The paper presents an interesting and relevant approach to error correction in river discharge forecasts, particularly for ungauged locations. Using data assimilation (DA) in a post-processing environment is appreciated. The authors employ the Localized Ensemble Transform Kalman Filter (LETKF) and state augmentation and provide tests on a large-scale, operational forecasting system (EFAS). In general, the research questions are well defined, particularly regarding the feasibility of DA in a post-processing framework.

However, I really struggled while reading the paper. I find it really hard to follow. While the method is interesting and promising, I have major concerns regarding the clarity of presentation and the justification of assumptions. There are several ad-hoc methodological choices such as inflation which feels like a workaround rather than a principled solution. I think there are also significant issues relating to the handling of the ensemble spread. Please find more details below. I recommend Major Revision.

1. The paper is difficult to follow due to dense mathematical notation and long, complex sentences. There is an overuse of jargon without sufficient introductory explanation for a broader hydrology audience. The structure could be more concise, with a clearer division between methodology and results.
2. The state augmentation approach is described in a way that makes the approach seem unnecessarily complex. The assumption of constant error propagation is not well justified. Also, related to this, the use of precomputed model outputs instead of an evolving state might introduce additional errors, which are not sufficiently discussed.
3. Inflation: (i) I find the inflation method to be heuristic with little to no mathematical rigor. For instance, the assumption that the hindcast variance is a proxy for error growth does not account for potential biases in the raw ensemble itself. (ii) Unlike RTPP, the proposed inflation blends analysis and “estimated” perturbation information without explicitly evolving them. What motivates such an approach? (iii) The inflation parameter, alpha, is computed from a 3 steps-average of the hindcast (eq. 28). Why this choice is appropriate? I recommend testing with different alpha values through sensitivity experiments. (iv) If inflation is not localized along the network, that should be clarified and justified.
4. Spread: It’s clear that the method tends to overcorrect at short lead times but yields underconfident ensembles at longer lead times (as shown in Figs. 4, 5). In general, one expects the ensemble spread to accurately represent the forecast uncertainty but the issues the authors face could be related to the ad-hoc inflation. I would also note that real hydrological errors are dynamic, but the paper assumes the errors to remain constant between cycles. A flow-dependent error propagation model and perhaps an adaptive inflation approach could address these issues.
5. Localization: The choice of the length scale (262 km) should be better justified. There is no sensitivity analysis to determine whether this choice is optimal or whether smaller/broader radius would improve the results. Also tangential to this, the authors need to revisit the equal error correction assumption in upstream and downstream locations. Overall, upstream locations are less dependent on distant downstream observations. Obviously, downstream conditions are often affected by accumulating upstream flows.
6. Figures: The figures are well-intended but too dense and overloaded with information, making them difficult to interpret and extract keys findings. I suggest splitting the complex ones (e.g., Figs. 3, 4, 6) and definitely simplify the annotations.

Other comments:

• Line 6: “Error vector for each ensemble members” seems vague and unclear.
• Line 12: The term “proxy” could mean a lot of different things. Clarify the nature of updates, whether that’s real data assimilation experiment or an OSSE.
• Line 160: I would use “cycled” instead of “iterated”
• Line 160: Replace “at each timestep” with “at each observation time”
• Line 178: Replace “weights” with “weighs”
• There are too many “see section xxx”. This made navigation frustrating; I kept going back and forth. Consider restructuring for better flow.
• The word “improved” is overused in my opinion. Consider other synonyms “enhanced”, “refined”, …
• Explain technical terms more clearly, for instance “spatiotemporal consistency”

Summary

This is an interesting and well-written paper attempting to address one of the thornier problems in hydrology: error propagation to ungauged areas. The authors combine data assimilation with several other techniques and apply these to a well established ensemble streamflow forecasting system (EFAS) to propagate errors to ungauged areas. The performance of their methods is assessed under spatial cross-validation, and the methods exhibit modest success. I commend the authors for attempting this difficult problem, and the paper is generally well presented. My two major issues with the paper are as follows:

1) The paper is too long, in particular the description of the methods. I appreciate the authors' diligence in describing their methods in sufficient detail to replicate them, but it makes the overall paper difficult to read. I suggest the authors move significant portions of the methods to appendices, and highlight those components that are most novel.

2) I am a bit concerned about how applicable these methods are outside the case study attempted (see specific comments below). For example, the use of a very large catchment is likely to allow the authors to make simplifying assumptions such as that residuals will be normally distributed, or that errors can be characterised using a 10-day window. I would be interested in some discussion of how generalisable these methods are.

Specific comments

L58 "ensemble Kalman Filters are common data assimilation methods for hydrological applications" this is true for hydrological research, but (to me at least) it remains a curiosity as to why data assimiliation within hydrological models - including with ensemble Kalman Filters - remains to my knowledge quite rare in operational streamflow forecasting systems.

L90 "Hydrological ensemble forecasts consist of N potential realizations referred to as ensemble members" I think it would be good to state explicitly which variable(s) you are discussing here, as it wasn't clear to me - I'm assuming streamflow (or runoff, as it's on a grid?)?

L107 I would have thought with a strongly skewed (and potentially zero bounded) variable like streamflow (assuming that is what is being assimilated?), an additive error only generally holds after a normalising transformation has been applied (and, if applicable, zero values have been dealt with).

L112 Similar to the above criticism at L107, Equation 6 appears to assume that errors are normal and homoscedastic. If my understanding of what is being assimilated is correct, this is highly unlikely to hold for streamflow, for which residuals are almost always non-normal and heteroscedastic. See e.g. Smith et al. 2015, among many others.

L145 "we adopt the common assumption that the error is constant" I would not have said this is common. I would say it's much more common to use autoregressive models (often AR1) to describe the autocorrelation between residuals in streamflow. I understand why this is a pragmatic simplification, but errors often do change with lead time as the value of forecast information decays.

L149 "define the propagation" I'm not sure what 'propagation' means here, given the error is assumed constant in time. Can the authors clarify? Nevermind - the authors do this in Section 3.2! The authors may want to flag that the explanation for this is coming.

L180 "(see Eqs. (8) and (9) in Bell et al., 2004)" I feel that if the authors need to specify equations from another study to describe these methods, the equations should be present in the paper (in an appendix is fine) - especially Eq (9) of Bell et al. which the authors later describe as 'key' to the method. (Unless they are included later?)

Figure 1 - this is a really nice, clarifying figure.

L223 "We enforce non-negativity by further adjusting the error ensemble members after the LETKF update step (Fig 1)." This indicates that zero values are present in output state, indicating that errors are not continuously distributed. I realise not everyone handles zeros, but it would be good to acknowledge the limitation of this assumption (as noted above). 

L265 "Eq. 4.10 in Gaspari and Cohn (1999)" - I think the authors should include this equation, as well as discussing (briefly) why they thought the form of this equation appropriate for this task. The regionalisation of errors is in my view the major contribution of the paper.

L272 "We propose instead for the localisation length scale to be defined as the maximum distance between any grid point and its closest observation." This seems like a sensible choice.

L325 "(here 10 days)" This is a long period over which to assess an error - some use periods of this length for bias correction (e.g. Bennett et al. 2021). I'm assuming this really only works for larger catchments where rivers have slower varying errors; I would have thought for small headwater gauges shorter periods would be more appropriate. It also explains why errors are assumed not to vary with lead time, above. This is all fine, but the authors may wish to mention this in their discussion.

L408 "we assume that the observation errors from different gauge stations are uncorrelated" I'm not suggesting a change here, and I think this is a reasonable suggestion without additional information. But I suspect the long-range nature of the errors (a 10 day period) may undermine the assumption somewhat. I'm also curious what happens when errors are propagated in space: what happens when you get a point equidistant (or close to equidistant) from two gauges, and the errors from the two gauges interact in some way (e.g. cancel each other, or sum).

L438 "forecast mean is decreased by the proposed method we use the Normalised Mean Absolute Error" It's preferable to apply measures of absolute error to the ensemble median. See, e.g., Taggart (2022).

L526 "However, this assumption is necessary to propagate the hindcast to the next time step without the use of a hydrogical model (Section 3.1)." Perhaps, but one application of ensemble predictions is to sum ensemble members through time (e.g. to assess cumulative inflows to reservoirs). From this figure, it seems this would result in highly unreliable accumulations. This may not be an application of EFAS (I don't know), but if the method is to have more general applicability this is a serious weakness.

L660 "Future work could look into applying anamorphosis to make the ensemble distribution more Gaussian-like" I'm not familiar with the concept of anamorphosis, but a conventional way of doing this is to use normalising transformations, of which many are available for hydrological variables.

Typos etc. 

L152 "of the precomputed the hindcast ensemble" should be "of the precomputed hindcast ensemble"

L212 "...for the hindcast analysis state were the component updated." There appears to be something wrong/missing in this sentence.

L306 "where k is the the current timesteps" - should be "timestep"

L311 "An inflation values of 0 implies" - should be "value"

L450 "we analyze the analysis increments of the mean" this might be reworded for clarity (i.e. avoiding using 'analyse/analysis' to mean two different things)

References

Bennett J.C., Robertson D.E., Wang Q.J., Li M. and Perraud J.-M. (2021) "Propagating reliable estimates of hydrological forecast uncertainty to many lead times", Journal of Hydrology, 126798. doi: https://doi.org/10.1016/j.jhydrol.2021.126798

Smith T., Marshall L. and Sharma A. (2015) "Modeling residual hydrologic errors with Bayesian inference", Journal of Hydrology, 528: 29-37. doi: https://doi.org/10.1016/j.jhydrol.2015.05.051

Taggart R. (2022) "Evaluation of point forecasts for extreme events using consistent scoring functions", Quarterly Journal of the Royal Meteorological Society, 148: 742, 306-320. doi: https://doi.org/https://doi.org/10.1002/qj.4206