Covariate and parameter uncertainty in non-stationary rainfall IDF curve

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. (27) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI:.2/joc.58 Covariate and parameter uncertainty in non-stationary rainfall IDF curve V. Agilan and N. V. Umamahesh* Department of Civil Engineering, National Institute of Technology, Warangal, India ABSTRACT: Since the substantial evidence of non-stationarity in the extreme rainfall series is already reported, the current realm of hydrologic research focusing on developing methodologies for a non-stationary rainfall condition. As the rainfall intensity duration frequency (IDF) curve is primarily used in storm water management and infrastructure design, developing rainfall IDF curves in a non-stationary context is a current interest of water resource researchers. In order to construct non-stationary rainfall IDF curve, the probability distribution s parameters are allowed to change according to covariate value and the current practice is to use time as a covariate. However, the covariate can be any physical process and the recent studies show that the direct use of time as a covariate may increase the bias. Moreover, the significance of selecting covariate in developing non-stationary rainfall IDF curve is yet to be explored. Therefore, this study aims to find the uncertainties in rainfall return levels due to the choice of the covariate (covariate uncertainty). In addition, since the uncertainty in parameter estimates (parameter uncertainty) is the major source of uncertainty in the stationary IDF curve, the relative significance of covariate uncertainty, when compared to the parameter uncertainty, is also explored. The study results reveal that the covariate uncertainty is significant, especially when a number of covariates produce significantly superior non-stationary model and, remarkably, it is nearly equivalent to the parameter uncertainty in such cases. KEY WORDS Bayesian inference; covariate uncertainty; non-stationarity; parameter uncertainty; rainfall IDF curve Received 28 November 26; Revised 29 March 27; Accepted 27 May 27. Introduction Quantification of extreme rainfall characteristics (i.e. intensity, duration and frequency) is crucial in hydrologic designs including the design of urban storm water collection systems and flood protection systems (Hassanzadeh et al., 24; Rupa et al., 25). The rainfall intensity duration frequency (IDF) curves contain characteristics of rainfall extremes. In specific, for a given duration and return period, the intensity of extreme rainfall events is described by the rainfall IDF curve. Therefore, rainfall IDF curves have been widely used as the primary input to approximate extreme rainfall for storm water management and other engineering design applications (Endreny and Imbeah, 29; Dourte et al., 23; Cheng and AghaKouchak, 24; Hassanzadeh et al., 24; Yilmaz and Perera, 24; Agilan and Umamahesh, 26). For developing rainfall IDF curve, observed extreme rainfall time series of different durations (i.e. h, 2 h, 3 h, 6 h, ) are fitted with a theoretical probability distribution. Note that the existing rainfall IDF curves are based on the concept of temporal stationarity, meaning, the parameters of the fitted probability distribution are not expected to change significantly over time. In other words, the exceedance * Correspondence to: N. V. Umamahesh, Department of Civil Engineering, National Institute of Technology, Warangal, India. mahesh@nitw.ac.in probability of extreme rainfall events is assumed to be constant over time (Jakob, 23; Cheng and AghaKouchak, 24). However, it is now widely recognized that the global climate change is intensifying the frequency and intensity of extreme rainfall events (Trenberth et al., 23; Emori and Brown, 25; Mueller and Pfister, 2; Tramblay et al., 22; Jena et al., 24; Xu et al., 25). For example, the mean surface temperature of the globe has increased during the last century due to various human activities such as excessive greenhouse gas emission by burning fossil fuels (Min et al., 2; IPCC, 23). Because of this additional temperature, the water holding capacity of the air is increasing 7% for every one-degree additional temperature and straightforwardly affecting rainfall (Trenberth, 2). More importantly, recent studies show that the higher atmospheric water vapour can cause more intense rainfall events (Lenderink and van Meijgaard, 28; Berg et al., 23; Kunkel et al., 23; Wasko and Sharma, 25). Therefore, the rise in global mean surface temperature and subsequent increase in moisture content of atmosphere may increase the occurrence probability of extreme rainfall (Trenberth et al., 23; Kunkel et al., 23). In recent years, the increasing trend in extreme rainfall characteristics is being detected in many regions of the world (Khaliq et al., 26; Westra et al., 23; Agilan and Umamahesh, 25; Asadieh and Krakauer, 25; Mondal and Mujumdar, 25; Vasiliades et al., 25). 27 Royal Meteorological Society

2 V. AGILAN AND N. V. UMAMAHESH Consequently, the extreme rainfall time series will have a non-stationary component. In other words, the parameters of the theoretical probability distribution fitted to the extreme rainfall series will change significantly over time and it violates the stationarity assumption which is made to derive the existing rainfall IDF curves. Hence, the rainfall IDF curves developed with the stationarity assumption will underestimate the return levels. As a result, the drainage networks which are designed using stationary IDF curve will fail more frequently than its actual design. Since the rainfall IDF curves developed with the stationarity assumption are not tenable in a changing climate, it is indispensable to update the rainfall IDF relationship developing methods for the non-stationary condition. In recent years, in the direction of coping non-stationarity in the extreme rainfall time series, researchers developed non-stationary rainfall IDF curves by modelling trend present in the observed extreme rainfall time series. For example, in order to develop non-stationary rainfall IDF curves, Cheng and AghaKouchak (24) introduced time-varying component in the location parameter of the generalized extreme value distribution (GEVD) using time covariate. The authors developed non-stationary IDF curves and analysed their uncertainties resulting from the estimation of GEVD parameters using the Bayesian method. From their study results, they reported that developing IDF curves under the stationary extreme value theory may lead to underestimation of extreme rainfall by as much as 6%. Similarly, Yilmaz and Perera (24) incorporated trend in the GEVD s location and scale parameters using time covariate for analysing non-stationarity in the IDF curves of Melbourne city, Australia. Further, more recently, Agilan and Umamahesh (26) developed non-stationary rainfall IDF curves of Hyderabad, India and Wilmington, USA by modelling non-linear trend in the extreme rainfall series using multi-objective genetic algorithm generated time-based covariate. In addition, they also reported that directly applying time covariate-based linear trend is sometimes increasing the bias of non-stationary model. Modelling linear trend using time covariate is simple and easy to implement because there is no need of any additional data and computation. However, it is already reported that directly applying time covariate-based linear trend is sometimes increasing the bias of non-stationary model. Moreover, even if the time covariate-based linear trend produces less bias non-stationary model, the covariate time (or directly applying any single covariate) may not be the best choice to model the non-stationarity in the extreme rainfall series. Further, the covariate can be any complicated form of time or any physical process, which influences extreme rainfall occurrence of the study area. In such a case, will the choice of covariate create any significant difference in estimating return levels in a non-stationary context? If so, are the uncertainties resulting from the choice of covariate higher/lower than those resulting from the estimation of non-stationary GEVD parameters? Till date, these two questions were not studied. However, it is indispensable to know the answers to these two questions for the accurate estimate of return levels. Therefore, the objectives of this study are: () to quantify the uncertainties in estimating non-stationary return levels due to the choice of covariate (covariate uncertainty); (2) to quantify the uncertainties resulting from the estimation of non-stationary GEVD parameters (parameter uncertainty) and (3) to analyse the significance of covariate uncertainties when compared to the parameter uncertainties. 2. Study area and data Since the aim of this study is to analyse the uncertainties in non-stationary rainfall IDF curves, the study areas where the non-stationarity in the rainfall IDF curves were already reported are chosen. Initially, Cheng and AghaKouchak (24) reported the non-stationarity in the rainfall IDF curves of Wilmington city, USA. Then, Agilan and Umamahesh (26) developed non-stationary IDF curves of Hyderabad city, India and Wilmington city, USA by modelling non-linear trend present in the extreme rainfall time series and reported the significance of non-stationary IDF curves for designing infrastructures of these two cities. Though there are many studies which show non-stationarity in the daily rainfall of many parts of the worlds (Khaliq et al., 26; Agilan and Umamahesh, 25; Asadieh and Krakauer, 25; Mondal and Mujumdar, 25; Vasiliades et al., 25), since the non-stationarity in the rainfall IDF curves of Hyderabad city and Wilmington city is already detected and modelled; these two cities are considered as study area for studying the uncertainties in non-stationary rainfall IDF curves. For more information on geographical and climatological details of Hyderabad city and Wilmington city, the interested reader is referred to Agilan and Umamahesh (26). For this study, the hourly rainfall observed at the centre of the Hyderabad city by the India Meteorological Department is procured for the period of January 972 to 3 December 23. From the hourly observations, different duration (, 2, 3, 6, 2, 24 and 48 h) rainfall series are generated using the moving window approach and the annual maximum series is extracted from each duration rainfall time series. In the case of Wilmington city, the annual maximum rainfall series of -, 2-, 3-, 6-, 2-, 24- and 48-h duration rainfall is available through the United States National Oceanic and Atmospheric Administration (NOAA) Atlas 4 (Bonnin et al., 26) and it is downloaded from hdsc.nws.noaa.gov/hdsc/pfds/pfds_series.html (Accessed on January 26) for the available period of 53 years (948 2). Then, for developing non-stationary rainfall IDF curves, there is a need for covariates. Since it is irrelevant to use covariates which are not having any relationship with extreme rainfall process, it is essential to identify a set of potential covariates. Recently, Agilan and Umamahesh 27 Royal Meteorological Society Int. J. Climatol. (27)

3 COVARIATE AND PARAMETER UNCERTAINTY IN NON-STATIONARY RAINFALL IDF CURVE (25) detected and attributed the non-stationarity present in the intensity and frequency of Hyderabad city extreme rainfall. For attributing non-stationarity in the Hyderabad city extreme rainfall characteristics, they have used four physical processes which are having a significant influence on Hyderabad city extreme rainfall, namely El Niño-Southern Oscillation (ENSO) cycle, urbanization, global warming and local temperature changes. Therefore, they are selected as possible covariates for developing non-stationary rainfall IDF curves of Hyderabad city. In addition to these four physical processes, Indian Ocean Dipole (IOD) has a significant influence on extreme rainfall of India (Ajayamohan and Rao, 28) and the greenhouse warming is also increasing the frequency of extreme IOD events (Cai et al., 24). Therefore, it is also added to the potential covariates list. Besides these five physical processes, the weather variables, namely geopotential height (hgt), sea level pressure (slp), relative humidity (rhum), specific humidity (shum), zonal wind velocity (uwnd), meridional wind velocity (vwnd) and total perceptible water (pr_wtr) are the general choices in many studies of downscaling precipitation from global climate models (Raje and Mujumdar, 29; Najafi et al., 2; Mujumdar and Kumar, 22), because they are linked with precipitation occurrence. Hence, these weather variables are also considered as potential covariates. Further, total rainfall (t-rain), total monsoon (June September) rainfall (m-rain) and time are also considered as potential covariates for developing non-stationary rainfall IDF curves. In the case of Wilmington city, due to unavailability of data, the covariates urbanization, local temperature changes, t-rain and m-rain are not considered. In addition, IOD is not considered for modelling non-stationarity in the Wilmington city extreme rainfall because there was no evidence in the literature showing the relationship between IOD and rainfall of North America. However, based on the available literature, other teleconnections that are having a significant influence on rainfall occurrence over North America are considered for Wilmington city. In particular, recently Mallakpour and Villarini (26) showed the possible relationship between the frequency of flooding over the United States and teleconnections, namely the North Atlantic Oscillation (NAO), the Pacific Decadal Oscillation, the Atlantic Multi-decadal Oscillation (AMO) and the Pacific-North American pattern (PNA). In addition, there are many previous studies which show the relationship between these large-scale climate variabilities and rainfall over the United States (Henderson and Robinson, 994; McCabe and Dettinger, 999; Enfield et al., 2; Sheridan, 23; Durkee et al., 28). Hence, these four physical processes are also considered for modelling non-stationarity in Wilmington city extreme rainfall. Then, the data sets that represent the aforementioned physical processes/weather variables need to be identified. The HadCRUT4 yearly global temperature anomaly with respect to mean is used to represent global warming and this data set is based on average surface air temperature observations [ (accessed on 5 July 25)]. The ENSO cycle is normally represented using three indices, namely Multivariate ENSO Index (MEI), Southern Oscillation Index (SOI) and sea surface temperature (SST). As one of the objectives of this study is to evaluate the uncertainties in return level estimation due to the choice of the covariate, we have considered all three indices to represent the ENSO cycle. The monthly SST anomaly over NINO 3.4 (7 E 2 W, 5 S 5 N) region with respect to 98 2 mean is used as SST index and it is available at Timeseries/Data/nino34.long.anom.data (accessed on March 26). The SOI is a standardized index based on the observed sea level pressure differences between Tahiti and Darwin, Australia and it is available at (accessed on March 26). The MEI is based on six main observed variables over the tropical Pacific, i.e. sea level pressure, zonal and meridional components of the surface wind, SST, surface air temperature and total cloudiness fraction of the sky and it is available at mei.ext/table.ext.html (accessed on March 26). The monthly values of SST, MEI and SOI are converted into yearly values [i.e. average of November March (Mondal and Mujumdar, 25)] and used as covariates representing the ENSO cycle in a yearly basis. The monthly Dipole Mode Index (DMI) (Saji et al., 999) based on HadISST data set is downloaded from frcgc/research/d/iod/data/dmi.monthly.txt (accessed on 5 March 26) and yearly (i.e. averaged from June to November) DMI is calculated and used as a covariate representing IOD. For representing weather variables, namely hgt, slp, rhum, shum, uwnd, vwnd and pr_wtr, the NCEP reanalysis data sets are downloaded from (accessed on March 26) and interpolated for both the cities. Then, the standardized yearly average value of each variable is used as a covariate. The yearly PNA index is downloaded from data_sets/pna/pnandjfm.ascii (accessed on 5 March 26) and the yearly NAO index is downloaded from (accessed on 5 March 26). Monthly POD and AMO indices are downloaded from (accessed on 5 March 26) and (accessed on 5 March 26), respectively, and the yearly average values are used as covariates. To represent local temperature changes of the Hyderabad city, the hourly temperature observed at the centre of the Hyderabad city by the India Meteorological Department is procured for the period of January 972 to 3 December 23. Then, three indices, namely local yearly mean temperature anomaly (lta-mean), local yearly maximum temperature anomaly (lta-max) and local yearly minimum temperature anomaly (lta-min) are calculated and used as covariates. Note that all three temperature anomalies are based on mean value. The urban growth of Hyderabad city is represented by a data set which was 27 Royal Meteorological Society Int. J. Climatol. (27)

4 V. AGILAN AND N. V. UMAMAHESH Table. List of covariates used to analyse the uncertainties in non-stationary rainfall IDF curves. S. no. Covariate name Hyderabad gta Yes Yes 2 urban Yes No 3 lta-mean Yes No 4 lta-max Yes No 5 lta-min Yes No 6 mei Yes Yes 7 sst Yes Yes 8 soi Yes Yes 9 dmi Yes No hgt Yes Yes slp Yes Yes 2 uwnd Yes Yes 3 vwnd Yes Yes 4 rhum Yes Yes 5 shum Yes Yes 6 pr_wtr Yes Yes 7 t-rain Yes No 8 m-rain Yes No 9 time Yes Yes 2 nao No Yes 2 pna No Yes 22 pdo No Yes 23 amo No Yes Wilmington developed by Agilan and Umamahesh (25). In details, they created urbanization data set 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 (25). As used in previous studies (Cheng and AghaKouchak, 24; Yilmaz and Perera, 24; Agilan and Umamahesh, 26), the standardized year values are used as covariate which represents time. From the hourly rainfall observations of Hyderabad city, the yearly total rainfall and monsoon season total rainfall are calculated. Then, these values are standardized and used as covariates for modelling non-stationarity in the Hyderabad city extreme rainfall. In summary, the list of covariates used to analyse the uncertainties in non-stationary rainfall IDF curves are given in Table. The number of covariates used to model the non-stationarity in the Hyderabad city and the Wilmington city extreme rainfall are 9 and 6, respectively. 3. Non-stationary IDF curve As stated before, the rainfall IDF curve describes the intensity of extreme rainfall events for a given duration and return period and it is developed by fitting an appropriate theoretical probability distribution to different duration extreme rainfall time series. In this study, the annual maximum approach is used to derive the extreme rainfall time series. Let M = m, m 2,, m n be the annual maximum rainfall series of d-h duration rainfall and the values in the series are n independent and identically distributed (iid) random variables. The GEVD is the asymptotic distribution of annual maximum series and the cumulative distribution function (CDF), F, of GEVD is given by Equation () (Coles, 2; Katz et al., 22). F (m; α,, k) exp = exp { [ { [ exp + k(m α) (m α) ] k }, >, + k(m α) >, k ]}, >, k = where, α is the location parameter, is the scale parameter and it characterizes the spread of the distribution and k is the shape parameter and it characterizes the tail behaviour of the distribution. This model is to fit the distribution of annual maximum series when the parameters are constant. In another sense, the annual maximum rainfall series is a stationary series. Once the parameters of the model are estimated using any parameter estimation method (such as method of maximum likelihood), the estimated parameters are used to calculate the return level of the d-h duration rainfall for the given return period in terms of the probability value. Estimation of return level for a Q year return period is given by Equation (2) (Coles, 2). In other words, the intensity of d-h duration rainfall with a /Q probability of exceedance is described by Equation (2). [ ( ( )) ] α + k log k Q, k, Z Q = [ ( ( ))] α + log log, k =, Q (2) where, α, k and are estimated values of α, k and, respectively. The current practice assumes that the extreme rainfall series is a stationary series and the parameters of the GEVD will not vary with respect to time. In other words, the (probability density function) PDF/CDF shape of the GEVD is constant over time. However, as mentioned in Section, the global climate change is creating a non-stationary component in the extreme rainfall time series. Therefore, it violates the assumption that the PDF/CDF shape of the GEVD is constant over time. For example, if the location parameter of GEVD increases with time, the PDF shape will shift forward and it will significantly change the estimated return level. In specific, the return period of the same intensity (Z) extreme rainfall will reduce and the situations similar to this are already reported in the literature (Cheng and AghaKouchak, 24; Agilan and Umamahesh, 26). Towards modelling the non-stationarity in the extreme rainfall time series, researchers allowed the parameters of the GEVD to vary with time and the CDF of a non-stationary GEVD is given by () 27 Royal Meteorological Society Int. J. Climatol. (27)

5 COVARIATE AND PARAMETER UNCERTAINTY IN NON-STATIONARY RAINFALL IDF CURVE Equation (3). { [ ] exp + k t(m t α k } t) t, F ( t ) m t ; α t, t, k t = t >, + k t(m t α t) >, k t { [ t ]} exp exp (m t α t), t t >, k t = (3) where, the parameters α t, t and k t are linked with some covariate(s) which vary with time. The covariate can be simply the year values (time) or any physical process (e.g. global warming) or any weather variable (e.g. local temperature). Once the time-varying parameters are defined and they are linked with any covariate(s), then the non-stationary GEVD parameters can be estimated and substituted in Equation (4) for estimating non-stationary return levels. Unlike the stationary model, the non-stationary GEVD parameters will vary with time. Consequently, the return level for the given design probability also varies with time; therefore, the return level for the given design probability is a set of values. { } Z Q = z Q, zq 2,, zq n = [ ( ( ] α t + k t log t k t Q)), ( α t + t [ log log k t (, t ))] =,2,, n, Q k t =, t =,2,, n As stated earlier, the goal of this study is to evaluate the uncertainties in estimating rainfall return levels (Equation (4)) due to the choice of covariate and due to estimating non-stationary GEVD parameters and they are carried out in the subsequent sections. 4. Covariate uncertainty 4.. Significant covariates In a non-stationary GEVD, the time-varying component is introduced in the parameters by linking covariates with them. Though all three parameters of GEVD can be allowed to vary with time, since the shape parameter k cannot be estimated precisely and assuming it as a smooth function of time is unrealistic (Coles, 2), the shape parameter is not allowed to vary with time and only the location and scale parameters are permitted to vary with time. The general form of incorporating time-varying component in the location parameter of the GEVD using a covariate is given by Equations (5) and (6) gives the same for the scale parameter. α t = [ ] [ ] α c t (5) α t = exp { [ ct ] [ ]} (4) (6) where, c is a covariate; α and are intercepts; α and are slope parameters. The exponential function in Equation (6) ensures the positive values of the scale parameter. To develop non-stationary GEVD for each duration extreme rainfall series, we have formulated two types of non-stationarity setting () type-: the time-varying component is introduced only in the location parameter while the scale and shape parameters are kept constant over time and (2) type-2: both location and scale parameters are allowed to vary with time by keeping shape parameter as time invariant. With each covariate (listed in Section 2), two non-stationary GEVDs are defined according to two types of non-stationarity setting. Then the parameters of these non-stationary GEVDs are estimated using the method of maximum likelihood. If M = m, m 2,, m n is an annual maximum rainfall series extracted from n years of data, the log-likelihood derived from Equation (3) is given by Equations (7) (). For k and type-, log L ( α,α,,k ) = nlog ( + k) [ ( n mi ( ))] α + α c i log + k i= [ ( n mi ( ))] k α + α c i + k, i= ( mi ( )) α + α c i + k > (7) For k and type-2, log L ( α,α,,, k ) = n ( ) + c i ( + k) i= [ ( n mi ( ))] α + α c i log + k i= exp ( ) + c i [ ( n mi ( ))] k α + α c i + k i= exp ( ), + c i ( mi ( )) α + α c i + k exp ( ) > (8) + c i For k = and type-, log L ( α,α, ) = nlog ( n mi ( )) α + α c i i= [ ( n mi ( ))] α + α c i exp i= (9) 27 Royal Meteorological Society Int. J. Climatol. (27)

6 V. AGILAN AND N. V. UMAMAHESH For k = and type-2, log L ( ) n ( ) α,α,, = + c i ( n mi ( )) α + α c i i= exp ( ) + c i [ ( n mi ( ))] α + α c i exp i= exp ( ) () + c i The most likely set of parameters is estimated by maximizing any of the above log-likelihood, according to the type of non-stationarity setting. Note that the method of maximum likelihood may not be the correct choice when the sample size is less than 25 because it estimates physically infeasible shape parameter with a small sample (Katz et al., 22; Sugahara et al., 29; Cannon, 2). As stated earlier, 9 and 6 are the number of covariates which are used to model the non-stationarity in the Hyderabad city and the Wilmington extreme rainfall, respectively. Since two types of non-stationarity settings are applied with each covariate, the total number of non-stationary GEVD for each duration extreme rainfall series of Hyderabad city and Wilmington city are 38 and 32, respectively. In order to avoid selecting two models which are from the same covariate, first, the best model for each covariate is selected (among two models) based on the corrected Akaike Information Criterion value. The Akaike Information Criterion with small sample size correction (AICc) is generally used to select the best model among different candidate models (Strupczewski et al., 2; Agilan and Umamahesh, 26) and it is given by Equation (). 2p (p + ) AICc = 2logL + 2p + () n p where, log L is the minimized negative log-likelihood of GEVD, n is the sample size and p is the number of parameters in the model. Once the best model for each covariate is identified, the uncertainties in rainfall return levels due to the choice of covariate are analysed. Though the covariates are carefully chosen, they may not necessarily produce a non-stationary model which is significantly superior to the stationary model. Therefore, it is unrealistic to analyse the uncertainties in return levels with all non-stationary models (9 or 6) and it is essential to identify the non-stationary models (the covariates), which are significantly superior to the stationary model of the corresponding duration extreme rainfall series. The statistical significance of the non-stationary model is estimated by testing the statistical significance of the trend parameter in the non-stationary model using the likelihood ratio (LR) test. In the LR test, the negative log-likelihood of the non-stationary and stationary model is compared to test the null hypothesis of no trend in a parameter. Under the null hypothesis of no trend in a parameter, based on twice the difference between minimized negative log-likelihood of the stationary and non-stationary model, the LR test statistic approximately follows chi-square i= distribution with dp degree of freedom (Katz, 23) and it is given by Equation (2). 2 [( log L s ) ( log Lns )] χ 2 (dp) (2) where, log L s is the minimized negative log-likelihood of the stationary model, log L ns is the minimized negative log-likelihood of non-stationary model and dp is the difference between a number of parameters in the non-stationary model and the stationary model. In this study, the non-stationary models that have LR test p-value less than.5 are considered for analysing the uncertainties in return levels due to the choice of the covariate Non-stationary return levels As aforementioned, two non-stationary GEVDs are constructed using two types of non-stationarity setting and the best model for each covariate is selected for further analysis. The best model (type) for each covariate with -h duration annual maximum rainfall series of Hyderabad city and Wilmington city are listed in Table 2. Once the best model for each covariate is identified, the list of covariates that produced non-stationary GEVDs which are significantly superior to the stationary GEVD are identified using the LR test p-value. The list of covariates which produced a significantly superior non-stationary model for -h duration annual maximum rainfall series of Hyderabad city and Wilmington city are highlighted (bold) in Table 2. Upon identifying these covariates for all selected durations of Hyderabad and Wilmington city, parameters of non-stationary GEVDs which are based on the identified covariates are substituted into Equation (4) for estimating rainfall return levels of different design probability values. Since the parameters of non-stationary GEVD vary with covariate value, the return level for a given return period (design probability) is a set of values, instead of a single value. Therefore, the changes in rainfall return level due to the choice of covariate are presented in the form of return levels CDF. First, the return levels PDF is obtained using kernel density estimation method and the return levels CDFs obtained from different non-stationary GEVDs are analysed. The -year return levels CDFs obtained from different non-stationary GEVDs which are significantly superior to the stationary GEVD of the corresponding duration extreme rainfall are plotted in Figures and 2 for Hyderabad city and Wilmington city extreme rainfall series, respectively. From Figures and 2, it is observed that the number of non-stationary GEVDs which are significantly superior to the stationary GEVD is not same for all selected durations of Hyderabad city and Wilmington city rainfall. In the case of Hyderabad city, the number of non-stationary models which are superior to the stationary model is relatively more for long-duration rainfall series when compared to the short-duration rainfall series. On the other hand, the Wilmington city results show totally different behaviour, i.e. the number of non-stationary models which are superior to the stationary model is relatively more for short-duration rainfall series when compared with 27 Royal Meteorological Society Int. J. Climatol. (27)

7 COVARIATE AND PARAMETER UNCERTAINTY IN NON-STATIONARY RAINFALL IDF CURVE Table 2. The best model (type) for each covariate with -h duration annual maximum rainfall series. Hyderabad Wilmington Covariate Type LR test p-value ΔAICc Covariate Type LR test p-value ΔAICc urban gta dmi sst.66.3 lta_mean mei.67.5 lta_max soi lta_min nao gta pna sst pdo mei amo soi time time hgt t_rain pr_wtr m_rain rhum..22 hgt shum pr_wtr slp rhum uwnd shum vwnd slp uwnd vwnd ΔAICc is the difference between AICc of non-stationary and the stationary GEVD. the long-duration rainfall series. Further, the number of significantly superior non-stationary models for the Wilmington city extreme rainfall series are less when compared with the Hyderabad city and it may be due to the fewer number covariates used to model the non-stationarity in the Wilmington city extreme rainfall (i.e. Hyderabad city = 9 covariates; Wilmington city = 6 covariates). In addition, whenever a number of non-stationary models qualified for the particular extreme rainfall series, the uncertainty in return levels also appears to be more (refer 2-h duration plot in Figure and -h duration plot in Figure 2). Further, in order to obtain the overall picture, the percentage change between the minimum and maximum return levels of each percentile is calculated for all return periods. Then, the boxplots of these differences are plotted for all selected durations in Figure 3. Similar to return levels CDFs (Figures and 2), Figure 3 reveals that the covariate uncertainty is more in Hyderabad city long-duration rainfall series and Wilmington city short-duration rainfall series. Since only one covariate produced a significantly superior non-stationary model for Wilmington city 2-, 24- and 48-h duration extreme rainfall time series, there is no covariate uncertainty for those time series. It is worth to note that the covariate uncertainty results of Hyderabad city and Wilmington city showed totally opposite trend, i.e. the covariate uncertainty is more in Hyderabad city long-duration rainfall series while Wilmington city has more uncertainty in the short-duration rainfall series. Therefore, it is clear that the covariate uncertainty is not controlled by the duration of the extreme rainfall series. Further, in the direction of understanding reasons for more covariate uncertainties in Hyderabad city long-duration rainfall series and Wilmington city short-duration extreme rainfall series, the magnitude of trend present in Hyderabad and Wilmington city extreme rainfall series of all selected durations are investigated using the Mann Kendall (M K) trend test (Mann, 945; Kendall, 962). The M K test is a non-parametric statistical test which is normally used to detect trends in time series and it has a long tradition of use in hydrology and has been applied in the case of rainfall extremes (Villarini et al., 29; Cheng and AghaKouchak, 24; Yilmaz and Perera, 24). The M K test Tau value of Hyderabad and Wilmington city extreme rainfall series of selected durations are given in Table 3. The interpretation of M K test Tau value is similar to the correlation coefficient, i.e. the Tau value varies from to + and the positive Tau value indicates an increasing trend in the time series, while the negative Tau value indicates the counterpart. From Table 3, it is observed that the number of covariates which produce a significantly superior non-stationary model is more when the magnitude of trend in the extreme rainfall series is relatively high. Consequently, the covariate uncertainty is more when a number of models (covariates) are qualified as a significantly superior model. Further, the significance of covariate uncertainty is also checked using low-risk approach (Cheng et al., 24). In a non-stationary context, Cheng et al. (24) defined 95 percentile of return level values as effective return level and called it as a low-risk (more conservative) approach. In addition, this method is also used by Cheng and AghaKouchak (24) and Agilan and Umamahesh (26) for developing non-stationary rainfall IDF curves. Therefore, the difference in 95th percentile return levels that are estimated with different covariates is analysed. In the case of Hyderabad city 2-h duration extreme rainfall series, the 95th percentile of -year return level estimated by time covariate and t-rain covariate-based non-stationary models are.73 mm h and 4.76 mm h. There is around 4mmh difference in estimated values and, even for a small catchment, this 48 mm (2 4) extra rainfall will create a significant difference in peak flow. Furthermore, 27 Royal Meteorological Society Int. J. Climatol. (27)

8 V. AGILAN AND N. V. UMAMAHESH h lta-mean t-rain Intensity mm h h urban lta-mean lta-max gta time t-rain m-rain Intensity mm h h lta-mean t-rain m-rain Intensity mm h h lta-mean lta-max lta-min gta t-rain m-rain Intensity mm h h urban lta-mean lta-max lta-min sst mei time t-rain m-rain Intensity mm h h lta-min sst mei t-rain m-rain Intensity mm h 2-h lta-mean sst mei t-rain m-rain Intensity mm h Figure. The -year return levels CDFs of Hyderabad city extreme rainfall. [Colour figure can be viewed at wileyonlinelibrary.com]. it is worth to note that the -year return level calculated with covariate t-rain is higher than 25-year return level calculated with covariate time, i.e mm h. Therefore, it is evident that the choice of covariate will make a significant difference in estimating rainfall return levels and IDF relationship. 5. Parameter uncertainty From the previous section, it is observed that the choice of the covariate in modelling non-stationarity in the extreme rainfall series will create a significant difference in the return level estimation. At the same time, it should be 27 Royal Meteorological Society Int. J. Climatol. (27)

9 COVARIATE AND PARAMETER UNCERTAINTY IN NON-STATIONARY RAINFALL IDF CURVE -h 2-h gta pna pdo time uwnd gta pna pdo time rhum uwnd Intensity mm h Intensity mm h.9 3-h.9 6-h gta pna gta pna pdo time Intensity mm h Intensity mm h 2-h 24-h.9.8 pna.9.8 vwnd Intensity mm h Intensity mm h 48-h pr-wtr Intensity mm h Figure 2. The -year return levels CDFs of Wilmington city extreme rainfall. [Colour figure can be viewed at wileyonlinelibrary.com]. 27 Royal Meteorological Society Int. J. Climatol. (27)

10 V. AGILAN AND N. V. UMAMAHESH Percentage change in return level 5 5 Percentage change in return level Hyderabad -h 2-h 3-h 6-h 2-h 24-h 48-h Duration Wilmington -h 2-h 3-h 6-h 2-h 24-h 48-h Duration has been previously used to address the parameter uncertainties in rainfall IDF curves which were developed under both stationary (Huard et al., 2; Rupa et al., 25) and non-stationary (Cheng and AghaKouchak, 24) conditions. Hence, in this study, the parameter uncertainty is addressed using the Bayesian method. 5.. Bayesian method The fundamental principle of the Bayesian method is to use Bayes theorem to update one s uncertainty about parameter expressed in terms of prior, before inclusion of additional information provided by data (Hao et al., 25). Unlike classical methods, in the Bayesian approach, parameters are treated as random variables and the posterior distribution of parameters is obtained by multiplying the prior distribution of parameters and the likelihood function of GEVD obtained from the data. The posterior density of parameters given data, π(θ m), is given by Equation (3). π (θ m) L (m θ) π (θ) (3) For type- non-stationary GEVD, Figure 3. Uncertainties in non-stationary rainfall return levels due to the choice of the covariate. [Colour figure can be viewed at wileyonlinelibrary.com]. also noted that the analysis of extreme rainfall is often hampered by lack of sufficient data (Rupa et al., 25) and it has been already reported that the uncertainties in parameter estimates due to insufficient quantity and quality of data is the major source of uncertainty in the stationary IDF curves (Rupa et al., 25). Therefore, for the accurate estimate of return levels in a non-stationary context, it is essential to know the relative significance of the covariate uncertainty when compared to the parameter uncertainty. In this study, the covariate uncertainty is addressed using the classical parameter estimation method (i.e. the method of maximum likelihood). But, the method of maximum likelihood (or any classical parameter estimation method) provides only point estimates and does not account the uncertainties in estimated parameters. On the other hand, the Bayesian method provides a coherent framework for incorporating the uncertainties in parameter estimates (Coles et al., 23; Rupa et al., 25) and it θ = { α,α, and k } (4) For type-2 non-stationary GEVD, θ = { α,α,, and k } (5) where, L(m θ) is the likelihood function of GEVD and π(θ) is the prior distribution of parameters. Solving the Equation (3) and obtaining π(θ m) gives the Bayesian inference about the non-stationary GEVD parameters. However, there are two common problems that come with the Bayesian method: () the choice of prior distribution of parameters and (2) solving Equation (3), because it is not possible analytically. The parameters prior distributions used in this study are discussed in Section 5... Since the method of Markov Chain Monte Carlo (MCMC) offers an effective solution to the second problem, it is used and discussed in Section Prior distribution In the Bayesian method, the choice of prior distribution of parameters is vital and it is normally obtained Table 3. The M K test Tau value and covariate uncertainties for all selected duration extreme rainfall series. Duration Hyderabad city Wilmington city Tau No. of covariate Covariate uncertainty Tau No. of covariate Covariate uncertainty Mean Max. Mean Max. -h h h h h NA NA 24-h NA NA 48-h NA NA 27 Royal Meteorological Society Int. J. Climatol. (27)

11 COVARIATE AND PARAMETER UNCERTAINTY IN NON-STATIONARY RAINFALL IDF CURVE through a general understanding of the physical rainfall process, and a specific knowledge of the rainfall characteristics within the vicinity of the study area (Coles and Tawn, 996). When such information is not available for the study area, in order to avoid prior ignorance, arbitrary distributions with large variance or non-informative distributions are used as a prior distribution of parameters (Coles et al., 23; Renard et al., 23). As the specific prior is not available for the study areas, non-informative/informative prior distributions are chosen from the previous studies which were aiming at developing non-stationary/stationary rainfall IDF curves using the Bayesian method. In detail, for developing stationary rainfall IDF curves, Huard et al. (2) used non-informative uniform and Jeffreys distributions as prior for the location and scale parameters of GEVD, respectively. For the shape parameter, authors used informative beta distribution with hyperparameters a = 6andb = 9 as a prior distribution. This choice is originally recommended by Martins and Stedinger (2) and Martins and Stedinger (2) based on their extensive study of rainfall depths observed worldwide. Later, Rupa et al. (25) used the same priors (i.e. uniform, Jeffreys and beta) for understanding parameter uncertainties in the stationary rainfall IDF curves. In a non-stationary context, Cheng et al. (24) used non-informative normal distribution as prior for the location and scale parameters components (i.e. slope and intercept) and, as suggested by Renard et al. (23), they have used informative normal distribution (mean = and standard deviation =.3) as prior distribution for the shape parameter. Further, Cannon (2) argued that the beta distribution with hyperparameters a = 6andb = 9 produce very narrow pdf with a mode at. and nearly 9% of its probability mass concentrated over k values between.3 and +.. Therefore, the author suggested another set of hyperparameter values, i.e. a = 2andb = 3.3, which result a broader pdf with a mode at.2 and approximately 9% of its probability mass concentrated between.4 and +.2. For more information about shape parameter s prior distribution, the interested reader is referred to Cannon (2). In this study, non-informative normal distribution is used as a prior distribution for the location and scale parameters components. For the shape parameter, as suggested by Cannon (2), the beta distribution with hyperparameters a = 2 and b = 3.3 is used as a prior distribution and it is given by Equation (6). The prior distribution of parameter vector, π(θ), is given by Equations (7) and (8). π (k) = Beta (k +.5; 2, 3.3) (6) For type- non-stationary GEVD, 3 π (θ) = π (k) f ( θ j ; μ, σ 2) (7) j= For type-2 non-stationary GEVD, 4 π (θ) = π (k) f ( θ j ; μ, σ 2) (8) j= where, f ( ( ( θ j ; μ, σ 2) θj μ ) 2 ) = exp 2πσ 2 2σ 2 (9) MCMC method MCMC is a simulation technique and it is used to draw samples from a more complex approximate distribution, which otherwise is impractical by conventional techniques (Rupa et al., 25). As the posterior distribution (Equation (3)) is a complex high-dimensional joint density, it is not possible to draw samples from this distribution using conventional techniques. Therefore, the MCMC method is used to solve Equation (3). The basic aim of MCMC is to simulate values from the posterior distribution π(θ m). If this achieved correctly, the Bayesian inference about the GEVD parameters is obtained from the MCMC simulated samples (samples of parameter sets). In this study, one of the most promising MCMC algorithms, i.e. the Metropolis Hastings algorithm is used. In the original Metropolis Hastings algorithm, the choice of a proposal distribution is critical for the mixing properties of the resulting Markov chain and it is difficult to determine an optimal proposal distribution for the given target posterior distribution π(θ m). To alleviate this problem, in this study, adaptive random-walk Metropolis Hastings algorithm is used and it is explained in the following steps. Step-: The algorithm starts with an initial set of parameters values (θ t ). In this study, parameters values estimated using the method of maximum likelihood are used as initial values. Step-2: Next step is to generate proposal value (θ*) as follows: θ = θ t + ε Z (2) where, ε is a positive scale parameter and Z is multivariate normal with mean vector and variance-covariance matrix V. Step-3: Then, calculate the acceptance probability of proposed value (θ*) using Equation (2). α ( { } ) θ π (θ m) θ t = min π ( θ t m ), (2) The values π(θ* m) and π(θ t m) are obtained by multiplying priors and GEVD likelihoods of θ*and θ t, respectively. Step-4: Next, draw a random number (u) from the uniform distribution defined on [,]; and let {θ, u α ( ) θ θ θ t = t θ t, u >α ( ) θ (22) θ t Upon repeating the steps 2 4 for N number times (iterations), samples of parameter sets can be obtained. There are two issues with the generated sample. () The sample is influenced by the starting value and (2) the values of parameters are correlated because they are generated 27 Royal Meteorological Society Int. J. Climatol. (27)

12 V. AGILAN AND N. V. UMAMAHESH Table 4. The best covariate, MCMC chain ar value and parameter uncertainties for all selected duration extreme rainfall series. Duration Hyderabad city Wilmington city Best covariate ar Parameter uncertainty Best covariate ar Parameter uncertainty Mean Max. Mean Max. -h lta_mean time h t_rain time h m_rain time h m_rain pna h t_rain pna h t_rain vwnd h t_rain pr_wtr by a Markov process. Though the method of maximum likelihood is used to assign the starting value of parameters, in order to reduce the influence of starting value, the first part of the sample is discarded and this part of the sample is called burn-in period. In this study, samples of parameter sets are generated and the first samples are discarded. Since the values are generated by a Markov process, a low degree of autocorrelation is allowed. However, the high degree of autocorrelation creates a problem (i.e. mixing of Markov chain is slow). To avoid excessive autocorrelation, the thinning interval is used. The process thinning refers to drawing samples from regular interval from the MCMC generated samples, excluding burn-in period. In this study, the thinning interval of 3 is used. Since the Bayesian inference is based on the MCMC sample, it is essential to check the MCMC chain and the inference is valid only if the MCMC chain has converged. First, the MCMC generated samples are visually checked using trace plot (the iteration number vs the simulated values for the parameter) and autocorrelation plot. In addition, the efficiency of MCMC chain is checked with acceptance rate. The ratio between a number of accepted proposal values (Step-2) and the total number of iterations is called the acceptance rate (ar). Too small (near to ) ar indicates that the MCMC chain failed to explore regions of appreciable posterior probability and too high ar (near to ) means that the MCMC chain stays in a small region and fails to explore the whole posterior domain (Gelman et al., 997). The MCMC chain which has neither too small nor too large ar and also has small autocorrelation is considered as an efficient MCMC chain Bayesian inference For the same duration extreme rainfall series, many covariates may produce a non-stationary model which is significantly superior to the stationary model of the corresponding duration (e.g. five covariates produced a significantly superior non-stationary model for Wilmington city -h duration extreme rainfall series). But, it is unrealistic to analyse the parameter uncertainty with all covariates which produced significantly superior non-stationary model. Therefore, the best covariate for each duration extreme rainfall series of Hyderabad and Wilmington city is first identified and the parameter uncertainty is calculated only for the best covariate-based non-stationary model. Among many non-stationary models which are significantly superior to the stationary model, the model which has the lowest AICc value is identified as the best model and the covariate which produced that particular model is referred as the best covariate for the corresponding duration extreme rainfall series. The best covariate for all selected duration extreme rainfall series of Hyderabad city and Wilmington city is given in Table 4 and the Bayesian inferences are obtained for the best covariates-based non-stationary GEVDs. As aforementioned, the Bayesian inference is valid only if the MCMC chain has converged. Therefore, it is essential to inspect MCMC chain of all selected durations. The trace plots and autocorrelation plots of Hyderabad city 6-h duration rainfall series non-stationary GEVD parameters are shown in Figure 4. The red portion in trace plot indicates a burn-in period of the MCMC chain. Further, the ar of MCMC chain for all selected duration extreme rainfall series of Hyderabad and Wilmington city is given in Table 4. The trace plot of a parameter that comes from a well-mixed MCMC chain should traverse the posterior domain rapidly and should have a nearly constant mean and variance. The autocorrelation of a parameter that comes from a well-mixed MCMC chain should become negligible fairly quickly. The trace plots in Figure 4 (top panel) show that the parameter values traverse the posterior domain rapidly with nearly constant mean and variance. The autocorrelation plots in Figure 4 (bottom panel) show that the autocorrelation of MCMC simulated values of every parameter decreases very quickly. Further, Roberts et al. (997) reported that.234 is the asymptotically optimal ar of a random-walk Metropolis Hastings algorithm. Recently, Hao et al. (25) reported that the ar of a well-mixed MCMC chain lies between.2 and.5. The ar value of 6-h duration Hyderabad city extreme rainfall series MCMC chain is.27 and it indicates that the MCMC chain of each parameter achieved a good mixing. The trace plots, autocorrelation plots and ar value indicate that the MCMC chain of 6-h duration Hyderabad city extreme rainfall series has converged. Similarly, from the trace plots, autocorrelation plots and ar values, it is observed the MCMC chains of remaining duration extreme rainfall series of Hyderabad and Wilmington city have converged. In the interest of brevity, the trace plots 27 Royal Meteorological Society Int. J. Climatol. (27)