The manuscript "Four-dimensional variational data assimilation with a sea-ice thickness emulator" by Durand et al presents an evaluation of a 4D-Var data assimilation framework using a data-driven sea ice thickness emulator of the neXtSIM sea ice model. The authors show how, through the emulator's back-propagation capabilities, sea ice thickness observations (both idealized and real) can be assimilated into the emulator using 4D-Var, effectively reducing the emulator bias. I very much enjoyed reading this manuscript and think it's a nice contribution to the literature. In fact, most of the comments I had noted down by the time of the discussion were then answered in the discussion, so thanks! My comments were overall minor, and I think the manuscript is almost ready for publication with a few small edits (see below).

General question regarding methodology

Could you just clarify something about the methodology for me. Are the EOFs used for the background covariance static over the course of the DA simulation? My concern early on was the ability of the EOF approach to capture flow-dependent processes, given the strong seasonal cycle of sea ice (you do mention this in the discussion). Is there some expectation that the minimization figures out which EOFs are most important and dynamically weights them (in time) according to w? I would be very interested to see how the approach compares to an Ensemble Kalman Filter (as you also say in the discussion).

Comments

L112:  I suggest adding a citation to show an example of where observations are typically log-normal. E.g Landy et al 2020.

L117 and elsewhere: change “In average,” to “On average,”

L129 - L132: Somewhere in this section it might be worth highlighting a recent paper (Nab et al. 2025) which quantified the effect on DA-derived analysis fields due to varying observational uncertainty on sea ice thickness measurements—Turns out to be quite sensitive.

Figures 3 and 7: Missing text in all labels

L258: Doesn’t the RMSE in Fig 5 peak in July? I guess the bias error peaks just before May and then rises again in December? Maybe changing L258 to “from Fig. 5 top” to make it clear which panel in Fig 5 we are looking at

L306-310 : Can you borrow some info from data-driven NWP models which retain sharpness by augmenting loss function

L350 : Might be worth highlighting here that there are ongoing developments in this space. For example Chen et al and Gregory et al both show ML-based approaches for deriving complete daily sea ice and ocean fields from satellite altimetry at 5 km grid resolution. Both of these approaches model the spatio-temporal covariance of daily fields, rather than simply averaging through time. Although these studies show sea ice freeboard, it is conceivable that daily sea ice thickness observations are on the horizon.

L400: I thought neXtSIM-F was initialized through nudging and not EnKF (L274/275)?

Appendix B: Can you quantify the time change in the 4D-var minimization when increasing the truncation index m? For example, on L324 you say it's 155 seconds for m=7000. What is the time if m is halved to 3500? I guess I'm wondering what is the cost-accuracy tradeoff.

References

Landy et al. Sea Ice Roughness Overlooked as a Key Source of Uncertainty in CryoSat‐2 Ice Freeboard Retrievals. JGR Oceans. 2020

Nab et al. Sensitivity to Sea Ice Thickness Parameters in a Coupled Ice‐Ocean Data Assimilation System. JAMES. 2025

Chen et al. Deep random features for scalable interpolation of spatiotemporal data. ICLR (arXiv). 2024

Gregory et al. Scalable interpolation of satellite altimetry data with probabilistic machine learning. Nature Communications. 2024

Please refer to the attachment.