Fluxes measured with the eddy covariance (EC) technique are subject to flux losses at high frequencies (low-pass filtering). If not properly corrected for, these result in systematically biased ecosystem–atmosphere gas exchange estimates. This loss is corrected using the system's transfer function which can be estimated with either theoretical or experimental approaches. In the experimental approach, commonly used for closed-path EC systems, the low-pass filter transfer function (

Vertical turbulent fluxes of momentum, energy, and gases between the atmosphere and the biosphere measured by the eddy covariance (EC) technique are subject to both low- and high-frequency losses

The EC sampling system acts as a low-pass filter on the flux, and the signal loss must be compensated for with the frequency response correction (FRC) during post-processing.
The first step in the FRC is the description of the effect of the low-pass filtering of the measurement system, and for this, the transfer function approach has been widely used since it was first proposed by

In the experimental approach for closed-path systems,

Interestingly, there has not been much debate to date whether to use power spectra or cospectra to determine the time constant of the

On the other hand, this noise is often assumed not to correlate with the fluctuations in

EC measurements conducted under low-flux conditions result in relatively high signal noise, i.e. low signal-to-noise ratio (SNR)

To our knowledge, the uncertainty in fluxes caused by the use of the PSA and the CSA has not been investigated systematically so far, motivating this study, which hypothesizes that the success of the PSA and CSA usage in FRC depends on the attenuation condition and the level of SNR. Consequently, we expect to see substantially different time constants, correction factors, and eventually different overall magnitudes of correction estimates with respect to the attenuation and SNR conditions. To test this hypothesis, we need a scalar dataset which represents different attenuation levels and noise conditions. Assuming spectral similarity between scalars,
we apply different levels of attenuation and noise to sonic temperature time series (

In order to calculate the true unattenuated (i.e. frequency-response corrected) covariance (

Another way to calculate

Regardless of the method chosen,

For the PSA,

Thus, prior to the calculation of Eq. (

For PSA,

In our study, we follow the same procedure (hereafter

Alternatively, for the CSA,

For CSA,

In the

Regarding the noise removal procedure, this approach can be problematic as we will show in Sect.

Here we introduce a new alternative approach (hereafter

Equation (

A diagram illustrating fitting procedures for PSA methods. Shown are the spectra of unattenuated and noise-free temperature (red line) and spectra of low-pass filtered and noisy scalar before (solid blue line) and after (black line) noise removal. For

Two datasets measured with sonic anemometer in an EC set-up from the Siikaneva fen site were mainly used in this study.
The site is located in southern Finland (61

The first dataset (D

The second dataset (D

Lastly, to demonstrate the performances of different methods in real-world data, we used

The data processing flow for all variants of PSA and CSA is summarized in Fig.

In practice there is no time lag between

In summary, for three methods (two PSAs and one CSA), we assessed 45 different conditions each, combining five different attenuation levels with nine different SNRs. We repeated the same procedure 100 times to account for the uncertainty associated with the white noise generation and thus obtained 100 different values for the time constant for all attenuation and SNR levels for both PSA and CSA. The relevant results are shown in Sect.

Flow chart of the data processing for time constant calculation using the dataset D

The dynamic performance of any EC measurement system can be approximated with a linear first-order non-homogeneous ordinary differential equation

The desired (i.e. low-pass filtered) output data in the frequency domain can be estimated as

We followed this procedure to filter

In this study we use Gaussian white noise, which has equally distributed spectral densities across all frequencies

Data processing steps when estimating the long-term budgets using the D

Flow chart of the data processing for the cumulative flux calculation using

We applied regular EC data processing, which included de-spiking, coordinate rotation, and de-trending. Next, the

Here we preferred using the square root to describe the true transfer function as it is a good approximation when maximization of the cross-covariance is used for the time-lag correction as shown by

The time constant (

Effect of several low-pass filtering (

In order to illustrate the important steps in the data analysis in PSA (e.g. low-pass filtering, noise superimposition, and noise removal only for

Results of the noise removal procedure applied in the power spectra approach following

Time constants calculated using the power spectrum approach, comparing

The results of the time constant estimation are shown in Fig.

For

We assumed that the noise contaminating the signal is white noise, which may not always be the case in real-world data. Thus, the accuracy of the

This also includes

Figure

Lastly, as expected, the level of noise does not affect the accuracy of the CSA method at any SNR level as the random noise does not correlate with

Normalized ensemble cospectra for the original and various attenuated time series (i.e.

Time constants calculated using the cospectral approach, i.e.

Figures

Relative biases of the cumulative fluxes derived with different PSA methods, i.e.

Relative biases of the cumulative fluxes derived with the

Correction factors (

The range of attenuation and SNR conditions reported in the literature is rather wide and varies depending on ecosystem type, the scalar of interest, data processing, a configuration of instruments, and set-up of the EC system.

In addition,

Given the wide range of SNR and attenuation conditions summarized above, we analysed only a limited range of SNR and attenuation. Also, the impact of the system response time depends on the position of the spectral peak frequency which changes not only with wind speed but also measurement height and surface roughness. Nevertheless, the analysis of the different approaches showed a systematic behaviour with respect to SNR level and attenuation conditions. This provides the opportunity to extend the results of this study beyond the examined values and guides the selection of the right method to find the relevant

The key constraint of the study was that we artificially simulated the various attenuation and SNR conditions. Thus, demonstrating the performance of the methods with real-world data is of great importance. Accordingly, here we provide an example via processing

Time constants obtained via different approaches (

Here we investigated the limitations of two commonly used approaches to empirically estimate the eddy-covariance (EC) transfer function needed for the frequency response correction of measured fluxes by analysing a temperature flux time series which was synthetically degenerated mimicking slow-response set-ups and noisy sensors. The first approach (PSA) is based on the ratio of measured power spectra, while the second (CSA) is based on the ratio of measured cospectra. For PSA, we examined two alternative approaches of accounting for the white noise contribution to the power spectra: i.e.

We then examined the effect of the different approaches to estimate the time constants on the cumulative fluxes: fluxes corrected using the

The SNR did not affect the accuracy of either PSA or CSA approaches, alleviating concerns on EC flux measurements with low SNR levels.

In summary, for the empirical estimation of parameters of

Finally, given the constraints of this study, we encourage additional studies based on the real attenuation and SNR conditions, investigating also other types of noise contamination to provide a step forward in efforts to standardize the EC method, which is of great importance to avoid systematic biases of fluxes and improve comparability between different datasets.

In this section, we derive the new approach for the PSA (i.e.

As described in Sect.

It is worth mentioning that this method can be used to retrieve the variance of noise as well since

Here we perform a short analysis to characterize the type of noise in an example high-frequency

Methane fluxes from an upland forest site can be considered to be greatly affected by instrumental noise because fluxes are small and near the detection limit. Therefore, we used a forest methane mixing ratio dataset to identify the type of noise by comparing its spectra with the spectra of white and blue noise which was artificially generated and with the same standard deviation of the methane dataset.

The data were collected at the SMEAR II station (Station for Measuring Forest Ecosystem–Atmosphere Relations), Hyytiälä, southern Finland (61

Power spectra of methane were calculated using measurements from 10 July at 12:00–14:30 UTC

Normalized spectra of methane concentration (red), white noise (black), and blue noise (blue) (

The cospectral model used in flux calculations for small fluxes (i.e. absolute value of sensible heat flux smaller than 15 W m

Data to reproduce Figs.

IM and OP designed the study, and TA processed and analysed the data. UR investigated the distortion caused by the quadrature spectrum in CSA. UR, TA, and AI investigated the limitations of the noise removal procedure of

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 through 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 (decision 315424) funded by the Academy of Finland, and Eiko Nemitz acknowledges support by the Natural Environment Research Council award number NE/R016429/1 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 Finnish National Board of Education (grant no. KM-18-10792), the Academy of Finland (decision no. 315424), the Natural Environment Research Council (award no. NE/R016429/1), and the Väisälän Rahasto.Open-access funding was provided by the Helsinki University Library.

This paper was edited by Glenn Wolfe and reviewed by George Burba and Marc Aubinet.