The paper by Lonsdale and Sun presents an interesting analysis of the effects of COVID-19 on NOx emissions in 54 cities over North America, Europe and East Asia, as well as the region- and city-specific features of such effects. The calculation is based on their previously developed “directional derivative approach” which considers the mass conservation under the influence of horizontal advection, emissions, chemical loss and terrain effect. The paper is well written and easy to read. I applaud their efforts to use a new emission inversion algorithm to study the COVID effects on air quality. There are a few major issues on the methodology and results in this paper that should be addressed.

Based on Eq. 1 (and Sun et al., 2022, GRL), the “wind divergence” term (Ω(∇ ⋅ ⃗𝑢) is assumed to be equal to vertical flux at z1. This assumption might be overly simple. Consider a flat terrain, in which the terrain term becomes zero, Eq. 1 apparently misses the wind divergence.

In Sun et al. (2022), there are a few key assumptions for which the validity is not (well) tested. For example, assuming no horizontal gradient of NO2 above PBL to derive Eq. 3, assuming surface concentration to be derived from the column using a single inverse scale height across a large domain, assuming the wind divergence to be equal to vertical flux at z1 (Eq. 10), assuming the chemical loss in the PBL to be the same as the loss for the whole column (Eq. 12). All these assumptions are necessary to finally derive the “directional derivative approach”, but these assumptions are subject to large uncertainties and should be tested rigorously to ensure the results for 54 cities here are robust. These tests are especially necessary, because there are few tests in Sun et al. (2022).

A few major assumptions are taken in addition. For example, the fixed NO : NO2 ratio, the fixed lifetime (for monthly climatology and for a large domain), and the fixed scale height (for a large domain). The NOx chemistry is highly nonlinear and its chemical lifetime varies greatly in space and time. The vertical mixing, PBLH and convection, which are related to the “scale height”, also vary substantially in space and time. How the assumptions will affect your city-specific results need clarification. There exist fast algorithms in the literature that have considered varying NO : NO2 ratio, as well as concentration-dependent lifetime to account for nonlinear chemistry, and these studies should be discussed and/or compared.

The resulting scale height is often very high (up to 5 km). It becomes difficult to interpret the exact physical meaning – it is too high for PBLH – and how it can be assumed constant across a large domain without causing major uncertainties.

The derived lifetime is also too high (tens of hours or even more than 100 hours), much longer than what one would expect for a city. What is the exact meaning of this derived lifetime? Why is it OK to use this high value to obtain a reasonable estimate of emission? Since the lifetime for each sub-region and climatological month is constant, what is the impacts on the derived interannual changes in emissions?

The derived emissions often show very strange seasonal patterns, e.g., emissions in winter could be a few times those in summer. The only anthropogenic source of NOx that exhibits strong seasonality of residential heating, but this source only contributes a small fraction of total emissions for a city. Better interpretation and explanations of the resulting seasonality should be given.

There are relatively detailed descriptions of the resulting emissions for each sub-region. However, it is not clear how the derived emissions are robust for each sub-region and city, given the abovementioned methodological weakness, and the fact that no independent data are used to compare with the emission results. There exist many social indices and other quantitative (proxy) data that can be used to represent the mobility change and lockdown stringency during COVID. Many studies of COVID have used these data, as should be used here.

The interpretation that the difference from 2019 is he effect of COVID should be cautious. For developing countries, their emissions often change from one year to another greatly, even without COVID, for example, due to economic growth and end-of-pipe control. Additional efforts should be made to assess (quantitatively or qualitatively) the non-COVID factors. There have been many COVID related studies that address this issue, for example, by taking advantage of the pre-COVID trend or variability.

This manuscript applies the flux divergence method to numerous cities on 3 continents in order to identify the changes in emissions over the 3 years impacted by COVID-19 lockdowns. The method includes refinements on the handling of terrain and lifetime that were reported in a prior publication. A clustering algorithm was applied to show that cities in North America, Europe and Asia had very different annual variability in NOx emissions over the last 4 years.

I believe that the method is sound and the results are valuable. The paper is clear and well written. I am happy to recommend it for publication.

My impression is that the height scale and the lifetime are at least partly numerical tuning parameters that have a loose connection with a physical interpretation. This would explain the particularly large values: the beta values are the inverse of the parameters, so large values suggest smaller than expected impact of the corresponding terms in the equations. I wonder if the question of the parameters would merit some more discussion and caveats in the analysis.

My other impression is that for each site the time series is robust in a relative sense. However, I think there are probably larger uncertainties in the absolute emission values of one city compared to another and of absolute estimates of emissions in the winter compared with the summer. Because the purpose of the paper is to look at lockdown-induced variability, I don’t think this is a major problem. However, I do think it should be discussed to prevent over-interpreting the data. A more detailed comparison of emission totals by city with published emission inventories is beyond the scope of this study, but would be interesting in the future.