1. Introduction
Evidence suggests that global warming has intensified extreme precipitation in most places, affecting the human and natural environment (Kirchmeier-Young and Zhang 2020; Min et al. 2011). Climate models predict future climate under different anthropogenic activities and natural phenomena. Studying these impacts allow for developing mitigation and adaptation plans, risk management, and successful infrastructure design. The Coupled Model Intercomparison Project (CMIP) aims to integrate climate modeling efforts from several organizations and institutes into a unified system. The sixth phase of CMIP (CMIP6) was launched after intensive discussions, meetings, and debates to bridge the previous CMIP5 gaps. It seeks to improve our understanding of climatic variability by increasing the number of experiments, global warming drivers, and the number of participating research communities (Eyring et al. 2016).
Climate indices are often used in the literature to represent extreme precipitation conditions (e.g., the ETCCDI indices; http://etccdi.pacificclimate.org/list_27_indices.shtml). The indices Rx1day and Rx5day refer to annual maxima precipitation at 1-day (hereafter, we use AMP) and 5-day scales, respectively. Globally, Rx5day is increasing by 6%, 10%, and 20% in the four representative concentration pathways (RCPs) of CMIP5: RCP2.6, RCP4.5, and RCP8.5, respectively (Sillmann et al. 2013). The projected 20-yr events (i.e., 5% exceedance probability) are increasing by 10% under the medium forcings (RCP4.5) and 30% under the high forcings (RCP8.5) over 1986–2005 levels (Kharin et al. 2013). Also, return periods of these levels are reduced almost everywhere in the globe. Recently, the CMIP6 projected events primarily are also increasing. However, considerable disagreement among simulations is evident in some dry areas, including the Sahara, northeastern Brazil, and some regions in Australia (Wehner et al. 2020). Noteworthy, Rx1day produced by CMIP5 and CMIP6 has no considerable differences on a global scale (Wehner et al. 2020). Additionally, the relative change of the 90th and 99th percentiles of precipitation distributions indicates an increase with heavier tails compared to the 1966–2005 period (Scoccimarro and Gualdi 2020).
Regionally, Rx5day is elevating mainly in high northern latitudes, northern Asia to the Tibetan Plateau and East Africa. Yet changes in precipitation extremes indices are not significant in the Sahara. CMIP5 models vary in simulating the sign of change for different regions and indices. In China, ETCCDI indices projected using CMIP5 multimodel median indicate an intensification of precipitation extremes. Specifically, the increase in Rx1day and Rx5day is more pronounced than the other indices (Zhou et al. 2014); such results hold for Xu et al. (2018). In the United States, ETCCDI indices show an increase in the frequency and intensity of extreme precipitation (Thibeault and Seth 2014; Wuebbles et al. 2014). In Zambia, projections from CMIP5 models show an increase in the magnitude of precipitation extremes (Libanda and Ngonga 2018). Further, 20-yr events of AMP estimated from generalized extreme value (GEV) distributions were projected over the Korean Peninsula using CMIP5 models (Lee et al. 2020). The results indicate relative increases in these events of 23% and 45% in RCP4.5 and RCP8.5, respectively, over the 1971–2005 levels. In Spain, the projected 10 and 100 events of AMP increase to 15% and 25%, respectively, for RCP4.5 and RCP8.5 in 2051–2100 compared to 1951–2000 (Monjo et al. 2016). Additionally, CMIP6 models predict inflation in precipitation extremes (ETCCDI indices) under different shared socioeconomic pathways (SSPs) in Iran (Zarrin and Dadashi-Roudbari 2021), Southeast Asia (Ge et al. 2021), East Africa (Ayugi et al. 2021), and China (Zhu et al. 2021). RCPs and SSPs are future scenarios for CMIP5 and CMIP6, respectively, which share the same radiative forcing by the end of the century. However, SSPs also include more drivers of radiation, such as aerosol impacts and land surface modules, in addition to accounting for the projected socioeconomic statuses [for more details, see O’Neill et al. (2016) and Stouffer et al. (2017)].
Most precipitation extreme studies did not use bias correction (BC). Usually, a suitable bias-correction technique is prompted for the outputs of climate models. Climate model projections encounter systematic biases from GCM modeling of different hydroclimatic variables, the variability in response to anthropogenic warming, and the stochastic internal natural variability (Eden et al. 2012). Typically, BC methods rely on quantile mapping transforming projection values to follow the distribution of the historical data. The basic quantile mapping (QM) technique inflates the trends of extreme precipitations (Cannon 2018; Cannon et al. 2015; Maraun 2013). The detrended quantile mapping (DQM), the quantile delta mapping (QDM; Cannon et al. 2015), and the semiparametric quantile mapping (SPQM; Rajulapati et al. 2022b) overcome some shortcomings of the basic QM. The DQM method preserves the projected trend of the time series, the QDM method preserves the relative changes between historical and projected time series, and the SPQM method preserves projected probabilities and trends. Other methods of BC include multivariate BC and regression transformation functions (Cannon 2018; Li et al. 2010; Piani et al. 2010). Note that different BC techniques have different assumptions that can affect the target analysis.
Although global warming is expected to lead to nonstationarity in climate extremes (e.g., AghaKouchak et al. 2020), few studies consider nonstationarity behavior in the projected precipitation extremes from climate models. For example, in Greece, 16 nonstationary GEV models were compared for 12 precipitation stations, to perceive the optimum model under two projection scenarios (Panagoulia et al. 2014). In the United States, 30 nonstationary GEV models were investigated for eight regions under the RCP4.5 and RCP8.5 in CMIP5 projections (Um et al. 2017). These studies indicate that stationary and nonstationary (linear change in location with time covariate) models best describe AMP. The literature lacks studies about nonstationarity in projected AMP from CMIP6 models.
There is no consensus in the literature about using nonstationary models while analyzing hydroclimatic variables, especially under climate change (Serago and Vogel 2018). However, many studies show that the nonstationarity of future extreme precipitation is more pronounced under global warming (De Paola et al. 2018; Singh and Chinnasamy 2021). Analyzing climate model projections under nonstationarity can improve the interpretation of extreme precipitation changes and assess large extremes beyond simulated levels, which helps design critical infrastructure. For example, the choice between stationary and nonstationary approaches can severely affect the estimated risk in real-world flood frequency applications. Applying stationary models to potentially nonstationary processes, such as precipitation extremes, can be hazardous, leading to catastrophic consequences for the design of critical hydraulic and water resources infrastructures.
Here, we use bias-corrected CMIP6 projections to estimate the design levels and return periods using the optimum model for each land grid, including stationary and nonstationary models. Further, we perform a fully controlled Monte Carlo (MC) experiment to understand and identify the deviations in design levels that stationary models estimate under nonstationary scenarios. We aim to identify the reasons for such deviations while understanding how stationary models perform when describing time series emerging from a nonstationary process. Overall, we aim to answer four questions: 1) How will the historical 33-yr levels change in the future? 2) How the return period of the historical 100-yr event is changing in the future? 3) What is the difference between future 100-yr events and estimated events from historical data? 4) What are the consequences of using stationary models instead of nonstationary models in describing projections with trends?
2. Data
a. Observational data
Gridded observational datasets are essential in bias correcting projections of climate models (Contractor et al. 2020; Liu et al. 2019; Wehner et al. 2020; Zhou et al. 2014). Here, we use the Global Precipitation Climatology Centre (GPCC) for daily precipitation since it depends only on rain gauge data. GPCC employs over 85 000 stations from all over the globe (Becker et al. 2013; Schneider et al. 2017). We checked the GPCC dataset for missing values because they can affect the estimated annual maxima (Papalexiou and Koutsoyiannis 2013), and we did not find any. The spatial resolution of the GPCC dataset is 0.5° × 0.5° and was regridded to 2° × 2° using the first-order area-conservative remapping (Jones 1999). This resolution is the average resolution of all used climate models (ranging from 0.5° to 2.5°). Spatial resolution should be common in both observations and simulations as the analyses were implemented grid-wisely. The analysis utilizes observations during 1982–2014 which is common between CMIP6 historical simulations and observation for applying BC techniques.
b. CMIP6 simulations
Here, we analyze AMP estimated from all available CMIP6 models (at the time of this study), that is, 34 models offering 284 historical simulations and 722 projections (see a brief description of simulations in online supplementary information Table S1). Each model may contain different simulations representing several parameterization schemes, forcing, initial conditions, and boundary assumptions. Different SSPs are considered in the analysis. The SSP1–2.6 assumes reduced emissions, SSP2–4.5 assumes the same trend of emissions as historical, SSP3–7.0 is a medium to high emission scenario, and SSP5–8.5 is optimal for economic development but with high emissions (O’Neill et al. 2016). Simulations were regridded to the average 2° × 2°, matching the observational dataset (Ahmadalipour et al. 2017).
3. Methods
a. Bias-correction technique
Here we bias correct the AMP using a semiparametric quantile mapping (SPQM). The SPQM method (first introduced and applied for minimum and maximum temperatures in Canadian cities and 199 megacities; Rajulapati et al. 2022a,b) preserves the trends in projections, the empirical probabilities of all projected quantiles, and the observed distribution. We apply this approach directly to AMP, correcting biases in extremes and not for daily time series. SPQM consists of five steps: 1) remove the linear regression trends in CMIP6 projections of AMP, 2) estimate the empirical distribution functions (CDFs) of the detrended projections using the Weibull plotting position, 3) fit a stationary GEV distribution to observed AMP [Mann–Kendall and KPSS (Kwiatkowski et al. 1992) tests resulted in 86% and 92% stationary time series of land grids, respectively], 4) transform the empirical CDF of the projections to follow the fitted GEV (quantile mapping), and 5) add the trend back to get the corrected AMP projections. In step 1, a linear trend is assumed in future projections to account for nonstationarity, following the detrended quantile mapping (DQM) method. The sole application of the DQM method reduces biases for precipitation (Cannon et al. 2015). We compare the results from SPQM to the QDM, which considers the relative change between historical and projected precipitation data and the projected trends of climate models (Cannon et al. 2015); see more discussions in appendix A.
We use the Mann–Kendall (MK) test (Kendall 1948; Mann 1945) in this analysis to 1) verify the existence of significant deterministic trends, 2) justify the selection of nonstationary models, and 3) monitor the impact of global warming on the nonstationarity of future AMP. To assess other forms of nonstationarity, the KPSS test (Kwiatkowski et al. 1992), which detects abrupt or unregular changes within the variance (unit roots), is implemented. Thus, this test can verify the ability of the SPQM technique in reproducing the nonstationarity of AMP projections.
An efficient bias-correction method should also preserve the nonstationarity in projections. SPQM preserves nonstationarity according to MK and KPSS tests (Table S2). For example, 51.83% and 53.77% of all samples have both trends and unit roots for SPQM and raw projections, respectively, for SSP5–8.5. Therefore, applying the following analysis and methods to projections preserves the original nonstationary of the projections and thus does not affect the results. Note that bias-correction techniques have limitations. For instance, past distributions are assumed to continue in the future, even when preserving trends or relative change (Maraun 2013).
b. Selection of the generalized extreme value models
The literature suggests different trends (linear and nonlinear) in extreme precipitation projections. For example, Panagoulia et al. (2014) and Um et al. (2017) suggested linear trends, Ouarda and Charron (2019) indicated that linear change in the GEV location parameter did not improve fitting, and Tian et al. (2021) and Vasiliades et al. (2015) suggested that log-linear and exponential change in both location and scale parameters can represent better nonstationary in AMP. Accordingly, we have tried several nonstationary GEV models with different combinations of location and scale parameters varying in log-linear and exponential relationships with time. Then, we have selected the models that were dominant in describing AMP time series (Fig. S1). Note that more complex models can increase the uncertainty of estimates (Serinaldi and Kilsby 2015). Specifically, here, we apply a stationary and three nonstationary GEV distributions with time-varying location α(t) and scale β(t) parameters described in Table 1.
Description of parameters in the applied nonstationary GEV models.
c. Estimation of return periods and levels
Finally, we demonstrate the implications of using the stationary GEV model in describing annual maxima samples with significant increasing trends. Specifically, implementing Monte Carlo simulations, we identify the percentage difference of the 100-yr return levels estimated from both model 0 (stationary) and model 1 (nonstationary) for samples with trends. The steps of these experiments are 1) selecting representative GEV parameters as α = 20, β = 10, and γ = 0.2, 2) generating 1000 random samples of the defined stationary GEV model with a sample size of 80, 3) adding a defined linear slope to the generated samples, 4) fitting these samples to the stationary model (model 0) and nonstationary model (model 1), 5) estimating GEV parameters and return levels, and 6) repeating steps 3, 4, and 5 for multiple slopes. This analysis adapts nine slopes (−0.2 to 0.2 mm day−1 yr−1, with an increment of 0.05). For each slope, we compare the stationary and nonstationary levels with each other and with the theoretical level. The theoretical level is estimated from a nonstationary model (model 1), reproducing the input slope and GEV parameters of the samples. We note that these experiments aim to understand and identify the level of deviation caused by stationary models for describing samples with trends of different magnitudes.
4. Results
a. Nonparametric analysis
Projected AMP time series show diverse regional variability in each continent (Fig. 1). The Southern Hemisphere seems to experience the highest historical variability during 1982–2014. The interquartile ranges of AMP values in the historical period are 71, 49, and 105 mm day−1 in Africa, Australia, and Latin America, respectively, whereas the ranges are 26.5, 16.5, and 30.5 in Asia, Europe, and North America, respectively. In Latin America, the historical variability (interquartile range is 105 mm day−1 during 1982–2014) is even higher than the projected variability (64 mm day−1 among all scenarios during 2015–2100). The Southern Hemisphere is potentially more vulnerable to natural climate extremes because oceans mostly surround it. Oceans and their variability, mainly El Niño–Southern Oscillation (ENSO), control most of the natural phenomena (e.g., tropical cyclones, thunderstorms, ocean mixing, and heat circulation) (Collins et al. 2010). Oceans also store about 90% of Earth’s energy, making them the main contributors to Earth’s energy balance (von Schuckmann et al. 2020). Overall, augmented by natural variability, the Southern Hemisphere has the highest projections at the end of the century for all scenarios. In contrast, the Northern Hemisphere seems more vulnerable to climate change caused by human influences (Fig. 1). Historical variability in AMP (interquartile ranges are 26.5, 16.5, and 30.5 mm day−1 in Asia, Europe, and North America, respectively) is lower than emission scenarios variability (39.0, 20.5, and 35.0 mm day−1, respectively). Note that individual projections (see, e.g., Figs. S2–S5) lead to similar interpretations. These findings can answer one of the addressed scientific questions of the CMIP6 project for which the ScenarioMIP experiment is developed. ScenarioMIP aims to identify the range of anthropogenic influences compared to the historical variability (O’Neill et al. 2016).
Annual maxima of daily precipitation (AMP) time series of observations (1982–2014) and bias-corrected projections (2015–2100) based on the SPQM technique showing the continental variations: (a) Africa, (b) Australia, (c) North America, (d) Asia, (e) Europe, and (f) Latin America. The lines show the means for all continents’ grids, and the shaded regions show the interquartile range. The bars indicate the interquartile ranges, with points indicating the mean values in 2100.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
We assess the empirical 33-yr level (i.e., the largest AMP in 33 years) for the near (2035–67) and far future (2068–2100). Nonparametric levels can offer valuable information about the effect of climate change on extreme precipitation in the short term. The far-future empirical 33-yr levels increase as warming increases, while they remain close to the historical levels for the near future (Fig. 2). In SSP5–8.5, the highest increase in the far future compared to the historical period occurs in Asia (26.5% difference compared to historical levels), while the lowest increase occurs in Latin America (12.3%). Latin America has the lowest increase because it has the highest historical variability compared to other regions (Fig. 1). Further, Asia and North America have the highest increase in far-future levels compared to the historical ones, especially in the coastal regions. The rest regions have average changes ranging from 12.30% to 16.86%. Africa and Latin America have similar regional variations; their northern regions are less affected by global warming than their southern regions. The 33-yr levels within the grids of each continent have large variability, and their distribution is positively skewed (attested by higher means than medians; Fig. S6).
Nonparametric 33-yr return level for each continent for all scenarios: (a) 181 simulations for SSP1–2.4, (b) 140 simulations for SSP2–4.6, (c) 177 simulations for SSP3–7.0, and (d) 224 simulations for SSP5–8.5. The points represent the mean value, and the bar represents the empirical 90% confidence interval. (right) The regional variations described by the percentage difference between the far-future and the historical levels.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
b. Parametric analysis
The percentage of projections with different forms of nonstationarity increases with global warming scenarios from SSP1–2.4 to SSP5–8.5 (Table S3). This finding agrees with previous studies (e.g., De Paola et al. 2018; Singh and Chinnasamy 2021; Um et al. 2017). The parametric analysis using various GEV models shows that parsimonious models are performing better over the more complex ones (Fig. S7). When significant trends exist, model 1 (which shows a linear change of location parameters with time) dominates. In absence of significant trends, the stationary GEV (model 0) dominates. Similar results held in Um et al. (2017) using CMIP5 AMP projections in the United States. Model 2 and model 3, which encompass a change of scale parameter with time, are mostly not selected. Based on the optimum fitted models, the return periods corresponding to the historical 100-yr return level are estimated for each grid for all scenarios (Fig. 3). As warming increases, the return periods decrease almost everywhere (i.e., the historical 100-yr levels become more frequent). The far-future frequency of the historical 100-yr level is higher than the near future. In SSP5–8.5, the frequency in the Northern Hemisphere becomes almost double in the near future (return periods equal 48, 54, and 57 years in Asia, Europe, and North America compared to 100 years in the historical period, respectively) and triple in the far future (25, 33, and 32, respectively). However, in the Southern Hemisphere, the frequency is around 1.5 times higher in the near future (return periods equal 67, 74, and 66 years in Africa, Australia, and Latin America, respectively) and around 2.5 times in the far future (39, 40, and 37 years, respectively). Although the Southern Hemisphere seems to be more vulnerable to natural variability and extremes (Fig. 1), it is anticipated to experience less frequent extreme events under the anthropogenic scenarios of potential warming compared to the Northern Hemisphere.
Return periods that correspond to the historical 100-yr return level, grouped by continents for all scenarios: (a) 181 simulations for SSP1–2.4, (b) 140 simulations for SSP2–4.6, (c)177 simulations for SSP3–7.0, and (d) 244 simulations for SSP5–8.5. The points represent the mean value, and the bar represents the empirical 90% confidence interval. (right) The regional variations described by the percentage difference of the 100-yr return period compared to the far-future return period, describing how frequent the historical 100-yr level becomes in the far future.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
Meanwhile, the change in 100-yr events relative to the historical ones is calculated based on the projection period of 2015–2100 (Fig. 4). The percentage differences increase as warming increases, showing an intensification of precipitation events resulting from global warming. Substantial regional variability is evident; Asia has the highest increase of an average of 21.05%, while Australia has the lowest of 7.5% in SSP5–8.5. Other continents have average changes within 10%–12.5%. However, some regions in Africa, Latin America, Australia, and dispersed localized grids over the globe have reductions (negative percentage difference) in the 100-yr level, which may indicate potential droughts in these regions. Considerable reductions in North Africa indicate minimal projected 100-yr levels compared to high historical ones, yet this may be attributed to dry regions. The regional variability shown by the average percentage differences of the 100-yr level holds for each continent’s 90% empirical confidence interval (Fig. S8). Noteworthy, variation among individual models of CMIP6 shows no vast differences between the projected 100-yr level of 2015–2100 (Fig. S9).
The average percentage differences of future (2015–2100) 100-yr level compared to the historical (1982–2014) level for all scenarios: (a) 181 simulations for SSP1–2.4, (b) 140 simulations for SSP2–4.6, (c) 177 simulations for SSP3–7.0, and (d) 224 simulations for SSP5–8.5.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
c. Stationary and nonstationary comparative analysis
Stationary levels are lower than nonstationary ones almost everywhere (Fig. 5, i.e., a negative percentage difference indicates a lower stationary level compared to the nonstationary one). Using stationary instead of nonstationary models reduces the 100-yr events on average by 3.4% globally. However, the range of percentage differences varies widely for different continents; Africa, Latin America, and Southeast Asia have high reductions when stationary models are used (Fig. 5). In SSP5–8.5, for example, Africa, Asia, Australia, Europe, Latin America, and North America have 90% empirical confidence intervals (i.e., percentage differences lying between 0.05 and 0.95 quantiles of all values in each region) of −22.4 to 8.4, −14.5 to 5.8, −15.1 to 7.8, −12.3 to 3.9, −21.9 to 5.4, and −16.7 to 4.5, respectively (Fig. S10). As warming increases, the reduction of stationary models increases, especially in the Southern Hemisphere, which is predicted to have higher return levels by the end of the century. Past studies (e.g., Read and Vogel 2015; Šraj et al. 2016; Tian et al. 2021; Wi et al. 2016) also depicted that stationary models give low return levels compared to nonstationary models for other extreme hydroclimatic variables. Here, we explain the reasons behind this underestimation in the following paragraphs and appendix B.
Percentage differences of stationary 100-yr levels compared to nonstationary ones for 2015–2100 in all scenarios: (a) SSP1–2.4, (b) SSP2–4.6, (c) SSP3–7.0, and (d) SSP5–8.5, showing a consistent reduction by stationary levels.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
Through extensive Monte Carlo experiments, we demonstrate the performance of stationary and nonstationary models under several nonstationary scenarios. We assess the sample variability and bias in GEV parameters and several design levels (100-, 200-, and 500-yr events). The results supported the underestimation of the stationary models. Percentage differences between stationary and nonstationary levels range from −2.9% to −8.6%, with increasing slopes ranging from 0.05 to 0.2 for the 100-yr levels. The estimated global percentage difference of −3.4% for 100-yr levels of AMP, with an average global slope of 0.12 for all scenarios, matches the results of the MC experiments. As return levels increase (200 and 500 years), the underestimation of stationary models increases for positive slopes more than for negative ones (Fig. 6). However, slight overestimations are observed for higher levels (500 years) estimated from time series with negative slopes. Also, different characteristics are observed for negative and positive trends because the GEV location parameter is not symmetric versus slope; in contrast, the scale and shape parameters are symmetric for negative and positive slopes (see Fig. B1). However, negative trends are not typical for AMP projections. Under the impacts of climate change, nonstationary models are expected to describe better extreme precipitation. To double-check, we compare levels from both models to a theoretical level that reproduces the input slope and GEV parameters. MC medians of the nonstationary levels for all slopes are almost the same as the theoretical levels (Fig. S11).
MC simulations investigating the effect of trends on stationary and nonstationary levels show percentage differences in stationary models compared to nonstationary ones for the (a) 100-, (b) 200-, and (c) 500-yr levels. Each violin plot shows the distribution of 1000 experiments, and the inner boxplot indicates the median and the interquartile range.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
5. Discussion
The nonstationary GEV analysis can describe changes in extreme variables, considering covariates. In the case of extreme precipitation, these covariates can be climate indices, temperature, and time. Climate indices affect the variability of global precipitation from annual to decadal variations (Gao et al. 2016). These indices include the East Asian Summer monsoon (EASMI), the western Pacific zonal and meridional variations of the East Asia jet stream entrance region (WP), El Niño–Southern Oscillation (SOI or ENSO), the Pacific–North American (PNA), and the Pacific decadal oscillations (PDO). Over China, GEV with time-varying location and scale parameters describe well nonstationary AMP time series with significant positive or negative trends (Gao et al. 2016). Only 4% of the study samples are invariant with time but are correlated with climate indices. In British Columbia, Canada, GEV models with a time-varying scale parameter and climate-indices-varying shape parameter perform the best (Ouarda and Charron 2019). However, using only shape parameters varying with climate indices does not improve performance, which indicates the significance of the time covariates. In Greece and Cyprus, several nonstationary GEV models were investigated in describing the nonstationarity of annual maxima of rainfall data. In these Mediterranean regions, the climate depends on low frequency and large-scale climate indices. The North Atlantic Oscillation (NAO) index and the Mediterranean oscillation index (MOI) greatly affected the precipitation in the study regions (Vasiliades et al. 2015). However, it has been challenging to detect the physical reasoning behind the correlation of AMP with climate indices. Some regions may have convective storms, which are not affected by large-scale cyclonic rainfall. For some other regions, the extreme precipitation can extend for more than one day. Several regional studies depicted that precipitation extremes last for more than three days, which could be extended to five and seven days (e.g., Eden et al. 2016; Philip et al. 2020; Risser and Wehner 2017). For instance, during Hurricane Harvey near Houston, United States, the average 3-day precipitation was the highest among all historical 9000-yr records of 3-day precipitation (van Oldenborgh et al. 2017). Thus, a 1-day extreme measure (i.e., AMP) cannot sufficiently reflect these extended extreme events represented by climate indices (Gao et al. 2016).
Temperature covariates have been investigated in nonstationary GEV models to attribute extreme precipitation to global warming. The Clausius–Clapeyron (CC) scaling describes the physical relationship between precipitation and temperature (e.g., Hobijn et al. 2004; van Oldenborgh et al. 2017). However, many studies have argued that the CC scaling is not the same everywhere (e.g., Scoccimarro et al. 2013; Utsumi et al. 2011). For subdaily time scales, super CC scaling has been detected probably because of the atmosphere’s fine temporal and spatial dynamics (Lenderink and Attema 2015). Also, the CC relationship may vary in different types of convective storms (Berg et al. 2013; Lenderink and van Meijgaard 2009) and be valid mainly in midlatitude regions (Pall et al. 2007). The change of extreme precipitation with temperature was investigated over the globe using CMIP5 (Kharin et al. 2013). GEV models with all parameters varying linearly with the mean temperature at each grid were used, then the return periods were calculated per unit of warming. The change of extreme precipitation with mean temperatures was not uniform globally, with an average of 6% °C−1 that conformed with the CC scaling. However, in northern extratropics, this scaling was lower (4% °C−1). Van Oldenborgh et al. (2017) investigated the annual maximum 3-day precipitation change with global mean temperature to attribute extreme precipitation changes to anthropogenic warming. The results indicated that the change of extreme precipitation with temperature was not consistent among various observational datasets. Among climate models, the change ranged from 4% to 16% °C−1 in the U.S. Gulf Coast. Linking regional or global mean temperatures with annual precipitation extremes is not necessarily informative because these temperatures may not reflect the weather conditions during the extreme period. However, global warming affects extreme precipitation generally, which has been simulated in climate models scenarios for the future through more complex climate systems and interactions.
Here, we do not use climate indices or temperature as covariates in GEV models for two reasons. First, climate models accredit different climate phenomena through complex simulations. Second, the variation of AMP with global temperature exists through the four adopted SSPs simulations. These simulations include various global warming scenarios and socioeconomic impacts by the end of the century. We also avoid using GEV shape parameters with any covariates as it can lead to inaccurate estimates of design level, coming from their high sample variability and the numerical instability of the fitting (for more details, see appendix B). Overall, a reliable description of extreme precipitation and an understanding of its variability using time covariates are of critical importance to the scientific and engineering communities (Kharin et al. 2013), which is the focus of this paper.
Our results indicate that Asia and North America have the highest 33- and 100-yr levels. Also, the maps show that the coastal regions facing the Pacific Ocean have the highest values among the continents. Wang et al. (2014) showed that combining ENSO and PDO as covariates can describe the distribution of wet and dry regions over the globe. The coastal regions of Asia and North America are vulnerable to such natural variabilities. The extent of this combination can also affect other continents, such as Africa and some parts of Australia, especially during the warm phase of PDO, causing severe dryness in these regions (as Fig. 2 shows a reduction in 33-yr levels, Fig. 3 shows a huge increase in the return periods of the historical 100-yr level, and Fig. 4 shows a reduction in the 100-yr level).
Further, North Africa and Sahara are dry regions that experience severe reductions in their design levels. Such regions might be affected by their severe aridity and dust transport in combination with the large tropical and subtropical variability (Bellprat et al. 2015; Cabré et al. 2016; Prospero et al. 2014). Also, these reductions could be because some local natural phenomena, such as cloud formalization, cyclones, or mountains aerodynamics, might be inaccurately captured by climate models (Bony and Dufresne 2005; Ngoma et al. 2021; Wengel et al. 2021).
Nonstationarity is inevitable for precipitation extremes, especially under climate change. However, three concerns should be considered when selecting a suitable model for extreme precipitation. First, assuming a specific pattern for change with time in the applied nonstationary model beyond the study period is hardly supported by a physical relationship explaining future nonstationarity in precipitation extremes (Serinaldi and Kilsby 2015). Second, the most common method for fitting nonstationary models is the method of likelihood which includes optimization techniques to maximize the likelihood function. These optimization techniques sometimes suffer from numerical errors that may give inaccurate estimates (Boomsma 1985). Such errors usually occur when using short samples or samples with missing values, which do not exist in climate models’ products. Third, stationary models should be used when nonstationarity forms are absent in time series, even if a general nonstationary model can be used for all samples. Stationary models are more parsimonious as they have fewer parameters than their counterparts of nonstationary models. Stationary GEV models yield more efficient parameters than the nonstationary models for samples with no trend. A more efficient parameter is a parameter which has a narrower variance. The interquartile ranges of Monte Carlo experiments show that the stationary model yields more efficient parameters for the case of no trend, especially for the location parameter (Fig. B1; slope = 0). Therefore, when nonstationarity is absent, stationary models are preferred. When trends appear in samples, parsimonious nonstationary models yield more efficient parameters (see appendix B for more discussion). Using stationary models for estimating design levels may lead to hydraulic structures and infrastructures with reduced capacities. This underdesign can lead to catastrophes of flooding in most cases.
6. Conclusions
Climate models are valuable in assessing future extreme precipitation. However, their outputs can include biases due to uncertainties in natural variability, human influences, and systematic errors. A new bias-correction technique [semiparametric quantile mapping (SPQM)] is adopted that corrects CMIP6 projected annual maxima of daily precipitation (AMP) based on observations while preserving the projected trends. In this paper, we aim to analyze variability in the projected precipitation extremes by answering four main questions: 1) What are the projections of the short-term extremes (33-yr events)? 2) What are the projected return periods for the historical 100-yr event? 3) What is the difference between the projected and historical 100-yr events? 4) What are the consequences of using stationary models for samples with trends? Further analyses are also implemented to understand these consequences and to check uncertainties for different nonstationary GEV models. The key findings of this study are as follows:
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The empirical 33-yr levels are increasing with warming, with absolute differences ranging from 0.2 to 33.2 mm day−1 under SSP5–8.5 compared to the historical period.
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In the Northern Hemisphere, the occurrence frequency of the historical 100-yr levels becomes double in the near future and triple in the far future, while in the Southern Hemisphere, they become 1.5 and 2.5 times in the near and far future, respectively.
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The projected 100-yr levels are higher than the historical levels, ranging from 7.5% to 21% under SSP5–8.5.
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Asia has the highest projected empirical 33-yr level (absolute difference with historical ones equal to 33.2 mm day−1), frequency of the historical 100-yr level (return periods equal to 48 years in the near future and 25 years in the far future), and 100-yr levels of 21% higher than the historical levels.
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Stationary models used in fitting samples with significant increasing trends underestimate design levels, which risk infrastructure and hydraulic structures (a global reduction of 3.4% on average).
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MC simulations explained the stationary models’ underestimation for the 100-, 200-, and 500-yr levels while showing the ability of simple nonstationary models to estimate design levels for nonstationary time series.
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High uncertainties are associated with including covariates for shape parameters and using complex models.
A proper understanding of the statistical characteristics of different GEV models is crucial before applying them in impact studies. Climate change caused by both natural and anthropogenic effects is expected to increase precipitation extremes in most of the globe. Using stationary models to describe projected extremes with trends leads to underdesigning critical hydraulic structures and infrastructures. Parsimonious GEV nonstationary models with location parameters varying linearly with time provide reliable estimates with reasonable uncertainties. Regional analysis of precipitation extremes with finer spatial resolutions is necessary for the next steps. For such regional analysis, coupling precipitation extremes with regional temperature before and during storms can provide valuable information about the effect of temperature on precipitation (e.g., van Oldenborgh et al. 2017; Philip et al. 2020; Risser and Wehner 2017).
Acknowledgments.
H.M.A. is funded by the Global Water Futures. S.M.P. is funded by the Global Water Futures program and by the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant: RGPIN-2019-06894). The authors acknowledge the World Climate Research Programme (WCRP), which coordinated and promoted CMIP6 through its Working Group on Coupled Modelling, the climate modeling groups for producing and making available their model output, the Earth System Grid Federation (ESGF) for archiving the data and providing access, and the multiple funding agencies who support CMIP6 and ESGF. The authors declare no competing interests.
Data availability statement.
CMIP6 simulations provided by ESGF can be found by the open-source link: https://esgf-node.llnl.gov/search/cmip6/. Users should select the variable as pr which stands for precipitation, the frequency as daily, select Experiment ID as historical, SSP126, SSP245, SSP370, and SSP585, and select CMIP6 models found in this study (see supplementary Table S1 for a brief description on the used CMIP6 models), and then download the NC files that appear as search outputs. GPCC “observations” dataset for daily precipitation (version 2020), provided by the Global Precipitation Climatology Centre daily precipitation through the open-source link: https://opendata.dwd.de/climate_environment/GPCC/html/fulldata-daily_v2020_doi_download.html, DOI: 10.5676/DWD_GPCC/FD_D_V2020_100.
APPENDIX A
Comparing QDM and SPQM
We use two BC methods (QDM and SPQM) to highlight the significance of using different BC methods on AMP projections. QDM depends on two main steps: 1) quantile mapping based on nonparametric quantiles of observations and 2) superimposing the relative change between historical and future simulations. The QDM is also applied directly to the AMP projections. Generally, there are no considerable changes between QDM and SPQM projections in most of the globe. However, QDM has higher projections in Africa and Latin America. For example, in SSP5–8.5, the average differences between QDM and SPQM projections are 1.5, 2.6, 2.9, 4.6, 13.5, and 20.2 mm day−1 in Europe, North America, Australia, Asia, Latin America, and Africa, respectively, in 2100. QDM preserves relative changes between projections and historical simulations in its application (Cannon et al. 2015). Therefore, a multiplicative factor is added to the projections after quantile mapping is done to preserve this relative change. Also, the applied QDM technique is nonparametric, which can yield unexpected patterns compared to a semiparametric or parametric technique. This procedure may lead to an overestimation, especially in regions that experience sensitive natural variability and highly diverse regional patterns in AMP time series, such as Africa (Lüdecke et al. 2021; Lyon and Vigaud 2017). QDM projections of all scenarios start from a low level and remain almost the same from 2015 until around 2050 (Fig. A1). In North and Latin America, SSP5–8.5 projections are even lower than SSP1–2.6 in the same period (2015–50), which seems unrealistic. In contrast, SPQM has increasing trends from 2015 to 2100 with reasonable differences between emission scenarios (Fig. 1). Eventually, differences between QDM and SPQM are comparable within −11 to 25 mm day−1; Latin America has the lowest difference, and Africa has the highest difference.
Annual maxima of daily precipitation (AMP) time series of observations (1982–2014) and bias-corrected projections (2015–2100) based on the QDM technique showing the continental variations: (a) Africa, (b) Australia, (c) North America, (d) Asia, (e) Europe, and (f) Latin America. The lines show the mean for all continent’s grids, and the shaded regions show the interquartile range.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
APPENDIX B
The Uncertainty of Stationary and Nonstationary GEV Models
We implement extensive Monte Carlo experiments to understand the influence of fitting stationary and nonstationary GEV models for time series with trends. Model 0 and model 1 represent stationary and nonstationary (with a location parameter that varies linearly with time) GEV models, respectively. In cases of nonzero trends, model 0 yields overestimated location and scale parameters and underestimated shape parameters (Fig. B1). The shape parameter mainly controls the extreme levels. The shape parameter of GEV distributions describes the properties of extremes (i.e., higher extremes are associated with heavier tails or equivalently high shape parameters). In contrast, model 1 produces well the given parameters for all slopes. The slope of the location parameter reflects the slope of the time series. In cases of zero trends, model 0 shows more efficient parameter estimates than model 1, especially in the location parameter (Fig. B1; slope = 0). Therefore, using stationary models when having samples with trends provides biased parameter estimates and thus underestimates design levels. However, using stationary models when having no trends is preferred because 1) they yield more efficient estimates (i.e., the variance in estimates is narrow), and 2) they are more parsimonious (i.e., they have fewer parameters and fit the data satisfactorily).
GEV parameters resulting from MC simulations for (top) stationary (model 0) and (bottom) nonstationary (model 1) models: (a) location parameter, (b) scale parameter, and (c) shape parameter. The dashed lines refer to the theoretical input values. The location parameter in model 1 refers to the fixed component of the location parameter.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
The uncertainty of the parameters also affects the estimated design levels. We note that the uncertainty of shape parameters is the highest, followed by scale parameters and location parameters for all tested slopes and models. The theoretical value of the shape parameter is given by 0.20, and the estimated shape parameters range from −0.08 to 0.42 for the stationary model and from −0.005 to 0.43 for the nonstationary model, which is huge variability (Fig. B1). Therefore, it is more robust to include covariates with location and scale parameters rather than shape parameters.
To reveal uncertainties associated with changing each of GEV parameters linearly with time, further MC experiments are implemented. We denote the applied GEV models by M1, M2, M3, and M4, where the location parameter, scale parameter, shape parameter, and all the parameters change linearly with time, respectively. The given theoretical slopes ensure realistic magnitudes of the parameters at all time steps (Fig. B2). Then, these samples are fitted to corresponding GEV models using the method of likelihood, and the fitted slopes of the parameters are estimated. Finally, we compare the fitted slopes of GEV parameters with the theoretical slopes (Fig. B2).
The uncertainty of linear slopes of GEV parameters with time. (top) Results of a nonstationary GEV model that has only one parameter that varies linearly with time: location (M1), scale (M2), and shape (M3) parameters. (bottom) Results of a nonstationary GEV model with all parameters changing with time, showing the slopes of the location (M4-1), scale (M4-2), and shape (M4-3) parameters. Filled boxplots show the interquartile range, while the bars/whiskers show the 90% confidence interval of 1000 experiments.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0467.1
For M1, the medians of the fitted slopes are precisely equal to the theoretical slopes of location parameters, and the 90% confidence intervals are at around 0.15 for all slopes (Fig. B2, M1). For M2, the medians also match with the theoretical slopes of scale parameters with increasing uncertainty with the slope (the 90% confidence intervals are −0.057 to 0.060 and 0.104–0.301 for slopes 0 and 0.20, respectively; Fig. B2, M2). The previous confidence intervals are very narrow for location and scale parameters. For shape parameters in M3, medians of the fitted slopes could not accurately capture the theoretical slopes with high uncertainty of −0.005 to 0.008 and −0.002 to 0.015 for theoretical slopes of 0 and 0.006, respectively (Fig. B2, M3). These results confirm that applying nonstationary GEV models with covariates for shape parameters is not robust due to their high sample variability besides the challenging computational demands.
Similarly, we apply a nonstationary GEV model with all parameters changing linearly with time (M4). When we compare theoretical and fitted slopes, none of the parameters’ rates is preserved (not even the medians), and the uncertainty is higher than in the case of individual parameter changes with time (the last row in Fig. B2, where M4-1, M4-2, and M4-3 indicate changes in location, scale, and shape parameters, respectively). These results verify that complex models are not preferred as their estimates are inaccurate with high uncertainty. Overall, this analysis highlights the importance of properly understanding the statistical characteristics of nonstationary and stationary GEV models before applying any of those in design studies.
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