Importance of the WPL correction for the measurement of small CO2 fluxes

The WPL (Webb, Pearman, and Leuning) correction is fully accepted to correct trace gas fluxes like CO2 for density fluctuations due to water vapor and temperature fluctuations for open-path gas analysers. It is known that this additive correction can be in the order of magnitude of the actual flux. However, this is hardly ever included in the analysis of data 10 quality. An example from the Arctic shows the problems, because the size of the correction is a multiple of the actual flux. As a general result, we examined and tabulated the magnitude of the WPL correction for carbon dioxide flux as a function of sensible and latent heat flux. Furthermore, we propose a parameter to better estimate possible deficits in data quality and recommend integrating the quality flag derived with this parameter into the general study of small carbon dioxide fluxes.


Introduction 15
The WPL (Webb, Pearman, and Leuning) correction (Webb et al., 1980) has been fully accepted in the international scientific community for many years. After initial discussions on the derivation of the correction up to the beginning of the 2000s, no further discussions took place after the clarification by Leuning (2007Leuning ( , 2004 and it is used in the textbook literature (Aubinet et al., 2012) and integrated in a uniform form in all software packages for the calculation of eddy-covariance measurements.
The correction is necessary because fluctuations in temperature and humidity cause fluctuations in trace gas concentrations 20 and that can simulate a flux for instance of CO2 or modify its size.
The size of the WPL correction on the CO2 flux is generally assumed to be a few 10 %, and on the latent heat flux only about 1 % Foken, 2003, 2004) but for Bowen ratios ≫ 1 up to 10 % (Foken, 1989) of the raw flux data. The absolute value of the correction is in the range from -2.5 to +10 µmol m -2 s -1 depending on the magnitude of the sensible and latent heat flux. This correction to the measured flux can be large, i.e. the additive correction may significantly change the CO2 25 flux calculated using the covariance of vertical velocity and partial density (Mauder et al., 2021). However, there is a lack of investigations under which circumstances the application of the WPL correction is uncritical and under which conditions it might produce physically impossible effects on the fluxes. However, it is obvious that CO2 fluxes on the order of 1 µmol m -2 s -1 can be significantly altered by the WPL-correction. Such low fluxes occur, for example, over burned areas (Oliveira et al., https://doi.org/10.5194/amt-2021-249 Preprint. Discussion started: 30 August 2021 c Author(s) 2021. CC BY 4.0 License. 2021) or over sandy deserts (Su et al., 2013). The Technical Note uses an example from the Arctic (Jentzsch et al., 2021) to 30 show that the WPL correction may well lead to misjudgements of water vapour and carbon dioxide measurements with an open path gas analyser. The conditions under which the WPL correction has a considerable influence on the CO2 flux are shown with possible consequences. Errors in CO2 flux measurements in cold climate conditions have often been associated with the Burba correction (Burba et al., 2008;Kittler et al., 2017). This correction is relevant when convective processes occur at heated windows of the gas analyser. Because of the application of a low window temperature and an inclined position of the 35 gas analyser this correction was not applied in the following study.
The WPL correction is relevant for the measurement of the partial density, e.g. of CO2, with open path gas analyzers and the basic equations are (Foken et al., 2012) with the corrected flux of scalar quantity (CO2) Fc, the mass density of a scalar (CO2) quantity , the mass density of dry air 40 ρd, the density of water vapor , the measured trace gas (CO2) flux ′ ′ ������� , the water vapor flux ′ ′ ������� , the sensible heat flux ′ ′ ������ , the mean temperature � , and with the molar masses of dry air md and of water vapor mw. Equation (1) can be simplified for the correction of H2O-flux (latent heat flux) 45 In (1)

Material and carbon dioxide fluxes under Artic conditions
In a recently published article on CO2 fluxes under Arctic conditions (Jentzsch et al., 2021), events with strong CO2 uptake and emission were investigated, which were also found by other authors (e.g., Lüers et al., 2014). It was found that these events may be artifacts due to the WPL correction because the CO2 fluxes during the events are very strongly correlated with the sensible heat flux. These investigations were the reason to investigate the WPL correction under conditions of small CO2 fluxes 60 in more detail.   In Fig. 2 both terms were explicitly calculated with the measured data and not by EddyPro with identical results. It is shown that temperature fluctuations have a much stronger influence on the size of the WPL correction.
In the case of a CO2 uptake with negative sensible heat flux, the absolute value of the flux would be even larger. In the case of positive sensible heat flux, the 3 rd term in Eq. (1) is also positive and an uptake would be reduced, while emissions would be 80 increased.
The influence of the WPL correction on the latent heat flow is not shown. It is known to be relatively small and in the present case for the period from 08:00 to 11:00 am about 3%, at all other times 1-2%. This is below the typical measurement error (Foken et al., 2012) of about 10%.

Size of the WPL correction 90
A variable CO2 density has a linear influence on the WPL correction, since it is part in both terms. This can lead to an increase of the correction up to about factor 3 (1000-1500 ppm CO2 concentration) over strongly respiring surfaces under stable stratification. With strong assimilation, on the other hand, the concentration is only slightly below the global average. Table 1 shows for standard atmospheric conditions (1013 hPa, 15 °C) and a CO2-concentration of 415 ppm the WPL correction as a function of the size of the sensible and latent heat flow. The table hardly differs from Fig. 1 in Webb et al. (1980), who 95 have already pointed out the considerable size of the correction in kg m -2 s -1 .
For small CO2 fluxes, the WPL correction is sometimes a multiple of the actual flux. Especially for fluxes < 5 µmol m -2 s -1 the interpretation of extraordinary events is very questionable. In the present case ( Fig. 1), an absolute value of the sensible heat flux increased by 20 W m -2 , which is still within the error range, would no longer show a clear emission event. Since the event cannot be plausibly explained physically, it should be marked as faulty and excluded by a MAD test (absolute deviation from 100 the median, Papale et al., 2006).
Small errors could have a large impact on cumulative fluxes, especially if the errors have a bias. Therefore, the interpretation of such fluxes has to be done very carefully. However, in the case of a carbon uptake and positive sensible heat fluxes, these fluxes would be reduced and, as the above example has shown, emissions with negative sensible heat fluxes would also be reduced. This could compensate the errors, but this would have to be shown in each individual case. 105 Furthermore, small CO2 fluxes often occur in connection with negative sensible heat fluxes. This is the case when evaporation takes place, but the energy required for this must be provided at least partially by a downwardly directed sensible heat flux. This is known as the so-called oasis effect (Foken, 2017;Stull, 1988), which occurs relatively often in the late afternoon. The case discussed here during the polar night can also be described as an oasis effect. Furthermore, a strong cooling of the surface by long-wave radiation causes a compensation by a downward directed sensible heat flux. This is particularly typical in the 110 first half of the night, when the stratification is stable. Small CO2 fluxes can always be considered reliable when the sensible heat flux is close to zero.

Quality flagging of the WPL correction
Since the specificity of possible errors in WPL-correction is not taken into account by usual error analyses of eddy-covariance data (e. g. Mauder et al., 2013), measurement data with very large WPL corrections should be marked. A possible parameter 120 could be the ratio of WPL correction and corrected CO2 flux: Similar parameters could be defined separately for the term of humidity fluctuations QFw and temperature fluctuations QFT 125 Fig. 3 shows these parameters for the example given in Fig. 1. Here QFw is < 1 and therefore has only little influence on the correction, which is essentially determined by QFT. The parameter QFWPL should be sufficient for the quality identification of WPL corrected CO2 flux. Following common systems with quality flags (Foken and Wichura, 1996;Foken et al., 2012), |QFWPL |≤ 0.5 could be classified as very good, 0.5 < |QFWPL |≤ 1 as good and all values |QFWPL| > 1 should be specially checked. It is 130 https://doi.org/10.5194/amt-2021-249 Preprint. Discussion started: 30 August 2021 c Author(s) 2021. CC BY 4.0 License. useful to define maximum limit values for the WPL-corrected CO2 flux if the flux is below the detection limit, because then the absolute |QFWPL| values can be significantly higher than 5 (see Fig. 1 and 3 after 18:00).  Due to the significant problems with the additive WPL correction, other possible influencing factors, such as spectral crosssensitivities between CO2 and water vapor (Kondo et al., 2014) are excluded from the discussion. Furthermore, the Burba correction (Burba et al., 2008), which is a modified WPL correction, should be re-examined under these aspects. This might explain Burba corrections in the summer half year that are difficult to interpret (Kittler et al., 2017). 140

Conclusion
According to the original work on the WPL correction (Webb et al., 1980), it should be known that the correction is expected to be in the same order of magnitude as the flux. This fact has never been given special attention in the error analysis of CO2 fluxes or other trace gas fluxes, to which the above statements apply analogously, measured with open-path gas analysers. (i) We strongly suggest that the WPL correction should be subjected to a special quality analysis. For this purpose, the quality flag of the correction expressed in Eq. (4) or/and (Eq. (6) could, for example, be explicitly output for all eddy-covariance software packages. At least for small fluxes it would have to be included in the quality considerations.
(ii) We propose that special events in time series of trace gas fluxes should only be interpreted if the WPL quality flag is below 150 a maximum limit values and they can be clearly assigned to physical processes, qualitatively and quantitatively. Otherwise, these data should be identified with a MAD analysis and interpolated if necessary.
(iii) We suggest that respiration data for the derivation of the gap-filling algorithm (Lloyd and Taylor, 1994) should be obtained at times when the sensible heat flux is near 0 W m -2 , i.e. if possible in the second half of the night and not at sunset because of the relatively large negative sensible heat fluxes. The quality criterion should be applied to these data. 155 (iv) If CO2 fluxes < 5 µmol m -2 s -1 are expected at a site, we recommend the use of a closed-path gas analyser instead of an open-path gas analyser. However, it must be guaranteed that complete isothermal conditions are achieved in the measurement volume of the closed path sensor, otherwise the same problems as described above will occur.
(v) In addition to the WPL correction we recommend that all corrections used in the eddy-covariance method, which are made by adding correction terms to the measured flux, should be critically reviewed. This applies, for example, to the Burba 160 correction for open-path gas analysers and the various corrections for closed-path devices. It may also be necessary to investigate whether errors in the application of the WPL correction and the above-mentioned corrections lead to a bias in accumulated fluxes, for which appropriate simulations will probably have to be performed.
(vi) These remarks suggest that a discussion on alternative measurement methods should be continued. This includes the direct measurement of the mixing ratio (Kowalski and Serrano-Ortiz, 2007), as realized in closed-path instruments, when complete 165 isothermal condition is guaranteed in the measurement volume. But also a density averaged measurement of CO2 fluctuations (Kramm et al., 1995) according to the Hesselberg averaging (Hesselberg, 1926) should be reconsidered.
Author contributions KJ made first investigations of the WPL term under Arctic conditions, guided by JB. These first analyses were extended by TF to the present paper. After the discussion of the results, KJ, JB, and TF completed the paper. 170 Competing interests. The authors declare that they have no conflict of interest.
Data availability. The eddy-flux data are openly available under CC-BY-4.0 in the FLUXNET data base (http://www.europefluxdata.eu/;ID:Sj-Blv). 175 Acknowledgements. This publication was funded by the German Research Foundation (DFG) and the University of Bayreuth in the funding programme Open Access Publishing.
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