the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Constructing a geography of heavy-tailed flood distributions: insights from common streamflow dynamics
Abstract. Heavy-tailed flood distributions depict the higher occurrence probability of extreme floods. Understanding the spatial distribution of heavy tail floods is essential for effective risk assessment. Conventional methods often encounter data limitations, leading to uncertainty across regions. To address this challenge, we utilize hydrograph recession exponents derived from common streamflow dynamics, which have shown to be a robust indicator of flood tail propensity across analyses with varying data lengths. Analyzing extensive datasets covering the Atlantic Europe, Northern Europe, and the continental United States, we uncover distinct patterns: prevalent heavy tails in the Atlantic Europe, diverse behavior in the continental United States, and predominantly nonheavy tails in Northern Europe. The regional tail behavior has been observed in relation to the interplay between terrain and meteorological characteristics, and we further conducted quantitative analyses to assess the influence of hydroclimatic conditions using Köppen classifications. Notably, temporal variations in catchment storage are a crucial mechanism driving highly nonlinear catchment responses that favor heavy-tailed floods, often intensified by concurrent dry periods and high temperatures. Furthermore, this mechanism is influenced by various flood generation processes, which can be shaped by both hydroclimatic seasonality and catchment scale. These insights deepen our understanding of the interplay between climate, physiographical settings, and flood behavior, while highlighting the utility of hydrograph recession exponents in flood hazard assessment.
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RC1: 'Comment on hess-2024-159', Anonymous Referee #1, 02 Jul 2024
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Comments on “Constructing a geography of heavy-tailed flood distributions: insights from common streamflow dynamics” by Wang et al., submitted to HESS for possible publication.
The authors provide empirical analyses on the pattern and drivers of the tail heaviness. They adopt hydrograph recession exponent as the indicator of watersheds with or without propensity for heavy tails of flood peak distributions. The contrasts in the tail heaviness across watersheds, climate regions, seasons, shed light on the importance of characterizing catchment storage in dictating flood regimes. The analyses are interesting and robust. A major of mine is that the manuscript is lengthy that obscure the new wisdom obtained. I would suggest the authors to further refine and remove unnecessary details.
Specific comments:
- The part that emphasize the utility of hydrograph recession exponents in characterizing tail heaviness is lengthy, and needs to be shortened. One question might be, as far as can be seen from the dataset, the record lengths are quite adequate despite of variance, some of other tail heaviness indicators would be able to perform as well.
- What is the rationale of using five days as the minimum duration, considering the vast variance in drainage areas?
- “The upper tail is defined by an optimized lower boundary of the discharge, determined by selecting the best fit based on the KS statistic”. This is not quite clear. How the upper tail is defined and statistically modelled is important. Section 3.2 also needs to be concise and informative. Please reconstruct.
- Line 269, by “majority” you use “50 %” as the threshold?
- From Figure 1 and Figure 2, we can see there are many overlaps between heavy tails and nonheavy tails. This is especially evident in Figure 2 where we see the scatters are well mixed. These results make me wonder the utility of recession exponent (a=2) as the criteria. I would suggest the authors to explain and discuss the limitation.
- Figure 4 and the text, please explain the rationale of using percentage. The absolute count of watersheds would matter, as can be seen that there is only one watershed in Bwk. This can be due to sampling uncertainties.
- What do the authors mean by “catchment storage”? Please clarify.
- Line 571, I would suggest the role of ET alone might not be that important. The ratio of ET to P worthwhile to be explored.
- Line 605-606, the three references use indicators that quantify the heaviness of upper tails, while in this study, the authors are in fact addressing “propensity”.
- Line 649-650, this is obvious.
- Section 5, I enjoy reading this section overall, but it can be further improved by explicitly highlighting what are found in this study, and what are proposed by previous studies, especially the review paper by Merz et al.
Citation: https://doi.org/10.5194/hess-2024-159-RC1 -
AC1: 'Reply on RC1', Hsing-Jui Wang, 24 Jul 2024
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Reply on 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 normal font.
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The authors provide empirical analyses on the pattern and drivers of the tail heaviness. They adopt hydrograph recession exponent as the indicator of watersheds with or without propensity for heavy tails of flood peak distributions. The contrasts in the tail heaviness across watersheds, climate regions, seasons, shed light on the importance of characterizing catchment storage in dictating flood regimes. The analyses are interesting and robust. A major of mine is that the manuscript is lengthy that obscure the new wisdom obtained. I would suggest the authors to further refine and remove unnecessary details.
Thank you for the summarized review and positive feedback. We will streamline the details, particularly addressing the specific comments below, to better highlight the key findings of this work as suggested by the reviewer.
Specific comments:
1) The part that emphasize the utility of hydrograph recession exponents in characterizing tail heaviness is lengthy, and needs to be shortened. One question might be, as far as can be seen from the dataset, the record lengths are quite adequate despite of variance, some of other tail heaviness indicators would be able to perform as well.
We will shorten the sections that describe the utility of hydrograph recession exponents in characterizing tail heaviness, as suggested by the reviewer. In particular, we will shorten section 3.1 by making larger use of references to a previous publication where this approach was first introduced (Wang et al., 2023). The dataset employed in this study spans 24 to 148 years. We acknowledge that other indicators could also be used; however, we are specifically interested in the recession exponent because it is a novel index that allows us to infer the propensity of rivers to experience extreme floods. This index enables us to identify potential risks even in the absence of recorded extreme floods, which is not possible with other indicators. Additionally, the recession exponent is suggested to mitigate the bias often introduced by the variance in dataset lengths across cases (Smith et al., 2018; Wietzke et al., 2020; Wang et al., 2023).
We have discussed the literature review on this topic in lines 64-84. We plan to supplement this with the following statement after line 84: “Nonetheless, we acknowledge that other indicators could also be used; however, we are specifically interested in the recession exponent because it is a novel index that allows us to infer the propensity of rivers to experience extreme floods. Such an index enables us to identify potential risks even in the absence of recorded extreme floods, which is not possible with other indicators. Its stability provides additional value to mitigate the bias often introduced by the variance in dataset lengths across cases.”
2) What is the rationale of using five days as the minimum duration, considering the vast variance in drainage areas?
Event-scale recession analyses typically choose a minimum of 3 to 5 days of recession for daily data (e.g., Shaw and Riha, 2012; Biswal and Marani, 2010) to minimize noise from short events (Ye et al., 2014) and ensure sufficient sample sizes for proper data quality (Shaw, 2016). We acknowledge that the recession period may vary depending on the drainage area. The recessions we identify and analyze have indeed different durations in different catchments. In this study, we select a fixed minimum number of 5 days (Dralle et al., 2017) to ensure sufficient sample size for suitably characterizing recession attributes.
3) “The upper tail is defined by an optimized lower boundary of the discharge, determined by selecting the best fit based on the KS statistic”. This is not quite clear. How the upper tail is defined and statistically modelled is important. Section 3.2 also needs to be concise and informative. Please reconstruct.
Thank you for your comment. We will modify lines 232-233 (“The upper tail is defined by an optimized lower boundary of the discharge, determined by selecting the best fit based on the KS statistic”) as below to improve the clarity:
“Empirical data following a power-law distribution (if applicable) typically do so above a certain lower bound, defining the analyzed tail (Clauset et al., 2009). Therefore, we employ the approach proposed by Clauset et al. (2007) to determine the optimized lower boundary. This method selects the boundary where the probability distributions of the data and the best-fit power-law model are most similar. If the optimized lower boundary is higher than the true lower boundary, the reduced data set size leads to a poorer match due to statistical fluctuations. If it is lower, the distributions differ fundamentally. The KS statistic is employed to quantify the distance between these distributions.”
We will also revise the remaining of section 3.2 with the goal of shortening it.
4) Line 269, by “majority” you use “50 %” as the threshold?
Yes we do. We will specify it in the revised paper.
5) From Figure 1 and Figure 2, we can see there are many overlaps between heavy tails and nonheavy tails. This is especially evident in Figure 2 where we see the scatters are well mixed. These results make me wonder the utility of recession exponent (a=2) as the criteria. I would suggest the authors to explain and discuss the limitation.
Ideally, the validation of the new index should include benchmarks for both heavy-tailed and non-heavy-tailed case studies. However, we can only statistically establish the benchmark for the former (power-law-tailed case studies, black dots in Figure 2), but not for the latter (uncertain case studies, gray dots in Figure 2). This is due to the latter category encompasses case studies that either do not follow a power-law distribution or whose underlying distributions cannot be determined due to high uncertainty from small sample sizes, and thus contributes to the ‘mixed pattern’ as indicated by the reviewer. Notice that several years of data are still a small sample to reliably characterize the tail of empirical and purely statistical distributions fitted on the data.
Due to this approach limitation, the effectiveness of the recession exponent shall be estimated only based on the former group (black dots), as highlighted in Figure 2. To rigorously validate the effectiveness of the recession exponent criteria, we need also Figure 1, where we statistically confirm the hypothesized heavy- and non-heavy-tailed groups using a=2 as the criterion.
This confusion seems unavoidable due to the inherent limitations of data analysis. Meanwhile, we acknowledge that misattribution can occur due to the recession exponent not always being able to properly distinguish between heavy and light tails, particularly when a is around the threshold value of 2. This issue is shown in the case studies in Norway, which we discuss in lines 356-363. We have addressed this in lines 242-246:
"We term such a case study a 'power-law-tailed case study,' while cases that don't meet these criteria are labeled as 'uncertain case studies' in subsequent analyses. The latter label acknowledges the awareness that we cannot definitively conclude whether these case studies are indeed not power-law-tailed or if their underlying distributions cannot be determined due to the high uncertainty caused by the small sample sizes of available observations."
Moreover, we will insert additional discussion after line 331 to improve clarity:
"We cannot conclude whether uncertain case studies (gray dots) represent cases that are indeed not power-law-tailed or if their underlying distributions cannot be determined due to the high uncertainty caused by small sample sizes. Therefore, we benchmark the recession exponent against the empirical power law exponent by focusing on the 'certain group,' i.e., power-law-tailed case studies (black dots)."
The following statement will also be inserted at the end of line 365:
“However, we acknowledge that misattributions can still occur, particularly when a is around the threshold value.”
6) Figure 4 and the text, please explain the rationale of using percentage. The absolute count of watersheds would matter, as can be seen that there is only one watershed in Bwk. This can be due to sampling uncertainties.
We recognize that the varying number of cases across climate types might introduce bias due to sample sensitivity (as we have mentioned in lines 490-493). Nonetheless, in order to estimate the propensity of tail behavior the ratio (i.e., percentage) of heavy- to non-heavy-tailed case studies in each climate region is considered to be one of the most direct approaches. We will revise lines 490-493 to emphasize this concern of the employed dataset and approach:
Original lines 490-493: “We acknowledge that these results are based on overarching conditions and do not encompass all climate types, and achieving an equal number of study sites across various climate regions might not always be feasible. Expanding the number of study sites could further enhance our understanding, especially for extreme cases.”
Modified version: “We acknowledge that these results are based on overarching conditions and do not encompass all climate types, and achieving an equal number of study sites across various climate regions might not always be feasible. We should be mindful of potential bias caused by sample sensitivity, particularly in regions with a limited number of cases (e.g., Csa, BSh, BWk in this study). Expanding the number of study sites could further enhance our understanding, especially for extreme cases.”
7) What do the authors mean by “catchment storage”? Please clarify.
Thank you for pointing this out. We will improve the clarity by adding the following description after lines 487-489 (“we have identified the conjunction of dry periods and higher temperatures as crucial meteorological factors significantly contributing to the dynamics of catchment storage, thereby influencing the nonlinearity of hydrological responses.”):
“We refer to catchment storage hereafter as the water contained in a catchment at a certain moment, which concurs to define its wetness status (this is chiefly the degree of saturation of the critical zone). This capacity is dynamic and depends on various factors, such as soil moisture states, precipitation, and evapotranspiration (Merz and Blöschl, 2009; Zhou et al., 2022).”
8) Line 571, I would suggest the role of ET alone might not be that important. The ratio of ET to P worthwhile to be explored.
We agree that, based on our analyses and findings, the temporal characteristics of rainfall and evapotranspiration collaboratively influence this seasonality, as discussed in detail in lines 539-548. We will therefore revise lines 569-571 as follows:
Original lines 569-571: “Regions with pronounced temperature variations across seasons, particularly with higher temperature in summer, tend to display such dynamics and highlight the role of evapotranspiration in catchments in driving this seasonality.”
Modified version: “Regions with pronounced temperature variations across seasons, particularly with higher temperatures in summer, and characterized by relatively evenly distributed rainfall throughout the year tend to display such dynamics. This highlights the importance of both evapotranspiration and the temporal characteristics of rainfall in shaping flood tail behavior across seasons, aligning with previous studies (Guo et al., 2014; Basso et al., 2023).”
9) Line 605-606, the three references use indicators that quantify the heaviness of upper tails, while in this study, the authors are in fact addressing “propensity”.
Thank you for pointing this out. We will improve the clarity as below:
Original lines 604-606: “These findings align with previous discussions on this matter (e.g., Merz and Blöschl, 2009; Villarini and Smith, 2010; Smith et al., 2018), which have suggested a relatively weak inverse correlation between catchment area and the occurrence of heavy-tailed flood behavior.”
Modified version: “In a similar context, previous studies, using different heavy-tailed flood indices, have suggested a relatively weak inverse correlation between catchment area and the occurrence of heavy-tailed flood behavior (e.g., Merz and Blöschl, 2009; Villarini and Smith, 2010; Smith et al., 2018).”
It is worth noting that the heavy-tail propensity identified in this study encompasses: 1) case studies confirmed to exhibit power-law tail behavior, 2) case studies where power-law tail behavior could not be confirmed due to insufficient samples, and 3) case studies that do not show power-law tail behavior based on historical data but are suggested to exhibit such behavior due to high catchment nonlinearity. The first type is likely to be identified in studies employing different heavy-tailed flood indices as well.
10) Line 649-650, this is obvious.
This sentence (lines 649-650) serves as a contrasting pattern with the following one (lines 650-652). We will revise the terms to tone it down and present a more neutral statement as follows:
Original lines 649-650: “Our findings first indicate that regions with relatively uniform hydroclimatic conditions (the Atlantic Europe and Northern Europe) tend to exhibit a single/dominant propensity of flood tail behavior.”
Modified version: “As expected, the results show regions with relatively uniform hydroclimatic conditions (the Atlantic Europe and Northern Europe) tend to exhibit a single/dominant propensity of flood tail behavior.”
11) Section 5, I enjoy reading this section overall, but it can be further improved by explicitly highlighting what are found in this study, and what are proposed by previous studies, especially the review paper by Merz et al.
Thank you for the suggestion. We will enhance the clarity of this section by highlighting the comparison between current understanding and the new findings contributed by this study in the revised version.
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
- Biswal, B., & Marani, M. (2010). Geomorphological origin of recession curves. Geophysical Research Letters, 37(24), 1–5. https://doi.org/10.1029/2010GL045415
- Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703. https://doi.org/10.1137/070710111
- Clauset, A., Young, M., & Gleditsch, K. S. (2007). On the Frequency of Severe Terrorist Events. Journal of Conflict Resolution, 51(1), 58–87. https://doi.org/10.1177/0022002706296157
- Dralle, D. N., Karst, N. J., Charalampous, K., Veenstra, A., & Thompson, S. E. (2017). Event-scale power law recession analysis: Quantifying methodological uncertainty. Hydrology and Earth System Sciences, 21(1), 65–81. https://doi.org/10.5194/hess-21-65-2017
- Guo, J., Li, H.-Y., Leung, L. R., Guo, S., Liu, P., & Sivapalan, M. (2014). Links between flood frequency and annual water balance behaviors: A basis for similarity and regionalization. Water Resources Research, 50, 937–953. https://doi.org/http://dx.doi.org/10.1002/2013WR014374
- Merz, R., & Blöschl, G. (2009). Process controls on the statistical flood moments - a data based analysis. Hydrological Processes, 23(5), 675–696. https://doi.org/10.1002/hyp
- Shaw, S. B. (2016). Investigating the linkage between streamflow recession rates and channel network contraction in a mesoscale catchment in New York state. Hydrological Processes, 30(3), 479–492. https://doi.org/10.1002/hyp.10626
- Shaw, S. B., & Riha, S. J. (2012). Examining individual recession events instead of a data cloud: Using a modified interpretation of dQ/dt-Q streamflow recession in glaciated watersheds to better inform models of low flow. Journal of Hydrology, 434–435, 46–54. https://doi.org/10.1016/j.jhydrol.2012.02.034
- Smith, J. A., Cox, A. A., Baeck, M. L., Yang, L., & Bates, P. (2018). Strange Floods: The Upper Tail of Flood Peaks in the United States. Water Resources Research, 54(9), 6510–6542. https://doi.org/10.1029/2018WR022539
- Villarini, G., & Smith, J. A. (2010). Flood peak distributions for the eastern United States. Water Resources Research, 46(6), 1–17. https://doi.org/10.1029/2009WR008395
- Wang, H., Merz, R., Yang, S., & Basso, S. (2023). Inferring heavy tails of flood distributions through hydrograph recession. Hydrol. Earth Syst. Sci, 27(24), 4369–4384. https://doi.org/10.5194/hess-27-4369-2023
- Wietzke, L. M., Merz, B., Gerlitz, L., Kreibich, H., Guse, B., Castellarin, A., & Vorogushyn, S. (2020). Comparative analysis of scalar upper tail indicators. Hydrological Sciences Journal, 65(10), 1625–1639. https://doi.org/10.1080/02626667.2020.1769104
- Ye, S., Li, H. Y., Huang, M., Alebachew, M. A., Leng, G., Leung, L. R., et al. (2014). Regionalization of subsurface stormflow parameters of hydrologic models: Derivation from regional analysis of streamflow recession curves. Journal of Hydrology, 519(PA), 670–682. https://doi.org/10.1016/j.jhydrol.2014.07.017
- Zhou, X., Sheng, Z., Yang, Y., Han, S., Zhang, Q., Li, H., & Yang, Y. (2022). Catchment water storage dynamics and its role in modulating streamflow generation in spectral perspective: a case study in the headwater of Baiyang Lake, China. Hydrology and Earth System Sciences, (November). Retrieved from https://doi.org/10.5194/hess-2022-357
Citation: https://doi.org/10.5194/hess-2024-159-AC1
Data sets
Global Runoff Data Centre (GRDC) Federal Institute for Hydrology (BfG) http://www.bafg.de/GRDC/EN
Shuttle Radar Topography Mission (SRTM) A. Jarvis et al. https://cgiarcsi.community/data/srtm-90m-digital-elevation-database-v4-1/
High-Resolution Present-Day Köppen Climate Map H. E. Beck et al. https://doi.org/10.1038/sdata.2018.214
High-Resolution Map of Derived Potential Evapotranspiration R. J. Zomer et al. https://doi.org/10.1038/s41597-022-01493-1
Global Dams and Reservoirs Dataset: GeoDAR v.1.0 J. Wang et al. https://doi.org/10.5281/zenodo.6163413
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