## 1. Introduction

Anthropogenic climate change poses potentially significant risks for the Indian Subcontinent through changes in extreme rainfall characteristics. General circulation models (GCMs) of climate have had only limited success in reproducing the key attributes of the intraseasonal and interannual variations in the Indian monsoon. Consequently, it is not clear whether GCM simulations forced with the Intergovernmental Panel on Climate Change (IPCC)-style anthropogenic change scenarios adequately represent changes in Indian rainfall extremes, especially for extreme rainfall that translates into floods or for multiday dry periods that impact crop yield. Recently, a few papers (Guhathakurta and Rajeevan 2008; Goswami et al. 2006) have investigated the trends in selected extreme rainfall attributes from a daily rainfall dataset that has become available through the Indian Meteorological Department (IMD). Such analyses provide a useful backdrop for assessing whether forced GCM simulations, such as those by May (2004), Kumar et al. (2006), among others, provide plausible scenarios for changes in extreme rainfall in the twenty-first century.

This paper presents an exploratory, spatially distributed analysis of the nature of monotonic trends in selected statistics of daily rainfall across India. The research presented differs from recent work on the issue (Goswami et al. 2006; Guhathakurta and Rajeevan 2008; Joshi and Rajeevan 2006; Rajeevan et al. 2008; Alexander et al. 2006; Klein Tank et al. 2006; Kumar et al. 2006; May 2004) in the specific statistics (frequency and intensity) of extremes considered, in the use of a nonparametric monotonic trend analysis (instead of a linear trend analysis, which is nonrobust to outliers, a concern in analyzing data on extremes), and in analyzing the complete spatially distributed dataset instead of an aggregate region. Further, we use methods for field significance analysis of spatial trends in each statistic and also document the concordance of trends across variables. Trends in the frequency with which daily rainfall exceeds selected thresholds, as well as in the intensity (magnitude of such rainfall events) are considered at each grid cell over India. The rainfall amount considered to define the exceedance events corresponds to fixed percentiles of the long-term rainfall data at that grid cell; hence, the threshold magnitude varies from grid cell to grid cell. Thus, changes in the local climatology of extremes, rather than the rate of occurrence of a fixed extreme magnitude across a region or even the entire country, are explored.

Further, analyzing trends in frequency and intensity separately is of interest since it is possible that the number of extreme events could increase without a corresponding increase in the intensity of each event (Trenberth 1999), and each measure provides information regarding different aspects of extreme rainfall. For instance, rainfall-indexed insurance is being introduced by several organizations in India (Gine et al. 2007) and the determination of a fair premium, and the associated payout structure, requires an assessment of whether the upper tail of the probability distribution of daily rainfall is changing at the specific location where contracts are likely to be written. The need to inform these and similar applications motivates our spatially distributed analysis of trends in the exceedance of specific percentiles of the local distribution of daily rainfall.

In the monsoonal setting (Indian or Asian monsoon), there have been a few studies focused exclusively on trends in extreme rainfall and most of these have been based on greenhouse-gas-forced model-based scenarios of the IPCC for the twenty-first century (Lal et al. 2000; Bhaskaran and Mitchell 1998; May 2004; Kumar et al. 2006). Keeping in mind the biases in the models (indicated in Kumar et al. 2006), we note that most models appear to predict enhanced summer monsoonal precipitation over parts of northwestern India, while predicting little or no change over much of peninsular India (Kumar et al. 2006). Climate-model-based studies appear to indicate an increase in the geographic extent of intense events but not necessarily an intensification of extreme events in areas already subject to high rainfall [which tend to be along the southwestern coast or the northeastern sub-Himalayan region (Kumar et al. 2006)]. Model results also indicate intensification of rainfall in most of India except parts of central and northeastern India (May 2004), with the most intense (maximum 24-h rainfall) rainfall events predicted to occur over northeastern and northwestern India.

The study by Goswami et al. (2006), using the same gridded dataset as here, reports an increasing trend in the frequency of extreme precipitation events, defined as events exceeding the thresholds of 100 and 150 mm, using pooled data from all grid cells over the central Indian region (the so-called monsoon belt), and also indicates an increase in the intensity of precipitation, as measured by the raw values of the 99.5th and 99.75th percentile of the rainfall distribution, over the same region.

Joshi and Rajeevan (2006) use station data (about 199 stations from 1901 to 2000) for India to carry out a linear, parametric trend analysis on various measures of extremes. They find increasing trends for certain regions (west coast and northwestern India) as well as an increase (as in Goswami et al. 2006) in the contribution of heaviest rains to total rainfall. Finally, Rajeevan et al. (2008), using a longer station-level dataset (1901–2004), carry out an analysis very similar to Goswami et al., over a slightly different region, and find increasing trends (after accounting for interdecadal variations in the extreme events) in both heavy and very heavy rainfall events (as defined in Goswami et al. 2006). They also make a preliminary attempt at linking such trends to ocean surface temperatures.

### a. Data

This study utilizes a recently available gridded daily dataset for India (Rajeevan et al. 2006), consisting of 1300 grid cells, each 1° latitude × 1° longitude, for 53 years (1951–2003), available from the IMD. Of these 1300 grids, 357 grids covering all of India’s land area were used for the analysis. This is the same dataset from which Goswami et al. (2006) draw their subset for analysis.

### b. Definition of statistics of extremes

We consider two measures of extremes, frequency and intensity, defined, respectively, as the number of days with rainfall events (each year) exceeding a threshold and the average daily rainfall (for each year) on the days on which rainfall exceeds the specific threshold.

*t*is the year,

*j*the grid box,

*P*the rain on day

_{itj}*i*in year

*t*at grid

*j*, and

*P**

_{j}is the rainfall threshold for grid

*j*;

For each grid cell, the number of nonzero precipitation events during each year was identified and the 90th (99th) percentile of this series estimated. The median of these 90th (99th) percentile values across all years was then selected to be the threshold for that particular grid cell. The spatially varying climatology of extreme rainfall across India is thus addressed (see also Joshi and Rajeevan 2006, p. 6).

We feel that this procedure better represents the spatial aspects of the monsoon process than a threshold fixed across grids, since the monsoon rainfall varies substantially across India, and we are interested in how the spatial pattern of extreme rainfall may have changed across the country. This is evident from Fig. 1, which illustrates that the spatial pattern of the thresholds are very similar to the monsoonal precipitation patterns, with the largest thresholds obtained at the southwestern, western coast, and the northeastern region.

The primary analyses were carried out separately for data for the monsoon season, June–September, and for the rest of the year. The monsoon season results are reported here.

### c. Trend analysis

The Mann–Kendall (MK) test is used for the detection of monotonic trends in the derived frequency and intensity data for each grid cell. An estimate of the Sen slope, a robust estimate of the monotonic trend, is also computed, along with its significance level. The MK test is a rank-based test, with no assumptions as to the underlying probability distribution of data (Helsel and Hirsch 1992, 326–327). The test statistic, computed based on pairwise comparison between the values of a series, is asymptotically normally distributed, independent of the distribution of the original series. A robust estimate of the magnitude of the slope of the trend is estimated using the method of Sen, as the median of pairwise slopes between elements of the series (Yue et al. 2002, 16–17).

For each grid cell, and separately for the frequency and the intensity data, we test (at the 10% significance level) (i) the null hypothesis of no trend, (ii) the null hypothesis of no increasing trend, and (iii) the null hypothesis of no decreasing trend. Recognizing that a certain number of rejections of the null hypothesis are to be expected, given the large number of tests conducted, we construct a field significance test (described in the next section) to assess whether the outcomes of the significance tests at the grid level may be consistent with what is expected purely by chance. Here, we examine the general features of the trends revealed by the MK test.

Figure 2 provides the spatial distribution of the trends for grids where the null hypothesis of no monotonic trend is rejected at the 10% significance level, while Table 1 provides a tabulation of the number of such grids.

For exceedances of the 90th percentile, the number of decreasing trends in frequency dominates the number of increasing trends. This observation runs counter to the assessments reported in the literature, where increasing trends in extremes are the focus. For instance, Goswami et al. (2006) and Kumar et al. (2006) find only increasing trends (in the first case over a restricted subset of the domain investigated here) with a fixed threshold of rainfall applied. Joshi and Rajeevan (2006), using thresholds varying with station, is the only study to report decreasing trends (at a few stations). Note from Table 1 that the number of increasing trends in intensity is higher than decreasing trends at the same threshold, which suggests that, when exceedance of the 90th percentile of grid rainfall occur, the amount of rain has been increasing—an observation likely to support the direction of trends reported in Goswami et al. (2006) (with a fixed rainfall threshold) and in Joshi and Rajeevan (2006) (with a spatially varying threshold).

A perusal of the trends in frequency and intensity of exceedance of the 99th percentile threshold supports such a speculation, given the dominance of increasing trends in both frequency and intensity at this threshold. However, contrary to much of the literature, a fair number of decreasing trends are noted in our analyses. From the figures it is clear that, while the details vary by threshold and metric (frequency and intensity), increasing trends dominate in the coastal regions and in the eastern region (west of Bangladesh), while decreasing trends appear to be more prevalent in the northern, central, and northeastern parts of India. Indeed, from these figures, it is difficult to argue that there has been an increase in the frequency and intensity of extreme rainfall across India.

The joint trends in frequency and intensity are investigated next. The motivation is to investigate whether, as hypothesized in Trenberth (1999), trends in both frequency and intensity increase or decrease jointly. The trends in frequency and intensity (in Figs. 3 and 4) that are deemed significant in the independent analyses agree completely for exceedances of the 99th percentile threshold. For exceedances of the 90th percentile, decreasing trends in frequency and intensity at the same location are much more prevalent than joint increasing trends in these two metrics. It is remarkable that at the higher threshold, there is not a single grid cell with opposite directions of trends in frequency and intensity, while at the lower threshold, grids with opposing trends are evident only in the monsoon belt. At the lower threshold, there are not very many grid cells for which trends in both frequency and intensity are significant. Most grids with trends significant in both are located in the eastern part of the country.

Given the spatial structure of the Indian monsoon, it is pertinent to ask if a similar (or some) structure is evident in the trends as well. To investigate this aspect, we plot the contours of the trends calculated at each grid point and note that, if there were some spatial structure in the trends, it would be evident in the contour plots. However, a perusal of Figs. 5 and 6 indicates that there is no evident structure to the trends, of either sign.

Having taken a broad look at the spatial distribution and direction of trends in the frequency and intensity of extreme rainfall across India, we next examine whether the number of statistically significant trends that appear to be different from zero at the 10% significance level could occur purely by chance, in an analysis of the spatially distributed dataset used here.

### d. Field significance test

The question addressed in this section is whether the number of increasing or decreasing trends deemed significant at the gridcell level analysis could occur purely by chance, taking into account the possibility that the rainfall data, and hence the trends, have an underlying spatial structure. The answer to this question depends on the specific area or domain considered [all of India or the core monsoon region, as identified in Goswami et al. (2006) and indicated as a box in all figures]. Results for the all-India data are presented first and those for the smaller, core monsoon region are discussed next.

The so-called field significance test (Livezey and Chen 1983) has been typically used to address the question posed in this section. The null hypothesis of the test is that the number *n* out of *N* total grid cells exhibiting a trend at the *α _{l}*% level of significance (local or at each grid cell) is not inconsistent with the value expected by chance, considering the potential for spatial correlation across the individual time series analyzed for trends. The null hypothesis is rejected if

*n*is larger than the number expected by chance at the

*α*% significance level (global or across the domain).

_{f}^{1}

*p*is greater than the (global) significance level

*α*, as below:

_{f}*N*(0, 1) under the null.

In most applications, the validity of the field significance test is compromised by the finiteness of the dataset used and by the spatial and/or temporal correlation between the series used (Livezey and Chen 1983; Elmore et al. 2006; Wilks 1997). We outline a procedure that addresses the spatial dependence of data. Spatial correlation reduces the degree-of-freedom of the test; that is, there are less than *N* individual realizations (of the test statistic or phenomenon; in this case, frequency and intensity) in a field of size *N*. Livezey and Chen (1983) estimate the effective number of realizations, *n* < *N*, by a Monte Carlo procedure involving generating the sampling distribution of *n* under the null (of field nonsignificance), obtaining a desired percentile (say, the 5th) of this distribution, and comparing it to the minimum number of effective degree of freedom for the significance of the field, *n*_{0}. The null hypothesis is rejected if *n* > *n*_{0}.

An alternative is to generate the sampling distribution of the test statistic under the null hypothesis, keeping intact the spatial structure of the dataset under consideration. The advantages of this approach include sampling the spatial correlation structure without formally specifying it (Wilks 1997).

The bootstrap is a nonparametric method that samples, with replacement, from the original data. It is applied here by resampling the spatial field associated with each year, which preserves the spatial structure but randomizes the temporal structure. The bootstrapping procedure for the field significance test was carried out by first generating 1000 random samples of 53 numbers, each from 1 to 53 (with replacement). Each of these 1000 samples is then treated as a realization of a time index corresponding to 53 years of data, with the frequency and intensity spatial fields then sampled for each of these generated years.

This approach preserves the correlation structure across space but not across time. Serial correlation across years in each grid cell was not found to be significant. For each of the 1000 samples generated, the MK test was repeated for each grid and for each sample. This leads to 1000 samples at each grid cell with a binary determination of trend significance at the *α _{l}*% level. The proportion of grid cells for which significant trends at the

*α*% level were found was calculated for each of the 1000 samples to obtain 1000 realizations of the proportion of grids,

_{l}*p*, exhibiting trends at the local significance level. This provides an estimate of the sampling distribution of the test statistic under the null hypothesis that the number of trends identified as significant at the

*α*% level, across the domain, is consistent with the number expected by chance. This computation was done separately for increasing and decreasing significant trends in frequency and intensity, leading to six different tests for each exceedance threshold. The procedure and results are summarized below.

_{l}- (i) Mann–Kendall trend test: Carry out the MK test; obtain the Sen slope and a count of the number of grids at which the null (of trend nonsignificance) was rejected (at the 10% level); compute the test statistic, denoted
*t*_{sample}. This step is carried out for each of three types of trends (trends in both directions, increasing trends and decreasing trends only), and steps (ii) and (iii) are then repeated separately for each of these three types of trends. - (ii) Field significance test:
- • Randomly sample, with replacement from the data, to obtain 1000 copies of the data matrix while retaining the spatial structure.
- • Obtain 1000 realizations of the proportion of the grid cells,
*p*, for which the hypothesis of no trend is rejected and the vector (of size 1000) of test statistics (denoted**t**_{bootstrap}). - • Construct the bootstrap estimate of the sampling distribution of the test statistic
**T**_{bootstrap}= (**t**_{bootstrap}−*t*_{sample}). Sort this vector and obtain its 100(1 −*α*)_{f}^{th}percentile (denoted*T**).^{2} - • Test
*T*=*t*_{sample}−*α*against_{f}*T** (recall that a one-sided test is employed)^{3}and reject the null hypothesis that the number of significant trends is what would be expected by chance if*T*>*T**.

- (iii) Repeat the analysis for different thresholds.

The results of the bootstrap procedure are summarized in Table 2. First, if we consider the total number of trends (of either sign), we observe that the null hypothesis is rejected for all tests. Next, if we consider increasing trends only in the case of frequency of exceedance of the 90th percentile is the null hypothesis not rejected. Changes in intensity indicate an increasing trend at both thresholds. The number of decreasing trends passes the significance test only for frequency at the 90th percentile. Thus, in summary, the hypothesis that overall there is an increasing trend in the frequency and intensity of extreme rainfall appears to have support with the caveat that, at the 90th percentile, the frequency of exceedance appears to be decreasing in the central and northern regions, while the intensity of these events is increasing. A contour plot of the slopes of frequency and intensity provides a smoother representation of the nature of these trends (Figs. 5 and 6).

Now consider the results over the region considered by Goswami et al. (2006), the main study with a similar analysis and dataset. Recall that Goswami et al. define extremes over a homogeneous region and use the number of days of rainfall above 100 and 150 mm and the intensity of rainfall for a fixed percentile (99.5th and 99.75th) as measures of extremes. They report an increase in the frequency of extreme rainfall events as well as an increase in the intensity of extreme rainfall. Carrying out the field significance test outlined in the preceding section over their domain, we find that the broad conclusions from the national analysis are essentially unchanged (Tables 3 and 4); that is, while increasing trends do exist, they are more predominant in the southwestern coast and northeastern regions, with decreasing trends being more prominent in the central regions.

Joshi and Rajeevan (2006), using a different dataset, find increasing trends (using somewhat different measures of extremes) in very similar regions; they also report negative trends (at only two stations).

Goswami et al. (2006) also performed a split sample analysis, splitting the data into two parts, pre- and post-1981, and find an increasing trend in the post-1981 sample. We repeated our analysis for the same two subperiods. The results of this analysis indicate that the conclusions obtained using the full sample are unaltered, unlike Goswami et al. who find a increasing trend only in the post-1981 sample. The importance of the spatially distributed analysis performed here is that, if the spatial differences noted represent nonhomogeneous aspects of the monsoon, then the spatial patterns identified in the trends would potentially help inform mechanism identification and model performance evaluation.

Kumar et al. (2006) find significant increases in intense precipitation in much of western, northwestern, and especially the southwestern regions. The results of the present analysis indicate trends mostly in parts of the southwestern coastal regions, similar to the results of Kumar et al., as well as in the eastern and central regions. May (2004) reports increases over much of the Indian peninsula, while the coarse spatial resolution of the model used does not provide detail over smaller regions of India. Further, our results that decreasing trends are likely in many areas are in concurrence with May (2004), who also finds similar decreases (in the scale and/or shape parameters of the gamma or the Generalised Pareto (GPA) distributions, which are fit to the rainfall data) for a small number of regions. However, the spatial coarseness of the model prevents a closer spatial comparison of the results.

## 2. Discussion and conclusions

An earlier examination of trends in extremes of Indian monsoon rainfall was developed further in this work. The analysis considered the spatial structure of changes in the extremes across the country rather than over a box (homogenous region) in central India. Broadly speaking, there is support for the hypothesis that the frequency and intensity of extreme rainfall over India may be increasing over the previous 53 years. However, there is considerable spatial variation as to the direction of change, and the spatial continuity of trends deemed statistically significant is weak. This is not unexpected since threshold crossings are a random process and the assessment of significance is also a threshold process. A visual examination of the spatial variation in the trends in frequency and intensity of extreme rainfall suggests that the north and central sections of the Indian Subcontinent have experienced a generally decreasing trend in the frequency and intensity of extremes, while the coastal regions in peninsular India and the region immediately west of Bangladesh have experienced increasing trends.

Even in central India, which was analyzed in aggregate by a previous study, there is some heterogeneity in the direction of the trends, and the larger-scale analysis performed here helps clarify the spatial structure of the changes in the region studied in Goswami et al. (2006).

We do not attribute the trends observed to anthropogenic climate change or to interdecadal climate variability, which may be of natural origin. Rather, we offer the results of this analysis as a benchmark to the climate community to consider more detailed studies of the spatial structure of changes in the Indian monsoon mechanisms so that a better informed attribution of change can be determined.

Generally, it is known that tropical depressions that form in the Bay of Bengal and then propagate westward or northward play a key role in extreme rainfall. These are associated with a mix of barotropic and baroclinic instabilities and their interaction with the mean monsoonal flow (Gadgil 2003). We suspect that the details of these interactions may be associated with the indicated changes in the spatial structure of the trends and, naturally, with the trends themselves. A recent study by Guhathakurta and Rajeevan (2008) notes a significant decreasing trend in the frequency of depressions and storms over the Bay of Bengal, lending support to our conjecture. Model-based studies and detailed analysis of individual extreme events are necessary to develop this intuition and are being pursued.

The design of our study was somewhat different from prior work. First, we considered changes in the climatology of extremes for each spatial location, rather than the frequency and intensity of exceedance of a fixed threshold, which is more meaningful for an analysis of the larger spatial scale considered. Second, instead of considering linear trends, we considered the more general case of monotonic trends (this would include, e.g., a step change in the process at some time or an exponential or logarithmic trend) and assessed the evidence for such a trend using robust, nonparametric methods. The field significance test was applied both nationally and regionally and essentially confirms that the number of cases for which the null hypothesis of no trend was rejected was statistically different from that obtained purely by chance (at the relevant level of significance).

Thus, our study lends credibility to previous assessments that report increasing trends in frequency and intensity of extreme rainfall over India, while identifying areas where there is a systematic departure from previous assessments. The issue of how these trends may be reinforced or reversed over the twenty-first century is not addressed here and is the subject of ongoing investigation. However, we note that the broad direction of trends identified here is consistent with the expectation from the model-based analysis for the twenty-first century scenarios for climate change, as reported by Kumar et al. (2006) and May (2004).

We wish to acknowledge the assistance of Dr M. Rajeevan, Director (NCC), Indian Meteorological Department, Pune, whose support was instrumental in obtaining the dataset on which the present analyses is based. We are also grateful to two anonymous reviewers for providing suggestions that helped improve the quality of the paper.

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Distribution of grids with statistically significant (at the *α _{l}* = 10% significance level) trends (357 grid cells in total).

Bootstrap test results [note i) *t*: raw statistic; *t**: critical value—the (1 − *α _{f}*) quantile of the bootstrap distribution ii)

*α*for increasing and decreasing trends is 0.05, while for all significant grids it is 0.1 and iii) null of field nonsignificance (

_{f}*p*=

*α*) is rejected if

_{f}*t*>

*t**].

Distribution of grids with statistically significant (at *α _{l}* = 10% significance level) trends in the region defined by Goswami et al. (2006) (74 grid cells in total).

Bootstrap test results for slopes in region defined by Goswami et al. (2006) [note i) *t*: raw statistic; *t**: critical value—the (1 − *α _{f}*) quantile of the bootstrap distribution ii)

*α*for increasing and decreasing trends is 0.05, while for all significant grids, it is 0.1 and iii) null of field nonsignificance (

_{f}*p*=

*α*) is rejected if

_{f}*t*>

*t**].

^{1}

Note that the local significance level *α _{l}* is always taken to be 10%, while the global significance level

*α*is either 10% or 5% depending on whether the field significance test pertains to both increasing and decreasing (significant) trends or only increasing or decreasing trends.

_{f}^{2}

This is known as the “percentile method” of bootstrap-based hypothesis testing (Davison and Hinkley 1997, 201–203).

^{3}

We note that this approach differs from the conventional one, involving testing **T**_{bootstrap} vs *α*; the approach adopted here is more efficient than the conventional one (Hall and Wilson 1991; Wilks 1997).