The manuscript „Horizontal distribution of tropospheric NO2 and aerosols derived by dual-scan multi-wavelength MAX-DOAS measurements in Uccle, Belgium” by E. Dimitropoulou is a highly relevant paper for the community. It is well-structured and thought through.

I am no expert on optimal estimation, therefore my contribution to the chapter 4.3 is limited.

I strongly recommend the publication, after considering the following comments:

General comments:

• In my opinion a distinction between the VCD_NO2 and c_NO2 measured in main viewing direction (retrieved by traditional profile inversion) and the values retrieved in other azimuthal directions would help a lot for clarification (e.g. page 9 line210, page 10 equation 1, page 11 equation (4))
• Inventing a new name for this new technique would help especially when comparing to other data-sets (e.g. in chapter 5.2). Usually, elevation scans are called “MAX-DOAS”. Something like “Mapping MAX-DOAS” would help to distinguish between the different kind of retrievals.
• Is a seasonal analysis really useful for NO2? An analysis sorted for different temperature regimes could be more useful. Also, an analysis sorted for several wind regimes could be interesting, especially as 3 huge NO2 emitters are in the region.
• The RIO data is available in an hourly model. It would be interesting to see a comparison to this data set.
• In a future study the comparison could also be performed for AOD with the 1km resolution data from MODIS.
• Harmonize axes and color bars of plots in one figure.

Specific comments:

• Page 2 line 29: If possible, restructure sentences for a better reading flow.
• Page 6 figure 2: Why are the reference lines not exactly at the maximum of O4 for all windows?
• Page 7: line 55: DDS is defined, what is the difference between DDS and DDS2?
• Page 7 table 1: Data products 01.03.00 and 01.03.01 have an overlap in time. What is used in this period?
• Page 8 figure 3: In yellow box last line: “1 EA, many TAA”
• Page 8 line 205: Is all data flagged for clouds or only the main viewing direction?
• Page 9 line 211: Add an equation, as this value is used in all future calculations.
• Page 10 line 235: Replace “:” by “,”.
• Page 10 line 252: State clearer, that this sentence refers to the old paper.
• Page 11 line 276: Equation (4) and the following sentence are confusing next to each other (I understood it after reading the next two pages). In the equation is written dSCD_NO2_simulated, in the text are mentioned simulated O4 DSCDs. Explain in more detail.
• Page 11 line 281: Was the impact of this fixed NO2 concentration investigated? If typical high values for Brussels are used, does the retrieval significantly change?
• Page 12 line 298: “are” instead of “is” (or restructure sentence)
• Page 13 line 307: Which order is the polynomial fit?
• Page 13 line 310: Is this statement also true for the high pollution cases in Brussels?
• Page 13 line 314: (see Fig. 6 (a))
• Page 16 equation (6) and (7): For readability it would be good to include the missing step between the two equations
• Page 17 equation (7): possible space character in subscript
• Page 22 equation (11), middle line: Confusing definition. If written like this, it should be x(first) >L_NO2 (the first value which extends over L_NO2).
• Page 23 line 536: If this is true, why do the lines differ in figure 13 and 14 for the first km from the a priori? And why is the information of this region used in the later comparisons if this information cannot be trusted?
• Page 25 figure 11: Add wavelengths and change colors (see https://www.color-blindness.com/coblis-color-blindness-simulator/). Add also time of the measurement.
• Page 26 figure 12: Same y-axis for all
• Page 27 figure 12: Mention RIO in caption
• Page 27 figure 13: Adjust both axes to the same values, add (a) and (b)
• Page 28 figure 14: Add a priori of aerosol distribution. Are errors available?
• Page 32 figure 16: For consistency it would be good to see the values corresponding to figure 13
• Page 33 figure 17: It is claimed before, that the measurement is insensitive for the first kilometers. It would be good to mark this range (e.g. draw a line). Why is the y-axis limited to 11km?
• Page 33 line 664: Is the wind direction and speed stable over the measurement period? What is the wind speed?
• Page 34 figure 18: The wind direction could be added to the plots. If possible, the latitude and longitude range could be shortened to make the plot bigger.
• Page 35 line 695: Mention typical overpass time of TROPOMI. Is every measurement from TROPOMI used? Even when several are available for one day? And is the exact time of the overpass used or the typical overpass time? The NO2 distribution can change quickly. Were also other time ranges used for comparison? (+/-30Min)?
• Page 35 line 698: How does this simplification (no weighting) influence the results?
• Page 35 line 706: Can this be explained/expected? (Different sensitivity…)
• Page 36 line 723: Are the retrieved values high/low, compared to typical values for this region?
• Page 37 figure 19: Add MAX-DOAS maps in original resolution for comparison with original APEX. Also, the RIO analysis could be added. Use the same color scale for all plots!
• Page 38 line 759: Point number 3 is unclear to me. A sketch could help to clarify.
• Page 39 line 768: Regridding routine is not completely clear to me. Are the averaged pixels weighted by their ratio by which they cover the reference pixel?
• Page 39 line 776: Agreement for summer seems to be worse than for other seasons. The maximum in TROPOMI is seen in the NW, in MAXDOAS it is south.
• Page 39 line 782: Can this be explained?
• Page 39 line 792: Explain SEAS, I guess season?
• Page 40 line 806: “… presented in Dimitropoulou et al. (2020), see table 4.”
• Page 41 figure 21: If possible, remove one color bar and use the free space to widen the plots in x-axis.
• Page 44 figure 23: Red line is hardly visible
• Page 45 table 4: Clarify R (seasonal) and s (seasonal). It probably corresponds to SEAS? Are the number of points the number of days or the number of pixels?
• Page 45 line 885: Add name of the improved FRESCO-S cloud retrieval.
• Page 45 line 887: “Fig. 24 (a) and (b) …”
• Page 46 line 910: Using MAX-DOAS profiles as a priori profiles for TROPOMI is suitable for a consistency check of the method. It should not be mistaken to be the “truth”.

• Supplement figure S1: Same y-axis for all plots, MLH could be in the title of the individual sub-plots.
• Supplement figure S3: Same color bar for all plots
• Supplement figure S4: Same y-axis, at least for AOD>0, this highlights the differences much stronger

General comments

Dimitropoulou et al. 2021 present an interesting and new way to retrieve horizontal trace gas and aerosol profiles from MAX-DOAS data measured at various geometries and wavelengths. The approach utilizes data from profiling algorithms, further RTM calculations, and makes several assumptions on the atmosphere and the spatial distribution of its absorbers.

I recommend the publication of this manuscript after addressing some critical comments, because of the interesting idea rather than the convincing approach/results.

Before I explain further on the issues below, please let me give you my personal opinion on the approach and its assumptions. You start with the calculation of vertical profiles of MAX-DOAS measurements and calculate MLH. Even though validated, a MAX-DOAS profiling algorithm is not the truth but a smoothed representation of the true atmosphere constrained by the limited vertical resolution. The calculation of MLH is again an assumption of a box like distribution even though it was shown with MMF before that the profile was most likely not box-like at all. Then, further assumptions about a homogeneity of the MLH, calculated effective light path lengths and AODs are made (e.g. aerosol parametrization, profile shapes with percentages of bulk load based on statistics, fix values of all input parameters). The estimation of VMR based on Sinreich et al. 2013 is again just an approximation which is only valid under certain conditions. And then, everything is used in another inversion step. Knowing all this, it appears to be a miracle that your correlations are still good! To remove some of my doubts, please assess the error budget and the propagation of errors between the individual steps in more detail.

Three further issues of this manuscript need to be solved before final publication and will be addressed in detailed below: 1. This manuscript is much too long. 2. The presented approach is not validated sufficiently. 3. The explanation of the approach in Section 4.2.1 needs a revision.

Details on the three main issues:

1. Nowadays, many paper show more and more results which are sometimes extremely insignificant or do not match the purpose of the publication. The main purpose of this manuscript is the introduction of a novel approach about the retrieval of horizontal absorber concentrations. A manuscript with this topic should introduce and explain the novel approach and show a thoroughly performed validation study. In this manuscript, you also address the question of how to optimize and validate Tropomi measurements. This has nothing to do with the content of your paper and is addressed in more detail elsewhere. So please remove Section 5.3.2 fully (or move it to the supplement) and remove parts of section 5.3.1! E.g. the three concluding points on page 40 can be removed as they are already content of the conclusions. When you add a proper validation (see point below), please move Section 4.2.3 to the supplement.
2. Validation of a novel approach is a necessary step but needs independent measurements! Validation can never be done with the same instrument. Since you already mentioned the in-situ air quality network in Brussels, I would recommend the use of this data or another independent data set. However, no validation of other instruments/data sets by your novel approach can be shown without first validating itself! This means of course that validation of Tropomi data with your approach is not appropriate. It would make more sense to validate/verify your approach with an already validated Tropomi data set.
3. It was difficult to understand your explanation in section 4.2.1 because it is not directly clear for the reader which quantities are fitted how and when. Please revise this section and explain specifically which measured or inverted quantity goes in which polynomial-fitting or RTM-calculation step. Maybe a small flowchart would help or adapt and refer to the existing flowchart in Fig. 3. Furthermore, Fig. 4 and Fig. 5 show simulated columns. Please add also an example with real measurements and the polynomial fit in the same figures.

Specific comments

P3, L88: Please write "telescope azimuth angle" so that the abbreviation (TAA) makes more sense.

P3, L89: Why did you use 2° instead of 1°? (see also P9, L219)

P3, L92: Remove "directions". It is not needed.

P3, L94: Integration time of 60s? Why was it set to such a long time? Depending on the wind speed, many things can happen within one minute in an area with that strong spatial inhomogeneities!

Figure 1: When I remember correctly, you will never again talk about the individual tests shown in this plot. So please remove these tests and show only the applied azimuthal directions. Please zoom slightly in and show the power plants, ring freeway, and MAX-DOAS site, similar to Fig. 18.

Section 2.2: Please explain how you decided on these wavelength intervals. Did you do some optimization for the shown fit settings? For example, I was wondering why the window 510-540nm was chosen like that even though large H₂O absorption is present at the start and end wavelengths. These absorption features might also explain slightly larger residual structures (compare Fig. 6).

P5, L116: Why are not all reference wavelengths in the peak center of their corresponding O₄ absorption bands?

P5, L123-L125: You said that the O₄ cross section of Finkenzeller improves results in the UV. Does it show any change in the visible range? Why did you decide against using it here as well?

P7, L172: "measured radiance spectra ... is analyzed" to "are analyzed"

P9, L195: Please cite some of the "several studies" you are referring to.

P9, L196: I would write of a scaling factor ≠ 1 because some studies suggest also larger scaling factors depending on spectral range, location and season.

P9, L208-L209: Why do you accept retrievals with homogeneous cloud coverage? The corresponding aerosol profile is wrong for sure! This means that your MLH is inaccurate because is is negatively affected by the wrong radiative transfer. All your RTM calculations of L_{NO_2} are wrong as well and, therefore, your horizontal profiles!

P9, L219: Why did you select an elevation angle of 2°? I would assume that 1°? is better suited for your purpose and assumptions about homogeneity are closer to the truth. (see also P3, L89)

P10, L251: Please move the full stop from the index of NO₂ to the normal level.

P11, L276: "Here, O4...". If "here" refers to the equation above it should be "Here, NO2..."

Section 4.2.1 Please revise according to the general comment.

P13, L324: "As discussed above, around 30%...". I would not call this a real discussion. You just mentioned it without showing any results of the analysis.

Fig. S4, S5: Please discuss the dependence on SZA and RAA for Fig. S1 and S2, respectively, as well. I would assume that a significant contribution of the dependence is due to aerosols and the applied phase-function.

P13, L335: "less pronounced Rayleigh scattering" to "less pronounced Rayleigh and Mie scattering"

Fig. 6: Please explain why the smaller deviations of L from the polynomial fit propagate into much larger deviations for the near surface concentration and vertical column density.

Eq 6 and 7: The step from Eq. 6 to 7 can not happen without further assumptions. I would rather prefer the calculation of the partial derivative than this assumption. Please explain this step in more detail. How is the uncertainty of the O₄ dSCD used?

Section 4.2.3: This section needs to move to the supplement and has to be replaced by a real validation. The sanity check is interesting but should not be content of the main manuscript. The validation part is unfortunately no validation but rather a verification. Validation only works with independent measurements. Using just a different azimuthal direction from the same instrument is not at all independent. For a study like this, I would assume comparison with e.g. in-situ instruments. Please add a validation study from an independent instrument as e.g. in-situ data from the air quality network in Brussels.

Eq 11: Please give more information on this approximation/definition. Is the weighting function for aerosol extinction coefficients defined in a similar way? With this definition, you suppress variability in the horizontal direction which means that you consider your effective light path lengths as perfect. I also have a problem with the fact that the weighting itself was arbitrarily defined as horizontal step width divided by whatever is found from your simulation of L. Is the information content not large enough to allow a more flexible implementation? Please discuss this further.

P22, L502: How large is this mean scaling factor? You already add a bias here by applying a mean factor. It would be interesting to know if the unscaled a priori profiles would lead to a better agreement with ancillary instruments or if it just destabilizes your retrieval.

Fig. 13 and 14: Please add two further examples for profiles with larger concentrations and examples with a small aerosol/NO₂ load. The readability should not suffer with two more curves in these plots. Please add the a priori profiles and errorbars for the aerosol horizontal profiles as well. In Fig. 14, you see a larger deviation of measured and simulated extinctions for the middle L_{O_4} values as well as for the first data point. Since the reader cannot see the a priori, it is hard to assess how this propagates into your retrieved profile. Please discuss this deviation together with the a priori profile and the retrieval errors.

Fig. 15: Since AK are the multiplication of the gain matrix G with your weighting function matrix K, your averaging kernels show the sensitivity based on your definition of the weighting function. Please show for this scenario the corresponding a priori profile and retrieval result. I would assume that the features of your AK matrix are strongly dominated by L_i and your a priori profile. It is difficult to understand why the sensitivity for blue and red are the highest at exactly the same distance (similar with purple and green). If this figure would be a good representation of the derivative of concentration with respect to the true concentration, individual peaks should be found at different distances. Please discuss!

P29, L601:  "close their" to "close to their"

P29, L610-L611: Why should this be the case? What about the smoothing error? Since you have many constraints due to your a priori assumptions, I would assume that the smoothing error has a significant contribution to your total error.

Table 3:  "Medium RMS" to "Median RMS". What do the DOFS and RMS values in brackets in the last row mean? If this refers to the total accepted retrievals, why did you write different thresholds in the text? Furthermore, it is hard to assess the range of RMS and DOFS values based on these numbers only. Please add a figure of the frequency distribution of RMS and DOFS values and discuss it together with this table!

Fig. 18: Please zoom in and increase the quality of this figure so that details can better be seen. Please add a similar plot showing the near-surface concentration/extinction in the supplement. Do near-surface values support your finding that air pollution in Brussels is mainly driven by the power plant and traffic emissions? Can larger values be found in the distance of the Ring-motorway?

P35, L698: Why are the segments not weighted? If there is just a tiny fraction of light path within a pixel, it should not be weighted with a similar factor as contributions with much longer light paths. Please change this or discuss why it is not possible/reasonable.

Fig. 19: Please change the range of colors for subfigure a) so that the plume is better visible. For the original data of APEX and the AEROMOBIL I would not assume the same color-scale. However, subplot b), c) and d), should have the same color-scale! Please change this.

P35, L706: How do you explain this intercept?

P35, L709: "channels" to "elevation angles" or "geometries"

Fig. 20b: Please correlate the not averaged data as well by comparing MAX-DOAS and AEROMOBIL data points which are close to each other. 6 data points are not statistically significant enough for such a comparison. Especially not when you can compare data in higher resolution.

Section 5.2: Please add the exact overpass times you used for the comparison of TROPOMI and add a reference to Fig. 18 in the text so that the reader can also compare the MAX-DOAS data in higher resolution at the overpass times with Fig. 19. Furthermore, add the start and end time for the AEROMOBIL measurement in the text and the caption of Fig. 19.

Section 5.3.1: In addition to my general remark about the purpose of this document, I would like to ask you to change this section either to a "validation of the new algorithm" or to a "comparison only" section.

Fig. 21: By just looking at this plot, I would say the agreement is not that good. Why is the difference in the SW direction so large in Summer and Winter? Why are the values close to the MAX-DOAS site much higher for the MAX-DOAS data in Spring? This is especially interesting because your algorithm was described of having a poor sensitivity close to the instrument (compare P26 L599-L600)!

P39, L780-L782: Similar to the comment above, I am wondering why there is a large negative Bias close to the instrument and a positive one for the pixels in large distance to the MAX-DOAS site? I was wondering if the values at the contour-legend are correct? Is the positive bias just smaller than 1%? If yes, this bias might be negligible. Please check the values and the figure and give an explanation for theses biases if possible.

P39, L790: I would not talk of an under-/overestimation by one of the instruments if it is not clear which instrument/algorithm shows the more accurate results.

P40 L818-828: This belongs to the conclusion and should be removed from this section.

Section 5.3.2: Remove this section from the manuscript or move it to the supplement (see general comments).

P49, L953-954: and AODs

Conclusions: Please change the conclusions according to the general and specific comments.

References

Sinreich, R., Merten, A., Molina, L., and Volkamer, R.: Parameterizing radiative transfer to convert MAX-DOAS dSCDs into near-surface box-averaged mixing ratios, Atmos. Meas. Tech., 6, 1521–1532, doi:10.5194/amt-6-1521-2013, 2013.