Sci.Int.(Lahore),25(2),333-336,2013 ISSN 1013-5316; CODEN: SINTE 333 MODELING OF MONSOON RAINFALL IN PAKISTAN BASED ON KAPPA DISTRIBUTION Ishfaq Ahmad*, Said Farooq Shah**, Iram Mahmood***, Zahoor Ahmad ++ *Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan. ** Department of Statistics Quaid-e-Azam University, Islamabad, Pakistan. ***School of Chemical and Materials Engineering, National University of Science and Technology, Islamabad, Pakistan ++Pest warning and Quality Control of Pesticides, DepalPur District Okara. + Supported by Higher Education Commission and International Islamic University, Islamabad, Pakistan under project No: PM-IPFP/HRD/HEC/2011/2254. * Corresponding author: ishfaq.ahmad@iiu.edu.pk Tel: 0092-51-9019733; 0092-345-4612656 ABSTRACT: Monsoon rainfall in Pakistan is of great importance because of its needs in agriculture and power generation. In this paper, we have analyzed the random behavior of monsoon rainfall in Pakistan through 4 parameter Kappa probability distribution at 27 meteorological stations for the period 1960- 2006. The parameters of this distribution have been estimated using method of L-Moments. Using these estimates we have calculated quantiles for different return periods from 2 to 500 years. The comparison of estimated quantiles with observed values of rainfall after five years is found to be in good agreement. Keywords: Monsoon, Bootstrapping, Kappa Distribution, L-Moments, Quantiles 1. INTRODUCTION Pakistan is located from southwest to Northwest at 23-37 degree north latitude and 61-76 degree east latitude. Pakistan faces much diversified climate pattern round the year. In northern areas the temperature is as low as 25 Celsius degree and in the southern areas it as high as 55 Celsius degree. Monsoon season is experienced in many parts of world for example northern Australia, Africa, and South America. But it is strong in South Asian countries including Afghanistan, Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan and Sri Lanka. Southwest monsoon season in Sri Lanka enters in late May just before it enters the Indian subcontinent. In India it enters in June and in Pakistan in early July and remains till the end of September. Summer monsoon season in Pakistan is of great importance for its agricultural, economic and social purposes. These rains are not only used for water needs of plants in agricultural sector but also to kill the insects by physical beating. By irrigating the fields, certain insect pests like crickets attacking the cotton seedlings and the white ants attacking cotton, sugarcane, chilies and other crops can be drowned in rain water and thus crops can be saved. Monsoon in Pakistan contributes almost 65-75 % of the total annual rainfall. Despite of some destruction in the forms of floods and droughts these rainfalls are welcomed in Pakistan. In Pakistan there are many studies which deal with rainfall data in different aspects for example, Rasul et al [1] carried out a diagnostic study of record heavy rain in twin cities of Pakistan as Rawalpindi and Islamabad. Karori and Zhang [2] investigated prospects of downscaling for seasonal precipitation prediction over Islamabad-Pakistan. Haroon and Rasul [3] applied the very common multivariate Principal Component Analysis (PCA) in order to identify the major modes of oscillations present in the data. Rasul et al [4] examined the heavy monsoon precipitation over the Indus plains of south Asia by non-hydrostatic numerical model MM5. But unfortunately there is not even a single study which determines underlying the probability distribution of the rainfall data and finds the extreme events after different return periods of time. In this paper we have focused on modeling the monsoon total rainfall in Pakistan across 27 different meteorological stations from 1960 to 2006 through four parameter Kappa distribution. This is the most suitable distribution for the given data set in the presence of extreme observations. We have also calculated different quantiles for different years and compared these estimated quantiles with observed values after five years. The results have been found to be in good agreement. The rest of the paper is as follows, section 2 is about the methodology about L-moments estimation and estimation of parameters of Kappa distribution from these moments. In section 3 we have applied the described methodology on Pakistan monsoon data. Section 4 presents the results and discussions. 2. Methodology For rainfall data usually we use Generalized Extreme Value (GEV) distribution to estimate extreme events. This distribution having three parameters is considered to be as limiting form of such extreme observations. This distribution gives unsatisfactory results when we have a finite sample Winchester [5]. We have used generalized form of GEV, named Kappa distribution developed by Hosking [6] to model the monsoon rainfall of Pakistan. Kappa Distribution has four parameters and gives quite satisfactory results when GEV or any other distribution having two parameters or three parameters becomes unsatisfactory to provide such results. Following Hosking [7] we have used method of L-moments for estimation of parameters of this distribution. This method is simply based on the linear functions of expected order statistics. This method is preferable as it provides reliable and robust estimates of the parameters of the given distribution and hence reliable quantiles when we have small samples Parida [8]. Let be the increasing order of monsoon rainfall considered as real valued random variable,