Investigating rainfall trend over India using the wavelet technique

Indian agriculture is rain-dependent, with almost two-thirds of the net cropped area being rain-fed. Therefore, a regular rainfall pattern is crucial for agricultural growth; too much or too little rain can have a devastating effect on agriculture. A regular pattern in rainfall may not be observed all the time. The series may behave erratically, with highs and lows. Recent evidence shows monsoon rainfall in India is becoming more erratic (Manton et al. 2001; Kumar et al. 2004; Goswami et al. 2006; Mall et al. 2006; Ghosh et al. 2010; Paul et al. 2011). Auffhammer et al. (2011) have shown that the Indian climate has become drier and warmer, adversely affecting the crop yields.

India is a large country, and regional patterns in rainfall and temperature are more important for agriculture. A few studies have analysed behaviour of rainfall at disaggregated spatial units, at the level of meteorological sub-division, state and district. Guhathakurta & Rajeevan (2006) found a sharp decrease in the number of rainy days and amount of rainfall in many sub-divisions of India. They reported a decline in rainfall in the months of June, July and September and an increase in August in a few sub-divisions. Goswami et al. (2006) studied rainfall behaviour over central India for the period 1951–2000 and observed a significant increase in heavy rainfall events and a decrease in moderate rainfall events. They also reported a rise in the frequency of extreme rainfall events such as droughts and floods. Krishnakumar et al. (2009) studied variations in monthly, seasonal and annual rainfall in the state of Kerala for the period 1871–2005, and found a significant decline in the south west monsoon (June–September) rains, especially in the months of June and July; and an upward trend in winter rainfall in the months of January, February and April. Soltani et al. (2012) computed rainfall and rainy day trends in Iran.

Some studies have also attempted to predict rainfall behaviour and estimation of trends using recently developed nonparametric techniques like wavelets. Kallache et al. (2005) applied discrete wavelet transform (DWT) analysis to river run-off data from gauges in Southern Germany to separate the variability of small and large time-scales, assuming that the trend component is part of the latter. Adamowski et al. (2009) proposed a method for trend detection and estimation based on continuous wavelet transform (CWT), which provides a very flexible and accurate tool for detecting and estimating complicated signals in hydro-climatologic series in Canada. Paul et al. (2011) have also used DWT for the estimation of trends in Indian monsoon rainfall and reported that that there is a significant decreasing trend in the same. Nalley et al. (2012) studied the DWT technique to analyse and detect trends in monthly, seasonally-based, and annual data from eight flow stations and seven meteorological stations in southern Ontario and Quebec during 1954–2008. Using wavelet and fuzzy logics to describe the frequency decomposition of annual rainfall and summer monsoon rainfall, Singh & Bhatla (2012) suggest that a combination of these two models provides a better forecast of rainfall behaviour. Ramana et al. (2013) used wavelet and artificial neural network (ANN) models for predicting monthly rainfall and found wavelet neural network models more effective in predicting rainfall than ANN models alone. Mahajan & Mazumdar (2013) applied non-linear regenerative predicting codes for rainfall forecasts, and predicted more than 1 year's rainfall trend with 90% or more accuracy. Paul et al. (2013) applied maximal overlap DWT (MODWT) for forecasting Indian monsoon rainfall. Adarsh & Reddy (2015) carried out trend analysis based on DWT on the post-monsoon rainfall time series of the Kerala subdivision, and the results show that there is a dominance of short-term periodicity of less than a decade in the subdivision.

Most of these studies have analysed rainfall behaviour either at national level or at state level. Estimates at these levels are too aggregative to provide regional variation in climate. India, being a large country, has considerable heterogeneity in agro-climatic conditions even within a state or sub-division, while an analysis of climate variables at a more disaggregated level, say the district, is desirable for a better understanding of their location-specific behaviour. In this paper, we have examined rainfall behaviour at the level of an agro-climatic zone by clustering the districts having similar agro-climatic conditions into homogeneous zones. The study of rainfall trends at the zone level may provide inputs to policymakers in designing regionally differentiated adaptation and mitigation strategies to cope with climate change. Moreover, in almost none of the previous studies, has MODWT been explored for estimation of trends in climate variables. In the present study, MODWT and DWT have been applied. The paper is organized as follows: the next section deals with the methodology, followed by the results and discussion section, and ending with the conclusions in the final section.

To identify the trend in rainfall we have used two nonparametric methods, viz. Mann-Kendall (M-K) and MODWT. The advantage of nonparametric methods is that these are based on fewer assumptions underlying the rainfall distribution and are not very sensitive to abrupt breaks, which are a common feature of a rainfall series. We have also used a parametric method, that is, the deterministic linear trend. A brief description of these is given below.

Wavelets are fundamental building block functions, analogous to the trigonometric sine and cosine functions. As with a sine or cosine wave, a wavelet function oscillates around zero. This property makes the function a wave. A description of wavelets can be found in Daubechies (1992), Ogden (1997) and Percival & Walden (2000). A completely different use of wavelets in statistical time series analysis is described in Rioul & Vetterli (1991) and Flandrin (1998). Wavelets, being time-scale representation methods, deliver a tool complementary to both classical and localized Fourier analyses. The search for localized time-scale representations of stochastic correlated signals leads to both analysing and synthesizing (i.e. modelling) mainly non-stationary processes with wavelets or wavelet-like bases (Nason & von Sachs 1999).

There are two main waves of wavelets; the first is known as a CWT designed to work with time-series defined over the entire real axis; another is the DWT which deals with series defined essentially over a range of integers. The DWT of a time-series is used to capture high and low frequency components. This enables modelling of time-series data through computation of the inverse DWT. Computation of the DWT is carried out by the ‘pyramid algorithm’, as described in Percival & Walden (2000). A brief description of this protocol can be written as follows.

The MODWT is a linear filtering operation that transforms a series into coefficients related to variations over a set of scales. It is similar to the DWT in that both are linear filtering operations producing a set of time-dependent wavelet and scaling coefficients. Both have basis vectors associated with a location t and a unit-less scale τ_j = 2^j − 1 for each decomposition level j = 1, …, J₀. The MODWT retains down-sampled values at each level of decomposition that would otherwise be discarded by the DWT. The MODWT is well defined for all sample sizes N, whereas for a complete decomposition of J levels, the DWT requires N to be an integer multiple of 2^J. In the MODWT, the output signal is never sub-sampled (not decimated). Instead, the filters are upsampled on each level. The redundancy of the MODWT facilitates alignment of the decomposed wavelet and scaling coefficients at each level with the original time-series, thus enabling a ready comparison between the series and its decomposition. Finally, the redundancy of the MODWT wavelet coefficients modestly increases the effective degrees of freedom on each scale, and thus decreases the variance of certain wavelet-based statistical estimates.

MODWT is a highly redundant non-orthogonal transform. For a redundant transform like the MODWT, an N samples input time-series will have an N samples resolution scale for each resolution level. Therefore, the features of wavelet coefficients in a multi-resolution analysis (MRA) will be lined up with the original time-series in a meaningful way.

The test statistic (N – N/2^J)/(N/2^J – 1) G follows F distributed with (N/2^J – 1) and (N – N/2^J) degrees of freedom (under the normality assumption of the scaling coefficients).

A preliminary investigation into rainfall behaviour across agro-climatic zones shows stable mean and variance, indicating that the series is stationary. The series is also not autoregressive as indicated by the values of autocorrelation and partial autocorrelation function at various lag orders (results are not reported here due to lack of space, but are available on request). The coefficients of the linear regression model and Theil–Sen slope estimates are given in the last two columns of Table 1, respectively. At the all India level, the trend in rainfall is positive but insignificant. It is positive and significant only in the arid zone. In other zones, the trend is negative and not significant. From these we find that over the years rainfall is becoming less, except in the arid zone where rainfall increased by 0.64 mm per year. The M-K test and Spearman's rank test confirm these observations except for the humid zone where one of the tests is significant. Here, the direction of change in rainfall in M-K and Spearman's test is opposite to that provided by linear regression. One possible reason could be the presence of some extreme events, i.e. very high or very low rainfall in some years.

For examining the local and global fluctuations in rainfall series, we computed MODWT coefficients along with the MRA for all the zones. The MODWT can accommodate any sample size N. In order to preclude decomposition at scales longer than the total length of the time-series, we select the highest scale (J₀) such that ⁠. In particular, for alignment of wavelet coefficients with the original series, condition (i.e. the width of the equivalent filter at the th level is less than the sample size) should be satisfied to prevent multiple wrappings of the time-series at level J₀. In our case, we found as 6. The choice of wavelet filter has a great significance on feature extraction of a time series. The decomposition would vary significantly, particularly for high frequency components with the choice of a different filter. A variety of ‘mother wavelet functions' is available that can be used for DWT analysis (e.g. the Haar wavelet, Dsaubechies wavelet). The mother wavelet function should reflect the type of features present in the time series. For time series with ‘steps’, one would choose a boxcar-like mother wavelet function such as the Haar wavelet, while for more smoothly varying time series one would choose a smoother mother wavelet function such as the Dsaubechies wavelet. In the present paper, both the filters have been used. Daubechies wavelets optimally capture the polynomial trends, while the Haar wavelet is discontinuous and resembles a step function. It is to be noted that for trend detection only, a low frequency component is used.

Coefficients plotted at the top of Figure 2(a) are low-frequency components, and at the bottom are high-frequency components. The wavelet coefficients do not remain constant over time, and reflect changes in the data series at various time points. The locations of abrupt jumps can be spotted by looking at vertical clustering of the relatively larger coefficients. To obtain a visual idea regarding this, the MODWT coefficients at 3–5 levels, i.e. W₃, W₄ and W₅, have been plotted in Figure 2(b). A perusal of Figure 2(b) represents that whenever there are abrupt changes in rainfall, it is depicted by clusters of long spikes on either side of the X-axis.

The behaviour of rainfall in the arid zone is in contrast with that in others. In this zone, the overall trend in rainfall is increasing, but the intervals of change in direction are greater, indicating more erratic rainfall behaviour. The intensity of rainfall decreases during 1901–1916, remains almost constant until 1931, increases until 1976 and then starts decreasing. At the all India level, rainfall has increased from 1950 onwards (Figure 3(e)).

The trend analysis studies by Guhathakurta & Rajeevan (2006), Goswami et al. (2006) and Krishnakumar et al. (2009) indicate that trends significantly depend upon the period and the locations. To see this, we analysed rainfall behaviour for the period 1939–2002, applying the same methods as for the entire period but instead of MODWT we applied DWT (the DWT requires that the number of observations (N) should be sufficient such that N = 2^^J, J takes a positive integer). Also, we used two wavelet filters, viz. Haar and Daubechies (D4).

The results are reported in Table 2. With application of the wavelet method, the direction of trend in rainfall remains as in the case of the entire period, but the trend coefficients are now significant. These results indicate that there has been a significant decline in rainfall in the past six decades or so (1939–2002) in all the zones, except in the arid zone and at the all India level, where it has increased. The wavelet analysis could determine a significant trend globally as well as locally that could not be captured through other methods.

Agriculture in India is highly sensitive to rainfall, therefore any change in rainfall will influence crop yields. An understanding of the spatial and temporal distribution and changing patterns in rainfall is important for planning and management of water resources. In this paper we examined behaviour of rainfall in different agro-climatic zones over a period of more than 100 years using wavelet techniques. The trends were tested for two periods, a long-term period (1901–2002) and for a shorter period (1939–2002). Findings indicate that at the national level there is no significant trend in rainfall in the long run, but there are pockets of change in the rainfall pattern. There was a significant increase in the rainfall in the arid zone, whereas in the humid, semi-arid tropics and semi-arid temperate zones the trend was downward but insignificant. In the short run, the direction of change in trend remains the same as in the long run, but the change is statistically significant. Since rainfall has been decreasing in three of the four agro-climatic zones, a possible sign of temporal change in Indian rainfall over the past century may be apparent.

This paper has been drawn from the project ‘Enhancing resilience of agriculture to climate change’, funded by the Indian Council of Agricultural Research under a mega project ‘National Initiative on Climate Resilient Agriculture’. The authors are grateful to the anonymous reviewers for providing useful comments which helped improve the paper.