General Comments

The authors introduce the kriging technique, originally from the geostatistics community, to the atmospheric science community. To properly compare point samples (such as Pandora spectrometer) and large satellite pixels (such as Ozone Monitoring Instrument (OMI) pixels), the authors take the following steps: 1) construct a semivariogram that takes account for spatial variances among the point samples, and subsequently produce a kriging estimate based on the point samples (as well as an error estimate) over a 2D grid, and 2) convolve the kriging estimate and error using a spatial response function that represents the large satellite pixel size. The authors first show detailed examples of this process using typical theoretical cases. Finally presented is an actual case comparing NO2 columns from point measurements from Pandora instruments and OMI.

This paper is well organized and fits well in the scope of Atmospheric Measurement Techniques. As this paper introduces a new, useful technique to the atmospheric science community, adding explicit statements to help readers’ better understanding will greatly benefit the community. I would recommend the paper for publication in AMT after addressing the specific comments listed below.

Specific Comments

Overall: I suggest using the term “grid box” rather than “grid” when it actually means a grid box.

Line 75: Studies to downscale satellite pixels using high-resolution model simulations worth to be mentioned, e.g., Kim et al., 2018, and Choi et al., 2010 (already referred to in the text).

Kim, H. C., Lee, S.-M., Chai, T., Ngan, F., Pan, L., and Lee, P.: A conservative downscaling of satellite-detected chemical compositions: NO2 column densities of OMI, GOME-2, and CMAQ, Remote Sensing, 10, 1001, https://doi.org/10.3390/rs10071001, 2018.

Line 115, Eq. (1) and Eq. (2): I think h in g(h) should be boldfaced, as in f(x+h) in the later part of the equation? Although this study deal with isotropic cases only, the length of vector h, i.e., h = |h|, needs to be explicitly defined or mentioned before being used in the text.

Line 122: Parallel to the above point, I suggest explicitly mentioning that γ(h) here is only dependent on the distance between samples, not direction/angle.

Line 130, Eq. (4) While the authors state that a_o, b_o, c_o=1.5, they fit paired samples into the given Gaussian function. Therefore, these coefficients cannot be fixed values. Moreover, red lines in all figures variate as well. Please check.

Line 140, Eq. (7): Does Z_o means Z(x₀) (true value at x₀)? Please define x₀ and Z_o.

Line 145, Eq. (9): Do γ_j1j2, γ_j1o, γ_oo mean γ(x_j1-x_j2), γ(x_j1-x₀), and γ(x₀-x₀), respectively? Please define them explicitly in the text.

Overall Sect.2.1: There are too many subscripts ₀ or _o. The coefficients of the Gaussian function are a_o, b_o, and c_o. A specific, random point we want to estimate Z is x₀. l₀ is a constant weight. Z_o is (probably) the true value at a point x₀. Some of these subscripts are relevant but others might not. Removing unnecessary 0 subscripts may help readers understand better.

Line 163-164: The authors take 200 samples and make 100 pairs. However, the maximum number of pairs seems ₂₀₀C₂, or _nC₂ (n=number of samples), according to the first row of Fig.2 and the second panel of Fig. 13. This is worth mentioned here.

Line 165 and Fig. 1: The authors mention that the semivariograms except C1 fit to the Gaussian function. I am not sure if the semivariogram of C2 is really Gaussian.

Line 223, Fig 2, and Fig. 3: The authors mention the relative error of C5 in line 223, but it is not shown in Fig 2 and Fig 3. Locations with large Z values will naturally exhibit larger error values as shown in the fourth columns of Fig. 2 and Fig.3. However, showing relative error in these figures might be meaningful, as reasonably illustrating the plumes is more important than the absolute value of the error in this case.

Line 248: I suggest replacing “realization” with “kriging attempts” for better understanding.

Line 337: The authors mention a two-dimensional super Gaussian spatial response function used in Sun et al. (2018) study. Although this function is a critical component in the actual OMI-Pandora case study (line 473), it is not used with the theoretical cases. Therefore, this sentence is hanging in this line and may confuse readers. It needs a better location in the paper.

Regarding the super Gaussian function: Again, although convolution with this function is a critical component in the actual OMI-Pandora case, no explanation or visual illustration has been made. Instead, only the uniform spatial response function/ideal box kernel is visually illustrated in Fig. 5. I suggest showing a figure showing the convolved C5opt with the super Gaussian function (possibly using various parameters, comparable to Fig. 5) when introducing the super Gaussian function. This figure may go into Supplement.

Line 339 and Fig.5: The authors suggest S[m,n] = 1/(m*n) J_m,n (J is the matrix of ones) as a uniform spatial response function. Also mentioned is that the panels in Fig. 5 are convolved with an ideal box kernel in the caption of Fig. 5. In summary, the authors mention that “If the spatial response function is assumed to be an ideal box, the resulting grid will represent the average.” Putting them all together, “S[m,n] = 1/(m*n) J_m,n (J is the matrix of ones)” is the “ideal box kernel”, and convolution with an ideal box kernel actually means taking the average within a grid box, which should be explicitly stated here.

Line 339-340, Eq. (14): Does S²[m,n] in Eq. (14) means the squared [m,n]-th element in the matrix S[m,n] in line 339? If so, please explicitly state it. Also, taking a matrix notation for the matrix S[m,n] will be helpful.

Line 358: Is there a reason for not showing the synthetic satellite measurements (upscaled truth)? I am interested to see how synthetic satellite measurements compare with the kriging estimate in the 30x30 resolution (although it is before converging) and the converged kriging estimate (1x1) resolution convolved into the 30x30 resolution.

Technical corrections:

Line 130, Eq. (4): In the equation, some subscripts look like alphabet o while others look like number 0. Please make them consistent.

Line 138 & Eq. (10): Also, while the subscript of x₀ in line 138 looks like a number 0, subscript of x_o in Eq. (10) looks like an alphabet o.

Fig. 10: Dots showing uniform sampling locations are barely visible. Can they be more visible?

Fig. 12: The resolution is 0.2^ox0.2^o (approximately 20x20 km²), instead of 20x20 km².

‘Dealing with Spatial Heterogeneity in Pointwise to Gridded Data Comparisons’ demonstrates a new methodology for comparing point measurements to those that are more spatially representative of an area (e.g., satellite or models) for the purpose of data product validation. This topic takes an important challenge with these types of data comparisons for the purposes of validation and attempts to lay out a solution going forward. The manuscript is very well written and organized and it fits well within the scope of AMT and should be considered for publishing after some minor changes.

General comments:

• The paper switches back and forth between the nomenclature of using the method in relation to validation vs. data comparisons. While the methodology and conclusions made related to using point measurements for validating grid-like data are completely valid and important, the language in this paper is dismissive of other potential reasons to compare point-like measurements with something like a model or satellite. Therefore, the suggestion is to clarify that this methodology is for validation and not generalize conclusions for data comparisons.
  • Examples of this type of language include, but are not necessarily limited to:
    • Line 80-81 with the question of whether a ‘comparison ever logical’ – in the right case the comparison could be logical when trying to learn how satellite data can be interpreted in relation to a ground-based measurements
    • Line 551: change ‘comparisons’ to something along the lines of validation
    • Line 572: ‘point-pixel comparisons’ should say pixel validation with point measurements or something along that line.
    • Figure 9: in the right orange box change ‘comparison between satellites and observations’ to ‘validation of satellites (models) with point observations’
  • Could the authors comment on the reality of ground-based networks that could actually contribute to satellite/model validation with this methodology? Does the required observational density exist anywhere? What are some paths forward/recommendations?
  • Section 4: Why is v3.0 OMI data using instead of the most up to date v4.0?

Lamsal, L. N., Krotkov, N. A., Vasilkov, A., Marchenko, S., Qin, W., Yang, E.-S., Fasnacht, Z., Joiner, J., Choi, S., Haffner, D., Swartz, W. H., Fisher, B., and Bucsela, E.: Ozone Monitoring Instrument (OMI) Aura nitrogen dioxide standard product version 4.0 with improved surface and cloud treatments, Atmos. Meas. Tech., 14, 455–479, https://doi.org/10.5194/amt-14-455-2021, 2021.

Specific comments:

• The first sentence in the abstract implies that the two communities have zero realization of the point vs grid problem, which is not true. This is a known problem with the lack of an easy solution.  Please consider rephrasing.
• Lines 28-29: This study demonstrates a method but it doesn’t actually prove that the only available method ‘must taking kriging variance…’etc. State what the paper demonstrates without implying there is no other alternative to this exact method.
• Line 50: consider adding some clarity to the hypothetic scenario by adding after ‘atmospheric model’ the phrase ‘simulating CO2 emissions’…
• Line 191: is there a reference for the terminology of ‘the sill’?
• Paragraph spanning lines 232-239: Use the word stratified somewhere to connect to the second row of the figure.
• Line 385: add that this is also a roadmap for model evaluation as well.
• Line 414: add a reference for the length scale of NO2 if connecting it to the range found in those months.
• Lines 433-440: More clearly explain the rational more clearly between 10 vs. 15 vs. 20. In the figure alone it isn't all that clear.  Was the choice of 15 quantified somehow as the best option or was it subjective?
• Line 449: Should be Herman et al. 2009
• Line 454-456: what resolution is OMI oversampled to?
• Lines 456-457 and Figure 12 caption. The wording states that the total column was subtracted from the stratospheric column, when it is the stratospheric column that should be subtracted from the total column. Please fix this wording.