884
BHAVANA ET AL
Environment & Ecology 30 (3B) : 884—887, July—September 2012
Modeling of Daily Rainfall using Gamma Probability Distribution C. P. BHAVANA, R. MUNIRAJAPPA, H. S. SURENDRA AND SANTOSHA RATHOD* Department of Agricultural Statistics, Applied Mathematics & Computer Science, University of Agricultural Sciences GKVK Bengaluru 560065, India *E-mail :
[email protected] (Received 26/03/2012 ; Accepted 12/05/2012) Abstract Gamma probability distribution was fitted for 30 years (1980—2010) daily rainfall data of southern dry zone of Karnataka. Southwest monsoon rainfall distributions of this zone are found to abide by gamma probability distribution function which is confirmed on the basis of chi-square tests. The rainfall sequences recorded for the last 30 years from June to September (southwest monsoon) are investigated statistically and gamma distribution parameters are calculated at existing stations of the zone. The shape and scale parameters are then regionalized and hence it becomes possible to find the parameter values at any desired location within the study area. Then the probability of occurrence of ≤50, 50—150, 150—250 and ≥250 mm of rainfall were obtained for the months of June, July, August and September. Key words : Rainfall patterns, Southern dry zone, Northern dry zone, Gamma distribution.
Water is an essential natural resource for the survival of life, a key input for plant growth and is instrumental in the upkeep of the environment. Although water is a renewable source, it is quite dynamic and scarce. The source of all water is annual precipitation/rainfall and it is affected by a number of factors. As a result, rainfall in India is highly variable, irregular and undependable with widespread variation among various meteorological sub-divisions in terms of distribution and amount. The highest and lowest annual average rainfall in India is 10,000 mm (Khasi-Jaintia Hill, Meghalaya) and 100 mm in Rajasthan, respectively (1—5). The primary source for agriculture production for most of the world is rainfall. The most important characteristic of rainfall is that it varies from place to place, day to day, month to month and also year to year. Agriculture has been the backbone of the Indian economy and it will continue to remain so for a long time. It is dependent on the amount of monsoon rains as a large part of the agricultural produce comes from the moonsoon fed crops. Good monsoon always means a good harvest and brings in cheers all around India. A weak or bad monsoon is always considered as a big setback to
India’s economy and always results in a big loss in the country GDP levels. Features of Study Area Southern dry zone consists of seven taluks of Mandya district, eight Eastern/Central taluks of Mysore, two taluks each of Tumkur and Hassan district. The zone is spread over a geographical area of 16.06 lakh ha forming 8.4% of TCA of Karnataka. The average annual rainfall of the zone is 717.8 mm. The percentage of area under forest, uncultivated land and fallow land is 19.11, 11.7 and 10.7 respectively and ramaining part is uncultivated land that can be brought under cultivation. The zone has a network of canal system (Krishnaraja Sagar, part of Kabini and Hemavathi) which provides a net irrigation to nearly 193,452 ha. The major crops grown in the zone are rice, ragi, sugarcane, jowar, oilseeds and a variety of vegetables and fruits. Methods The rainfall data were collected from the Direc-
885
BHAVANA ET AL
Table 1. Estimates of parameters of gamma distribution and goodness of fit statistics.
Months June Jul Aug Sep Southwest monsoon
α
Mandya Estimate Calculated β χ2
1.847 1.337 1.595 2.371 6.572
38.452 48.800 43.325 53.700 50.637
α
0.203 NS 0.633 NS 0.441 NS 0.226 NS 0.289 NS
Mysore Estimates Calculated β χ2
2.562 2.849 1.551 2.088 8.195
29.008 20.783 46.318 51.478 38.181
Chamaraj nagar Estimates Calculated β χ2
α
0.917 NS 0.129 NS 0.0002 NS 0.003 NS 0.809 NS
2.077 1.317 1.473 2.580 5.643
31.082 46.715 43.035 56.325 59.328
0.671 NS 0.292 NS 0.325 NS 0.759 NS 0.658 NS
Table 1. Continued. Hasan Months
Estimates β
α
Jun Jul Aug Sep Southwest monsoon
2.122 1.938 1.876 2.921 9.598
33.669 35.569 37.725 44.076 35.418
Tumkur Calculated χ2
Estimates α
0.534 NS 0.080 NS 0.100 NS 0.572 NS 0.424 NS
torate of Economics and Statistics, MS Building, Bangalore and data on other weather parameters like temperature, RH and sunshine hours for the southern dry zone (zone-6) of Karnataka were collected from Zonal Agricultural Research Station (ZARS), VC Farm, Mandya are used in this study.
1.787 1.867 2.524 3.111 7.495
0.420NS 0.868NS 0.047NS 0.752NS 0.709NS
39.611 48.285 47.307 54.851 60.175
waiting time until next raining is a random variable that is frequently modeled with a gamma distribution. The gamma distribution is defined by two parameters i.e., the shape (α) and scale (β) parameters, in the following formula : –
α –1
Gamma Distribution f
The gamma distribution is a two-parameter distribution classified under continuous probability distributions. It has a scale parameter ‘β’ and a shape parameter ‘α’. If ‘α’ is an integer then the distribution represents the sum of p independent exponentially distributed random variables, each of which has a mean of ‘β’. Application of gamma distribution based on intervals between events which derive from it being the sum of one or more exponentially distributed variables. In this form, illustration of its use include queuing models, the flow of items through manufacturing and distribution processes and the load on web servers and the many and varied forms of telecom exchange. Due to its moderately skewed profile, it can be used as a model in a range of disciplines, including climatology where it is a workable model for rainfall. The gamma distribution is frequently a probability model for waiting times; in case of rainfall, the
Calculated χ2
β
x
e (β) ( xβ ) (x) = ––––––––––––––––
,α > 0, β > 0, x > 0
βαΓ α
Where, •α a value of the standard mathematical function is termed as gamma function and is defined as ∞
Γα=∫t
α–1
e – t dt
0
The shape parameter (α) essentially determines the level of positive skew. This value found is always greater than 0 and for the particular case of α = 1, the distribution is exponential. When α > 1, the shape of the distribution alters to approach the y-axis at the origin. The scale parameter (β) determines the spread of values, stretching or squeezing the distribution when large or small, respectively. The product of the shape and scale parameters (αβ) gives the mean of the distribution. As the results deals with the distribution of monthly rainfall values only, this gives the mean monthly amount. The distribution variance is
886
BHAVANA ET AL
Table 2. Probabilities of rainfall amounts of south-west monsoon months based on gamma distribution in southern dry zone of Karnataka (1980—2010). Amount of rainfall (mm)
Jun
< 50 51—150 151—250 > 250
0.360 0.351 0.225 0.064
Mandya 0.375 0.437 0.125 0.063
0.464 0.321 0.144 0.071
0.194 0.387 0.290 0.129
< 50 51—150 151—250 > 250
0.322 0.419 0.193 0.066
Mysore 0.214 0.469 0.250 0.07
0.375 0.345 0.218 0.062
0.267 0.234 0.332 0.167
< 50 51—150 151—250 > 250
Chamarajnagar 0.290 0.337 0.486 0.332 0.128 0.272 0.096 0.039
0.366 0.294 0.265 0.066
0.303 0.272 0.274 0.151
< 50 51—150 151—250 > 250
0.285 0.356 0.25 0.109
Hassan 0.5 0.266 0.20 0.034
0.233 0.534 0.133 0.1
0.259 0.185 0.407 0.149
< 50 51—150 151—250 > 250
0.416 0.166 0.25 0.168
Tumkur 0.326 0.355 0.213 0.106
0.178 0.322 0.323 0.177
0.142 0.216 0.393 0.249
Jul
Aug
Sep
given by α β . 2
Fitting of Gamma Distribution The gamma distribution is fitted using method of moments for the south-west monsoon season rainfall and rainfall of individual months of the season by using the following estimators for the shape (α) scale (β) parameters. x–2 α=— s2 s2 β=— –x
Where, x– : Sample mean, s2 : Variance.
– The sample mean (X) and sample variance (s2) are calculated for monthly rainfall data of the southwest monsoon and overall south-west monsoon rainfall data for thirty years, using these sample mean and sample variance estimators of shape (α) and scale (β) parameters were computed. Using these estimators corresponding expected values were determined.
Testing the Goodness of Fit for Gamma Distribution The chi-square test is used to test if a sample of data emerged from a population with a specific distribution. Chi-square test is used to test the goodness of fit for the gamma distribution of rainfall data for different months of the season and overall season rainfall data for thirty years. The chi-square test is defined for the hypothesis : The rainfall data follow the gamma distribution; and the rainfall data do not follow the gamma distribution. For the chi-square goodness of fit computation, the statistical data are divided into ‘k’ classes and the test statistic is defined as K
2
X =
Σ[
i=1
( Oi – Ei )2 ––––––––– Ei
]
~ X2
(α, k – c ) d f
Where, observed frequency for class i , Expected frequency for class i. The expected frequency is calculated by using the estimates of scale and shape parameters computed by sample observations through statistical density function of gamma distribution. The test statistic follows, approximately, a chisquare distribution with (k – c) degrees of freedom where, k = Number of non-empty cells and c = Number of estimated parameters (including scale parameters and shape parameters) for the distribution + 1 and is a shape parameter. Therefore, the hypothesis is rejected, if calculated value of X2 > table value of X2 (α , k – c) d f Results and Discussion Estimates of parameters of gamma distribution
887
BHAVANA ET AL
and calculated X2 value for goodness of fit for different months namely June, July, August, September and for overall south-west monsoon season are presented in Table 1 and in all the cases values of α > 1, this implies that the distribution is less skewed and the probability density function will be shifted to the right. The value of scale parameter (β) is more than one for all the months and also for overall south-west monsoon season. Thus higher values of β imply that distribution shows a stretching pattern. For all the months and for the season as a whole the calculated chi-square values are found to be nonsignificant which indicates that the null hypothesis that rainfall data fallows gamma distribution cannot be rejected, which indicates that the distribution of rainfall data for the months of June, July, August, September and amount of rainfall of overall southwest monsoon season follows the gamma distribution. Based on the fitted distribution, probabilities of occurrence of rainfall for selected ranges of rainfall are presented in Table 2. The months of June, July, August, September and overall south west monsoon for the southern dry zone that includes the taluks come under of Mandya, Mysore, Chamarajnagar, Tumkur and Hassan districts. Table 2 indicates that, in Mandya district, the probability of occurrence of 51—150 mm rainfall is high in June, July and August while the probability of occurrence of 250 mm rainfall is low in June, July and August. In Mysore district, the probability of occurrence of 51—150 mm rainfall is high in June and July while the probability of occurrence of 250 mm rainfall is low in almost all the months. While in Chamarajnagar, the probability of occurrence of 51—150 mm rainfall is high in June and
July while the probability of occurrence of 250 mm rainfall is low in June, July and August. In Hassan district, the probability of occurrence of 51—150 mm rainfall is high in July and August while the probability of occurrence of 250 mm rainfall is low in July. The data also indicate that in Tumkur district, the probability of occurrence of 51—150 mm rainfall is high in July and August while the probability of occurrence of 250 mm rainfall is highest in September. Conclusion Non-significant chi-square values indicate the goodness of fit of the gamma distribution. Based on the fitted distribution, probabilities of occurrence of rainfall for selected ranges ( ≤ 50, 50—150, 150—250 and ≥250 mm) are presented for June, July, August and September. References 1. 2.
3.
4.
5.
Hafzullah Aksoy. 1999. Use of gamma distribution in hydrological analysis. J. Hydrol. 219 : 20—33. Lars S. H. and R. Vogel. 1999. The probability distribution of daily rainfall in the United States. Hydrol. Sci. J. 44 : 65—69. Burn D. H. and M. A. Hagelnur. 2002. Detection of hydrologic trends and variability. J. Hydrol. 255 : 107—122. Adugna L. 2005. Rainfall probability and agricultural yield in Ethiopia. Eastern Africa Soc. Sci. Res. Rev. 21 : 57—96. Das P. K., N. Subash, A. K. Sikka, V. N. Sharda and N. K. Sharma. 2006. Modeling of weekly rainfall using gamma probability distribution and Markov chain for crop planning in a sub-humid (dry) climate of central Bihar. Ind. J. Agric. Sci. 76 : 358—361.
