Large variability is inherent to turbulent flux observations. We review different methods used to estimate the flux random errors. Flux errors are calculated using measured turbulent and simulated artificial records. We recommend two flux errors with clear physical meaning: the flux error of the covariance, defining the error of the measured flux as 1 standard deviation of the random uncertainty of turbulent flux observed over an averaging period of typically 30 min to 1 h duration; and the error of the flux due to the instrumental noise. We suggest that the numerical approximation by Finkelstein and Sims (2001) is a robust and accurate method for calculation of the first error estimate. The method appeared insensitive to the integration period and the value 200 s sufficient to obtain the estimate without significant bias for variety of sites and wide range of observation conditions. The filtering method proposed by Salesky et al. (2012) is an alternative to the method by Finkelstein and Sims (2001) producing consistent, but somewhat lower, estimates. The method proposed by Wienhold et al. (1995) provides a good approximation to the total flux random uncertainty provided that independent cross-covariance values far from the maximum are used in estimation as suggested in this study. For the error due to instrumental noise the method by Lenschow et al. (2000) is useful in evaluation of the respective uncertainty. The method was found to be reliable for signal-to-noise ratio, defined by the ratio of the standard deviation of the signal to that of the noise in this study, less than three. Finally, the random uncertainty of the error estimates was determined to be in the order of 10 to 30 % for the total flux error, depending on the conditions and method of estimation.

The eddy covariance (EC) method is the most direct and defensible way to
measure vertical turbulent fluxes of momentum, energy and gases between the
atmosphere and biosphere. Considering an optimal measurement setup and a
standardised scheme for post-field processing of the measured EC raw data, we
can assume that the systematic error is minimised, and then the random error
of the fluxes is typically dominating the EC flux measurement uncertainty at
short timescales. The accuracy of flux random error estimates becomes
important for interpretation of measurements especially when detecting small
fluxes in terms of turbulent exchange or signal-to-noise ratio (SNR) of the
instrumentation. Moreover, it is desirable to estimate the total random
uncertainty for each averaging period as well as to separate it into the main
components, e.g. one-point sampling error and instrumental noise (Businger et
al., 1986). For the uncertainty due to instrumental noise, the method
proposed by Lenschow et al. (2000) has been recently applied to EC
measurements not only for energy and CO

Recently Langford et al. (2015) analysed in detail the uncertainties related to flux detection from the EC data with low SNR. The authors evaluated the impact of the time-lag determination and called for caution since under low SNR condition the traditional methods of maximising the cross-covariance function can lead to a systematic bias in determined fluxes. The study also reviewed the approaches for estimation of flux random errors. For quantifying the flux uncertainty Langford et al. (2015) suggest using the method by Wienhold et al. (1995) and, following Spirig et al. (2005), suggest multiplying the flux error standard deviation by a factor of three to obtain the limit of detection at 99 % confidence level. The method by Lenschow et al. (2000) to calculate the effect of instrumental noise on the flux error was also validated for data with low SNR by Langford et al. (2015). They compared the method with estimates derived from the root-mean-square (RMS) deviation of covariance of white noise and vertical velocity records and found that the error was not sensitive to the type of distribution of the noise and the RMS approach was consistent with the method by Lenschow et al. (2000).

In the current study we review available methods for the random error estimation of turbulent fluxes, which are widely used by the flux community. We perform calculation and analysis of flux errors by considering different error formulations described in Sect. 2.

We use the measured natural turbulent records for (i) quantitative comparison of the error estimates by Finkelstein and Sims (2001), Salesky et al. (2012), Wienhold et al. (1995), Lenschow et al. (2000) and Billesbach (2011) and (ii) evaluation of sensitivity of error estimates on numerical approximations and calculation details. Based on the analysis we provide recommendations regarding the choice of the flux random error estimates, together with calculation guidelines for numerical evaluation.

In addition, we generate artificial records with predefined statistical properties characteristic to atmospheric turbulence to (iii) evaluate the flux error estimates with high accuracy. Numerically evaluated error estimates are compared with the analytical predictions to validate the theoretical expressions for different error estimates. From simulated time series the calculated error estimates allow us also to (iv) evaluate the uncertainty of the flux random errors.

Turbulent fluxes averaged over a limited time period have random errors because of the stochastic nature of turbulence (Lenschow et al., 1994; Rannik et al., 2006) as well as due to noise present in measured signals (Lenschow and Kristensen, 1985).

The random error of the flux defined by

For stationary time series, in the limit

The integral timescale (ITS) of

Rannik et al. (2009) estimated the timescale

Perhaps the most frequently used method to estimate the flux error is the
method equivalent to the spectral method proposed by Lenschow and Kristensen (1985) in the time domain as

The Eq. (

Salesky et al. (2012) introduced filtering method for error evaluation. By
using the box filtering operation with filter width

At the limit

The value

All the error estimates presented in Sect. 2.1 are different methods for
evaluation of the same flux error provided that the averaging period

Random uncertainty of the observed covariance due to presence of noise in
instrumental signal, assuming the white noise with variance independent of
frequency, gives essentially the lowest limit of the flux that the system is
able to measure. Such uncertainty estimate can be expressed in its simplest
form as (e.g. Mauder et al., 2013)

The noise level of the sonic anemometers is typically a few hundredths of m s

In most practical cases the instrumental noise is of importance for
atmospheric compounds with low concentrations and fluxes. Therefore, we
further assume negligible contribution of anemometer noise and express the
flux error due to gas analyser noise only by

For the white type of noise, which is typical for measurement
instrumentation, the signal noise at frequency

Wienhold et al. (1995) use a method to calculate “the error in the flux
determination, the flux detection limit”, calculating the standard
deviation of the covariance function

Billesbach (2011) proposed a method to calculate the flux error estimate,
which according to the authors was “designed to only be sensitive to random
instrument noise”. The error is calculated according to

Random shuffling of the time series

Lenschow and Kristensen (1985) have shown that the auto-covariance function
for the Poisson type of noise has the form

If an average over fluxes

Measurements from three different and contrasting sites with different
surface properties and observation heights of about 23 m (forest site in
Hyytiälä, SMEAR II), 2.7 m (Siikaneva fen site) and 1.5 m (Lake
Kuivajärvi) above surface were used to evaluate the flux error estimates
for June 2012 (July 2012 at Kuivajärvi) for measured temperature, carbon
dioxide (CO

The first set of measurements was done at the SMEAR II station (Station for
Measuring Forest Ecosystem-Atmosphere Relationships), Hyytiälä,
Southern Finland (61

The second dataset is taken from Lake Kuivajärvi (61

The third dataset was collected at Siikaneva fen site (61

Turbulent fluxes and other statistics reported in the study were calculated over 30 min averaging period by block averaging approach (i.e. no detrending was applied if not mentioned otherwise) using the EddyUH software (Mammarella et al., 2016).

Prior to flux calculation, raw data despiking, conversion of CO

The measured data (wind speed and concentration records) were quality
screened for spikes (all 30 min periods with a single data point exceeding
physically meaningful value excluded) and, according to Vickers and Mahrt (1997), by applying the following statistics and selection thresholds: data
with concentration skewness outside (

SNR is defined in the current study as SNR

We generated artificial records with pre-defined statistical properties characteristic to atmospheric turbulence. Gaussian probability density functions were assumed for vertical wind speed and concentration time series. The atmospheric surface layer (ASL) similarity relationships were assumed for the variances of the records and the timescales of the auto-correlated processes were defined via normalised frequencies (Appendix A).

The analysis was carried out as following. First, we calculated the flux
errors according to analytical expressions Eqs. (A4), (A5) and (

Second, we calculated from the repeated simulated artificial records time
series (

Third, we evaluated the error estimates

Flux error estimates

Dependence of relative flux random errors on

The random errors of the covariance time series as presented in Appendix A
can be derived analytically as the total error (Eq. A4) and as the error for
the Wienhold et al. (1995) method (Eq. A5). Note that the flux error
estimates by Finkelstein and Sims (2001) and Salesky et al. (2012) represent
theoretically the same error given in general case by Eq. (

According to ASL similarity functions the relative flux error is largest at
near-neutral stability and decreases both for unstable and stable
conditions. In stable cases the errors are smaller because the turbulent spectrum
is shifted towards higher frequencies, resulting in more efficient averaging
(over the same period

Flux error estimates

We further analysed in detail the flux errors for the conditions
characterised by

The variability of the error estimates (calculated according to Eq. 22) is
around 10 to 20 % for

The flux uncertainty estimates increase approximately linearly with the flux
magnitude at all sites (Fig. 2 and Table 2). As expected from theory, the
error estimate

Linear fitting parameters between absolute value of the flux and
flux uncertainty estimates (uncertainty

Based on the linear regression statistics presented in Table 2, a few
findings can be emphasised. The intercept values are small (compared to the
flux error magnitudes) and imply that flux uncertainties tend to vanish with
no turbulent exchange. Generally the relative flux error is larger over the
forest (slope 0.14 and 0.18 for CO

The error due to instrumental noise (

For qualitative comparison with the behaviour of our theoretical model based
on ASL similarity theory we constructed a plot similar to the one presented
in Sect. 4.1 (Fig. 1); see Fig. 4. The figure illustrates that the observed
behaviour holds: the relative flux errors increase with increasing

Error estimate calculated based on the Salesky et al. (2012)
method (

Dependence of relative errors on

Frequency distribution of the ratio between the flux instrumental
noise error (the error estimate

Relative difference between the estimated (

Three example cases for application of the Lenschow et al. (2000)
method with presented spectra (upper panels) and auto-correlation functions
(lower panels). Case 1 (

The method

In most cases the flux random uncertainty is dominated by the stochastic
nature of turbulence and the instrumental noise is a minor part of the total
uncertainty. The relative contribution of instrumental noise to the total
uncertainty can be assessed by comparing the flux error estimates

The calculated

If the temperature signal was low-pass filtered before superimposing the
signal with Gaussian noise, the accuracy of the noise estimation was
improved (cf. Fig. 6d–f). When the SNR was below 5, the noise
variance was estimated successfully (relative error within

Error estimate calculated based on the Billesbach (2011) method
(

On the whole, the accuracy of the Lenschow et al. (2000) method depends on how strong is the signal relative to noise at high frequencies, since the noise is estimated using small time shifts close to the auto-covariance peak. The signal-to-noise ratio at high frequencies decreases if (i) the total SNR decreases, (ii) the ITS increases (power spectrum shifts to lower frequencies) or (iii) the high-frequency variation in the signal is dampened. Thus for instance for signals measured at a tall tower with a closed-path analyser, the method should work well in estimating the instrumental noise, since most of the turbulent signal is at relatively low frequencies (high measurement height) and the high-frequency variation in the signal is dampened. However, for measurements close to the ground with an open-path analyser the method does not perform equally well under all conditions. The reliability of the Lenschow et al. (2000) method in estimating the instrumental noise from the signal is determined by the SNR, as illustrated in the Fig. 6. Therefore, a priori knowledge on the instrumental precision characteristics is needed when analysing the outcome of the method.

Amount and magnitude of reliable noise estimates obtained by using
Lenschow et al. (2000) method. The reported values are estimates of 1 SD at
10 Hz sampling rate (i.e.

We analysed in more detail the performance of the Lenschow et al. (2000) method in Fig. 7. In case of low SNR and high ITS the method estimates the true noise with relatively small bias (Fig. 7a, d). For the same SNR but low ITS value the variance is strongly overestimated (Fig. 7b, e). We argued earlier that low-pass filtered signals enable to obtain better noise variance estimates (Fig. 6). However, in case of high SNR (Fig. 7c, f) the method leads to significant underestimation of the true variance. Thus filtered signals (instruments with not perfect frequency response) are not always preferred in terms of the method's ability to determine the signal noise.

Normalised errors

In Table 3 we report the estimated signal noise statistics for the
instruments used in the current study by defining reliable values according
to criterion SNR < 3. The fraction of reliable estimates is low for
the instruments with high precision characteristics. For example, the method
does not typically work for estimation of the precision of horizontal wind
speed measurements of the sonic anemometers. However, the method performed
to produce the noise characteristics for the vertical wind speed components
at all sites. This is at least partly due to the fact that the vertical wind
speed variance is smaller than that of the horizontal component, resulting
in lower SNR for the vertical component. The estimated signal noise
characteristics are in good correspondence with instrument specifications
(where available) except for the CH

Billesbach (2011) introduced the so-called “random shuffle” method to
estimate the instrumental noise from EC measurements. However, as argued in
Sect. 2.3 this method does not estimate the flux error due to the
instrumental noise since it mixes turbulent variation with noise and thus
the error corresponding to the sum of the variances of the turbulent signal
and the noise is deduced by the method. This is exemplified by Fig. 8a and b: the error estimated with the “random shuffle” method (

Normalised uncertainty estimate according to Finkelstein and Sims (2001),

Under different wind speed and stability conditions the ITS of turbulence
varies and therefore it becomes relevant what would be the appropriate
integration time for the method by Finkelstein and Sims (2001). The
integration time of the flux error

Plots for different sites (varying the observation level and surface type)
and wind speed and stability influences as reflected by ITS classes indicate
that integration time 200 s serves as an optimal choice for all conditions
(Fig. 10). This would guarantee less than 10 % systematic underestimation
of the flux error even in case of 25 % largest ITS values (ITS > 75th percentile) for fen and lake sites. The figure also
illustrates that the cross-covariance term in Eq. (

Commonly applied random error estimates of turbulent fluxes were tested and
compared in this study. The methods proposed by Finkelstein and Sims (2001)
and Salesky et al. (2012), the error estimates

The methods by Finkelstein and Sims (2001) and Salesky et al. (2012) do not require estimation of the ITS and have an advantage compared to the methods which do so because of uncertainty of the estimation. It was demonstrated by Salesky et al. (2012) that, under non-stationary conditions, their method performed better than the other methods involving direct evaluation of the ITS, namely the methods by Lumley and Panofsky (1964) and Lenschow et al. (1994). The methods proposed by Lenschow and Kristensen (1985) and in a discrete form by Finkelstein and Sims (2001) require just integration of the products of the covariance functions over sufficiently long time exceeding the ITS. While avoiding direct calculation of the ITS, the impact of non-stationarity is indirectly embedded in the method by Finkelstein and Sims (2001). As discussed above, we believe that the impact of non-stationarity is reduced due to the filtering in Salensky et al. (2012) method. The theory of turbulent flux and flux error calculation relies on the assumption of the stationarity. Violation of this assumption introduces additional uncertainty in time-average statistics including the flux random errors. Therefore superior performance of any of these two methods discussed above is not evident, and we suggest that both methods are reliable alternatives for flux error evaluation.

Wienhold et al. (1995) defined an error estimate (

An alternative to one-point statistical estimation of the flux random errors as described in this study (Sect. 2.1) is the two-tower approach, where the flux random error is evaluated by using the difference of the fluxes measured at two EC towers (e.g. Hollinger et al., 2004). The method assumes statistically similar observation conditions with independent realisations of turbulence at the two towers. Since the conditions are difficult to realise because of spatial correlation in measurements (e.g. Rannik et al., 2006), we suggest that the one-point statistical approach provides rigorous but more convenient method to estimate the flux random errors. Nevertheless, the two-tower approach was shown to give close results to the method by Finkelstein and Sims (2001) when similar weather conditions at the two sites were included in the analysis (Post et al., 2015).

The error estimates with very clear physical meaning are the total error
resulting from stochastic nature of turbulence due to limited sampling in
time and/or space, the methods by Finkelstein and Sims (2001) and Salesky
et al. (2012), and to a good approximation the method by Wienhold et al. (1995), as well as the random error due to instrumental noise only. To estimate the
latter from the field measurements (not from laboratory experiment) Lenschow
et al. (2000) suggested calculating the signal noise variance from the
difference between the signal auto-covariance at zero lag and the
extrapolated value of the auto-covariance function to zero lag. The noise
variance enables to calculate the flux error according to Eq. (

Billesbach (2011) proposed the flux error estimate based on the product of
vertical wind speed and concentration fluctuations, randomly re-distributing
one of the series (denoted by

Different flux error estimates have been assigned the meaning of the flux
detection limit. For example, Wienhold et al. (1995) called their method as
“detection limit of the flux”. Billesbach (2011) suggested that the method
they introduced “was sensitive to random instrument noise”. The method by
Lenschow et al. (2000) estimates the flux value that the system is able to
detect within an averaging period

The flux detection limit has been used also in conjunction with other flux measurement techniques. For example, in the case of chamber measurements the flux detection limit has been used to denote the flux error arising from all possible error sources. The traditional way to perform chamber measurements is to determine the gas concentration at several time moments during the chamber operation. In such data collection the sources of uncertainty are the imprecision related to gas sampling (either manual or automatic) as well as instrumental uncertainty (e.g. Venterea et al., 2009), leading to a measurement precision which is called a detection limit of chamber-based flux measurement system. It has to be noted that the flux detection limit of the chamber systems depends on several factors such as the type of the chamber and respective sampling method, the precision of the instrument, chamber dimensions and operation time. Therefore the flux detection limit of the chamber-based systems (which accounts for all possible sources of uncertainty) is comparable to the total stochastic error of the EC fluxes.

We also studied the performance of the Lenschow et al. (2000) and Finkelstein and Sims (2001) flux error estimation methods over different ecosystems and observation conditions. The performance of the Lenschow et al. (2000) method is affected by the SNR and the ITS of turbulence. We established that the method provides reliable estimates for SNR < 3 (in the statistical sense, single biased values can occur). However, no criterion based on the ITS could be provided as the results deviated among sites.

Application of the EC method requires stationarity of time series within
averaging period (e.g. Foken and Wichura, 1996). Non-stationarity results in
higher random uncertainty of the flux value and therefore the stationarity
requirement has to be fulfilled if each 30 min or 1 h average value is
expected to be statistically significant. We tested sensitivity of the flux
errors derived by the Finkelstein and Sims (2001) method on integration time
and high-pass filtering of the fluxes performed by mean removal, linear
detrending and auto-regressive filtering. It was observed that the flux
error increased with integration time up to about 300 s, revealing the
influence of the low-frequency (possibly non-stationary) signal variance on
the flux estimates. The high-pass filtered time series were less affected.
For consistency, the flux errors should be calculated based on the same time
series (in terms of filtering) as used for flux calculation. Apart from the
impact of the low-frequency contribution to flux errors (and fluxes), which
we believe is related to non-stationarity of the conditions, we observed
that, in order to obtain

The EC fluxes are uncertain due to stochastic nature of turbulence by about 10 to 20 % under typical observation conditions. By using simulation of time series with statistical properties similar to natural records we deduce that the flux error estimates in turn are uncertain by about 10 to 30 %.

Data are freely available upon request from the authors.

The wind speed and scalar concentration time series were simulated as

Here

The theoretical random error estimate for the flux calculated from described
artificial time series is given according to Eq. (

In addition, the timescales were related to wind speed and stability via

The study was supported by EU projects InGOS and GHG-LAKE (project no. 612642), Nordic Centre of Excellence DEFROST, and National Centre of Excellence (272041), ICOS (271878), ICOS-FINLAND (281255), ICOS-ERIC (281250) and CarLAC (281196) funded by Academy of Finland. This work was also supported by institutional research funding (IUT20-11) of the Estonian Ministry of Education and Research. O. Peltola is grateful for Vilho, Yrjö and Kalle Väisälä foundation for funding. Edited by: H. Chen Reviewed by: two anonymous referees