Analysis of spatial and temporal variation in rainfall trend of Madhya Pradesh, India (1901–2011)

Abstract

Monthly rainfall data from 1901 to 2011 for 45 stations of Madhya Pradesh were used to analyze the changed rainfall trend of the region. Spatial distribution and temporal variation of rainfall were observed in the seasonal and annual series. Mann–Kendall test and Sen’s slope were used in the study for the trend analysis. Standard normal homogeneity test and Mann–Whitney–Pettitt test were applied for detection of break point in the series. Change percentage was calculated for seasonal and annual series for 111 years. Trend analysis was then done by dividing the entire series into two divisions before and after the break point. The impact of extreme events of rainfall on the trend was evaluated on the whole and partial series (before and after the break point). Variation in the spatial distribution of annual and monsoon rainfall was observed, which showed decreasing trend. The probable break point in the series was 1978 and extreme negative event of rainfall became frequent in the years after the break point from 1979 to 2011. The annual decrease of −6.75 % of rainfall is observed from 1901 to 2011.

Appendices

Appendix 1: Serial correlation and pre-whitening

The serial correlation coefficient of ρk for a discrete time series for lag-k is given as (Yue et al. 2003)

$$ \rho_{k} = \frac{{\sum\limits_{t = 1}^{n - k} {(x_{t} - \bar{x}_{t} )(x_{t + k} - \bar{x}_{t + k} )} }}{{\left[ {\sum\limits_{t = 1}^{n - k} {(x_{t} - \bar{x}_{t} )^{2} \times \sum\limits_{t = 1}^{n - k} {(x_{t + k} - \bar{x}_{t + k} )^{2} \times } } } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}, $$

(1)

where $\bar{x}_{t}$ and Var (x _t ) stand for the sample mean and sample variance of the first (n − k) terms, respectively, $\bar{x}_{t + k}$ and Var (x _t  + k) are the sample mean and sample variance of the last (n − k) terms, respectively. Further, the no correlation hypothesis is checked 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 represents the Student’s t-distribution with (n − 2) degrees of freedom (Cunderlik and Burn 2004). If |t| ≥ t _α/2, then the null hypothesis of no serial correlation is declined at the α level of significance.

The pre-whitening method removes the serial correlation effect on the MK test (Storch 1993) and it was applied on the data series with some modification in the technique (Yue et al. 2002).

$$ Y_{i} = x_{i} - (\beta \times i) $$

(3)

where $\beta$ is the Theil–Sen’s estimator. The r₁ (lag-1 serial correlation coefficient) is computed for the new series. In case the r₁ do not vary significantly from zero, the data are used without serial correlation and the MK test in that case will be applicable directly to the sample data. Otherwise, the method of pre-whitening will be used before trend analysis.

$$ Y^{\prime}_{i} = Y_{i} - r_{1} \times Y_{i - 1} . $$

(4)

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

$$ Y^{\prime\prime}_{i} = Y^{\prime}_{i} + (\beta \times i), $$

(5)

where $Y^{\prime\prime}_{i}$ represents the final pre-whitened series.

Mann–Kendall test and Theil–Sen’s estimator

The MK test is given as

$$ Z_{c} =\left\{ \begin{array}{ll} \frac{S - 1}{{\sqrt {Var(S)} }} &\quad if \, S > 0 \\ 0&\quad if \, S = 0 \\ \frac{S + 1}{{\sqrt {Var(S)} }} &\quad if \, S < 0 \\ \end{array}\right . $$

(6)

where

$$ S = \sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\text{sgn} (x_{j} - x_{i} )} } . $$

(7)

Here, $x_{j}$ and $x_{i}$ are the data values which 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 and 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).

$$ \beta = {\rm median}\left( {X_{i} - X_{j} /i - j} \right),\,\,\,\,\,\,\forall j < i $$

(8)

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

Change magnitude as percentage of mean

The change percentage is computed by approximating it with the linear trend. So, it is equivalent to the median slope multiplied by the length of the period and the whole is divided by the corresponding mean value that is given in percentage (Yue and Hashino 2003):

$$ {\rm Percentage \,\, change} \left( \% \right) = \frac{\beta \times {\rm length \,of\,year}}{\rm mean} \times 100. $$

(9)

Mann–Whitney–Pettitt method (MWP)

A time series {X1, X2…, Xn} with the length n is considered. Let t be considered as the time of the most expected change point. Two samples such as {X1, X2, …, Xt} and {Xt + 1, Xt + 2, …, Xn} can then be attained by dividing the time series at t time. The U _t index is developed in the following way:

$$ U_{t} = \sum\limits_{i = 1}^{t} {\sum\limits_{j = t + 1}^{n} {\text{sgn} (X_{i} - X_{j} )} } , $$

(10)

where

$$ {\text{sgn}}(x_{j} - x_{i} ) = \left. {\left\{ {\begin{array}{*{20}l} {1 \ldots if(x_{j} - x_{i} ) > 0} \hfill \\ {0 \ldots if(x_{j} - x_{i} ) = 0} \hfill \\ { - 1 \ldots if(x_{j} - x_{i} ) < 0} \hfill \\ \end{array}\!\!\!} \right.} \right\}. $$

(11)

There will be constantly increasing value of |Ut| if it is plotted against t in a time series with no change point. But, if there is presence of a change point (even a local change point) then |Ut| will rise up to the level of change point and then start decreasing. The major significant change point t is considered as the point where the value of |Ut | remains highest:

$$ _{{K_{T} = }} \mathop {\hbox{max} }\limits_{1 \le t \le T} \left| {U_{\tau } } \right|. $$

(12)

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

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

(13)

When probability p(t) surpasses (1−α), then the change point becomes statistically significant at t time with the α significance level.

Appendix 2: Serial correlation

See Tables 3, 4, 5, 6 and 7.

Table 3 Serial correlation

Table 4 Z statistic and change percentage (1901–2011)

Table 5 Z statistic and change percentage (1901–1978)

Table 6 Z statistic and change percentage (1979–2011)

Table 7 Extreme events