The northern area of Pakistan is surrounded by the Himalayan, Karakoram, Hindu Kush, and Pamir mountain ranges which with high peaks of between 6500 m to 8600 m. The snowmelt from these mountains, combined with glacier melt and monsoon rainfall, contribute to the many rivers, most notably the Indus River, that Pakistan has relied on to develop an advanced irrigation canal network However, the distribution and quantity of monsoon rainfall varies widely throughout the year, and occurs due to seasonal winds and western disturbances. In northern areas, such as Khyber Pukhtonkhuwa and Balochistan provinces, the maximum rainfall occurs during December to March, and in Punjab and Sindh, the maximum rainfall (50-75%) occurs during the monsoon season [10-15].

The 24-hour annual maximum rainfall data from six rainfall-gauging stations in northern Pakistan were used in this study. The locations of these stations are shown in Figure 1. A summary of the statistics is presented in . These statistical parameters are used to calculate the estimated 24-hour annual maximum rainfall in different return periods using different probability distributions. Of the six selected stations, Oghi has 46 years of rainfall data, spanning from 1961 to 2010. Three stations, including Kalam, Daggar and Mardan, have 44 years of rainfall data, spanning from 1962 to 2009, 1963 to 2010 and 1963 to 2010, respectively. Two stations, including Puran and Besham Qilla, have 38 years of rainfall data, spanning from 1963 to 2004 and 1969 to 2010, respectively.

Fig. 1 Locations of selected rainfall gauging stations.

Table 1 Summary of statistics from the selected rainfall gauging stations.

The distribution of 24-hour maximum rainfall observed during different months of a year is shown in Figure 2. Figure 2 shows that Kalam and Besham Qilla received 42% and 21%, respectively, of observed rainfall in March. Oghi, Daggar and Puran received 37%, 32% and 23%, respectively, of observed rainfall in July. Mardan received 37% of observed rainfall in August. These results suggest that the maximum rainfall at these selected stations occurred between March and August.

Fig. 2 Distributions of 24-hour annual maximum rainfall in a year.

Four probability distributions (normal, log-normal, log-Pearson type-III and Gumbel) were used in this study. The parameters of probability distributions were calculated using the method of moments and are given in .

Table 2 Parameters of probability distributions at rainfall gauging stations.

The four probability distributions were subjected to three goodness of fit tests (Kolmogorov Smirnov Test, Chi-Squared Test and Anderson Darling Test) to determine the best-fitting probability distribution model at each rainfall gauging station. A standard procedure was followed for application of goodness of fit tests that was described earlier by several authors [16-18].

The goodness of fit tests was ranked from one (bestfit) to four (least-fit) for all probability distributions.

Selection of the best-fit probability distribution is based on the total score from all the goodness of fit tests. The results of goodness of fit tests at each selected rainfall gauging station and for each probability distribution used in this study are shown in . Based on the results of the goodness of fit tests, the best-fit probability distribution and mathematical expression for the calculation of rainfall in different return periods at each gauging station are shown in .

Table 3 Results of goodness of fit tests.

Table 4 Best-fit distributions and mathematical expressions.

The normal distribution provides the best-fit at the Mardan rainfall gauging station, while log-Pearson type-III provides the best-fit at the other rainfall gauging stations analyzed in this study. Probability density functions (PDF) and cumulative distribution functions (CDF) at the rainfall gauging stations were calculated using the best-fit distribution, i.e., the normal distribution at Mardan and the log-Pearson type-III distribution at the rest of the rainfall gauging stations, and are shown in Figures 3 and 4.

Fig. 3 PDFs of probability distributions at rainfall gauging stations.

Fig. 4 CDFs of probability distributions at rainfall gauging stations.

The rainfall estimates or maximum values of expected rainfall (mm) for return periods of 2, 5, 10, 20, 50, 100 and 200 years at the rainfall gauging stations were calculated using the best-fit distribution. The rainfall estimates are given in .

Table 5 Rainfall estimates at the rainfall gauging stations using the best-fit distribution.

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