Journal of Hydrology

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1 Journal of Hydrology 54 (206) Contents lists available at ScienceDirect Journal of Hydrology journal homepage: Research papers Is the covariate based non-stationary rainfall IDF curve capable of encompassing future rainfall changes? V. Agilan, N.V. Umamahesh Department of Civil Engineering, National Institute of Technology, Warangal, Telangana , India article info abstract Article history: Received 5 June 206 Received in revised form August 206 Accepted 26 August 206 Available online 30 August 206 This manuscript was handled by Andras Bardossy, Editor-in-Chief Keywords: Climate model Extreme rainfall IDF curves Non-stationarity Physical covariate Storm water management and other engineering design applications are primarily based on rainfall Intensity-Duration-Frequency (IDF) curves and the existing IDF curves are based on the concept of stationary Extreme Value Theory (EVT). However, during the last few decades, global climate change is intensifying the extreme precipitation events and creating a non-stationary component in the extreme rainfall time series. Subsequently, in recent years, advancements in the EVT helped the researchers to propose a method for developing non-stationary rainfall IDF curve by modelling trend present in the observed extreme rainfall series using covariate. But, is it capable of encompassing future rainfall changes? Towards answering this question, the Hyderabad city, India non-stationary rainfall IDF curves are compared with the IDF curves of two future time periods ( and ). Using 24 Global Climate Models (GCMs ) simulations and K Nearest Neighbor (KNN) weather generator based downscaling method, the IDF curves are developed for two future time periods and they are compared with covariate based non-stationary rainfall IDF curves of the Hyderabad city. The results of this study indicate that the return of period of an extreme rainfall of the Hyderabad city is reducing. In addition, it is noted that the non-stationary IDF curve developed by modelling trend in the observed extreme rainfall with covariate is an appropriate choice for designing the Hyderabad city infrastructure under climate change. Ó 206 Elsevier B.V. All rights reserved.. Introduction The rainfall Intensity-Duration-Frequency (IDF) curves are generally used in storm water management and other engineering design applications (Endreny and Imbeah, 2009; Cheng and AghaKouchak, 204) and these curves are constructed using observed rainfall time series data by fitting a appropriate theoretical probability distribution to partial duration series or annual maximum rainfall series (Cheng and AghaKouchak, 204). The concept of stationary extreme value theory (i.e. exceedance probability of extreme rainfall event is not expected to change significantly over time (Jakob, 203)) was used to develop the existing IDF curves. However, it is now widely recognized that the global climate change is intensifying the extreme rainfall events and creating a non-stationary component in the extreme rainfall time series (Allen and Ingram, 2002; Trenberth et al., 2003; Emori and Brown, 2005; Tramblay et al., 202; Cavanaugh et al., 205; Xu et al., 205; Milly et al., 2008; Wasko and Corresponding author. addresses: agilanvensiv@gmail.com (V. Agilan), mahesh@nitw.ac.in (N.V. Umamahesh). Sharma, 204). In particular, during the last century, the global temperature is increased due to human activities (Min et al., 20; IPCC, 203) and this additional temperature increases the air s water holding capacity by around 7% for every C warming, in this manner straightforwardly influencing rainfall (Trenberth, 20). In addition, the recent studies prove that more intense rainfall events may occur due to the high atmospheric water vapor (Berg et al., 203; Kunkel et al., 203; Wasko and Sharma, 205; Lenderink and van Meijgaard, 2008). Moreover, the probable maximum precipitation or the expected extreme precipitation may increase due to rising temperatures and subsequent increases in atmospheric moisture content (Trenberth et al., 2003; Kunkel et al., 203). In addition, greenhouse warming is also increasing the frequency of extreme Indian Ocean Dipole (IOD) events (Cai et al., 204) and this IOD has a significant influence on extreme rainfall of India (Ajayamohan and Rao, 2008). Furthermore, the recent studies demonstrate the effect of El Niño-Southern Oscillation (ENSO) cycle on extreme rainfall at regional and local scale (Kenyon and Hegerl, 200; Zhang et al., 200; Villafuerte and Matsumoto, 205; Franks and Kuczera, 2002; Kiem et al., 2003; Pui et al., 202) and the frequency of El Niño events (part of ENSO /Ó 206 Elsevier B.V. All rights reserved.

2 442 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) cycle) will increase, if the concentration of future greenhouse-gas is more (Timmermann et al., 999). In addition to global warming, IOD and ENSO effects on extreme rainfall, there is another process which also affects the extreme rainfall of urban area i.e. urbanization. In specific, in urban areas, artificial surfaces that have different thermal properties (e.g., thermal inertia and heat capacity) are replacing the natural land surfaces. Typically, such surfaces are capable of storing solar energy and transforming it into sensible heat. The temperature of the air in urban areas tends to be 2 0 C higher than neighboring nonurban areas because the sensible heat is transferred to the air (Shepherd et al., 200). Therefore, through the creation of an Urban Heat Island (UHI), urban areas alter boundary layer processes. Consequently, the mesoscale circulations and resulting convection are significantly influenced by UHI (Shepherd et al., 200). During the previous decade, the possible changes in rainfall due to urbanization are identified (Shepherd et al., 200; Shepherd and Burian, 2003; Burian and Shepherd, 2005; Lei et al., 2008; XiQuan et al., 2009; Zhang et al., 204; Yang et al., 205). Particularly, the recent studies show that the extreme rainfall events are significantly influenced by urbanization (Lei et al., 2008; Kishtawal et al., 2009; Miao et al., 20; Agilan and Umamahesh, 205). Hence, it is clear that the intensity, duration and frequency of rainfall extremes are expected to change over time due to various physical processes (discussed in the above paragraphs). Therefore, the time series will have a non-stationary component in it and the stationary extreme value theory based rainfall IDF curves may underestimate the extreme event. In other words, the frequency of future extreme rainfall events that exceed the capacity of current drainage networks will increase if the drainage is designed based on the concept of stationary extreme value theory (Langeveld and Schilperoort, 203; Willems, 203; Zahmatkesh et al., 205). To cope with climate change, researchers have developed future rainfall IDF curves with the help of Regional Climate Models (RCMs) or Global Climate Models (GCMs) future rainfall simulations (Mailhot et al., 2007; Prodanovic and Simonovic, 2007; Willems, 203; Rodriguez et al., 204; Rupa et al., 205; Hassanzadeh et al., 204). In detail, Prodanovic and Simonovic (2007) developed IDF curves of London for the future wet scenario using K Nearest Neighbor (KNN) based weather generator and they reported 30% increase in the intensity of future return level. Rodriguez et al. (204) constructed IDF curves of Barcelona, Spain for different scenarios and reported that the daily rainfall with a return period longer than 20 years will increase at least 4%. All these studies compared rainfall IDF curves constructed with observed rainfall under stationary assumption and IDF curves constructed with simulated future rainfall. On the other hand, to design infrastructure in a changing climate, researchers have developed a non-stationary rainfall IDF curves by modelling trend present in the observed extreme rainfall. In particular, Cheng and AghaKouchak (204) constructed a non-stationary rainfall IDF curves by introducing linear trend in the Generalized Extreme Value (GEV) distribution s location parameter. Yilmaz and Perera (204) investigated the nonstationarity in the IDF curves of Melbourne, Australia by introducing linear trend in the GEV distribution s location and shape parameter. However, till date, it is not clear that the nonstationary rainfall IDF curves developed by modelling trend present in the extreme rainfall series are capable of encompassing future rainfall changes or not. Therefore, in this study, the Hyderabad city, India future IDF curves are developed using 24 GCM outputs and KNN weather generator based downscaling method. Further, these IDF curves are compared with the covariate based non-stationary IDF curves of the Hyderabad city. 2. Study area and data Recently Agilan and Umamahesh (205) detected and attributed non-stationarity existing in the Hyderabad city extreme rainfall frequency and intensity, and they reported that the stationary statistical model is not even qualified as an adequate model when compared to non-stationary statistical model for modelling extreme rainfall of the city. Therefore, Hyderabad city is chosen as a study area to compare covariate based non-stationary IDF curve with climate model based future IDF curve. The Hyderabad city is the fourth biggest city in India. 796 mm/year was the average precipitation of Hyderabad city during , and it is 840 mm/year amid The hourly rainfall observed at the centre of the Hyderabad city by the India Meteorological Department is procured for the period of 0-January-972 to 3-December GCM (Table ) precipitation flux outputs for historical and future time periods are downloaded from CMIP5 website org/sim/gcm_monthly/ar5/reference-archive.html (accessed during September and October 205). In addition, for developing nonstationarity IDF curves, five physical processes, namely, ENSO cycle, urbanization, global warming, local temperature changes and IOD are considered as covariates. In order to represent global warming, with respect to the mean, the HadCRUT4 yearly Global Temperature Anomaly (GTA) is used. The GTA is directly downloaded from (Accessed on 5-July- 205) and it is based on average surface air temperature observations. Similarly, based on India Meteorological Department observed hourly temperature of the Hyderabad city, yearly Local Temperature Anomaly (LTA) with respect to the mean is used to represent local temperature changes. Southern Oscillation Index (SOI), Multivariate ENSO Index (MEI) and Sea Surface Temperature (SST) are the indices which are used to represent the ENSO cycle. To model the non-stationarity, different ENSO indices are used by different studies, i.e. SOI (Katz et al., 2002), SST (Mondal and Mujumdar, 205). Mondal and Mujumdar (205) used SST index averaged over the winter season (November to March) as a covariate for modelling the non-stationarity in intensity, duration and frequency of daily extreme rainfall over India. In this study, the approach of Mondal and Mujumdar (205) is considered due to their local relevance. The monthly sea surface temperature anomaly over NINO 3.4 (7 E-20 W, 5 S-5 N) region with respect to mean is the more common SST index and it is downloaded from ncep.noaa.gov/data/indices/sstoi.indices (Accessed on 9-July- 205). Then the average November to March NINO 3.4 SST anomalies is used as a covariate representing the ENSO cycle in a yearly basis. Indian Ocean Dipole (IOD) is quantified with Dipole Mode Index (DMI) (Saji et al., 999). Monthly DMI derived from HadISST dataset is downloaded from (Accessed on 5-June- 205) and yearly (i.e. averaged from June to November) DMI is calculated and used as a covariate which represents IOD. For this study, the urbanization of Hyderabad city modelled by Agilan and Umamahesh (205) is used. In order to know the urbanization pattern, Agilan and Umamahesh (205) used high-resolution remote sensing data to model the urban growth of the Hyderabad city. In particular, they studied the growth in urban built-up land using remote sensing data and supervised image classification algorithm. For more information on preparing urbanization data set, the interested reader is referred to Agilan and Umamahesh (205).

3 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) Table Details of GCM and future projections considered for each GCM. Model name Modelling centre RCP scenarios considered ACCESS_0 Commonwealth Scientific and Industrial Research Organization and Bureau of Meteorology, RCP 4.5, RCP 8.5 Australia BCC-CSM- Beijing Climate Center, China Meteorological Administration RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 BCC-CSM--M Beijing Climate Center, China Meteorological Administration RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 BNU-ESM College of Global Change and Earth System Science, Beijing Normal University RCP 2.6, RCP 4.5, RCP 8.5 CanESM2 Canadian Centre for Climate modeling and Analysis RCP 2.6, RCP 4.5, RCP 8.5 CCSM4 National Center for Atmospheric Research RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 CMCC-CMS Centro Euro-Mediterraneo per I CambiamentiClimatici RCP 4.5, RCP 8.5 CNRM-CM5 Centre National de RecherchesMeteorologiques/Centre Europeen de Recherche et Formation RCP 2.6, RCP 4.5, RCP 8.5 Avancees en CalculScientifique CSIRO-MK3-6-0 Commonwealth Scientific and Industrial Research Organization RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 GFDL-CM3 NOAA Geophysical Fluid Dynamics Laboratory RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 GFDL-ESM2G NOAA Geophysical Fluid Dynamics Laboratory RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 GFDL-ESM2M NOAA Geophysical Fluid Dynamics Laboratory RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 GISS-E2-R NASA Goddard Institute for Space Studies RCP 4.5 HadGEM2-CC Met Office Hadley Centre RCP 4.5, RCP 8.5 HadGEM2-ES Met Office Hadley Centre RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 INMCM4 Institute for Numerical Mathematics RCP 4.5, RCP 8.5 IPSL-CM5A-LR Institut Pierre-Simon Laplace RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 IPSL-CM5A-MR Institut Pierre-Simon Laplace RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 MIROC5 Atmosphere and Ocean Research Institute, National Institute for Environmental Studies, and RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 Japan Agency for Marine-Earth Science and Technology MIROC-ESM Japan Agency for Marine-Earth Science and Technology, Atmosphere and Ocean Research RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 Institute, and National Institute for Environmental Studies MIROC-ESM-CHEM Japan Agency for Marine-Earth Science and Technology, Atmosphere and Ocean Research RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 Institute, and National Institute for Environmental Studies MPI-ESM-LR Max Planck Institute for Meteorology RCP 2.6, RCP 4.5, RCP 8.5 MRI-CGCM3 Meteorological Research Institute RCP 2.6, RCP 4.5, RCP 6.0, RCP 8.5 NorESM-M Norwegian Climate Centre RCP 2.6., RCP 4.5, RCP 6.0, RCP Methodology The aim of this study is to compare covariate based nonstationary IDF curves with climate model based future IDF curves. The methodology adopted to achieve the aim of this study comprises of following three sections,. Developing future rainfall IDF curves with the help of Global Climate Model (GCM) simulations and KNN weather generator based downscaling method. 2. Developing non-stationary rainfall IDF curves by modelling trend present in the annual maximum rainfall series using the best possible covariate. 3. Comparison of these two IDF curves in terms of return levels and suggesting an appropriate method for urban infrastructure design. The general schematic diagram of developing different IDF curves is plotted in Fig Climate model based future IDF curve 3.. based downscaling Global Climate Model (GCM) simulations offer possibilities of what might happen if the future development follows a certain course of action, i.e. scenarios (Prodanovic and Simonovic, 2007) and it can be used to develop rainfall IDF curves for future rainfall conditions (Mailhot et al., 2007; Prodanovic and Simonovic, 2007; Willems, 203; Rodriguez et al., 204; Rupa et al., 205). For this study, 24 GCM outputs of different Representative Concentration Pathways (RCPs) scenario are considered. Table provides the list of GCMs and scenarios used in this study. To study the impact of climate change at the smaller scale, rainfall data from GCMs cannot be used directly because they are available only at the low spatial resolution (Willems et al., 202; Bi et al., 205). Downscaling of climate variable (in this case rainfall) is a technique in which high-resolution climate data is derived from low-resolution GCM data. Downscaling methods are generally grouped into two families: dynamical downscaling and statistical downscaling. The information about different downscaling methods can be obtained from Willems et al. (202) and Bi et al. (205). In this study, the change factor method (also called delta change method) is used to downscale future rainfall of Hyderabad city because, change factor methods are commonly used to assess the impacts of climate change at hydrological scale (Boé et al., 2007; Bi et al., 205). Constant scaling and daily scaling are two commonly used change factor methods. In constant scaling method, to derive estimations of future rainfall at the local scale, a constant factor is applied to the observed data (Bi et al., 205). The daily scaling method applies different scale factor for different percentiles of observed data. As daily scaling method describes changes in different levels of rainfall intensity, it is better than the constant scaling method (Olsson et al., 2009; Bi et al., 205). Therefore, in this study, the daily scaling method is adopted to estimate future rainfall changes of the Hyderabad city. In the daily scaling method, for a given time period and emission scenario, the ratio of percentiles of the simulated future precipitation distribution and corresponding values for the historical period is the distribution of change factors. In detail, the ratio between x th quantile value in future rainfall and the same x th quantile value in historical rainfall is the change factor for the observed rainfall events which have intensity greater than (x ) th quantile (if x = 50th quartile, (x ) = 49th quartile) value of observed rainfall and less than or equal to x th quantile value of observed rainfall. CF x ¼ Q xðr fut Þ Q x ðr his Þ here, R fut is the x th percentile REA averaged future rainfall of a given time period and scenario, R his is the x th percentile REA averaged historical rainfall. CF x is change factor and it is to be applied (multiplied) only with observed rainfall events which have an intensity greater than (x-) th quantile value of observed rainfall and less than ðþ

4 444 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) Fig.. Schematic diagram of developing different IDF curves. or equal to x th quantile value of observed rainfall. Calculating CF for each quartile (00 divisions: st to 00 th quartile) is more appropriate than calculating only for some quartiles (Lafon et al., 203). Therefore, in this study, CF is calculated for each quartile. For this study, two future time slices: and are considered for constructing future rainfall scenarios and the time slice is considered as the historical time period. The GCM outputs are interpolated to the Hyderabad city rain gauge location. Further, the intermodal uncertainty resulting from the ensembles of projections generated with multiple GCMs is addressed using the Reliability Ensemble Average (REA) method (Appendix A.). Table 2 provides the weights assigned using the REA method to GCMs for historical and for four scenarios of two future time periods. For each future time slice, the CF is calculated for each RCP scenario using REA averaged GCM output. The climate model outputs are available only for the daily timesteps. But, for developing IDF curves, there is a need of short duration rainfall. One possible way of getting short duration rainfall is using statistical disaggregation methods. In this study, the method followed by Prodanovic and Simonovic (2007) is used to obtain future short duration rainfall. i.e. From the available hourly observations ( ), the -, 2-, 3-, 6-, 2-, 8- and 24-h duration Table 2 Weights assigned to GCMs for the historical time period and four future scenarios using REA technique. Model name Historical ( ) Future- ( ) Future-2 ( ) RCP2.6 RCP4.5 RCP6.0 RCP8.5 RCP2.6 RCP4.5 RCP6.0 RCP8.5 ACCESS_ BCC-CSM BCC-CSM--M BNU-ESM CanESM CCSM CMCC-CMS CNRM-CM CSIRO-MK GFDL-CM GFDL-ESM2G GFDL-ESM2M GISS-E2-R HadGEM2-CC HadGEM2-ES INMCM IPSL-CM5A-LR IPSL-CM5A-MR MIROC MIROC-ESM MIROC-ESM-CHEM MPI-ESM-LR MRI-CGCM NorESM-M

5 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) rainfall are calculated using the moving window approach (i.e., H-hour duration rainfall are calculated using a moving window with the size of H that slides hour-by-hour). Then for-the-day maximum (one value per day) rainfall is extracted from the original -, 2-, 3-, 6-, 2-, 8- and 24-h duration rainfall series and this modified dataset are used for further analysis. In order to get future short duration rainfall, the modified -, 2-, 3-, 6-, 2-, 8- and 24-h duration rainfall series are multiplied with the same CF. The output of this step produces input for simulating future short duration rainfall (-, 2-, 3-, 6-, 2-, 8- and 24-h) for four RCP scenarios and two time slices i.e. 8 sets of short duration (-h to 24-h) rainfall. These datasets are then used as input to KNN based weather generator to produce future rainfall data. 3.. KNN weather generator Weather generators are stochastic simulation methods that are used to generate a long record of plausible data using mathematical algorithms and locally observed rainfall (any climate variable) pattern (Prodanovic and Simonovic, 2007). Generally, weather generators are classified into two groups: non-parametric and parametric methods (Sharif and Burn, 2007). Parametric weather generators make distribution assumption and they require site specific parameters. In addition, parametric weather generators have difficulties in representing persistent events such as prolonged rainfall (Sharif and Burn, 2007). In other hand, nonparametric weather generators do not make distribution assumption and they do not need site specific parameters. Therefore, in this study, one of the non-parametric weather generators, i.e. The KNN weather generator is chosen to produce future rainfall data of the Hyderabad city. The input of KNN weather generator is the change factor applied for-the-day maximum rainfall (say RCP 2.6 scenario time period -h duration modified rainfall). The KNN weather generator re-shuffles the input and produces future forthe-day maximum -h duration rainfall series and the output is non-identical to the observed for-the-day maximum -h duration rainfall series. The advantages of KNN algorithm are: () is capable of modelling non-linear dynamics of geophysical processes; (2) preserves the temporal and spatial correlation of generated data; and (3) do not require knowledge of probability distributions or variables (Prodanovic and Simonovic, 2007). For this study, the KNN weather generator of Lall and Sharma (996), Sharif and Burn (2007) and Prodanovic and Simonovic (2007) has been modified and coded in the R programming language. The flowchart of KNN weather generator is shown in Fig. 2 and step-by-step procedure of the KNN weather generator is discussed as follows: Step : For-the-day maximum, change factor applied, rainfall of any duration, any future time period and any RCP scenario is given as input to the KNN weather generator (for example, time period = ; scenario: RCP 2.6; rainfall duration: 2-h). This input data consists T number of days of N years (in this case N = 42). X ¼ x i ; 8i 2f; 2; 3;...; Tg ð2þ Step 2: Initially, a set of potential neighbors is selected to represent current day rainfall x i. To begin this step, a temporal window of size w is selected. If we are generating n th day rainfall, the rainfall values between [n (w/2)] th day and [n + (w/2)] th day from all years are selected as neighbors and the current day rainfall is removed from the selected neighbors. Because the current day rainfall cannot be considered as a neighbor to itself. Then the size of potential neighbors is: L ¼ f½ðw þ ÞNŠ g. For example, if w is 4 days and simulation is for August 5 th, then all days between August 8 th and August 22 nd (except August 5 th of the current year) are selected from all N years of record as neighbors. Most of the previous studies used window size (w) of 4 days (Yates et al., 2003; Sharif and Burn, 2007; Prodanovic and Simonovic, 2007). In this study, initial KNN simulations with a window size of 4 days produced a poor performance (in terms of total rainfall and extreme rainfall intensity) in most of the cases. In particular, more deviation is found in August and September months KNN simulations. Therefore, the changes in Hyderabad city monthly rainfall is analyzed using observed rainfall. The total observed rainfall time period (42 years) is divided into two time slices i.e and and the average monthly rainfall for these two time periods are calculated and plotted in Fig. 3. From the Fig. 3, it is observed that the average August month rainfall is increasing significantly and the average September month rainfall is decreasing significantly and this may affect the window size. Therefore, in order to select the optimal window size, window size of 4 days, 28 days and 42 days are evaluated and, for this study, window size of 28 days found to be optimal. Step 3: Next, the absolute distance (d) (Prodanovic and Simonovic, 2007; Sharma and Mehrotra, 204) is calculated between current day rainfall (x n ) and rainfall values of all neighbors. d j ¼ x n x j ; j ¼ ; 2;...; L: ð3þ r where, x n is current day rainfall value, x j is rainfall value of neighbor j, r is the standard deviation of L neighbors. Step 4: The absolute distance (d) is sorted in ascending order, and the first K neighbors are retained out of L sorted potential neighbors for further sampling. These K neighbors are referred pffiffi as the nearest neighbors. The recommended value of K is L (Rajagopalan and Lall, 999; Yates et al., 2003). Step 5: In order to select current day rainfall from K nearest neighbors, a discrete probability distribution is used to assign weight (w k ) to all K neighbors. The weight of a neighbor is directly proportional to its distance from the current day rainfall (i.e. highest weight is assigned to the closest neighbor and vice-versa). Weights are assigned to each of these K neighbors according to Eq. (4). w k ¼ P =k K ; k ¼ ; 2;...; K ð4þ i¼=i The cumulative probabilities (p k ) are given by Eq. (5). p k ¼ Xk i¼ w k The output of this step is K nearest neighbors and their corresponding cumulative probabilities (p k ). Step 6: Now, a random number (r) between 0 and is generated. In order to select current day rainfall from the K nearest neighbors, generated random number is compared with the cumulative probability of all K neighbors. If the cumulative probability of the first neighbor is greater than or equal to the generated random number (p P r), the rainfall value of the first neighbor is selected as the current day rainfall. Otherwise, the location (k) of neighbor is identified using Eq. (6). p k < r 6 p k ð5þ ð6þ

6 446 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) Fig. 2. Flowchart of the KNN weather generator. Then the rainfall value of k th neighbors is assigned as the current day rainfall. Average monthly rainfall (mm) Month Fig. 3. Average monthly rainfall of the Hyderabad city during two time periods. Step 7: Step 2 6 is repeated T times to calculate rainfall of all T days. As the performance of KNN weather generator is based on generated random number value, sometimes, the KNN may produce a very poor performance. Therefore, the performance evaluator (PE) given by Eq. (7) is calculated between input and KNN simulated values to evaluate the simulation. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P N i¼ PE ¼ ðkr i MR i Þ 2 N where, KR is sorted annual maximum series from KNN simulated values, MR is sorted annual maximum series from input (Step ). Step 8: Step 2 7 is repeated for 50 times (50 runs) and the run which has minimum PE value is the final output. ð7þ

7 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) Step 8 is repeated for all durations modified for-the-day maximum rainfall series. Then the extreme rainfall series (annual maximum) is extracted from the KNN simulations and they are used to develop IDF curve of different RCP scenarios of two future time periods Future IDF curve development Consider an annual maximum series of any duration of future time period generated in the previous Section. This annual maximum series is considered as n independent and identically distributed (iid) random variable x,x 2,...,x n. Asymptotically, the annual maximum series converges to GEV distribution and the cumulative distribution function is given by Eq. (8) (Coles, 200; Katz et al., 2002; Khaliq et al., 2006; Nogaj et al., 2007). 8 h i =n >< exp þ nðx lþ r ; r > 0; þ nðx lþ r > 0; n 0 Fðx;l;r;nÞ¼ n h io >: exp exp ðx lþ r ; r > 0; n ¼ 0 ð8þ where l, n and r are the location, shape and scale parameters of the GEV distribution respectively. The distribution is heavy tailed, unbounded light tailed and bounded upper tailed for positive, zero and negative shape parameter values respectively (Katz et al., 2002). The scale parameter r characterizes the spread of the distribution. In this study, the parameters of the GEV distribution are estimated by the method of maximum likelihood. Let the values X=x,x 2,...,x n be the n years of annual maximum series. The log likelihood, which is derived from Eq. (8), is given as For n 0, log Lðl; r; njxþ ¼ n log r ðþ=nþ Xn h log þ n x i li r For n =0, Xn i¼ log Lðl; rjxþ ¼ n log r Xn h þ n x i l r i¼ i¼ x i l Xn r i =n; x i l þ n r i¼ h > 0 exp x i l r i ð9þ ð0þ The maximum likelihood estimate with respect to the entire GEV family can be estimated through maximizing Eqs. (9) and (0) with respect to the parameter vector (l, r, n) (Coles, 200). In this study, in order to estimate the parameters, the negative log likelihood ( logl(b/x); b = (l, r, n)) is minimized using Nelder-Mead (Nelder and Mead, 965) optimization algorithm. Note that the method of maximum likelihood estimates physically infeasible shape parameter values for small samples (i.e. when the value of n is less than 25) and it is not suggested in such situations (Martins and Stedinge, 200; Sugahara et al., 2009; Katz et al., 2002). Once the parameters of GEV model are estimated, the rainfall intensity of different return period is calculated. Estimation return level for the given p (probability of occurrence) is provided by Eq. () (Coles, 200; Cheng et al., 204). 8 >< bl z T ¼ þbr ð logð pþþ bn ; b n 0 bn ðþ >: bl þ br½ logð logð pþþš; b n ¼ 0 The m year rainfall intensity refers to the annual maximum rainfall of specified intensity and duration having a probability of exceedance of /m (Cheng and AghaKouchak, 204). In this way, rainfall intensity is calculated as a function of the return period T, T = /p Covariate based non-stationary IDF curve Based on recent theoretical developments in the EVT, Cheng and AghaKouchak (204) developed a non-stationary rainfall IDF curves by modelling trend exist in the observed extreme rainfall series. In particular, they have incorporated linear trend in the GEV distribution s location parameter using Time covariate. In this study, the non-stationary IDF curves are developed based on the best possible covariate. In particular, instead of modelling trend directly using time covariate, a set of covariates for modelling trend in each duration extreme rainfall (annual maximum) is analyzed and non-stationary IDF curve of Hyderabad city is developed based on the best possible covariate. For developing non-stationary IDF curves based on the best possible covariate, there is a need for identifying possible physical covariates and data to represent the covariates. As mentioned in the introduction section, the physical processes, namely, global warming, local temperature changes, ENSO cycle, IOD and urbanization are having a significant influence on the Hyderabad city extreme rainfall characteristics, therefore, non-stationarity in the Hyderabad city extreme rainfall can be modelled using these physical processes. As the previous studies have used Time as a covariate (Sugahara et al., 2009; Cheng and AghaKouchak, 204; Yilmaz and Perera, 204), in addition to these five physical covariates, Time is also considered as one of the possible covariates. As used by other researchers (Yilmaz and Perera, 204; Cheng and AghaKouchak, 204), the years of annual maximum series are standardized and used as a covariate which represents Time. Once the covariates and data to represent the covariates are identified, the non-stationarity can be modelled by introducing trend in the GEV parameters. Since the precise estimation of GEV s shape parameter is difficult, assuming it as a smooth function of time is unrealistic (Coles, 200; Yilmaz and Perera, 204). Therefore, the non-stationarity is introduced only in location and scale parameter of GEV while the shape parameter is kept constant. In this study, five physical covariates and one Time covariate are considered. Eq. (2) and (3) provides the non-stationary setting for the parameters of the GEV as a function of covariates. Case-I: Non-stationary location; Stationary scale; Stationary shape (NSS). lðtþ ¼l 0 þ l c rðtþ ¼r nðtþ ¼n ð2þ Case-II: Non-stationary location; Non-Stationary scale; Stationary shape (NNS). lðtþ ¼l 0 þ l c rðtþ ¼expðr 0 þ r cþ nðtþ ¼n ð3þ where, c is any covariate (i.e. ENSO cycle (E), Urbanization (U), IOD cycle (D), Global warming (G), Local temperature changes (L) and Time (T)). For the stationary GEV distribution, the covariate c value is zero. The trend in the location parameter due to the effect of the covariate c is represented by the slope parameters l. The slope parameter r represents the trend in the scale parameter due to the effect of the covariate c. In order to ensure the positive value of the scale parameter, the exponential function is used in Eq. (3). In this study, based on six covariates, twelve non-stationary

8 448 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) GEV models are constructed using two cases of non-stationary setting (i.e., NSS and NNS). In addition, one stationary model is also constructed for comparison. The details and the descriptions of thirteen GEV models (i.e., 2 non-stationary GEV models and one stationary GEV model) are listed in Table 3. The covariate of the best GEV model is the best covariate(s) for developing non-stationary rainfall IDF relationship for the corresponding duration. In this study, the parameters of non-stationary models are estimated using the method of maximum likelihood because it can be easily extended to the non-stationary case (Coles, 200; Katz, 203). In the case of non-stationary GEV, the location and scale parameters in the Eqs. (9) and (0) are replaced with Eqs. (2) and (3) based on the case of non-stationary setting. Once the parameters are estimated, selecting the best model is a critical step. To select the best model among all candidate models, the Akaike Information Criterion (AIC) can be used because it penalizes the minimized negative log likelihood (Eqs. (9) and (0)) for the number of parameters estimated (Burnham and Anderson, 2004; Katz, 203). However, in this study, the small sample version of AIC, called corrected Akaike Information Criterion (AICc), is used Table 3 AICc value of GEV models constructed with -h duration observed rainfall. Model ID Description AICc AICc-min(AICc) GEV- X GEVðl; r; nþ GEV-2 X GEVððl 0 þ Ul Þ; r; nþ GEV-3 X GEVððl 0 þ Ll Þ; r; nþ GEV-4 X GEVððl 0 þ Gl Þ; r; nþ GEV-5 X GEVððl 0 þ El Þ; r; nþ GEV-6 X GEVððl 0 þ Dl Þ; r; nþ GEV-7 X GEVððl 0 þ Tl Þ; r; nþ GEV-8 X GEVððl 0 þ Ul Þ; e ðr0þurþ ; nþ GEV-9 X GEVððl 0 þ Ll Þ; e ðr0þlrþ ; nþ GEV-0 X GEVððl 0 þ Gl Þ; e ðr0þgrþ ; nþ GEV- X GEVððl 0 þ El Þ; e ðr0þerþ ; nþ GEV-2 X GEVððl 0 þ Dl Þ; e ðr0þdrþ ; nþ GEV-3 X GEVððl 0 þ Tl Þ; e ðr0þtrþ ; nþ to select the best GEV model because it outperforms AIC, in such way that it helps in avoiding over-fitting the data more than conventional AIC (Sugahara et al., 2009). In this study, the GEV model which has lowest AICc value is considered as the best model. After identifying the best model, the quality of a fitted model can be checked by the probability-probability (PP) and quantile-quantile (QQ) plots (Coles, 200; Katz et al., 2002; Sugahara et al., 2009). Once the best non-stationary model is identified and its quality is checked, the nonstationary rainfall intensity is estimated using the model parameters of the best non-stationary model. Unlike stationary model, the location and scale parameter value of nonstationary model will vary over the time. In this study, the low risk (more conservative) approach (Cheng et al., 204) is used to calculate the location and scale parameter value. i.e. the ninety-five percentiles of the parameter values in historical observation and it is given by Eqs. (4) and (5). The 95 percentile of the location parameter value gives the effective return level (Cheng et al., 204). bl 95 ¼ Q 95 ðbl t ; bl t2 ;...; bl tn ; Þ br 95 ¼ Q 95 ðbr t ; br t2 ;...; br tn ; Þ ð4þ ð5þ The location and scale parameter estimated using Eqs. (4) and (5) are substituted in Eq. () to calculate nonstationary rainfall intensity of different return levels. 4. Results and discussion 4.. Climate model based future IDF curve As mentioned before, the change factors have been calculated for two future time periods, i.e and and for four RCP scenarios using REA averaged GCM outputs. The change factors for two future time periods and for four RCP scenarios are shown in Figs. 4 and 5. From the Figs. 4 and 5, it is noted that the change factors are mostly greater than for all RCP scenarios of two future time periods and change factors are comparatively higher than changes factors. It indicates (a) (b) (c) (d) Fig REA technique based change factors for (a) RCP 2.6, (b) RCP 4.5, (c) RCP 6.0 and (d) RCP 8.5 scenario.

9 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) (a) (b) (c) (d) Fig REA technique based change factors for (a) RCP 2.6, (b) RCP 4.5, (c) RCP 6.0 and (d) RCP 8.5 scenario. that the Hyderabad city total rainfall is increasing. Further, it is also noted that the sign of increasing trend in the Hyderabad city total rainfall is detected in observed rainfall i.e. the average precipitation of Hyderabad city during was 796 mm/year and it has increased to 840 mm/year amid This observation agrees with Changnon et al. (97) study results, i.e. the urbanization leads to increased precipitation. Further, from the Figs. 4 and 5, it is also observed that the change factors of extreme rainfall (high percentile rainfall) are much greater than low and medium percentile rainfall of all RCP scenarios of two future time periods and it indicates that the extreme rainfall is going to be more severe than normal rainfall. During , a maximum of 7% increase in 98 th percentile rainfall is observed and during , a maximum of 37% increase in 00 th percentile rainfall is observed. The change factors of two future time periods of four RCP scenarios are then applied to for-the-day maximum rainfall of different durations. The change factors applied for-the-day maximum rainfall series of different durations are used as input of the KNN weather generator to simulate future rainfall. The performance of KNN weather generator with observed for-the-day maximum rainfall of different durations is shown in Fig. 6. Fig. 6(a) shows the scatter plot between simulated and observed for-the-day maximum rainfall of different durations. Q-Q plots between annual maximum of observed for-the-day maximum rainfall of different durations and annual maximum of corresponding KNN simulation are plotted in Fig. 6(b). From the Fig. 6(a), it is seen that the number of KNN underestimated events are higher than the overestimated events. However, in the case of annual maximum value, the KNN simulation is nearly matched with the observed. Overall, the performance of KNN weather generator developed in this study is satisfactory. Therefore, future rainfall is simulated using the KNN weather generator for four RCP scenarios of each future time period using corresponding input. Further, the KNN simulated rainfall values are used to develop IDF curves of four RCP scenarios of each future time period. Return levels for two future time periods are presented in Table 5. The future return levels and observed stationary and non-stationary return levels are compared and discussed in Section Covariate based non-stationary IDF curve Towards developing covariate based non-stationary IDF curve, the best possible covariate for modelling trend in each duration extreme rainfall (annual maximum) of Hyderabad city is analyzed. In particular, thirteen GEV models are constructed for each duration annual maximum rainfall series using the method discussed in Section 3.2. The AICc value of thirteen GEV models constructed with -h duration observed annual maximum rainfall series is given in Table 3. From Table 3, it is observed that the GEV-3 has the lowest AICc value among all models. Therefore, GEV-3 is chosen as the best model for Hyderabad city -h duration observed annual maximum series. GEV-3 is a non-stationary model and it has a time varying location parameter which is based on local temperature changes. The PP and QQ plots of GEV-3 for -h duration observed annual maximum rainfall series is plotted in Fig. 7. From the Fig. 7, it is noted that the model-derived and empirical probabilities and quantiles show a good match with each other. Therefore, GEV-3 is an appropriate choice for the -h duration observed annual maximum rainfall series. Similarly, the best model for the Hyderabad city each duration extreme rainfall series is analyzed and the quality of the best model is checked with PP and QQ plots. Similar to -h duration rainfall series, from the PP and QQ plots of annual maximum rainfall series of all other durations, it is observed that the selected models are appropriate choice for the corresponding durations. Moreover, the significance of non-stationary model when compared to the stationary model is checked using the likelihood ratio test and for more information on this test, the interested reader is referred to Katz (203). The likelihood ratio test p-values (Table 4) indicate that the selected non-stationary models are significantly superior to the stationary models. For brevity, the AICc values and PP and QQ plots of all other durations are not shown in this paper. Table 4 provides the best model for each duration observed annual maximum rainfall series and time varying parameter of the best model and corresponding covariate. The models listed in Table 4 are then used to construct non-stationary IDF curves of Hyderabad city using Eqs. (), (4) and (5). The stationary models (GEV-) of all duration annual maximum rainfall are used to

10 450 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) Fig. 6. (a) Scatter plot between observed for-the-day maximum rainfall of different durations and corresponding KNN simulation; (b) Q-Q plot between annual maximum of observed for-the-day maximum rainfall of different durations and annual maximum of corresponding KNN simulation.

11 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) Fig. 7. The PP and QQ plots of GEV-3 for -h duration observed annual maximum rainfall series. For obtaining these plots, annual maximum data have to be standardized to the Gumbel distribution (Refer Katz et al. (2002) for more information). Table 4 Best GEV model for modelling Hyderabad city extreme rainfall and p-values of likelihood ratio test between the best non-stationary model and the stationary model. Duration Model ID Covariate Time varying parameter Likelihood ratio test p-value -h GEV-3 Local temperature changes Location h GEV-3 Local temperature changes Location h GEV-3 Local temperature changes Location h GEV-3 Local temperature changes Location h GEV-3 Local temperature changes Location 7.4e 03 8-h GEV- ENSO cycle Location and scale 5.8e h GEV- ENSO cycle Location and scale 4.4e 03 Table 5 Return levels in mm/h. S NS RCP2.6 RCP4.5 RCP6.0 RCP8.5 RCP2.6 RCP4.5 RCP6.0 RCP8.5 2-Year -h h h h h h h Year -h h h h h h h Year -h h h h h h h Year -h h h h h h h Note: S-Stationary; NS-Non-Stationary. construct the stationary IDF curves of Hyderabad city. The stationary and non-stationary return levels of the Hyderabad city are listed in Table 5 and they are compared with future return levels in the next Section.

12 452 V. Agilan, N.V. Umamahesh / Journal of Hydrology 54 (206) Comparison As the study area of this paper is an urban catchment and most of the urban infrastructure is designed for return level of 25 year or less, the stationary, non-stationary and future IDF curves are compared in terms of 2, 5, 0 and 25 year return levels. The stationary, non-stationary and future time period 2, 5, 0 and 25 year return levels are presented in Table 5 and the 0-year return period of stationary, non-stationary and future Intensity-Duration curves are plotted in Fig. 8. Overall, from Table 5, it is found that the return period of an extreme rainfall of the Hyderabad city is reducing. In particular, during and under RCP 8.5 scenario, the 0 year return period future extreme rainfall of -h duration is mm/h and it is nearly equal (59.64 mm/h) to the 25 year return period observed extreme rainfall (from the observed stationary IDF curve). Similarly, during and under RCP 8.5 scenario, the 5 year return period future extreme rainfall of -h duration is mm/h and it is more than (59.64 mm/h) the 25 year return period observed extreme rainfall (from the observed stationary IDF curve). Moreover, this change is also observed when comparing observed-stationary and observed-non-stationary IDF curves. For example, for an event with a return period of 0 years and -h duration, the non-stationarity and stationarity extreme rainfall are 6.72 mm/h and mm/h respectively. The nonstationary extreme rainfall of 5 year return period and -h duration is mm/h and it is nearly equal to the stationary extreme rainfall of 0 year return period. From the results, it is observed that the return of period of an extreme rainfall of the Hyderabad city is reducing. If such an IDF curve (developed from the stationary model) is used for an infrastructure design, the drainage networks will fail more frequently than its actual design. In other hand, from Table 5 and Fig. 8, the non-stationary IDF curves developed by modelling trend in the observed extreme rainfall is estimating return levels reasonably to cope climate change for at least the next fifty years. In particular, the observed nonstationary IDF curves are having higher values (return level) than IDF curves developed using future rainfall of all RCP scenarios (see Table 5) and the IDF curves developed using future rainfall of RCP 2.6 scenario (see Table 5). Nevertheless, the IDF curves developed using future rainfall of RCP 4.5, 6.0 and 8.5 scenarios are showing higher return levels than observed non-stationary return levels. It indicates that the IDF curve developed by modelling trend in the observed rainfall is an appropriate choice for designing Hyderabad city infrastructure if the design life of the infrastructure is less than 50 years. Moreover, under RCP 2.6 scenario, the observed non-stationary IDF curve is reasonable choice for designing infrastructure which has design life more than 50 years. As mentioned before, we have used the ninety-five percentiles of the parameter values in historical observation to calculate the non-stationary return levels and they are compared with the future IDF curves. However, for the better understanding of nonstationary IDF curves, the suitable percentile for designing Hyderabad city infrastructure is further analyzed. In particular, for the Fig year return period stationary, non-stationary and future Intensity-Duration curves of (a) RCP 2.6, (b) RCP 4.5, (c) RCP 6.0 and (d) RCP 8.5 scenario.