Spatial and temporal analysis of rainfall and temperature trend of India

Abstract

Climate change is a serious issue resulting in global variation in the temperature and precipitation pattern. In this study, changes in rainfall trend in India for 141 years (1871–2011) and temperature trend for 107 years (1901–2007) were analysed. The annual, seasonal and monthly changes in different regions of India were investigated to see the climate change in different parts of the country, and the net excess or deficit of rainfall and temperature in India was obtained. Statistical non-parametric tests were performed to see the trend magnitude with the Mann-Kendall (MK) test and Sen’s slope. Mann-Whitney-Pettitt (MWP) test was used for probable break point detection in the series, and change percentage was calculated over 30 sub-divisions and 7 broad regions. The results indicate decreasing annual and monsoon rainfall of India in most of the sub-divisions, and temperature fluctuations were observed in all the places. Temperatures (minimum, maximum and mean) were showing a significant increase, particularly in the winter and post-monsoon time. Wide variation was noticed all over India in the case of the minimum temperature. Variation was also observed at different spatial scales of sub-divisions and regions. This study gives the net impact of climate change in India which shows net excess of temperature and net deficit of rainfall.

Appendix

1.1 Method

1.1.1 Serial correlation and pre-whitening

Detection of trend in a series is affected by the presence of a positive or negative autocorrelation (Hamed and Rao 1998; Yue et al. 2003). The autocorrelation coefficient of ρk for a discrete time series for lag-k is given as

$$ {\rho}_k=\frac{{\displaystyle \sum_{k=1}^{n-k}\left({x}_t-{\overline{x}}_t\right)\left({x}_{t+k}-{\overline{x}}_{t+k}\right)}}{{\left[{\displaystyle \sum_{k=1}^{n-k}{\left({x}_t-{\overline{x}}_t\right)}^2{\left({x}_{t+k}-{\overline{x}}_{t+k}\right)}^2}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} $$

(1)

where ${\overline{x}}_t$ and Var (x _t ) are represented as the sample mean and sample variance of the first (n − k) terms, respectively; ${\overline{x}}_{t+k}$ and Var (x _t  + k) stand for the sample mean and sample variance of the last (n − k) terms correspondingly. Again, the hypothesis of no correlation is examined by the lag-1 autocorrelation coefficient as H ₀: ρ ₁ = 0 against H ₁: |ρ ₁| > 0:

$$ t=\left|{\rho}_1\right|\sqrt{\frac{n-2}{1-{\rho}_1^2}} $$

(2)

Here, the t test is the Student’s t distribution with (n − 2) degrees of freedom (Cunderlik and Burn 2002, 2004). If |t| ≥ t _α/2, the null hypothesis about no serial correlation is rejected at the significance level α.

Pre-whitening method is used to remove the serial correlation effect on the MK test (Storch 1993). Pre-whitening method with no trend was applied by Yue et al. (2002) with modification in the technique.

$$ {Y}_i={x}_i-\left(\beta \times i\right) $$

(3)

Here, β is Theil-Sen’s estimator. The r ₁ (lag-1 serial correlation coefficient) has been computed for new series. If r ₁ does not vary significantly from zero, then the data will be used without serial correlation, and the MK test will be applicable to the sample data directly. But, if it is opposite, then method of pre-whitening will be applied before the testing of trend.

$$ {Y}_i^{\prime }={Y}_i-{r}_1\times {Y}_{i-1} $$

(4)

The β × i value is added to the residual data set of Eq. 4

$$ {Y}_i^{{\prime\prime} }={Y}_i^{\prime }+\left(\beta \times i\right) $$

(5)

This Y ^″ _i is the final pre-whitened series.

1.1.2 Mann-Kendall test and Theil-Sen’s estimator

The statistic of the MK test is given as

$$ {Z}_c=\left\{\begin{array}{l}\frac{S-1}{\sqrt{Var(S)}} if,S>0\\ {}0 if,S=0\\ {}\frac{S+1}{\sqrt{Var(S)}} if,S<0\end{array}\right. $$

(6)

where

$$ S={\displaystyle \sum_{i=1}^{n-1}{\displaystyle \sum_{j=i+1}^n\operatorname{sgn}\left({x}_j-{x}_i\right)}} $$

(7)

Here, x _j and x _i are data values that are in sequence with n data; sgn (θ) is equivalent to 1, 0 and −1 if θ is more than, equal to or less than 0 respectively. If Z _c appears to be greater than Z _α/2, then the trend is considered as significant, where α represents the level of significance (Xu et al. 2003).

The rainfall trend magnitude is calculated by Theil-Sen’s estimator (Theil 1950; Sen 1968).

$$ \beta = median\left({X}_i-{X}_j/i-j\right),\kern1.5em \forall j<i $$

(8)

where 1 < j < i < n and β estimator stands for the median of the entire data set of all combination of pairs and is resistant to the effect of extreme values (Xu et al. 2003).

1.1.3 Percentage of mean

The change percentage is calculated by its approximation with linear trend. So, change percentage is equal to the median slope multiplied with the length of the period and the whole divided by the corresponding mean value which is given in percentage (Yue and Hashino 2003b).

$$ \mathrm{Percentage}\ \mathrm{change}\ \left(\%\right)=\frac{\beta \times \mathrm{data}\ \mathrm{length}}{\mathrm{mean}}\times 100 $$

(9)

1.1.4 Mann-Whitney-Pettitt method (MWP)

A time series {X ₁, X ₂…, X _n} with length n is considered. Let t be taken as the time of the most expected change point. Two samples of {X ₁, X ₂, …, X _t} and {Xt _+ 1, X _t + 2, …, X _n} can be then obtained by dividing the time series at t time. The U _t index is derived in the following way:

$$ {U}_t={\displaystyle \sum_{i=1}^t{\displaystyle \sum_{i=1}^n\operatorname{sgn}\left({X}_i-{X}_j\right)}} $$

(10)

where

$$ \operatorname{sgn}\left({x}_j-{x}_i\right)=\left\{\begin{array}{c}\hfill 1....... if\left({x}_j-{x}_i\right)>0\hfill \\ {}\hfill 0....... if\left({x}_j-{x}_i\right)=0\hfill \\ {}\hfill -1....... if\left({x}_j-{x}_i\right)<0\hfill \end{array}\right\} $$

(11)

Plotting the U _t value against t in a time series with no change point will result in a continuously increasing value of |U_t|. Nevertheless, if there is a presence of change point (even a local change point), then |U _t | will increase up to the level of the change point, and then, it will begin to decrease. The main significant change point t is considered as the point where the value of |U _t | remains highest:

$$ {}_{K_T=}\underset{1\le t\le T}{ \max}\left|{U}_{\tau}\right| $$

(12)

The estimated significant probability p(t) for a change point (Pettitt 1979) is given as:

$$ p=1- \exp \left[\frac{-6{K}_T^2}{n^3+{n}^2}\right] $$

(13)

The change point becomes statistically significant at t time with the significance level of α when probability p(t) surpasses (1 − α).