The manuscript describes three different approaches to analyze the dependency of open-path FTIR trace gas measurements on path length and retro-reflector array size. These approaches are: 1. simulation of absorption spectra for different path lengths, 2. dedicated measurements of spectral signal-to-noise-ratio (SNR) with the reflector array at different distances, and 3. analysis and comparison of two field deployments, which primarily differ in the size of the utilized reflector array. While this study focuses on the trace gas formaldehyde (HCHO), it aims also for more general results, especially where signal return and spectral SNR is discussed. The data presented here has the potential to provide important insights on how the choice of reflector array size and path length influence the performance of open-path FTIR instruments, and, further, might provide some results applicable to other open-path techniques. However, in my view, the manuscript in its current form contains several ambiguities, inaccuracies and also a few errors throughout the results, which in some cases impact the results and their interpretation significantly. Hence I recommend to publish this work in AMT, since it fits the scope of the journal perfectly, but only after major revisions, so it fits the journals technical standards. In the following, I address in depth my major points of critique and provide a detailed list of general remarks and line-by-line comments and technical corrections. I hope my thoughts and input on this matter is helpful for the authors moving forward.

Major point: Quantification of SNR dependency on path length (L. 20f, L. 224f, Figure 6, L. 247, Figure 7)

I strongly disagree with the linear fits in Figure 6. Fitting a line to this data is neither supported by the data itself, nor by prior knowledge of the system.

In a simplified model describing the overfilling of the reflector array you could model the diameter of your collimated beam by the telescope diameter d₀ plus the increase due to divergence 2 * α * s, where α is your divergence half angle and s your distance from the telescope. As long as this is smaller than your reflector diameter, all light should be returned. This is also what your data supports quite clearly for Franklyn Street and also somewhat for Otter Lake. The signal only decreases, once the diameter of the beam is equal to the diameter of the reflector array. According to your 1 mrad divergence, this should be the case for a distance of 150m, so right at 300m two-way path length. What happens after that is less clear. From our simplified model, we should assume a quadratic drop of returned signal (d_r / d_s)² with d_r being the reflector diameter and d_s the beam diameter at distance s. Your data however seems to indicate a linear drop, so I could understand if you fit a line to it for empiric reasons, but in my opinion this should only be done for data points where overfilling is clearly happening. It would be interesting to get behind the reason for this linear behavior and where the simplified model from above breaks down, but I understand that this might not your focus or main interest for this publication. I would assume that some "fuzziness" in the transition region between the two regimes takes place, making the transition less harsh. This might be caused by the combination of turbulent seeing effects and the until here disregarded displacement of the reflector cubes. The Franklyn street results might just be a combination of this blurring in combination with only 3 data point in the region of overfilling. For Otter Lake, the model obviously breaks down, because the reflector array is not close to circular and the ratio of the beam which hits the reflector is not just (d_r / d_s)² as above. But the overall shape of the data still fits the idea, since a real decrease of signal only happens slightly later as for Franklyn street and the reflector array is slightly wider (in its shorter dimension) as for Frankly street. In a bit of a stretch, the data could also show an accelerated signal drop from roughly 700m onward, when the beam should now be large enough to overfill the reflector array in the larger dimension, but I would think that this is difficult to argue without better knowledge of the uncertainties of each data point.

Further, I think the absolute values in Figure 6, partially Figure 7, and especially for you slopes (if you stick to them) are not that helpful here. Such numbers are already highly dependent on the concrete setup (beam divergence, light throughput, reflector array) but if you would use relative numbers (so norm everything by your starting intensity or SNR and give your slopes in relative intensity drop or relative SNR drop per distance) it would at least be independent of your concrete measurement mode, i.e. the measurement time, resolution, co-addition, etc.

Lastly the data points in the lower panel of Figure 7 puzzle me, since they seem inconsistent with Figure 6: While the signal is the same for the first two data points for Franklyn Street (Fig. 6) the SNR differs by more than 10%. This means, that there is a sudden noise increase by more than 10% even though the signal stays the same. This should not be the case if your instrument works properly and is (somewhat) shot noise limited. But even if the noise is dominated by another (instrumental) noise source, this should not change between measurements. But if there is some external reason for this, the magnitude is on the same order as your trend (see the two data points), making it difficult to disentangle this effect from the trend you are interested in. At list I would ask for a proper treatment of this uncertainty in the plot and the fit. It is also not only these two data points. Throughout both datasets the trend is way less clear then for Fig. 6, indicating a large uncertainty as a result of the noise estimation.

General remarks

Detection limit as a function of path length: The detection limit is only directly proportional to the path length if the SNR is independent of the path length, since the detection limit (how I think you define it here) is basically the point where the spectral response is on the order of the noise level. As you address yourself at several points within this manuscript, the spectral response (due to interfering species, and the exponential nature of absorption) and the SNR (due to a loss of light) can both show a complex, non-linear dependency on path length. Since you like to analyse exactly the interaction of these two dependencies, I would try to avoid formulations which proclaim a direct dependency (such as “directly proportional”) between the detection limit and the path length.

Cube Corners shading themselves: Ad multiple points you mention, that large flat arrays of cube corners lead to increased shading between the cube corners. I don’t think that this is true for the discussed situations, where the beam divergence stays the same (no modifications of the instrument or telescope) but the array has a larger distance to the instrument and, thus, the beam is larger in diameter. Even at this larger distance the maximum divergence of all the “rays” within this beam stays the same and the cube corners are hit from the same angles independent of the distance to the instrument. Thus the shading should not change with distance, even with larger arrays (of course assuming the telescope points vertically at the center of the array).

First report of retroreflector array size in open-path measurements: Considering FTIR open-path you might be right concerning the explicit testing, but I am not so sure about general open-path measurements. You might want to check out the DOAS open-path measurements in the UV and VIS spectral region, which go back at least to the 1990s, and where a lot of work on open-path optics was done. While I am no expert in this field and do not have an extensive overview, publications like Merten et al. (2011) addressed similar questions.

Detection limit results in simulation: I can easily follow your description of your spectral simulations in subsections 2.1 and 3.1, but I do not understand to which extend and how you added noise to the spectra and treated it as a function of path length. So to me it is unclear what you refer to (for example in L. 207 “the random noise at these separations”) when you draw your conclusions on the detection limit.

Inconsistencies of numbers: At a couple of places your numbers are inconsistent. This might include rounding errors, but sometimes also wrongly calculated quotients of numbers. This of course leads to unnecessary ambiguities concerning the real results and I encourage you to fix that.

In a few points, your interpretation of your results conflicts with my personal understanding of the underlying physics. In these cases, I can see how your data might indicate your interpretations, but since it conflicts with my understanding of some fundamental principles I would ask you to consider some provided possible reasons for your differing results and, after that, if you still stand firm to your interpretation address these conflicts. The two most relevant examples for this are addressed in detail below in my remarks concerning L. 206ff and L. 359f.

 Technical Corrections/Remarks

L. 10: Please refer to the general comment on detection limit as a function of path length.

L. 20: Please refer to the major point on Quantification of SNR dependency on path length.

L. 62: it should be a translation of “up to ~6cm”, since a central beam does not undergo a translation.

L. 90f: Please refer again to the general comment on detection limit as a function of path length.

L. 98f: The 1 mrad beam divergence fits quite well to your data presented in Figure 6, so I assume it is true. But with the given values for aperture, focal length, and telescope reduction I arrive at 2.4 mrad. I don’t know where the error lies, but maybe you can double check these numbers.

L. 102: Please refer to the general comment on cube corners shading themselves.

L. 123ff: Please refer to the general comment on the first report of retroreflector array size in open-path measurements.

L. 139: Why do you cite Rothman et al. (2013)? Did you use HITRAN 2012 and not the more recent versions? If so, why?

L. 157: While the difference between separation and total optical path length should be clear to every reader after the first mentioning, I would encourage you to stick either to optical path length or separation consistently in the paper when describing your setups.

L. 160: An interesting piece of information on the conduction of the experiment would be the timescale on which all of this happened. Did you move the array one increment roughly every 10 minutes, every hour, per day?

L. 162: See comment on L. 157. I think you use separations and optical path here inconsistently yourself (separations, but two-way?).

L. 179: Concerning the exclusion of IR intensity <0.15 arb. Unit: Shouldn’t this lead to a bias, since you would only keep the best measurements for long distances (where IR intensity is generally lower) but would include more measurements for shorter distances? I might just lack context here concerning your average IR intensity in these arbitrary units, since this might be a really low bar which filters mostly close to zero intensity spectra.

L. 183: How representative is your metrological data if it is from 7 km away? If it is flat, the pressure should be fine, but could you give some context concerning the temperature?

L. 185: What parameters of “phase and shift” are you referring to? Maybe a “phase shift” in the parameterization of the instrument line shape (like often done for FTIR instruments) or a spectral shift?

Figure 4: Your label of the color bar is inconsistent with your labels of the x and y axis, where you give you units in parentheses. I think it should be “Absorption(%)” then.

L. 205: With your provided numbers of 0.06% and 0.015% it should be “~4x lower”.

L. 206ff: I do not necessarily agree with your interpretation here. If you have a maximum absorption for HCHO of 0.12% you would be in a close to linear regime of Beer-Lambert’s law. This means, that you would expect 5x the maximum absorption signal for 1500 m than for 300 m (you seem to observe x2). I could only explain this if there actually is an interfering species which significantly obscures the absorption features of HCHO, by reducing transmission significantly and bringing you in a way more saturated region of Beer-Lambert’s law.

L. 224f, Figure 6: Please refer to the major point on Quantification of SNR dependency on path length.

L. 229: Concerning your explanation of the low signal levels. This would mean a drop to 80% of the initial reflectivity of the cube corners on average, even including the 10% globar drop. And since you have a pristine array in the center, maximize for signal return and, thus, should only look at this pristine center for short distances, I find this difficult to believe. I do not consider the absolute value of your arbitrary signal fundamentally important to the results of your study, but this explanation seems lackluster. There are multiple settings for a Bruker Spectrometer which could change the level of the arbitrary signal, are you sure there is no better explanation?

L. 237: I do not understand what you mean with “outside of instrumental response”.

L. 247: If the quoted number of -3.6/m stems from the fit in Figure 7 it should be rounded as -3.7/m.

L. 247, Figure 7: Please refer to the major point on Quantification of SNR dependency on path length.

L. 256: “lager” should be “larger”.

L. 257ff: I do not really see why you differentiate in this way between “a larger retroreflector array (and higher SNR)” and “different acquisition times”. In the first order both (array size and acquisition time) just influence SNR, which then influences the retrieved concentration precision.

L. 260ff: You probably also could just have averaged 2 interferograms or spectra for the 1 minute measurement mode (if each has half of the number of scans, which is actually the critical information) and performed the evaluation from there. This would make the two modes of operation comparable even if you have systematic differences/drifts between two consecutive measurements.

L. 265, Figure 8: Why an “arbitrary time index”? Isn’t this just something like “days since start or measurements” (ignoring the offset)?

L. 269f: If the quality of measurements is doubled by the larger retroreflector, this means that your instrument is not shot noise limited. I don't know if this is typical for such MIR instruments, but in a shot noise limited case double the reflector surface should mean double the signal (in case of overfilling) and sqrt(2) larger noise, meaning sqrt(2) larger SNR. If your instrument noise is dominated by other sources (detector electronics/thermal noise maybe) than this behaviour would be expected. Could you comment on that?

L. 290: I do not understand how this higher intensity in 2021 is consistent with your finding in Figure 6 and Line 228, where your signal with the larger reflector array is suddenly smaller. Could you clarify that?

L. 307ff, “even though there are four times fewer spectra in each hour in 2021”: The here underlying assumption/interpretation is (in my opinion) wrong. Precision of the individual measurement results within an hour is a property of the spectra, which should be better for 2021 due to larger reflector array and longer measurement time (both increase spectral SNR). The number of measurements within an hour do not influence the precision of the individual measurements, but would only influence the precision of the mean over an hour - basically the difference between a standard deviation and the standard deviation of the mean which you mixed up here, I think.

L. 309ff, Figure 9: The notion, that there is no discernible diurnal pattern in 2018 but in 2021 due to the difference in precision seems like a stretch to me. Also the 2018 data shows clear signals/drifts over the day on a similar magnitude. This might be caused by something else, but is still larger than the mentioned diurnal pattern in the 021 data. Attributing a lack of clear diurnal pattern to the precision does not seem correct. Furthermore, the visual comparison between the two panels is not really fair, since the upper panel shows a higher time resolution, resulting in more, but less precise measurement points.

Figure 10: Concerning the textbox at the top of each panel: giving the not only month and days, but especially the year of your data might be more important, since this is what you actually use to differentiate the datasets and refer to them.

Figure 11: Text is way too small and hard to read, even on a digital device with appropriate zoom.

L. 354f: Should “Fig. 10” be “Fig. 11” in both cases?

L. 359f: I do not agree with the statement, that (gaseous) water amount reduces SNR, unless in your spectroscopic window the total water absorption reduces the IR signal throughput (which I could not gather from your previous data in Figure 4, 5, and 6 for example). Rather, I assume that your calculated noise is inflated for the longer path due to systematic errors when fitting water lines. In the spectral window where you determine your noise (Fig. 7) are some water absorption features (even though no particular strong ones) but they might be strong enough so that for higher amounts of water the fit residual might reach the magnitude of the noise level. Or, if you are only taking the standard deviation from a polynomial background in this spectral region, the lines itself would cause an error once they are deep enough. I did not fully understand your process here. But I would encourage you to double check that this is no artefact of the way you calculate your noise value. Or maybe I misunderstood completely how you ended up at your conclusion.

Figure 12: Why do you plot this as a function of relative humidity and not specific humidity or even total water column if you consider the strength of the absorption features the relevant cause?

L. 370ff: As mentioned above, I do not agree with the generalized formulation of path length being inversely correlated to the detection limit.

L. 389ff: If this refers to the results from Figure 7, I think you would need to compare the relative drop in SNR, not the absolute ones. Since they are on the same order of magnitude, this detail results not in a dramatic difference, but for your numbers in Figure 7 it would then be pretty exactly a factor of 2 (slightly below).

L. 424f: You say that there is an optimum array size and path length combination for each observation. What is it for the ones you discussed?

Summary and Conclusion in general: of course many points above apply to the respective parts in summary and conclusion where they are picked up again.

References

André Merten, Jens Tschritter, and Ulrich Platt, "Design of differential optical absorption spectroscopy long-path telescopes based on fiber optics," Appl. Opt. 50

General comments:

This paper describes a nice study to look at the impact of compensating for increasingly long pathlengths with larger arrays of retro-reflectors with the aim of improving the detection limits and precision of measurements of formaldehyde at typical ambient concentrations of 1ppb in the study area.

I have marked this as a minor revision but to address my concerns some extra analysis as well as discussion is required.

The paper in its current form lacks some important discussion points, such as:

1. The data of formaldehyde retrievals include a large proportion of negative concentrations, but the paper does not discuss any possible retrieval strategies that could be applied to help fit the interfering gases and minimise these negative retrievals.
2. Regardless of whether a better retrieval strategy could be designed, the formaldehyde retrievals could be analysed to determine an actual limit of detection for each of the measurement set-ups described so that a time-series of measurements of ambient concentrations above this detection limit could be provided. This really is required (in my opinion) to make this manuscript publishable.
3. The discussion of the differential absorption appears to be discussed only as a %, whereas by my understanding MALT will fit the absorption area (not depth). Simply using a higher spectral resolution would provide a greater absorption depth and probably improve the detection limit and precision. Whilst this may not be possible with the equipment owned by the authors – it should be included in the discussion (since it will also benefit the problem of interfering gases by improving the selectivity as the absorption lines become separated at higher resolutions).

Specific comments:

I provide some more specific comments for some different sections of the paper below:

Abstract

Line 16: “We demonstrate that two-way path lengths > ~300 m are necessary for robust HCHO spectral signatures (at typical random plus systematic noise levels.” But this MUST also depend upon the ambient concentration!! Some discussion about typical concentrations of formaldehyde in different environments would help set this context.

Line 19: “We demonstrate that the larger retroreflector array resulted in a smaller decrease in the signal-to-noise ratio as a function of measurement path, ~1.5 m-1 for the larger array as compared to ~3.6 m-1 for the smaller array” – Won’t this depend upon the field of view of the individual spectrometer and telescope? If so then this is quite a specific detail that probably doesn’t belong in an Abstract.

Line 23 : ‘(average standard deviation of 0.352 ppb for 2021 and 0.678 ppb for 2018 in hourly formaldehyde data bins over two days), - This is also detail that doesn’t belong in an Abstract. And referring to the data by the different years is a strange choice – when it is the path length and cube-corner array size that it what matters not the date.

Quoting the standard deviation to 3 significant figures seems over the top.

Experimental Design

Line 130 “(p, T, precipitation, which causes IR beam extinction), Use of parenthesis is confusing here as the latter clause refers only to precipitation not temperature or pressure. Rephrase to clarify?

Results

Line 229: Could the detector efficiency and/or pre-amp and amplifier gains also have decreased as the equipment aged? What is the difference in signal to noise at a part of the spectrum near to the formaldehyde absorption? Some of these factors will decrease the signal but not necessarily impact the signal to noise?

Figure 7: I am trying to understand the decision to calculate noise from a part of the spectrum outside of the detector response. I am not sure if this measure of signal to noise is the same as calculating it at a point in the spectrum with no absorption features but close to the wavenumber region where the gas of interest absorbs. Does this produce the same S:N? (Sorry if I am being slow – it has been a long week!) Maybe clarify this point in the text in any case??

As well as the spectral signal to noise as discussed in and around Figure 7, there is the retrieval signal to noise. i.e. the spectrum to spectrum retrieved values give some idea of precision in a stable atmosphere. Given the values shown in Figure 9, I expect a discussion of the LOD here in terms of concentrations 3 x the retrieved value “noise” – to determine where you have a clear detection of formaldehyde in the atmosphere.

Line 306 and 308: Are 3 and 4 significant figures justified here?