The authors have certainly addressed all the specific criticisms and by that improved the manuscript. Also, I do not have the impression that there are any differences between authors and reviewers on the mathematics.
That said, I would still encourage the authors to make stronger recommendations for change to "established" or "widespread" eddy flux processing procedures. I will elaborate more in the paragraphs below. I believe such strengthening of the messages would increase the impact of the paper and would be fruitful for the "flux community".
There seems to be some remaining disagreement between me and the authors on the relative merits of the cross-covariance maximisation method. While I would call it "dubious" and "flawed", the authors still describe it as a "practical solution". I do not doubt that the authors have understood my line of reasoning, so it comes down to a matter of opinion what degree of known imperfection is still considered acceptable. Ultimately it is the authors' call how to put this. I can only appeal to them to consider a more strongly critical wording, on the following grounds.
I have many years of experience with EC measurements with closed-path analysers near the ground, i.e. 2 m above agricultural surfaces. There, lag times are typically of order 0.3 to 3 s, and I have always found the cross-covariance maximisation method causing more trouble than a fixed lag time, even a somewhat inaccurate one, ever would. For all gases I've looked at (H2O, CO2, CH4, N2O, NH3), covariance maximisation introduces frequent unrealistic, widely oscillating, lag time estimates that have nothing to do with instabilities in the sampling system. The reason for that is that correlation coefficients of w with a scalar variable are not huge to begin with (typically 0.2 to 0.4) and when one of the variables is attenuated, the peak of the correlation function becomes rather flat and thus hard to detect. Worse, because it is a "maximisation" procedure, the method selects against near-zero fluxes even when these are true. For example, measuring N2O fluxes after a long period without any nitrogen inputs to the soil should yield a single-peak flux histogram with a near-zero median. However, with the maximisation method, the histogram becomes double-peaked (one positive, one negative) because zero flux is actively avoided by the method. But this is not a reflection of separate source and sink processes, it is simply due to bias in the individual flux estimates (because random non-zero correlations, of either sign, get "detected" and locked on by the method). Gas fluxes that change sign twice daily, such as that of CO2, are equally affected by this bias around the sign-change periods.
I would concede that cross-covariance maximisation can be useful as a diagnostic tool, because it allows early detection of clear trends (such as two clocks drifting apart). But still, actual flux values should be calculated with a lag time that is based on the technical flow parameters of the system and not automatically varying from run to run (apart from corrections for known clock drifts).
Hence, my opinion remains that the use of the cross-covariance maximisation method (in its present widespread form) should be discouraged, and if the present authors do not include a recommendation to this effect, they are missing a chance to influence the thinking of the "flux community" in this regard.
The following remarks are deliberately provocative. Please do not take them as personal criticism!
Can the authors perhaps ask themselves why they are reluctant to recommend abandoning the cross-covariance maximisation method? The fact that it is "widespread" is no good reason. If a method is known to be poor, it should be replaced with better ones. In this case, do the authors consider such change too laborious for users? Do they not wish to make this recommendation because, if followed widely, it would remove the need to use their here-presented correction procedure in the future? Do they fear conflict in the scientist networks they are involved in (in particular ICOS with its drive to standardise procedures)? Do they worry about changes in the FLUXNET databases that could ensue from revised processing?
I do not expect answers to these questions. They are merely intended to encourage the authors to be bolder in the paper's Discussion and Conclusions sections.
Specific comments (line numbers refer to the tracked-changes version)
P 5 L 23 and P 17 L 1
The authors have added information on flow and tube dimensions for Hyytiälä. The tube volume is so small that the total physical lag time is affected at a comparable rate by the time it takes to exchange the air in the volume of the Li-Cor measurement cell. In addition, there is a known processing lag if the LI-7550 Interface Unit is used. The three effects (tube, cell, processing) need to be added to estimate physical lag time.
P 21 L 20-21
I believe the reader should be warned more clearly that the "approximation" is limited to the cases of relatively minor attenuation presented in this study. In particular, it should be noted that in cases where the phase effects cause sign reversals in the cospectrum, the sqrt(H) approach must fail.
P 21 L 32
In line with my General Comments, I find the inserted "Hence, investigating..." too weak. Why not say something like "Hence, other means for estimating the physical lag time should be used whenever possible." (Please do not use "signal travel time", that sounds like an electromagnetic phenomenon.)
This paper discusses different spectral corrections procedures for low pass filtering effects in eddy covariance systems. Despite eddy covariance has become the most common approach to determine, among others, CO2, water vapour or greenhouse gas budgets of ecosystems, the method still suffers from uncertainties due to random and, more worryingly to my opinion, to systematic errors. From this point of view each study providing a better understanding of measurements errors and improving correction procedure is welcome.
In that respect, the paper by Peltoli et al is important for at least two reasons: first it points a systematic error actually made by some eddy covariance data treatment softwares (including EddyPro, cf Sabbatini et al., 2018) which definitely requires a correction; secondly it clarifies the question of the cospectra transfer function shape and reconcile theory and observations. It also provides a new method to correct low pass filtering effects but I think that it’s robustness and applicability to routine measurements needs still to be proven.
For these reasons, I think that the paper deserves publication. As it is generally well written and structured, I think that only minor revision is required before acceptation.
I would add that this study comforts me in the opinion that, despite the great interest of theoretical studies that help understanding the causes and modalities of low pass filtering by eddy covariance systems, empirical approaches relying as little as possible on theoretical hypotheses remain the most robust ones to apply frequency corrections on routine measurements. In particular, in the present study, the approach deducing a transfer function from cospectra rather than from power spectra (Method 2) remains one of the most robust. The fact that the shape of the transfer function and the time constant are not exact is not very problematic to my opinion, as it does not affect critically the values of the correction factor, which is the target. The Method 4 proposed by the authours could be an interesting alternative, as it also relies on cospectra but uses a different transfer function shape. However, it is more complex as it requires the determination of two parameters (against one for Method 2) and, if the method worked well in the present case where the high frequency attenuation was artificially introduced, I suspect (and they confirm on P16L9) that disentangling the two time constants could be sometimes difficult, even impossible.
My regret is that the authors do not detail an implementation procedure of Method 4 for routine measurements.
The paper is the second of a series of two papers on spectral corrections. I was first asked to review the first of them (Aslan et al, also available on AMT discussions) but had to wait the submission of this one to really understand some issues and methodical choices of the Aslan paper. As the present paper appears to me more “standing alone”, I suggest to place this one in the first place and the Aslan paper in the second place. This is the order I followed for my reviews.
The introduction offers a review of the knowledge about spectral corrections. It is clear and highlights the most important points. I have no specific comment about this except two small remarks :
P2L13: I think there’s a typo (“contribute” rather than “contributed”)
P3L7: Reference to Aubinet is not relevant I think as it refers to the chapter “night flux correction” in my book. I suggest to rather refer to the book itself or to a specific chapter (time lag is evoked in Ch 2 – Munger et al., 2012; Ch 3 – Rebmann et al., 2012 and Ch 4 – Foken et al., 2012).
I liked this chapter as it helps me to understand the issue of the cospectral transfer function shape. I must say honestly that I overlooked the debate about the presence or not of a square root in the cospectral transfer function shape (for my defense, I was more concerned in the past by the cospectral – Method 2 - approach than by the spectral approach – Method 3) but, when comparing recently spectral and cospectral approaches on crop sites data, I found a better agreement when applying the square root (Method 3) than not (Method 1), which contradicted the theoretical predictions by Horst (1997), among others. I thus found the explanations given by this paper clever and convincing.
Two remarks, anyway:
P5L27-P6L2: I don’t see the interest of presenting the approximation on L29. I tested the equation on L29 and found it fitted quite loosely equation 6. In addition, as I understand, this equation was not used in the paper and equality between tlpf and tau was not assumed further. Maybe could you consider to skip this.
P5L29: I’m wondering about the equality (and below, the proportionality) between tlpf and tau. Indeed, these two time constants are a priori not physically linked (except when both result from tube attenuation, which is of course an important case) and I’m wondering if you don’t loose generality by introducing this dependency. This question is discussed below but, in the end, there is no clear description on how you really implement the transfer function computation: do you fit an equation for H Hp based on equations (5) and (6) ? On equation (5) and those of P5L29? Do you consider tau and tlpf as independent parameters or do you relate them in some way?
Material and methods:
No specific comments. Clear and well presented. It is important to keep in mind (Sect 3.2.2) that the results presented below are not based on real measurements (I mean the attenuation is artificially provoked and does not reflect real attenuation processes), which is a limitation of the study (but this is well stated in the discussion).
P9L13-24: Same remark as above: the proportionality between tau and tlps is clear here as both time constant result from an artificial attenuation but how would this relation look like in the case of measurements with a real attenuation and a real time lag, possibly independent ?
P9L25-27: I was not sure to understand well: is it an approach that mimics the covariance maximisation procedure? If yes, it could be worth specifying it explicitly.
P10L2: What’s the meaning of s in Eq 14 (second, I suppose, but I would not mix symbols and units in a formula).
P10L3-4: This sentence let me hunger. As high attenuation could occur often (especially for gases other than CO2) this question should be clarified. Which attenuation levels do you consider? what is the order of magnitude of the bias? what is the impact of this bias on the next steps (correction factor estimation)?
P13 Fig 4: As I understand, the red curve corresponds to Method 1, the blue one to Method 3 and the black one to method 4. Is it correct? A direct reference to the method could facilitate figure reading (and why is method 2 absent from the figure?)
P14L5: isn’t it rather by the ratio of cospectral peak frequency to the cut off frequency ?
P15Fig6: The legend is not fully clear. I suppose that the symbols refer to the sites and the colours to stability conditions. This should be stated more clearly.
P15Fig6: I’m intrigued by the curve of Hyytialla in unstable conditions for methods 2, 3 and 4. Why is the bias positive, contrary to other site/conditions? Can you comment on this?
I’m also intrigued by the fact that the Method 4 more overestimates the correction factor than Methods 2 and 3 (and thus seems to work less good) at Hyytialla in unstable conditions. Here also I would expect a comment.
P15L2-6: I think that the figure shows clearly that the Method 1 gives different results from the three other methods. To my opinion Methods 2, 3, 4 provide all reasonable estimates of the correction factors while Method 1 biases the correction factors due, as you showed in the theory section, to a misinterpretation of the theory. In this sense, giving a relation to quantify the bias introduced by Method 1 is maybe not very useful. It could appear more clearly that this method is wrong and should be definitely not recommended (which notably implies a modification of the ICOS protocol).
P16L33: Same remark as above: don’t mix symbols and units in a formula.
P16L34: the meaning of x and y is not fully clear to me. Could you express the relation between these variables and time constants presented above?
P17L2, L5 and elsewhere: rather than referring to Section numbers, it would be more easy for the reader if you referred directly to the figures or tables presenting the results.
P17L7 and elsewhere: use a uniform notation to present the different methods (“Method X” is fine to me).
P17L9: one word is missing.
P17L10: As Hyytialla is equipped with a LI7200 and Siikaneva with a LI7000, I would have expected the inverse: a lower tau value at Hyytialla. Could you comment ?
P17L12 and foll: This section (and the legend of Table 3) should be clarified: In the text, are you presenting difference between correction factors? between half hourly fluxes? between cumulated fluxes? On which period? I finally supposed that you were comparing cumulated fluxes but this should be specified.
P17L12 and foll: I’m not convinced by the relevance of comparing relative differences on cumulated flux values. Indeed, relative values depend strongly on flux values (I suppose that H2O flux values at night should be low and in these conditions larger relative errors do not mean much). In addition, the low error on cumulated values may also result from partial compensation of errors (for example during day and night). I have the same problem when I try to compare different correction methods on my data set and I’m not sure to have the best solution. I prefer comparing the fluxes by looking at the slope between the fluxes submitted to different corrections. Anyway, in view of the preceding remarks, I’m not sure that the fact that Method 3 gives the biggest difference at both sites (L16) is really relevant.
P22L1; I feeled (of course!) concerned by the remark on our paper about the impact of dead volumes on the frequency response of gas sampling system. I could recognise that the fact that we didn’t distinguish physical time lag from attenuation induced time lag led to cut off frequencies that are probably not really representative of the attenuation. However, the general decrease of the cut off frequency with increasing dead volumes (our Figures 5 and 6) and the need for reducing these volumes in the gas sampling system were important results that we showed in this paper, along those of Metzger et al. And this again reinforces my opinion that transfer functions based on observed cospectra and taking thus account of all attenuation processes affecting the system (even if in some cases we do not fully understand all of them) are to be preferred for routinely correcting measurements, as they provide more robust estimates of fluxes.