The paper focus on the instrument spectral response functions estimation based on sparse methodologies. This is an interesting paper with a new proposed method that reveals to be competitive and outperforms the state of the art, specifically, the parametric based strategies as the Gaussian and Generalized Gaussian.

Please find below some remarks:

How confident are you in the reference spectrum obtain using a radiative transfer model and/or the ground characterization for each instrument and what would be the impact of a possible mismatch on such transfer model/ ground characterization?

What is the number of representatives ISRF examples used in the simulation part? Can you discuss (at least numerically) the minimum number that leads to an error below of the required 1%?

From fig 1 it is clear that the plotted ISRF cannot be accurately modeled by bell-shaped Gaussian distribution nor the generalized Gaussians one. Nevertheless, it seems that a mixture of Gaussians can be a good fit. Can you elaborate more on this option (the number of mixtures can be estimated using BIC or AIC)?

The authors said that the analysis of concentrations from two spectrometers could provide a better understanding of the carbon cycle. Nevertheless, it seems that the analysis in the paper has been done separately (ie., per instrument). Is it possible to use data fusion in this case in order to have a more accurate results?

The authors should add a supplementary material, or annex in order to elaborate more on the algorithm, specifically, in page 6, it is not clear how the matrix S is constructed, how we compute the ‘’appropriate’’ error matrix … A pseudo code would be much appreciated from the readers

In page 7, the authors want to assess the robustness of the proposed ISRF. What do you mean exactly by robustness (is it w.r.t. the noise level, a possible mismatch, a possible presence of outliers …), can you be more specific?

The caption of some figures is too short and needs more explanation, eg, Fig 1, Fig. 2,  Fig 9

Typo in eq 5 (=)

Summary

The authors present a novel method to estimate the Instrument Spectral Response Function (ISRF) of pushbroom-like spectrometers using a data driven approach. They propose to model the ISRF as sparse linear combination of an over-complete set (dictionary) of basis functions (atoms) derived from laboratory characterization measurements. The proposed method is applied to estimate the ISRF of one ground-based (Avantes) and 5 space-borne imaging spectrometers (OMI, TROPOMI, GOME-2, OCO-2 and MicroCarb) designed for remote sensing of the Earth's atmospheric composition. The results obtained for these spectrometers are compared to a state-of-the-art reference ISRF model (fit of a generalized Gaussian). The proposed new algorithm outperforms the reference method in all cases presented by the authors.

General Comments

Accurate post-launch ISRF estimation is a prerequisite for the delivery of accurate atmospheric remote sensing products and the proposed approach is a novel and creative contribution which should be of interest to the community. 

The presented investigation seems thorough and the main ideas are presented clearly.

Despite some minor issues in some equations, the mathematical basis is outlined sufficiently well.

The results shown generally support the authors' claims.

There are two major issues, which need to be addressed in my opinion:

Firstly, the authors claim (quite correctly), that a continuous post-launch monitoring of the ISRF is required, because instruments change over time (e.g. due to thermal breathing) and because the ISRF is usually scene dependent, because an inhomogeneous along-track illumination of the spectrometer entrance slit leads to a different effective ISRF than the one typically measured during on-ground pre-flight characterization with a homogeneous along track illumination. However, according to my understanding, the training data set used to compile the dictionary of atoms suggested by the authors consists exclusively of laboratory measurements obtained under homogeneous illumination conditions in the laboratory (except for a very limited study for the MicroCarb instrument in chapter 5.4.3). As such, I would expect these measurements to neither contain effects caused by thermal drift nor those caused by inhomogeneous along track illumination (within one pixel). As the usefulness of the proposed method greatly depends on its behavior under these typical conditions, I would recommend to add results obtained with synthetic data (e.g. by changing the FWHM of the ISRFs of the simulated spectra) to analyze the performance of the proposed algorithm under real-world conditions. 

Secondly, I have difficulties understanding, how exactly the training data set was obtained from the laboratory measurements and how the spectra used for validation were generated. Simply stating that the ISRFs were "obtained by spline interpolation" is not sufficient in my opinion without explaining in which dimension and at which nodes the interpolation was carried out. Additionally, I am not sure, whether the ISRFs chosen for the validation were part of the training data (modulo noise) or whether additional changes were introduced (e.g. those mentioned above, leading to shapes and FWHMs typically not encountered during on-ground CAL). As I would expect the behavior of the proposed algorithm to depend strongly on the choice of training data (see chapter 5.4.3), this is a critical issue which has to be addressed before publication in my opinion as one strength of the reference (Gaussian) model is its independence of prior knowledge in this regard. Consequently, a fair comparison should investigate these scenarios, which are of high practical importance.

Additionally, it is my impression, that the authors have a very comprehensive knowledge of the MicroCarb mission, which clearly emerges when discussing results related to this specific instrument. Maybe it is worth considering whether the suggested publication could be dedicated entirely to MicroCarb? Demonstrating that the proposed ISRF retrieval method works under a broad variety of conditions for this instrument alone would convince me of its value. I feel, that the studies presented for the other instruments add little substance to an already convincing demonstration in this regard. This might also help to shorten the manuscript and allow to add currently incomplete or ambiguous information.

Notation for the following sections:

[Page]/[Line(s)]: [Comment]

Specific Comments

2/4: The optical layout described here is basically a pushbroom spectrometer. In principle, other designs are in use as well (e.g. FTIR), which have to be treated differently. Maybe clarify to which spectrometer types your method applies. Additionally, the telescope creating the virtual image in the slit does not necessarily image the spectrally dispersed scene on the detector (MicroCarb is rather an exception than the rule in this regard). Maybe add a sketch showing the essential principle design you are investigating?

4/1-4: What exactly do you consider similar w.r.t. the ISRF and how do you formalize this mathematically? The following unnumbered equation seems to suggest that neighboring ISRFs are assumed to be equal (not just similar) inside a window of N_obs bands. Within an accuracy of 1% I would challenge this assumption for the instruments under consideration.

Additionally, could you simply solve this problem (a constant ISRF for multiple bands) using Fourier Transform / Wiener Filter without further assumption on the shape of the ISRF ?

Why do you need one equation per band (l) instead of a single equation / matrix including all bands simultaneously ?

4/24-26: It is not obvious to me, why an ISRF model of two (generalized) Gaussians with slightly shifted center wavelengths would be insufficient to model the displayed ISRF. Please elaborate. Are the shown ISRFs obtained under homogeneous illumination or are they part of the "ISRF Scene" examples shown in fig. 9? I would suggest to specify the band / channel / geom. pixel combination for all shown ISRFs to eliminate ambiguity.

Entire Chapter 4: I think more details regarding the generation of the reference spectra is required here:

1.  Which parameters are chosen for the radiative transfer simulations (trace gas concentration profiles, aerosols, scattering, surface albedo, ...) ?
2. Which SNR was assumed to generate the noise ? Does the SNR change with sensor and/or wavelength? (ref. section 5.4.1)
3. Except from adding noise, are you only using ISRFs included in the training data set (dictionary) or are you also creating reference spectra with ISRFs slightly narrower or wider than the training data (e.g. to simulate sharpening or blurring caused by thermal breathing of the instrument)? If so, how are those modeled ?
4. Do the simulations include the effect of (along-track) surface albedo inhomogeneity within one pixel ? If so, on which length scale / sampling distance ?
5. How exactly (along which dimension and at which sampling points) do you interpolate the ISRFs for each instrument?
6. Why does N_λ differ from the number of spectral bands for some instruments?
7. I think the description of each instrument in a separate sub-section does not add a lot of relevant information beyond what can be found in the cited literature. Have you considered summarizing the relevant information (number of spectral bands, source for the ISRFs, reference citation for the instrument/mission) in a table (and remove sections 4.2 to 4.7)? I think this might enhance clarity and readability of the manuscript.

10/second equation: Why are you including the entire sliding window into the error measure? Would it not be sufficient and more meaningful to compute the difference between measured spectrum s and simulated spectrum r \hat I at the center wavelengths λ_l of each channel l? The N_obs nodes left and right of λ_l are only used as computational aid as far as I understand and may e.g. increasingly suffer from boundary effects when approaching the limits of the window. Why do you include these effects into the error measure? In order to support a direct comparison with the assumed SNR, I think a relative measure would be desirable as well (ref table 1).

12/fig(3e): How is it possible, that the Gaussian (as special case of the super-Gaussian) fits the data that much better than the super-Gaussian here? Are you sure the fit converged properly?

How are the ISRFs in this figure normalized (not unit area) ? 

Chapter 5.2 and 5.3.x: Lacking the information listed above, I feel I cannot comment on the authors' claims here in a meaningful way, as it is of fundamental importance how the reference spectra were chosen and whether they are included in the training data or not. 

Chapter 5.4.1:

1. Does SNR=50 dB imply SNR=100 000? (which seems quite high to me). In this case I would expect a log₁₀ residual for a "perfect fit" around -5 if sufficiently many atoms are chosen, but the values in fig. 6 (b) are significantly higher. Could you elaborate on this?
2. Looking at table 1, it seems to me that the SPIRIT approach works significantly less effective for SNR = 100 (20 dB), which is not uncommon for many earth observing instruments. Could you comment on the usability of SPIRIT in these scenarios?

Chapter 5.4.3: Why do you choose across-track binning into these exact pixel groups? Are the ISRFs for the bands in these geometric regions similar?

How exactly are the ISRFs for inhomogeneous scenes obtained? Any simulation would require knowledge of the ISRFs for an inhomogeneously illuminated pixel/slit I presume. Is this knowledge inferred from measurements or simulations based on an optical instrument model? 

Technical Corrections

1/2-3:  optical elements "induce errors in the measurement"? Then why not leave them out ;-) ? Maybe rephrase ?

2/9: Is an ISRF not rather associated with a channel than a wavelength? The (center) wavelength of said channel can then be chosen based on (mean, max or median of) the ISRF?

2/14: ... "ISRF wavelength variations exceed this threshold" ? Which wavelength varies here? How does this affect the ISRF error budget ? 

2/17: I would argue, that all pushbroom instruments are susceptible to effective ISRF changes if the illumination varies within a pixel (along-track), as partial illumination of the spectrometer entrance slit is equivalent to a narrower slit and thus (usually) a smaller FWHM. Unless optically mitigated (e.g. by means of a slit homogenizer or optical fibers), this effect has to be taken into account in the error budget. Does the 1%-requirement include the associated uncertainties?

3/9: Technically each channel / geometric pixel combination has an individual ISRF. This ISRF can then be used to define a center wavelength for each channel of each geometric pixel. Have you considered associating each ISRF with the number (l) of a channel instead of the center wavelength? This might simplify the notation in many equations in my opinion.

3/eq(1): The convolution is usually defined over the entire space of real numbers. Also, the ISRF I is mirrored along the wavelength axes in this notation, as I(u) is the sensitivity to the wavelength λ_l - u, which might be unexpected for many readers. It has no practical effect on your results of course.

3/eq(2): I think a Δ is missing in front of the sum. Also: Does λ_l - nΔ equal λ_l-n ? If not: How do you choose Δ?

3/21: "A major difficulty with the inverse problem ...": Maybe also mention another very fundamental problem: Eq. 1 is a Fredholm equation, the solution of which is the classical example of an ill-posed problem. Additionally the constraint of a single measurement could be removed experimentally using e.g. the sun and a spectrally tunable on-board calibration source, albeit at extra cost.

4/eqs(3&4): Why x ∈ Δ (i.e. the sampling interval) ? Could you not use any real number for x ? (For most atmospheric spectrometers the sampling ratio is greater 2, so the FWHM exceeds one sampling interval in most cases.)

4/eq(5): I think there is one superfluous equation symbol following the summation symbol and the summation index n should probably occur somewhere in the equation (unless the summation is part of the higher dimensional 2-norm)?

10/5: "carbon" without trailing "e"

10/first equation: If the ISRFs are normalized to unit area, can the denominator have values different from one?

12/fig(3): The font is barely legible and my aging eyes can hardly discriminate the two LASSO variants. Please increase the size of these plots. The Avantes SVD/K-SVD fits seem to have linear segments, which I would not expect in a "real" ISRF. Are these linear interpolation artifacts? Maybe add markers at the computational nodes for clarity?

13/fig(4): Even at a zoom level of 190 % I can barely read this figure! Please increase the size of this figure and the font size.

20/fig(9): Please indicate for which band / channel / pixel combination these ISRFs are valid. 

20/fig(10): Do the figures on the left-hand side show a single pixel IFOV in along-track / across-track direction? Otherwise please indicate the pixel limits. How do these images enter into the ISRFs shown on the right? Are these ISRFs simulated or measured ? 

17/table 1: I think it would be helpful to indicate the expected minimum error for a given SNR. Considering an ideal Gaussian with synthetic noise at an assumed SNR of 1000 I would e.g. expect a mean residual error of approx. 0.1 %, but I do not observe a relation of this kind in the data. Could you elaborate on the relationship between SNR and normalized approximation error a little bit more in the text?

18/fig(7) & 19/fig(8): Considering the limited range of the error values, I would recommend a linear scale for the y-axis.

21/fig(11): This figure is also quite small. Maybe also consider a linear scale. Is it necessary to resolve differences smaller than the noise level?