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 χ (e.g. CO2, H2O), and this induces a time lag
(tlpf) between vertical wind speed (w) and scalar χ time series which is additional to the travel time of the gas signal in the
sampling line (tube, filters). Time-lag estimation via cross-covariance maximisation inadvertently accounts also for tlpf and hence
overestimates the travel time in the sampling line. This results in a phase shift between the time series of w and χ, which distorts the
measured cospectra between w and χ and hence has an effect on the correction for the dampening of the EC flux signal at high frequencies. This
distortion can be described with a transfer function related to the phase shift (Hp) which is typically neglected when processing EC
flux data. Based on analyses using EC data from two contrasting measurement sites, we show that the low-pass-filtering-induced time lag increases
approximately linearly with the time constant of the low-pass filter, and hence the importance of Hp in describing the high-frequency
flux loss increases as well. Incomplete description of these processes in EC data processing algorithms results in flux biases of up to 10 %, with
the largest biases observed for short towers due to the prevalence of small-scale turbulence. Based on these findings, it is suggested that spectral
correction methods implemented in EC data processing algorithms are revised to account for the influence of low-pass-filtering-induced time lag.
Introduction
The eddy covariance (EC) is the standard micrometeorological technique for measuring vertical turbulent fluxes of momentum, heat and gases in the
atmospheric surface layer . Under certain conditions (flat terrain, homogeneous and stationarity turbulent flows, source/sink
homogeneity, absence of chemical sources/sinks), the measured EC fluxes represent direct and continuous estimates of the surface exchange of energy
and matter between the surface and the atmosphere and are then widely used to estimate energy and water balances, as well as the carbon budget of
different type of ecosystems .
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 . All EC systems, including open-path, enclosed-path or
closed-path gas analysers, act as low-pass filters and thus cause a systematic bias to flux estimates. Flux loss at high frequency is due to the
incapability of the measurement system to detect small-scale variation. The main reasons for co-spectral attenuation are inadequate frequency response
of the sensor, sensor separation and line averaging, as well as, in closed-path systems, the sampling of air through tubes and filters.
The frequency response correction is usually performed based on a priori knowledge of the system transfer function and the unattenuated cospectrum:
CF=∫f1f2Cow,χdf∫f1f2Cow,χHlpfdf.
Here CF is the estimated spectral correction factor, Cow,χ the normalised unattenuated cospectrum between scalar (χ) and vertical
wind speed (w), f the frequency, Hlpf the total transfer function describing the low-pass filtering, and f1 and f2 the
integration limits that are determined by the length of the averaging period and Nyquist frequency, respectively. The correction, performed by
multiplying the measured covariance by the factor CF, always increases the absolute flux magnitude. Note that the high-pass transfer function
is not included in Eq. () as here we focus only on the low-pass filtering effect of the EC system. The low-pass
filtering transfer function Hlpf can be derived either theoretically or empirically . The correction is in general
different for momentum flux, sensible and latent heat fluxes, and the various gas fluxes and is specific to each EC system. In the theoretical
approach, the atmospheric surface layer co-spectral models are used, and Hlpf is calculated as a convolution of
specific transfer functions representing different causes of flux loss, the equations for which can be found in and
. This approach works well for correcting the fluxes of momentum and sensible heat, as well as gas fluxes measured by open-path
systems. Further, and have proposed alternative theoretical approaches, providing an analytical estimation of
correction factors.
Alternatively, the empirical approach can be used, in which the model cospectra and Hlpf are estimated using in situ measurements. This is
the preferable approach for closed-path systems, in which additional effects related to the inlet dust filters and rain caps
, potential variations in the sampling line flow rate, and absorption and/or desorption processes inside the sampling tube
may contribute to the EC system cut-off frequency, introducing transfer functions which are difficult to
estimate a priori.
For this approach, different methods have been proposed for retrieving Hlpf from the measured power spectra or cospectra of the sonic
temperature T and the target gas dry mole fraction . While the use of power spectra is the
method recommended by the ICOS methodology and is implemented in the EddyPro software package, other studies and
software packages (EddyUH) have supported the use of measured cospectra for the empirical retrieval
of Hlpf. The companion paper has investigated methodological issues and flux uncertainty related to these two methods
under different attenuation conditions and signal-to-noise ratio scenarios.
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 . The standard procedure is to determine this time delay for each
averaging period (typically 30 min) by maximising the related cross-covariance function (in absolute terms) and taking the corresponding lag as the
true signal delay. However, this time shift between the two signals depends not only on the system set-up (e.g. air sample travel time from the inlet
to the infrared gas analyser, IRGA, sampling cell, separation distance between the inlet and the centre of the sonic anemometer path), but it can additionally reflect the
low-pass filtering of the measured signal .
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 Hlpf time constants are empirically derived from measured power spectra or cospectra and when related CFs values
are applied to covariances estimated from the cross-covariance maximum. Finally, we present a method to take the phase shift effect into account when
processing EC data. Towards these aims, we use EC data collected above a forest canopy (Hyytiälä) and a wetland (Siikaneva) in Finland
covering a large range of attenuation conditions and integral turbulent timescale characteristics.
TheoryTransfer functions
The measured cross-spectrum (Crm) between the fluctuations of vertical wind speed and the attenuated and lagged scalar can be
described by (e.g. )
Crm=ZwhχZχ*e-jϕphys,
where Zw and Zχ are Fourier transforms of the time series of vertical wind speed (w′) and scalar (χ′)
fluctuations which are perfectly in phase and not attenuated, hχ is the Fourier transform of the filter function that describes the response
of the instrument measuring the scalar time series, j=-1, and ϕphys=ωtphys, where tphys is a
constant time shift (s) between w′ and χ′ and ω=2πf is the angular frequency, with f representing the
frequency (Hz). The superscript * denotes complex conjugation. Equation () describes the cross-spectra measured with a
typical EC system employing a closed-path gas analyser in which the scalar fluctuations are attenuated (hχ) and delayed in time
(tphys) with respect to w′ due to the gas sampling system (tubes, filters) and horizontal sensor separation. The signal travel
time in the gas sampling line (tphys) could be approximated from the volume of the sampling line and flow rate. Note that we assume that
the sensor measuring w′ is perfect without any distortions in the measured w′ time series. This assumption does not limit the
generality of the findings below. We also neglect any influence of sensor separation or high-pass filtering on flux attenuation.
Following, for example, and , the scalar sensor is approximated to be a linear first-order sensor for which hχ can
be written as
hχ=11-jωτ,
where τ is the time constant (s) of the filter, also called response time. Note that ZwZχ* is the true cross-spectrum
(Cr) between unattenuated w′ and χ′ time series which are in phase, and it has a real (cospectrum Co) and an
imaginary (quadrature spectrum Q) part (Cr =ZwZχ*=Co+jQ). Recall also that the integral of
Co equals the unattenuated covariance w′χ′‾, i.e. the vertical turbulent flux of scalar χ. Following
previous studies , the quadrature spectrum (Q) can be assumed to be zero for stationary turbulent flow and hence
Cr =ZwZχ*≈ Co. Now, using Euler's formula and some derivation (see Appendix ), we find
Crm=cosϕphys-ωτsinϕphys1+ω2τ2Co-jsinϕphys+ωτcosϕphys1+ω2τ2Co=Com+jQm,
where the first term on the right equals the measured cospectrum (Com) and the second term the measured quadrature spectrum
(Qm). Note that whilst Q was assumed to be negligible, Qm is in fact not equivalent to zero even when
ϕphys=0, i.e. when the travel time of scalar χ′ signal in the sampling system is zero. This means that w′ and
the attenuated χ^′ (the ∧ denotes attenuation throughout the text), depending on frequency, can be out of phase due to low-pass
filtering of Zχ with hχ. This has already previously been noted e.g.. Now, simply based
on the real part of Crm, we can define a transfer function for the first-order system (H) as
H=11+ω2τ2
and a transfer function (Hp) for a generic phase shift ϕ as Hp=cosϕ-ωτsinϕ.
These two transfer functions together describe how the measured cospectrum (Com) deviates from the cospectrum calculated from
ideal measurements free from any attenuation and time shifts (Co).
Similarly, forming the cross-spectrum of the attenuated scalar with itself yields
7Crm,p=hχZχhχZχ*8=11+ω2τ2ZχZχ*=HZχZχ*,
where ZχZχ* is the unattenuated power spectrum. From this it follows that the same transfer function (i.e. H) applies to both power
spectrum and cospectrum in the case that there is no phase shift between w′ and χ′ (i.e. Hp equals 1) and the
quadrature spectrum Q is zero.
Time-lag determination via cross-covariance maximisation
Ideally the time lag between w′ and χ^′ would be estimated from the known dimensions of the gas sampling system and flow
rate, in addition to other components causing time delay between signals. However, it is in practise difficult to keep track of all the components
causing the time delay, and hence a practical (yet sometimes flawed; e.g. ) solution has been to estimate the time lag between
w′ and χ^′ via cross-covariance maximisation. This can be considered to be equivalent to finding a time shift t (and
hence ϕ=ωt) that maximises the integral of Com (i.e. maximises the covariance between w′ and
χ^′).
∫0∞cos(ωt)-ωτsin(ωt)1+ω2τ2Codω=w′χ^′‾(t)
The original aim of this cross-covariance maximisation is to account for the time lag between the time series w′ and χ^′
that is induced by sampling lines (air filters, tubing) and horizontal sensor separation, in other words to find such a time lag that removes the
e-jϕphys term in Eq. (). After successfully accounting for the travel time (tphys), the
attenuation of the measured cospectrum Com (and hence flux) would be described only by H instead of H and Hp
(see Eq. ) since the time-lag estimation sets Hp=1. However, the filter described by Eq. () results
in a phase shift and hence produces an additional time lag between w′ and χ^′, i.e. tlpf. This can be seen
from Eq. (): Qm differs from zero when ϕ=0 (see also ). Cross-covariance
maximisation inadvertently derives the sum of tphys and tlpf, which implies that the time lag determined by cross-covariance
maximisation is not the desired transport time lag. Using the sum of tphys and tlpf instead of tphys induces a
non-negligible negative ϕ in Eq. (). Hence when shifting the time series according to the cross-covariance maximisation, the
transfer function related to the phase shift (Hp) can no longer be neglected and should be calculated using ϕ=-ωtlpf, where tlpf is the bias in the time lag when estimated via cross-covariance maximisation, in other words the low-pass-filtering-induced time lag (note the minus sign since Hp accounts for the overestimated time lag).
As the maximal possible cospectral energy content per unit frequency can be described with the amplitude spectrum (Am=Com2+Qm2), it can be assumed that after the cross-covariance maximisation (time series shifted by
tphys+tlpf), Com can be approximated by Am. It is straightforward to derive Am
from Eq. ():
Am=Com2+Qm2=Co1+ω2τ2=HCo.
Hence Am is attenuated with H instead of H which describes the attenuation of Com in the case that there
is no phase shift between w′ and χ^′ as shown above. This suggests that after cross-covariance maximisation H
approximates HHp and the correct transfer function for cospectrum calculated after cross-covariance maximisation. However, H is
not equivalent to HHp. This relates to the fact that Qm cannot be nullified with a constant time shift since low-pass
filtering induces a frequency-dependent time shift (arctan(-ωτ)ω; see ). If Qm could be
nullified, then Com=Am (see Eq. ) and the attenuation of cospectra could be described accurately with
H. Note also that Eq. () applies universally, independent of ϕ.
The dependence between low-pass-filtering-induced time lag (tlpf) and filter response time (τ) was estimated using the approximation
HHp≈H or Hp≈1H. For this analysis the phase transfer function Hp was
approximated by using the series expansion of the terms in Hp (Eq. ). This approximation resulted in
Hp≈1-12(ωtlpf)2+ωτ(ωtlpf). Similarly, 1H can be
approximated by 1+12(ωτ)2. Equating these two approximations up to the second order yields a quadratic equation in which the
frequency dependence cancels out: tlpf2-2τtlpf+τ2=0. The solution of this simple equation gives
tlpf=τ. Thus, up to the second-order approximation, which holds roughly up to the frequency value ω=1, the low-pass-filtering-induced time lag equals the time constant of the filter. Note, however, that at this frequency range (ω between 0 and 1) the filtering effect
is very weak. The dependence between tlpf and τ can be determined also by numerically solving
Hp≈1H (which equals cos(-2πftlpf)-ωτsin(-2πftlpf)≈1+2πfτ2) and using an assumption that tlpf is proportional to τ, i.e. tlpf=Cτ, where C is
a proportionality constant. These analyses are continued in Sect. . Note that the dependency between tlpf
and τ derived here and in Sect. was not used in the estimation of correction factors (see Sect. ) but
merely to analyse the factors controlling the dependency.
Materials and methodsMeasurement sites
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∘51′ N, 24∘17′ E; 181 ma.s.l., above sea level). The station is
surrounded by extended areas of coniferous forests, and the EC tower is located in a 57-year-old (in 2019) Scots pine (Pinus sylvestris L.)
forest with a dominant tree height of ca. 19 m. The EC measurements were performed with 10 Hz sampling frequency at 27 m
height, whereas the zero plane displacement height (d) was 14 m. The wind speed components and sonic temperature were measured by a three-dimensional ultrasonic anemometer (Solent Research HS1199, Gill Ltd, Lymington, UK), while carbon dioxide and water vapour mixing ratios were measured by an
infrared gas analyser (LI-7200, LI-COR Biosciences, Lincoln, NE, USA). A 0.77 m long tube (4 mm inner diameter) was used to sample air
for the gas analyser with a nominal flow rate of 10 Lmin-1. The centre of the sonic anemometer was displaced 20 cm horizontally
and 1 cm vertically from the intake of the gas analyser. The measurement set-up was designed to closely follow the ICOS EC measurement
protocols . The measurements were conducted between May and August 2019.
The Siikaneva fen site is located in southern Finland (61∘49.9610′ N, 24∘11.5670′ E; 160 ma.s.l.),
consisting mainly of sedges (Eriophorum vaginatum, Carex rostrata, C. limosa) and Sphagnum species, namely
S. balticum, S. majus and S. papillosum, with a height of ca. 10–30 cm. Further details about the site can be found
in , and . The EC measurements were conducted between May and August 2013. The data used in this
study were measured with 10 Hz sampling frequency using a three-dimensional sonic anemometer (Metek USA-1, GmbH, Elmshorn, Germany) and a closed-path
analyser (LI-7000, LI-COR Biosciences, Lincoln, NE, USA). The sonic anemometer and the gas inlet were situated 2.75 m above the peat
surface, and the air was drawn to the analyser through a 16.8 m long heated inlet sampling line. The centre of the sonic anemometer was
displaced 25 cm vertically above the intake of the gas analyser.
A short dataset (hereafter DS1) measured between 09:30 and 11:30 (UTC + 2) on 16 June 2013 at the Siikaneva site was used in the method
validation (see Sect. ), whilst long-term datasets from both Siikaneva (hereafter DS2) and Hyytiälä (hereafter
DH) were used to evaluate the accuracy of different spectral correction methods and their effect on CO2 and H2O
fluxes (see Sect. ).
Data processing
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 (CO2 and H2O) flux estimates at these sites. The processing steps followed commonly accepted routines: (1) data were
despiked using a method based on the running median filter , (2) coordinates were rotated using double-rotation which
aligned one of the horizontal wind components with the mean wind, setting the mean vertical and cross wind components to zero, (3) gas (CO2 and
H2O) mole fractions were converted point-by-point to be relative to dry air if not already done internally by the gas analyser (LI-7200),
(4) turbulent fluctuations were extracted from the measurements using block-averaging, and (5) time lags between gas or filtered T (see
Sect. ) and vertical wind time series were accounted for using cross-covariance maximisation. Power spectra and cospectra were
calculated after these processing steps. Spectral corrections and the related correction factors (CFs) were calculated using four methods
described below; see Sect. . After processing, the flux time series were quality filtered by removing periods with unrealistic sensible
heat fluxes or friction velocities and highly non-stationary fluxes . Site-specific friction velocity thresholds (0.17 and
0.3 ms-1 for Siikaneva and Hyytiälä, respectively) were used to discard gas flux data during low-turbulence periods. Furthermore, at
Hyytiälä wind directions between 150 and 230∘ were discarded since here the mast construction disturbed the airflow. At Siikaneva an
omnidirectional sonic anemometer was used mounted on the top of a mast, and hence clearly disturbed wind directions were not identified. After quality
filtering 4210 and 4722 30 min periods were available for analysis from Hyytiälä and Siikaneva, respectively.
Atmospheric stability was evaluated using the Obukhov length (L):
L=-u*3vkgw′θv′‾θv‾,
where u* is the friction velocity, vk=0.4 the von Karman constant, g the acceleration due to gravity, and θv virtual potential temperature. Overline denotes temporal averaging and primes deviations from the mean. From the Obukhov length, the stability
parameter was calculated as ζ=z-dL, where z is the measurement height and d is the zero plane displacement height.
Empirical estimation of the low-pass filtering of the signal
The total transfer function of an EC set-up can be estimated empirically from measured w–χ^ and w–T cospectra:
He,cs=Cow,χ^Cow,Tw′T′‾w′χ^′‾Fn,
or, alternatively, from the power spectra of the measured scalar and T:
He,ps=Sχ^,χ^ST,TσT2σχ^2Fn,
where Cow,χ^ denotes the cospectral density between w and χ^, and w′χ^′‾
represents the covariance between w and χ^. Sχ^,χ^ and σχ^2 denote the power spectral density and
variance of χ^, respectively, and the corresponding terms are defined for T. When estimating He,cs and
He,ps, the covariances and variances were calculated from unattenuated frequencies (between 5 × 10-3 and
5 × 10-2Hz) . An additional normalisation factor (Fn) was also incorporated in
order to take into account the fact that the variances and covariances in Eqs. () and () may be subject to various high-frequency losses . Here, w and T were approximated to be free from any low-pass filtering effects, and thus Cow,T and
ST,T could be used as a reference. He,cs and He,ps were calculated from ensemble-averaged spectra from
measurement periods that fulfilled the following criteria: flux stationarity was below 0.3, u*> 0.1 ms-1, and
w′T′‾> 0.02 Kms-1. For CO2 it was also required that its turbulent flux was below
-0.05 µmolmol-1ms-1, and for H2O it was required that the flux was directed upwards. For H2O the transfer
functions were determined in relative humidity (RH) bins, and the τ and tlpf values used in calculating CF were estimated
from exponential fits made to the values obtained in the RH bins (similar as for τ in ). In the case of CO2 and
H2O fluxes, the power spectra were subjected to noise removal prior to utilising them in Eq. () by assuming that the signal was
contaminated by white noise and that at the highest frequencies the power spectra contained only noise. See the influence of different noise removal
techniques on the estimation of high-frequency attenuation in our companion paper .
Often both He,cs and He,ps are approximated by H with a value for τ that describes the low-pass
filtering of the EC set-up according to Eq. (). Then the measured flux is corrected for low-pass filtering by multiplying it with CF (see
Eq. ). There are different ways to estimate Cow,χ for Eq. (). Here we limited the analysis only to periods when
the absolute sensible heat fluxes were higher than 15 Wm-2 and estimated Cow,χ in Eq. () with the measured
Cow,T following .
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 H and Hp in Eqs. () and (), respectively. Values for τ (and tlpf in Method 4) were estimated with non-linear least squares fit to He,ps or He,cs, in which Fn and τ (and tlpf in Method 4) were used as fitting parameters. The additional normalisation factor Fn took into account any inaccuracies in estimation of variances and covariances in the calculation of He,ps and He,cs (Eqs. and ), respectively. The fits were weighted with temperature cospectra in order to give more weight to frequencies where the scalar fluxes were high.
Estimation of τFitting parametersHlpf used in the estimation of CF, Eq. ()ReferenceMethod 1Fit H to He,psFn, τHMethod 2Fit H to He,csFn, τHMethod 3Fit H to He,psFn, τHMethod 4Fit HHp to He,csFn, τ, tlpfHHpThis study
We used four methods to estimate EC system response times (and additionally tlpf in Method 4) and then to calculate CF (see
Table ). All the four methods are subject to similar limitations; i.e. they are applicable only when the attenuation is not strong
at frequencies close to the cospectral peak frequency. Method 1 follows the ICOS EC data processing protocol and is implemented, for example, in EddyPro based on (see also https://www.licor.com/env/support/EddyPro/topics/whats-new.html, last access: 24 June 2021). Method 2 follows and is implemented in EddyUH . Method 3 follows and is
implemented in older versions of EddyPro, and Method 4 is the only one that explicitly accounts for the temporal lag caused by the low-pass filtering
action of the system and is introduced in this study. In Method 4, both parameters τ and tlpf, were estimated by fitting
HHp to He,cs. Throughout the study, cross-covariance maximisation was used to account for the lag between scalar
and vertical wind speed as typically done in the global flux measurement network. Note that also other techniques have been used to estimate the time
lag e.g. and that the selection of the time-lag estimation method may have an effect on the correct shape of the cospectral
transfer function (see Sect. ). For instance, the cross-covariance moving average method to estimate the time lag introduced in
should be related to Method 4 as it is also based on cross-covariance maximisation, whereas for the approach in the
correct cospectral transfer function is H (with the caveat that the physical time lag is estimated accurately).
Low-pass filtering of temperature data
In order to evaluate the performance of the different spectral correction methods presented in Sect. , high-frequency T time series
were attenuated with different values of τ in order to mimic attenuated scalar time series. Similar to the procedure used by ,
T time series were converted to frequency domain via a Fourier transform, multiplied by Eq. () and converted back to time domain
with the inverse Fourier transform. Three different values for τ were used: 0.1, 0.3 and 0.7 s. The correct value for CF was
obtained from the ratio between w′T′‾ and w′T^′‾, where T^ was the time series
attenuated with the methodology described above. Note that here w′T^′‾ was calculated using cross-covariance
maximisation in order to mimic regular EC gas flux processing, meaning that the lag caused by low-pass filtering was taken into account. This correct
value for CF was then compared against CF estimated with the four methods described in Sect. in order to evaluate the
accuracy of the methods.
Top: the value for the proportionality constant C (tlpf=Cτ) obtained when approximating Hp≈1/H at different frequency ranges. Bottom: Hp and 1/H calculated using the proportionality tlpf=0.63τ. See Sect. for more details.
Results and discussionTime-lag dependence on attenuation and turbulence timescales
In Sect. , we explained why the following approximate relationship must hold up to some bound at the high-frequency end:
Hp≈1/H. Using the assumption that tlpf=Cτ, this equation was solved numerically for the constant C as
a function of frequency f. Figure (upper panel) illustrates that such a solution is dependent on frequency and varies over a
relatively large interval up to the frequency 23πτ (the relationship holds well up to such a frequency for any τ). This
suggests that tlpf and τ are not directly proportional, but their relation depends on frequency. In accordance with the
approximations with the Taylor expansion (Sect. ), at low frequencies the constant C equals 1. For illustration, the lower panel of Fig. presents the left and right side terms of the approximate equality (Hp≈1/H) calculated using
a value of 0.63 for the coefficient C. In spite of the large variation of the exact coefficient with frequency shown in the upper panel in
Fig. , the selected constant value produces very good correspondence for the whole frequency range, suggesting that C can be
approximated to be constant, although strictly speaking it is not. The value of C=0.63 was obtained by estimating the average time lag in a least
square sense over equally spaced frequencies in logarithmic scale over the interval from τ=0.01πτ to
23πτ for a range of time constants from 0.05 to 1 s. We observed that the optimum coefficient was independent
of τ and equal to 0.63 (with two-digit accuracy). Note that these analyses were independent of the actual turbulent signal as they were obtained
using the transfer functions only (H and Hp).
Low-pass-filtering-induced time lag (tlpf) plotted against response time (τ). The tlpf values were estimated by numerically integrating the integral in Sect. with various time-lag values and evaluating which value resulted in maximum value for various τ and (z-d)/U combinations. The dashed line shows 1:1 correspondence. See Sect. for more details.
In order to evaluate the influence of turbulent signal on the dependence between tlpf and τ, the integral in Sect. ,
i.e. Eq. (), was solved numerically with various combinations of time lag, τ and (z-d)/U in order to evaluate which time-lag
value resulted in maximum value for the integral for a given τ and (z-d)/U combination. This approach is similar to finding the time lag
between signals using cross-covariance maximisation, and it identified the dependence of the low-pass-filtering-induced lag (tlpf)
on τ and the turbulence timescale ((z-d)/U). For this numerical experiment, τ varied between 0.05 and 1 s, (z-d)/U between 1
and 10 s, and Co was calculated based on a model (similar to the one in , but fitted to Siikaneva
observations). Based on this analysis, tlpf depended approximately linearly on τ (Fig. ) in accordance with the
analysis above, yet the dependence had an additional weak τUz-d term: tlpf=aτ-bτUz-d+c, where
a=0.73, b=0.15s and c=0.02s. However, this numerical experiment was repeated also for very high-attenuation levels; the
dependence found above failed to fully recover tlpf values at high attenuation, and this bias increased with τ and inverse of
turbulence timescale (i.e. U/(z-d)). For example at τ=2s and (z-d)/U=1s the equation above gave
tlpf=1.2s, whereas based on the numerical integration value 1.0 s was obtained, meaning that the dependence above was not
able to accurately describe the low-pass-induced time lag at these high-attenuation levels. This is likely due to the fact that also the shape of the
cospectrum Co in Eq. () has an effect on tlpf dependence on τ. At very high-attenuation levels the peak
of the cospectrum is also attenuated, and this results in a different tlpf dependence on τ than the equation above which describes the
dependence when attenuation takes place at high frequencies (i.e. inertial subrange).
The discrepancy between the two dependencies between tlpf and τ obtained above was likely due to the fact that the latter took into
account also the variability in the turbulent flux with frequency, whereas the former was based purely on transfer functions. Nevertheless, these
results indicate that, for a given site, tlpf can be approximated to be constant at a specific attenuation (τ) level, hence
supporting the estimation of tlpf as a fitting parameter in Method 4 when fitting HHp to He,cs (see
Sect. and Table ).
Cross-covariance between w and T^ calculated from data measured between 09:30 and 11:30 on 16 June 2013 at the Siikaneva site. T data were attenuated with Eq. () and three different values for τ in order to demonstrate the effect of filtering on cross-covariance. All the cross-covariance values were normalised with a maximum of unattenuated cross-covariance (i.e. unattenuated covariance). Dots highlight the maxima. Horizontal dashed lines show the attenuation factor (inverse of correction factor, 1/CF) values calculated with Method 4, and dash-dot lines show the attenuation factors calculated with Method 1. Note that the dash-dot lines match the cross-covariance values at zero lag, whereas dashed lines agree with the cross-covariance maxima.
Transfer functions (left y axes) and frequency-weighted and normalised cospectrum (right y axes) for the same data shown in Fig. . Note that here H, H and HHp were not fitted but calculated using the known values for τ and tlpf resulting from the low-pass filtering of the T time series (Sect. ). Nevertheless, H is related to Method 1, H0.5 to Method 3 and HHp to Method 4 (see Table ). Note also that the cospectrum shown in the figure was calculated from the unattenuated T time series in order to demonstrate the frequency range making the biggest contribution to the flux.
Assessment of the flux loss correction methods using an attenuated T time series
The four methods to estimate the CF (Table 1) were evaluated using an artificially attenuated turbulent T time series comparing different values
for τ. Figure shows example cross-covariance functions from the Siikaneva site with three different values of τ applied to
the same 2 h time series.
The peak of the cross-covariance between w and T^ shifted to longer positive times (i.e. T^ lags w), and the
cross-covariance maximum decreased as τ increased. This is related to the phase shift and flux attenuation caused by the low-pass filter,
respectively. The low-pass filtering was also evident in the corresponding power spectra and cospectra. The resulting empirical transfer functions in
Fig. show that power spectra were more attenuated than the cospectra when cospectra were calculated from data in which the time lag
derived from cross-covariance maximisation was used to shift the time series (as is typically done when processing EC data). Moreover, the shape of
the empirical transfer function derived from the cospectra differed from Eq. (). This difference in shape was related to
Hp and the phase shift caused by the low-pass filter. Note that the phase shift resulted in negative values for HHp and
He,cs at high frequencies, meaning that the related attenuated cospectrum had both positive and negative values. This implies
that attenuated cospectra should not be presented on log–log scales (negative values cannot be presented on log scale). Note also that
HHp is not exactly equal to H at all frequencies. In particular, HHp shows negative values at high frequencies,
whereas H does not. Hence the approximation of HHp with H will likely fail under very strong attenuation. When the
correction factor was calculated with Method 4, which takes into account the low-pass-filtering-induced phase shift, the attenuation factors
(1/CF) agree with the normalised cross-covariance maxima (see Fig. ). This indicates that the attenuation of
cross-covariance maxima was accurately estimated with this method. However, when Hp was neglected and the attenuation factors were
calculated with H only, then they agreed with cross-covariance values at zero lag like those predicted by theory (Sect. ). Note that
the time lag discussed here only represents the low-pass-filter effect and not other effects that also alter the time lag. This is only possible because
we work with degraded temperature time series, in which phase shifts through low-pass filtering are the only cause for time delays.
Comparison of estimated correction factors and response times to the values used to attenuate the T time series. The reference values for the correction factors were calculated as the ratio between w′T′‾ and w′T^′‾, where T^ is the temperature time series attenuated according to Sect. . Note that w′T^′‾ was calculated using cross-covariance maximisation. The time lags obtained from cross-covariance maximisation and from fitting HHp to He,cs are shown in panel (c). The same data were used as in Figs. and .
The value for τ used in attenuating the T time series was further varied between 0.05 and 1 s and used for the same 2 h time
series. The bias in CF calculated with Method 1 increased with τ (Fig. a). The other methods did not produce as significantly
biased estimates for CF when using this example dataset, although methods 2 and 3 somewhat underestimated CF (approximately 4 %
underestimation when τ= 0.9 s), and this underestimation increased with τ. These findings are in agreement with
Fig. : H was above and H below the empirical transfer function calculated from the cospectrum
(He,cs), meaning that they underestimated and overestimated the attenuation, respectively. These findings are also in accordance
with the results of with noisy measurements. Note that Method 2 estimated close to correct value for CF (Fig. a)
despite biased estimates for the response time (Fig. b) due to the compensation of two errors (biased τ and incorrect shape for
the cospectral transfer function). Method 4 was able to reproduce the correct value for τ from He,cs
(Fig. ), indicating that the difference between He,cs and He,ps was indeed related to
Hp and the low-pass-filtering-related phase shift. The time lags estimated by Method 4 (i.e. by fitting HHp to
He,cs) agreed with the ones estimated by maximising the cross-covariance (Fig. c), also corroborating this
conclusion. It should be noted that the step changes in the data that are very visible in Fig. c but to some extent also in
Fig. a and b are caused by the finite resolution of the underlying data of w and T of 0.1 s.
Relative difference between the mean sensible heat fluxes obtained with the different correction methods ((cor-ref)/ref⋅100%) and the reference (unattenuated sensible heat flux). All values are in per cent (%).
In order to evaluate the four methods to estimate CF further, turbulent T data were low-pass filtered with three values of τ (0.1, 0.3 and
0.7 s) over several summer months (May–August) at two contrasting EC sites, Hyytiälä and Siikaneva. Then the related flux attenuation
was corrected with the four methods, and the results were compared against a reference flux calculated from unattenuated
T data. Table summarises the relative differences between the four correction methods and the reference. In general, the
values obtained with this extended dataset are in line with Fig. a for Siikaneva, although slight differences can be noted. The
performance of all four methods decreases with increasing τ at both sites, and Method 1 showed the worst performance, which is in line with the
example shown in Fig. and follows the predictions from theory (Sect. ). Of the analysed cases, the biggest bias
(+8.6 %) was found when applying Method 1 at Siikaneva with τ= 0.7 s. In general, the performance of the methods was worse at
Siikaneva than at Hyytiälä. This was likely related to the lower measurement height at Siikaneva (z-d= 2.7 m) than at
Hyytiälä (z-d= 13 m) and thus the comparably larger contribution of high frequencies to the covariance. At Siikaneva, smaller
eddies dominated the turbulent transport, and hence the effect of inaccuracies in describing the high-frequency attenuation of the signal was amplified
compared with Hyytiälä.
Relative bias in the estimated correction factor (CFest) against τ/ITS, where ITS is the integral timescale of w′T′. Data were divided into unstable (ζ<-0.1), stable (ζ>0.1) and near-neutral (ζ<0.1) cases before bin-averaging and plotting. Reference values for the correction factor (CF in the figure) were calculated from the ratio between unattenuated and attenuated covariances. Note that the different colours refer to different stability conditions and markers to the different sites.
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 τ/ITS (Fig. ), where ITS is the integral timescale of w′T′ calculated from the
autocovariance of w′T′ time series by integrating the normalised autocovariance function until its first zero crossing
e.g.. This ratio describes the ratio of attenuation and turbulent transport scales. It is also worth noting that this ratio can
alternatively be approximated by the ratio of the cospectral peak frequency (fm=nmU/(z-d), where nm is the normalised
cospectral peak frequency) to the cut-off frequency (fco=1/(2πτ)) . High values for τ/ITS describe
periods when the attenuation timescale is large relative to the timescale of turbulent scalar transport, whereas low values correspond to cases when
the attenuation timescale is small relative to this turbulence scale. When CF was estimated with methods 2 or 3, the relative bias in CF was
typically within ± 3 % and increased linearly with τ/ITS at Siikaneva. The different behaviour of the two sites in
Fig. b and c may be related to differences in spectral characteristics of turbulent transport at these two sites since measurements in
Hyytiälä were made above forest (roughness sublayer flow) and in Siikaneva above short vegetation (boundary layer flow). For Method 4 the bias
was within ± 2 % at both sites and independent of the τ/ITS ratio. The remaining small bias obtained with Method 4 can be
speculated to be related to the quadrature spectra, which was assumed to be zero, although strictly speaking it is not (not shown). For Method 1
the relative bias in CF scaled with τ/ITS and, after normalisation with ITS, the data from both field sites collapsed into one common
relationship (Fig. a). derived a simple formula to estimate CF for different situations as a function of
(τ/ITS)α, where α=7/8 in neutral and unstable cases and α=1 in stable cases. Similar dependencies could be derived
for the bias in CF derived with Method 1. However, these dependencies should not be used to rectify the biased CF values, but rather
CF should be estimated with methods that do not result in biased CF values in the first place.
Since the relative bias in CF when estimated with Method 1 scales with τ/ITS, this dependence can be used to assess how much scalar fluxes
are biased with different EC set-ups when Method 1 is utilised in the EC data processing. For the following example calculations we approximate a
linear dependence between ITS and (z-d)/U and used the fit to estimate ITS for different z-d and U combinations. For a short tower
(z-d= 1.5 m) with moderate wind speed (U= 2 ms-1) and attenuation (τ= 0.2 s), fluxes are biased
by 4 % based on the dependency shown in Fig. a. For the same tower, higher attenuation (τ= 0.8 s) results in
larger bias (9 %) if Method 1 is utilised. For a tall tower the biases are not as significant since larger eddies dominate the transport, and
hence τ/ITS is smaller for a given τ than in the case of a shorter tower. For a tall tower (z-d= 20 m) with moderate wind speed
(U= 3 ms-1) and attenuation (τ= 0.2 s) the bias is 1 %–2 %, and with higher attenuation
(τ= 0.8 s) the bias is 3 %–4 %. These example calculations demonstrate the magnitude of the flux bias resulting from biased
spectral corrections at contrasting EC sites.
CO2 and H2O fluxes corrected with the four methods
The applicability of the results acquired with attenuated T was analysed by processing CO2 and H2O fluxes from summer months at the
two sites (Sect. ). The gas fluxes were corrected with the four spectral correction methods in order to assess the systematic biases
stemming from biased spectral corrections. For CO2 the fitting of H to He,ps, of H to He,cs and
of HHp to He,cs estimated response times of 0.05, 0.10 and 0.13 s at Hyytiälä and of 0.13, 0.12 and
0.18 s at Siikaneva, respectively. The low-pass-induced time lag (tlpf) estimated as a fitting parameter when fitting
HHp to He,cs yielded close to 0.1 s at both sites for CO2. For reference, the time lags estimated
via cross-covariance maximisation which are combinations of signal travel times in the sampling line (tphys) and tlpf yielded
on average 1.4 and 0.2 s at Siikaneva and Hyytiälä, respectively. For Hyytiälä the signal travel time calculated from the
sampling tube dimensions and flow rate was 0.06 s. However, this calculation neglects the additional time lag caused by the LI-7550 interface unit
and hence can be assumed to slightly underestimate tphys.
Empirical transfer functions estimated for CO2 from ensemble-averaged cospectra (He,cs; Eq. ) and power spectra (He,ps; Eq. ) and three fits to the empirical transfer functions. Also the corresponding frequency-weighted normalised average cospectrum related to heat flux is shown (grey line) in order to illustrate the frequency range with the highest contribution to the scalar fluxes. Data from Siikaneva were used.
Based on theoretical considerations (Sect. ) and results obtained with attenuated T time series, H fit to
He,cs (Method 2) was projected to estimate smaller response times than the other two methods that were projected to agree (see, for example, Fig. b). Instead, here for CO2 the HHp fit to He,cs gave longer response times than
the other two methods (Fig. ). For signals with small attenuation both τ and tlpf are likely difficult to estimate
accurately by fitting HHp to He,cs. The low-pass-filtering-induced lag (tlpf) can attain values only
with the temporal resolution of the data being processed (here 0.1 s), and hence for time series exhibiting small attenuation tlpf
may well be zero. This would cause He,ps and He,cs to be similar, and hence they would result in similar
estimates for the response time, as observed here for CO2. This granular estimation of tlpf might explain why the response times
for CO2 did not follow the expectations. It should be also noted that the methods to estimate the response time have a limited accuracy of
estimation stemming from the algorithms used in the estimation, instrument limitations and meteorological conditions under which the observations are
made . Furthermore, the empirical transfer function derived from cospectra (He,cs) contains the attenuation
related also to sensor separation , whereas the empirical transfer function estimated from power spectra (He,ps)
is related only to gas-analyser-induced low-pass filtering.
Similar figure as Fig. but for H2O in moderate relative humidity conditions (59 % < RH < 62 %). Note that τ estimated from He,ps agrees with τ estimated from He,cs with HHp. Data from Siikaneva were used.
For H2O, τ depends on RH , and hence the analysis was done in RH bins. Figure
shows an example of empirically derived transfer functions for H2O at Siikaneva in moderate relative humidity
conditions. He,cs showed less high-frequency attenuation than He,ps due to cross-covariance maximisation as
predicted by the theory (Sect. ). HHp fit to He,cs resulted in a similar value for τ as the one
obtained by fitting H to He,ps. The other fitting parameter in HHp fit (tlpf) was 0.14 s,
indicating the low-pass filtering of H2O time series caused an additional lag to the H2O time series. Note that H does not perfectly
describe the shape of He,ps. This is due to adsorption and desorption of H2O to sampling tube walls, a process which
cannot be described accurately with Eq. () . Hence different transfer functions have been proposed for H2O.
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 τ and tlpf values for the CF calculation (see Sect. ).
Figure shows results obtained for H2O in different RH bins at the two sites. The H fit to He,cs
(Method 2) gave smaller values for τ in agreement with the results obtained with attenuated T (Fig. b) and expectations based
on the theory (Sect. ). The HHp fit to He,cs (Method 4) was able to reproduce the H2O time-lag dependency on RH, yet it estimated systematically smaller values for the time-lag difference between H2O and CO2 than those obtained
with cross-covariance maximisation. The time-lag difference between H2O and CO2 can be assumed to be related to tlpf of
H2O since H2O is more attenuated than CO2. The difference between H2O and CO2 time lags obtained with
cross-covariance maximisation showed linear dependence on estimated response times in relative humidity bins with a similar dependency at both sites
(y=ax+b, where a=0.47 and b=-0.05 s for Siikaneva and a=0.44 and b=-0.07 s for Hyytiälä, and where
y is the time-lag difference and x the response time. Fits were based on orthogonal linear regression). Difference between H2O and CO2
time lags resulted from stronger low-pass filtering of H2O than CO2 signal, and this analysis assumes that the mere travel times
through the sampling lines (i.e. tphys), which do not depend on the low-pass filtering effects, were the same for H2O and
CO2. These results are in line with Fig. where it was shown that the low-pass-filtering-induced time lag was primarily
determined by a linear dependence on τ with a secondary dependence on turbulence timescale. However, the slopes obtained above were different
from the ones derived in Sect. .
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 > 80 % the four methods gave on average 1.39, 1.29, 1.23 and 1.37 at the Hyytiälä site, respectively. These results are in
line with the ones obtained with attenuated T time series (Sect. and Fig. a): when τ was large, Method 1
overestimated and Method 3 underestimated CF, similar as here for H2O at high RH (and hence large τ). In contrast, for
Siikaneva Method 1 gave systematically higher values for the correction factor regardless of RH, whereas the other three methods gave more similar CF
RH dependency. This difference to Hyytiälä was likely due to lower τ values also at high RH (Fig. ). Note also that
the EC measurement set-up (measurement height, gas sampling system) was not identical at the two sites.
Relative difference between the mean CO2 and H2O fluxes obtained with the different correction methods ((cor-ref)/ref⋅100%). Fluxes corrected with Method 4 were used as a reference for the other three methods. All values are in per cent (%).
On average, the differences between the four different methods to calculate CF were small, typically within ± 3 % at both sites for
both gases (CO2 and H2O) (Table ). The biggest mean relative difference (4.1 %) was found for H2O
measurements at the Siikaneva site for Method 1. This was likely due to the combination of low measurement height and high attenuation which resulted in
biased CF values, in accordance with the findings obtained with the attenuated T time series (Fig. a). Interestingly, for
CO2 the biggest difference at both sites was obtained with Method 3, yielding smaller fluxes than the reference (Method 4). The difference was
bigger than the one obtained with Method 1 which contradicts the findings presented above with attenuated T (Sect. and
Table ) and expectations based on theory (Sect. ), although the differences found between the methods were
generally small. Note that if tlpf≈0, then both power spectra and cospectra are attenuated similarly (i.e. with H) resulting in
methods 1, 2 and 4 being similar. Infrared gas analysers for CO2 and H2O can often be mounted close to the inlet, and in both set-ups
considered here inlet lines are fairly short. The measurement of other gases, including CH4 and N2O, usually requires larger equipment
which often has to be operated on the ground and/or some distance away from the mast, requiring longer inlet lines with slow time response. In these
situations the correct shape of the cospectral transfer function matters a lot more than for the measurement set-ups included here, and hence the
variability between methods is expected to be larger.
Correct form for cospectral transfer function
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
described the flux attenuation related to the scalar sensor with H (see Eq. ), which was later deemed erroneous, for example, by
, , and others. Partly based on similar derivations as shown here in Sect. , the correct form for
the cospectral transfer function was argued by and to be H instead of H . Lately, both forms
(H and H) have been utilised in the literature e.g. without clear
consensus on which one is correct. Part of this confusion is likely related to the fact that some of the past studies have been using incorrect
terminology by using the term cospectral transfer function (Eq. ) even though they have derived/utilised a transfer function for the
amplitude spectrum (Eq. ).
Briefly summarising the findings in this study, it was shown in Sect. in accordance with, for example, that in the case the
travel time in the gas sampling system could be accurately estimated and accounted for, the attenuation of cospectrum (and hence flux) could be
described with H. However, cross-covariance maximisation inadvertently accounts also for the scalar low-pass-filtering-related time shift
(tlpf) and consequently induces a phase shift between the lag-corrected attenuated scalar and vertical wind time series. Hence in the case
that cross-covariance maximisation is used to align the two time series, the correct form for cospectral transfer function is HHp, in which
the part related to the phase shift (Hp) is calculated with tlpf. It was shown above that HHp can be
approximated with H. Therefore the debate about which is the correct form for cospectral transfer function (H or H) relates to the
low-pass-filtering-induced phase shift as discussed already by and , albeit with a different conclusion. Note,
however, that H is an approximation and not an exact representation of the cospectral transfer function.
Summary and conclusions
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 (τ) with a small additional dependence on turbulence timescale. Hence, other means for estimating the physical
time lag might be warranted, especially in the case of noisy measurements . Further research on this topic is required.
Both power spectra and cospectra are attenuated with the same transfer function (H, as noted also by ) in the case that the
travel time of the scalar signal in the sampling line can be accurately estimated. However, if cross-covariance maximisation is used, then
attenuation of cospectra follows HHp in which Hp accounts for the bias in the estimated travel time caused by
cross-covariance maximisation, and it was shown that HHp can be roughly approximated with H. Therefore, all flux calculation
algorithms which rely on cross-covariance maximisation and at the same time use H to describe the attenuation of the cospectra
e.g. can be projected to be biased in the same way as Method 1 used here if the response time (i.e. τ) is
accurately estimated (e.g. from power spectra). In these cases the bias in EC fluxes can be 10 % or above for short tower measurements with
moderate attenuation. Hence, data reprocessing is warranted in order to rectify this bias.
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 H to cospectra (Method 2) biased the first-order response time due to the use of the incorrect
cospectral transfer function, it resulted only in a small bias for the flux correction factor since the method describes the attenuation of
cospectra accurately once the cross-covariance maximisation has been applied. By contrast, fitting of H to power spectra (Method 1) correctly
estimates the response time, but it nevertheless yields biased flux corrections since it utilised a transfer function that is incorrect when
estimating the flux using cross-covariance maximisation.
The theoretical framework proposed in this paper was able to describe the changes in H2O time lag as H2O response time increased
with relative humidity. This suggests that the findings derived here with attenuated temperature time series are applicable also in real world
situations for EC gas flux measurements. However, results for CO2 deviated from the expectations based on the theory. Reasons for this
remained unclear; however, low-pass filtering of inert gases (such as CO2) in the gas sampling line may deviate from what was described here
in such a way that the low-pass filter acting on CO2 may not induce a phase shift. This should be investigated by using CO2 flux
data from an EC set-up with pronounced attenuation of CO2 fluctuations in the sampling line or alternatively in a controlled environment in a
laboratory .
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.
Derivation of Eq. ()
Here we derive Eq. () starting from the cross-spectrum (Eq. ):
Crm=ZwhχZχ*e-jϕphys,
where hχ can be described with Eq. (), which gives
A2Crm=ZwZχ1-jωτ*e-jϕphys,A3Crm=ZwZχ*1+jωτe-jϕphys,A4Crm=ZwZχ*11+jωτe-jϕphys,A5Crm=Co+jQ1+jωτe-jϕphys,A6Crm=(Co+jQ)(1-jωτ)1+ω2τ2e-jϕphys,A7Crm=(Co-jωτCo+jQ+ωτQ)1+ω2τ2e-jϕphys,
where ZwZχ*=Co+jQ was utilised. Next the quadrature spectrum (Q) was assumed to be zero and hence
Crm=Co-jωτCo1+ω2τ2e-jϕphys.
Now we can use Euler formula (e-jϕphys=cosϕphys-jsinϕphys):
A9Crm=Co-jωτCo1+ω2τ2(cosϕphys-jsinϕphys),A10Crm=Cocosϕphys-jωτCocosϕphys-jCosinϕphys-ωτCosinϕphys1+ω2τ2,A11Crm=cosϕphys-ωτsinϕphys1+ω2τ2Co-jsinϕphysωτcosϕphys1+ω2τ2Co,
which equals Eq. () in Sect. .
Code and data availability
Data and MATLAB codes to reproduce Figs. , and , in addition to time series processed with the four methods (Table ) at the two sites, have been uploaded to the open repository Zenodo (10.5281/zenodo.4769420, ).
Author contributions
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.
Competing interests
The authors declare that they have no conflict of interest.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
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.
Financial support
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).
Review statement
This paper was edited by Glenn Wolfe and reviewed by Johannes Laubach, Marc Aubinet, and one anonymous referee.
ReferencesAslan, T., Peltola, O., Ibrom, A., Nemitz, E., Rannik, Ü., and Mammarella, I.: The high-frequency response correction of eddy covariance fluxes – Part 2: An experimental approach for analysing noisy measurements of small fluxes, Atmos. Meas. Tech., 14, 5089–5106, 10.5194/amt-14-5089-2021, 2021.
Aubinet, M., Grelle, A., Ibrom, A., Rannik, U., Moncrieff, J., Foken, T., Kowalski, A. S., Martin, P. H., Berbigier, P., Bernhofer, C., Clement, R., Elbers, J., Granier, A., Grunwald, T., Morgenstern, K., Pilegaard, K., Rebmann, C., Snijders, W., Valentini, R., and Vesala, T.:
Estimates of the annual net carbon and water exchange of forests: The EUROFLUX methodology,
Adv. Ecol. Res.,
30, 113–175, 2000.
Aubinet, M., Vesala, T., and Papale, D.: Eddy covariance: a practical guide to measurement and data analysis, Springer Atmospheric Sciences, Springer, Dordrecht, 2012.Aubinet, M., Joly, L., Loustau, D., De Ligne, A., Chopin, H., Cousin, J., Chauvin, N., Decarpenterie, T., and Gross, P.: Dimensioning IRGA gas sampling systems: laboratory and field experiments, Atmos. Meas. Tech., 9, 1361–1367, 10.5194/amt-9-1361-2016, 2016.Baldocchi, D. D.:
Assessing the eddy covariance technique for evaluating carbon dioxide exchange rates of ecosystems: past, present and future,
Glob. Change Biol.,
9, 479–492, 10.1046/j.1365-2486.2003.00629.x, 2003.Brock, F. V.:
A Nonlinear Filter to Remove Impulse Noise from Meteorological Data,
J. Atmos. Ocean. Tech.,
3, 51–58, 10.1175/1520-0426(1986)003<0051:ANFTRI>2.0.CO;2, 1986.De Ligne, A., Heinesch, B., and Aubinet, M.:
New Transfer Functions for Correcting Turbulent Water Vapour Fluxes,
Bound.-Lay. Meteorol.,
137, 205–221, 10.1007/s10546-010-9525-9, 2010.Eugster, W. and Senn, W.:
A Cospectral Correction Model for Measurement of Turbulent NO2 Flux,
Bound.-Lay. Meteorol.,
74, 321–340, 10.1007/BF00712375, 1995.
Foken, T. and Wichura, B.:
Tools for quality assessment of surface-based flux measurements,
Agr. Forest Meteorol.,
78, 83–105, 1996.Foken, T., Aubinet, M., and Leuning, R.:
The Eddy Covariance Method, chap. 1,
in: Eddy Covariance,
edited by: Aubinet, M., Vesala, T., and Papale, D., Springer Atmospheric Sciences, Springer Netherlands, 10.1007/978-94-007-2351-1_1, 1–19, 2012a.Foken, T., Leuning, R., Oncley, S. R., Mauder, M., and Aubinet, M.:
Corrections and Data Quality Control BT – Eddy Covariance: A Practical Guide to Measurement and Data Analysis,
Springer Netherlands, Dordrecht, 10.1007/978-94-007-2351-1_4, 85–131, 2012b.Franz, D., Acosta, M., Altimir, N., Arriga, N., Arrouays, D., Aubinet, M., Aurela, M., Ayres, E., López-Ballesteros, A., Barbaste, M., Berveiller, D., Biraud, S., Boukir, H., Brown, T., Brömmer, C., Buchmann, N., Burba, G., Carrara, A., Cescatti, A., Ceschia, E., Clement, R., Cremonese, E., Crill, P., Darenova, E., Dengel, S., D'Odorico, P., Filippa, G., Fleck, S., Fratini, G., Fuß, R., Gielen, B., Gogo, S., Grace, J., Graf, A., Grelle, A., Gross, P., Grönwald, T., Haapanala, S., Hehn, M., Heinesch, B., Heiskanen, J., Herbst, M., Herschlein, C., Hörtnagl, L., Hufkens, K., Ibrom, A., Jolivet, C., Joly, L., Jones, M., Kiese, R., Klemedtsson, L., Kljun, N., Klumpp, K., Kolari, P., Kolle, O., Kowalski, A., Kutsch, W., Laurila, T., De Ligne, A., Linder, S., Lindroth, A., Lohila, A., Longdoz, B., Mammarella, I., Manise, T., Jiménez, S., Matteucci, G., Mauder, M., Meier, P., Merbold, L., Mereu, S., Metzger, S., Migliavacca, M., Mölder, M., Montagnani, L., Moureaux, C., Nelson, D., Nemitz, E., Nicolini, G., Nilsson, M., De Beeck, M., Osborne, B., Löfvenius, M., Pavelka, M., Peichl, M., Peltola, O., Pihlatie, M., Pitacco, A., Pokorný, R., Pumpanen, J., Ratié, C., Rebmann, C., Roland, M., Sabbatini, S., Saby, N., Saunders, M., Schmid, H., Schrumpf, M., Sedlák, P., Ortiz, P., Siebicke, L., Šigut, L., Silvennoinen, H., Simioni, G., Skiba, U., Sonnentag, O., Soudani, K., Soulé, P., Steinbrecher, R., Tallec, T., Thimonier, A., Tuittila, E.-S., Tuovinen, J.-P., Vestin, P., Vincent, G., Vincke, C., Vitale, D., Waldner, P., Weslien, P., Wingate, L., Wohlfahrt, G., Zahniser, M., and Vesala, T.: Towards long-Term standardised carbon and greenhouse gas observations for monitoring Europe's terrestrial ecosystems: A review, Int. Agrophys., 32, 439–455, 10.1515/intag-2017-0039, 2018.Fratini, G., Ibrom, A., Arriga, N., Burba, G., and Papale, D.:
Relative humidity effects on water vapour fluxes measured with closed-path eddy-covariance systems with short sampling lines,
Agr. Forest Meteorol.,
165, 53–63, 10.1016/j.agrformet.2012.05.018, 2012.
Horst, T. W.:
A simple formula for attenuation of eddy fluxes measured with first-order-response scalar sensors,
Bound.-Lay. Meteorol.,
82, 219–233, 1997.Horst, T. W.:
On Frequency Response Corrections for Eddy Covariance Flux Measurements,
Bound.-Lay. Meteorol.,
94, 517–520, 10.1023/A:1002427517744, 2000.Horst, T. W. and Lenschow, D. H.:
Attenuation of Scalar Fluxes Measured with Spatially-displaced Sensors,
Bound.-Lay. Meteorol.,
130, 275–300, 10.1007/s10546-008-9348-0, 2009.Hunt, J. E., Laubach, J., Barthel, M., Fraser, A., and Phillips, R. L.: Carbon budgets for an irrigated intensively grazed dairy pasture and an unirrigated winter-grazed pasture, Biogeosciences, 13, 2927–2944, 10.5194/bg-13-2927-2016, 2016.Ibrom, A., Dellwik, E., Flyvbjerg, H., Jensen, N. O., and Pilegaard, K.:
Strong low-pass filtering effects on water vapour flux measurements with closed-path eddy correlation systems,
Agr. Forest Meteorol.,
147, 140–156, 10.1016/j.agrformet.2007.07.007, 2007a.Ibrom, A., Dellwik, E., Larsen, S. E., and Pilegaard, K.:
On the use of the Webb-Pearman-Leuning theory for closed-path eddy correlation measurements,
Tellus B,
59, 937–946, 10.1111/j.1600-0889.2007.00311.x, 2007b.
Kaimal, J. C. and Finnigan, J. J.:
Atmospheric boundary layer flows: their structure and measurement,
Oxford University Press, New York, 1994.Kristensen, L., Mann, J., Oncley, S. P., and Wyngaard, J. C.:
How Close is Close Enough When Measuring Scalar Fluxes with Displaced Sensors?,
J. Atmos. Ocean. Tech.,
14, 814–821, 10.1175/1520-0426(1997)014<0814:HCICEW>2.0.CO;2, 1997.Langford, B., Acton, W., Ammann, C., Valach, A., and Nemitz, E.: Eddy-covariance data with low signal-to-noise ratio: time-lag determination, uncertainties and limit of detection, Atmos. Meas. Tech., 8, 4197–4213, 10.5194/amt-8-4197-2015, 2015.Mammarella, I., Launiainen, S., Gronholm, T., Keronen, P., Pumpanen, J., Rannik, U., and Vesala, T.:
Relative Humidity Effect on the High-Frequency Attenuation of Water Vapor Flux Measured by a Closed-Path Eddy Covariance System,
J. Atmos. Ocean. Tech.,
26, 1856–1866, 10.1175/2009JTECHA1179.1, 2009.Mammarella, I., Nordbo, A., Rannik, U., Haapanala, S., Levula, J., Laakso, H., Ojala, A., Peltola, O., Heiskanen, J., Pumpanen, J., and Vesala, T.:
Carbon dioxide and energy fluxes over a small boreal lake in Southern Finland,
J. Geophys. Res.-Biogeo.,
120, 1296–1314, 10.1002/2014JG002873, 2015.Mammarella, I., Peltola, O., Nordbo, A., Järvi, L., and Rannik, Ü.: Quantifying the uncertainty of eddy covariance fluxes due to the use of different software packages and combinations of processing steps in two contrasting ecosystems, Atmos. Meas. Tech., 9, 4915–4933, 10.5194/amt-9-4915-2016, 2016.Massman, W. J.:
A simple method for estimating frequency response corrections for eddy covariance systems,
Agr. Forest Meteorol.,
104, 185–198, 10.1016/S0168-1923(00)00164-7, 2000.
Massman, W. J. and Lee, X.:
Eddy covariance flux corrections and uncertainties in long-term studies of carbon and energy exchanges,
Agr. Forest Meteorol.,
113, 121–144, 2002.Metzger, S., Burba, G., Burns, S. P., Blanken, P. D., Li, J., Luo, H., and Zulueta, R. C.: Optimization of an enclosed gas analyzer sampling system for measuring eddy covariance fluxes of H2O and CO2, Atmos. Meas. Tech., 9, 1341–1359, 10.5194/amt-9-1341-2016, 2016.
Moncrieff, J. B., Massheder, J. M., deBruin, H., Elbers, J., Friborg, T., Heusinkveld, B., Kabat, P., Scott, S., Soegaard, H., and Verhoef, A.:
A system to measure surface fluxes of momentum, sensible heat, water vapour and carbon dioxide,
J. Hydrol.,
189, 589–611, 1997.
Moore, C. J.:
Frequency-Response Corrections for Eddy-Correlation Systems,
Bound.-Lay. Meteorol.,
37, 17–35, 1986.Nemitz, E., Mammarella, I., Ibrom, A., Aurela, M., Burba, G., Dengel, S., Gielen, B., Grelle, A., Heinesch, B., Herbst, M., Hörtnagl, L., Klemedtsson, L., Lindroth, A., Lohila, A., McDermitt, D., Meier, P., Merbold, L., Nelson, D., Nicolini, G., Nilsson, M., Peltola, O., Rinne, J., and Zahniser, M.: Standardisation of eddy-covariance flux measurements of methane and nitrous oxide, Int. Agrophys., 32, 517–549, 10.1515/intag-2017-0042, 2018.Nordbo, A., Kekäläinen, P., Siivola, E., Lehto, R., Vesala, T., and Timonen, J.:
Tube transport of water vapor with condensation and desorption,
Appl. Phys. Lett.,
102, 194101, 10.1063/1.4804639, 2013.Nordbo, A., Kekäläinen, P., Siivola, E., Mammarella, I., Timonen, J., and Vesala, T.:
Sorption-Caused Attenuation and Delay of Water Vapor Signals in Eddy-Covariance Sampling Tubes and Filters,
J. Atmos. Ocean. Tech.,
31, 2629–2649, 10.1175/JTECH-D-14-00056.1, 2014.Peltola, O., Mammarella, I., Haapanala, S., Burba, G., and Vesala, T.: Field intercomparison of four methane gas analyzers suitable for eddy covariance flux measurements, Biogeosciences, 10, 3749–3765, 10.5194/bg-10-3749-2013, 2013.Peltola, O., Aslan, T., Ibrom, A., Nemitz, E., Rannik, Ü., and Mammarella, I.: Dataset for “The high frequency response correction of eddy covariance fluxes. Part 1: an experimental approach and its interdependence with the time-lag estimation”, Zenodo [data set], 10.5281/zenodo.4769420, 2021.
Rannik, Ü. and Vesala, T.:
Autoregressive filtering versus linear detrending in estimation of fluxes by the eddy covariance method,
Bound.-Lay. Meteorol.,
91, 259–280, 1999.Rannik, Ü., Peltola, O., and Mammarella, I.: Random uncertainties of flux measurements by the eddy covariance technique, Atmos. Meas. Tech., 9, 5163–5181, 10.5194/amt-9-5163-2016, 2016.Rebmann, C., Aubinet, M., Schmid, H., Arriga, N., Aurela, M., Burba, G., Clement, R., De Ligne, A., Fratini, G., Gielen, B., Grace, J., Graf, A., Gross, P., Haapanala, S., Herbst, M., Hörtnagl, L., Ibrom, A., Joly, L., Kljun, N., Kolle, O., Kowalski, A., Lindroth, A., Loustau, D., Mammarella, I., Mauder, M., Merbold, L., Metzger, S., Mölder, M., Montagnani, L., Papale, D., Pavelka, M., Peichl, M., Roland, M., Serrano-Ortiz, P., Siebicke, L., Steinbrecher, R., Tuovinen, J.-P., Vesala, T., Wohlfahrt, G., and Franz, D.:
ICOS eddy covariance flux-station site setup: a review,
Int. Agrophys.,
32, 471–494, 10.1515/intag-2017-0044, 2018.Rinne, J., Tuittila, E.-S., Peltola, O., Li, X., Raivonen, M., Alekseychik, P., Haapanala, S., Pihlatie, M., Aurela, M., Mammarella, I., and Vesala, T.:
Temporal Variation of Ecosystem Scale Methane Emission From a Boreal Fen in Relation to Temperature, Water Table Position, and Carbon Dioxide Fluxes,
Global Biogeochem. Cy.,
32, 1087–1106, 10.1029/2017GB005747, 2018.Riutta, T., Laine, J., Aurela, M., Rinne, J., Vesala, T., Laurila, T., Haapanala, S., Pihlatie, M., and Tuittila, E. S.:
Spatial variation in plant community functions regulates carbon gas dynamics in a boreal fen ecosystem,
Tellus B,
59, 838–852, 10.1111/j.1600-0889.2007.00302.x, 2007.Runkle, B. R. K., Wille, C., Gazovic, M., and Kutzbach, L.:
Attenuation Correction Procedures for Water Vapour Fluxes from Closed-Path Eddy-Covariance Systems,
Bound.-Lay. Meteorol.,
142, 401–423, 10.1007/s10546-011-9689-y, 2012.
Sabbatini, S., Mammarella, I., Arriga, N., Fratini, G., Graf, A., Hörtnagl, L., Ibrom, A., Longdoz, B., Mauder, M., Merbold, L., Metzger, S., Montagnani, L., Pitacco, A., Rebmann, C., Sedlák, P., Šigut, L., Vitale, D., and Papale, D.:
Eddy covariance raw data processing for CO2 and energy fluxes calculation at ICOS ecosystem stations,
Int. Agrophys.,
32, 495–515, 10.1515/intag-2017-0043, 2018.Starkenburg, D., Metzger, S., Fochesatto, G. J., Alfieri, J. G., Gens, R., Prakash, A., and Cristóbal, J.:
Assessment of Despiking Methods for Turbulence Data in Micrometeorology,
J. Atmos. Ocean. Tech.,
33, 2001–2013, 10.1175/JTECH-D-15-0154.1, 2016.Taipale, R., Ruuskanen, T. M., and Rinne, J.: Lag time determination in DEC measurements with PTR-MS, Atmos. Meas. Tech., 3, 853–862, 10.5194/amt-3-853-2010, 2010.Vesala, T., Kljun, N., Rannik, Ü., Rinne, J., Sogachev, A., Markkanen, T., Sabelfeld, K., Foken, T., and Leclerc, M. Y.:
Flux and concentration footprint modelling: State of the art,
Environ. Pollut.,
152, 653–666, 10.1016/j.envpol.2007.06.070, 2008.Wintjen, P., Ammann, C., Schrader, F., and Brümmer, C.: Correcting high-frequency losses of reactive nitrogen flux measurements, Atmos. Meas. Tech., 13, 2923–2948, 10.5194/amt-13-2923-2020, 2020.