General remarks

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This article presents an interesting comparison between the airmass factors

computed with the three models included in SAKSTRAN, as well as an interesting 

application to a specific case.

More details should be given about the derivation of the AMF in the specific cases

of HR and DO.

It is interesting to see the effect of wavelength changes  on the differences

HR/MC and HR/DO. For these comparisons, I would suggest including aerosols

layers in the model atmospheres. It is needed, since, in section 6, 

aerosols are considered in the sensitivity study.

Specific comments:

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* Section 2.1

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Line 85 : it could be recalled that the radiance is the sum over n

of the radiances scattered n times (even if it seems obvious).

Line 101: I agree with the fact that, when the state of the atmosphere depends

only on the SZA and the altitude, only these two spatial coordinates need to be

considered. However, I disagree with the reason you give: there is no rotational

symmetry around the solar direction, since there is an other privileged

direction: the vertical (the grounds acts as a radiance source when the albedo

is nonzero).

Lines 111-112: 

- if the built-in weighting function are analytical, could you mention it ?

- could you precise what approximations make built-in AMF unsuitable

for precise AMF computations ?

Table 1: I like the idea of recalling the definitions. However, they should

be accompanied by figures to make the geometry more clear.

* Section 2.2

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Line 121: Could you define mathematically x_n ? It is a function s -> (z,lon,lat)(s) ?

* Section 3.1

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Equation 8 defines the AMF for the MC method only, it should be specified.

For definition for the other methods should

* Section 3.2

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Line 189: "equivalent (up to a sign)" should be replaced by "equal to the opposite of".

Wouldn't it be possible to homogenize the notations between equations 10 and 11 ?

(k(z)+ epsilon * phi(z) in 10, kbar + Delta_k(z) in 11 ?

* Section 3.3

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equation 15 rather defines an effective layer geometrical thickness

A effective layer height would be

\int_{0}^{H} z\phi_i(z) dz

* Section 3.4

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To be as general as possible, it would have been preferable to write the

equations in the 3D case with longitude and latitude.

* Section 3.5

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Lines 259 and 260: is the assumption that the change in sigma(z) is negligible

valid whatever the molecules considered ? This point should be detailed in the paper.

Equation 35:

For HR method, is this equation applied for the radiance at every order of scattering ? For their sum ?

* Section 4.1

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Lines 290-291: what is the complete list of molecules considered ? No H2O ?

Line 295: What is a moderate geometry ?

Lines 300-301 and 305-306 : Is there a reason why there are more downward facing than upward facing

        directions ?

307: Precise what is an extreme geometry. 

In figure 1, right graph. What is the meaning of a (e.g. "a 0.05") ?

* Section 4.2

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In figure 4, the effect of the SZA is not very clear: the difference in transparency

is difficult to distinguish. I would advise less SZA, with curves for every SZA distinguished

otherwise (dashed lines, ...)

Line 336: "small zenith angle" :  SZA or VZA 

Lines 342-348: Changing the radius is a simpler way to change the sphericity. Could

               you try it ?

* Section 6

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A 3D study would have been more realistic. 

What order of scattering (max  value of n in eq 1) was used ? 

Table 3: what are the nature of  cloud and aerosol particles ? SSA ? What size distributions where used ?

A plot of their phase function would be interesting. Where the aerosol / cloud properties dependent

on the altitude ?

In figure 6, NO2 VMR, is used, in figure 7, NO2 number density. You should use the same quantity

in both figure. 

Could you plot the 2 NO2 profiles vertical chosen  (clear/polluted ) ?

Figure 8 should be completed, to have the same graph for the four cases shown in figure

9 (homogeneous/inhomogeneous albedo,  homogeneous/inhomogeneous NO2 ).

Figure 8, graph at the right: could you plot both the 1D and the sum of 2D box-AMF ?

Figure 9: could you precise the unit of total and partial SCD ?

Technical corrections

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Line 112:  AMF acronym should be defined here (first occurrence), not in line 163

table 2: "9 profile" => "9 profiles" (2 occurrences)

In figure 6, the blue box is difficult to distinguish in the right graph. A change of color (maybe

using dashed line) would make it easier to read.

In figure 6, some white patches appear in the left graph. You should specify what they mean (white = no data ?)

The paper presents several developments to the radiative transfer framework developed at the University of Saskatchewan to calculate air mass factors. The framework includes three radiative transfer models, one of which is based on the Monte Carlo method that is used to evaluate the other two models employing the successive order scattering and the discrete ordinate models. 

The paper should be of interest to researchers who do remote sensing of trace gases and probably even aerosols. It is fairly well written, and I appreciate the “gentle” introduction to the specific subject of the paper, but I feel that some simple drawings or something similar illustrating the various AMFs in section 3 would make the material even more accessible to readers less familiar with the topic.

The calculations are carried out on scenarios with increasing complexities. It is not entirely clear to me whether aerosols have been included in the atmosphere. Reading the bulk of the paper I’ve got the impression they have not, but Table 3 lists aerosol optical depth. Do the results presented in Figures 1-4 include the impact of aerosols? Due to their phase function being distinctly different from those of molecules the size of the area of pixels with varying surface reflectances that impact radiances in the viewing direction from a pixel inside the area would be different depending on whether or not the atmosphere includes aerosols.

Minor comments or suggestions:

Line 35: Define VCD here since I think this is the first time it is mentioned.Line 43: TEMPO was already launched.  It was launched on April 7, 2023 so the authors may want to revise the sentence.

Line 75: Perhaps it would be better to write “modelling horizontal inhomogeneities” instead of “modelling horizontally inhomogeneities”.

Table 1: Do “Layers” and “Streams” also apply to the DO method in addition to MC?

Line 131: It may be useful to define box-AMF here or point the reader to the place where it is defined.

Equation (10): Please define epsilon. I did not see it defined before this equation, but it is possible I missed it.

Figure 1: In the caption, you may want to add “…  and surface albedos, a, 0.05 and 0.8”. This would inform the reader what “a” in the legend is without searching the text that comes after the figure (at least it is the case in the manuscript).

Lines 298-301: Would it possible to also include a cartoon showing the arrangement described in the text?

Line 374-375: This should say “… added one or more …”.

Line 388: Should this read “ … homogeneity …” instead of “ … inhomogeneity …”?