the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Development of flexible double distribution quantile mapping for better bias correction in precipitation of GCMs
Abstract. The double gamma quantile mapping (DGQM) can outperform single gamma quantile mapping (SGQM) for bias correction of global circulation models (GCMs) using two gamma functions for two segments based on 90^{th} quantile. However, there are two ambiguous points: the 90^{th} quantile and considering only the Gamma probability function. Therefore, this study introduced a flexible dividing point, δ (%), which can be adjusted to the regionally observed values at the station and considered the combination of various probability distributions, Weibull, lognormal, and Gamma, for two separate segments. The newly proposed method, flexible double distribution quantile mapping (FDDQM), was employed to correct the bias of 8 GCMs of Coupled Model Intercomparison Project Phase 6 (CMIP6) to correct bias at 22 stations in South Korea. The results clearly showed higher performance of FDDQM than DGQM and FlexibleDGQM (FDGQM) by 25 % and 5 %, respectively, in root mean square error. The FDGQM also showed better performance in replicating probability distribution, spatial variability and extremes of observed precipitation than other methods. This study contributes to improving the bias correction method for the better projection of extreme values.
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RC1: 'Comment on hess2022107', Anonymous Referee #1, 24 Apr 2022
This manuscript developed a new bias correction method that improved the popularly used double gamma quantile mapping method. It showed the out performances of the flexible double distribution quantile mapping method over single and double gamma quantile mapping methods in terms of various performance indicators. In addition, this study showed the intermediate improved method, the flexible double gamma quantile mapping method. Therefore, readers could easily follow the strengths of the new method developed in this study. In my opinion, it has clear originality compared to any other previous studies and has shown the distinct reasons why the flexible double distribution quantile method should be used in the bias correction of general circulation models. Therefore, it is worthwhile being published in HESS. However, the miscellaneous following must be checked and revised carefully.
 A few notes on annual rainfall value might help in assessing the typical climate patterns over the study area. Describing trend of rainfall values would be nice. Also, where was the climate data collected? [L91]
 3.3 Flexible double gamma quantile mapping (FDGQM)  > bold [L156]
 I wonder why authors use RMSE to determine delta. For example, several studies use AIC or BIC to find a suitable distribution. However, the authors determined the distribution using only RMSE. Please tell us why you used RMSE.
 Because JSD and KLD evaluated the performance of biascorrected precipitation, Sections 37 should be included in Sections 35.
 As shown in Figure 5, the most suitable deltas are at both extremes. Authors should add a discussion of the determined delta to section 5 or section 41.
 The authors should elaborate on the metrics results in Figures 6 and 11. For example, MD is more sensitive to extreme values ââthan NSE.
 Figure 6: bais  > bias
 In the scatter plot of Figure 8, it isn't easy to discern the difference between FDGQM and DGQM. Therefore, the authors should remove figure 8 as these results have already demonstrated a difference with the evaluation indices. It would also be nice to present an annual time series figure, but it is unnecessary to add it.
 My comment on Figure 10 is like Figure 5. Authors need to improve their results.
 My comment on Figure 14 is like Figure 8.
 To avoid confusion about what the difference is in Figure 15, the authors need to indicate in the title.
 The authors should add about this study later in section 6 for the improvement of this study.
 The authors need to improve the resolution for the legend in Figure 1.
 How about moving Figure 1 to Section 21? It doesn't fit with section 22.
 Authors should specify the number of grids in the study area [L132].
 Improve the resolution for Figure 16. Or find another way to show the results clearly.
 Authors should state the limitations of the study in the conclusion. For example, authors may use a variety of metrics.
 double gamma quantile mapping [L494]  > DGQM
 The shapes of distributions in Figure 2 and 3 looks same. The shape in Figure 3 can be different because various distributions can be used here.
 L8284 Check the grammar.
 The lowest value of delta is 80% in this study. How about considering lower values below 80%?
 L330331: Make a paragraph.
 What is SD in Table 2?
 L448449 Check the meaning.
 The future study should be described in the end of conclusion. I want to know the future plan based on this new technique. For example, more distributions can be considered. The dataset for more stations. More GCMs can be used.
 The format of references must be checked. ex.) Heo et al. (2019), ...
 Check the following errors.
L189 add “and” after “outputs”.
L213 add “and” before “k”.
L433 add “in” after “boxplots”
L434 change “The” by “It” or “This result”.
Citation: https://doi.org/10.5194/hess2022107RC1  AC4: 'Reply on RC1', EunSung Chung, 19 Jun 2022

RC2: 'Comment on hess2022107', Anonymous Referee #2, 24 Apr 2022
This study proposed a new flexible double distribution quantile mapping (FDDQM) method to correct the bias of global climate models (GCMs), through determine the varied δ value to divide the probability distribution of precipitation into two segments and fit these segments with three distributions rather than only the gamma distribution to improve the performance of bias correction. The results show that the FDDQM method performs best in correcting bias in extreme precipitation compared with the flexible double gamma quantile mapping (FDGQM), the double gamma quantile mapping (DGQM) and the single gamma quantile mapping (SGQM) methods.
From my perspective, this topic is not very innovate using different δ values to determe precipitation extremes and using different distribution for different segments. Using the monthly temporal scale data to validate the effectiveness of this proposed FDDQM is not strongly persuasive.
My main concerns are:
 The section of Introduction is not well written, and the logic is not so clear. For example, what is the relationship between the paragraph 2 and 3? In Paragraph 2, you introduced the advantage of QDM method that can address the drawbacks of QM and also cited several different categories of methods developed based on QDM, thus according to the normal logic, it should describe the method of QDM in Paragraph 3 rather than the QM. In addition, without giving the reasons that “QM does not always outperform other biascorrection methods at all locations”, how did you get the conclusion of “this emphasizes choosing an appropriate probability distribution function for successful bias correction”? Furthermore, the objectives/problems aimed to be achieved/addressed are not properly stated in the last paragraph.
 In this study, using the monthlyscale precipitation to test the effectiveness of the newly proposed flexible double distribution quantile mapping method in correcting the bias of GCM is not proper and it cannot well reflect the extreme precipitation characteristics. The authors must valid the performance of this method in biascorrecting of dailyscale GCMs, the daily data present larger spatial variability and are more useful for climate change studies.
 This paper is written casually, and there exist many grammar and tense problems, which needs to be polished by native English speakers.
Specific comments:
Line 1415: It is not appropriate to directly use the 90^{th} quantile to reflect the question, because in many references they may also using the 95^{th} or 99^{th} quantile rather than only the 90^{th} quantile.Line 15: “Gamma probability distribution function” instead of “Gamma probability function”.
Line 17: “consider” instead of “considered”.
Line 18: add “e.g.” before “Weibull, lognormal…..”, add “the” before “ two separate segments”.
Line 20: delete “to correct bias”.
Line 21 and 23: the tense is wrong. “show” instead of “show”.
Line 25: delete “the” before “better projection of extreme values”.
Line 45: why use “but” when there is no turning point?
Line 55: “and” < “with” or “using”.
Line 75: “is aimed to propose” instead of “proposed”.
Line 77: Why do you choose these three PDFs since you mentioned that the most appropriate distribution can be different for different regions in Line 72? Whether the only three PDFs are too few? Are these three PDFs suitable for precipitation extremes, i.e., the segment larger than the given threshold like 90^{th} quantile? Are these three PDFs suitable for precipitation extremes, i.e., the segment larger than the given threshold like 90^{th} quantile?
Line 79: Why only use RMSE to select the dividing point? Are there any indicators that are more suitable to select the dividing point?
Line 81: add “method” after “The performance of the proposed”.
Line 8384: What do you mean “the performance …………based on GEV distribution? It doesn’t seem to be a complete sentence.
Line 126: “based on” instead of “for”.
Line 127134: What’s the meaning of those variables in Eq. (1) – Eq. (2)? They are not stated properly and could not be well understood. How do you determine the surrounding grids close to one specific location that are used in Eq. (1) – Eq. (2)?
Line 144: Is the F^{1}_{g} is Eq. (3) correct? It is very easy to consider F^{1}_{g} as the converse function of F_{g}. Please explain the variables in a proper way.
In Section 3.33.4: These two parts are the core contents of this study, but the relevant information is two little. The detailed calculation process should be described here.
In Line 160: Why do you choose the δ values between 80%95%? It seems also very random like other studies. In addition, how do you use RMSE to determine the δ has not been clearly given.
Line 161: “determine” instead of “determined”. Add “the” before “optimal RMSE”.
Line 169: Add “the” before “Gamma distribution”. There are so many places where “the” has not been properly used or not been added. Please check it in the whole paper.
In Figure 2 and 3, what are the differences? What are your points?
Line 179: delete “the” before “other climate variables”.
Line 195: add “by” before “positive value”.
Line 202: Please check the correction of the Eq. (8).
Line 208219: In Section 3.6, the aim of using the Generalized extreme value distribution in this paper should be firstly explained.
Line 218: “biascorrected precipitation” rather than “precipitation biascorrected”.
Line 221: add the abbreviation KLD and JSD in the subtitle.
Line 245: The most selected quantile is the 80^{th} that can be seen from Fig. 4, is this related with the lower bound of the δ values you set in this paper? This means whether the most selected δ value will be smaller than the 80^{th} quantile if the lower bound of δ values is set lower than the 80^{th} quantile. Similar for the second most selected 95^{th} quantile.
Line 255256: Please rewrite the title of Figure 5, same for Figure 10.
Line 301: The tense should be the present tense when describing the founded results. Please check in the whole paper.
Line 312: add “distribution” after “Weibull”. Please check the tense.
For all figures, delete the “The” at the beginning place of the corresponding title.
Line 372: “based on” instead of “based”.
Line 418419: Please ensure the sentence is complete.
Line 433: Add “in” before “Figure 17”. There are many places that the sentences are not complete.
Line 435: What do you mean by this sentence?
Line 447: Since you mentioned future projection, how do you determine the δ value for extreme precipitation in the future period using the methods in this study?
Line 466 and Line 475: Where are the figures for the performances of different fitted distributions, like the gamma distribution and the Weibull distribution?
Line 441491: In the section of Discussion, the discussion should be strengthened rather than repeating describing the results in the part of results.
Citation: https://doi.org/10.5194/hess2022107RC2  AC2: 'Reply on RC2', EunSung Chung, 19 Jun 2022

CC1: 'Comment on hess2022107', Qin Zhang, 18 May 2022
General comments:
This manuscript proposed two bias correction methods (FDDQM and FDGQM) based on DGQM for monthly precipitation. The results show that FDDQM method performs best in correcting bias in precipitation compared with FDGQM, DGQM and SGQM methods. Recently, our team conducted a similar work, which has been accepted at International Journal of Climatology (Paper title: Piecewise‐Quantile Mapping Improves Bias Correction of GCM Daily Precipitation towards Preserving Quantiles and Extremes; DOI: https://doi.org/10.1002/joc.7687).
In my opinions, there are some questions need to be explained and addressed for this manuscript.
 It is not really appropriate to use monthly precipitation to validate the proposed methods in correcting precipitation extremes from my perspective. For the precipitation extremes, the daily or subdaily scale precipitation data is required.
 What is the relationship between FDDQM and FDGQM authors proposed? Since the FDDQM performs best because of the consideration of diverse distribution function, what is the point of the FDGQM's existence?
 Segmenting the precipitation series to two fragments by selecting an optimal threshold, is a good idea. But in my opinions, it is not optimal to use QM based on the theoretical distribution function approach for two different sequences. For the nonextreme series, the nonparametric transformation, i.e., interpolation method in QQ plot, is able to capture more precipitation information compared with parametric transformation method and theoretical distribution function method. For the extreme series, when the future precipitation extremes lie outside the domain of historical model data, the simple extrapolation algorithm, such as linear, cubic, and spline interpolation, might lead to great bias. So, in this situation, the theoretical distribution function can be applied due to its advantage of extending the data reasonably.
 For equation 3, the F_{g} corresponds the GCM outputs and the F_{g}^{1} corresponds the observed data. I think this is wrong. Different letters subscript should be used.
 I don’t know why the authors used the GEV to fit the corrected models data compared with observation. A direct comparison of the empirical distribution functions of the extreme value series seems more appropriate. Additionally, for the extreme series obtained by POT model, GP distribution is generally more appropriate, rather than GEV.
Citation: https://doi.org/10.5194/hess2022107CC1  AC3: 'Reply on CC1', EunSung Chung, 19 Jun 2022
Status: closed

RC1: 'Comment on hess2022107', Anonymous Referee #1, 24 Apr 2022
This manuscript developed a new bias correction method that improved the popularly used double gamma quantile mapping method. It showed the out performances of the flexible double distribution quantile mapping method over single and double gamma quantile mapping methods in terms of various performance indicators. In addition, this study showed the intermediate improved method, the flexible double gamma quantile mapping method. Therefore, readers could easily follow the strengths of the new method developed in this study. In my opinion, it has clear originality compared to any other previous studies and has shown the distinct reasons why the flexible double distribution quantile method should be used in the bias correction of general circulation models. Therefore, it is worthwhile being published in HESS. However, the miscellaneous following must be checked and revised carefully.
 A few notes on annual rainfall value might help in assessing the typical climate patterns over the study area. Describing trend of rainfall values would be nice. Also, where was the climate data collected? [L91]
 3.3 Flexible double gamma quantile mapping (FDGQM)  > bold [L156]
 I wonder why authors use RMSE to determine delta. For example, several studies use AIC or BIC to find a suitable distribution. However, the authors determined the distribution using only RMSE. Please tell us why you used RMSE.
 Because JSD and KLD evaluated the performance of biascorrected precipitation, Sections 37 should be included in Sections 35.
 As shown in Figure 5, the most suitable deltas are at both extremes. Authors should add a discussion of the determined delta to section 5 or section 41.
 The authors should elaborate on the metrics results in Figures 6 and 11. For example, MD is more sensitive to extreme values ââthan NSE.
 Figure 6: bais  > bias
 In the scatter plot of Figure 8, it isn't easy to discern the difference between FDGQM and DGQM. Therefore, the authors should remove figure 8 as these results have already demonstrated a difference with the evaluation indices. It would also be nice to present an annual time series figure, but it is unnecessary to add it.
 My comment on Figure 10 is like Figure 5. Authors need to improve their results.
 My comment on Figure 14 is like Figure 8.
 To avoid confusion about what the difference is in Figure 15, the authors need to indicate in the title.
 The authors should add about this study later in section 6 for the improvement of this study.
 The authors need to improve the resolution for the legend in Figure 1.
 How about moving Figure 1 to Section 21? It doesn't fit with section 22.
 Authors should specify the number of grids in the study area [L132].
 Improve the resolution for Figure 16. Or find another way to show the results clearly.
 Authors should state the limitations of the study in the conclusion. For example, authors may use a variety of metrics.
 double gamma quantile mapping [L494]  > DGQM
 The shapes of distributions in Figure 2 and 3 looks same. The shape in Figure 3 can be different because various distributions can be used here.
 L8284 Check the grammar.
 The lowest value of delta is 80% in this study. How about considering lower values below 80%?
 L330331: Make a paragraph.
 What is SD in Table 2?
 L448449 Check the meaning.
 The future study should be described in the end of conclusion. I want to know the future plan based on this new technique. For example, more distributions can be considered. The dataset for more stations. More GCMs can be used.
 The format of references must be checked. ex.) Heo et al. (2019), ...
 Check the following errors.
L189 add “and” after “outputs”.
L213 add “and” before “k”.
L433 add “in” after “boxplots”
L434 change “The” by “It” or “This result”.
Citation: https://doi.org/10.5194/hess2022107RC1  AC4: 'Reply on RC1', EunSung Chung, 19 Jun 2022

RC2: 'Comment on hess2022107', Anonymous Referee #2, 24 Apr 2022
This study proposed a new flexible double distribution quantile mapping (FDDQM) method to correct the bias of global climate models (GCMs), through determine the varied δ value to divide the probability distribution of precipitation into two segments and fit these segments with three distributions rather than only the gamma distribution to improve the performance of bias correction. The results show that the FDDQM method performs best in correcting bias in extreme precipitation compared with the flexible double gamma quantile mapping (FDGQM), the double gamma quantile mapping (DGQM) and the single gamma quantile mapping (SGQM) methods.
From my perspective, this topic is not very innovate using different δ values to determe precipitation extremes and using different distribution for different segments. Using the monthly temporal scale data to validate the effectiveness of this proposed FDDQM is not strongly persuasive.
My main concerns are:
 The section of Introduction is not well written, and the logic is not so clear. For example, what is the relationship between the paragraph 2 and 3? In Paragraph 2, you introduced the advantage of QDM method that can address the drawbacks of QM and also cited several different categories of methods developed based on QDM, thus according to the normal logic, it should describe the method of QDM in Paragraph 3 rather than the QM. In addition, without giving the reasons that “QM does not always outperform other biascorrection methods at all locations”, how did you get the conclusion of “this emphasizes choosing an appropriate probability distribution function for successful bias correction”? Furthermore, the objectives/problems aimed to be achieved/addressed are not properly stated in the last paragraph.
 In this study, using the monthlyscale precipitation to test the effectiveness of the newly proposed flexible double distribution quantile mapping method in correcting the bias of GCM is not proper and it cannot well reflect the extreme precipitation characteristics. The authors must valid the performance of this method in biascorrecting of dailyscale GCMs, the daily data present larger spatial variability and are more useful for climate change studies.
 This paper is written casually, and there exist many grammar and tense problems, which needs to be polished by native English speakers.
Specific comments:
Line 1415: It is not appropriate to directly use the 90^{th} quantile to reflect the question, because in many references they may also using the 95^{th} or 99^{th} quantile rather than only the 90^{th} quantile.Line 15: “Gamma probability distribution function” instead of “Gamma probability function”.
Line 17: “consider” instead of “considered”.
Line 18: add “e.g.” before “Weibull, lognormal…..”, add “the” before “ two separate segments”.
Line 20: delete “to correct bias”.
Line 21 and 23: the tense is wrong. “show” instead of “show”.
Line 25: delete “the” before “better projection of extreme values”.
Line 45: why use “but” when there is no turning point?
Line 55: “and” < “with” or “using”.
Line 75: “is aimed to propose” instead of “proposed”.
Line 77: Why do you choose these three PDFs since you mentioned that the most appropriate distribution can be different for different regions in Line 72? Whether the only three PDFs are too few? Are these three PDFs suitable for precipitation extremes, i.e., the segment larger than the given threshold like 90^{th} quantile? Are these three PDFs suitable for precipitation extremes, i.e., the segment larger than the given threshold like 90^{th} quantile?
Line 79: Why only use RMSE to select the dividing point? Are there any indicators that are more suitable to select the dividing point?
Line 81: add “method” after “The performance of the proposed”.
Line 8384: What do you mean “the performance …………based on GEV distribution? It doesn’t seem to be a complete sentence.
Line 126: “based on” instead of “for”.
Line 127134: What’s the meaning of those variables in Eq. (1) – Eq. (2)? They are not stated properly and could not be well understood. How do you determine the surrounding grids close to one specific location that are used in Eq. (1) – Eq. (2)?
Line 144: Is the F^{1}_{g} is Eq. (3) correct? It is very easy to consider F^{1}_{g} as the converse function of F_{g}. Please explain the variables in a proper way.
In Section 3.33.4: These two parts are the core contents of this study, but the relevant information is two little. The detailed calculation process should be described here.
In Line 160: Why do you choose the δ values between 80%95%? It seems also very random like other studies. In addition, how do you use RMSE to determine the δ has not been clearly given.
Line 161: “determine” instead of “determined”. Add “the” before “optimal RMSE”.
Line 169: Add “the” before “Gamma distribution”. There are so many places where “the” has not been properly used or not been added. Please check it in the whole paper.
In Figure 2 and 3, what are the differences? What are your points?
Line 179: delete “the” before “other climate variables”.
Line 195: add “by” before “positive value”.
Line 202: Please check the correction of the Eq. (8).
Line 208219: In Section 3.6, the aim of using the Generalized extreme value distribution in this paper should be firstly explained.
Line 218: “biascorrected precipitation” rather than “precipitation biascorrected”.
Line 221: add the abbreviation KLD and JSD in the subtitle.
Line 245: The most selected quantile is the 80^{th} that can be seen from Fig. 4, is this related with the lower bound of the δ values you set in this paper? This means whether the most selected δ value will be smaller than the 80^{th} quantile if the lower bound of δ values is set lower than the 80^{th} quantile. Similar for the second most selected 95^{th} quantile.
Line 255256: Please rewrite the title of Figure 5, same for Figure 10.
Line 301: The tense should be the present tense when describing the founded results. Please check in the whole paper.
Line 312: add “distribution” after “Weibull”. Please check the tense.
For all figures, delete the “The” at the beginning place of the corresponding title.
Line 372: “based on” instead of “based”.
Line 418419: Please ensure the sentence is complete.
Line 433: Add “in” before “Figure 17”. There are many places that the sentences are not complete.
Line 435: What do you mean by this sentence?
Line 447: Since you mentioned future projection, how do you determine the δ value for extreme precipitation in the future period using the methods in this study?
Line 466 and Line 475: Where are the figures for the performances of different fitted distributions, like the gamma distribution and the Weibull distribution?
Line 441491: In the section of Discussion, the discussion should be strengthened rather than repeating describing the results in the part of results.
Citation: https://doi.org/10.5194/hess2022107RC2  AC2: 'Reply on RC2', EunSung Chung, 19 Jun 2022

CC1: 'Comment on hess2022107', Qin Zhang, 18 May 2022
General comments:
This manuscript proposed two bias correction methods (FDDQM and FDGQM) based on DGQM for monthly precipitation. The results show that FDDQM method performs best in correcting bias in precipitation compared with FDGQM, DGQM and SGQM methods. Recently, our team conducted a similar work, which has been accepted at International Journal of Climatology (Paper title: Piecewise‐Quantile Mapping Improves Bias Correction of GCM Daily Precipitation towards Preserving Quantiles and Extremes; DOI: https://doi.org/10.1002/joc.7687).
In my opinions, there are some questions need to be explained and addressed for this manuscript.
 It is not really appropriate to use monthly precipitation to validate the proposed methods in correcting precipitation extremes from my perspective. For the precipitation extremes, the daily or subdaily scale precipitation data is required.
 What is the relationship between FDDQM and FDGQM authors proposed? Since the FDDQM performs best because of the consideration of diverse distribution function, what is the point of the FDGQM's existence?
 Segmenting the precipitation series to two fragments by selecting an optimal threshold, is a good idea. But in my opinions, it is not optimal to use QM based on the theoretical distribution function approach for two different sequences. For the nonextreme series, the nonparametric transformation, i.e., interpolation method in QQ plot, is able to capture more precipitation information compared with parametric transformation method and theoretical distribution function method. For the extreme series, when the future precipitation extremes lie outside the domain of historical model data, the simple extrapolation algorithm, such as linear, cubic, and spline interpolation, might lead to great bias. So, in this situation, the theoretical distribution function can be applied due to its advantage of extending the data reasonably.
 For equation 3, the F_{g} corresponds the GCM outputs and the F_{g}^{1} corresponds the observed data. I think this is wrong. Different letters subscript should be used.
 I don’t know why the authors used the GEV to fit the corrected models data compared with observation. A direct comparison of the empirical distribution functions of the extreme value series seems more appropriate. Additionally, for the extreme series obtained by POT model, GP distribution is generally more appropriate, rather than GEV.
Citation: https://doi.org/10.5194/hess2022107CC1  AC3: 'Reply on CC1', EunSung Chung, 19 Jun 2022
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