Dear Dr. Riegger,

Thank you for the detailed responses to the referee comments. All referees recommend major revisions before the manuscript can be reconsidered for publication, and I will follow this recommendation. While following up on the referee comments and your responses, I stumbled over a very recent paper by one of your former co-authors:

Tourian, M. J., Reager, J. T. and Sneeuw, N.: The Total Drainable Water Storage of the Amazon River Basin: A First Estimate Using GRACE, Water Resources Research, 54(5), 3290–3312, doi:10.1029/2017WR021674, 2018.

This a very relevant paper that uses the same data and gets to very similar results in terms of estimating the total drainable water storage of the Amazon basin. The paper goes even further and analyses water table data and sub-catchments within the Amazon basin. I am aware that Tourian et al. (2018) use the data in a different way to your work and that it was published after submission of your manuscript, but in order for the advantages of your approach to be understood, the manuscript will need to be put into relation to Tourian et al. (2018) and other recent papers, as pointed out by the referees.

The referees raised two general concerns related to the applicability of your approach, related to (a) the representation of the catchment as 2 linear storage compartments in series and (b) the absence of significant uncoupled storages. I agree with you that a model should be as simple as possible for a desired task and therefore, for the purpose of estimating drainable storage volume (DSV), fitting two time constants likely has advantages over fitting the parameters of a full-blown hydrological model to GRACE and streamflow data. However, given that Tourian et al. (2018) have already demonstrated a way to estimate DSV from GRACE and streamflow data in the Amazon, your manuscript will need to illustrate more clearly how it advances current understanding beyond this. Referee #2 made a very valid point that water in the unsaturated zone would constitute time-varying uncoupled storage, which would violate your own criteria for applicability of your approach. The reason why the model still performs well might be that the relative magnitudes of total DSV are much larger than the maximum variations of water in the unsaturated zone at the monthly time scale. This should be adequately discussed and analysed.

Maybe a good way to address concerns about limited applicability of the appraoch would be to apply it to more than one basin and test whether the cascade storage model can be calibrated using GRACE and then used to predict variations in streamflow and inundated areas.

Language: I agree with the referees that the language and use of specific terms is confusing and needs significant improvements. Precipitation surplus more accurately expresses P-ET than recharge, and it would also be good to distinguish between drainage (or runoff) and streamflow, as suggested by one of the referees. I think that runoff may not be a good term, as it is often expected to include infiltration excess runoff, which would not be a function of storage. I would recommend "drainage" for subsurface flow into the channel network, and streamflow for the flow at the outlet. The abstract has to be shortened substantially, as suggested by the reviewers, and made understandable without reading any other portions of the paper. I agree with the referees that its current form is more like a summary, which would be read after reading the paper, not before.

For all the other referee comments, please consider them as typical thoughts readers might have and modify the revised manuscript in a way to not leave any of these questions open. I would actually propose to include a separate Discussion section for this purpose, followed by a short Conclusions section. With your revised manuscript, please provide also a version with all changes highlighted and explain how and where the different referee comments were addressed in the revised manuscript.

Below, I list some more comments that I had while re-reading your paper.

RANDOM TECHNICAL COMMENTS:
- Title is misleading, as it implies that you can use the approach for the whole globe, not just for very large catchments. In fact, you only showed application in the Amazon, so unless you add more catchments to the analysis, generalisation may not be warranted. You may also consider emphasising the difference to Tourian et al. (2018) in the title.
- Please do not forget considering Mark Bierken's annotations.
- Please clarify the origins of the different equations, e.g. Eq. 1 is the result of a linear storage function, provide reference.
- Please put each labelled equation on a separate line (e.g. P9).
- Please include more information in the figure captions. Figures and captions should be informative without having to read the main text, and contain all information needed to reproduce the figures (e.g. which data set, what model/equation, what parameter set).
- Fig 1: what does "phase-adapted" mean?
- P5L10-15: Why not just state that extrapolation of curve to 0 runoff gives the 0-point for drainable storage?
- Table 1: Please describe a-f in the caption.
- P5L22: "Opposite to parallel storages..."
- Please include label for table with abbreviations
- P8L13: There is no RC in Eq. 10.
- Eq. 11: Please provide reference for this general solution.
- Fig. 2: N is a flux, the others are state variables. Need two axes! Please specify what parameter values were used to generate these plots.
- Eq. 24: Is this an arbitrary function that can be fitted to the simulation results? Then it would only be valid for these model runs, not for real conditions with non-sinusoidal forcing, right? Please clarify.
- P14L7: "lambda = 3.2"?
- P15L9: Isn't this limited to sinusoidal forcing?
- Eq. 28: Where does this come from, what are the sigmas?
- Figs 8-10: How is separation between signal and residual performed?
- Fig. 9: Typo: mass deviation
- Fig. 11: Caption unclear. Mention symbols and colours in the caption. How was everyhting scaled to 0, i.e. how was dMT and dMR computed? What does the hysteresis mean and in what direction is it?
-P20L10: This should be moved to figure caption.
- Table 1: Describe meaning of columns a-f in the caption.

Dear Dr. Rieger,

Both referees consider your manuscript valuable but still requiring improvements before publication. I am very grateful for the reviewers' extremely constructive approach and their contributions to help improve your article further, despite your partly confrontational responses to previous referee comments. As suggested by Referee #4 and myself at an earlier stage, the arguments brought forward by the referees are likely thoughts that other readers will have as well and therefore it is not helpful to spend your effort in lengthy replies to the referees, but preferable to address the main points in the manuscript itself, for the benefit of the readership. As an example, Referee #1 pointed out that global hydrological models (GHMs) are mentioned in the abstract, but no-where else. This is surprising, given your detailed discussion of this point in the previous round of review. Why was this discussion not included in the manuscript? Please include a concise discussion of the pros and cons of your approach in comparison to using GHMs for a similar analysis, as requested by Referee #1. Both Referee #2 in the previous round of review and Referee #4 in the current round pointed out that references to groundwater in this manuscript are misleading as groundwater may be disconnected from the hydrological processes considered here. Therefore I would like to ask you to carefully revise the article and remove or clarify any potentially misleading statements. Again, please do not rebut this point in your response but address it in the manuscript.

After conferring with Referee #4, we agree that your discussion of Tourian et al. (2018) on P6L13-20 is adequate, so please disregard this comment. All the other comments seem straightforward to address. The inclusion of "application to the Amazon basin" in the title, as suggested by Referee #4, would also alleviate concerns about the untested claim of global applicability in the previous round of review. At the same time, I agree with the suggestion to shorten the title and abstract and I am also not convinced that "phasing" and explicit mentioning of "Both the Catchment and the River Network" are crucial in the title.

Please pay attention to a clean organisation of the document, as pointed out by both referees. Indeed, Fig. 1 is part of the introduction, whereas it contains results generated as part of the present work if I am not mistaken, so it should be part of the results, and re-organised to be consistent with the flow of the main text. It is also not obvious to me or referee #4 what data and theory Fig. 1 is actually based on. This would be much easier to clarify if it were part of the results, after all the concepts, equations and data needed for this figure have been explained. If Fig. 1 is needed to discuss Tourian et al. (2018), then this discussion might be better placed in the dicussion section, instead of the introduction. Conclusions have to be placed in a separate section, as described in the manuscript preparation instructions (see link below), so the last section has to be separated into two sections, Discussion followed by Conclusions, as has been pointed out before. The conclusions may contain elements of the Summary and Outlook section, but there is no place for an explicit summary in HESS. As pointed out by both referees, Section 6, "Data driven Determination of Drainable Storage volumes" is out of place. The equations given here are not referred to in the main text as far as I can see, so what is their relevance? Please revise and re-organise the manuscript accordingly, while respecting the clear instructions given here: https://www.hydrology-and-earth-system-sciences.net/for_authors/manuscript_preparation.html. For example: "Figure captions: Each illustration should have a concise but descriptive caption. The abbreviations used in the figure must be defined, unless they are common abbreviations or have already been defined in the text. Figure captions should be included in the text file and not in the figure files."

Dear Dr. Rieger,

One referee recommends acceptance of the article after minor revision, the other is asking for a major revision, mainly to increase clarity. Both referees make valuable suggestions for improvement of the article before publication. I agree with Referee #1 that the reader should be pointed to up-to-date literature on global hydrological models and the two papers suggested seem indeed very relevant for this work. Refree #4 is still not happy with the abstract and also repeats for a third time the concern that the use of the term "groundwater" may be misleading. When I pointed to this issue in my previous decision, you responded that this has been "checked and changed". However, the revised manuscript still only distinguishes between "groundwater" and "surface water". Why don't you use the term "soil water" instead, as suggested by the referee to remove confusion with groundwater studies dealing mainly with a water compartment that is not permanently linked to river flow? With respect to the use of "basin" instead of "catchment", I find the current distinction between "Amazon basin" and "sub catchments of the Amazon basin" adequate, but please check for yourself if you find places where "basin" would be a better term. Actually, since you subdivide between the basin and the catchments therein, you could leave out the word "sub".
Please also add a clarification about the meaning of the terms "catchment runoff" and "runoff river network". I agree with the referee that "river discharge" may be a clearer term for R_R. I am not sure about the meaning of the referee's comments on Eqs. 1, 2 and 23. Could you please verify an add appropriate clarifications in the text? As said earlier, I cannot ask you to extensively discuss Tourian et al. (2018) as it appeared long after the first submission of your manuscript, but if you see that such a discussion could address the missing validation of the method as suggested by Referee #4, it would be very helpful if you could add such a paragraph. Thank you.

Best regards,
Stan

Dear Dr. Riegger,

Thank you for the revision. I think that the manuscript can now be brought to a publishable form without a further review from my side. I would still like you to consider the following points before the manuscript enters the copy-editing process:

- Eqs 1 and 2: Thinking about the comment by Referee #4 that "an introductory discussion of how basin storage can drive river flow" is missing, I realised that Eq. 1 comes a bit out of the blue. It would be important to state clearly that by a "linear runoff-storage (R-S) relationship" you mean that R(M) = M/tau, as implied in Eq. 2. This, with the definition of R(t) = Q(t)/A and assuming that there is no recharge or other losses of water, notably no ET, then yields Eq. 1. In Eq. 2 then, you only need to show that M(t0) is equal to the infinite integral of Q(t) from t0 to infinity. This would make it clear that M refers only to the part of the storage that has an effect on R and Q, as requested by Referee #4. Please also discuss that the assumption underlying Eq. 1 is that there are no processes affecting M other than Q.

- P4L21: This should be tau=M/R, for consistency with Eq. 2.

- Eqs. 4-7: Please follow the HESS convention where each equation stands on its own line with its own label, e.g.:
"The total system behaviour is described by two balance equations, one for catchment storage (Eq. 4) and one for river storage (Eq. 6):"
followed by Eqs. 4-7, where the word "with" before 5 and 7 would be on a separate line before the equation.

- Eq. 8: Why do you call the "time dependent recharge" N(w), not N(t), as in Eq. 9?

- P28L11-: In the track-changes file, this is labelled "Discussion", but the discussion has to come before the conclusions. Please fix the formatting of P28L11 and move this Discussion section before the Conclusions. I would also remove the first sentence of the Discussion: "Comparing global hydrological or land surface models with the presented Cascaded storage approach it can be summarized:"

Best regards,
Stan Schymanski