection of $K$ using techniques from the bootstrapping methodology. With highly resolved data from laminar and turbulent velocity field measurements we will evaluate different strategies for this statistical bootstrap optimization.