General comments

The authors present a very interesting and promising study about spatial snow data assimilation for high-resolution simulations. The study shows how information from sparse snow depth observations can be used to improve spatially complete simulations obtained using a physically-based snow model at very high spatial resolution. The manuscript is easy to read and well-written, the results are clearly shown and discussed in depth. Overall, the paper is a strong contribution to existing literature on snow data assimilation since few studies have addressed the problem of propagating information from sites with observations to locations lacking measurements. My comments on the manuscript are only minor, and listed below.

Specific comments

Abstract: The study would benefit from a shorter and more concise abstract.

L 150-152: The sentence is difficult to read. Please reformulate.

L 173: Is optimal interpolation only occasionally used in operational data assimilation? Is not many very important weather forecasting models using this method, such as ECMWF?

ECMWF: IFS Documentation CY45R1 – Part II: Data assimilation, in: IFS Documentation CY45R1, IFS Documentation, ECMWF, https://www.ecmwf.int/en/elibrary/80893-ifs-documentation-cy45r1-part-ii-data-assimilation (last access: 16 November 2022), 2018.

Data and methods: How were the snowpack layers in FSM2 updated during the assimilation experiments? If the assimilation step adjusts the total depth of the snowpack, also the number of modelled snow layers may change. How was this handled? Please clarify.

Data and methods: Please specify the total computation time of running one assimilation experiment, and also add information about the computer resources that were used. This is interesting for potential applications of the methods developed in this study.

Table 1: Would it be possible and more visually appealing to show these metrics as time series plots?

Technical comments

L 119: A space is missing before the reference.

L190: Error in spelling. “Spare” to “sparse”.

L405: Is the sentence correct? (“which shows is nearly flat”)

L 518: “e.g.” misplaced.

Spatio-temporal information propagation using sparse observations in hyper-resolution ensemble-based snow data assimilation

HESS, Alonso-Gonzalez et al. 2023

This paper shows how sparse high-resolution snow depth observations can be integrated into a hyper-resolution model that is by itself producing rather homogenous snow depths over a small catchment in the Pyrenees. The assimilated observations are a subsample taken from some drone images during the peak and melt season of snow. Three different prior (spatial, non-diagonal) error covariance matrices are constructed to use in a 3D EnKF scheme and to allow the propagation of information in space. The novelty lays in defining such error covariance matrices, using other measures than the distance between grid cells, i.e. e.g. using topography or observed (as opposed to similated) snow pack similarity. The strength of this paper is in its technical innovation and rigor, and the paper is very well written -  a pleasure to read. Some details are not entirely clear and questions for clarification are listed below, together with minor suggestions to improve the presentation of the paper.

1. The ensemble generation is at the heart of this paper and 3 methodological points are a bit unclear:
   • the perturbations are applied (as usual) to meteorological input, but not to snow depth or any snow state variable. Do you maintain enough spread after the assimilation events this way? Most often, some extra state perturbation is desirable, either to reflect some parameter uncertainty (if the parameters were not perturbed) or just to do some covariance inflation. In addition, it could actually help to impose the right error structure in the prior state ensemble in your case – see next comment.
   • the prior state error correlations are technically derived from the ensemble snow depth members, which in turn were obtained by propagating ensembles of precipitation and temperature through the FSM2. Right? The newly developed error structures are thus imprinted in the forcings, not directly in the state (only after propagation through the model). Yet, the forcings are nonlinearly transformed to snow depth, and on top of that the error correlations between the various forcings is assumed to not exist, meaning that the different forcing error structures will interact and might destroy or amplify each other locally. Furthermore, I did not see (may have overlooked) any temporal autocorrelation in the error structures. Therefore, I do not really understand how the resulting snow depth error correlation would have the same error structure as that of its input. Have you verified if the diagnosed ensemble state error covariance matrix (truly derived from the ensembles) effectively has the structure that it was meant to have? (From Fig 7, I get that the snow depth pattern itself (not the errors) indeed was reproduced with Exp III DA.)
   • Exp I, II or III: the distance, topography or similarity in snow depth are assigned to error structures in the meteorological input. Could they not equally originate from error structures in model parameters instead?
2. Perturbing precipitation w/ a logit-normal distribution and a mean of -1.6 seems to introduce a bias in precipitation. Is there a reason for this choice?
3. FSM2 is run with MicroMet data. Can you explicitly say how much variability you expect at the scale of this small catchment? I would expect almost none, perhaps truly nothing at all for precipitation, but I wonder if there is any (other than some radiation variability mentioned in the results). Is there a reason why the wind distribution is not turned on?
4. The text mentions both ensemble Kalman filter and smoother (e.g. L. 244 and further). The time dimension (smoothing) is unclear to me, if it is applied. Which exact technique is applied?
5. Not sure about the technical details for the computational implementation: could you not use halos around a good radius of influence to parallelize in space as is done in 3D global DA systems?
6. Evaluation: it is a pity that there are no in situ data available of any sort, but I agree that your setup does its job for the application at hand. Would be nice to try to assimilate e.g. lidar (e.g. ASO?) or radar (Sentinel1?) data and have drone data as reference data.

Details:

• The abstract is long, and the first 7 lines can be removed. This is a rather technical paper, and there is no need for an extensive introduction in the abstract.
• L. 119: space before reference
• L. 215: should d be d_ij in line w/ Eq. 7, and should d_ij on L. 223 be redefined as some D matrix? I think d_ij refers to a distance for a single pair of grid cells.
• L. 296: I agree that it is fine to do this; we also call that the use of statistical “signatures”.
• Algorithm 1 box - line 13 bis: Y(i) is not defined, should it just be y(i)?
• L. 411: typo “horizontal”
• Fig 8: units of snow volume are right for an area of 55 ha, but I would just write them in average snow depth [m].