Review of the manuscript  "Improved retrieval of SO2 plume height from TROPOMI using an iterative Covariance-Based Retrieval Algorithm" by They et al.

The manuscript presents a new and appealing method to retrieve the SO2 plume height from TROPOMI UV measurements even for low SO2 VCDs. The paper is very well written and I suggest publication after minor revision.

The suggested covariance-based method is an extension of the existing COBRA algorithm that has been developed by the authors to retrieve the SO2 SCD from TROPOMI measurements and uses a method that was so far only used for IASI SO2 retrievals. It allows for the retrieval of SO2 LH even for SO2 VCDs as low as 5DU with a precision better than 2km.

I have only a few minor points, which I would like the authors to address in a revision of the manuscript:

Sect. 2.1:

•  You are using the Bogumil et al 2003 SO2 cross-section for LIDORT. Have you considered using improved SO2 XS by Birk et al from the ESA IAS SEOM project?
• Since you apply an I0 correction to the SO2 cross-section, please mention which value you have used. Could you give numbers of how strong the effect is when (not) using the I0 correction? Does it have an effect for low SO2 VCDs?
• Since you are using LIDORT to calculate SO2 OD spectra, why do you also take into account O3 absorption? Maybe I am missing something here, but you are only interested in SO2 OD spectra itself, right?

Table 1:

• Is there a reason why you have chosen a coarser SO2 LH grid at high altitude for generating your LUT? What is the effect when using a finer grid?
• Is there a reason why you only extend the LH grid until 25km and not higher, e.g. up to 30km?
• At the nadir point of the TROPOMI swath, the RAA shows a strong jump by up to 90° from one pixel to the neighboring pixel along the scanline. Does this cause problems/jumps in the retrieved SO2 LH/VCD between pixels in the center of the swath since the OD is interpolated at strongly different RAAs?

Sect 2.2: When you construct the covariance matrix S, how is the whole retrieval affected by clouds? In detail, when the SO2-free spectra contain a huge number of cloudy pixels, whereas the retrieved SO2 pixels are cloud-free (or vice-versa), is the covariance matrix still well-posed? The same applies to ash clouds in the SO2-free pixels used to construct S. For these cases I would like to see some more discussion on the effect of the SO2 LH/VCD retrieval, if possible

Technical corrections:

Sect. 3.1, Acknowlegdements and Reference sections  show some strange bold uppercase 'K' letters (at least in my pdf viewer). Please correct

Reference to Koukouli et al 2021 is officially published: Atmos. Chem. Phys., 22, 5665–5683, 2022, https://doi.org/10.5194/acp-22-5665-2022. Please update the reference

The manuscript of Theys et al. 2022 "Improved retrieval of SO₂ plume height from TROPOMI using an iterative Covariance-Based Retrieval Algorithm" presents an interesting new retrieval approach to retrieve both SO2 VCD and LH based on UV Earthshine spectra.

Although the paper is clearly written, I am not sure to understand how the retrieval itself works:

In Section 2, equation 3 you write that x_i is the state vector representing VCD_i and LH_i and k_i contain the Jacobians. So is k_i in the end a 2d vector containing the two Jacobians as a function of wavelength that you multiply with your covariance matrix S?

How dependent are your results on the apriori VCD_0 and LH @7km? I.e. what happens if this initial estimate is completely wrong and the plume is at 20km in reality (with much lower true VCD_0)?

In your construction of the covariance matrix S you write that you follow the approach of your Theys 2021 paper and filter out pixels with VCD > 2.5 x VCD SNR? Do you also put an upper limit of the LH SNR to filter out pixels?

The study ``Improved retrieval of SO2 plume height from TROPOMI using an iterative Covariance-Based Retrieval Algorithm'' by Theys et al. introduces an optimal estimation retrieval algorithm to retrieve SO₂ column and layer height of volcanic injections from nadir measurements between 309 and 329 nm by TROPOMI. First, the retrieval is described and its theoretical uncertainties are assessed. Then examples of the retrieval for various volcanic plumes are presented. The layer heights are verified by comparison with IASI measurements and reconstructed plume heights from a backward trajectory approach. In general I found this paper well structured and well written. Hence I recommend this paper for publication after addressing the minor comments below.

General comments:

This paper contains many abbreviations, which often hamper the reading flow (e.g. ``The LER albedo is retrieved by matching the measured mean radiance to a LUT of radiances (generated in parallel to the SO2 OD LUT), and which depends on SZA, VZA, RAA, surface height and albedo, with the same grid definition as in Table 1.``). I suggest using less abbreviations, i.e. write out LH, LER, LUT, OD, VCD.

Please note, usually it is called a volcanic ``injection'' if a volcanic eruption reaches the upper troposphere and stratosphere. Volcanic ``emission'' is rather used for low altitude degassing. Please consider replacing ``emit, emission,...'' by ``inject, injection,...'' (e.g. p2ll6,9,11).

For the setup of the LUTs the US standard atmosphere was used. This is rather representative of midlatitude conditions. Could this have a (negative) impact on the retrievals of plume heights close to the tropical tropopause? Did you perform sensitivity tests for tropical, midlatitude, and polar atmospheric conditions?

Specific comments:

p2l4: Please specify what ``difficult observation conditions'' means.

p6l6: Please explain ``Lambertian Equivalent Reflectivity'' or add a reference.

p6l7: ``More details are given below.'' Please specify details on what are given? Simulation setup? Simulation output parameters?

p6l13: Please add one sentence describing the ``so-called solar I₀ effect''.

p6l14-15: It is not clear to me if 450 refers to the across-track pixel position or to the ISRF parameters.

p6l30: Why does the LER approach work only well in principle? How is it in practice? Please provide a reference.

p8l21: Is there a default value for VCD0? Which one?

p9l9: Is minimum height + 1 km = surface height 1 km?

p9l10: Is maximum height - 1 km = 24 km?

p9l18: How much is 300 pixels roughly in km?

p9l20-22: Is there an upper limit in terms of distance to the central pixel?

p10: How did you predefine the Jacobians and the covariance matrix?

Figure 2: Is this the ``sweet spot'' configuration where the layer height error is smallest for low DU? For which surface type (i.e. ocean, land, ice) is an albedo of 5% representative? In the text you mention that you tested the impact of the albedo and SZA. What about the impact of surface height, ozone, RAA and VZA?

p13l22: What are S-5P FP_ILM results?

p13l23-24: How do you know from TROPOMI data that the plume had multiple layers?

Fig. 3: Could you please add contours to DU>20 to all panels or show contours for DU>5 to the DOAS panel? I see the point that you want to compare the layer heights, but I find the different shapes of the plumes confusing. If I understood correctly the DOAS method can retrieve lower DU, but has a larger uncertainty for <20 DU. Also, please add the orbit footprint to make the gap between two orbits better visible, especially in the region around the volcano.

Fig. 4: Please add the orbit footprint here too. Also I think it makes sense to cut Fig b and d at 20 km, as the color bar in a and b is also cut at 20 km.

p17l7: Could it be that the larger difference to CALIOP aerosol height is due to the fact that for CALIOP the aerosol layer top altitude is compared to the TROPOMI layer height that is more representative of the center of the SO₂ layer?

Fig. 5: See general comment on using only the US standard atmosphere instead of representative profiles for the tropics and midlatitudes. How well do the TROPOMI and IASI results compare to CALIPSO aerosol heights?

p18l9: Please specify ``difficult conditions''.

p19l10-15: The method of using backward trajectories from satellite observations of plumes back to the volcano has been also used in other studies, e.g. Wu et al., 2017, Cai et al. 2022.

p19l20: Does PlumeTraj only rely on position and time, or does it also consider the DU at each pixel to weight the reconstructed heights (e.g. high weight for large DU, low weight for small DU)?

p19l25-29: Could this difference in plume height at the plume edges be also due to low DUs at the edges? How can you tell there is no underlying low altitude plume? Which altitude range is considered by PlumeTraj? Would this ``edge-effect'' still be there if you used e.g. 10 DU as a selection criterion for the layer height retrieval?

Fig. 6: According to Fig. 7 Fig. 6 shows the best case. What does a bad case, e.g. 3 March or 4 March look like? Is there also an ``edge-effect'' visible?

p22l12: How old is ``old''? Days, weeks?

Fig. 8: The figure quality needs to be improved. I don't understand why there are lines for TROPOMI.

p25l8-10: How were the deviations calculated? On a daily basis? There are lines for TROPOMI and points for IASI data. Sometimes there are data points for IASI, but not TROPOMI data.

Technical comments:

p2l4: expect -> except

p3l27: remove ``obviously''

p5l16: x_l -> x_i

p6l28: aerosols layer -> aerosol layers

p7l2: aerosols -> aerosol particles

p7l5: Please introduce the abbreviation OCRA/ROCINN CRB.

p7l9: Please introduce the abbreviation RAA.

p13l26: Please introduce the abbreviation CrIS.

p13l31: sensors -> measurements

p17l20: observed a -> observed an

Fig. 5 caption: 2018 -> 2019

p22l29: Example -> Examples

References:

Wu et al., 2017; https://doi.org/10.5194/acp-17-13439-2017

Cai et al. 2022; https://doi.org/10.5194/acp-22-6787-2022

This paper describes a new algorithm using TROPOMI UV measurements to retrieve effective volcanic SO2 height. This is an extension of the COBRA SO2 algorithm previously described by the authors, with the addition of SO2 height (in addition to SO2 VCD) to the retrieved state vector. The authors presented several examples of the TROPOMI SO2 height retrievals, including both degassing volcanoes and large explosive eruptions (Raikoke 2019). Comparisons with thermal IR SO2 height retrievals (IASI), CALIPSO lidar measurements, and trajectory-based height estimates show generally good agreement between TROPOMI retrieved SO2 heights and those datasets, even for relatively small SO2 loading. Overall, the paper is well written and the results are quite impressive for a rather challenging problem. The SO2 height retrievals also can have a host of different applications and should be of interest to the readers of AMT. I'd recommend that the paper be accepted for publication after minor revisions.

Specific comments

My main concern is that the nodes in the LUT (Table 2) appears to be relatively coarse especially for larger SO2 amounts. This may lead to interpolation errors at shorter wavelengths. Can the authors compare the interpolated Jacobians with model calculated ones to assess the uncertainty from interpolation?

Page 9, lines 8-12: would an out-of-bound value in VCD or SO2 height generally lead to failed retrieval?

Page 9, lines 22-23 - is the SZA limit also applied to the selection of pixels to construct the covariance matrix?

Page 10, line 22: “referred as” should be “referred to as”.

Figure 3: there appears to be more structure in Figure 3a LUT-COBRA results - is there an explanation for this?

Page 17, line 15: suggest changing “as high as” to “as large as” to avoid confusion.

Page 19, lines 27-29: there appears to be some striping in Figure 4 even for pixels that are near the center of the plume. Could this have anything to do with the initial conditions based on the operational SO2 product?

Figure 7: it would be useful to have some statistics of the comparison (e.g., r, RMSE).