This is a well-written 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 well-known 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 heavy-tailed 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 heavy-tailed 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 pan-Canadian investigation. Advances in Water Resources 145, 103720. https://doi.org/10.1016/j.advwatres.2020.103720

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 lag-time 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 low-level 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