Y =at + b (4.1) Where, t = time (year) a = slope coefficient; and b = least-squares estimate of the intercept The slope coefficient indicates the annual average rate of change in the hydrologic characteristic. If the slope is significantly different from zero statistically, it is entirely reasonable to interpret that there is a real change occurring over time. The sign of the slope defines the direction of the trend of the variable: increasing if the sign is positive, and decreasing if the sign is negative. Magnitude of trend The magnitude of trend in a time series was determined using a non-parametric method known as Sen’s estimator (Sen 1968). This method assumes a linear trend in the time series and has been widely used for determining the magnitude of trend in time series (Lettenmaier 1994; Yue & Hashino 2003; Partal & Kahya et al. 2006). In this method, the slopes (Ti) of all data pairs are first calculated by for i = 1,2,…,N (4.2) where and are data values at time and (j > respectively. The median of these N xj xk j k k) values of is Sen’s estimator of slope which is calculated as Ti (4.3) A positive value of indicates an upwards (increasing) trend and a negative value indicates a β downwards (decreasing) trend in the time series. Significance of To ascertain the presence of a statistically significant trend in hydrologic climatic variables such as temperature, precipitation and streamflow with reference to climate change, the non-parametric Mann–Kendall (MK) test has been employed by a number of researchers (Yu 1993, 2003; Douglas 2000; Burn 2004; Singh 2008a, b). The MK et al. et al. et al. et al. method searches for a trend in a time series without specifying whether the trend is linear or non-linear. The MK test was also applied in the present study. The MK test checks the null hypothesis of no trend versus the alternative hypothesis of the existence of an increasing or
Y =at + b (4.1) Where, t = time (year) a = slope coefficient; and b = least-squares estimate of the intercept The slope coefficient indicates the annual average rate of change in the hydrologic characteristic. If the slope is significantly different from zero statistically, it is entirely reasonable to interpret that there is a real change occurring over time. The sign of the slope defines the direction of the trend of the variable: increasing if the sign is positive, and decreasing if the sign is negative. Magnitude of trend The magnitude of trend in a time series was determined using a non-parametric method known as Sen’s estimator (Sen 1968). This method assumes a linear trend in the time series and has been widely used for determining the magnitude of trend in time series (Lettenmaier et al. 1994; Yue & Hashino 2003; Partal & Kahya 2006). In this method, the slopes (Ti) of all data pairs are first calculated by for i = 1,2,…,N (4.2) where xj and xk are data values at time j and k (j > k) respectively. The median of these N values of Ti is Sen’s estimator of slope which is calculated as (4.3) A positive value of β indicates an upwards (increasing) trend and a negative value indicates a downwards (decreasing) trend in the time series. Significance of To ascertain the presence of a statistically significant trend in hydrologic climatic variables such as temperature, precipitation and streamflow with reference to climate change, the non-parametric Mann–Kendall (MK) test has been employed by a number of researchers (Yu et al. 1993, 2003; Douglas et al. 2000; Burn et al. 2004; Singh et al. 2008a, b). The MK method searches for a trend in a time series without specifying whether the trend is linear or non-linear. The MK test was also applied in the present study. The MK test checks the null hypothesis of no trend versus the alternative hypothesis of the existence of an increasing or