FUNCTIONS OF PROBABILITY FOR FITTING MONTHLY RAINFALL IN SITES OF MATO GROSSO DO SUL

The identification of the probability distribution function for the representation of the monthly rainfall is relevant in agricultural planning, mainly regard to the establishment of crops. The aim of this work was to verify the probability distribution (exponential, gamma or normal) which best fits to data monthly rainfall of 14 sites in the state of Mato Grosso do Sul. Rainfall data of 14 stations (sites) of the State of Mato Grosso do Sul it were obtained from the National Water Agency (ANA) database, collected in the period 1975 2013. At each of the 168 time series of monthly rainfall was applied the Kolmogorov-Smirnov test to assess the fit to probability distributions exponential, gamma and normal. The normal probability distribution presented the best fit to monthly rainfall series of Mato Grosso do Sul and it can be used for the estimation the monthly rainfall, especially in the rainy season months (October to March). The exponential probability distribution can be used for the estimation of monthly rainfall in the driest months of the year (May to September). Thus, we recommend that these distributions be used in future research, aimed to estimate the probable rainfall for the Mato Grosso do Sul State.


INTRODUCTION
The State of Mato Grosso do Sul (MS) covers an area of approximately 350,000 km², of which 13,000 km² are explored in agriculture, being the crops of higher expression soybean, maize, cotton, sugarcane and irrigated rice (CONAB, 2014).Agriculture has the rainfall as its main source of water, which may compromise the crop production due to its uneven behavior, sometimes with long periods of drought, sometimes with high intensity rains that exceed the water retention capacity of the soil, triggering floods (SILVA et al., 2007;SOCCOL et al., 2010;VIEIRA et al., 2010;CORRÊA et al., 2014).Besides the influence in agriculture, very long periods of droughts affect the water level of water sources and reservoirs of hydroelectric plants, bringing problems to the urban water supply and electric power generation (RODRIGUES et al., 2013;TEODORO et al., 2015a;TEODORO et al., 2016).Teodoro et al. (2015b), by using the algorithm Ward to evaluate the spatial and temporal variability of rainfall in the state of Mato Grosso do Sul, identified the main systems leading producers of rainfall systems.The central region of that State is influenced by South Atlantic Sub-tropical Anticyclone, Chaco's Low, Bolivia's Upper, Low Levels Jet, Madden-Julian Oscillation and South Atlantic Convergence Zone.The northern region is is affected by Upper Tropospheric Cyclonic Vortex, Front Systems and South Atlantic Convergence Zone.The southern region is influenced by Upper Tropospheric Cyclonic Vortex and Front Systems.
The impossibility of know the exact evolution of rainfall values over time and space according to their random nature is highlighted by Sampaio et al. (2007).From these difficulties, probabilistic models are used to describe the behavior both the expected rainfall as the precipitation period, because obtaining the correct temporal distribution of rainfall is important in agricultural planning, and enable the appropriate use of water resources for given region (RODRIGUES et al., 2013).
Most of the probability density functions have from one to three parameters and the multiparameter functions generally show superior fit to rainfall when compared to a single parameter (LYRA et al., 2006).However, due to the complexity of the processes involved in space-time variation of rainfall, are selected these functions according to the criteria of best fit with historical data, ease of estimates of its parameters and computational flexibility (DUAN et al., 1998;TEODORO et al., 2015b).
Thus, the aim of this study was undertaken to verify the probability distribution (exponential, gamma or normal) which best fit to monthly rainfall data from 14 sites of State of Mato Grosso do Sul.

MATERIAL AND METHODS
Rainfall daily database of 14 stations (sites) of Mato Grosso do Sul, from 1975 to 2013, it were obtained of the Database of the Agência Nacional de Águas (ANA, 2014).At each site and year, it were added up the rainfall daily data to obtain the monthly rainfall (mm.monthly -1 ), of each month of the year, as performed by Teodoro et al. (2015 b).Thus, were formed 168 time series (12 months × 14 sites), with different numbers of years of observations in each series, defined according to the availability of meteorological data (Table 1).Processing of the data removing the outliers, which are observations that deviate markedly from the others in the sample in which they occur, causing inconsistencies was carried out.About 10 % historical series showed failures (outliers), which are filled by the climatological normal of each of the 11 micro-regions of the state.-51º25'26'' 1983-2013Três Lagoas 313 -20º47'41'' -51º42'46'' 1975-2013 The exponential distribution generally fits well to the data that have marked asymmetry, as histograms in the shape of "J" inverted (THOM, 1958).Its probability density function f (x) is express as follows by Equation 1: x is the random variable (monthly rainfall, mm) and λ is the inverse of the average.
For total rainfall of monthly periods or smaller, the distribution gamma has been one of the most used (ASSIS et al., 1996), being represented its probability density function as follows by Equation 2: with β, α, τ(α) > 0 and f(x) = 0 for x < 0 being: β scale parameter, α the shape parameter and τ(α) the incomplete gamma function of the parameter α, defined by Equation 3 (THOM, 1958): The normal probability distribution, also called Gaussian curve presents two parameters and its probability density function is define by Equation 4(HASTINGS; PEACOCK, 1975): wherein: µ is the average and σ the standard deviation.
In each month and site, it were calculated the probabilities (P) of the monthly rainfall (RAINFALL) to be less than 50 mm [P (RAINFALL < 50 mm)], between 50 mm and 100 mm [P (50 mm ≤ RAINFALL < 100 mm)] and greater than 100 mm [P (RAINFALL ≥ 100 mm)].The choice of these intervals was based on the recommendation of Assis et al. (1996) which recommends for the design of these agricultural projects.These probabilities were calculated from the parameters of the distribution (exponential, gamma and normal) of best fit, ie, the parameter with the lowest value of the statistic D maximum of the Kolmogorov-Smirnov adhesion test (KS).Statistical analyzes were performed with the use of the application Microsoft Office Excel ® and of the Statistica 7.0 ® software (STATSOFT, 2005).
The largest of adhesions of normal distribution was observed in the months of the rainy season, identified by Teodoro et al. (2015b) among October to March, agreeing with Lyra et al. (2006) and Junqueira Júnior et al. ( 2007) of which the same fits well to values of weekly, monthly and seasonal rainfall that do not show many dry periods.
In the dry season, identified by Teodoro et al. (2015b) among May to September, the exponential distribution showed 31 series with fit in the p-value class ≥ 0.20, being higher than the normal distribution (29 fits), and gamma (21 fits) (Table 2).This can be explained by the higher frequency observed during the initial classes (lower value of rainfall), decreasing mildly in the shape of "J" inverted, with strong positive asymmetry.Lyra et al. (2006) andJunqueira Júnior et al. (2007) obtained similar results, concluding that in the estimating the probability of monthly rainfall, the exponential distribution showed the best results in the dry season.
According to Dourado Neto et al. (2005), one major problems of estimating the probable rainfall with gamma distribution function is the estimation of α and β parameters, due to the complexity and extension of the involved calculations.The exponential and normal distribution functions have less difficulty for obtaining the parameters and ease in the probability estimates.
Thus, estimates of the parameters λ, α, β, µ and σ in each month and site (Tables 3 and 4) enable to estimate the probabilities above or below any monthly rainfall value, in order to minimize risks and facilitating the planning of various agricultural activities.This enable the prediction of better time for the tillage, harvest, sowing, fertilizer application and defensive (SILVA et al., 2007;ÁVILA et al., 2009;VIEIRA et al., 2010).

Table 1 .
Altitude (m), latitude (º), longitude (º) and observation period of the monthly rainfall of 14 sites in the State of Mato Grosso do Sul, Brazil.

Table 2 .
and Uliana et al. (2013)obtained similar results on researches about the probable rainfall in the States of Paraná, Minas Gerais and Espírito Santo, respectively.D maximun statistic and p-value classes(1)(in brackets) of the Kolmogorov-Smirnov adhesion test, applied to check the fit of time series of monthly rainfall (mm) in 14 sites of the State of Mato Grosso do Sul, to the functions of exponential, gamma and normal probability distributions.

Table 3 .
Estimate parameters for the functions of exponential and gamma probability distributions of time series for monthly rainfall (mm) in 14 sites of the State of Mato Grosso do Sul.Brazil.