EGUSPHERE-2023-660: Inferring heavy tails of flood distributions from common discharge dynamics

Response to Reviewer 1

We thank the Reviewer for providing valuable comments and suggestions. We have addressed each point below and will incorporate the comments into the revised manuscript after considering feedback from other reviewers. The Reviewer's comments are marked in italic font, while our replies are indicated in bold font. For ease of reference, we have included line numbers from the preprint manuscript (L).

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In this work, the authors investigate a new heavy-tail (i.e., extreme power-law tail) index a originating from the PHEV riverflow hydrograph model, which is compared to common practices in the literature of extreme-tail fitting, and it is applied to daily streamflow records in Germany. In my opinion, the manuscript needs multiple major and minor revisions before it can be evaluated for publication, and then, I believe it can serve as a review paper that compares several methods in the literature for extreme-tail oriented probability distribution fitting.

Thank you for the summarized review. In this study, we have identified the primary factors influencing heavy-tailed flood behavior through theoretical analysis and substantiated our findings with empirical data. We have also discussed literature encompassing various methodologies for detecting heavy-tailed flood behavior. This discussion serves as a valuable point of reference to highlight the significance of our work, going beyond a mere survey of different approaches. In response to specific comments from the reviewer, we have provided our replies below each comment.

1) The a index seems to be based on some assumptions related to precipitation probability distribution and to rainfall-runoff model. Although the Poisson distribution has been used in the literature to fit precipitation records (e.g., see Cox and Isham, 1988), rainfall extremes are shown to exhibit heavy-tail behaviour. Please see one of the first studies in the literature that suggest to use the EV2 distribution for rainfall extremes, through theoretical (i.e., physically-based) and empirical (I think is the first global-scale analyses on rainfall extremes) reasoning. This selection of distribution has been adopted by many researchers, and has been verified by many studies in the literature. Please see a recent extensive review on the rainfall-extremes in Koutsoyiannis (2022), where methods are also described how to adjust this to perform well in even short records.

Cox, D.R. and V. Isham, A simple spatial-temporal model of rainfall, Proceedings of the Royal Society London A415, 317–28, 1988.

Koutsoyiannis,D., Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575-590, doi:10.1623/hysj.49.4.575.54430, 2004a.

Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 2, Empirical investigation of long rainfall records, Hydrological Sciences Journal, 49 (4), 591-610, doi:10.1623/hysj.49.4.591.54424, 2004b.

Koutsoyiannis, D., Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, Εκδοση 2, ISBN: 978-618-85370-0-2, 346 pages, doi:10.57713/kallipos-1, Kallipos Open Academic Editions, Athens, 2022.

We sincerely appreciate the reviewer for bringing attention to the significant contributions made by previous studies in advancing the simulation of observed rainfall extremes through the application of specific probability distributions. In contrast to the simulation of observed rainfall extremes, the aim of this study is to statistically mimic rainfall-triggered runoff events. It is crucial to note that rainfall extremes do not necessarily translate into streamflow extremes, as demonstrated in numerous studies (e.g., McCuen and Smith, 2008; Pall et al., 2011; Hall et al., 2014; Archfield et al., 2016; Rossi et al., 2016; Zhang et al., 2016; Hodgkins et al., 2017; Sharma et al., 2018). For instance, McCuen and Smith (2008) identified this fact and proposed the influence of catchment responses and storage. Sharma et al. (2018) supported this notion, providing additional evidence and discussion that highlighted the limited correspondence between increased rainfall extremes and flooding. They found that factors such as decreased antecedent soil moisture, reduced storm extent, and declining snowmelt work in conjunction with rainfall to modulate flood events. In summary, while rainfall is the primary contributor to runoff, the emergence of extreme floods is largely determined by catchment responses and water balance (Merz et al., 2022). Given these premises, an appropriate approach for describing runoff and its extremes should be rooted in the dynamics of soil moisture within catchments. This necessitates the use of typical rainfall patterns (such as daily rainfall, as employed in this study) rather than solely focusing on extreme rainfall events.

Particularly, the framework used in this work is accomplished by stochastically representing the effective contribution of rainfall, which refers to the fraction of rainfall that infiltrates into the hillslopes, eventually reaching the outlet through the channel network. The representation incorporates classical hydrological processes, i.e., soil moisture dynamics in the top layer storage primarily involve rainfall-related processes, including infiltration as a positive contribution and loss through evapotranspiration, direct surface flow, and/or deep percolation, which reduce rainfall's contribution to the storage. The Poisson distribution has been commonly employed to model precipitation records, as noted by the reviewer. Consistent with this, we characterize daily precipitation (not rainfall extremes) as a Poisson process. Specifically, we utilize a Poisson process to simulate the temporal occurrence of rainfall events, while modeling rainfall depths as exponentially distributed, following the approach proposed by Botter et al. (2007). The effective contribution from rainfall to runoff is determined by applying an exceedance threshold for the soil moisture.

This method offers a physically-based description of the role played by nonlinear hydrological responses and catchment water balance, which are highly relevant to the occurrence of heavy-tailed floods (Merz et al., 2022). As a result, it facilitates the investigation of the objectives outlined in this study.

2) Additionally, the proposed distribution by the authors for the streamflow process (that combines exponential and power-type expressions) resembles the Pearson-III distribution, which has been used to fit streamflow records (e.g., Buckett and Oliver, 1977), but again it is not always recommended for the streamflow extreme-tail (e.g., Anghel and Ilinca, 2023), whereas, it has been shown that, for the parent distribution of streamflow (and by accounting for the impact of correlation among streamflow records through higher-order moments), a pareto-distribution type (i.e., the so-called Pareto-Burr-Feller probability distribution, which is a generalized form of the Pareto IV or Burr XII distribution) seems to adequately fit streamflow records even in a global-scale (please see such analysis with thousand of streamflow gauges in Dimitriadis et al., 2021).

Anghel, C.G., and C. Ilinca, Evaluation of Various Generalized Pareto Probability Distributions for Flood Frequency Analysis, Water, 15, 1557, 2023.

Buckett, J., and F.R. Oliver, F.R., Fitting the Pearson type 3 distribution in practice, Water Resources Research, 13(5), 851–852, 1977.

Dimitriadis, P., D. Koutsoyiannis, T. Iliopoulou, and P. Papanicolaou, A global-scale investigation of stochastic similarities in marginal distribution and dependence structure of key hydrological-cycle processes, Hydrology, 8 (2), 59, doi:10.3390/hydrology8020059, 2021.

Thank you for the feedback. We would like to emphasize that the proposed streamflow description (Equation (1)) is distinct from the Pearson-III distribution. While the Pearson-III distribution comprises a power law and an exponential distribution, our approach incorporates a power law and two stretched exponential distributions. It is important to note that the stretched exponential distribution offers greater flexibility in terms of tail behavior compared to the exponential distribution. Depending on its parameters, the stretched exponential distribution can exhibit either a light-tailed or heavy-tailed behavior, whereas the exponential distribution always exhibits a light-tailed behavior. This highlights a significant difference in the utilization of our proposed streamflow description (PHEV equation) compared to the Pearson-III distribution, particularly regarding tail behavior, which is a focal point of this analysis.

Furthermore, it is essential to clarify that this study does not rely on assuming a specific probability distribution to represent streamflow. Instead, we conduct an analysis based on classical hydrological processes, utilizing the PHEV equation as a general outcome derived from the representation of these processes (as explained in the first comment). In this regard, the PHEV approach shares similarities with the consideration of large simulation ensembles of deterministic hydrological models, which are employed to account for diverse realizations of hydro-meteorological scenarios and boundary/antecedent catchment conditions. Thanks to the low-dimensionality of PHEV,  this approach operates differently by offering a convenient analytical solution.

We acknowledge that the depiction of precipitation and runoff generation mechanisms incorporated in PHEV does not encompass the entirety of potential rainfall-runoff processes. However, the chosen representation is firmly rooted in established scientific frameworks that have undergone extensive testing through numerous case studies over the past decades (see L61-L74). Notably, we emphasize its representation of high flows, which has been validated and documented in Environmental Research Letters (Basso et al., 2021), and Nature Geoscience (Basso et al., 2023).

3) By observing the results from the fitting illustrated in the Figure 3, it can be observed that there is a smaller variability on the ξ-index than in the a-index for longer lengths; please consider discussing this observation and in what cases the a-index can offer a higher statistically significant extreme-tail fitting (perhaps only in the small-length records?).

Thank you for your comment. We appreciate the feedback and acknowledge that our work, in line with previous studies, provides support for the stability of the ξ-index when the data length is adequately long. We have duly recognized this finding in our manuscript (L329-L333). In the revised version, we plan to enhance the description by explicitly mentioning that numerous existing studies have also demonstrated similar results, specifically highlighting that the Generalized Extreme Value (GEV) approach exhibits reliable performance for long data lengths.

However, it should be noted that Figures 3a and 3c do not demonstrate that the a-index provides a poorer estimation than the ξ-index when the data length is long. The width (whiskers) of the boxplots for different indices are not on the same scale, making them incomparable. It is important to understand that the estimated values of each index can only be compared within the same index. Figure 3a illustrates that the a-index consistently exhibits a stable width of estimated values across various data lengths, indicating good stability in both short and long data length analyses. Conversely, the ξ-index displays varying widths of estimated values in short and long data length analyses, indicating instability. We apologize for any confusion caused by the presentation of the figures and assure you that we will improve their clarity in the revised version.

4) It is my understanding that the whole (not peaks-over-threshold and maxima) streamflow timeseries are fitted to equation (1) from where the a-index is estimated. If this is true, please discuss the method used to fit this distribution to data. For example, is it through the method of moments, where an n-number of moments is estimated from data to estimate the n-number of parameters of the distribution or through the method of curve-fitting, where one estimates the parameters of the distribution that result in the smaller small-square-error between the theoretical and empirical distribution?

We would like to address a misunderstanding and clarify that we do not fit the observed streamflow to Equation (1). Consequently, the value of a is not estimated through this fitting process. Equation (1) serves as a physically-based framework that helps us understand the role of a in determining the tail behavior. The estimation of the a value follows a classical and widely applied approach (Brutsaert and Nieber, 1977; see also L115-L116). In particular, a power law is used to represent hydrograph recessions of a single event i,

Where q denotes the streamflow, t denotes the unit time, K_i and a_i denote the estimated coefficient and exponent of hydrograph recessions for event i, respectively. The median value of all the a_i is the estimated value of a considered in this study and here used to represent the average nonlinearity of catchment response.

We apologize for any confusion caused by our previous statement and assure you that we will enhance our description of a estimation in the revised version.

References

Archfield, S. A., Hirsch, R. M., Viglione, A., & Blöschl, G. (2016). Fragmented patterns of flood change across the United States. Geophysical Research Letters, 43(10), 10232–10239. https://doi.org/10.1002/2016GL070590

Basso, S., Botter, G., Merz, R., & Miniussi, A. (2021). PHEV! The PHysically-based Extreme Value distribution of river flows. Environmental Research Letters, 16(12). https://doi.org/10.1088/1748-9326/ac3d59

Basso, S., Merz, R., Tarasova, L., & Miniussi, A. (2023). Extreme flooding controlled by stream network organization and flow regime. Nature Geoscience, 16(April), 339–343. https://doi.org/10.1038/s41561-023-01155-w

Botter, G., Porporato, A., Rodriguez-Iturbe, I., & Rinaldo, A. (2007). Basin-scale soil moisture dynamics and the probabilistic characterization of carrier hydrologic flows: Slow, leaching-prone components of the hydrologic response. Water Resources Research, 43(2), 1–14. https://doi.org/10.1029/2006WR005043

Brutsaert, W., & Nieber, J. L. (1977). Regionalized drought flow hydrographs from a mature glaciated plateau. Water Resources Research, 13(3), 637–643. https://doi.org/10.1029/WR013i003p00637

Hall, J., Arheimer, B., Borga, M., Brázdil, R., Claps, P., Kiss, A., et al. (2014). Understanding flood regime changes in Europe: A state-of-the-art assessment. Hydrology and Earth System Sciences, 18(7), 2735–2772. https://doi.org/10.5194/hess-18-2735-2014

Hodgkins, G. A., Whitfield, P. H., Burn, D. H., Hannaford, J., Renard, B., Stahl, K., et al. (2017). Climate-driven variability in the occurrence of major floods across North America and Europe. Journal of Hydrology, 552, 704–717. https://doi.org/10.1016/j.jhydrol.2017.07.027

McCuen, R. H., & Smith, E. (2008). Origin of Flood Skew. Journal of Hydrologic Engineering, 13(9), 771–775. https://doi.org/10.1061/(asce)1084-0699(2008)13:9(771)

Merz, B., Basso, S., Fischer, S., Lun, D., Blöschl, G., Merz, R., et al. (2022). Understanding heavy tails of flood peak distributions. Water Resources Research, 1–37. https://doi.org/10.1029/2021wr030506

Pall, P., Aina, T., Stone, D. A., Stott, P. A., Nozawa, T., Hilberts, A. G. J., et al. (2011). Anthropogenic greenhouse gas contribution to flood risk in England and Wales in autumn 2000. Nature, 470(7334), 382–385. https://doi.org/10.1038/nature09762

Rossi, M. W., Whipple, K. X., & Vivoni, E. R. (2016). Precipitation and evapotranspiration controls on daily runoff variability in the contiguous United States and Puerto Rico. Journal of Geophysical Research: Earth Surface, 128–145. https://doi.org/doi:10.1002/2015JF003446

Sharma, A., Wasko, C., & Lettenmaier, D. P. (2018). If Precipitation Extremes Are Increasing, Why Aren’t Floods? Water Resources Research, 54(11), 8545–8551. https://doi.org/10.1029/2018WR023749

Zhang, X. S., Amirthanathan, G. E., Bari, M. A., Laugesen, R. M., Shin, D., Kent, D. M., et al. (2016). How streamflow has changed across Australia since the 1950s: Evidence from the network of hydrologic reference stations. Hydrology and Earth System Sciences, 20(9), 3947–3965. https://doi.org/10.5194/hess-20-3947-201

EGUSPHERE-2023-660: Inferring heavy tails of flood distributions from common discharge dynamics

Response to Reviewer2

We thank the Reviewer for providing valuable comments and suggestions. We have addressed each point below and will incorporate the comments into the revised manuscript after considering feedback from the editor. The Reviewer's comments are marked in italic font, while our replies are indicated in bold font. For ease of reference, we have included line numbers from the preprint manuscript (L).

In this article, the author investigated a new method for representing the probability distribution function of flood with a heavy tail. A new method is to fit a power raw function to recession parts of hydrographs and then the derived power parameter a is utilized for the probability distribution function of flood as the reviewer’s understanding. The result showed a high capability of power parameter a derived from recession parts of hydrographs to represent the power parameter of the probability distribution function for streamflow data including flood data. The advantage of this bland new approach is to utilize recession parts of hydrographs, which means usage of restricted data but numerous numbers of data. Instead of using all the data, this approach enables us to handle a huge amount of data to reduce uncertainty of flood estimation. However, another uncertainty, which is the representativeness of the power parameter from the recession hydrograph to the probability distribution function of the flood, rises. Moreover, it must be well discussed whether a power parameter from a recession hydrograph could be considered as a power parameter of a probability distribution function.

We thank the Reviewer for the positive feedback and suggestions, which we will address point by point below.

1) Please discuss how the power parameter which is fitted to hydrograph recession can be applied to the power parameter of the distribution function of flood.

Thank you for your comment. In this paper, we present a mathematical demonstration of why the hydrograph recession exponent serves as an indicator of heavy-tailed flood behavior. This demonstration is based on the description of runoff generation provided by physically-based extreme value distributions. Particularly, the tail behavior of flow distribution is shown to be determined solely by a power law function (i.e., heavy-tailed behavior) in this mathematical description when the hydrograph recession exponent is above two. We discuss the case of flow distributions in the main text (L60-L93) and emphasize that the same critical value of the recession exponent can also be applied to identify heavy-tailed behavior for flood distributions, with detailed explanations provided in Appendix A (L94-L98). To address the reviewer's comment, we intend to integrate the mathematical explanation regarding flood distributions into the main text to enhance its visibility. Additionally, we highlight that the observations support the theoretical findings for all three cases proposed in this work, i.e., streamflow distributions, ordinary peak distributions, and flood distributions, which are discussed in figures 1 and 2 in the main text.

We also intend to better discuss the justification of the hydrograph recession exponent representing heavy-tailed flood behavior by emphasizing the nonlinearity of catchment response as a plausible control of heavy-tailed behavior. Recent publications in WRR (Merz et al., 2022) and Nat. Geoscience (Basso et al., 2023) have supported this idea, and our findings align with it. We will add this discussion to Section 5, particularly in the paragraph from L294 to L305.

2) In the text, a parameter is mentioned as “a physically-based index”. However, this index was derived by fitting the raw power function to the recession part of the hydrograph. In this sense, using the word “physically” may lead to confusion. Rather, “recession-derived” is suited, the reviewer thinks (for example, “a recession-based exponent”).

Thank you for the suggestion. In response to the Reviewer's comment, we will enhance the clarity of the writing by avoiding phrases like "physically-based index." Instead, we plan to improve our description by explicitly expressing the finding of this work, e.g., for the description in L98-L99, we intend to modify the description as “we propose the hydrograph recession exponent a as a suitable indicator of heavy-tailed flood behavior, based on the description of hydrological processes embedded in the physically-based extreme value model.”

3) In L147, the skewness of the histogram of KS statistics was mentioned. What does it mean in this study? More explanation is effective for the readers.

Thank you for the suggestions. In this study, we used the KS statistic to assess how well the power law describes the empirical data, indicating whether the data exhibits heavy-tailed behavior or not. A skewness of the KS statistics towards lower values, therefore, suggests a stronger presence of heavy-tailed behavior in those cases.

We found that this was indeed the case for instances with recession exponents above two. This finding aligns with the suggestion based on the recession exponent, demonstrating a consistency between the suggested index and the observed heavy-tailed behavior in the data.

Furthermore, we observed significant differences in the skewed patterns between cases with recession exponents above two and those with recession exponents below two. The former group, which is associated with recession exponents above two, is more likely to exhibit heavy-tailed behavior, as supported by the data. Conversely, the latter group, with recession exponents below two, is less likely to show heavy-tailed behavior, as indicated by the data.

These results further support the effectiveness of our proposed index in identifying heavy-tailed behavior in the data. In response to the Reviewer's comment, we intend to make improvements in the discussion (L142-153) to enhance clarity for readers.

4) For figs. 1d-1f, why does the accuracy index become one when the threshold of KS statistics becomes larger? The large threshold of KS statistics means considering almost all cases. However, the index shown on the vertical axis is the probability of a>2, which is never all the cases.

We believe there is a misunderstanding regarding the accuracy calculation when considering the threshold of KS statistics. It appears that the accuracy does not become one when the threshold of KS statistics becomes larger, as indicated by the inverse x-axis in Figure 1 and its second axis legend. In fact, the accuracy tends to approach one when the threshold of KS statistics becomes smaller.

By lowering the threshold of KS statistics, we impose a stricter criterion for including cases in the calculation of the conditional probability of accuracy (described in L157-L159). As depicted in Figures 1d-1f, the accuracy approaches one when the threshold of KS statistics is set to a very small value. This indicates that only cases with very high reliability (i.e., cases with KS statistics below this very small value) are taken into account for the accuracy calculation. Among these cases, almost every one exhibits a recession exponent above two, as evident from the accuracy being close to one.

Upon recognizing the potential for misleading interpretations, we plan to enhance the clarity of the current description in L160-161.

References

Basso, S., Merz, R., Tarasova, L., & Miniussi, A. (2023). Extreme flooding controlled by stream network organization and flow regime. Nature Geoscience, 16(April), 339–343. https://doi.org/10.1038/s41561-023-01155-w

Merz, B., Basso, S., Fischer, S., Lun, D., Blöschl, G., Merz, R., et al. (2022). Understanding heavy tails of flood peak distributions. Water Resources Research, 1–37. https://doi.org/10.1029/2021wr030506