The authors have presented a methodology to construct FDCs for missing portions of the hydrologic record at ungaged sites. The topic is very interesting and relevant. The authors have deeply considered all comments from the reviewers. To me, there is still a significant amount of work that needs to be completed before this manuscript can be published. There is nothing factually incorrect about the method, but I believe that it over-complicates itself and misses a few major elements. In particular, the derivation I have provided shows how this method is almost identical to previous publications. The revision should seek to explore the novelty of what is presented here.
As you can tell from the length of my review, I found this manuscript particularly engaging. (Sorry!) The main contribution of this work, in my opinion, is the hypothesis that the API can be used to produce the temporal sequence of nonexceedance probabilities of streamflow at a partially gaged site. In a related way, the authors hypothesize the existence of a unique, functional relationship between API and Q. This argues that the FDC is a function of weather, not just a function of the basin. These three points are phenomenally interesting to me, but the manuscript seems to get bogged down in the overly complex methodology rather than these hypotheses, which, when demonstrated, could be used to develop simplified methods. I would focus my revision on the novelty. I see the novelty as: (1) The FDC is a function of weather and location, not just location. (2) There exists a unique, one-to-one function relating API and streamflow. (3) For a given period, the temporal sequence of nonexceedance probabilities of streamflow is identical to the temporal sequence of nonexceedance probabilities of API. [NOTE: This is (I) on page 10.] (4) The temporal sequence of nonexceedance probabilities of API is identical at all sites in a region. [Note: This is (III) on page 10.] [NOTE FURTHER: Novelties 3 and 4 combine for an interesting result: QPPQ, described below.]
My main concerns are included below and pertain to the presentation and analysis of the methods. While these is nothing factually incorrect, the underpinnings and the implications of the method need to be explored and analyzed more fully.
To discuss the method, I’ll begin by rephrasing the assumptions on page 10 in probabilistic terms, as they are central to this method:
I. P(Q_{XY} < Q_{i,XY}) = P(API_{XY} < API_{i,XY}), where X and Y designate the basin and time period, respectively. X and Y can therefore take a value of either D (donor) or T (target). The i indicates an arbitrary day of the record at the designated location and in the designated period.
II. P(API_{DY} < API_{i,DY}) = P(Q_{TY} < Q_{i,TY}) where all variables have been previously defined.
III. P(API_{DY} < API_{i,DY}) = P(API_{TY} < API_{i,TY}) where all variables have been defined previously.
As an aside: It should be noted that (II) is an implication of (I) and (III). That is, if (I) and (III) are true, then II must be true. To be explicit: P(Q_{TT} < Q_{i,TT}) = P(API_{TT} < API_{i,TT}) by (I), and P(API_{TT} < API_{i,TT}) = P(API_{DT} < API_{i,DT}) by (III), so P(Q_{TT} < Q_{i,TT}) = P(API_{DT} < API_{i,DT}). This last equality is (III).
The method proposed by the authors relies on (I) and (III) and the assumption that there exists a one-to-one function H_X at each site that maps API to Q.
The method proposed by the authors is as follows: (The method is accurate, though I’m repeating it to put us in the same language.)
Given, P(Q_{TT} < X) = 10.11%, find X.
By (I), P(Q_{TT} < X) = P(API_{TT} < Y).
By (III), P(API_{TT} < Y) = P(API_{DT} < Z).
So we know that 10.11% = P(Q_{TT} < X) = P(API_{TT} < Y) = P(API_{DT} < Z), and Y and Z are knowable by observation. The method presented here ignores Y and focuses on Z.
From the appendix example, Z = 37.72 mm.
Because H_D(…) is a unique, one-to-one function that ingests API and transforms it to Q we can say that we are curious to know what Q is produced by the API Z at the donor site.
The method answers this question by looking at the donor basin in the donor period. That is, by (III), P(API_{DD} < Z) = P(API_{TD} < Y). (Note, because H_D(…) and H_T(…) are one-to-one, unique, time-invariant functions there exists a constant mapping from Y to Z based on (III).)
By (I) we can then say further that P(API_{TD} < Y) = P(Q_{TD} < X). Now, because we are in the donor period, X is knowable by observation given that we know P(API_{DD} < Z).
From the appendix example, P(API_{DD} < 37.72mm) = [7.52%, 7.54%].
The value X is therefore knowable by observation such that, P(Q_{TD} < X) = [7.52%, 7.54%].
Because H_D(…) and H_T(…) are one-to-one, unique, time-invariant functions, Y-->X regardless of time period. So X is the value we were seeking. By observation, the value of X is 3.21mm.
There is nothing factually inaccurate about this method, but it does over-complicate the problem at a partially gaged site. If the target site (T) is partially gauged, then the assumptions provided above make the use of a donor unnecessary. This would proceed as:
Given, P(Q_{TT} < X) = 10.11%, find X.
By (I), P(Q_{TT} < X) = P(API_{TT} < Y).
Because all APIs at the target site in the target period, we can observe the value of Y.
We can then go to the donor period at the target basin with a known Y.
Because we assumed there existed a one-to-one, unique, time-invariant function H_T(…), and all the APIs and Qs of the donor period at the target basin are known, we can interpolate the value Y along H_T(…) as observed in the donor period to determine X. So there is no need for the donor basin when the target is partially (and sufficiently) gaged.
A minor aside, the authors have no clearly defined that a partially gaged basin is. Is it any site with at least one value of streamflow? Is there some threshold for sufficiency? The method, whether as presented by the author or simplified here, requires a substantial donor period at the target catchment, but it might be important to define some length of record that would be required.
The need for a donor basin will be essential when you move to a completely ungaged basin. However, with the ungaged basin, it is impossible to apply this method. This method relies on the donor period for an approximation of the FDC, though embodied through the function H_T(…), the proposed transferability of API across sites, and the proposed equality of probabilities for API and Q.
The reason that this method won’t work at the fully ungaged site is because it relies on the donor period at the target site to estimate the FDC at the target site and uses the assumed H_T(…) to translate that across time periods (from donor to target).
To apply this method at the ungaged site, we will need an alternative method to approximate H_T(…). Arriving at this conclusion, it is obvious that this method, when applied to ungaged basins, is no different than the method presented by Smakhtin and his team in the mid to late 1990s. I will show this now, abbreviating their method as QPPQ:
Let’s begin as we did before:
Given, P(Q_{TT} < X) = 10.11%, find X.
By (I), P(Q_{TT} < X) = P(API_{TT} < Y).
By (III), P(API_{TT} < Y) = P(API_{DT} < Z).
Whereas earlier we stopped here, we can use (I) again to say that P(API_{DT} < Z) = P(Q_{DT} < W).
We can then summarize this into a new implication of (I) and (III), namely P(Q_{TT} < X) = P(Q_{DT} < W).
More generally, P(Q_{DY} < Q_{i,DY}) = P(Q_{TY} < Q_{i,TY}).
Now, if we had a first-order approximation of the relationship between Q and P at the target site, we could estimate X as a day in a simulated hydrograph.
The method presented by the present authors, uses API and H_T(…) in the donor site to make this approximation. Smakhtin et al., and following work, use something like a regional-regression FDC.
The QPPQ school uses this first-order FDC to produce a sequence of daily flows based on the probabilistic implication described here (P(Q_{DY} < Q_{i,DY}) = P(Q_{TY} < Q_{i,TY})). The QPPQ stops here.
You could move further to take the simulated hydrograph and compute a new FDC.
Because of inaccuracies used to produce the first-order FDC at the target site, inaccuracies stemming form estimation error, for example, and the evolving sequence of probabilities in the target period (for example, it could be overly weighted toward high flows). There is no guarantee that the first-order FDC used in QPPQ will be the same as the FDC that would be built from the resulting simulated hydrograph.
Moving to this second-order FDC would produce a time-dependent FDC at the ungauged site, much in the spirit of the methods presented by these authors.
These two processes, that of simplifying the method presented for partially ungaged sites and the extension of the QPPQ method, show that the work presented by these authors misses the novelty of their own method. I believe that the work of these authors has merit and the potential to advance the field. With further consideration of the implications and theoretical underpinnings of the proposed methodology, this manuscript could prove impactful. I will follow now with a few less-major concerns I have with the manuscript.
At the opening, it was necessary for me to rephrase the assumptions outlined on page 10. This is because those assumptions are not clear, and the tests provided to demonstrate their veracity test something different. In (I) it is assumed that the CDFs “correlate”, and the test shows a temporal correlation between nonexceedance probabilities. While this isn’t incorrect, it is ambiguous. For example, one could equally compute the correlation of API and Q quantiles to say that the CDFs correlate, but that would be very different than what the authors are seeking to assume or prove. Similarly, (III) states that the CDFs of API are identical across sites, but the proof shows only that the exceedance probabilities correlate in time. Identical CDFs would have the same probabilities associated with the same values of the proxy. These assumptions need to be less ambiguous.
As notes above, these assumptions are also redundant. Really, only (I) and (III) are needed, as (II) is a natural extension. Another natural extension, as shown above, is P(Q_{DY} < Q_{i,DY}) = P(Q_{TY} < Q_{i,TY}). This last one raises a point of confusion in the manuscript.
On page 13, line 34, it is stated that “the FDC of a donor site cannot be transferred to another site”. First, it is unclear what is mean by “the FDC”: Do you mean the FDC in its entirety, including probabilities and quantiles, or the correlation as “proven” for (I), (II) and (III). In any case, the extension that P(Q_{DY} < Q_{i,DY}) = P(Q_{TY} < Q_{i,TY}) contradicts this statement. Furthermore, the authors then use this extension to transfer the FDC of a donor to a target in the German catchments. This confusion should be clarified. (In point of fact, I recommend removing the German catchments altogether.)
While this revision did a great job of supporting its assumptions, I feel that a basic one was left out. The assumption that these exists the function H_T(…) (page 9, line 7). As shown above, this is the crux of the methodology. I think it would be worthwhile to show that the interpolation of values implies that this relationship (between Q and API) is site dependent and time independent through a cross-validation exercise.
Stepping away form the methodology for a moment, I still struggled with the structure of the manuscript. Here, I have two recommendations. First, I suggest removing the section on energy and water limited catchments (2.3). We know that there are different limiters, and we know this is important, but I don’t think this section adds to the manuscript. This is evidenced by the fact that the information in this section is never mentioned again in the manuscript: Not to explain the results or adapt the methodology. In a similar vein, I suggest removing the presentation and analysis of the German catchments. The results for the German catchments, starting on page 18, provide little value. The methodology is substantially different (confusingly using a streamflow proxy, as described above). The authors provide little discussion as to what new information is presented by these catchments. Obviously, with some additional discussion, both sections could prove useful, but it may be simpler just to remove them from the manuscript.
Finally, before considering minor comments, I was quite surprised that the authors did not address the large literature on the appropriateness of donor catchment selection. Instead, the examples presented here use the same donor for all gages and treat all gages as donors. Could it be that the method is sensitive to donor selection? In truth, the partially gaged method is probably not sensitive to donor selection, as demonstrated by the simplification where a donor is not needed, but the ungaged approach (QPPQ) is certainly dependent on donor selection.
MINOR COMMENTS:
Page 1, line 10: It seems odd to say that FDCs are “set up”. They are a product of the record, they aren’t established. Maybe something like, “The FDC of streamflow at a specific site provides knowledge on the distribution and characteristics of streamflow at that site.”
Page 1, line 12: “In spite of its importance,…”
Page 1, line 14: What do you mean by partially gaged? The ability to build an FDC depends on this definition. Certainly, a site that has a record from 1980-2000 and 2002-2018 is partially gauged, but it still has more than enough information to build an FDC. I see your point that it is weather dependent, but it still leaves the definition of partially gaged ambiguous.
Page 1, line 14: “…among the other streamgages.”
Page 2, line 3: I'm not sure that FDCs provide information on the severity because they don't talk about duration (length) of drought. So two sites could have a similar frequency of low flows, but one might see all low-flows consecutively (long drought), while the other might see intermittent lows (episodic drought). One could argue that drought is more severe in the former case.
Page 2, line 12: “…to meet intake requests…”
Page 2, line 15, “As the FDC…”
Page 2, line 27: “In spite of its importance, the FDC…”
Page 3, line 27, “In the following…”
Page 3, line 28, “…divided into water…”
Page 3, line 28: Are you making a meaningful distinction between basins and catchments? If so, explain. If not, please use consistent terminology throughout.
Page 3, line 40: Check citation style.
Page 3, line 40: Revise to “The soil type can strongly affect the impact of climate on the water balance.”
Page 3, line 41: I think the terms are karst and non-karst and should be throughout.
Page 4, line 2: “…karst regions makes it difficult to transfer information from ..."
Page 4, line 13: Please provide citation for degree-day approach.
Page 4, line 15: Keep in mind for formatting: This is an odd page break for a header.
Page 5, line 21: “…the catchment allows for infiltration and stores…”
Page 5, line 21: “…runoff production declines since…”
Page 5, line 23: “However, the climatic timing…”
Page 5, line 24: “They show that the difference…”
Page 8, line 30: I think maybe an extra sentence or two of explanation is needed here. Why would you expect two different time periods at the same site to be independent (as assumed by the KS test)? I see the point about auto-correlation, but I find it hard to imagine that the distribution of Fall 2012 streamflow and Fall 2013 stream flows are wholly independent at the same site. I think I follow your logic, and I would love to see it spelled out more fully. It may be tied to ambiguous definition of partial gaging.
Page 8, line 38: “…Figure 4 shows the magnitude of the difference between…”
Page 9, line 10: “…hydrological modelling as modelling often introduces additional errors…”
Page 9, line 11: “…errors and may be…”
Page 9, line 15: As k, defined above, denotes location, what location is being addressed here?
Page 10, line 10: This paragraph cites discussion about why the alpha should decay with time. Does it not decay here? It’s fixed at 0.85?
Page 10, line 17: See discussion above. Here you need to clarify that you are talking about the correlation between p(Q<Qi) and P(API < APIi) for the day i. This is different than the correlation of Qi and APIi of rank i.
Page 10, line 27: Are these Pearson correlations? Please specify?
Page 13, line 5: Here you are using "," to denote a decimal point and "." to denote figures to the right of the decimal. This is not the style of the journal. I suggest, "X.yz, where y and z are the digits to the right to the decimal" Perhaps still better: In binning by percentiles, all percentages were rounded down to the nearest whole number.
Page 13, line 9: Shouldn't there be an absolute value for this statement to be true? Bias is defined here as an average. If, for example, we looked at three days producing these values of the thing in parenthesis: -3, 0, 3. In this case, the BIAS would be 0, but this is certainly not a perfect fit.
Page 13, line 13: “…and so is the MAE. It …”
Page 13, line 29: I think it would be better to show the p-value of D*. Looking at Figure 8 (page 14, line 12), the D may be bigger, but it may not be as significant because of changing sample size.
Page 17, line 13: I find it very confusing that the color scale changes from figure to figure (e.g. 9 and 10). The change makes it hard to compare across figures.
Page 18, line 6: Here the proxy is donor streamflow. I discuss this above, but this strikes me as odd when page 13, line 34 seems to say you should not do this. (As you know, I think it might be best to just remove this section.)
Page 19, line 2: The correlation between the donor and the target is problematic and cannot be known a priori (line 12). This needs to be contextualized in an ungaged example that is not discussed here. The ambiguity here makes the conclusion on line 39 of page 40 rather tenuous.
Page 23, line 18: The probability statement should be in terms of API, right?
CONCLUDING REMARK:
Thank you so much for such an interesting work! Your revision demonstrates a dedication to this work, and I look forward to further discussion. I’ve spilled a lot of ink on this page, but it is nowhere near as much thought as you have put into the work. With that in mind, please do not hesitate to reach out to me if you would like to discuss any of this analysis further. I’m more than happy to discuss the work, and especially happy to make changes to my comments if further discussion finds them to be inaccurate. I strongly support the novelties of this work, which I described earlier, and look forward to seeing your responses.
Thanks,
William Farmer |

Specifically, all reviewers commented on the need for more robust testing of the method and point to other similar published methodologies that I agree are not adequately contrasted in the current manuscript. It is not clear from the author responses that these papers will be discussed. I acknowledge that the proposed revisions note that the authors will include these references but it is not clear these references will be more deeply discussed. Therefore, I request that a discussion of this work in the context of other papers on this topic be made clear in the revision.

I am also unclear as to whether the authors are proposing to complete the additional testing and interpretation requested by the review comments. This needs to be made more clear in the revision and final response stage.

I hope that my additional comments will help guide the revision of the discussion paper to fully address the reviewer comments. The manuscript will be sent for re-review once the revision is submitted.