The eddy covariance method is commonly used to calculate vertical turbulent exchange fluxes between ecosystems and the atmosphere. Besides other assumptions, it requires steady-state flow conditions. If this requirement is not fulfilled over the averaging interval of, for example, 30 min, the fluxes might be miscalculated. Here two further calculation methods, conditional sampling and wavelet analysis, which do not need the steady-state assumption, were implemented and compared to eddy covariance. All fluxes were calculated for 30 min averaging periods, while the wavelet method – using both the Mexican hat and the Morlet wavelet – additionally allowed us to obtain a 1 min averaged flux.

The results of all three methods were compared against each other for times
with best steady-state conditions and well-developed turbulence. An excellent
agreement of the wavelet results to the eddy covariance reference was found,
where the deviations to eddy covariance were of the order of

At a glance, the Mexican hat wavelet flux offers the possibility of a detailed analysis of non-stationary times, where the classical eddy covariance method fails. Additionally, the Morlet wavelet should be used to provide a trustworthy flux in those 30 min periods where the eddy covariance method provides low-quality data due to instationarities.

The eddy covariance technique is a common method to measure vertical
turbulent exchange fluxes between ecosystems and the atmosphere. It has the
great advantage of being a direct and in situ measurement method

The application of wavelet analysis became popular in geoscience and
atmospheric turbulence at the beginning of the 1990s

Conditional sampling is furthermore applicable under non-steady conditions
and was proposed by

Until now no direct comparisons of the eddy covariance method with results
obtained by wavelet analysis and conditional sampling for long-time periods
of methane fluxes had been conducted. Therefore, the challenge of this paper is
not only the comparison of both already often-applied methods in relationship
to the eddy covariance method but also to show with two examples that the
quality test for eddy covariance data

The data used for this work were obtained from June to September 2014 at a
study site located approximately 15 km south of the settlement Chersky
(68.613

An eddy covariance system has been running continuously since July 2013. The
measurements were conducted using the heatable 3-D sonic anemometer USA-1
(Metek GmbH, Elmshorn, Germany) combined with a closed-path setup, where the
inlet of the gas tube was fixed directly below the sonic anemometer. This
tube (Eaton Synflex decabon, length 13.8 m, Reynolds number

The raw data from the sonic anemometer and the closed-path analyser were
collected by the software EDDYMEAS

As the present work aimed in a methodological comparison of different flux
calculation methods, the early preprocessing for all three methods was done
identically using the software TK3

In order to obtain the finalized flux, a number of additional corrections
should be applied to the calculated covariance, e.g. the transformation of
the measured buoyancy flux into the sensible heat flux

The eddy covariance method is based on the turbulent Navier–Stokes equation

Particularly important assumptions are fully developed turbulent flow as well
as horizontal homogeneity of the surface and thus the flow field

For this study the program TK3, version 3.11

The conditional sampling method – also known as eddy accumulation – is
based on

In the present study the conditional sampling flux was calculated following
Eq. (

The wavelet transform allows the decomposition of a time series into the
frequencies that represent the signal without losing information about its
localization in time

A continuous wavelet transform of a discrete time series

In this study wavelets with a sinusoidal form were used, where especially the
complex-valued Morlet wavelet has been proven to be an appropriate choice for
atmospheric turbulence

The expression

Summary evaluation of eddy covariance, conditional sampling and wavelet analysis method.

For two simultaneously recorded time series

As the intention of this study was on short events as well as a comparison to
the traditional eddy covariance method, the wavelet cross-spectrum was
calculated for both averaging intervals

Although the eddy covariance method has been proven as the highly accurate
standard, the wavelet analysis allows us to neglect two main requirements of
eddy covariance: at first, the time averaging can be smaller than 10 to
30 min due to wavelet decomposition in time and frequency domain without
ignoring flux contributions in the low-frequency range. Secondly, wavelet
transform does not require steady-state conditions, but can also be applied
on time series containing non-stationary power

In order to compare the three calculation methods, no more corrections were
applied, but a second run of the TK3 routine was executed to provide quality
assessments. It included double rotation, spectral correction in the high-frequency range

In order to detect short-time turbulent events, a MAD spike test similar to

For result validation of the methane flux a statistical evaluation using the
concept of linear orthogonal regression

An evaluation of the quality of the calculated results of both newly
implemented methods for conditional sampling and wavelet fluxes was necessary
to be sure that they are reliable. In this section all results were validated
against each other for times with best steady-state conditions and well-developed turbulence as described in detail in
Sect.

In comparison to wavelet analysis, only for conditional sampling 17 (0.7 %)
outliers were found and consequently removed by adaption of the MAD test
(

Besides the found outliers, the very good regression slope of

Scatterplot of conditional sampling against eddy covariance methane
flux for times with best steady-state conditions and well-developed
turbulence. The dashed line follows the function

The main difference between the Mexican hat and Morlet wavelet is the
excellent resolution in the time domain at the first (see also
Fig.

About 99.5 % of the variance in the results of each method can be explained
by the linear relationship. Theoretically deciding for a predictor variable,
the standard deviations for

Scatterplot of Morlet against Mexican hat wavelet methane flux
(right) for times with best steady-state conditions and well-developed
turbulence. The dashed line follows the function

Scatterplot of Morlet (left) and Mexican hat wavelet methane flux
(right) against eddy covariance for times with best steady-state conditions
and well-developed turbulence. The dashed line follows the function

Case study of 23 July 2014. The colours in the wavelet
cross-scalograms between

Case study of 2/3 August 2014. The colours in the wavelet
cross-scalograms between

In contrast to the comparison of the two wavelet methods, the validation of
both against the eddy covariance flux determined the actual quality of the
calculated results, because the latter is considered the reference standard
in the context of this study. For each wavelet
(Fig.

To sum up, the method developed and implemented in this study to obtain
methane fluxes using wavelet analysis agreed very well with the eddy
covariance results under best steady-state conditions and well-developed
turbulence. Both methods resulted in a small, but detectable underestimation
of the flux, where the use of the Morlet wavelet marginally showed better
results. This is due to its excellent frequency resolution and in agreement
with other authors who also applied or recommended the Morlet wavelet on
atmospheric turbulence time series

In order to discover a situation under well-developed turbulence and best
steady-state conditions, the afternoon of 23 July 2014 from 13:00 to 16:00
was chosen as a random example (Fig.

In contrast to the very good agreement of wavelet and eddy covariance fluxes,
the conditional sampling results showed a non-systematic deviation from the
latter flux type by

Filtering the 1 min averaged wavelet flux as described in
Sect.

In the same time during the event the eddy covariance flux quality was
determined to be very low following the overall quality flag system by

As wavelet analysis does not require steady-state conditions, the obtained
wavelet results are the most trustworthy fluxes. The wavelet flux over
30 min was about

As this study aims on a methodological comparison, the meteorological and ecological discussion of this event will be presented in a future paper.

The aim of the present study was to develop a software, which calculates the
flux from 20 Hz wind and methane concentration data in order to resolve and
investigate peaks in flux of only short duration within minutes properly.
Under best steady-state conditions and well-developed turbulence it was found
that the 30 min averaged results of the developed routine based on wavelet
analysis were in very good agreement with eddy covariance. This also implies
that the wavelet results itself might be used as reference flux in future
studies. For conditional sampling a high sensitivity regarding the correct
choice of

Eddy covariance is the standard method for flux investigation on ecosystem scale. But in the case of short-time turbulent events, it typically results in a flux of bad quality due to a violation of the steady-state assumptions over the averaging period. Exactly in such situations, the wavelet method provided a more trustworthy flux, because it does not require steady-state conditions. The Mexican hat flux allowed an exact localization of the event in time, while the Morlet flux resolves the flux contributions in frequency domain best. If the Morlet wavelets indicates large flux contributions for low frequencies, these time series should be controlled or even corrected with the ogive method. Therefore, the Mexican hat wavelet flux offers the possibility of a detailed analysis of non-stationary times, where the classical eddy covariance method fails. Additionally, the Morlet wavelet should be used to provide a trustworthy flux in those 30 min periods where eddy covariance led to low quality due to instationarities.

In the next stage of this project, we will evaluate the performance of eddy
covariance and wavelet methods to detect fluxes under different types of
non-steady-state events, which are typically observed during long-term flux
monitoring campaigns for CH

The dataset containing all necessary data to calculate methane fluxes for both case studies is publicly available at:

The authors declare that they have no conflict of interest.

This work has been supported by the European Commission (PAGE21 project, FP7-ENV-2011, grant agreement no. 282700, and PerCCOM project, FP7-PEOPLE-2012-CIG, grant agreement no. PCIG12-GA-2012-333796), the German Ministry of Education and Research (CarboPerm project, BMBF grant no. 03G0836G), and the AXA Research Fund (PDOC_2012_W2 campaign, ARF fellowship M. Göckede). Furthermore the German Academic Exchange Service (DAAD) gave financial support for travel expenses. We thank Fanny Kittler, who greatly assisted the research.The article processing charges for this open-access publication were covered by the Max Planck Society. Edited by: S. Malinowski Reviewed by: two anonymous referees

^{th}Symosium on Boundary Layer and Turbulence, American Meteorological Society, 245–246, 1997.