We are grateful for the comments from the first reviewer which we find constructive and useful, and hope that we have managed to revise our paper to meet any critical feedback. More detailed account is provided below. The reviewer comments are repeated here in italics.

This manuscript is addressing an important topic and is well-written. However, I would recommend major revisions to give more details on the main body of the manuscript rather than in the SI. I think that the manuscript should be rewritten for the broad readership of HESS, or alternatively be submitted to a more specialized journal."

Thank you for this comment. The paper is intended for the broad readership of HESS, and we hope the revised manuscript will make it accessible to most readers.

"The essence of the approach presented is to build a statistical model that relates large-scale information (EA5 or CMIP6) to station-level data. Then, this model is used to estimate the future climate at stations locations, based on CMIP6. I am not sure this approach can be termed downscaling, as in my mind downscaling is the passage of data from a coarse grid to a finer grid. Here the correct term, in my understanding, would be something like change of support. In particular, the data are only produced at the stations locations, meaning that it is less effective in areas with a lower station density (such as northern Norway/Finland)."

Thank you for this comment - there are different ways of thinking of downscaling, and the revised paper will explain these different positions, as the passage to a finer grid is not the only correct one. It’s also beneficial with an increased awareness about what we mean by downscaling, and there are many examples of the use of downscaling in the scientific literature, some which are in line with our definition. We interpret all station data as a quantification of local small-scale conditions and there are some cases where global climate model simulations have been downscaled to a single point.

"Moreover, the details of the methodology are very condensed and not comprehensible for a reader that is not already familiar with it. The method is summarized very shortly in section 2.2, and mostly referred to through citations to other papers by the authors, with whom the reader is not necessarily familiar. Therefore, it is difficult to grasp it by only reading the manuscript. A schematic of the methodology would be helpful."

We are grateful for this constructive feedback. The revised manuscript will include a new appendix that lays out the method in more details and will present schematics illustrating the method. The reason why we go for an appendix is to keep the preamble to a minimum for readers already familiar with the methodology and still make the detailed information available for the readers without the need to read the supporting material. There will still be some supporting material with additional analysis and plots as well as a complete transcript of the analysis in the shape of an R-markdown script and its output (PDF-document).

"Very few details are given on the kriging procedure used. Which variogram parameters were used, and what does this tell us about the spatial dependency - which is more discussed from the angle of the variance carried by the principal components, a very indirect way of looking at spatial correlation. Furthermore, I expect the spatial dependancy to be non-stationary, e.g. very different in western Norway than in other parts of the domain. How is this addressed?"

The revised paper will provide more details about the kriging in a new appendix. The kriging routine was developed at UCAR in the US for gridding climate data and is a multiresolution kriging method based on Markov random fields. It uses a large set of basis functions to estimate spatial estimates that are comparable to standard families of covariance functions (Nychka et al., 2016). “LatticeKrig: Multiresolution Kriging Based on Markov Random Fields.” doi:10.5065/D6HD7T1R <https://doi.org/10.5065/D6HD7T1R>). We used this technique as provided and the gridding aspect was not the main focus of our paper as our main thrust of our work is the downscaling of precipitation statistics for the locations with rain gauge data. Any spatial non-stationarity will be captured by different modes of the PCA - it’s essentially a machine learning method that adapts to the information hidden in the data.

"For instance, section 2.2 mentions the use of EOFs for the large-scale data and PCA for the station data. It is not clear why a different analysis was done for each data type."

Thanks for this comment, and this will be clarified in the revised paper. We reserve the term EOF for regularly gridded data as most cases in literature, using area-based weights for the grid boxes. PCA on the other hand is reserved for irregular groups of coordinates typical for stations and does not involve area weighting. Hence, PCA tends to give more weight to regions with dense networks of measurements. 

"The validation of the results is mostly done visually, with a few assessment metrics given along the text in section 2.3. I would expect results of the cross-validation to be given as detailed tables."

Good point. We will include tables of cross-validation results in the revised paper, in the appendix explaining the method. 

"I guess that many of the comments above can be addressed by referring to the extensive supplementary material, but I believe that an article should be self-contained and should not require readers to go through some 160 pages of annexes to understand the main points."

We agree, and will provide all necessary information in the appendices. However, we will also make the output of our R-markdown script available from FigShare for the sake of transparency and replicability. 

"Figure 5: are these statistics calibrated? In other terms, for a Normal distribution I would expect about 64% of the data to fall with the 1-standard deviation confidence interval. This does not seem to be the case here. Can this be commented on?"

Thanks for this question. The results were not calibrated but were based on probabilities estimated from downscaled annual wet-day frequency and annual wet-day mean precipitation for Oslo-Blindern. The results are slightly biased, but the approximate match also indicates that the method does provide sensible numbers. This will be explained more carefully in the revised version.

"The discussion section is long and windy. Some of it could be moved to the introduction (e.g. ll.290-300), or to the conclusion, or removed."

Good idea. Some of the text has been moved to appendices.

"l.83: Russian -> Russia": Thanks - fixed in revised version

"l.91.92: Specify the cause of that: some runs are not available at a daily resolution":  The data stored on ESGF appears to include a large number of runs, but the shear data volume makes it practically impossible to use all of it to compute monthly wet-day frequency and wet-day mean precipitation. So since these monthly statistics are not readily available, we kept the selection of runs to include only one per model type. Also, our processing of the data revealed some problems that appeared during the data processing for some runs and SSPs, and some of the runs had suspect data that we excluded.  This is perhaps not surprising with vast data volumes that contain data not often visited.

"l.257: …the presence *of* various…": Thanks - fixed in revised version.

"l.304 (and other places): references to unpublished work should be removed": The two references to unpublished work has been left out in the revised paper.

"Figures 3 and 4: the projection seems incorrect, resulting in deformed maps": We understand this comment to regard the geographical projection rather than the projection of future estimates. The figures are a bit squashed in the present version, but will be fixed in the revised paper.

"Figure 6: The legend does not match the figure": Thanks for noticing this, the figure caption is fixed in the revised version. 

We are grateful for the comments from the second reviewer which we find constructive and useful, and hope that we have managed to revise our paper to meet any critical feedback. More detailed account is provided below. Our response will be in bold font and we will repeat the comments in italics.

"The subject is valuable, the data used is pertinent, and the paper is interesting and well-written, but major revisions are necessary. The document is not self-sufficient, and relies too much on the supplementary material."

We understand this objection and a new appendix will provide detailed information in the revised version so that it does not rely on the supplementary material.

"An example is the difficulty to assess the method's quality due to insufficient description of the methodology and evaluation parts."

Thanks for this feedback. We will provide this information in the new appendix of the revised paper.

"These sections should be enhanced by adding more details, including figures."

We think this is a good idea and a new appendix will be included in the revised paper with infographics illustrating the method and with more details about the method. We opted for an appendix that is accompanying the main text to reduce the preamble for readers already familiar with the method while providing explanations and necessary information for other readers without the need to look up separate sources.

"Key details to add in the method part should include the amount of data used to calibrate the regression model (i.e., train the model), the amount of data used to evaluate the model's performance, the size of the data used as predictors (inputs), and the size of the data used as predictands (outputs)."

Good point. This will be provided as part of the detailed method description in a new appendix. 

"The evaluation on ERA5 data is crucial and should be comprehensively included in the main document. References to supporting material should be limited to more detailed evaluations. For example, clarifications are needed for the statement about the “close match between the rain gauge data and ERA5” (Line 134). It should specify whether this evaluation was done on data used to calibrate the regression model."

Good point. A separate appendix will be dedicated the evaluation of ERA5 as well as the GCMs. We think the use of appendices makes the paper easier to read for readers who are not so focused on the ERA5 quality, but readily accessible for all readers nevertheless.

"L71-77 : It seems the authors are describing a bias/variance tradeoff. What I understand is that it is desired to avoid fitting the predictands too closely with statistical techniques to avoid overfitting (high variance, where models become overly sensitive to data used for training/calibrating). It would be helpful to clarify if the exactitude (precision and accuracy in capturing detailed patterns - in calibration data, akin to high variance ?) versus robustness (the ability to generalise to new data, capturing essential patterns) tradeoff is similar to a bias/variance tradeoff. Additionally, the authors describe their strategy as “obtaining a model with high robustness at the cost of precision” but do not detail what happens after this choice. I would expect some specifics on the chosen model in this introduction."

Thanks for asking this. This is not really about bias/variance tradeoff, but using a simpler expression with less parameters for estimating probabilities. In that sense, it’s less advanced than other expressions with a larger set of parameters to fit, but also there is less room for getting a closer match than more advanced expressions. The simple expression is more approximate but also less sensitive to errors. We have tried to rephrase the text to improve it in the revised version.

"L144:  “The evaluation also involved testing the ability of the GCMs in reproducing the predictors in a skillful way” needs clarification. I assumed the GCMs were the predictors used for projections. From the context, it seems the GCMs are compared against ERA5 to verify if the model trained with ERA5 data could be applied using GCM predictors."

Thanks for this question. Yes, predictors from GCMs were used for projections and the point was to ensure that the GCMs were able to reproduce them in a credible manner. They were indeed evaluated against ERA5, which again was evaluated against the rain gauge data. We will revise the paper to make this point more clear.

"L83 : “the highest point being 2062 above” missing unit": Thanks, unit is m. This is fixed in the revised version.

"L257: missing “of” between various and meteorological": Thanks, this is fixed in the revised version.

We are grateful for the comments from the second reviewer which we find constructive and useful, and hope that we have managed to revise our paper to meet any critical feedback. More detailed account is provided below. Our response will be in bold font and we will repeat the comments in italics.

"The subject is valuable, the data used is pertinent, and the paper is interesting and well-written, but major revisions are necessary. The document is not self-sufficient, and relies too much on the supplementary material."

We understand this objection and a new appendix will provide detailed information in the revised version so that it does not rely on the supplementary material.

"An example is the difficulty to assess the method's quality due to insufficient description of the methodology and evaluation parts."

Thanks for this feedback. We will provide this information in the new appendix of the revised paper.

"These sections should be enhanced by adding more details, including figures."

We think this is a good idea and a new appendix will be included in the revised paper with infographics illustrating the method and with more details about the method. We opted for an appendix that is accompanying the main text to reduce the preamble for readers already familiar with the method while providing explanations and necessary information for other readers without the need to look up separate sources.

"Key details to add in the method part should include the amount of data used to calibrate the regression model (i.e., train the model), the amount of data used to evaluate the model's performance, the size of the data used as predictors (inputs), and the size of the data used as predictands (outputs)."

Good point. This will be provided as part of the detailed method description in a new appendix. 

"The evaluation on ERA5 data is crucial and should be comprehensively included in the main document. References to supporting material should be limited to more detailed evaluations. For example, clarifications are needed for the statement about the “close match between the rain gauge data and ERA5” (Line 134). It should specify whether this evaluation was done on data used to calibrate the regression model."

Good point. A separate appendix will be dedicated the evaluation of ERA5 as well as the GCMs. We think the use of appendices makes the paper easier to read for readers who are not so focused on the ERA5 quality, but readily accessible for all readers nevertheless.

"L71-77 : It seems the authors are describing a bias/variance tradeoff. What I understand is that it is desired to avoid fitting the predictands too closely with statistical techniques to avoid overfitting (high variance, where models become overly sensitive to data used for training/calibrating). It would be helpful to clarify if the exactitude (precision and accuracy in capturing detailed patterns - in calibration data, akin to high variance ?) versus robustness (the ability to generalise to new data, capturing essential patterns) tradeoff is similar to a bias/variance tradeoff. Additionally, the authors describe their strategy as “obtaining a model with high robustness at the cost of precision” but do not detail what happens after this choice. I would expect some specifics on the chosen model in this introduction."

Thanks for asking this. This is not really about bias/variance tradeoff, but using a simpler expression with less parameters for estimating probabilities. In that sense, it’s less advanced than other expressions with a larger set of parameters to fit, but also there is less room for getting a closer match than more advanced expressions. The simple expression is more approximate but also less sensitive to errors. We have tried to rephrase the text to improve it in the revised version.

"L144:  “The evaluation also involved testing the ability of the GCMs in reproducing the predictors in a skillful way” needs clarification. I assumed the GCMs were the predictors used for projections. From the context, it seems the GCMs are compared against ERA5 to verify if the model trained with ERA5 data could be applied using GCM predictors."

Thanks for this question. Yes, predictors from GCMs were used for projections and the point was to ensure that the GCMs were able to reproduce them in a credible manner. They were indeed evaluated against ERA5, which again was evaluated against the rain gauge data. We will revise the paper to make this point more clear.

"L83 : “the highest point being 2062 above” missing unit": Thanks, unit is m. This is fixed in the revised version.

"L257: missing “of” between various and meteorological": Thanks, this is fixed in the revised version.

We are grateful for the review comments from all three reviewers and hope that our revised manuscript addresses all the points made by the reviewers. The revised paper has a different structure with extensive use of appendixes to provide more details and discussion about downscaling. It is now stated that we define downscaling as a process that utilises large-scale information that the global climate models skillfully reproduce to derive local information, a definition that also has been used elsewhere in the scientific literature (we provide a couple of prominent examples of such references in our paper). One suggestion that we didn’t follow was to assess the approximation based on the exponential distribution, as it has already been done for numerous sites in North America and Europe and the results are available in the cited open-access paper. However, we have explained in the revised paper that the method has already been evaluated and the results are published in an open-access paper. The formula for estimating intensity-duration-frequency (IDF) curves has also recently been assessed for sites in Norway, and we cite the open-access paper with those results rather than repeating them in our paper. Apart from these points, we have revised the paper according to the points raised by the reviewers. We think this paper should be of interest to the HESS readership and the hydrology community in general, as we present an approach for downscaling robust information on heavy precipitation statistics.

We are grateful for the review comments from all three reviewers and hope that our revised manuscript addresses all the points made by the reviewers. The revised paper has a different structure with extensive use of appendixes to provide more details and discussion about downscaling. It is now stated that we define downscaling as a process that utilises large-scale information that the global climate models skillfully reproduce to derive local information, a definition that also has been used elsewhere in the scientific literature (we provide a couple of prominent examples of such references in our paper). One suggestion that we didn’t follow was to assess the approximation based on the exponential distribution, as it has already been done for numerous sites in North America and Europe and the results are available in the cited open-access paper. However, we have explained in the revised paper that the method has already been evaluated and the results are published in an open-access paper. The formula for estimating intensity-duration-frequency (IDF) curves has also recently been assessed for sites in Norway, and we cite the open-access paper with those results rather than repeating them in our paper. Apart from these points, we have revised the paper according to the points raised by the reviewers. We think this paper should be of interest to the HESS readership and the hydrology community in general, as we present an approach for downscaling robust information on heavy precipitation statistics.