Abstract
In this study, an uncertainty analysis of extreme precipitation return levels was performed for the Chaliyar river basin, India, under representative concentration pathways (RCPs) 4.5 and 8.5. Weighted average projections of various climate models (for RCPs 4.5 and 8.5) using reliability ensemble averaging were used in the analysis for projecting the future extremes. To start with, the presence of nonstationarity in the observed annual maximum precipitation (AMP) series and the future ensemble averaged AMP projections were investigated. For this purpose, three generalized extreme value (GEV) models—one stationary model with constant parameters and two nonstationary models with trends in location and scale parameters—were applied to assess the goodness of fit using Akaike information criterion and likelihood ratio test. The best fit model was used in the uncertainty analysis, and the confidence bounds of extreme precipitation return levels were estimated. A nonparametric bootstrapping approach was followed in the uncertainty analysis. Results of the study suggest that a nonstationary GEV distribution with linear trend in location parameter and constant scale and shape parameters are the best fit distribution for the AMP series under the RCP scenarios, whereas the stationary GEV distribution fits the observed AMP series the best. The expected values and confidence bounds of return levels obtained from the uncertainty analysis reveal that precipitation extremes in the river basin would intensify under the projected climate change scenarios. Compared with the RCP4.5 scenario, the confidence intervals of return levels under the RCP8.5 scenario were wider, implying that uncertainty in the latter scenario is higher.
Introduction
Drastic increase in the concentration of greenhouse gases in the atmosphere is considered as one of the dominant causes of climate change. This is believed to have resulted in the changes in pattern and occurrence of extreme weather and climatic events in different parts of the world [1]. An increase in temperature induced by the greenhouse gases will increase the rate of evaporation and thereby enhance the moisture content in the atmosphere [2]. This can cause more frequent and heavier rainfall events. According to the Clausius–Clapeyron relationship, the intensity of extreme precipitation events is expected to increase at the rate of 7% per 1 °C rise in temperature [3]. Analyses of the global trends in extreme rainfall events reveal increasing trends in some regions and decreasing trends in some other regions. Countries such as the USA, UK, Australia, South Africa, and India have been experiencing an increasing trend whereas a decreasing trend has been observed over Western Australia, Northern and Eastern New Zealand, parts of Central Pacific, and South East Asia [4,5,6,7,8,9].
India witnessed a number of unprecedented extreme precipitation events in the recent past which resulted in massive flooding. The floods in Kerala (2018 and 2019), Uttarakhand (2013), Chennai (2015), Bengaluru (2016), and Mumbai (2005 and 2017) are some among them [10, 11]. Based on the analysis of observed annual maximum precipitation (AMP) from 1979 to 2015, an increase in AMP is reported over many parts of India except the Gangetic Plain, northeastern India, and Jammu and Kashmir; the increase in AMP is found to be more prominent in Southern India than in Northern India. Furthermore, the frequency of precipitation extremes is projected to increase more prominently in southern and central India in the mid and end of the twentyfirst century under the representative concentration pathway (RCP) 8.5 [11]. An earlier study using the special report on emissions scenarios, SRES A2 and B2, also projected an increase in the intensity and frequency of heavy precipitation events toward the end of the twentyfirst century over many parts of India [12]. The present study is conducted in the Chaliyar river basin which is located in Kerala, the southernmost state of the Indian Union. This was one of the worst affected river basins in Kerala during the floods of 2019; it was badly affected during the 2018 floods as well. Frequent floods have adversely affected all important sectors of the society including socioeconomy, infrastructure, transportation, agriculture, etc. [13]. Since frequent extreme precipitation events with increased magnitudes can be expected with climate change, studies focusing on the impact of climate change on these extremes would be very useful in the formulation of mitigation and adaptation strategies to cope with and reduce the negative impacts of such events on the society.
Climate extremes, being rare in occurrence, follow different laws and are harder to predict when compared with the averages [14]. Extreme value theory forms the foundation of analysis of extremes and determination of the likelihood of occurrence of extreme events. Gumbel distribution, generalized extreme value (GEV) distribution, logPearson Type III distribution, and generalized Pareto distribution (GPD) are the four commonly used extreme value distributions in the analysis of hydrometeorological extremes [15,16,17,18]. The first three distributions follow a block maximum approach based on the annual maximum series, whereas the last one follows a peak over threshold approach which is based on the peak values over high thresholds [15, 19]. The distributions of classical extreme value theory are based on the assumptions of independence and stationarity of the extreme data series [20,21,22]. Later, many studies considered extending this theory to suit dependent and nonstationary extreme sequences [23, 24].
In recent years, several studies have reported that hydrometeorological extremes violate the concept of stationarity [25, 26]. Nonstationarity in hydrologic time series can be identified if there exist some trends, periodicity, or shifts. Kundzewicz and Robson [27] suggested that shortterm variability in the climate arising from very short records of data should not be confused with nonstationarity. Scientists are attributing the underlying causes of nonstationarity in hydrological extremes to climate change, lowfrequency internal variability, and anthropogenic influences such as urbanization, deforestation, change in catchment characteristics, etc. [28,29,30,31,32]. Mondal and Mujumdar [33] advocated that in a climate change context, future extremes which are obtained from climate model projections may exhibit nonstationarity irrespective of the observed extremes in the past which may be stationary. Vogel et al. [34] established that the magnitude of a future flood tends to increase whereas its average recurrence interval tends to reduce if an increasing trend is observed in the flood series, i.e., a flood event associated with a high return period at present may become more frequent in future under nonstationary conditions.
The assumption of stationarity can reduce the complexity of modeling the extremes, but non stationary modeling is advised if there exists a significant trend in the extreme series [35]. In recent studies, for investigating and modeling the nonstationarity in the extreme series, nonstationary extensions of extreme value distributions are used, where the parameters are expressed either as functions of time or physical covariates [19, 32, 36, 37]. In case of a GEV distribution, trends in time are incorporated either in the location/scale parameter or both the location and scale parameters to incorporate the nonstationarity in the extreme series [25, 36]. In case of a GPD, the scale parameter is treated as a function of time [38]. The shape parameter is usually treated as a constant, independent of time, since varying it as a function of time is difficult [39]. Covariates related to physical processes such as urbanization, local temperature changes, global warming, El NiñoSouthern Oscillation (ENSO) cycle, etc. are used in some studies to account for the nonstationarity in the extreme rainfall series [28, 40]. In some studies, generalized additive models for location, scale, and shape (GAMLSS) have been used in nonstationary flood frequency analysis [41]. This approach is based on semiparametric regression, in which a parametric distribution belonging to a general distribution family such as a highly skewed/kurtotic distribution is used for the annual maximum flood series and the parameters of the distribution are modelled as linear, nonlinear, or nonparametric smoothing functions of the dependant variables [42]. Also, in some studies, nonlinear trends in nonstationary series are modeled using a conditional density network, which is a probabilistic extension of the multilayer perceptron neural network [43] and by a multiobjective genetic algorithm generated timebased covariate [44].
Quantification of uncertainty in an extreme quantile is of paramount importance in the context of assessment of risk and decision making. Uncertainty yields a range of values for the extreme quantile, which otherwise is estimated as a single value (point estimate) corresponding to a particular return period. Methods for quantification of uncertainty are well developed for a stationary case [45, 46]. Since hydrological extremes may exhibit nonstationarity in a changing climate, methods used to perform uncertainty analysis should be capable of taking nonstationary conditions into account. One of the premier works on the quantification of uncertainty of design quantile under nonstationarity is by Obeysekera and Salas [47]. They extended the delta method, profile likelihood method, and bootstrap method to nonstationary conditions for dealing with uncertainty in the parameters as well as in the return levels [47, 48]. Serinaldi and Kilsby [49] reported that the bootstrap method is the most practical one among these methods since it relies only on the available information, does not rely on a particular estimation method, and could be easily implemented for complex models. Return level uncertainty as well as parameter uncertainty can be estimated by Bayesian analysis too [25, 50]. Like parameter uncertainty, covariate uncertainty could also have some significance in the case of a nonstationary model with a number of covariates [40].
The objectives of the present study are (i) to investigate nonstationarity in the annual maximum precipitation series—observed and corresponding to the future climate (RCPs 4.5 and 8.5)—of the Chaliyar river basin by identifying an appropriate frequency distribution from stationary and nonstationary GEV models and (ii) to quantify the uncertainty in return levels as confidence intervals by performing frequency analysis in a bootstrapping framework.
Materials and Methods
Study Area
Chaliyar is a river in the south western Indian state of Kerala which is located between the Western Ghats in the east and the Arabian Sea on the west (Fig. 1). The state, most areas of which are located in the humid tropical zone, is endowed with 44 rivers, Chaliyar being the fourth longest (170 km). River Chaliyar originates from the Elambalari hills at an altitude of about 2067 m above mean sea level (MSL). The river basin spreads over an area of 2933km^{2}, in the states of Kerala and Tamil Nadu. The basin of this Virgin River can be divided into highland, midland, lowland, and coastal plain based on topographic features. The predominant land uses are agriculture (60.04%) and forests (38.74%). Small patches of urban areas, pastures, waste lands, and rocky areas are also present. The major soil type is loam (42.74%), followed by clay (28.66%), clay loam (24.18%), and sandy loam (4.42%) [51]. The climate is humid tropical with hot summer and heavy monsoon rainfall. March and April are the hottest months, and December and January are the coolest months. The average annual precipitation in the river basin is 3012.61 mm and the average maximum and minimum temperatures are 34 °C and 24 °C, respectively [51].
Data
Rainfall data at three rain gauge stations, namely, Kottamparamba, Manjeri, and Nilambur, were collected. Daily rainfall data at the Kottamparamba station for the period 1981–2005 was obtained from the Centre for Water Resources Development and Management (CWRDM), Kunnamangalam, Kerala, whereas the daily rainfall data for Manjeri and Nilambur rain gauge stations for the period 1976–2004 were collected from the India Meteorological Department (IMD). To study the precipitation extremes due to projected climate change, daily rainfall data (interpolated to station point using inverse distance weighting method) for the RCPs 4.5 and 8.5 were obtained from eight highresolution general circulation models of the Coordinated Regional Climate Downscaling Experiment (CORDEX) under the South Asia domain. The dynamically downscaled output of a general circulation model (GCM) is referred to as highresolution GCM output in some studies, since the coarse resolution of the GCM output is brought down to a finer resolution by downscaling with a nested regional climate model (RCM) [50, 52]. In the present study also, the usage “highresolution GCM” is adopted to represent the GCM which is nested with an RCM. Details of the highresolution GCMs used in this study are presented in Table 1. CORDEX datasets are available from the climate data portal of the Centre for Climate Change Research (CCCR), Indian Institute of Tropical Meteorology (IITM), Pune, India (http://cccr.tropmet.res.in).
Methodology
The overall methodology used in the study is presented in Fig. 2. To start with, the AMP series were extracted from the observations and the dynamically downscaled projections of global climate models for the RCPs 4.5 and 8.5. The weighted averaged projections of the AMP series using reliability ensemble averaging (REA) were used to obtain a comprehensive realization of projections from multiple climate models. REA weights were assigned for each climate model simulation based on the ability of the model to emulate the present climate as well as the degree with which its simulations converged with the simulations of other models. Disparities of the weighted projections with the observations were compensated by a bias correction technique. The relationship between the magnitude and frequency of extremes was obtained by performing frequency analysis using extreme value distributions. In order to fit an extreme value distribution to the series of extreme precipitation or stream flow, it should satisfy the conditions of independence and stationarity [28, 63, 64]. In the present study, nonstationarity was investigated by detecting the trend in the AMP series using nonstationary extensions of the GEV distribution. For this purpose, a stationary GEV model with constant parameters and two nonstationary GEV models with timedependent parameters were evaluated for goodness of fit on the AMP series using the Akaike information criterion (AIC). In order to validate the AIC test, a significance test called likelihood ratio test was also performed [52, 65, 66]. If the best fit model happens to be any one of the two nonstationary GEV models, the annual maximum precipitation series can be said to possess a nonstationary behavior. The best fit model can be used for the uncertainty analysis of extreme precipitation return levels. For this purpose, nonparametric bootstrapping was carried out, and the confidence intervals of return levels were obtained. Detailed discussions on these are presented in Sects. 2.3.1 to 2.3.6.
REA for Weighted Ensemble Projections
Projections of climate models are subjected to various sources of uncertainties. It includes uncertainties arising from anthropogenic forcing estimates, initial conditions, boundary conditions, model structure, model response, etc. As a result, projections of the future by different climate models yield different results. Using an ensemble of climate model projections is an accepted option for regional analysis [67]. In the present study, REA, a weighted averaging procedure based on reliability weights was used to obtain the ensemble averaged projection of the highresolution GCMs listed in Table 1. Historical data of the highresolution GCMs for the three stations were taken corresponding to the period of observed data, i.e., 1981–2005 for Kottamparamba and 1976–2004 for Manjeri and Nilambur. Future projections of the highresolution GCMs were taken for the period 2006–2099.
Xu et al. [68] investigated the relevance of weighting schemes of REA by comparing it with a simple averaging procedure using several performance metrics and observed that REA is capable of providing finer variations of climate change signals than simple averaging for regionalscale applications. In REA, two reliability criteria, namely, model performance and model convergence criteria, are used to assign the weight to each climate model projection. Model performance criterion deals with the ability of the climate model projection to replicate the presentday climate (i.e., a measure of degree of bias) whereas model convergence criterion takes into account the convergence of the climate model projections to the projections of other models in the ensemble. In this study, the methodology proposed by Riano [69] for applying REA on the AMP series was used. Reliability (\(r_{i,s}\)) of the projection by the climate model i in a particular scenario s (e.g., RCP4.5) can be obtained as:
where \(rb_{i}\) is the reliability factor for bias of the model i, and \(rc_{i,s}\) is the reliability factor for convergence of the model i in the scenario s.
The reliability factor for bias \(rb_{i}\) can be computed as:
where \(AMP_{o,j}\) is the observed annual maximum precipitation for the jth year, \(AMP_{h,i,j}\) is the historical annual maximum precipitation for the ith model in the jth year, and t is the number of years.
The reliability factor for convergence \(rc_{i,s}\) can be computed as
where \(AMP_{s,i,j}\) is the annual maximum precipitation projection for a particular scenario s for the ith model in the jth year, \(\overline{AMP}_{s,j}\) is the REAweighted averaged projection of all the models in a particular scenario s for the jth year, and t is the number of years. The calculation of \(rc_{i,s}\) requires \(\overline{AMP}_{s,j}\), and this can be computed as
in which \(r_{i,s}\) is unknown. Therefore, \(rc_{i,s}\) can be computed using a trialanderror procedure, starting with an initial assumed value of \(r_{i,s}\) (say \(rb_{i}\)). After several iterations, a converged solution can be obtained for \(rc_{i,s}\). \(\overline{AMP}_{s,j}\) can also be written as
where \(w_{i}\) is the weight of the \(i^{th}\) model in the ensemble represented by the term \(\frac{{r_{i,s} }}{{\sum\limits_{i = 1}^{n} {r_{i,s} } }}\) in Eq. 4.
A model will be less reliable if the values of \(rb_{i}\) and \(rc_{i,s}\) are higher, since the calculations of these terms are based on mean square errors. For easy interpretation, these factors can be converted to a score system in which the best model is represented by the highest score. This can be achieved by normalizing \(rb_{i}\) and \(rc_{i,s}\) [70] as
where \(RB_{i}\) and \(RC_{i,s}\) represent normalized values of \(rb_{i}\) and \(rc_{i,s}\), \(rb_{\max }\) is the maximum of the \(rb_{i}\) values of all models, and \(rc_{\max ,s}\) is the maximum of the \(rc_{i,s}\) values of all models for a scenario s.
Application of a Modified Scaling Factor Approach for Bias Correction
Climate model projections are affected by various sources of errors such as incorrect assumptions in model parameterization, incorrect initial conditions, incorrect representation of physical processes, etc. So, the direct application of these projections results in erroneous prediction of different climaterelated processes [71]. Usually, bias correction methods are used to reduce the bias in climate model projections and enhance the consistency and quality of these projections. The commonly used bias correction techniques include delta change approach, linear transformation, local intensity scaling, variance scaling, power transformation, and distribution mapping [72]. In this study, the ensemble averaged AMP projections, after performing REA, were corrected for bias. For this purpose, a methodology recently developed by Mehr and Kahya [73] to correct the biases in the annual maximum series was followed. Using this method, the annual mean of the ensemble averaged AMP series from climate model projections for the historical (reference) period is matched with that of the observed AMP series by using a scaling factor. Correction is applied to the ensemble averaged AMP series under the scenario \(s\) (\(AMP_{s}\)) as
where \(AMP_{c}\) is the bias corrected annual maximum precipitation, \(\frac{{\eta (AMP_{o} )}}{{\eta (AMP_{r} )}}\) is the scaling factor in which \(\eta\) is the longterm mean of the bracketed series, \(AMP_{o}\) is the observed AMP series, and \(AMP_{r}\) is the ensemble averaged AMP series from the highresolution GCM output for the same reference period as that of the observed series.
Detection of Nonstationarity in the Annual Maximum Precipitation Series
Detection is the process of identifying statistical changes in the climate or climaterelated systems [74]. Since extreme events are being reported more frequently with transient changes in intensity, the presence of nonstationarity has to be detected before performing frequency analysis. In order to handle the nonstationarity of hydrological extremes in frequency analysis, trends in the extreme series should be accounted [32]. In some earlier studies, the significance of trend in the extreme series was investigated either by performing a nonparametric test such as MannKendall test which was not designed particularly for dealing with the extremes or by employing a linear regression which assumed the data as normally distributed [75, 76]. Clarke [75] and Katz [76] pointed out that the latter approach is not logical since it involves the assumption of normal distribution for the series of extremes, although it actually follows an extreme value distribution. In recent years, extensions of extreme value distributions that are capable of shifting over time have been developed for the detection and analysis of nonstationarity in hydrological extremes [19, 32, 76, 77]. In some studies, the trends in block maximum series have been detected by fitting a nonstationary generalized extreme value distribution with timevarying parameters on the block maximum series [78, 79]. The goodness of fit of the modified extreme value distributions was checked with methods for model selection or with tests for significance of the trend parameters [66]. Zhang et al. [80] compared the MannKendall test, linear regression, and the nonstationary extreme value distribution fitting method for detecting the trend in the extreme values using Monte Carlo simulations and found that the latter always outperformed the other two methods in trend detection. Therefore, in the present study, the GEV distributions with timedependent parameters were used to detect nonstationarity in the annual maximum precipitation series.
The cumulative distribution function (CDF) of a classical GEV distribution [19, 77] can be expressed as
where \(\phi = \left\{ {\mu ,\,\sigma ,\,k} \right\}\) denotes a parameter set in which \(\mu\), \(\sigma\), and k are the location, scale, and shape parameters respectively. Under nonstationary conditions, the parameters of the GEV distribution become time dependent and the CDF changes to
where \(\phi_{t} = \left\{ {\mu_{t} ,\,\sigma_{t} ,\,k_{t} } \right\}\) denotes a parameter set in which \(\mu_{t}\), \(\sigma_{t}\), and \(k_{t}\) are the timedependent location, scale, and shape parameters, respectively [76]. Usually, the location and scale parameters are treated as functions of time, and the shape parameter is considered as a constant, independent of time, since modeling the shape parameter as a function of time is complicated [39, 80]. Zhang et al. [80] pointed out that any variation of tail behavior of the distribution represented by a shape parameter can be approximately handled by variations of the scale parameter. In this study, three cases of GEV distributions were considered:

1.
A stationary GEV distribution with location, scale, and shape parameters as constants
(\(\mu\), \(\sigma\), k)—GEV1.

2.
A nonstationary GEV distribution with location parameter as a linear function of time (\(\mu_{t} = \mu_{0} + \mu_{1} \,t\)), scale, and shape parameters as constants (\(\sigma_{t} = \sigma\), \(k_{t} = k\))—GEV2.

3.
A nonstationary GEV distribution with location parameter as a linear function of time (\(\mu_{t} = \mu_{0} + \mu_{1} \,t\)), scale parameter as loglinear function of time (\(\log (\sigma_{t} ) = \sigma_{0} + \sigma_{1} \,t\)) and shape parameter as constant (\(k_{t} = k\))—GEV3.
In these models, \(\mu_{1}\) and \(\sigma_{1}\) are the slopes of the location and scale parameters which could represent the transient nature of the probability distribution with the covariate time t. The scale parameter is logtransformed to maintain it as a positive value [19, 76]. The AMP series of the observations and the RCPs 4.5 and 8.5 were fitted with these three models by the maximum likelihood method. This is one of the widely used methods of distribution fitting owing to its simplicity, ability to generate unbiased estimates, and the flexibility to be used in a nonstationary condition [32, 77].
The best model from among the three GEV models considered was selected by the model selection criteria discussed below. Using the selected model, inferences can be drawn about the characteristics of the data to which it is fitted. Also, it can be used to perform frequency analysis for estimating the return levels.
Model Selection
The main aim of model selection is to identify a model that adequately fits the data. It shall maintain a balance between the goodness of fit of the model and the number of parameters, with due consideration of the sample size [81]. Forward, backward, and stepwise selection procedures, cross validation, information criteria such as Akaike information criterion, Bayesian information criterion, and likelihood ratio test are some of the widely used statistical methods for model selection [82]. The process of model selection should be treated as crucial since the predictive ability of the model depends on the selected model [83].
In this study, a stationary GEV model and two nonstationary GEV models were fitted to the observed annual maximum series as well as to the ensemble averaged annual maximum series from future projections under the RCPs 4.5 and 8.5. To select the best model among these, Akaike information criterion was used along with the likelihood ratio test.
AIC
Akaike information criterion was developed by Akaike [84] to identify an optimal model that fits the data from a class of competing models. The model with minimum AIC value can be considered as the best predictive model. AIC reduces the possibility of overfitting of a model with the use of a penalty term. AIC can be computed as
where L is the loglikelihood value and 2p is the penalty term with p representing the number of parameters of the model.
For easy interpretation, the AIC value of each model i can be rescaled with respect to the minimum AIC (\(AIC_{\min }\)) of the competing class. The rescaled value \(\Delta_{i}\) is given by
From different ranges of \(\Delta_{i}\) values, inferences can be drawn about the relative reliabilities of competing class of models. The model with \(\Delta_{i}\) value equal to 0 is regarded as the best and that with \(\Delta_{i}\) value greater than 10 is regarded as the worst. Though not the best, models that have small \(\Delta_{i}\) values (less than or equal to 2) can be considered as substantially reliable [85].
Likelihood Ratio Test
This is a hypothesis test that can be used to compare the goodness of fit of a simple model (with few parameters) and a complex model (with more parameters) if these are nested. Two models can be called nested when one model contains all the terms of the other model along with some additional terms [86]. Likelihood ratio (LR) test checks whether the use of a complex model with additional parameters could improve the goodness of fit when compared with what is obtained with a simple model. Often, the simple model is called a restricted model, and the complex model is called an unrestricted model. The null hypothesis of the test is in favor of the simple model, and the alternative hypothesis is in favor of the complex model. The null hypothesis of the test is rejected when the likelihood ratio test statistic is greater than the critical value at a certain significance level [86, 87]. The likelihood ratio test statistic (\(L_{r}\)) is given by
where \(L_{c}\) is the loglikelihood of the complex model, and \(L_{s}\) is the loglikelihood of the simple model.
When there are more than two models, this test can be implemented by successive pairwise evaluations of the nested models in a hierarchical manner [65, 88]. In the present study, the order of complexity (based on the number of parameters) of GEV models is GEV1 < GEV2 < GEV3. Hence, by using the LR test, two possibilities can be checked:

1.
Whether the addition of an extra parameter (\(\mu_{1}\)) to GEV1, i.e., the GEV2 model, could give any significant improvement in the goodness of fit than that obtained using a simple model, i.e., GEV1

2.
Whether the addition of an extra parameter (\(\sigma_{1}\)) to GEV2, i.e., the GEV3 model, could give any significant improvement in the goodness of fit than that obtained using a simpler model, i.e., GEV2
If in the first step the null hypothesis is not rejected, i.e., the simple model GEV1 is the best model for the data, it can be inferred that the use of a complex model with an additional parameter is a wrong choice, since it cannot provide any significant improvement in the goodness of fit. In this case, the second step can be avoided. If in the first step the null hypothesis is rejected and the complex model GEV2 is found to be the best model for the data, the second step can be performed to check whether significant improvement in the goodness of fit can be obtained by the addition of an extra parameter.
Estimation of Return Levels
In classical flood frequency analysis, the return level corresponding to a return period (T) is the extreme quantile for which the exceedance probability is equal to the reciprocal of the return period (i.e., 1/T) [77]. The inverse of Eq. 9 gives the expression for calculating the return level (\(x_{T}\)):
In recent times, a number of methods have been developed to estimate the return levels under a nonstationary paradigm. In one of the methods, nonstationary return level for a T year return period event was derived from the effective return level concept [19, 36, 89]. The nonstationary counterpart of Eq. 14 is the expression for calculating the timevarying extreme quantile, which is usually referred to as the effective return level (\(\mathop x\nolimits_{{\mathop T\nolimits_{t} }}\)):
A value representative of the effective return level, generally the one corresponding to the median value or the 95^{th} percentile of the timevarying location parameter, is taken as the nonstationary return level [25]. Parey et al. [90] estimated the return level in a nonstationary context by redefining it as a level such that the expected value of the number of exceedances over that level in the next T years will be one. Later, Cooley [36] and Obeysekera and Salas [30, 47] came up with the extensions of expected waiting time theory for return period estimation under nonstationary conditions. In the present study, the expected waiting time approach was followed for the estimation of nonstationary return levels.
Expected Waiting Time Method
In this method, the return period is regarded as the expected time till the occurrence of the next extreme event [91]. The expressions of return period and timevarying exceedance probability proposed by Obeysekera and Salas [30, 47] were used in this study for the estimation of extreme precipitation return level. This method is based on the exceedance probabilities of a fixed threshold with time. Let \(x_{{T_{0} }}\) be a fixed threshold and Y be the random variable representing the waiting time till the first exceedance of \(x_{{T_{0} }}\) at Y = y. The probability distribution of Y is given by
Since return period is considered as the expectation of waiting time till the occurrence of the next extreme event, T can be obtained as
The expected waiting time \(E(Y)\) is a function of timevarying exceedance probability \(p_{t}\) (i.e., the probability of the given threshold being exceeded each year) and is given by
where \(F(x,\phi_{t} )\) is presented in Eq. 10.
Uncertainty in the Return Levels from Bootstrapping
Bootstrapping is a statistical method developed by Efron [92] in which the original sample is resampled and statistical inferences are derived. Bootstrapping is widely used in frequency analysis for the estimation of uncertainties in distribution parameters and return levels [93, 94]. There are two approaches in bootstrapping, namely, parametric and nonparametric. Parametric approach can be used when an appropriate probability distribution of a sample is known, and bootstrap samples are derived from the known distribution. This method is preferred when the sample size (n) is very small (\(n \le 60\)), since the coverage performance of the confidence interval reduces with increase in sample size [94]. The nonparametric approach is based on sampling with replacement from the original sample and is more accurate for sample sizes above 40 [94]. Compared with the parametric approach, the nonparametric approach being simple and requiring less information is widely used in frequency analysis [47, 95]. In this study, uncertainties in distribution parameters and return levels were estimated using the nonparametric approach proposed by Obeysekera and Salas [47] for use under nonstationary conditions. The steps involved in this method are as follows:
Step 1. Initially, the nonstationary GEV model was fitted to the AMP series \(x_{t}\). The return level (\(x_{{T_{0} }}\)) corresponding to return period T was obtained from Eqs. 17 and 18. Using the parameters of the fitted model and the AMP series \(x_{t}\), the standard residuals \(\overline{x}\) were calculated using the equation:
Step 2. These residuals were resampled with replacement, and a new set of \(\overline{x}\) with a length equal to that of the original AMP series was obtained. This new set of \(\overline{x}\) was back transformed and a new maximum series of \(x_{t}\) was obtained using Eq. 19. The nonstationary GEV model was then fitted to the new \(x_{t}\) series, and the new set of parameters and return level were obtained.
The processes in step 2 were repeated a large number of times; in this study, 1000 iterations were performed.
For the stationary case, the uncertainty in distribution parameters and return levels were obtained in the same manner but with a stationary GEV distribution with constant parameters.
Results and Discussion
REA of Projections as an Average Climate Realization
The ensemble averaged AMP series for the historical and future periods were obtained by applying REA on the AMP projections of the eight highresolution GCMs listed in Table 1. The models, whose projections for the historical period were comparable with the observations and whose future projections converged, were identified by assessing the reliability weights. Normalized values of reliability factors for bias, convergence, and the final weights of each model in the ensemble are presented in Table 2. Details regarding the number of iterations taken for the REA procedure are also presented in Table 2.
Since the factors \(RB_{i}\) and \(RC_{i,s}\), listed in Table 2, are normalized values, a higher \(RB_{i}\) value indicates lesser model bias, and a higher \(RC_{i,s}\) value indicates that the model projections exhibit a higher degree of agreement to the projections of other models in the ensemble. Therefore, a climate model with the highest weight is the best model. At Kottamparamba, the performance of ACCESS(CCAM) and CanESM(RegCM4) was found to be poor for the historical period and the RCP scenarios. IPSL(RegCM4) performed the best for the historical period and under RCP8.5, whereas the performance of CSIRO(RegCM4) was the best under RCP4.5. The best models for a particular scenario for the other stations can be identified from Table 2. The weighted averaged AMP projections obtained from REA was used in the further analyses.
Bias Correction Statistics
Even though the bias in historical projections with respect to the observed data was accounted for in the weighted averaging procedure, some amount of bias would still be left uncorrected. So, bias correction was applied on the ensemble averaged AMP series for better results. In this study, bias correction of the AMP series was performed following the methodology outlined in Sect. 2.3.2. Scaling factors of 1.07, 1.13, and 1.14 were obtained for Kottamparamba, Manjeri, and Nilambur, respectively. The ensemble averaged AMP series during the historic (raw) and future periods were multiplied by these scaling factors to obtain the biascorrected AMP series during the respective periods. In order to assess the performance of the bias correction technique adopted, the ensemble averaged AMP series for the historic period was compared with the observed AMP series after bias correction, and the results are presented in Table 3. It can be seen that the statistics such as mean, median, 25th percentile, and 75th percentile of the biascorrected AMP series exhibit a much better match with the observations than that obtained with the uncorrected AMP series.
Appropriate Probability Distributions for the AMP Series
Nonstationarity modeling of the AMP series was implemented by introducing time trends in the parameters of the generalized extreme value distribution. The three GEV models discussed in Sect. 2.3.3 were fitted to the annual maximum precipitation series derived from the observations as well as the ensemble averaged projections for the future under RCPs 4.5 and 8.5. The model performance was evaluated using the AIC and LR tests. AIC values, \(\Delta_{i}\) values, and p values of likelihood ratio test statistics obtained from the fit of the annual maximum series of observations and ensemble averaged RCP projections with each of the three GEV models are presented in Table 4.
According to the Akaike information criterion, the model with minimum AIC value is considered to be the best fit model for the data. For the observed AMP series at the three stations, it can be inferred that GEV1, i.e., the stationary GEV distribution, fits the best since AIC is the least (\(\Delta_{i} = 0\)) for this model. For the AMP series under the RCP scenarios, the GEV2 model with the location parameter having a timevarying linear trend and constant scale and shape parameters yielded the best fit at all the three stations. The stationary model, GEV1, was found to be the worst model under both RCPs at all the stations since their \(\Delta_{i}\) values are greater than 10. Under the RCP8.5, for the Kottamparamba and Manjeri stations, the AIC values of GEV2 and GEV3 are very close (\(\Delta_{i}\) is very low). In some other cases also, \(\Delta_{i}\) values are less than 2, and these models too have substantial predictive reliability. However, in this study, the best model with \(\Delta_{i}\) equal to 0 was chosen to obtain better accuracy in the estimation of return levels.
In order to support the results of the AIC test, the likelihood ratio test was also performed. At 10% significance level, the probability associated with the LR test statistic (p value) was reviewed. As mentioned in Sect. 2.3.4, the LR test was performed in two steps. In step 1, the null hypothesis was considered in favor of GEV1 against the alternative of GEV2. If the p value is less than 10% (i.e., 0.1), the null hypothesis is rejected. It can be seen from Table 4 that for the observed AMP series, the p values at all the stations are greater than 0.1. This indicates that the null model GEV1 cannot be rejected, i.e., the stationary model is the best model for the observed series. But for the RCPs 4.5 and 8.5, the null model GEV1 is rejected since the p values are less than 0.1 at all the stations. GEV2, the nonstationary model with timevarying location parameter, was found to be the best fit model for both the RCPs in the first step.
In the second step, the suitability of a more complex model, i.e., GEV3, for the AMP series was investigated by keeping the null hypothesis in favor of GEV2 (against the alternative of GEV3). Since for the observed AMP series the stationary model GEV1 is the best fit model, there is no need for checking the suitability of a model which is more complex than GEV2. Hence, the second step was implemented only for the ensemble averaged AMP series derived for RCPs 4.5 and 8.5. The p values obtained were greater than 0.1, which means that the null model GEV2 cannot be rejected against the alternative of GEV3.
From the twostepped LR test, the GEV2 model with timevarying location parameter was found to be the best model for the AMP series under both the RCPs, and the GEV1 model was found to be the best model for observed AMP series. The results of the LR test complemented the results of AIC test.
Uncertainty Estimates of Parameters and Return levels
A nonparametric bootstrapping algorithm was used to quantify the uncertainty associated with the return levels. Uncertainty in the distribution parameters as well as uncertainty in the return levels can be obtained from this method. To obtain the uncertainty in the observed return levels, GEV1 with constant parameters was employed; for the projected climate change scenarios, GEV2 with timevarying location parameter was used in the bootstrap method. Uncertainties in the distribution parameters obtained from 1000 runs of the bootstrapping algorithm are presented as boxplot in Figs. 3 and 4.
The return levels were estimated using the parameter sets obtained from bootstrapping. It yielded the uncertainty in return levels, and these are presented in the form of empirical frequency curves for the three stations Kottamparamba, Manjeri, and Nilambur (Figs. 5, 6, and 7 respectively) for return periods 10, 25, 50, and 100 years. The empirical frequency curves for RCPs 4.5 and 8.5 exhibited a shift to the right with respect to the frequency curves for the observed AMP series at all the three stations. This implies that the magnitude of extreme precipitation return levels would increase under projected climate change. The shift is more for the curve corresponding to RCP8.5 than that for the curve of RCP4.5, as the extremes projected by the former are comparatively higher.
For a better understanding of the ranges of return levels, percentilebased confidence intervals for the return levels are presented in Table 5. A 90% confidence interval (\(CI_{90}\)) is obtained from the confidence limits corresponding to the 5th percentile and the 95th percentile values of the empirical cumulative distribution function. The expected values and confidence bounds of the return levels were higher under both the RCPs when compared with that for the observed AMP series. Furthermore, compared with RCP4.5, wider confidence intervals were obtained for the return levels under RCP8.5, especially for higher return periods. The finding that increase in extreme precipitation return levels and the associated uncertainty is higher under the high emission scenarios is consistent with the results of some studies in which the extremes were defined using climate indices [96, 97].
Conclusions
Uncertainty analysis of extreme precipitation return levels was performed for the Chaliyar river basin in Kerala, India, under projected climate change. Future extremes were derived from the reliability ensemble averaged projections of eight highresolution GCMs. A stationary GEV model and two nonstationary GEV models—one with linear trend only in the location parameter and the other with linear trend in the location parameter and loglinear trend in the scale parameter—were evaluated for goodness of fit of the observed AMP series and the ensemble averaged AMP series obtained from the highresolution projections. The stationary GEV model was obtained as the best fit model for observed AMP series. The nonstationary GEV model with linear trend only in location parameter and constant scale and shape parameters was found to be the best model among all the three models for future ensemble averaged AMP projections. Uncertainties in the distribution parameters and the return levels were obtained by a nonparametric bootstrapping approach. The expected values of return levels and the confidence bounds were found to be higher under climate change (RCPs 4.5 and 8.5) when compared with the corresponding values estimated from the observations. Results of the study point toward the intensification of precipitation extremes under projected climate change, and this indicates that the likelihood of occurrence of floods in the river basin would be enhanced. The expected return levels of precipitation extremes and associated uncertainties could be used for the optimal planning of adaptation and mitigation strategies and for the proper management of water resources in the river basin. Hydraulic infrastructure, designed based on traditional frequency analysis, should be appropriately rehabilitated/modified/reconstructed to withstand the extremes with time varying trends under a changing climate.
Data Availability
The climate model projections used in this study are from Coordinated Regional Climate Downscaling Experiment (CORDEX). These can be accessed through the climate data portal of Centre for Climate Change Research (CCCR), Indian Institute of Tropical Meteorology (IITM), Pune, India (http://cccr.tropmet.res.in).
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The manuscript was prepared based on the discussion between Ms. Ansa Thasneem S, Dr. Chithra N R, and Dr. Santosh G Thampi. Ms. Ansa Thasneem S carried out all the numerical simulations included in the manuscript. The manuscript was prepared by Ms. Ansa Thasneem S, and this was reviewed and corrected by Dr. Santosh G Thampi and Dr. Chithra N R.
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Ansa Thasneem, S., Chithra, N.R. & Thampi, S.G. Assessment of Nonstationarity and Uncertainty in Precipitation Extremes of a River Basin Under Climate Change. Environ Model Assess (2021). https://doi.org/10.1007/s1066602109752y
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Keywords
 Climate change
 Nonstationarity
 Extreme precipitation
 Uncertainty