Systematic deviation in wavelet-based covariance estimation and implications for eddy-covariance flux calculations

Abstract. Accurate estimation of variance and covariance is central to the analysis of geophysical time series and underpins key diagnostics such as turbulent fluxes in eddy-covariance (EC) measurements. Continuous wavelet transform (CWT) methods are widely used for this purpose due to their ability to represent non-stationary and multi-scale processes. However, the consistency of commonly used formulations with the theoretical properties of the CWT has not been fully clarified. In this work, we show that the widely used formulation of Torrence and Compo can lead to systematic deviations in covariance estimation, which can be traced to normalization choices that are not fully consistent with the CWT reconstruction framework. These deviations, of the order of 10–15 % under typical conditions, are observed across a range of signal types, including deterministic, white-noise, and long-range dependent processes. We derive an alternative formulation directly from the CWT reconstruction identity, ensuring consistency with wavelet energy conservation. This approach reduces bias and improves convergence properties. Application to EC data shows that the proposed formulation recovers classical covariance estimates under stationary conditions while providing a more consistent framework for the analysis of non-stationary signals. These results emphasize the importance of scale-consistent estimation of second-order moments for interpreting geophysical variability and turbulent fluxes, and provide a methodological basis for multi-scale analysis of environmental time series.