Interrelationship of rainfall, temperature and reference evapotranspiration trends and their net response to the climate change in Central India

Abstract

The monthly rainfall data from 1901 to 2011 and maximum and minimum temperature data from 1901 to 2005 are used along with the reference evapotranspiration (ET₀) to analyze the climate trend of 45 stations of Madhya Pradesh. ET₀ is calculated by the Hargreaves method from 1901 to 2005 and the computed data is then used for trend analysis. The temporal variation and the spatial distribution of trend are studied for seasonal and annual series with the Mann-Kendall (MK) test and Sen’s estimator of slope. The percentage of change is used to find the rate of change in 111 years (rainfall) and 105 years (temperatures and ET₀). Interrelationships among these variables are analyzed to see the dependency of one variable on the other. The results indicate a decreasing rainfall and increasing temperatures and ET₀ trend. A similar pattern is noticeable in all seasons except for monsoon season in temperature and ET₀ trend analysis. The highest increase of temperature is noticed during post-monsoon and winter. Rainfall shows a notable decrease in the monsoon season. The entire state of Madhya Pradesh is considered as a single unit, and the calculation of overall net change in the amount of the rainfall, temperatures (maximum and minimum) and ET₀ is done to estimate the total loss or gain in monthly, seasonal and annual series. The results show net loss or deficit in the amount of rainfall and the net gain or excess in the temperature and ET₀ amount.

Appendix

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 ρ _k for a discrete time series for lag-k is given as

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

(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 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 r₁ (lag-1 serial correlation coefficient) has been computed for new series. If r₁ do not vary significantly from zero, then the data will be used without serial correlation and 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 ${Y}_i^{{\prime\prime} }$ is the final pre-whitened series.

Mann-Kendall test and Theil-Sen’s estimator

The statistic of 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,

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

(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).

$$ \beta = median\left({X}_i-{X}_j/i-j\right),\kern0.5em \forall j<i $$

(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).

Change rate as 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 length of the period and the whole divided by the corresponding mean value which is given in percentage (Yue and Hashino 2003).

$$ \mathrm{Percentage}\kern0.5em \mathrm{Change}\left(\%\right)=\frac{\beta \times \mathrm{lengthofyear}}{\mathrm{mean}}\times 100 $$

(9)

Net change using areal average method

The net change is calculated by the areal average method for three parameters as given by Sen (1998).

$$ \overline{P}=\frac{{\displaystyle \sum_{i=1}^n{A}_i{P}_i}}{{\displaystyle \sum_{i=1}^n{A}_i}} $$

(10)

Where $\overline{P}$ is the estimated average areal value, P _i is the variable amount of station i in the corresponding subarea A _i , N represents the number of station.