All eddy-covariance flux measurements are associated with random
uncertainties which are a combination of sampling error due to natural
variability in turbulence and sensor noise. The former is the principal
error for systems where the signal-to-noise ratio of the analyser is high,
as is usually the case when measuring fluxes of heat, CO

Surface layer fluxes of gases such as carbon dioxide (CO

Increasingly, eddy covariance is now being applied to measure fluxes of
pollutants which are more difficult to measure precisely. Examples include
measurements of volatile organic compounds (VOCs; Karl et al., 2002;
Langford et al., 2010; Park et al., 2013), ozone (O

In addition, for many of these systems, co-location of anemometer and sensor
is not possible. Closed-path sensors require inlet lines that create a
time lag (

With so many options available, it is clear that the calculated flux may differ depending on the chosen time-lag method. For example, in their study Taipale et al. (2010) confirm that using a prescribed time lag may result in a systematic underestimation of the flux, as the “true” time lag is likely to vary over time due to fluctuations in pumping speed but also due to the degree of absorption/desorption with the inlet wall and its effect on the effective transport time through the tube. Especially for the more water-soluble compounds this may change with humidity and the aerosol coating of the inlet. Similarly, systematically searching for a maximum within a noisy cross-covariance with multiple local maxima may well bias fluxes towards more extreme values (Laurila et al., 2012). The AVG method offers something of a compromise between the two approaches, but some systematic bias may still remain.

We hypothesise that the bias induced by using methods that search for a maximum in the cross-covariance is closely linked to the random error in the flux, which is in part a function of the SNR of the analytical instrumentation and in some cases may be greater than the systematic error induced from using a prescribed time lag. In order to address this hypothesis, an appropriate method is needed to quantify the random error in the flux and separate it into sampling and instrument error components.

Assuming the time lag is known, the random error (RE) of an eddy-covariance
flux can be estimated in a variety of ways (Lenschow and Kristensen, 1985).
One traditional method is mainly used to estimate the flux error due to the
limited sampling of the stochastic nature of turbulence. It is based on the
variance of the instantaneous values of

Mahrt (1998) offered an alternative method that negates the use of the
integral timescale by splitting the time series into sub-records and
formulating the error as the standard error between sub-records (Mahrt,
1998; Rannik et al., 2009):

Another, quite widely used, method to determine the random uncertainty in
eddy-covariance flux measurements was devised by Finkelstein and Sims (2001). Their approach is based on the variance of a covariance between two
variables which are first auto- and cross-correlated. The random error is
approximated through the integration of the auto-covariance and
cross-covariance functions of the vertical wind velocity and scalar
concentration as

In cases where the time lag is unknown, random flux errors are often
assessed based on the statistical properties of the cross-covariance
function used to identify the time lag. This technique, first conceived by
Wienhold et al. (1995) and developed further by Spirig et al. (2005),
involves taking the standard deviation of the cross-covariance function at a
distance far from the zero time lag (typically several times the integral
timescale). In theory, the cross-covariance in this region reflects both
random sensor noise and variability of the (genuine) atmospheric
signal/concentration; thus, multiples of the standard deviation can yield a
random flux error at a given confidence interval (e.g. 1.96

While the cross-covariance method is widely used, few studies have attempted to go beyond this and isolate the effects of random sensor noise, mainly due to its negligible influence in many conventional eddy-covariance systems. Noticeable exceptions include the work of Shurpali et al. (1993) who proposed a new technique for estimating random instrument uncertainty which was popularised in the mid-nineties (Clement et al., 1995; Billesbach et al., 1998). In this approach the flux of a tracer is measured while sampling air with a constant mixing ratio e.g. directly sampling from a gas standard, and hence any observed flux, is purely a reflection of the random instrumental noise. This method has proved extremely robust, but has the obvious disadvantage of requiring routine data acquisition to stop while the random instrument uncertainty is assessed.

Billesbach (2011) proposed a more practical approach for quantifying random
uncertainties from sensor noise. The so-called “random shuffle approach”
involves random reshuffling of one of the variables in time and thereby
removing any covariance between source/sink terms and transport, leaving
only accidental correlations which can mostly be attributed to instrument
noise. This is an intriguing option, yet, if we consider a measured time
series

More recently, Mauder et al. (2013) approximated errors associated with
random instrument noise by first calculating the signal-to-noise ratio of
the concentration time series using an auto-covariance function and then
using a basic error propagation to estimate the contribution of that noise
to the uncertainty in the cross-covariance:

In this study we explore a further possibility for estimating the portion of the random error that is attributable to sensor noise by combining the ideas of Billesbach (2011) and Mauder et al. (2013), focusing in particular on the interplay between random instrument uncertainty, cross-covariance peak width and the systematic flux bias induced when determining the flux through the use of a cross-covariance function (Taipale et al., 2010; Laurila et al., 2012). In understanding this linkage, our aims are to (i) validate the use of Eq. (5) for use with EC and DEC data sets, (ii) outline an optimal strategy for calculating and reporting random errors, (iii) identify the optimal strategy for determining time lags for eddy-covariance data with low SNR and (iv) to draw conclusions on the validity of flux measurements made with low SNR.

Instrumental noise comes in both structured and unstructured forms. For
example, the 50–60 Hz signal from a mains AC power supply might introduce a
structured noise into a time series, and optical fringes often introduce
periodic features in optical spectroscopic approaches. By contrast,
uncorrelated white noise can result from minor fluctuations in the mechanics
of instrument components, or fluctuations in temperature, pressure or
humidity. Here we focus our attention on unstructured, white noise only, and
define the SNR for a given time series

Using the auto-covariance function as opposed to the auto-correlation
function (i.e. the normalised auto-covariance) means the calculated signal
and noise are variances and retain their original units. Taking the square
root gives the standard deviation of both the signal (

Illustration of the two methods for determination of analyser (temperature measurements,

The auto-covariance is a convenient method when working in the time domain,
but alternatives are available when analysing the data in the frequency
domain. Figure 1b shows the variance spectrum for a time series of
temperature measurements (

Throughout this paper we utilise the auto-covariance method in the time domain as it is readily applicable to both eddy-covariance and disjunct eddy-covariance data sets.

As discussed previously, the precision with which a flux can be measured is
commonly approximated from the properties of the cross-covariance function
between

For time lags much different from the true time lag (

Multiplying this measure of the random error by

A modification of the LoD

Cross-covariance functions for sensible heat fluxes (

Figure 2, panel c shows the LoD for sensible heat flux data calculated using
both the LoD

Analysis of the statistical properties of the cross-covariance function
seems to offer a practical approach for approximating the total random error
of the flux, because the variability of the cross-covariance function
comprises both instrument noise and the variability of the (genuine)
atmospheric concentration. Yet, as discussed above, isolating the
instrumental component of the total random error remains a challenge. Here,
we attempt to untangle the two errors using an approach similar to the
“random shuffle” method of Billesbach (2011). Rather than shuffling the
measured scalar time series to remove any covariance between

The four steps of this method are summarised as follows:

Perform an auto-covariance of

Generate a time series of white noise (

Calculate the cross-covariance

Apply the RE

Although this proposed technique does not make any assumption about the
distribution of

With this in mind we performed several tests to determine if the covariance
between the vertical wind velocity and a time series of white noise differs
depending on the distribution of that noise (e.g. whether it is Gaussian,
Poisson or log-normally distributed). For a single 30 min averaging
period the covariance between

Panel

These findings confirm the theoretical considerations of Lenschow and Kristensen (1985), that, if the time lag is known, the presence of uncorrelated noise induces a random uncertainty in the flux but does not induce a systematic bias. Nonetheless, this conclusion does not consider the interplay of this noise with the determination of a time lag, which is vital for sensors that are spatially separated from the vertical wind velocity measurement, and its potential to introduce a bias that is a function of the random error.

In order to investigate the influence of unstructured white noise from
analysers and the method of time-lag determination on calculated fluxes, a
series of simulations were performed using 31 days of sensible heat flux
data (see the Supplement). Time lags were determined using the
three main methods outlined above, MAX, AVG and PRES. For the AVG method, a
further ten scenarios were implemented, whereby the running mean applied to
the cross-covariance was increased from 0.5 to 5 s in 0.5 s intervals. In
all scenarios the time lag was sought within a 10 s window which ranged from

The covariance between the genuine signals of

Simulating the effects of different time-lag determination methods and the effects of SNR is a useful exercise, but it is important to verify that these are representative of real world data. We assessed the performance of each lag method on example data from a variety of analysers operated in field experiments with varying levels of signal-to-noise, bearing in mind that the SNR for a given application will depend on concentrations and instrument operation. The analysers included an ultrasonic anemometer (Gill HS-50), a condensation particle counter (CPC, TSI Model 3776), an ultra-high-sensitivity aerosol spectrometer (UHSAS, PMS, Boulder, USA), a tuneable diode laser (TDL, Aerodyne Research Inc.) as well as disjunct data from a proton transfer reaction mass spectrometer (PTR-MS; Ionicon, Innsbruck, Austria). Figure 4 shows the frequency distribution of the SNR of 30 min averaging periods for each of the analysers.

A detailed description of each of the data sets used is supplied in the Supplement.

Signal-to-noise ratios of typical instruments used for the flux
measurement of various trace gases and aerosols. Analysers include a sonic
anemometer (temperature) and PTR-MS (isoprene, methanol and acetone)
operated above a mixed oak forest at a height of 32 m and above a city
(benzene). Particle number concentrations were measured by a CPC and UHSAS
(single size bin) above a Douglas fir forest and N

The procedure for calculating random errors using the auto-covariance
approach was applied to EC fluxes of sensible heat (A) and DEC fluxes of
isoprene (B) and acetone (C), and the results are shown in Fig. 5. The error
bars denote the total random error obtained from the cross-covariance
function (e.g. RE

The lower panels show scatter plots of the random instrument error calculated using the numerically calculated Gaussian white noise flux versus the analytical approximation of Mauder et al. (2013; Eq. 5). The two methods give consistent results to within a few percent for both eddy-covariance and disjunct eddy-covariance data sets. Therefore, implementation of either method can enable operators to estimate the minimum detectable flux for their analyser under a given turbulence regime.

The simulations applied to sensible heat flux data reveal a distinct relationship between the signal-to-noise ratio of the raw temperature data and the relative flux bias for both the MAX and AVG lag determination methods. Figure 6a shows the results for 10 Hz eddy-covariance data. It is immediately apparent that methods that systematically search for a maximum (red trace) induce an average positive bias (towards more extreme emission or deposition) to the reported flux which increases linearly as the analyser signal deteriorates. For this data set, the relative bias can be as much as 18 %. Adopting the AVG method can significantly reduce this error provided the applied running mean is of a suitable length. However, selection of an inappropriate running mean may allow the bias to persist and can also become negative when overly long. The reason for the negative result lies in the fact that the shape of the peak in the covariance spectrum tends to be skewed, while the running average of the AVG peak fit is symmetrical. However, theory cannot currently explain the skewness which is therefore difficult to predict. By contrast, the use of a prescribed time lag (for the anemometer temperature data, the time lag is known to be zero), uncertainty increases as the signal is deteriorated more and more, but to a smaller degree, and its sign is random.

Sensible heat

Figure 6b and c show the same set of simulations for fluxes calculated using the disjunct eddy-covariance method using sampling intervals of 2.5 s (panel b), 5 s (see Supplement) and 7.5 s (panel c), respectively. It is well understood that adopting a disjunct sampling approach reduces the statistical sample size and thus increases the random error (Lenschow et al., 1994; Rinne and Ammann, 2012). However, it is frequently assumed that the increased random error does not translate to a systematic bias in the measured fluxes, but our simulations show this not to be the case. The poor sampling statistics and high instrument noise combined with the MAX method for time-lag identification can potentially lead to 100 or even 200 % overestimation in the mean flux.

The method used to determine the time lag is a key factor in accurately
resolving the flux as already demonstrated with the 10 Hz eddy-covariance
data. Additional random uncertainty incurred from disjunct sampling
amplifies the bias at signal-to-noise ratios less than 100, and in this
instance resulting in relative errors of about 300 % at SNR

Panel

These findings come with the caveat that in these simulations the prescribed time lag was a known quantity. When applied to real world data the adopted time lag must be a well-defined parameter that does not drift significantly over time. Failure to meet this requirement would undoubtedly result in a systematic underestimation of the flux, the magnitude of which would become a function of the cross-covariance peak width. This is discussed further in Sect. 3.2.2.

Of equal importance is the magnitude of the expected flux. Fluxes may be large even if the scalar mixing ratios are very noisy and thus the relative error is dependent on both the signal-to-noise ratio and the magnitude of the flux. Thus, although Fig. 6 describes the behaviour of the bias, the exact values depend on the magnitude of the fluxes and also the structure of the underlying turbulence data. However, these simulations do serve to highlight those aspects that make flux data more vulnerable to systematic errors.

Figure 7 shows the results of the peak width simulations on two artificially
generated, multi-frequency signals. As well as reiterating the increase in
the relative error associated with the analyser signal-to-noise ratio, this
plot serves to demonstrate that the FWHM of the covariance peak is an
equally important parameter. Broader covariance peaks, reflecting slower
turbulence/larger eddies, result in a higher probability of an extreme
maximum (i.e. the true cross-covariance between

Image plot depicting the relationship between the average relative bias and both the signal-to-noise ratio of the analyser and the full width half maximum (FWHM) of the cross-covariance function peak for simulated eddy-covariance flux data calculated using the MAX time-lag method.

When using a prescribed time lag, the attenuation of samples through long
inlet lines, adsorption/desorption effects and fluctuations in pump flow
rate are not typically considered. More often, the prescribed value is
chosen purely on the basis of the inlet dimensions and a spot measurement of
the flow rate, or through a single test where a pulse in concentration and
wind speed is created near the anemometer/inlet. In these cases the
potential for underestimating the flux is large. It is therefore good
practice to initially search for the time lag using the AVG or MAX method
and to plot the results as a histogram or time series. This may confirm that
the time lag was indeed constant or it may reveal a clear peak or trend in
time lags which can be used to set the prescribed value. For instruments
that measure multiple species (e.g. mass spectrometers, optical
spectrometers), it may be suitable to use the average time lag of a species
that shows a clear flux (and thus clear time lags) as a proxy for the other
compounds being measured. However, difference between gases in terms of
solubility and therefore adsorption/desorption characteristics need to be
considered. For example, it is well known that for closed-path sensor
measurements of CO

When measuring trace gas and aerosol fluxes, the fast sampling requirements
of eddy covariance can result in low SNRs. Working in this region can see
the random flux error equal or even exceed the magnitude of the flux,
potentially introducing a bias as discussed above. In addition to these
effects, where a maximum in the cross-covariance is still sought, the
derived flux may switch between emission and deposition values of similar
magnitude. This phenomenon, which we term “mirroring”, is observed in the
example flux data shown in Fig. 8 which were obtained using TDL, UHSAS and
PTR-MS instruments and occurs because the random error in the flux is
sufficiently large to span the zero line. It may be tempting to remove the
negative (positive) fluxes on the basis of biophysical implausibility. For
example, when measuring aerosol fluxes where only deposition fluxes are
expected, it would be easy to dismiss positive fluxes as artefacts.
Nevertheless, removal of these points is clearly incorrect and would introduce a positive bias to the reported average data
(cf. Nemitz et al., 2002). Under such circumstances the reported flux is predominately driven by
fluctuations in the amount of turbulence which evolves throughout the day to
give a diurnal cycle which might be mistaken for a flux. Adopting the MAX or
AVG methods exaggerates the mirroring by systematically choosing the furthest
point away from zero which in the extreme case can result in the very
unnatural flux distributions shown in Fig. 9. Adopting the AVG method with 5 s running mean limits this effect to a certain extent, but a noticible dip
around zero remains. Importantly, the use of a prescribed time lag
eliminates the splitting of data from either side of zero to give a much
more natural looking flux distribution. For many compounds an assessment of
the frequency distribution of flux data evaluated with the MAX method will
highlight whether mirroring occurs and whether this approach is therefore
not applicable. Care needs to be taken, however, when making this type of
assessment on CO

An example of “mirroring” in eddy-covariance data with low SNR
processed with the MAX time-lag method. The data were obtained by TDL (panel

Distributions for benzene fluxes calculated using the MAX, AVG (5 s) and PRES time-lag methods. The benzene concentration data had an average signal-to-noise ratio of 0.09 and ranged between 0.007 and 0.24 at the 5th and 95th percentiles.

In addition to the calculated fluxes, the red time traces in Fig. 8 show the
Gaussian white noise flux (

For data where mirroring is observed, there are either no fluxes present or
insufficient statistics to resolve them. If these data are to be utilised at
a 30 min time resolution then they are of little use and should be
rejected. In some cases, extending the averaging period may provide the
additional statistical information required for resolving the flux, but it
is also increasingly likely to violate the requirements for stationarity.
Yet, in the literature, measured fluxes are seldom utilised at the resolution
with which they are collected, but are more typically aggregated either to
establish longer term budgets, by time of day or by a meteorological
parameter such as light or temperature, in order to establish robust
relationships for model parameterisations. Where data are averaged,
presented and utilised in this way, the statistical significance of the
average can be evaluated against the LoD of the ensemble average
(

Figure 10a shows the averaged diurnal fluxes of the acetone time series
shown in Fig. 8c for 1, 7, 14 and 21 day periods (Acton et al., 2015). The
shaded areas represent the averaged LoD

Figure 10b shows the same plot for the much longer time series of benzene
(Valach et al., 2015). In this case we observe how the averaged fluxes
eventually exceed the

Benzene flux measurements calculated using a prescribed time lag and averaged as a function of traffic density. Error bars represent the ensemble LoD. In cases where the error bars intersect zero (blue points), the flux cannot be considered significantly different from zero.

In this study we have carefully examined several key factors affecting the analysis of flux data with high noise level. Clearly, the effect of instrument noise on flux measurements has been studied before. Here we have developed a technique to quantify the uncertainty due to sensor white noise by first quantifying the amount of noise and then calculating a flux with this noise level. This numerical approach has been used to validate the approximations of Mauder et al. (2013) and shows consistent results when applied to both EC and DEC data sets. Both these methods can be easily implemented into eddy-covariance processing software and share the key advantage over the more traditional experimental approach of Shurpali et al. (1993) that measurements do not need to be interrupted for the assessment to take place. Nonetheless, it is important to reiterate that both of these approaches are not sensitive to the effects of structured noise, which cannot be quantified using the auto-covariance method (e.g. Eq. 7) upon which each of these approaches are based.

Most of the earlier analyses of random errors have been carried out under the assumption that the time lag between wind and concentration measurement is known. To our knowledge, the systematic bias introduced through the interplay between random sensor noise and the techniques used to determine the time lag has so far not been studied very systematically, although the problem has been highlighted in general terms in several publications and textbooks. Taipale et al. (2010) studied the effect of routines that are based on maximising the absolute value of the cross-covariance for disjunct data produced by PTR-MS and introduced the AVG approach to reduce this effect. We show here that, in general, the effectiveness of this approach depends on the length of the running mean chosen and the shape of the peak in the cross-covariance function.

Our work highlights the benefit of constraining the time lag of the air sampling and quantifying it as precisely as possible by external means when working with noisy sensors. In practical terms, this might mean controlling the inlet flow carefully and heating the inlet line to minimise adsorption/desorption effects, or deriving the time lag from a simultaneously measured compound with better SNR. Here we compile a list of general recommendations for the collection and processing of eddy-covariance data with limited SNR.

Where possible, log anemometer and scalar data to a single computer. This eliminates uncertainty in time lags due to clock drift and should restrict time lags to positive time shifts.

In-line flow meters should be used to monitor and record fluctuations in pumping speeds.

Use pressure controllers to limit fluctuations in sample flow rate (this may not always be possible when high flow rates are required to maintain turbulent flow and increases the required pumping capacity and therefore power consumption).

For water-soluble trace gases, heating of the entire inlet line should be considered to limit adsorption/desorption effects. Care needs to be taken not to generate aerosol evaporation artefacts for trace compounds that are distributed between the gas and aerosol phase according to a temperature dependent equilibrium.

Online monitoring of sample humidity may be necessary to account for adsorption desorption effects.

Use of the MAX method to generate an initial histogram or time series of time lags, followed by a second analysis using the PRES approach with a thus informed predefined time lag may be preferable to either estimating the time lag based on sampling flow rates alone or using the MAX method for final processing. This is because time lags estimated from the sampling flow do not consider the potential for a phase shift in the cross-covariance due to either signal attenuation or limited response of the instrumentation (Massman, 2000; Hörtnagl et al., 2010).

When reporting processed fluxes, results should be reported even if they are below the single-flux LoD, as long as they fulfil other quality control criteria. However, each individual flux value should be reported with its own quantification of the random uncertainty, so that uncertainties can be combined when fluxes are averaged.

We thank the members of the ECLAIRE flux community and three anonymous reviewers for their useful comments and recommendations. This work was funded by the EU FP7 grant ECLAIRE (no. 282910) and through the NERC grants ClearfLo (NE/H003169/1) and CLAIRE-UK (NE/I012036/1). W. J. Acton acknowledges financial support from BBSRC and Ionicon Analytik GmbH through the award of an industrial CASE studentship. Edited by: S. Malinowski