Journal of Hydrology 521 (2015) 217–231

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Journal of Hydrology

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Modeling non-stationarity in intensity, duration and frequencyof extreme rainfall over India

http://dx.doi.org/10.1016/j.jhydrol.2014.11.0710022-1694/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: Department of Civil Engineering, Indian Institute ofScience, Bangalore 560012, India. Tel.: +91 80 2293 2669; fax: +91 80 2360 0290.

E-mail address: pradeep@civil.iisc.ernet.in (P.P. Mujumdar).

Arpita Mondal a, P.P. Mujumdar a,b,⇑a Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, Indiab Divecha Center for Climate Change, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o s u m m a r y

Article history:Received 26 May 2014Received in revised form 22 September2014Accepted 24 November 2014Available online 10 December 2014This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of Uwe Haberlandt, AssociateEditor

Keywords:Extreme rainfallNon-stationarityExtreme Value TheoryNatural variabilityGlobal warmingLocal temperature changes

Significant changes are reported in extreme rainfall characteristics over India in recent studies thoughthere are disagreements on the spatial uniformity and causes of trends. Based on recent theoreticaladvancements in the Extreme Value Theory (EVT), we analyze changes in extreme rainfall characteristicsover India using a high-resolution daily gridded (1� latitude � 1� longitude) dataset. Intensity, durationand frequency of excess rain over a high threshold in the summer monsoon season are modeled bynon-stationary distributions whose parameters vary with physical covariates like the El-Nino SouthernOscillation index (ENSO-index) which is an indicator of large-scale natural variability, global averagetemperature which is an indicator of human-induced global warming and local mean temperatureswhich possibly indicate more localized changes. Each non-stationary model considers one physical covar-iate and the best chosen statistical model at each rainfall grid gives the most significant physical driverfor each extreme rainfall characteristic at that grid. Intensity, duration and frequency of extreme rainfallexhibit non-stationarity due to different drivers and no spatially uniform pattern is observed in thechanges in them across the country. At most of the locations, duration of extreme rainfall spells is foundto be stationary, while non-stationary associations between intensity and frequency and local changes intemperature are detected at a large number of locations. This study presents the first application of non-stationary statistical modeling of intensity, duration and frequency of extreme rainfall over India. Thedeveloped models are further used for rainfall frequency analysis to show changes in the 100-yearextreme rainfall event. Our findings indicate the varying nature of each extreme rainfall characteristicand their drivers and emphasize the necessity of a comprehensive framework to assess resulting risksof precipitation induced flooding.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Extreme rainfall over India has been of interest to researchersover the past few years. Analyzing patterns of change in extremerainfall over India is challenging because of the variability associ-ated with Indian rainfall (Krishnamurthy and Shukla, 2000). More-over, extreme events are rare by their very definition, making itdifficult to analyze or model them (Katz, 2013). At the same time,global climate change is expected to intensify the hydrologicalcycle leading to an increase in hydroclimatic extremes (IPCC,2012). Since such extremes significantly affect societies (IPCC,2012), it becomes necessary to understand and analyze changesin them.

Different studies provide different, often conflicting, conclusionsabout the nature of changes in rainfall extremes over India. Forexample, Goswami et al. (2006) concluded that extreme rainfallmagnitude and frequency is increasing over a large region markedas ‘central India’ using a fixed threshold to define the extreme.Rajeevan et al. (2008) also reached similar inferences using past100 years’ data. On the other hand, increasing spatial variabilityin extreme rainfall was reported by Ghosh et al. (2011) who usedthe theoretical basis of EVT and emphasized the importance of afine-resolution analysis. This study was followed by Vittal et al.(2013) who showed that the post-1950 trends are significantly dif-ferent from the pre-1950 trends. Kulkarni et al. (2012) showed thatconsideration of spatial correlation reduces the significance of thedetected trends in extreme rainfall events. Kishtawal et al. (2010)hypothesized the possible role of urbanization in changes inextreme rainfall. Vittal et al. (2013) used a change point detectiontechnique and established that urbanization, in terms of change

218 A. Mondal, P.P. Mujumdar / Journal of Hydrology 521 (2015) 217–231

in population density, is a possible cause of change in extreme rain-fall. While Ghosh et al. (2011) defined extreme rainfall as the grid-specific annual maximum, Vittal et al. (2013) used the more com-prehensive peak-over-threshold (POT) approach to include multi-ple extreme events in a year. In both these studies, stationaryextreme value distributions were fitted to a 30-year moving-win-dow to compute trends assuming that the climate is shifting to anew stationary regime.

Changes in climate extremes are often reported to be theoreti-cally more robustly detectable than changes in mean (Hegerl et al.,2004; Min et al., 2009). Globally, the model-simulated signal ofhuman-induced climate change is detected in observed intensifica-tion of rainfall extremes over the Northern Hemisphere land areasby formal fingerprint-based detection and attribution analyses(Min et al., 2011; Zhang et al., 2013). The unique advantages ofnon-stationary extreme value distributions can be applied to studytransient extremes both to enhance the rigor and to make the anal-ysis more physically meaningful (Katz et al., 2002; Gilleland andKatz, 2011). Non-stationary extensions of the EVT have onlyrecently been implemented to analyze changes in hydroclimaticprocesses, particularly those influenced by rapid climate change(Katz et al., 2002; Katz, 2010; Kharin and Zwiers, 2005; Brownet al., 2008; Towler et al., 2010; Sillmann et al., 2011; Kharinet al., 2013).

Detection of trends in hydrologic extremes using the non-sta-tionary Generalized Extreme Value (GEV) distribution for the blockmaximum is attempted by some recent studies (Westra et al.,2013; Clarke, 2013; Westra and Sisson, 2011; Zwiers et al.,2011). In particular, Westra et al. (2013) use non-stationary GEVto study the scaling relationship between annual maximum rain-fall and global average temperature at the global scale and reportincreasing trends at about two-thirds of the stations. For extremerainfall over India at a fine spatial resolution, we extend thisapproach to the detection and analysis of changes in the intensity,duration and frequency of POT rainfall because of each of the threephysically-based covariates – namely, global average surface airtemperature which has been established to contain discernible sig-nals of anthropogenic climate change (Hegerl et al., 2007), theENSO-index which represents large-scale natural climate variabil-ity and local temperature changes which are possible indicators oflocalized effects.

Although the fundamental probabilistic theory of extreme val-ues has been long developed, its applications for statistical model-ing of extremes have been of interest only in the recent years (Katzet al., 2002). Though there have been studies analyzing changes inextreme rainfall over India (Goswami et al., 2006; Rajeevan et al.,2008; Dash et al., 2009; Kulkarni et al., 2012). Ghosh et al. (2011)and Vittal et al. (2013) are the only two studies which have usedEVT to statistically model extreme rainfall over India using ablock-maximum and a peak-over-threshold approach, respec-tively. Our study is the first one to apply non-stationary extensionsto the EVT models for the statistical modeling of extreme rainfallover India. Here, we use recent theoretical developments (Katzet al., 2002; Katz, 2010, 2013) in the EVT for detection and model-ing of changes in the intensity, duration and frequency of extremerainfall spells due to non-stationary effects of one or morecovariates.

Significance of the association between these physical covari-ates and extreme rainfall characteristics (intensity, duration andfrequency) is tested using appropriate statistical tests. At each gridpoint, for intensity, duration or frequency of extreme rainfall, thecovariate in the chosen best non-stationary statistical model repre-sents the most significant physical driver. A field significance test isalso used to assess whether the fraction of grid points with non-stationary effects in extreme rainfall intensity is beyond the rangeexpected from random variability. The developed models are

further used for rainfall frequency analysis to show changes inthe 100-year extreme rainfall event.

The remainder of the paper is organized as follows: Section 2describes the data, justification of choice of covariates and providesthe definition of extreme rainfall spells. Section 3 elaborates themodels and discusses the analyses of strength of the associationsbetween extreme rainfall characteristics and their possible drivers.Section 4 provides a summary of the findings with a mention of thelimitations and possible extensions of this work.

2. Data

2.1. Extreme rainfall data

High-resolution gridded (1� latitude � 1� longitude) daily rain-fall data is available for India for 1901–2004 (Rajeevan et al.,2008). This dataset is prepared from quality-controlled observedrainfall data from more than 1800 gauges and is increasingly beingused in studies on the Indian monsoon (Mani et al., 2009; Neenaand Goswami, 2010; Kulkarni et al., 2012; Vittal et al., 2013).High-resolution (1� latitude � 1� longitude) observed daily meantemperature is also available for India for 1969–2009 (Srivastavaet al., 2009). The signals of climate change are reported to be visi-ble only toward the end of the twentieth century and majorchanges in urbanization in India came in around 1971–1981(Crook and Dyson, 1982; Mohan and Pant, 1982). These factorsmay affect extreme rainfall characteristics. For this reason, we con-sider 1969–2004 for our analysis as it is also the time period forwhich both observed high-resolution rainfall and local mean tem-perature, which is considered as a covariate, are available. Dailydata for the summer monsoon months only (June, July, Augustand September) are considered. The summer monsoon seasonbrings most of the annual rainfall in India, resulting in non-zerorainfall values on many days. Moreover, the threshold at each gridpoint can be reasonably assumed to be constant for these fourmonths. At each location, 90th percentile rainfall is chosen as thethreshold for getting the excesses or the POT series. The meanexcess plot (Coles, 2001) has been suggested to be a more technicaland objective way for threshold selection. In this study, instead ofcomputing the mean excess plot for each of the 349 grid points, astandard definition of threshold is chosen, following guidelines ofKatz (2013) who prescribe that the threshold should be highenough for the values to be extreme, but not so high as to retainonly a few grid points in the analysis. Vittal et al. (2013) choosethe 95th percentile rainfall as the threshold in their POT analysisof extreme rainfall over India, following studies on other parts ofthe world (Haylock and Nicholls, 2000; Khan et al., 2007). In thisstudy, the 90th percentile of rainfall has been chosen as a thresholdfollowing other studies on POT rainfall analysis (Begueria et al.,2011; Gemmer et al., 2011; Groisman et al., 2005) as changingthe threshold to 95th percentile resulted in loss of a few datapoints for some grids, reducing the sample size for parameterestimation.

Similar to Vittal et al. (2013) who used the same definition ofPOT as Furrer et al. (2010), we define an extreme rainfall spell(cluster) as consecutive days when rainfall is higher than thethreshold. The spells are separated from each other by one or moredays when rainfall is lower than the threshold. To address tempo-ral dependence within each extreme rainfall spell, some form ofdeclustering is necessary (Coles, 2001). We decluster the POT rain-fall series by retaining only the maximum rainfall in each spell(cluster maximum) and we call it the intensity of extreme rainfall,similar to Vittal et al. (2013). Frequency denotes the number ofextreme rainfall spells, while duration of the extreme rainfall spell

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is the cluster length, that is, the number of consecutive days whenrainfall is above the threshold.

Although it is argued in some studies that global warmingaffects thermodynamic conditions only on a large scale(Goswami et al., 2006), local water resources decision makingrequires rainfall to be analyzed at finer scales (Ghosh et al.,2009). Analysis of extreme rainfall over India within the ExtremeValue Theory (EVT) revealed increasing spatial variability ofextreme rainfall over India, thereby emphasizing the necessity ofa fine-resolution analysis (Ghosh et al., 2011; Vittal et al., 2013).This justifies the use of a fine-resolution rainfall data for our anal-ysis. Moreover, averaging extremes such as annual maximum orpeak rainfall over large areas will alter the tail behavior ofextremes.

For each grid point, the extreme rainfall days in each year arefirst identified as the days when rainfall exceeds the threshold atthat grid point. The sum of the rainfall on all extreme rainfall daysfor each year is thereafter calculated for each grid point. The leastsquares linear trend of this yearly series is plotted grid-wise inFig. 1(a). The total extreme rainfall in a year has increasing trendsat many grid points pointing toward the existing non-stationaritiesin peak-over-threshold extreme rainfall, as can be observed inFig. 1(a). This pattern is consistent with that reported by Vittalet al. (2013) for trends in volume of extreme rainfall over Indiafor the post-1950 period. In Fig. 1(a), at about 39% of the gridpoints, the increase in total annual extreme rainfall volume is sig-nificant at 90% statistical confidence as inferred from the non-para-metric Mann–Kendall trend test. These non-stationarities arefurther explored in the subsequent sections, by fitting non-station-ary distributions to the intensity, duration and frequency of declu-stered daily extreme rainfall (POT series) at individual grids. Thespatial variability of extreme rainfall over India is shown in termsof the coefficient of variation (Cv) values of the declustered POTseries at each grid point in Fig. 1(b). Both the patterns of trendsin total annual extreme rainfall volume and Cv of the declusteredPOT rainfall series show wide variations from one grid to anotherpointing toward large spatial variability in the extremes in Indianmonsoon rainfall. This lack of uniformity makes detection of trendsin extreme rainfall a difficult problem for the Indian region. Thetrends in average duration and frequency of extreme rainfall spellsare shown in Fig. 2. For each grid point, the number of extremerainfall spells in each year is calculated. The trend of this yearlyseries is the trend in frequency of extreme rainfall spells at that

Fig. 1. Changes in extreme rainfall intensity. (a) Trends (mm/year) in total extreme rathreshold) show that about 39% of the grids have increasing trends which are significant(b) The coefficient of variation of the declustered extreme rainfall POT series points tow

grid point. Similarly, for each grid point, the average duration ofextreme rainfall spells in each year is calculated as the ratio ofthe total number of extreme rainfall days to the number of extremerainfall spells in that year. The trend of this yearly average durationseries is denoted as the trend in average duration at that grid point.This figure also shows the spatial non-uniformity in the extremerainfall characteristics. It is thus not unexpected that differentgrids may have different significant physical drivers without anycoherent spatial patterns. The statistical models fitted to theextreme rainfall characteristics allow the parameters to vary asfunctions of the physical covariates instead of simple time depen-dence such that not just the linear trend, but the non-linear varia-tions are also captured.

2.2. Data for covariates

The El-Nino Southern Oscillation affects extreme precipitationat local and regional scales significantly in some parts of the world(Kenyon and Hegerl, 2010; Zhang et al., 2010; Min et al., 2011)(Supplementary Information of Min et al., 2011). The effects ofENSO on heavy precipitation events are studied in various regionssuch as the United States (Gershunov and Barnett, 1998; Schubertet al., 2008), Southern India (Revadekar and Kulkarni, 2008) andSouth America (Grimm and Tedeschi, 2009). Though the influenceof ENSO on extreme precipitation at lower scales such as at indi-vidual sites has not been extensively studied (Shang et al., 2011).El Adlouni et al. (2007) established ENSO influences on return lev-els of extreme precipitation at a station in California. Zhang et al.(2010) has also fitted non-stationary extreme value distributionsto precipitation block-maxima at several individual sites in NorthAmerica using ENSO as a covariate. Using EVT, Katz et al. (2002)showed the influence of ENSO on peak streamflow in a river basin– another example of a local response, emphasizing ‘‘downscaling’’from global to local scales. Thus, the choice of the ENSO-index as acovariate for fine-resolution extreme rainfall characteristics isjustified.

Monthly Sea Surface Temperature (SST) anomaly series withrespect to 1981–2010 mean, over the NINO3.4 region (17�E–120�W, 5�S–5�N) (served by NOAA, available at: http://www.cpc.ncep.noaa.gov/data/indices/) is used to compute theENSO-index. ENSO is a large-scale phenomenon resulting from nat-ural internal variability of climate. NINO3.4 SST anomalies havebeen used as an ENSO-index in earlier studies on changes in mean

infall volume (sum of rainfall on all the days in a year when rainfall exceeds theat 90% statistical confidence, as inferred by the non-parametric Mann–Kendall test.ard the spatial non-uniformity across the country.

Fig. 2. Trends in average duration and frequency of extreme rainfall spells. The average duration and total number of extreme rainfall spells in a year are first computed andtheir trends in terms of numbers per 10 years are plotted.

220 A. Mondal, P.P. Mujumdar / Journal of Hydrology 521 (2015) 217–231

Indian rainfall (Gershunov et al., 2001; Maity and Nagesh Kumar,2006, 2008). Revadekar and Kulkarni (2008) show that there arestrong correlations between the intensity and frequency of heavyrainfall events in winter in Southern India and NINO3.4 SST anom-alies 4–6 months in advance, while no such relationship could beobserved for wet or dry spell lengths. Several ENSO-indices havebeen used as a covariate for extreme precipitation in recent litera-ture; for example, El Adlouni et al. (2007) used the Southern Oscil-lation Index (SOI) derived from monthly sea level pressureanomalies at Tahiti and Darwin, Katz et al. (2002) used SOI-aver-aged over the fall season as a covariate for peak streamflow, whilein the two-component Poisson model for hurricane damage, Katz(2002) considered an index derived from NINO3.4 SST anomalies,which takes the values of �1, 0 and +1 for La-Nina, normal andEl-Nino years respectively. Kenyon and Hegerl (2010) report thatthe effect of ENSO on extreme precipitation is the strongest in win-ter; following this study, Min et al. (2011) (Supplementary Infor-mation) used SOI index averaged over the winter season(November to March) as a covariate for gridded observed annualmaximum rainfall over the Northern Hemisphere land areas.Zhang et al. (2010) used SOI averaged over the winter season andalso showed that the average SST anomalies over the NINO3.4region yield similar results. We choose the NINO3.4 SST anomaliesas the ENSO-index following earlier studies which investigate theeffects on ENSO on the Indian monsoon (Gershunov et al., 2001;Maity and Nagesh Kumar, 2006; 2008; Revadekar and Kulkarni,2008).

We choose the approach of Min et al. (2011; SupplementaryInformation) and consider the average November to MarchNINO3.4 SST anomalies as a covariate representing ENSO for theintensity, duration and frequency of extreme rainfall over India.Additionally, we have also followed the approach of Katz (2002)and considered as covariate an ENSO index which takes values of�1, 0 and +1 respectively for La-Nina, normal and El-Nino years,which are taken from Kothawale et al. (2008). The second approachdoes not change our conclusions and hence is excluded subse-quently. The nature of the covariate representing natural variabil-ity in both the cases is periodic, without pronounced trends, unlikeglobal climate change (which has a marked increasing trend) orlocal temperature changes (which have increasing trends at severallocations). If any other indicator of large-scale variability of climatehas similar periodicities and no trends, it intuitively follows thatthe association between that covariate and the extreme rainfallcharacteristic is not likely to get altered in a major way.

Can global warming influence fine-resolution extreme precipi-tation characteristics? Atmospheric moisture content, the principaldriver of extreme precipitation (Trenberth et al., 2003), is reportedto scale with temperature following the Clausius–Clapeyron rela-tionship, with the possible exceptions of drier land regions wherethe scaling is lower (O’Gorman and Muller, 2010; Simmons et al.,2010). Thus, it is expected that extreme precipitation intensifieswith global warming in most regions across the globe (Westraet al., 2013). Based on climate model simulations, the intensity ofextreme precipitation is projected to increase under global warm-ing even in regions where mean precipitation decreases (Kharinand Zwiers, 2000, 2005), though local extreme precipitation sensi-tivity may deviate from the expected rate (Kharin et al., 2013).Fowler and Wilby (2010) computed detection times for griddedextreme precipitation in the UK assuming that regional precipita-tion changes in proportion to projected change in global meantemperature. Another example of possible local response to globalwarming can be observed in the work of Westra et al. (2013) whohave also considered global average temperature as a covariate forstudying changes in extreme rainfall at each of the 11,391 observ-ing stations across the globe. They argue that specific humidityglobally has risen in accordance with global warming, and henceglobal mean temperature can serve as a proxy for total precipitablewater that can cause extreme precipitation. Additionally, followingTrenberth (2011, 2012), they also mention that the physical mech-anisms linking local temperatures with precipitation may not bethe same as those linking global warming to extreme precipitationchanges. Hence, local and global temperatures are chosen sepa-rately as covariates while modeling intensity, duration and fre-quency of extreme rainfall in our study.

HadCRUT4 monthly observed global average surface air tem-perature anomaly series (tas) with respect to the 1961–1990 mean(served by the UK Met Office Hadley Centre, available at: http://www.metoffice.gov.uk/hadobs/hadcrut4/) is used as a covariateas an indicator of global warming. Global average temperature isthe most robust indicator of human-induced climate change withthe strongest attribution of causes (Hegerl et al., 2007). Addition-ally, monthly mean temperature anomaly series (tmean) basedon the 1969–2009 mean at each grid point is considered as thethird covariate. Local temperatures may be influenced by otherlocalized processes in addition to natural climate variability orgreenhouse gas-induced climate change. Only those rainfall gridpoints are retained in the analysis where local mean temperatureis also available to avoid additional uncertainty due to any

A. Mondal, P.P. Mujumdar / Journal of Hydrology 521 (2015) 217–231 221

interpolation. Thus, 349 grids are considered in the subsequentanalysis.

The physical covariates are plotted in Figs. 3 and 4. In Fig. 3, itcan be observed that both the ENSO-index and the global averagesurface air temperature have distinctive individual periodicities,with the global average surface air temperature showing a definitepositive trend toward the end of the twentieth century. Fig. 4shows the trends in the local mean temperature in the monsoonmonths at each grid. For each grid point, the average monthly tem-perature for the monsoon months in each year is first calculated asthe mean of temperature (mean temperature on a day, not themaximum or the minimum temperature on the day) on all daysof the months June, July, August and September for that year.The average of these four temperature values per year gives theaverage monsoon local mean temperature for that grid. The trendof this yearly average monsoon local mean temperature is plottedgrid-wise in Fig. 4(a). The correlations between local mean temper-ature and the monthly NINO3.4 SST anomalies (b) or global meantemperature (c) are also shown. The trends in local mean temper-atures are increasing at many grid points. There could also be otherforms of non-stationarity in the local mean temperatures such asintra or inter-annual cycles not shown in this figure. The localmean temperature is not found to be significantly correlated withthe monthly NINO3.4 anomalies or global average temperature atmost of the grid points. Thus, it is appropriate to consider theeffects of each of these covariates separately for obtaining informa-tion on the most significant driver for each extreme rainfallcharacteristic.

3. Non-stationary models for extreme rainfall characteristics

The threshold-exceedance approach of the EVT has two compo-nents – the Generalized Pareto (GP) distribution which is theapproximate distribution for excesses over a high threshold anda one-dimensional Poisson process that describes the rate ofexceedance. The Point Process approach (Katz, 2013), which mod-els the threshold exceedances and the excesses simultaneously as atwo-dimensional non-homogeneous Poisson process can also beused for the declustered monsoon POT rainfall series. We use themore common Poisson-GP approach (Katz, 2013) to model theintensity and frequency of extreme rainfall separately. Addition-ally, the length of an extreme rainfall spell is approximated bythe Geometric distribution. Intensity, duration and frequency ofextreme rainfall are modeled separately since one of our aims isto test whether there is change in each of these characteristics

Fig. 3. (a) November to March average NINO3.4 SST anomalies used as the ENSO

and what is the physical covariate responsible for the change.The packages ‘extRemes’ (http://cran.r-project.org/web/packages/extRemes/) and ‘MASS’ (http://cran.r-project.org/web/packages/MASS/index.html) for the open source statistical programming lan-guage ‘R’ are used for obtaining the models.

Non-stationary statistical models are fitted to each of the inten-sity, duration and frequency of the declustered POT series such thata statistical model considers linear effects of only one covariate.Appropriate statistical tests are then conducted to obtain the bestmodels at each grid for each extreme rainfall characteristic. Thecovariate in the best chosen model represents the most significantphysical driver of the extreme rainfall characteristic at that gridpoint. Could random variability have resulted in the fraction of gridpoints where significant non-stationarity exists in extreme rainfallintensity due to each covariate? We explore this question througha field significance test using resampling. The following sectionsdescribe the EVT-based models, the statistical tests and their out-comes in details.

3.1. Modeling changes in extreme rainfall intensity

If the monsoon daily rainfall, denoted by {X1, X2, . . ., Xn}, is iidwith common cumulative distribution function (cdf) F, and n isthe total number of days, then, for a sufficiently high threshold u,the excesses Yi = Xi � u, conditional on Xi > u, follow a GP distribu-tion whose cdf is given by (Coles, 2001):

H½y; rðuÞ; n� ¼ 1� 1þ ny

rðuÞ

� ��1=n

; y > 0; 1þ ny

rðuÞ > 0 ð1Þ

where r(u) > 0 and n are the scale and the shape parameters respec-tively. Positive, zero and negative values of the shape parameterresult in the heavy-tailed, unbounded light-tailed and boundedfinite tail extreme value distributions, respectively. The scaleparameter r(u) is a function of the chosen threshold u. As men-tioned before, grid-wise 90th percentile monsoon daily rainfall ateach grid is chosen as the threshold in this study.

In the non-stationary setting, the parameters are expressed as afunction of covariates in the general form

logrðu; iÞ ¼ r0 þ r1VnðiÞ ¼ n

ð2Þ

where i denotes day and V represents any covariate. The shapeparameter is kept constant since precise estimation of this parame-ter is difficult and it is unrealistic to assume that it is a smooth

-index; (b) monthly global average surface air temperature (tas) anomalies.

Fig. 4. (a) Trends in annual average local mean temperature in the monsoon months (shown in degree C per 10 years). At about 80% of the grid points, the trends areincreasing. Correlation coefficients between monthly local mean temperature (tmean) and (b) monthly NINO3.4 SST anomalies and (c) the global average surface airtemperature (tas) for the monsoon months are also shown. In (b) and (c) only those grids are shown where the correlation coefficient is statistically significant at 95%confidence level. Local monsoon monthly mean temperatures are significantly correlated to the ENSO-index and tas at only about 12% and 25% grid points respectively.

222 A. Mondal, P.P. Mujumdar / Journal of Hydrology 521 (2015) 217–231

function of time (Coles, 2001). V can be zero (stationary model), orthe ENSO-index (November to March average NINO3.4 SST anoma-lies), tas, or tmean for the month corresponding to i. The logarithmin Eq. (2) is taken to ensure positive values of the scale parameter.The slope parameter r1 represents the trend due to effects of covar-iate V and our aim is to test the significance of this parameter. Addi-tionally, another model of the form log rðu; iÞ ¼ r0 þ r1 ENSOþr2 tasþ r3 tmean is also considered to test for significance ofsimultaneous effects of all the three covariates. At no grid pointall the three parameters – r1, r2 and r3 are found to be significant.Keeping this in view and that our goal is to find out significant indi-vidual physical factors that influence rainfall extremes over India,for brevity, we do not test whether a combination of any twocovariates is better than any one individual covariate. Along withnon-stationary statistical modeling of extreme rainfall, one of theprimary aims of this study is to assess which physical driver signif-icantly affects each of the extreme rainfall characteristics – inten-sity, duration and frequency. A linear dependence model issufficient in this regard as the significance of each of the covariatescan be tested through this model for each grid point and for eachextreme rainfall characteristic. Linear dependence of the parame-ters on the covariates has been considered in several studies onnon-stationary extreme value modeling of hydrometerological vari-ables (Katz et al., 2002; El Adlouni et al., 2007; Towler et al., 2010;Sillmann et al., 2011; Katz, 2013; Kharin and Zwiers, 2005; Burkeet al., 2010; Coles, 2001; Zhang et al., 2010). The shape parameteris kept constant in all these studies as well. Trends in u are alsonot considered for computational simplicity in obtaining returnlevels. The method of maximum likelihood is used to estimate theparameters of the GP distribution since it can be easily extendedto the non-stationary case (Katz, 2013). If h is the pdf of the GP dis-tribution, i.e. the derivative of H in Eq. (1) with respect to y, and b isthe vector of the parameters such that b = (r0, r1, n), then the like-lihood function is given by

LðbÞ ¼Yk

i¼1

hðyi; bÞ ð3Þ

where k is the total number of exceedances and yi is the excess rain-fall on day i. Minimization of negative log-likelihood (nllh) (Katz,2013) through numerical techniques is often adopted (Coles,2001) to arrive at the estimates of r0, r1, and n.

The significance of the slope parameter r1 can be tested by thelikelihood ratio test for large samples (Coles, 2001). The nullhypothesis of no-trend is tested by comparing nllh for the twomodels – with and without trend (Katz, 2013). For example, for

extreme rainfall intensity, if the two models are denoted by (1)and (2), and

nllhð1Þ ¼ � log Lðr; nÞnllhð2Þ ¼ � log Lðr0;r1; nÞ

ð4Þ

then, under the null hypothesis of r1 = 0, the test statistic, based onthe difference between nllh(1) and nllh(2) approximately followschi-squared distribution with degree of freedom = differencebetween number of parameters in models (1) and (2) = 1. Thus,the test statistic follows a Chi-square distribution with 1 degreeof freedom and is given by:

2½nllhð1Þ � nllhð2Þ� � v2ð1Þ ð5Þ

For multiple candidate models, we use one of the popular crite-ria – the Akaike Information Criteria (AIC) which penalizes theminimized nllh for the number of parameters estimated (Katz,2013). The best statistical model is the one that minimizes (Katz,2013)

AICðpÞ ¼ 2nllhðpÞ þ 2p ð6Þ

where p is the number of parameters estimated. Thus, for theextreme rainfall intensity, the most significant covariate is theone whose corresponding non-stationary model minimizes theAIC. Whether the non-stationary effects (increasing or decreasing)of that particular covariate are indeed significant is further testedby the likelihood ratio test against the null hypothesis of stationarydistribution described above.

The grid-wise best statistical models chosen by the AIC forintensity of rainfall are shown in Fig. 5. It can be observed thatthe most significant physical driver is different from one grid pointto another. This is because of the large spatial variability inobserved extreme rainfall over India that is discussed earlier. Forintensity of extreme rainfall, the physical covariate that is signifi-cant at the highest number (38%) of locations is the local meantemperature, though there is no uniform spatial pattern in its influ-ence. The stationary model is the best model for extreme rainfallintensity at only about 31% of the grids implying that the tradi-tional assumption of stationarity in assessing the risk of extremerainfall events are not valid at 69% of the locations (240 gridpoints). One of the methods to test whether the chosen modelsare appropriate is the residual probability or the quantile plots(Coles, 2001). For obtaining the quantile plots in the non-stationarycase, it is necessary to transform the data into a standardized form(Coles, 2001). Standard exponential distribution is used in thisstudy. For brevity, all the 240 probability or the quantile plots

Fig. 5. Grid-wise best statistical models for extreme rainfall intensity: (a) spatial patterns (b) percentage of locations falling under each category of models.

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are not shown in this paper. All the calculations and the diagnosticchecks for one example grid location are given toward the end ofthis section. We noted that the probability and the quantile plotsshow a reasonably good match between the observed andmodel-predicted quantiles at all the ‘non-stationary’ locations.The goodness of fit of the chosen models is further tested by thelikelihood ratio test at these 240 grid points. The null hypothesisof a stationary GP distribution is tested against an alternativeGP-distribution of Eq. (2) where the parameter is a linear functionof the grid-specific covariate from the best chosen model. Thus, thelikelihood ratio test evaluates the null hypothesis of r1 = 0 againstthe alternative that it is not zero. At 202 out of the 240 ‘non-sta-tionary’ grids, this null hypothesis can be rejected at a statisticalconfidence of 90% (p-value < 0.1). At 80% confidence, this nullhypothesis can be rejected at all the 240 grid points. This showsthat the chosen non-stationary models are acceptable.

3.2. Modeling changes in extreme rainfall duration

The duration of the extreme rainfall spell is modeled by a Geo-metric distribution which is an adequate and simple model forspell length that allows easy introduction of trends (Furrer et al.,2010). The probability mass function of the distribution is givenby (Furrer et al., 2010):

PðlÞ ¼ ð1� hÞl�1h l ¼ 1;2; . . . ð7Þ

with the mean being the reciprocal of the parameter h. The param-eter h can be estimated by the method of moments which is equiv-alent to the method of maximum likelihood for this distribution(Furrer et al., 2010). In the case of extreme rainfall spells, h = h(u)which is a function of the chosen threshold u. Using a GeneralizedLinear Model (GLM) framework, trend is introduced into h such that

hðu; iÞ ¼ h0 þ h1 ENSOþ h2 tasþ h3 tmean ð8Þ

The GLM uses a Negative Binomial family for this distribution.For the duration of the extreme rainfall spell, the significance ofeach of the parameters h0, h1, h2 and h3 is assessed by conductingtests on them individually in the GLM framework to evaluate thenull hypothesis that each of them is zero. It is found that the inter-cept parameter h0 is significant at all the locations. The most signif-icant trend parameter out of h1, h2 and h3 determines the mostsignificant physical factor causing the trend. If none of the trendparameters are significant at a very high confidence level, the dura-tion of the extreme rainfall spells is reported to be stationary. Fig. 6shows the most significant drivers for the duration of extremerainfall spell. Most locations are found to be stationary in termsof extreme rainfall duration, unlike the pattern in intensities.

3.3. Modeling changes in extreme rainfall frequency

Following the Law of Small Numbers, threshold exceedancescan be approximated by a one-dimensional Poisson process(Katz, 2013) with the rate parameter k(u) > 0 which depends onthe chosen threshold u. Within time T, the expected number ofexceedances is approximately Poisson with mean k(u)T.

Trend is introduced into the parameter k(u) through the GLMframework. Thus, the non-stationary Poisson rate parameter isgiven by

log kðu; iÞ ¼ k0 þ k1 ENSOþ k2 tasþ k3 tmean ð9Þ

The significance of each of the parameters k0, k1, k2 and k3 isassessed in the GLM by the p-values of the tests conducted individ-ually on each of them to test the null hypothesis that it is zero. It isfound that the intercept parameter k0 is significant at all the

Fig. 6. Grid-wise best statistical models for extreme rainfall duration: (a) spatial patterns (b) percentage of locations falling under each category of models.

Fig. 7. Grid-wise best statistical models for extreme rainfall frequency: (a) spatial patterns (b) percentage of locations falling under each category of models.

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Fig. 9. The 100-year extreme rainfall intensity return levels (RLs) (a) observed maximumreturn level (mm) with stationary Poisson-GP model for rainfall intensity at all locations(including both stationary and non-stationary Poisson-GP models); for a randomly chostationary models lead to varying return levels due to significant effects of the corresponlevel in 2004 (d) is greater than that in 1969 (c), though it is to be noted that there could bwhich are not shown here. The stationary 100-year RLs (b) more or less follows the spa

Fig. 8. Distribution (histograms) of the percentage of locations at which rainfall isnon-stationary due to (a) ENSO, (b) global warming and (c) local temperaturechanges from the 1000 resampled data matrices for the field significance test for theintensity of rainfall. The observed percentage of locations at which trends inextreme rainfall intensity due to each of these covariates are significant, is shownby the red dot. For each of (a), (b) and (c), for each percentage of locations, ‘Count’represents the number of times out of the 1000 samples when that particularpercentage of locations where non-stationarity due to the corresponding covariateis significant. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

A. Mondal, P.P. Mujumdar / Journal of Hydrology 521 (2015) 217–231 225

locations. The most significant trend parameter other than k0

hence determines the most significant physical factor causing thetrend. If none of the trend parameters (k1, k2 and k3) is significantat a high confidence (say, level of significance = 0.05), then the rateparameter is stationary. Fig. 7 shows the significant physical driv-ers of non-stationarity in the frequency of extreme rainfall spellsover India. It is found that the frequency of extreme rainfall spellsis mostly stationary or is significantly associated with local meantemperatures.

The global-scale variables – ENSO and tas are thus noted to havesome effects on extreme rainfall intensity, but no noteworthyeffects on the frequency or duration. At many locations, tmean isnoted to play a role in the intensity and frequency. Also, it is foundthat the intensity, duration and frequency of extreme rainfall overIndia behave differently from each other. Spatial non-uniformity inthe patterns of changes in these extreme rainfall characteristics isobserved. It is to be noted that the chosen statistical models allowthe trends in extreme rainfall characteristics due to each of thecovariates to be either increasing or decreasing depending on thenature of observed changes. Also, periodicities or non-linear varia-tions in the covariates are automatically incorporated in the modelin terms of the covariates though the model itself is linear; that is,the model allows the non-linear variations in the covariates affectthe changes in the parameters of the distributions. The estimatedreturn levels of extreme rainfall will thus have the same natureof variation as the most significant covariate. This is further shownlater for one example grid point where the estimated 100-yearextreme rainfall intensity is shown to have non-linear variations.However, our aim is to find out the significant factors influencing

extreme daily rainfall intensity (mm) in the time period 1969–2009; (b) 100-year; 100-year return level (mm) with the chosen best statistical model at each locationsen day in July in 1969 (c), and the same day in July in 2004 (d). The chosen non-ding covariate. At about 29% of the locations, the magnitude of the 100-year returne other forms of non-stationarity induced due to the covariates (such as seasonality)tial variability exhibited by the maximum observed intensity (a).

Fig. 10. (a) Observed declustered extreme rainfall intensities and 100-year return levels at 10.5�N 78.5�E; Physical covariates considered for this location on the extremerainfall (cluster maximum) days: (b) the ENSO-index, (c) global average surface air temperature anomalies and (d) local mean temperature anomalies. If a stationary GPdistribution is fitted to the declustered extreme rainfall intensities, a fixed return level (horizontal line) is obtained (a). On the other hand, the non-stationary Poisson-GPmodel leads to transient return levels as shown (‘+’ markers in (a)).

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each extreme rainfall characteristic at a fine resolution over Indiaand not to arrive at empirical relationships between these factorsand rainfall, although that can be achieved by an extension of thisstudy.

3.4. Field significance test

To assess whether the non-stationarities in extreme rainfallintensities are due to random variability, we use a resampling-based field significance test that preserves the spatial dependencein the observed data. Spatio-temporal modeling and detection oftrends in extreme rainfall can be performed using max-stable pro-cesses (Westra and Sisson, 2011); however, that approach is com-putationally expensive (Westra et al., 2013). We use a fieldsignificance test technique used by Westra et al. (2013) to evaluatewhether the test statistic is significant under the null hypothesis.The test statistic chosen is the percentage of locations where a par-ticular covariate is the most significant. A resampled data matrix isobtained from the original declustered daily extreme rainfall inten-sity data matrix by randomly drawing, with replacement, the samenumber of rows as that of the original data, where rows represent

grid locations and columns represent time. This is repeated 1000times to get a distribution of percentage of locations where a par-ticular covariate is most significant. The observed percentage oflocations where that particular covariate is significant is comparedwith the distribution so obtained. In the field significance test, wedo not account for temporal autocorrelation by use of a movingblocks bootstrap approach (Westra et al., 2013) as the block sizeof one (no autocorrelation) is noted to be sufficient by Westraet al. (2013) for extreme rainfall. We therefore conduct the conven-tional spatial bootstrap technique described above.

The field significance test results for extreme rainfall intensityare shown in Fig. 8. It is found that the observed percentage oflocations (red dot) where the effects of local mean temperatureare most significant (Fig. 8c), is most unlikely to be associated withrandom variability. This implies that the changes in extreme rain-fall that are induced by more local processes, especially at the gridswhere local temperatures are not correlated with tas or ENSO, arethose beyond the range expected from natural variability of cli-mate alone. Although in the field significance test, we do notaccount for temporal dependence (Westra et al., 2013) of extremerainfalls, since the intensity represents a declustered extreme

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rainfall and each spell is separated from the other by definition,independence of extreme rainfall spells is not an unrealisticassumption.

3.5. Estimation of return levels

For the Poisson-GP model described above, the m-observationreturn level of one-day rainfall that is exceeded on average onceevery m observations, conditional on ym > u, is given by (Coles,2001):

ym ¼uþ r

n ½ðmkðuÞÞn � 1� for n – 0

uþ r logðmkðuÞÞ for n ¼ 0

(ð10Þ

where k(u) = Pr{Y > u} is the rate of exceedance. For the daily mon-soon extreme rainfall, there are 122 (number of days in JJAS = 122)rainfall values per year, so the 100-year return level would corre-spond to m = 100 � 122. In the non-stationary setting, ym represents‘effective return level’ (Katz, 2013) that varies with each day ofextreme (denoted by i) is given by

ymðiÞ ¼uþ rðu;iÞ

n ½ðmkðu; iÞÞn � 1� for n – 0

uþ rðu; iÞ logðmkðu; iÞÞ for n ¼ 0

(ð11Þ

Here, the rate of exceedance is estimated by the non-stationaryPoisson rate parameter in Eq. (9). The transient return levels at eachgrid are computed from the best statistical model for that grid. Thenature of changes in the return levels will be according to thechanges in the covariate in the selected model. For example, ifthe selected best model is one with local tmean as the covariatewhich is increasing and has periodicities, the transient return levelsare also increasing with periodicities. We can assess whether thereis long-term trend in the return levels for any grid by comparing theeffective return level on a particular day toward the beginning ofthe time period and that on the same day toward the end of thetime period.

The maximum observed daily rainfall intensity (maximumintensity out of all the extreme rainfall spells for 1969–2004) isshown in Fig. 9(a). If a stationary Poisson-GP model is fitted to

Table 1Details of the statistical models for the extreme rainfall characteristics at the chosen grid peach extreme rainfall characteristic is shown in bold letters. SE = standard error of estimathypothesis of no trend. Minimum AIC gives the best model for intensity. For frequency acovariates.

Models AIC Likelihoo

Model for extreme rainfall intensityModel (1) 2137.78 –Y � GP(r, n)Model (2) 2137.97 0.79Y � GP(exp(r0 + r1 ENSO), n)Model (3) 2139.36 0.52Y � GP(exp(r0 + r1 tas), n)Model (4) 2135.54 0.03Y � GP(exp(r0 + r1 tmean), n)Model (5) 2139.54 0.94Y � GP(exp(r0 + r1 ENSO + r2 tas + r3 tmean), n)

Parameter Estimate SEModel for frequency of extreme rainfall spellsFrequency � Poisson(exp(k0 + k1 ENSO + k2 tas + k3 tmean))k0 �1.87 0.11k1 �0.043 0.06k2 �0.371 0.33k3 �0.644 0.08

Model for duration of extreme rainfall spellsDuration � geometric(h0 + h1 ENSO + h2 tas + h3 tmeanh0 3.29 0.07h1 �0.028 0. 04h2 �0.103 0.22h3 �0.098 0.05

POT series at all the grids, a constant value of 100-year return levelwill be obtained at each grid, as shown in Fig. 9(b). The maximumobserved daily rainfall intensity is mostly found to be smaller thanthe 100-year return level assuming stationary models at each gridpoint. At 22% of the grid points, though, the maximum observedintensity is larger than the 100-year return level which means thatthe 100-year extreme rainfall event is exceeded at least once in theobserved record at these grid points.

The chosen best non-stationary Poisson-GP model at the ‘non-stationary’ locations leads to changes in the 100-year return levelwith time. Thus, the transient 100-year return levels will haveone value corresponding to each declustered POT rainfall intensity.The changes in 100-year return level from the starting year 1969 tothe ending year 2004, is shown in Fig. 9(c) and (d), reflectingchanges due to the corresponding most significant covariate. Atabout 29% of the grid point locations, the 100-year return levelextreme rainfall on a random day in July, for example, in 2004 isgreater than that on the same day of July in 1969, as reflected bythe non-stationary Poisson-GP model. If annual maxima were usedinstead of the POT series, the 100-year return level would changeevery year. Increases in extreme rainfall return levels are criticalas they may imply greater risk of pluvial flooding. Also, the non-stationarity is shown only in terms of increase or decrease in the100-year return level from Fig. 9(c) to (d). There could be season-alities in the covariates which may induce seasonal changes inthe return levels which are not shown in this figure. However,the models do account for those changes in terms of changes inthe values of the corresponding covariates.

Estimation of confidence intervals for the transient return levelsin the non-stationary case can be carried out by the Delta method(Oehlert, 1992) involving derivatives of the quantile estimator withrespect to the parameters of the statistical model. The Deltamethod also has implicit assumptions of normality of the esti-mated quantiles (Cooley, 2013). Other methods such as the profilelikelihood method or the Bayesian method have rarely been usedfor the non-stationary threshold exceedance models and can betaken up in future studies. Nonetheless, estimation of confidenceintervals for the transient return levels for each grid point is not

oint 10.5�N 78.5�E. Y denotes the daily declustered POT rainfall. The chosen model fore. The likelihood ratio test is conducted for extreme rainfall intensity to test the nullnd duration, the GLM p-value (<0.05) for each parameter determines the significant

d ratio test p-value for the trend parameter Parameter estimates (SE)

16.30, �0.045(1.31, 0.05)2.79, 0.013, �0.044(0.08, 0.053, 0.056)2.81, �0.2, �0.05(0.09, 0.31, 0.05)3.01, �0.19, �0.06(0.13, 0.09, 0.06)3.002, �0.03, �0.18, �0.06(0.13, 0.33, 0.09, 0.06)

p-value

<2e�160.4820.2674.8e�14

<2e�160.440.630.064

Fig. 11. Diagnostic plots of the stationary GP model fitted to the declustered extreme rainfall intensities. The empirical and model quantiles in the quantile plot are in mm.Visually, the stationary GP appears to have captured the quantiles of the extreme rainfall intensities (except very high quantiles); however, the likelihood ratio testestablishes that non-stationary changes in the scale parameter with local mean temperature as the covariate are significant.

Fig. 12. Residual probability and quantile plots for the fitted non-stationary Poisson-GP model with local mean temperature as the covariate. Standardization to theexponential distribution is required for obtaining the diagnostic plots. The empirical and model-derived probabilities and quantiles show a good match with each other,pointing out that the chosen model is appropriate. Similar plots are obtained for all of the 240 ‘non-stationary’ grid points.

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carried out further since our major aim is to investigate the natureof variability of the extreme rainfall characteristics and their signif-icant physical drivers from the appropriate statistical models.

3.6. Example calculations for one grid point

As an example, all the computations are shown below for a par-ticular grid point. We consider a grid point at 10.5�N 78.5�E in theSouthern Peninsular India. The declustered extreme rainfall inten-sities are shown by ‘o’ markers in Fig. 10(a). The covariates corre-sponding to the extreme rainfall intensities are also plotted inFig. 10(b)–(d). The details of the statistical models for the intensity,duration and frequency of extreme rainfall for this grid point aregiven in Table 1. The standard error (SE) of estimate of each param-eter is obtained by MLE, which can be used to compute confidenceintervals on each parameter, assuming the parameters to be nor-mally distributed. The SE of each parameter estimate is also pro-vided in Table 1 (the SE of the parameter estimates are obtainedfor all the 349 grid points which are not provided in the manu-script for brevity). For this example grid point, for intensity and fre-quency, the best statistical model is the one that considers localmean temperature as the covariate. The duration is found to be sta-tionary. The diagnostic plots of the stationary GP distribution fittedto the declustered extreme rainfall intensities are shown in Fig. 11.A stationary GP model leads to a constant 100-year return level ofextreme rainfall, shown by the horizontal line in Fig. 10(a).However, the trend parameter due to local mean temperaturechanges is found to be significant against the null hypothesis of astationary GP at 95% confidence (p-value = 0.03). Therefore, the

non-stationary Poisson-GP model with tmean as the covariate forthe scale parameter is the more correct model for the extremerainfall intensity at this grid point. This is also established by theresidual probability and quantile plots from the non-stationaryPoisson-GP model, given in Fig. 12. The empirical quantiles andprobabilities show a good match with those predicted by themodel (standardized to exponential distribution). The transientreturn levels from the non-stationary Poisson-GP model are shownby ‘+’ markers in Fig. 10(a). These return levels follow the patternof changes in tmean shown in Fig. 10(d). Though the 100-yearreturn levels do not have a marked increasing/decreasing lineartrend, there is a considerable periodicity which renders consider-ation of a constant scale parameter incorrect. It is thus establishedthat the non-stationary effects of local temperature changes aresignificant for extreme rainfall characteristics at this grid point.Similar analysis is done for all the 240 ‘non-stationary’ grid points,while the stationary models are retained at the remaininglocations.

4. Summary and conclusions

We analyze changes in daily extreme rainfall in India asexcesses over grid-specific thresholds and examine the causes ofchanges using non-stationary extensions of the EVT. For each ofthe extreme rainfall characteristics, viz. intensity, duration andfrequency, at a fine resolution, the most significant physical driversare obtained by testing for the significance of their associations interms of trends in the parameters of the non-stationary statisticaldistributions. Three physical factors or covariates are considered as

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possible causes of changes in extreme rainfall – ENSO representinglarge-scale natural climate variability, global average surface airtemperature representing global climate change and local temper-ature change representing more localized processes. Trends due tothese covariates are detected based on the best statistical modelrepresenting effects of the corresponding covariate, which are alsoshown to be significant by a likelihood ratio test. Intensity, dura-tion and frequency are found to exhibit different non-stationarybehaviors. Large spatial variability of the associations betweenextreme rainfall characteristics and the physical covariates are alsoreported. Through a field significance test, we show that randomvariability is very unlikely to have lead to the observed percentageof locations where significant non-stationarity, particularly thatdue to local temperature changes, exists in rainfall intensities.Local temperature changes have been found to be more signifi-cantly affecting the extreme rainfall characteristics, particularlythe intensity and frequency of extreme rainfall spells, as comparedto the global processes.

Extreme precipitation changes are expected to occur in accor-dance with the Clausius–Clapeyron relationship in regions whereatmospheric circulation is expected to alter only slightly(O’Gorman and Muller, 2010). Increase in water vapor adds on tothe greenhouse effect thereby providing a positive feedback to cli-mate change thereby providing more moisture to the storms(Trenberth, 2011). Increasing land-sea contrast and more moistureover the Indian Ocean can cause increase in extreme precipitationover India. However, it is to be noted that the scaling associationbetween extreme rainfall and increasing temperatures are differ-ent for land and ocean and may become weaker or noisier underweak greenhouse gas forcings, such as that observed in the secondhalf of the twentieth century (Min et al., 2011, SupplementaryInformation). In particular, for the Indian monsoon, global warm-ing is shown to decrease the predictability of monsoon weather(Turner and Slingo, 2009).

Changes in atmospheric and ocean circulations are also knownto affect monsoon rainfall in the Indian region. For example, theimpact of ENSO can be observed as a suppression of convectionin the Indian Ocean and the Indian region in the El-Nino phaseand an enhancement of convection in the La-Nina phase (Gadgilet al., 2004). Earlier in the 80s, a strong link between the IndianSummer Monsoon Rainfall (ISMR) and ENSO was established withincreasing droughts in the El-Nino or warm phase and excess rain-fall in the La-Nina phase of this oscillation (Rasmusson andCarpenter, 1983). Later, however, it was suggested that the linkbetween ISMR and ENSO had weakened (Kumar et al., 1999) andthat deficits/excesses occur in the absence of El-Nino or La-Nina.It is to be noted, though, that all of these studies consider the aver-age monsoon rainfall over the country and fine-scale effects maybe different.

At regional or local scales, increasing aerosols may causechanges in shortwave forcing modifying the energy balance inthe atmospheric column and alter microphysical properties thatmay affect precipitation. Local convective instabilities causeshort-duration extreme rainfall in a moist atmosphere. Synopticdisturbances trigger heavy rainfall events which can align aroundtracks of monsoon lows and depressions, though a significant frac-tion of rainfall coming from large thunderstorms may be spatiallyuniform (Goswami et al., 2006). High variability in rainfall overIndia makes it difficult to observe a homogeneous large-scaleenvironment and it is very difficult to detect changes in extremeprecipitation at very fine resolutions (Goswami et al., 2006). Fineresolution extreme rainfall may be affected by local featureslike topography, vegetation or non-uniform population growth orother local processes that influence precipitation (Ghosh et al.,2011). Such processes are likely to differ from one grid to another(100 km) and hence, the behavior of extreme rainfall

characteristics is also expected to change. Ghosh et al. (2011;Fig. 2) and Vittal et al. (2013; Fig. 2) also show that the trendsin return levels and the trends in intensity, duration and fre-quency of extreme rainfall over India vary from one grid point toanother. Thus, the more significant influences of local tempera-tures in the fine-resolution extreme rainfall characteristicsand the general spatial non-uniformity across the country interms of the behavior of extreme rainfall characteristics andtheir significant physical drivers, as reported in our study, are notunexpected.

It can also be noted that there exist alternative definitions ofextreme rainfall. We have performed a fine-resolution analysisusing the standard definition of POT (Furrer et al., 2010; Katz,2013; Vittal et al., 2013) and choosing appropriate family of distri-butions, thereby ensuring that our results are reasonably reliable.Nonetheless, the non-stationary extension to the EVT has its ownlimitations; for example, the choice of the threshold is subjectiveand the method of maximum likelihood estimation (MLE) maynot be the best one for estimating the parameters of the distribu-tions. However, MLE is presently the most used method since itallows easy introduction of trends in the parameters (Katz,2013). In the stationary situation, a comparison between thesemethods can be easily carried out (Martins and Stedinger, 2000,2001); however, in the non-stationary setting such a comparisonmay not be possible since not all methods can be extended to con-sider the presence of covariates in the first place (for example,method of moments, probability-weighted moments and L-moments cannot be easily extended to consider the effects ofcovariates).

It may be noted that 40-years’ data may raise some concernsabout the uncertainty in the parameters of the distributions fittedto each of intensity, duration and frequency of extreme rainfall. Weare constrained by data availability since the gridded local meantemperature, which is considered as a covariate in our analysisand is found to be significantly associated with the extreme rainfallcharacteristics, is only available from 1969. In MLE, numerical opti-mization schemes (such as the Newton–Raphson method; thepackage used in this study also allows more sophisticated heuristicsearch methods such as simulated annealing) are used to arrive atthe parameter estimates. The MLE method may diverge only whenthe sample size is small out (Martins and Stedinger, 2000, 2001; ElAdlouni et al., 2007). For example, problems in maximum likeli-hood parameter estimation arise only with small samples625 years for annual maxima (Hosking et al., 1985; Supplementaryinformation of Min et al., 2011). In particular, absurd or physicallyinfeasible values of the shape parameter may arise only in smallsamples (El Adlouni et al., 2007; Martins and Stedinger, 2000),and in the threshold-exceedance model, Martins and Stedinger(2001) point out that suitable alternatives to MLE, such as the Gen-eralized Maximum Likelihood Method (GMLE), can lead to betterestimates of the shape parameter for small or moderate samplesizes, also depending on the range of shape parameter values,whether two or three-parameter partial duration series model isconsidered and also on the Poisson rate of occurrence. We considerPOT rainfall series for 40 years at each grid point, and there aremultiple rainfall spells within each year; our sample size, for eachgrid point, is thus much larger than 40 (ranging up to 100–200 ormore, depending on grids). Thus, the numerical problems of smallsample size or those resulting in absurd values of the shape param-eter are not expected in our study. It is to be also noted that GMLEconsiders a similar optimization of the likelihood function as MLE,except that it also integrates prior information on the shapeparameter which is subjective and may vary for hydroclimatologicvariables across regions. Though the GMLE has recently beenextended to the non-stationary setting for the block maxima rain-fall at a station (El Adlouni et al., 2007), such a formulation for the

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non-stationary threshold-exceedance approach has not adopted inany earlier study on hydrometeorological applications.

Large spatial variability of the Indian monsoon rainfall leads tohighly non-uniform and noisy patterns in the non-stationarybehavior. A regional trend detection technique (Panthou et al.,2013) can be employed to improve the signal to noise ratio in afuture study. Also, other physical covariates can be considered tostudy the changes in extreme rainfall characteristics. The field sig-nificance test can be further modified to account for both spatialand temporal dependence. However, our aim is to build non-stationary statistical models for each of intensity, duration and fre-quency of fine-resolution extreme rainfall over India using the EVT,which was not done before. Moreover, such additional analyses areunlikely to broadly alter our major conclusions about reporting themost significant physical drivers for each of the extreme rainfallcharacteristics.

The non-stationarities in the extreme rainfall characteristics atmany locations across India may affect the hydrologic cycle in away that restoration to its steady state might not happen naturally.This necessitates improved measures to deal with the changingrisk of extreme rainfall events. Modified risk assessment tech-niques such as that of Rootzén and Katz (2013) that are tailor-made for the non-stationary extremes can be used in hydrologicdesigns as part of adaptive measures.

Acknowledgments

The authors thank Rick Katz, Eric Gilleland, Chris Paciorek andP.N. Vinayachandran for helpful clarifications. They also thankthe associate editor and two anonymous reviewers whose valuablecomments helped to improve the manuscript significantly.

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