Global Changes in 20‐Year, 50‐Year, and 100‐Year River Floods

Concepts like the 100‐year flood event can be misleading if they are not updated to reflect significant changes over time. Here, we model observed annual maximum daily streamflow using a nonstationary approach to provide the first global picture of changes in: (a) the magnitudes of the 20‐, 50‐, and 100‐year floods (i.e., flows of a given exceedance probability in each year); (b) the return periods of the 20‐, 50‐, and 100‐year floods, as assessed in 1970 (i.e., flows of a fixed magnitude); and (c) corresponding flood probabilities. Empirically, we find the 20‐/50‐year floods have mostly increased in temperate climate zones, but decreased in arid, tropical, polar, and cold zones. In contrast, 100‐year floods have mostly decreased in arid/temperate zones and exhibit mixed trends in cold zones, but results are influenced by the small number of stations with long records, and highlight the need for continued updating of hazard assessments.

• We provide the first global nonstationary assessment of changes in the magnitude, return period, and probability of observed extreme floods • We find increasing 20-year and 50-year floods in temperate climate zones but mostly decreases in arid, tropical, polar, and cold zones • For the 100-year floods, at a smaller sample of sites, we find decreases in arid and temperate zones and mixed results elsewhere used to estimate changes in the magnitude of extremes, such as the annual/seasonal series of maximum streamflow (Faulkner, Warren, Spencer, & Sharkey, 2020;Prosdocimi et al., 2015). In contrast, the GAMLSS framework (Generalized additive models for location, scale, and shape) has the advantage of being able to fit a broad range of distributions to the response variable, and allows the distribution parameters to be modeled as linear or smooth functions of explanatory variables (Rigby et al., 2019). GAMLSS models are a popular alternative for explaining changes in design floods (e.g., Yan et al., 2017), flood flashiness (Saharia et al., 2017), and flood magnitudes as a function of explanatory variables such as time, urbanization (e.g., , or even reservoir indices (López & Francés, 2013).
Application of nonstationary methods is beginning to extend beyond research settings and become adopted by environmental management authorities, for instance in the design of flood mitigation schemes (European Cooperation in Science and Technology, 2013;. A range of technical approaches has been proposed to improve flood risk management, including the design of engineering structures such as dikes, dams, sewers or bridges. These include evaluating time-evolving changes in: the return period or "expected waiting time" (EWT) of an event (Luke et al., 2017;Obeysekera & Salas, 2016;Salas & Obeysekera, 2019;Salas et al., 2018); the return period of the stationary 100-year event ; the design life level (DLL) of structures (Rootzén & Katz, 2013); the short-term risk (Towe et al., 2020); the expected number of exceedances or equivalent reliability of an event (Yan et al., 2017). However, there has not yet been a global empirical assessment of nonstationarity in different flood extremes and how it differs across climate regions.
Here, we provide the first global examination of changes in the return periods and probabilities of observed extreme peak river flows and a discussion of the sensitivity of the results to different approaches, including site selection. We use a nonstationary distributional regression (Stasinopoulos et al., 2018) approachsometimes referred to as a time-varying moments model -where the series' distribution parameters are expressed as functions of time. We: (1) quantify how the magnitudes of flows of a given exceedance probability (e.g., 20-, 50-, and 100-year return period) have changed over the historical record in each stream gauge; (2) estimate how the return periods of flows of given magnitude (e.g., 20-, 50-, and 100-year flood as assessed in the 1970s) have changed over time, from the 1970s to today; and (3) assess corresponding changes in flood probabilities. We visualize these changes, describe the uncertainties involved, and discuss the advantages and drawbacks of different nonstationary measures for describing changes in extreme floods.

Data and Methods
We obtained 10,093 river gauge records from a combination of global and national streamflow archives extending from before 1980 to after 2007 (see Methods in the supporting information for full details and filtering criteria; Figure S1 for entire record length; recent data extend to 2019). The 1970s are selected as a reference point to compare changes in flood return periods because this decade provides a good compromise for observed streamflow record length and availability worldwide. All stream gauges are used, not only those in benchmark or near natural conditions, to obtain a realistic picture of changing flood hazard, whether "natural" or modified by anthropogenic influences like flood defences. The entire record length is employed at each site to reduce uncertainties in the detection of changes in extremes. We require a strict minimum of 30, 50, and 70 complete years of data to estimate changes in the 20-, 50-, and 100-year return periods, respectively. These selected record lengths represent a trade-off between statistical requirements for robustness (ideally, the record length should exceed the magnitude of the extreme under consideration) and global data availability (few countries actually have observed flow records longer than a few decades). Stations with insufficient record length are dropped from the analysis (e.g., if a site has a 30-year record, we only compute changes in the 20-year flood but not changes in 50-or 100-year return periods). Increasing the record length requirement for extremes is technically more robust and allows us to assess consistency in the magnitude and direction of changes in 20-, 50-, and 100-year extremes when using fewer sites with long records.
At each site, we fit both a stationary model (i.e., constant parameters) and nonstationary models (i.e., one or both parameters dependent on time), and select the best-fitting GAMLSS model (Table S1, Figure S2).
At sites where the nonstationary model indicates the best fit, that is, 35% of the 9350 sites where models could be fitted (Methods in the supporting information), we estimate year-to-year changes in magnitudes of the 20-, 50-, and 100-year floods ( Figure 1a). We extract the value of the 20-, 50-, and 100-year flood in the 1970s (from a nonstationary model fitted to the whole record, setting the time covariate to the earliest water year available in the 1970s at each site), along with the corresponding confidence intervals (Figure 1b) to compute the changing return periods and uncertainties by today (where "today" means the most recent complete water year available since 2007; Figure 1c). The uncertainties tend to be smaller at sites with very long (e.g., >100-year) records, but otherwise vary considerably from site-to-site, irrespective of record length ( Figure S3). Sites where the flood magnitude has decreased considerably over time tend to have the largest uncertainties in the recent period ( Figure S4 and Methods in the supporting information).

Flood Magnitudes
The magnitudes of the 20-, 50-, and 100-year flood flows estimated using nonstationary models have both increased and decreased in different regions of the world since the 1970s (Figures 2 and S5). Grouping the SLATER ET AL.
Estimates of change depend on the observational data used to assess each flood extreme (the number of available sites decreases as the record length requirement increases). If the same sites were employed, then results would be relatively consistent across all extremes (20-/50-/100-year floods). However, as we restrict detection of changes in 50-/100-year floods to increasingly small site samples with long records, the average trends expressed for different climate zones may vary (e.g., Figures 2b vs. 2f). In arid and polar regions, flood magnitudes are decreasing on average for all three extremes: 20-, 50-, and 100-year floods (Figures 2b, 2d, and 2f). In tropical and cold regions, we find a majority of decreasing 20-/50-year floods, but different patterns for the 100-year flood. Conversely, in the temperate regions, we find a majority of increasing 20-/50year floods, but decreasing 100-year floods (e.g., the UK). These differences are not surprising and depend on the smaller numbers of sites used to evaluate increasingly rare extremes.
Evidently, at each individual site, the direction and magnitude of change depend inherently on the choice of beginning and end dates (Harrigan et al., 2018) as well as the dominant flood-generating mechanisms and human influences. Changes in extreme floods are likely to be driven by a combination of climatic changes, land cover changes such as urbanization (e.g., Blum et al., 2020;Prosdocimi et al., 2015), agriculture, and anthropogenic influences on river regimes. The next step of such an analysis, inevitably, would be to tease out the underlying causes of the nonstationarity seen in these data (Salas et al., 2018). However, a detailed analysis of the covariates driving these changes in extreme floods is beyond the scope of this work.
To date, there has been limited evaluation of changes in extreme flood properties across climate regions. An assessment of trends in the 30-year flood found overall increases in their frequency and magnitude over Europe and the USA, but weaker increases over Brazil and Australia (Berghuijs et al., 2017). Trends in annual maximum daily streamflow (as opposed to rarer extremes like the 20+-year flood) were found to be mostly decreasing over western North America, northeastern Brazil, and eastern Europe; but increasing over parts of western Europe, western Brazil and parts of eastern North America (Do et al., 2017). Another study of trends in the 90 th percentile and annual maximum of daily streamflow over 1951-2010 (Gudmundsson et al., 2019) found increases over the central USA and northwestern Europe, but decreases over southern Australia, eastern Brazil and western north-America. In Australia, analysis of changes in the annual maximum flow (Ishak et al., 2013) found mostly decreases in floods along eastern and southeastern Australia, and some increases in the north. In China, significant decreases in flooding have been found over large parts of the Yellow River over 1990-2010 (Bai et al., 2016). Our work broadly supports results found for lower-magnitude floods, but also suggests that national/regional averages should be discussed with care when countries span multiple climatic zones (e.g., China, Brazil, and the USA).

Flood Return Periods
Following the assessment of changes in extreme peak river flow magnitudes, we also measure changes in return periods. There has not yet been an evaluation of changes in return periods of extreme floods at the global scale. Changes in return periods help communicate whether environmental extremes are occurring more or less frequently relative to any chosen point in the past. At each site, we estimate the magnitude of the 20-, 50-, and 100-year flood in the 1970s. Using the time-varying distribution parameters at each site (Table S1), we estimate the changing return period of the 1970s floods in every year (Figures 1c and 3).
Changes in flood return periods mirror the changes in flood magnitude and vary greatly around the world (Figures 3 and S6), but some distinct regional patterns can be seen. The 20-, 50-, and 100-year floods of the 1970s have become, on average, ∼41-, 152-, and 358-year floods under present-day conditions (at 3174/2164/1262 sites with nonstationary records, respectively). These overall decreases in flood frequency at nonstationary sites may be due to a combination of flood control measures, climatic changes, and long-term water storage/antecedent conditions (Rodell et al., 2018;Sharma et al., 2018). However, this global average masks considerable spatial redistribution of flood hazard across climate zones (Figure 3). In  change rather than precise estimates, since averages depend on site selection, and more information may be available locally than when using a global data set.

Flood Probabilities
One alternative way to communicate the flood hazard is to employ the concept of annual exceedance probability (AEP) instead-that is, the probability that a flow of a given magnitude will be exceeded in a given year. An annual flood probability conveys the same information as a return period, defined on the annual maximum scale (since it is merely the inverse of that return period), but it expresses the time scale explicitly. Arguably, probabilities simplify expressions of change: for example-"our chance of being flooded this year is 10%, up from 1% in the past." As a result, some institutions such as the Environment Agency in England and the U.S. Geological Survey (USGS) have moved toward expressing extremes in terms of annual probabilities instead of return periods (Fleming, 2002;Holmes & Dinicola, 2010). However, event probabilities are equally likely to be flawed if they are not updated, and perhaps more important than choosing between the two, is using recent statistics that more realistically reflect the true hazard.
To illustrate changing AEPs, we compute an "updated stationary" assessment (e.g., Luke et al., 2017), which describes how the probability of exceeding a given flow during a planning horizon varies over time. We estimate the exceedance probability of the T-year flow representative of 1970 conditions over a fixed, 30-year period, which may be relevant for a mortgage (Read & Vogel, 2015) or infrastructure design horizon (Olsen, 2015). We then compute the updated 30-year exceedance probability estimate representative of recent conditions, using the same flow rate but an updated annual probability. The 30-year exceedance probability is 1-(1-p) 30 , where p is the fixed annual probability over the 30-year period.
At the global scale, consistent with results for flood magnitude and return periods, we find a decrease in flood probabilities, which contrasts with global increases in the frequency of extreme precipitation (Papalexiou & Montanari, 2019). The 20-year flood representative of conditions in 1970, which had a 79% chance of occurring over a 30-year horizon under conditions equivalent to the 1970s, now has a 52% chance of occurring (median of 3174 sites). The 50-year flood, which had a 45% chance of occurring over a 30-year horizon under conditions equivalent to the 1970s, now has an 18% chance of occurring (median of 2164 sites). Similarly, the 100-year flood, which had a 26% chance of occurring, now has an 8% chance of occurring (median of 1262 sites), based on an updated, but constant 30-year annual event probability over the lifetime of a 30-year mortgage or design horizon. For comparison with the analysis of return periods, we provide maps of these updated event probabilities in Figure S7.

Conclusions
Here, we provide a global assessment of changes in the magnitude, return period, and probability of extreme river floods. For the 20-and 50-year return periods, we find a majority of increasing floods in temperate climate zones, but a majority of decreasing floods in arid, tropical, polar, and cold climate zones. For the 100-year return periods (a smaller sample of sites with at least 70 years of data), we obtain slightly different results, with decreases in arid and temperate zones; mixed trends in cold zones; and increases at a small sample of tropical sites. These differences serve as a reminder that the regional results of nonstationary analyses depend inherently on site selection and different return periods (flood quantiles) of interest.
This work does not evaluate the flood drivers that led to regional patterns of change, whether tied to climate or land use change. Additional analyses could classify the sites by their catchment characteristics (climate, land cover, or reservoir storage) and use physically based covariates such as time series of the percent land cover (Blum et al., 2020;Prosdocimi et al., 2015), or climate and reservoir indices (López & Francés, 2013). Extrapolation of flood probability estimates into the future also requires knowledge of the causes of the change. One growing avenue is to bridge the gap between statistical modeling of past trends and physics-based modeling of future changes in the event-generating mechanisms using hybrid dynamical-statistical models (Slater & Villarini, 2018;Vecchi et al., 2011;Wang et al., 2009) and machine learning. Such approaches take advantage of the ability of physical models to predict and explain large-scale phenomena and the strengths of the nonstationary statistical models to estimate probabilities of extreme events conditioned on observed data.
Overall, distributional regression models provide a convenient approach to detect changes in the magnitudes, return periods, and probabilities of extremes. However, such results, and their uncertainties, should always be treated with care. Our findings indicate that in presence of nonstationarity, local flood hazard assessments should be updated regularly (Luke et al., 2017;Towe et al., 2020), for example, in the aftermath of significant floods, or at a fixed interval such as every five years (UK Cabinet Office, 2015), to ensure regular provision of the most up-to-date estimates of changing flood probabilities.

Acknowledgments
The authors thank the authors of the gamlss R package; the GRDC for global streamflow data; Robert Holmes of the U.S. Geological Survey for a helpful critical review; and the associate editor and manuscript reviewers. The authors gratefully thank the developers and maintainers of rnrfa, dataRetrieval, and tidyhydat R packages (cited in the manuscript), which enabled us to access comprehensive data for three countries (UK, United States, and Canada, respectively). Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government.