Thank you for your review and for your comments. The issue of length is a good point and I will try to make any future work shorter. However it would be hard to cut anything out of the current manuscript at this stage; many aspects are already covered with less detail than would be ideal.

On the minor points, I will address all the typos in a revised manuscript, and I am very grateful for the pointer to the historic work with the T-28 storm-penetrating aircraft. The recent work using this dataset to describe hail PSDs (reaching up to at least 5 cm, Field et al., 2019) would clearly be a good future alternative for the PSD representing the “convective snow” category, rather than repurposing PSDs intended for other hydrometeor types, as done in the current work. In the revised manuscript I will adapt the relevant discussion on lines 228 - 235 to reflect this, and to acknowledge that some direct observations inside convective cores do exist; also perhaps mentioning the recent balloon-borne measurements of Waugh et al. (2020) if space permits. I will also try to fit this point into the discussion or conclusion of the revised manuscript, if at all possible.

References

Field, P.R., Heymsfield, A.J., Detwiler, A.G. and Wilkinson, J.M., 2019. Normalized hail particle size distributions from the T-28 storm-penetrating aircraft. Journal of Applied Meteorology and Climatology, 58(2), pp.231-245.

Waugh, S.M., Ziegler, C.L. and MacGorman, D.R., 2020. In Situ Microphysical Observations of a Multicell Storm Using a Balloon‐Borne Video Disdrometer During Deep Convective Clouds and Chemistry. Journal of Geophysical Research: Atmospheres, 125(8), p.e2020JD032394.

Thank you for your review and for your comments. The main issue of supercooled liquid water (SLW) in deep convection is an important point and I appreciate your introduction to the literature on this. As suggested, I will add some discussion, likely in the conclusion of the revised manuscript. The basic point is well taken that the existence of lightning in deep convection, and the rimed particles that generate it, requires significant mass of SLW above the freezing level, likely both raindrops and cloud liquid (e.g. Kumjian et al., 2012; Fuchs et al., 2018). The IFS model broadly does not represent SLW within convection. This issue has already come up, for example in the need to detrain a higher proportion of SLW into stratiform cloud in order to correctly represent shallow convection in cold air outbreaks (Forbes et al., 2016). It would be possible to use the parameter estimation framework to explore alternative partitioning between ice and liquid, as you propose in minor comment 4, however I would like to leave such a substantial additional piece of work to a future study. 

My guess is that properly representing SLW will make the representation of strong scattering brightness temperature depressions in deep convection even harder. As illustrated in Geer et al. (2021), above 100 GHz, the bulk optical properties of rain provide a single scattering albedo (SSA) of around 0.5, as compared to around 0.95 from completely frozen particles. This means that liquid droplets above the freezing level will likely provide strong absorption and emission, so that they will be able to increase the brightness temperature of the cloud towards the physical temperature (e.g. 200K) and away from the scattering-dominated brightness temperature of the cloud (e.g. 100K or less). These mechanisms are described in a little more detail in Geer et al. (2021). I will try to make this point compactly in the conclusion of the revised paper.

On the minor points:

1) Noise in observed radiances is not considered at any point in this work. Currently, noise is irrelevant to the modelling of cloud and precipitation-affected radiances, where the modelling errors may be at least 10 K in brightness temperature, and the instrument noise may be around 0.5 K (e.g. MHS, https://space.oscar.wmo.int/instruments/view/mhs ). This point is explored in more detail in Geer and Bauer (2010); I will see if it is possible to mention this in the description of the cost function in section 4.2 of the revised manuscript; certainly there is no need to represent the true observation error explicitly in the parameter estimation.

2) On the mixing ratio adjustment, yes this is done simultaneously at all altitudes. This point will be clarified in the revised manuscript.

3) On the validity of the assumption of linearly additive perturbations, yes, the larger errors of 20 K are predominantly in convective areas over land (as judged by plotting them on a map, for example). These are the locations where the revision to the cloud overlap has had most effect on the brightness temperatures, seen also in Fig. 8 in the submitted manuscript. Another way to quantify these errors is as a fractional error: by this measure, the assumption of linear additive perturbations underestimates the larger TB (>10 K) increases by typically around 30%. Also, on revisiting these results, I discovered one mistake in the text: the overall standard deviation of the differences is 0.96 K, not 0.2 K as stated. These numbers and additional discussion will be added to the revised manuscript. Another way to judge the validity of the assumption is to compare maps of the mean and skewness measures that are used in the cost function; these are nearly identical to a visual inspection (based on figures equivalent to Figure 7 in the submitted manuscript). The assumption is not perfect, but it was vital as a way of making this work practically achievable. Based on this discussion, I would also like to add a line or two to the conclusion to say that future work should include further testing of the validity of this assumption; it should not be just relied upon by future investigators. 

4) Yes, as mentioned above, it would be possible to explore alternative partitioning of the ice and liquid in convection. Some discussion around the importance of supercooled liquid water in convection will be added, probably in the conclusion of the revised manuscript.

5) This is a good suggestion to be more precise about the model resolution and the satellite field of view in the first sentence of the abstract; I will include it in the abstract of my revised submission (word limit permitting).

References

Forbes, R., Geer, A., Lonitz, K., and Ahlgrimm, M.: Reducing systematic errors in cold-air outbreaks, ECMWF newsletter, pp. 17–22, 2016. 

Fuchs, B. R., & Rutledge, S. A. (2018). Investigation of lightning flash locations in isolated convection using LMA observations. Journal of Geophysical Research: Atmospheres, 123, 6158–6174. https://doi.org/10.1002/2017JD027569

Geer, A. J. and Bauer, P.: Enhanced use of all-sky microwave observations sensitive to water vapour, cloud and precipitation, Tech. Memo. 620, ECMWF, Reading, UK, 2010.

Geer, A. J., Bauer, P., Lonitz, K., Barlakas, V., Eriksson, P., Mendrok, J., Doherty, A., Hocking, J., and Chambon, P.: Hydrometeor optical properties for microwave and sub-mm radiative transfer in RTTOV v13.0, Geophys. Mod. Dev., submitted, 2021.

Kumjian, M.R., Ganson, S.M. and Ryzhkov, A.V., 2012. Freezing of raindrops in deep convective updrafts: A microphysical and polarimetric model. Journal of the atmospheric sciences, 69(12), pp.3471-3490.

Thank you for your review and for your comments. The point about the length of the paper, consistent with reviewer 1, is taken onboard for future work, but it would be hard to adjust the manuscript now without cutting important details. Responding to the three concerns:

1) The point is taken that it is hard to get a sense of the improvement going from the control to the improved configurations, based on the global measures of cost. However, Figure 5 is intended to show the shape of the cost function and to illustrate the technique, whereas Figures 7 and 8 are already provided to try to illustrate the changes, such as improvements in simulating extratropical convection over land, and slightly worse fits in tropical convection over land. I will try to add the PDF of simulated and observed brightness temperatures to the revised manuscript, as suggested. However, although it shows improvement, this is not very large. The problem is that there is not a big overall improvement in fits to observations (and I hope the manuscript does not imply this). The reduction in cost from control to the “best” configuration is only from 0.925 to 0.908, as already mentioned on lines 391-392 of the submitted manuscript (note that the “final” configuration, from Table 7 is a little separate from the main part of this work so I will concentrate on the “best” configuration from Table 4, which is what Figure 5 refers to). 

It is also worth restating that the achievement of the global parameter search was not so much to provide an overall improvement in fit to observations but to provide a more physically realistic underpinning for the simulations that should support future developments (e.g. the more physical cloud overlap, the use of a non-spherical particle for cloud ice for the first time, the use of ARTS database particles to support future sub-mm work)

2) I acknowledge that the “snow mixing ratio” dimension is not a physically-justified way to improve the results. However, it was never intended to build a fudge factor into the upgraded version of the forward model, but rather to use this as a straightforward way to investigate a number of interesting things: (i) it underpins Figure 9, which shows the stability of the results to potentially large errors in the modelled convective mixing ratio; (ii) it is also helpful in the situation-dependent parameter search (Table 6) and discussion (section 6.1). Here, the need to increase it in tropical convective areas to better fit the observations might indicate an error in the modelled convective mixing ratio, but it might also point to a need for broader convective cores, thicker anvils, or bigger and/or more scattering particles, as discussed in the text. Hence, the “snow mixing ratio” dimension is an important part of the work and I would like to keep it. However, its purpose could be introduced better in section 3.2, where I will try to add a few sentences in the revised manuscript.

3) This comment suggests that one way to improve the representation of convective anvils could be to treat them as a microphysically distinct cloud type, separate from the large-scale ice cloud generated by frontal uplift; hence one practical solution could be to treat anvil cloud as a separate hydrometeor type. This is a good suggestion and will be added to the conclusion of the revised manuscript.