| 17 Jul 2026
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.
This study revisits wavelet-based variance/covariance estimation for eddy-covariance flux calculations. The authors identify that the Torrence and Compo formulation introduces 10–15% systematic bias due to normalisation issues, and propose a corrected CWT-derived estimator. The theoretical derivation is rigorous, synthetic validation is comprehensive, and the EC data application demonstrates practical relevance. The work addresses an important methodological gap in atmospheric flux measurements. Overall, the paper is well structured and clearly written. However, several minor issues require attention before publication.
1. The authors claim that the TC formulation exhibits a systematic bias of approximately 10–15%, whereas the proposed CWT-based estimator is essentially unbiased. For synthetic signals, this claim is well supported. However, in the EC application, the authors treat the classical time-averaged EC covariance as the reference benchmark (page 12: "Under stationary conditions... covariance reconstructed with the CWT-based formulation closely matches EC estimates... TC covariance exhibits larger deviations..." and "Under non-stationary conditions... the agreement decreases substantially"). The concern is that the classical EC estimator itself relies on Reynolds decomposition and strict stationarity assumptions; it is not an absolute reference, particularly under non-stationary conditions. Using a compromised reference to evaluate wavelet methods in the non-stationary regime risks circularity. The observed discrepancy may reflect the breakdown of the bulk EC average's representativeness rather than a limitation of the wavelet approach. A clearer distinction should be drawn between (i) the theoretical truth (synthetic signals) and (ii) the EC reference value (a conventional, operationally defined estimate, not a gold standard). The non-stationary results would be better framed as a diagnostic of multi-scale covariance structure and the limitations of ergodicity assumptions, rather than as a "deviation from the true flux."
2. Synthetic experiments show Morlet outperforms DOG2 in convergence and frequency localisation (page 10, Figure 2), suggesting theoretical superiority. Yet in the EC application under stationary conditions (Figures 3a, 3c), DOG2 shows closer 1:1 agreement than Morlet—an apparent reversal not adequately explained by invoking DOG2's "stronger temporal localisation." Possible hypotheses: (i) DOG2's broad frequency response may compensate for the 2/3 inertial subrange slope; (ii) Morlet's wider temporal support may increase sensitivity to residual non-stationarities in "stationary" EC periods. Further investigation (e.g., residual spectral decomposition) would clarify whether this advantage is physically meaningful.
3. In the TC formulation (page 3), the reconstruction factor is denoted as Cδ, while in the proposed formulation (page 4), the admissibility constant is denoted as Cψ. Although the authors indicate that these are conceptually distinct—the former being an empirical discretisation factor and the latter the theoretical admissibility constant—their numerical values differ substantially for a given wavelet. This notational proximity is likely to cause confusion. An explicit statement is needed, where Cδ is first introduced, clarifying that it is not equivalent to Cψ, but rather an empirical normalisation scalar arising from the discretisation scheme of Torrence and Compo (1998). A small table listing the numerical values of both constants for the two wavelets used would further aid clarity.
4. On page 6, the authors state: "Stationary and non-stationary conditions were identified using the combined steady-state and integral turbulence characteristics test (SSITC), with SSITC=0 indicating stationary conditions and SSITC=2 indicating non-stationary conditions (Foken et al., 2004)." However, the threshold values or criteria used to assign these classes are not described. It would be helpful to specify, either here or in a supplementary section, the exact numerical thresholds employed for the steady-state test and the integral turbulence characteristics test, as different implementations of the SSITC use different tolerance levels.
5. The reduction from 44 (DOG2) to 42 (Morlet) non-stationary periods (page 6) is modest given Morlet requires 5× longer extensions. It should be clarified whether this reflects data availability or other filtering criteria, and the edge-effect exclusion criteria for each wavelet should be briefly explained.
6. The term "pseudo-ensemble" (page 6) is non-standard. An ensemble typically means multiple realisations from the same parameter setting; here it appears to mean discrete parameter sampling (frequency, amplitude, variance) instead. For stochastic signals, a single realisation itself contains sampling variability, so the "pseudo-ensemble" mixes parametric and stochastic variability. Clarification is needed on whether each parameter combination corresponds to one or multiple realisations, and justification of the statistical representativeness.
7. In Figure 3, a legend is provided only in subpanel (d); subpanels (a)–(c) lack one. This hinders interpretation when comparing across conditions or wavelet types. A consistent legend should be added to all subpanels, or a single comprehensive legend placed outside the panel array.