Revealing intermittency in experimental data with steep power spectra

The statistics of signal increments are commonly used in order to test for possible intermittent properties in experimental or synthetic data. However, for signals with steep power spectra [i.e., $E(\omega) \sim \omega^{-n}$ with $n \geq 3$], the increments are poorly informative and the classical phenomenological relationship between the scaling exponents of the second-order structure function and of the power spectrum does not hold. We show that in these conditions the relevant quantities to compute are the second or higher degree differences of the signal. Using this statistical framework to analyze a synthetic signal and experimental data of wave turbulence on a fluid surface, we accurately characterize intermittency of these data with steep power spectra. The general application of this methodology to study intermittency of experimental signals with steep power spectra is discussed.

💡 Research Summary

The paper addresses a fundamental limitation in the statistical analysis of signals whose power spectra decay steeply, i.e., (E(\omega)\sim\omega^{-n}) with (n\ge 3). In the standard approach, intermittency – the presence of rare, intense fluctuations – is probed through the scaling of low‑order increments (first‑order differences) (\delta\eta(t,\tau)=\eta(t+\tau)-\eta(t)). Under the assumption of a power‑law spectrum, the second‑order structure function (S_{2}(\tau)=\langle\delta\eta^{2}\rangle) scales as (\tau^{\beta}) with (\beta=n-1). When (n\ge 3), however, the signal is at least twice differentiable, causing (\delta\eta) to behave almost linearly with (\tau). Consequently, (S_{2}(\tau)) saturates at a (\tau^{2}) scaling, breaking the phenomenological link between structure‑function exponents and spectral exponents. As a result, higher‑order structure functions built from first‑order increments fail to reveal the non‑linear (\zeta(p)) curves that signal intermittency.

To overcome this, the authors propose using higher‑order differences (or “m‑th degree differences”) defined as
\