The eddy covariance (EC) technique has emerged as the prevailing method to observe the ecosystem–atmosphere exchange of gases, heat and momentum. EC
measurements require rigorous data processing to derive the fluxes that can be used to analyse exchange processes at the ecosystem–atmosphere
interface. Here we show that two common post-processing steps (time-lag estimation via cross-covariance maximisation and correction for limited
frequency response of the EC measurement system) are interrelated, and this should be accounted for when processing EC gas flux data. These findings
are applicable to EC systems employing closed- or enclosed-path gas analysers which can be approximated to be linear first-order sensors. These EC
measurement systems act as low-pass filters on the time series of the scalar

The eddy covariance (EC) is the standard micrometeorological technique for measuring vertical turbulent fluxes of momentum, heat and gases in the
atmospheric surface layer

Gas flux measurements are made using a three-dimensional sonic anemometer and a gas analyser, which are able to provide fast-response measurements of
turbulent fluctuations of vertical wind velocity and gas concentration

The frequency response correction is usually performed based on a priori knowledge of the system transfer function and the unattenuated cospectrum:

Here CF is the estimated spectral correction factor,

Alternatively, the empirical approach can be used, in which the model cospectra and

For this approach, different methods have been proposed for retrieving

In EC systems, particularly those using closed-path gas analysers, the measurements of vertical wind velocity and gas concentration are not
co-located, and a time delay between the two signals exists

In this study, we investigate how the low-pass-filtering-induced phase shift affects the estimation of the high-frequency flux loss, and we show the
implications that occur when

The measured cross-spectrum (

Following, for example,

These two transfer functions together describe how the measured cospectrum (

Similarly, forming the cross-spectrum of the attenuated scalar with itself yields

Ideally the time lag between

The original aim of this cross-covariance maximisation is to account for the time lag between the time series

As the maximal possible cospectral energy content per unit frequency can be described with the amplitude spectrum (

Hence

The dependence between low-pass-filtering-induced time lag (

Measurements from the Siikaneva fen and the Hyytiälä forest site (SMEAR II) were used. Both stations are part of the Integrated Carbon Observation System (ICOS) measurement station network.

The SMEAR II station is situated in southern Finland (61

The Siikaneva fen site is located in southern Finland (61

A short dataset (hereafter

High-frequency EC data from the two sites were processed in order to evaluate (1) the accuracy of different spectral correction methods and (2) their
effect on gas (

Atmospheric stability was evaluated using the Obukhov length (

The total transfer function of an EC set-up can be estimated empirically from measured

Often both

Methods used in this study to estimate flux losses and related correction factors (CFs) due to low-pass filtering of the scalar signal. See the definitions for

We used four methods to estimate EC system response times (and additionally

In order to evaluate the performance of the different spectral correction methods presented in Sect.

Top: the value for the proportionality constant

In Sect.

Low-pass-filtering-induced time lag (

In order to evaluate the influence of turbulent signal on the dependence between

The discrepancy between the two dependencies between

Cross-covariance between

Transfer functions (left

The four methods to estimate the CF (Table 1) were evaluated using an artificially attenuated turbulent

The peak of the cross-covariance between

Comparison of estimated correction factors and response times to the values used to attenuate the

The value for

Relative difference between the mean sensible heat fluxes obtained with the different correction methods (

In order to evaluate the four methods to estimate CF further, turbulent

Relative bias in the estimated correction factor (CF

The importance of turbulence scale on the performance of the four methods was evaluated by stratifying the data according to stability and plotting
them against

Since the relative bias in CF when estimated with Method 1 scales with

The applicability of the results acquired with attenuated

Empirical transfer functions estimated for

Based on theoretical considerations (Sect.

Similar figure as Fig.

For

Response times (top row), low-pass-filtering-induced time lags (middle row) and correction factors (bottom row) as a function of relative humidity. Results for Hyytiälä pine forest are on the left and for Siikaneva open peatland on the right. The lines describe the fits used to obtain

Figure

The CF values estimated with the four methods agreed at Hyytiälä for low RH periods, but they diverged at high RH periods. For
instance when RH

Relative difference between the mean

On average, the differences between the four different methods to calculate CF were small, typically within

There has been a long-standing debate about what the correct form for cospectral transfer function is when the scalar measurements are done with
a first-order linear sensor and vertical wind speed is not attenuated. The seminal paper on EC frequency-response corrections by

Briefly summarising the findings in this study, it was shown in Sect.

The influence of low-pass-filtering-induced phase shift on estimation of high-frequency response of an EC set-up was analysed. The analysis assumed
that the EC set-up consisted of a fast-response anemometer and a linear, first-order-response scalar sensor. Spectral corrections aiming at correcting
the EC fluxes for low-pass filtering were estimated with four methods: three widely used methods and one newly proposed method which specifically
accounts for the interaction between the low-pass-filtering-induced phase shift and attenuation. Based on theoretical considerations and experimental
results we come to the following conclusions:

Cross-covariance maximisation overestimates the signal travel time in the EC sampling line since it inadvertently accounts also for the lag
caused by low-pass filtering of scalar time series caused by the non-ideal measurement system. The bias in the estimated time lag depends linearly on
low-pass-filter response time (

Both power spectra and cospectra are attenuated with the same transfer function (

In order to estimate and correct for the flux attenuation correctly, it is vital to accurately describe the attenuation of the cospectra in the
correction procedure. Hence, while fitting

The theoretical framework proposed in this paper was able to describe the changes in

In summary, it is suggested that the spectral correction methods implemented in EC data processing software are revised so that the influence of low-pass-filtering-induced phase shift is recognised following the findings presented above. In particular, the usage of Method 1 is discouraged as it resulted in clearly biased flux values.

Here we derive Eq. (

Now we can use Euler formula (

Data and MATLAB codes to reproduce Figs.

OP devised the original concept for the study, and all the authors provided further ideas. OP did the data analysis, with input also from ÜR. OP wrote the first draft of the manuscript, with contributions also from TA, ÜR and IM. All the authors commented on the first draft and made improvements.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study was supported by ICOS and the European Commission. Toprak Aslan is grateful to the Finnish National Agency for Education and the Vilho, Yrjö and Kalle Väisälä foundation for their kind support for funding. Olli Peltola is supported by the postdoctoral researcher project funded by the Academy of Finland, and Eiko Nemitz acknowledges support by the Natural Environment Research Council as part of the UK-SCAPE programme delivering national capability.

This research has been supported by the European Commission, H2020 Research Infrastructures (RINGO; grant no. 730944), the Väisälän Rahasto (grant to Toprak Aslan), the Academy of Finland (grant no. 315424) and the Natural Environment Research Council (award no. NE/R016429/1).

This paper was edited by Glenn Wolfe and reviewed by Johannes Laubach, Marc Aubinet, and one anonymous referee.