A mixed distribution approach for low-flow frequency analysis &ndash; Part 1: concept, performance and effect of seasonality

Laaha, Gregor

doi:https://doi.org/10.5194/hess-2022-195

Preprints

https://doi.org/10.5194/hess-2022-195

Preprints

03 Jun 2022

Status: a revised version of this preprint was accepted for the journal HESS and is expected to appear here in due course.

A mixed distribution approach for low-flow frequency analysis – Part 1: concept, performance and effect of seasonality

Gregor Laaha Gregor Laaha Gregor Laaha

Show author details

University of Natural Resources and Life Sciences, Vienna, Department of Landscape, Spatial and Infrastructure Sciences, Institute of Statistics, Peter-Jordan-Strasse 82/I, 1190 Vienna, Austria

Received: 21 May 2022 – Discussion started: 03 Jun 2022

Abstract. In seasonal climates with a warm and a cold season, low-flows are generated by different processes so that the annual extreme series will be a mixture of summer and winter low-flow events. This leads to a violation of the homogeneity assumption for all statistics derived from the annual series and give rise to inaccurate conclusions. In this paper we propose a mixed distribution approach to perform frequency analysis in catchments with mixed low-flow regime. We formulate the theoretical basis of the mixed distribution approach for the lower extremes based on annual minima series. The main strength of the model is that it allows the user to estimate return periods of summer low-flows, winter low-flows and annual return periods in a theoretically sound and consistent way. Using prototypical examples, we show how the model behaves for a range of low-flow regimes, from distinct winter and summer regimes to mixed regimes where seasonal occurrence in summer and winter is equally likely. The examples show in a qualitative way the errors we have to expect with conventional extreme value statistics performed with the annual extremes series. The model is then applied to a comprehensive Austrian data set to quantify the expected gain of using the mixed distribution approach compared to conventional frequency analysis. Results indicate that the gain of using a mixed distribution approach is indeed large. On average, the error is reduced by 21, 39 and 63 % when estimating the low-flow with a 20-, 50- and 100-year return period. For the 100-year event, 75 % of stations show a performance gain of > 10 %, 41 % of stations > 50 % and 25 % of stations > 80.6. This suggests a much broader relevance of the approach that goes beyond highly mixed seasonal regimes to include the strongly seasonal ones. We finally correlate the performance gain with seasonality indices in order to show the expected gain conditional to the strength of seasonality expressed by the ratio of average summer and winter low flow SR. For the 100-year event, the expected gain is about 70 % for SR =1.0, 20 % for SR = 1.5, and 10 % for SR = 2.0. The errors are further allocated to the spatial patterns of SR in the study area. The results suggest that the mixed estimator is relevant not only for mountain forelands but for a broad range of low flow regimes. The mixed distribution approach provides one consistent approach for summer, winter, and annual probabilities and should be used by default in seasonal climates with a cold winter season where summer and winter low flows can occur.

Gregor Laaha

Status: closed

RC1:
'Comment on hess-2022-195', Anonymous Referee #1, 08 Jul 2022
This work presents an innovative formulation for the low-flow frequency analysis which accounts for the seasonal behavior of the low-flow that characterizes some regions.

The manuscript is very well written and organized and the research is of significant scientific interest. To my opinion, the work can be published after some minor revisions that could make it clearer.

Please, read in the following my suggestions.

The title suggests that a part 2 exists. If so, please refers to it in the manuscript. What does it deal with?

I suggest to include a figure that synthesizes the seasonal characteristics of the analyzed dataset, e.g. empirical cumulative distribution of the observed low-flow values, mean monthly series, so to appreciate the effectiveness of the mixed proposed approach.

Sometimes variables are not explicitly defined. Please, make sure of that. For example, at line 129, define clearly the utilized circular seasonality index (r) and the seasonality ratio (SR). I see the reference to Laaha nd Blooschl (2006b), but I suggest to make the manuscript self-structured.

The values of the relative frequency in Figure 3 are very low; the discussion cannot be appreciated very much. Maybe I would add the cumulative relative frequency or the CDF

Comments to table 2 at lines L241-245 are a bit confusing. What does it mean 32 to 34 cases? I don’t have the total number of gauges to verify the 10%. Please, refer also to the syntax used in the header of the table, i.e. 1^st and 3^rd

Please, make clear in the text among which variables the correlations are computed (section 4.2.1)

Please, note that Figure 4 is neither recalled nor discussed in the text. Add comments. Define the Relative error. Which return period does it refer to?

Figure 6: be consistent with the shown variables (relative error not defined, performance gain not defined)

TECHNICAL CORRECTIONS

Please see in the following technical observations:

L70: “Extreme events”

Figure 1: make consistent the red colors (line and circles)

Figure 2: please add the total number of the analyzed gauges, even in the text.

Table 1: please add the symbol rd_T of the relative deviation.

I would swap Table 1 and Table 2, since this last is discussed earlier.
Citation: https://doi.org/10.5194/hess-2022-195-RC1
- AC2: 'Reply on RC1', Gregor Laaha, 14 Oct 2022
  
  I thank the reviewer for the very positive and constructive feedback. Please find my responses below.
  1. The title suggests that a part 2 exists. If so, please refers to it in the manuscript. What does it deal with?
  
  Thanks, now mentioned in the abstract and in the introduction (L 60):
  
  In a two-part series we aim to fill this research gap. In this first part we propose a mixed distribution approach for low-flows to perform frequency analysis in catchments with a mixed summer/winter regime. The method is based on an independence assumption which is explored in the second part of the series \citep{Laaha_copula_2022}, where we address possible seasonal dependency using a copula-based estimator.
  2. I suggest to include a figure that synthesizes the seasonal characteristics of the analyzed dataset, e.g. empirical cumulative distribution of the observed low-flow values, mean monthly series, so to appreciate the effectiveness of the mixed proposed approach.
  
  The seasonal characteristics are characterized in Figure 5 and a detailed assessment is given in Laaha and Blöschl 2006ab. I am therefore reluctant to add “more of the same” information to this paper. However, I see your point and will add an additional Figure (attached) with regime plots (monthly mean and monthly low flow regime) of the four example gauges of Fig. 1.
  
  3. Sometimes variables are not explicitly defined. Please, make sure of that. For example, at line 129, define clearly the utilized circular seasonality index (r) and the seasonality ratio (SR). I see the reference to Laaha and Blooschl (2006b), but I suggest to make the manuscript self-structured.
  
  Added to the beginning of the section, which is now formulated in the following way:
  
  Using archetypal examples, we demonstrate how the model behaves for a range of low-flow regimes: from weakly seasonal ones where summer and winter occurrences are equally likely, to strongly seasonal ones where low-flows in one season predominate. Seasonality is characterized by two indices: the seasonality ratio (SR), where SR > 1 indicates a winter and SR < 1 a summer low flow regime, and the circular seasonality index (r), where r=0 indicates the weakest and r=1 the strongest possible seasonality for the definition of indices see {laaha_seasonality_2006}).
  
  Seasonality is characterized by two indices: the seasonality ratio (SR), where SR > 1 indicates a winter and SR < 1 a summer low flow regime, and the circular seasonality index (r), where r=0 indicates the weakest and r=1 the strongest possible seasonality (for the definition of indices see{laaha_seasonality_2006}).
  
  4. The values of the relative frequency in Figure 3 are very low; the discussion cannot be appreciated very much. Maybe I would add the cumulative relative frequency or the CDF
  
  Will be added.
  
  5. Comments to table 2 at lines L241-245 are a bit confusing. What does it mean 32 to 34 cases? I don’t have the total number of gauges to verify the 10%. Please, refer also to the syntax used in the header of the table, i.e. 1st and 3rd
  
  The number of catchments is mentioned in L. 178. This number is also now added to L. 241, and to the caption of Fig. 2, and reference to the syntax in the header is given.
  
  6. Please, make clear in the text among which variables the correlations are computed (section 4.2.1)
  
  Between the accuracy gain and three seasonality indices (see first sentence of this paragraph).
  
  7. Please, note that Figure 4 is neither recalled nor discussed in the text. Add comments. Define the Relative error. Which return period does it refer to?
  
  Thanks, will be included in the first sentence of the paragraph.
  
  8. Figure 6: be consistent with the shown variables (relative error not defined, performance gain not defined)
  
  The term error was defined as a synonym for the deviation when we place emphasis on the inferior model (L. 194). Analogously, the term gain was defined as the change in performance of the superior model compared to the inferior model (L. 192). All have been used throughout the text.
  Technical corrections:
  
  Many thanks, will be amended in the revised version of the MS.
  
  Citation: https://doi.org/10.5194/hess-2022-195-AC2
RC2:
'Comment on hess-2022-195', Anonymous Referee #2, 12 Oct 2022

This study develops a mixed distribution approach for low flow frequency analysis, particularly focused on regions that potentially have a distinct winter low flow period, due to freeze/snowfall/lack of moisture, and a summer flow flow period, due to high ET and low precipitation. The approach is built off similar mixed distribution statistics used for flood frequency analysis, but applies it in a unique way, due to the nuances of low flow. The paper first develops the concepts/statistical underpinning and then applies it to characteristic sites in Austria to show how the model responds when hydrologic drought is dominated by one season or a mix of both seasons.

Overall, I believe this is a useful and well conceived study. I list a few major comments that I believe should be addressed before this study could be published. With that said, I do believe this study has the novelty and value to ultimately be published in HESS.

I ultimately recommend a major revision.

Major issues:

1. Line 85 states that summer and winter events are independent of one another. This is a relatively strong assumption that underlies the method. I question this assumption. Given that baseflow has long persistence along with the climate drivers that create hydrologic drought, I imagine potential for strong temporal autocorrelation between summer and winter events.

Please test the temporal correlation of Summer and Winter, lagged in either direction.

To further eliminate the potential for temporal autocorrelation, I recommend checking the dates of each hydrologic drought and potentially including a buffer period. If November 1 is the division between Summer and Winter (see comment 3 below), please make sure there are no years where Summer drought occurs on Oct 25 and Winter drought occurs Nov 7, for example.

2. Overall issue - use of the word "gain" and, for example line 12 of the abstract "the error is reduced by ...". I am not convinced, if the underlying assumptions are not checked (above), that the new mixed distribution estimates are exactly correct. Therefore, this phrasing that the estimates are improved by XXX% or that error is reduced by XXXX% should be softened or modified. In the derivation of gain (Eqs 10-12), this is referred to as "change in return period" or "relative deviation". Those terms, e.g. Delta T, are more neutral, and in my opinion more accurate, than stating that error is reduced by XXXX. I would be more willing to accept the stronger wording only with centuries of simulated data.

3. Line 36 - You state that many studies have suggested defining summer from about April or May to November, and winter as remaining. But, you never state how you are defining the two seasons for the analysis in Sections 2.4 and 3. Is this the division? Is it April or May? Please provide the exact day.

4. Please provide how many years of data are used for each river (n = ?).

5. Line 242 - Please explain how the deviation can be exactly zero. I find this highly unlikely given the fitting of two distributions and then comparisons of the extreme tails of each distribution. If this occurs because the Mixed Model defaults to the single season, explain this. Although, I doubt it would be exactly the same to 1 or 2 decimal places.

This shows up in Figure 4 as well and is confusing.

Minor issues:

Line 14 - misspelling "broade"

Line 70 - misspelling "extreme events"

Line 130 - I appreciate the link to Laaha and Bloschl 2006, but the first time you introduce SR, please insert the sentence from the caption of Fig 5 "SR > 1 indicates ..., SR < 1 indicates." Perhaps add in "Seasonality ratio ranges from XXX to YYYY".

Line 216 - spelling of Pearson

Line 270 - "Despite there is a large". Please revise for grammar

Line 285 - It looks like there are some words missing "as the have an "

Line 301 - misspelling "one"

Citation: https://doi.org/10.5194/hess-2022-195-RC2
- AC1: 'Reply on RC2', Gregor Laaha, 14 Oct 2022
  
  We thank the reviewer for the positive and constructive feedback. Please find my responses below.
  
  1. Line 85 states that summer and winter events are independent of one another. This is a relatively strong assumption that underlies the method…
  
  Response: I fully agree with this statement that the methods proposed in the paper rely on the independence assumption and there will be certainly catchments where the assumption is not strictly fulfilled. This sis similar to the iid assumption of other estimators that is often not strictly fulfilled in practice, yet useful for model development.
  
  I have therefore structured the research in a two-part paper series where the first part develops methods assuming there that dependency will have no significant effect. The second part extends the scope will explore seasonal dependency in detail. It examines the value of an extended estimator that accounts for the seasonal correlation of low-flow events and assesses its value in a hydrological context. This companion paper (hess-2022-358 A mixed distribution approach for low-flow frequency analysis – Part 2: Modeling dependency using a copula-based estimator) was submitted to HESS on 13 Oct, and is currently in the state of editor assignment.
  
  In the MS of this paper, the effect of possible seasonal correlation has been discussed at the end of the paper and I see from your comment that it should come earlier, where the reader starts to question the assumptions of this paper. We will therefore add at Line 90:
  
  It should be noted that the assumption of strict seasonal independence will only be met in part of the catchments, while there may be cases where some dependency of seasonal minima exists. This is explored in detail in the second part of this two-paper series \citep{Laaha_copula_2022}, where we examine the value of an extended estimator that accounts for the seasonal correlation of low-flow events.
  
  I need to note here that the results of the second part show that there is indeed some correlation in a part of the catchments, but this has only impact on the estimates of mild (e.g. 2-year) low flow events, which are usually of little interest. For severe events the difference is negligible what supports the validity of the estimates of this study (and supports the interpretation of “deviation” being a “gain” based on being a valid generalization of the common estimator).
  2. Overall issue - use of the word "gain" and, for example line 12 of the abstract "the error is reduced by ...". I am not convinced, if the underlying assumptions are not checked (above), that the new mixed distribution estimates are exactly correct…
  
  I see your point, but argue from the findings of the companion paper (where not only the assumptions, but also their impact on the estimates have been checked) that for the return periods we are usually interested in, the differences between the mixed and the extended mixed copula estimator are negligible so that the mixed distribution approach is indeed a valid and accurate estimator of the low-flow event. Given that the terminology is also set out in Section 3.2.1, I would like to keep this wording, but suggest to add the following sentence (Line 212):
  
  It should be noted again that the interpretation of the deviation as a gain depends on the (reasonable) assumption that the model is superior to the alternative model, and the terminology of the study should be understood as such.
  3. Line 36 - You state that many studies have suggested defining summer from about April or May to November, and winter as remaining. But, you never state how you are defining the two seasons for the analysis in Sections 2.4 and 3. Is this the division? Is it April or May? Please provide the exact day.
  
  Thanks – The period in the Austrian study starts in April and this will be added to the MS.
  4. Please provide how many years of data are used for each river (n = ?).
  
  Thanks – will be added.
  5. Line 242 - Please explain how the deviation can be exactly zero. I find this highly unlikely given the fitting of two distributions and then comparisons of the extreme tails of each distribution. If this occurs because the Mixed Model defaults to the single season, explain this. Although, I doubt it would be exactly the same to 1 or 2 decimal places.
  
  Yes, it occurs as you said because the Mixed Model defaults to the single season and this will be added.
  Minor issues:
  
  Many THANKS, they will be amended.
  
  Citation: https://doi.org/10.5194/hess-2022-195-AC1

Status: closed

RC1:
'Comment on hess-2022-195', Anonymous Referee #1, 08 Jul 2022
This work presents an innovative formulation for the low-flow frequency analysis which accounts for the seasonal behavior of the low-flow that characterizes some regions.

The manuscript is very well written and organized and the research is of significant scientific interest. To my opinion, the work can be published after some minor revisions that could make it clearer.

Please, read in the following my suggestions.

The title suggests that a part 2 exists. If so, please refers to it in the manuscript. What does it deal with?

I suggest to include a figure that synthesizes the seasonal characteristics of the analyzed dataset, e.g. empirical cumulative distribution of the observed low-flow values, mean monthly series, so to appreciate the effectiveness of the mixed proposed approach.

Sometimes variables are not explicitly defined. Please, make sure of that. For example, at line 129, define clearly the utilized circular seasonality index (r) and the seasonality ratio (SR). I see the reference to Laaha nd Blooschl (2006b), but I suggest to make the manuscript self-structured.

The values of the relative frequency in Figure 3 are very low; the discussion cannot be appreciated very much. Maybe I would add the cumulative relative frequency or the CDF

Comments to table 2 at lines L241-245 are a bit confusing. What does it mean 32 to 34 cases? I don’t have the total number of gauges to verify the 10%. Please, refer also to the syntax used in the header of the table, i.e. 1^st and 3^rd

Please, make clear in the text among which variables the correlations are computed (section 4.2.1)

Please, note that Figure 4 is neither recalled nor discussed in the text. Add comments. Define the Relative error. Which return period does it refer to?

Figure 6: be consistent with the shown variables (relative error not defined, performance gain not defined)

TECHNICAL CORRECTIONS

Please see in the following technical observations:

L70: “Extreme events”

Figure 1: make consistent the red colors (line and circles)

Figure 2: please add the total number of the analyzed gauges, even in the text.

Table 1: please add the symbol rd_T of the relative deviation.

I would swap Table 1 and Table 2, since this last is discussed earlier.
Citation: https://doi.org/10.5194/hess-2022-195-RC1
- AC2: 'Reply on RC1', Gregor Laaha, 14 Oct 2022
  
  I thank the reviewer for the very positive and constructive feedback. Please find my responses below.
  1. The title suggests that a part 2 exists. If so, please refers to it in the manuscript. What does it deal with?
  
  Thanks, now mentioned in the abstract and in the introduction (L 60):
  
  In a two-part series we aim to fill this research gap. In this first part we propose a mixed distribution approach for low-flows to perform frequency analysis in catchments with a mixed summer/winter regime. The method is based on an independence assumption which is explored in the second part of the series \citep{Laaha_copula_2022}, where we address possible seasonal dependency using a copula-based estimator.
  2. I suggest to include a figure that synthesizes the seasonal characteristics of the analyzed dataset, e.g. empirical cumulative distribution of the observed low-flow values, mean monthly series, so to appreciate the effectiveness of the mixed proposed approach.
  
  The seasonal characteristics are characterized in Figure 5 and a detailed assessment is given in Laaha and Blöschl 2006ab. I am therefore reluctant to add “more of the same” information to this paper. However, I see your point and will add an additional Figure (attached) with regime plots (monthly mean and monthly low flow regime) of the four example gauges of Fig. 1.
  
  3. Sometimes variables are not explicitly defined. Please, make sure of that. For example, at line 129, define clearly the utilized circular seasonality index (r) and the seasonality ratio (SR). I see the reference to Laaha and Blooschl (2006b), but I suggest to make the manuscript self-structured.
  
  Added to the beginning of the section, which is now formulated in the following way:
  
  Using archetypal examples, we demonstrate how the model behaves for a range of low-flow regimes: from weakly seasonal ones where summer and winter occurrences are equally likely, to strongly seasonal ones where low-flows in one season predominate. Seasonality is characterized by two indices: the seasonality ratio (SR), where SR > 1 indicates a winter and SR < 1 a summer low flow regime, and the circular seasonality index (r), where r=0 indicates the weakest and r=1 the strongest possible seasonality for the definition of indices see {laaha_seasonality_2006}).
  
  Seasonality is characterized by two indices: the seasonality ratio (SR), where SR > 1 indicates a winter and SR < 1 a summer low flow regime, and the circular seasonality index (r), where r=0 indicates the weakest and r=1 the strongest possible seasonality (for the definition of indices see{laaha_seasonality_2006}).
  
  4. The values of the relative frequency in Figure 3 are very low; the discussion cannot be appreciated very much. Maybe I would add the cumulative relative frequency or the CDF
  
  Will be added.
  
  5. Comments to table 2 at lines L241-245 are a bit confusing. What does it mean 32 to 34 cases? I don’t have the total number of gauges to verify the 10%. Please, refer also to the syntax used in the header of the table, i.e. 1st and 3rd
  
  The number of catchments is mentioned in L. 178. This number is also now added to L. 241, and to the caption of Fig. 2, and reference to the syntax in the header is given.
  
  6. Please, make clear in the text among which variables the correlations are computed (section 4.2.1)
  
  Between the accuracy gain and three seasonality indices (see first sentence of this paragraph).
  
  7. Please, note that Figure 4 is neither recalled nor discussed in the text. Add comments. Define the Relative error. Which return period does it refer to?
  
  Thanks, will be included in the first sentence of the paragraph.
  
  8. Figure 6: be consistent with the shown variables (relative error not defined, performance gain not defined)
  
  The term error was defined as a synonym for the deviation when we place emphasis on the inferior model (L. 194). Analogously, the term gain was defined as the change in performance of the superior model compared to the inferior model (L. 192). All have been used throughout the text.
  Technical corrections:
  
  Many thanks, will be amended in the revised version of the MS.
  
  Citation: https://doi.org/10.5194/hess-2022-195-AC2
RC2:
'Comment on hess-2022-195', Anonymous Referee #2, 12 Oct 2022

This study develops a mixed distribution approach for low flow frequency analysis, particularly focused on regions that potentially have a distinct winter low flow period, due to freeze/snowfall/lack of moisture, and a summer flow flow period, due to high ET and low precipitation. The approach is built off similar mixed distribution statistics used for flood frequency analysis, but applies it in a unique way, due to the nuances of low flow. The paper first develops the concepts/statistical underpinning and then applies it to characteristic sites in Austria to show how the model responds when hydrologic drought is dominated by one season or a mix of both seasons.

Overall, I believe this is a useful and well conceived study. I list a few major comments that I believe should be addressed before this study could be published. With that said, I do believe this study has the novelty and value to ultimately be published in HESS.

I ultimately recommend a major revision.

Major issues:

1. Line 85 states that summer and winter events are independent of one another. This is a relatively strong assumption that underlies the method. I question this assumption. Given that baseflow has long persistence along with the climate drivers that create hydrologic drought, I imagine potential for strong temporal autocorrelation between summer and winter events.

Please test the temporal correlation of Summer and Winter, lagged in either direction.

To further eliminate the potential for temporal autocorrelation, I recommend checking the dates of each hydrologic drought and potentially including a buffer period. If November 1 is the division between Summer and Winter (see comment 3 below), please make sure there are no years where Summer drought occurs on Oct 25 and Winter drought occurs Nov 7, for example.

2. Overall issue - use of the word "gain" and, for example line 12 of the abstract "the error is reduced by ...". I am not convinced, if the underlying assumptions are not checked (above), that the new mixed distribution estimates are exactly correct. Therefore, this phrasing that the estimates are improved by XXX% or that error is reduced by XXXX% should be softened or modified. In the derivation of gain (Eqs 10-12), this is referred to as "change in return period" or "relative deviation". Those terms, e.g. Delta T, are more neutral, and in my opinion more accurate, than stating that error is reduced by XXXX. I would be more willing to accept the stronger wording only with centuries of simulated data.

3. Line 36 - You state that many studies have suggested defining summer from about April or May to November, and winter as remaining. But, you never state how you are defining the two seasons for the analysis in Sections 2.4 and 3. Is this the division? Is it April or May? Please provide the exact day.

4. Please provide how many years of data are used for each river (n = ?).

5. Line 242 - Please explain how the deviation can be exactly zero. I find this highly unlikely given the fitting of two distributions and then comparisons of the extreme tails of each distribution. If this occurs because the Mixed Model defaults to the single season, explain this. Although, I doubt it would be exactly the same to 1 or 2 decimal places.

This shows up in Figure 4 as well and is confusing.

Minor issues:

Line 14 - misspelling "broade"

Line 70 - misspelling "extreme events"

Line 130 - I appreciate the link to Laaha and Bloschl 2006, but the first time you introduce SR, please insert the sentence from the caption of Fig 5 "SR > 1 indicates ..., SR < 1 indicates." Perhaps add in "Seasonality ratio ranges from XXX to YYYY".

Line 216 - spelling of Pearson

Line 270 - "Despite there is a large". Please revise for grammar

Line 285 - It looks like there are some words missing "as the have an "

Line 301 - misspelling "one"

Citation: https://doi.org/10.5194/hess-2022-195-RC2
- AC1: 'Reply on RC2', Gregor Laaha, 14 Oct 2022
  
  We thank the reviewer for the positive and constructive feedback. Please find my responses below.
  
  1. Line 85 states that summer and winter events are independent of one another. This is a relatively strong assumption that underlies the method…
  
  Response: I fully agree with this statement that the methods proposed in the paper rely on the independence assumption and there will be certainly catchments where the assumption is not strictly fulfilled. This sis similar to the iid assumption of other estimators that is often not strictly fulfilled in practice, yet useful for model development.
  
  I have therefore structured the research in a two-part paper series where the first part develops methods assuming there that dependency will have no significant effect. The second part extends the scope will explore seasonal dependency in detail. It examines the value of an extended estimator that accounts for the seasonal correlation of low-flow events and assesses its value in a hydrological context. This companion paper (hess-2022-358 A mixed distribution approach for low-flow frequency analysis – Part 2: Modeling dependency using a copula-based estimator) was submitted to HESS on 13 Oct, and is currently in the state of editor assignment.
  
  In the MS of this paper, the effect of possible seasonal correlation has been discussed at the end of the paper and I see from your comment that it should come earlier, where the reader starts to question the assumptions of this paper. We will therefore add at Line 90:
  
  It should be noted that the assumption of strict seasonal independence will only be met in part of the catchments, while there may be cases where some dependency of seasonal minima exists. This is explored in detail in the second part of this two-paper series \citep{Laaha_copula_2022}, where we examine the value of an extended estimator that accounts for the seasonal correlation of low-flow events.
  
  I need to note here that the results of the second part show that there is indeed some correlation in a part of the catchments, but this has only impact on the estimates of mild (e.g. 2-year) low flow events, which are usually of little interest. For severe events the difference is negligible what supports the validity of the estimates of this study (and supports the interpretation of “deviation” being a “gain” based on being a valid generalization of the common estimator).
  2. Overall issue - use of the word "gain" and, for example line 12 of the abstract "the error is reduced by ...". I am not convinced, if the underlying assumptions are not checked (above), that the new mixed distribution estimates are exactly correct…
  
  I see your point, but argue from the findings of the companion paper (where not only the assumptions, but also their impact on the estimates have been checked) that for the return periods we are usually interested in, the differences between the mixed and the extended mixed copula estimator are negligible so that the mixed distribution approach is indeed a valid and accurate estimator of the low-flow event. Given that the terminology is also set out in Section 3.2.1, I would like to keep this wording, but suggest to add the following sentence (Line 212):
  
  It should be noted again that the interpretation of the deviation as a gain depends on the (reasonable) assumption that the model is superior to the alternative model, and the terminology of the study should be understood as such.
  3. Line 36 - You state that many studies have suggested defining summer from about April or May to November, and winter as remaining. But, you never state how you are defining the two seasons for the analysis in Sections 2.4 and 3. Is this the division? Is it April or May? Please provide the exact day.
  
  Thanks – The period in the Austrian study starts in April and this will be added to the MS.
  4. Please provide how many years of data are used for each river (n = ?).
  
  Thanks – will be added.
  5. Line 242 - Please explain how the deviation can be exactly zero. I find this highly unlikely given the fitting of two distributions and then comparisons of the extreme tails of each distribution. If this occurs because the Mixed Model defaults to the single season, explain this. Although, I doubt it would be exactly the same to 1 or 2 decimal places.
  
  Yes, it occurs as you said because the Mixed Model defaults to the single season and this will be added.
  Minor issues:
  
  Many THANKS, they will be amended.
  
  Citation: https://doi.org/10.5194/hess-2022-195-AC1

Gregor Laaha

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Latest update: 11 Jan 2023

Short summary

Knowing the return period of an extreme event is of particular importance to hydrology and water policy. In this paper we propose a mixed distribution approach for low-flow frequency analysis. It provides one consistent approach to measure the severity of summer, winter, and annual low flows. We show that the mixed estimator is much more accurate than the general frequency analysis and should be used by engineers and water agencies for a wide range of seasonal low-flow regimes.


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