22 Feb 2021
22 Feb 2021
Impact of Flood Control Systems on the Probability Distribution of Floods
 ^{1}Dipartimento di Ingegneria Civile, Edile e Ambientale, Università degli Studi di Napoli Federico II, 80125 Napoli, Italy
 ^{2}Dipartimento di Scienze Agro Ambientali e Territoriali, Università degli Studi di Bari Aldo Moro, 70126 Bari, Italy
 ^{1}Dipartimento di Ingegneria Civile, Edile e Ambientale, Università degli Studi di Napoli Federico II, 80125 Napoli, Italy
 ^{2}Dipartimento di Scienze Agro Ambientali e Territoriali, Università degli Studi di Bari Aldo Moro, 70126 Bari, Italy
Abstract. Detention dams are one of the most effective practices for flood mitigation. Therefore, the impact of these structures on the basin hydrological response is critical for flood management and the design of flood control structures. With the aim to provide a mathematical framework to interpret the effect of flow control systems on river basin dynamics, the functional relationship between inflows and outflows is investigated and derived in a closedform. This allowed the definition of a theoretically derived probability distribution of the peak outflows from inline detention basins. The model has been derived assuming a rectangular hydrograph shape with a fixed duration, and a random flood peak. In the present study, the undisturbed flood distribution is assumed to be Gumbel distributed, but the proposed mathematical formulation can be extended to any other floodpeak probability distribution. A sensitivity analysis of parameters highlighted the influence of detention basin capacity and rainfall event duration on flood mitigation on the probability distribution of the peak outflows. The mathematical framework has been tested using for comparison a Monte Carlo simulation where most of the simplified assumptions used to describe the dam behaviours are removed. This allowed to demonstrate that the proposed formulation is reliable for small river basins characterized by an impulsive response. The new approach for the quantification of flood peaks in river basins characterised by the presence of artificial detention basins can be used to improve existing flood mitigation practices, support the design of flood control systems and flood risk analyses.
Salvatore Manfreda et al.
Status: closed

RC1: 'Comment on hess202147', Anonymous Referee #1, 04 Apr 2021
This is a wellwritten paper and easy to follow that highlights the need to update flood mitigation plans in a changing world. The topic is relevant and timely and suitable for HESS. I enjoyed reading the paper and there are not much to add, thus I will provide a brief review. The key idea is to provide a theoretically derived distribution to describe the flood peaks affected by reservoirs. Deriving theoretically such a distribution is not easy given the many factors affecting flood attenuation. Thus, the authors wish to form a mathematical framework to interpret functional relationships among variables such as flood wave shape and duration, storage capacity and geometric parameters of the detention basin.
The authors are very honest stressing the strong simplifying assumptions used and this is clear throughout the text. Yet as reader I strongly felt that these assumptions and their effects should be discussed in more detail. Starting with equation 4 (I might be missing soothing here) it is not clear to me why Equation 3 cannot be used, meaning that is not much more complicated. Would the use of Eq.3 complicate that much the analytical calculations and make impossible the analytical formulations?
Could you please offer more details on the impacts of the rectangular hydrograph assumption? Yes, it can significantly overestimate the flood volume but how much and under what conditions? How strong in the linearity assumption leading to the same exponent values (eq. 7)?
The symmetric assumption leading to equation 10 how realistic can it be? It is wellknown if I am correct the volume is not symmetrically distribution around the peak.
Could you provide some extra details on the nature of the tails of the derived distribution in Eq 15? It is well accepted in the literature that floods peaks are described by heavy tails (see for example Vogel & Wilson (1996), Villarini and Smith (2010) and recently Miniussi et al. (2020) and Zaghloul et al. (2020)).
Finally, I believe the readers would be very curious to see how the results would be modified if a heavytailed distribution was used instead of the Gumbel which has exponential tail. The exponential tails offer “no surprises” in the generation of random discharge values and thus the good results shown might be case specific and only for the Gumber distribution. What would be the performance if really heavy tailed distributions were used, e.g., a GEV with shape parameter close to 0.5?
Please double check your equations, for example, Equation 13 is not correct, it should be dg^1(y)/dy f(g^1(y))
Overall, I believe this is a good paper and deserves publication pending some corrections, additional discussion and clarifications, and maybe an extra analysis with a heavytailed distribution.
References
Miniussi, A., Marani, M., Villarini, G., 2020. Metastatistical Extreme Value Distribution applied to floods across the continental United States. Adv. Water Res. 136, 103498. https://doi.org/10.1016/j.advwatres.2019.103498
Villarini, G., Smith, J.A., 2010. Flood peak distributions for the eastern United States. Water Resour. Res. 46 (6). https://doi.org/10.1029/2009WR008395
Vogel, R.M. , Wilson, I. , 1996. Probability distribution of annual maximum, mean, and minimum streamflows in the United States. J. Hydrol. Eng. 1 (2), 69–76.
Zaghloul, M., Papalexiou, S.M., Elshorbagy, A., Coulibaly, P., 2020. Revisiting flood peak distributions: A panCanadian investigation. Advances in Water Resources 145, 103720. https://doi.org/10.1016/j.advwatres.2020.103720

AC1: 'Reply on RC1', Salvatore Manfreda, 11 May 2021
We would like to thank the first reviewer for the significative and constructive suggestions, that allowed us to improve the quality of the manuscript and clarify some concepts. We carefully considered all observations and reviewed the manuscript accordingly. Specific comments are addressed in the following using the same order adopted by the reviewer.RC: The authors are very honest stressing the strong simplifying assumptions used and this is clear throughout the text. Yet as reader I strongly felt that these assumptions and their effects should be discussed in more detail. Starting with equation 4 (I might be missing something here) it is not clear to me why Equation 3 cannot be used, meaning that is not much more complicated. Would the use of Eq.3 complicate that much the analytical calculations and make impossible the analytical formulations?AC: Basically, equations 4 and equations 3 differ only for the assumption that the outflow in eq. 4 is assumed equal to the inflow as long as h<hf. This assumption helps in the theoretical derivation of the probability distribution of the peak outflows and affect the shape of the distribution only for the low flows. Moreover, such an assumption may be considered reasonable for the scope of the present manuscript, which is more focused on the right tail of the derived probability distribution of the outflows.RC: Could you please offer more details on the impacts of the rectangular hydrograph assumption? Yes, it can significantly overestimate the flood volume but how much and under what conditions? How strong in the linearity assumption leading to the same exponent values (eq. 7)?AC: Rectangular hydrographs have been used for design purposes in several design applications. As demonstrated in the graph of figure 7, the assumptions is quite reasonable as long as the flood hydrograph has a relatively short duration. This means that the assumption can be used in small river basins with a lagtime of less than one hour.RC: The symmetric assumption leading to equation 10 how realistic can it be? It is wellknown if I am correct the volume is not symmetrically distribution around the peak.AC: The use of a synthetic hydrograph symmetric respect to the peak is an approximation which have a limited impact on the dynamic of the process. It is introduced to provide an estimate of the impact of nonuniform discharge on the lamination process. We could potentially use a different shape of the hydrograph, but this would have increased the number of modelling parameters, while this form represents the simplest form known.RC: Could you provide some extra details on the nature of the tails of the derived distribution in Eq 15? It is well accepted in the literature that floods peaks are described by heavy tails (see for example Vogel & Wilson (1996), Villarini and Smith (2010) and recently Miniussi et al. (2020) and Zaghloul et al. (2020)).AC: The choice of a Gumbel distribution is not mandatory in the proposed schematization. It is absolutely true that the Gumbel distribution is not necessarily the best option for the description of floods, but it represents a reference distribution for flood distributions. As we stated in the text, the choice of the flood distribution can be any of the available in the literature, because the derived laminated discharge can be used to obtain a derived distribution for any flood distribution chosen. We have included some examples of applications based on the assumption of floods distributed according to a Fréchet and a Weibull distribution.RC: Finally, I believe the readers would be very curious to see how the results would be modified if a heavytailed distribution was used instead of the Gumbel which has exponential tail. The exponential tails offer “no surprises” in the generation of random discharge values and thus the good results shown might be case specific and only for the Gumbel distribution. What would be the performance if really heavy tailed distributions were used, e.g., a GEV with shape parameter close to 0.5?AC: In order to satisfy readers’ curiosity, we provide in the following few examples where the peak outflow distributions are derived using flood peaks distributed according to the three types of GEV distributions (Gumbel, Fréchet and Weibull). Reliability of the derived distribution has been tested using a numerical simulation, which demonstrates the good agreement with the theoretical functions.Figure 1. Comparison between three different derived probability density functions of the peak outflows obtained using three different flood peak distributions and the empirical pdfs obtained via numerical hydraulic simulation (red dots for inflows, blue dots for outflows). The three graphs are obtained modifying the shape parameter, Imageof the GEV distribution, which is equal to 0 for Gumbel distribution (A), 0.5 for Fréchet distribution (B) and  0.5 for Weibull distribution (C). Remaining parameters are: the scale parameter of GEV α=30m3/s; the location parameter of GEV β=120 m^{3}/s; w_{1}=5000; h_{s}=4m; b=1m; d=1m; n=1.9; h_{f}= d/2; μ_{f}=0.85; μ_{s}=0.385; L=3m; t_{p}=1h.RC. Please double check your equations, for example, Equation 13 is not correct, it should be dg^1(y)/dy f(g^1(y))AC: We thank the referee for this suggestion. The text has been modified accordingly.

AC1: 'Reply on RC1', Salvatore Manfreda, 11 May 2021

RC2: 'Comment on hess202147', Anonymous Referee #2, 18 Apr 2021
The paper presents a mathematical framework for deriving the probability distribution of peak outflows from a detention basin from the probability distribution of incoming flood peaks. The authors first derive a relationship between the peak outflow from a detention basin and the peak inflow assuming a rectangular inflow hydrograph. This involves the definition of an equivalent delay constant that depends on the properties of the detention basin, and an equivalent flood duration that depends on the lagtime of the river basin. The mathematical framework is applied assuming that the undisturbed flood peaks follow a Gumbel distribution, but the formulation can potentially be applied to any other probability distribution of inflows. The authors also present results from Monte Carlo simulations in which most of the simplifying assumptions are relaxed, and show that the proposed formulation provides a good approximation of the mitigated flood peak distribution in river basins with relatively small concentration time.
The paper is well written and organized, and the mathematical derivations are presented in a clear and logical manner. To the best of my knowledge, the proposed approach is novel, and can be useful for design and environmental impact assessment of flood detention basins.
I recommend some minor changes before publication.
Specific comments:I believe the reference to "flood control systems" in the title raises the expectations beyond what is presented in the paper, as there are other types of flood control systems that cannot be treated with the same mathematical framework proposed by the authors. I therefore recommend changing the title to refer more specially to "flood detention basins".
The symbol D is not defined in the notation list.
Given that the symbol q is used for discharge, I recommend using a different symbol for the height of the lowlevel opening instead of qf.
P80,90: Use italic style for mathematical variables.
P10: "the undisturbed flood distribution is assumed to be Gumbel distributed" => "the undisturbed flood peaks are assumed to be Gumbel distributed"
P50: "see i.e., Manfreda et al." => "e.g." not "i.e."
P165: "... mathematically inverting Equation..." => "mathematically by inverting Equation..."
P140: "computed comparing... and setting..." => "computed by comparing... and imposing..."
P255: Remove "realized"
P235: "the impact due to the approximation adopted by the rectangular hydrographs" => I suggest changing this as follows: "the impact of the assumption of rectangular inflow hydrograph"
P235: "allowed to reproduce correctly the flood mitigation that looks very similar to those..." => "produced probability distributions of the outflow that look very similar to those...".
L375: 15,000.00 => 15,000

AC2: 'Reply on RC2', Salvatore Manfreda, 11 May 2021
we would like to thank also the second referee for his/her effort and constructive suggestions, that allowed us to improve the quality of the manuscript. We have carefully considered all his/her observations and reviewed the manuscript accordingly.RC: I believe the reference to "flood control systems" in the title raises the expectations beyond what is presented in the paper, as there are other types of flood control systems that cannot be treated with the same mathematical framework proposed by the authors. I therefore recommend changing the title to refer more specially to "flood detention basins". The symbol D is not defined in the notation list. Given that the symbol q is used for discharge, I recommend using a different symbol for the height of the lowlevel opening instead of qf.AC: We agree with the reviewer. Title will changed in … "Impact of Detention Dams on the Probability Distribution of Floods". We will also add the flood event duration D in the notation list. Finally, we will change qf in lf.RC: List of minor correctionsP80,90: Use italic style for mathematical variables.P10: "the undisturbed flood distribution is assumed to be Gumbel distributed" => "theundisturbed flood peaks are assumed to be Gumbel distributed"P50: "see i.e., Manfreda et al." => "e.g." not "i.e."P165: "... mathematically inverting Equation..." => "mathematically by invertingEquation..."P140: "computed comparing... and setting..." => "computed by comparing... andimposing..."P255: Remove "realized"P235: "the impact due to the approximation adopted by the rectangular hydrographs" =>I suggest changing this as follows: "the impact of the assumption of rectangular inflowhydrograph"P235: "allowed to reproduce correctly the flood mitigation that looks very similar tothose..." => "produced probability distributions of the outflow that look very similar tothose...".L375: 15,000.00 => 15,000AC: All these suggestions have been already implemented in the revised version of the manuscript.

AC2: 'Reply on RC2', Salvatore Manfreda, 11 May 2021
Status: closed

RC1: 'Comment on hess202147', Anonymous Referee #1, 04 Apr 2021
This is a wellwritten paper and easy to follow that highlights the need to update flood mitigation plans in a changing world. The topic is relevant and timely and suitable for HESS. I enjoyed reading the paper and there are not much to add, thus I will provide a brief review. The key idea is to provide a theoretically derived distribution to describe the flood peaks affected by reservoirs. Deriving theoretically such a distribution is not easy given the many factors affecting flood attenuation. Thus, the authors wish to form a mathematical framework to interpret functional relationships among variables such as flood wave shape and duration, storage capacity and geometric parameters of the detention basin.
The authors are very honest stressing the strong simplifying assumptions used and this is clear throughout the text. Yet as reader I strongly felt that these assumptions and their effects should be discussed in more detail. Starting with equation 4 (I might be missing soothing here) it is not clear to me why Equation 3 cannot be used, meaning that is not much more complicated. Would the use of Eq.3 complicate that much the analytical calculations and make impossible the analytical formulations?
Could you please offer more details on the impacts of the rectangular hydrograph assumption? Yes, it can significantly overestimate the flood volume but how much and under what conditions? How strong in the linearity assumption leading to the same exponent values (eq. 7)?
The symmetric assumption leading to equation 10 how realistic can it be? It is wellknown if I am correct the volume is not symmetrically distribution around the peak.
Could you provide some extra details on the nature of the tails of the derived distribution in Eq 15? It is well accepted in the literature that floods peaks are described by heavy tails (see for example Vogel & Wilson (1996), Villarini and Smith (2010) and recently Miniussi et al. (2020) and Zaghloul et al. (2020)).
Finally, I believe the readers would be very curious to see how the results would be modified if a heavytailed distribution was used instead of the Gumbel which has exponential tail. The exponential tails offer “no surprises” in the generation of random discharge values and thus the good results shown might be case specific and only for the Gumber distribution. What would be the performance if really heavy tailed distributions were used, e.g., a GEV with shape parameter close to 0.5?
Please double check your equations, for example, Equation 13 is not correct, it should be dg^1(y)/dy f(g^1(y))
Overall, I believe this is a good paper and deserves publication pending some corrections, additional discussion and clarifications, and maybe an extra analysis with a heavytailed distribution.
References
Miniussi, A., Marani, M., Villarini, G., 2020. Metastatistical Extreme Value Distribution applied to floods across the continental United States. Adv. Water Res. 136, 103498. https://doi.org/10.1016/j.advwatres.2019.103498
Villarini, G., Smith, J.A., 2010. Flood peak distributions for the eastern United States. Water Resour. Res. 46 (6). https://doi.org/10.1029/2009WR008395
Vogel, R.M. , Wilson, I. , 1996. Probability distribution of annual maximum, mean, and minimum streamflows in the United States. J. Hydrol. Eng. 1 (2), 69–76.
Zaghloul, M., Papalexiou, S.M., Elshorbagy, A., Coulibaly, P., 2020. Revisiting flood peak distributions: A panCanadian investigation. Advances in Water Resources 145, 103720. https://doi.org/10.1016/j.advwatres.2020.103720

AC1: 'Reply on RC1', Salvatore Manfreda, 11 May 2021
We would like to thank the first reviewer for the significative and constructive suggestions, that allowed us to improve the quality of the manuscript and clarify some concepts. We carefully considered all observations and reviewed the manuscript accordingly. Specific comments are addressed in the following using the same order adopted by the reviewer.RC: The authors are very honest stressing the strong simplifying assumptions used and this is clear throughout the text. Yet as reader I strongly felt that these assumptions and their effects should be discussed in more detail. Starting with equation 4 (I might be missing something here) it is not clear to me why Equation 3 cannot be used, meaning that is not much more complicated. Would the use of Eq.3 complicate that much the analytical calculations and make impossible the analytical formulations?AC: Basically, equations 4 and equations 3 differ only for the assumption that the outflow in eq. 4 is assumed equal to the inflow as long as h<hf. This assumption helps in the theoretical derivation of the probability distribution of the peak outflows and affect the shape of the distribution only for the low flows. Moreover, such an assumption may be considered reasonable for the scope of the present manuscript, which is more focused on the right tail of the derived probability distribution of the outflows.RC: Could you please offer more details on the impacts of the rectangular hydrograph assumption? Yes, it can significantly overestimate the flood volume but how much and under what conditions? How strong in the linearity assumption leading to the same exponent values (eq. 7)?AC: Rectangular hydrographs have been used for design purposes in several design applications. As demonstrated in the graph of figure 7, the assumptions is quite reasonable as long as the flood hydrograph has a relatively short duration. This means that the assumption can be used in small river basins with a lagtime of less than one hour.RC: The symmetric assumption leading to equation 10 how realistic can it be? It is wellknown if I am correct the volume is not symmetrically distribution around the peak.AC: The use of a synthetic hydrograph symmetric respect to the peak is an approximation which have a limited impact on the dynamic of the process. It is introduced to provide an estimate of the impact of nonuniform discharge on the lamination process. We could potentially use a different shape of the hydrograph, but this would have increased the number of modelling parameters, while this form represents the simplest form known.RC: Could you provide some extra details on the nature of the tails of the derived distribution in Eq 15? It is well accepted in the literature that floods peaks are described by heavy tails (see for example Vogel & Wilson (1996), Villarini and Smith (2010) and recently Miniussi et al. (2020) and Zaghloul et al. (2020)).AC: The choice of a Gumbel distribution is not mandatory in the proposed schematization. It is absolutely true that the Gumbel distribution is not necessarily the best option for the description of floods, but it represents a reference distribution for flood distributions. As we stated in the text, the choice of the flood distribution can be any of the available in the literature, because the derived laminated discharge can be used to obtain a derived distribution for any flood distribution chosen. We have included some examples of applications based on the assumption of floods distributed according to a Fréchet and a Weibull distribution.RC: Finally, I believe the readers would be very curious to see how the results would be modified if a heavytailed distribution was used instead of the Gumbel which has exponential tail. The exponential tails offer “no surprises” in the generation of random discharge values and thus the good results shown might be case specific and only for the Gumbel distribution. What would be the performance if really heavy tailed distributions were used, e.g., a GEV with shape parameter close to 0.5?AC: In order to satisfy readers’ curiosity, we provide in the following few examples where the peak outflow distributions are derived using flood peaks distributed according to the three types of GEV distributions (Gumbel, Fréchet and Weibull). Reliability of the derived distribution has been tested using a numerical simulation, which demonstrates the good agreement with the theoretical functions.Figure 1. Comparison between three different derived probability density functions of the peak outflows obtained using three different flood peak distributions and the empirical pdfs obtained via numerical hydraulic simulation (red dots for inflows, blue dots for outflows). The three graphs are obtained modifying the shape parameter, Imageof the GEV distribution, which is equal to 0 for Gumbel distribution (A), 0.5 for Fréchet distribution (B) and  0.5 for Weibull distribution (C). Remaining parameters are: the scale parameter of GEV α=30m3/s; the location parameter of GEV β=120 m^{3}/s; w_{1}=5000; h_{s}=4m; b=1m; d=1m; n=1.9; h_{f}= d/2; μ_{f}=0.85; μ_{s}=0.385; L=3m; t_{p}=1h.RC. Please double check your equations, for example, Equation 13 is not correct, it should be dg^1(y)/dy f(g^1(y))AC: We thank the referee for this suggestion. The text has been modified accordingly.

AC1: 'Reply on RC1', Salvatore Manfreda, 11 May 2021

RC2: 'Comment on hess202147', Anonymous Referee #2, 18 Apr 2021
The paper presents a mathematical framework for deriving the probability distribution of peak outflows from a detention basin from the probability distribution of incoming flood peaks. The authors first derive a relationship between the peak outflow from a detention basin and the peak inflow assuming a rectangular inflow hydrograph. This involves the definition of an equivalent delay constant that depends on the properties of the detention basin, and an equivalent flood duration that depends on the lagtime of the river basin. The mathematical framework is applied assuming that the undisturbed flood peaks follow a Gumbel distribution, but the formulation can potentially be applied to any other probability distribution of inflows. The authors also present results from Monte Carlo simulations in which most of the simplifying assumptions are relaxed, and show that the proposed formulation provides a good approximation of the mitigated flood peak distribution in river basins with relatively small concentration time.
The paper is well written and organized, and the mathematical derivations are presented in a clear and logical manner. To the best of my knowledge, the proposed approach is novel, and can be useful for design and environmental impact assessment of flood detention basins.
I recommend some minor changes before publication.
Specific comments:I believe the reference to "flood control systems" in the title raises the expectations beyond what is presented in the paper, as there are other types of flood control systems that cannot be treated with the same mathematical framework proposed by the authors. I therefore recommend changing the title to refer more specially to "flood detention basins".
The symbol D is not defined in the notation list.
Given that the symbol q is used for discharge, I recommend using a different symbol for the height of the lowlevel opening instead of qf.
P80,90: Use italic style for mathematical variables.
P10: "the undisturbed flood distribution is assumed to be Gumbel distributed" => "the undisturbed flood peaks are assumed to be Gumbel distributed"
P50: "see i.e., Manfreda et al." => "e.g." not "i.e."
P165: "... mathematically inverting Equation..." => "mathematically by inverting Equation..."
P140: "computed comparing... and setting..." => "computed by comparing... and imposing..."
P255: Remove "realized"
P235: "the impact due to the approximation adopted by the rectangular hydrographs" => I suggest changing this as follows: "the impact of the assumption of rectangular inflow hydrograph"
P235: "allowed to reproduce correctly the flood mitigation that looks very similar to those..." => "produced probability distributions of the outflow that look very similar to those...".
L375: 15,000.00 => 15,000

AC2: 'Reply on RC2', Salvatore Manfreda, 11 May 2021
we would like to thank also the second referee for his/her effort and constructive suggestions, that allowed us to improve the quality of the manuscript. We have carefully considered all his/her observations and reviewed the manuscript accordingly.RC: I believe the reference to "flood control systems" in the title raises the expectations beyond what is presented in the paper, as there are other types of flood control systems that cannot be treated with the same mathematical framework proposed by the authors. I therefore recommend changing the title to refer more specially to "flood detention basins". The symbol D is not defined in the notation list. Given that the symbol q is used for discharge, I recommend using a different symbol for the height of the lowlevel opening instead of qf.AC: We agree with the reviewer. Title will changed in … "Impact of Detention Dams on the Probability Distribution of Floods". We will also add the flood event duration D in the notation list. Finally, we will change qf in lf.RC: List of minor correctionsP80,90: Use italic style for mathematical variables.P10: "the undisturbed flood distribution is assumed to be Gumbel distributed" => "theundisturbed flood peaks are assumed to be Gumbel distributed"P50: "see i.e., Manfreda et al." => "e.g." not "i.e."P165: "... mathematically inverting Equation..." => "mathematically by invertingEquation..."P140: "computed comparing... and setting..." => "computed by comparing... andimposing..."P255: Remove "realized"P235: "the impact due to the approximation adopted by the rectangular hydrographs" =>I suggest changing this as follows: "the impact of the assumption of rectangular inflowhydrograph"P235: "allowed to reproduce correctly the flood mitigation that looks very similar tothose..." => "produced probability distributions of the outflow that look very similar tothose...".L375: 15,000.00 => 15,000AC: All these suggestions have been already implemented in the revised version of the manuscript.

AC2: 'Reply on RC2', Salvatore Manfreda, 11 May 2021
Salvatore Manfreda et al.
Salvatore Manfreda et al.
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