Estimate of intense rainfall equation parameters for rainfall stations of the Paraíba State , Brazil 1

distributed are essential in studies related to irrigation needs, water availability for domestic and industrial uses, soil erosion, flood control and water projects, among others (Oliveira et al. 2005, Araújo et al. 2008, Santos et al. 2010, Castro et al. 2011). According to Mello et al. (2008), the 21st century should experience a great frequency of extreme temperature and rainfall events. These maximum rainfalls, also denominated intense rainfalls


PALAVRAS-CHAVE
To characterize rainfall, it is necessary to know its duration, intensity and frequency of occurrence.These relationships are commonly denominated intensity-duration-frequency (IDF) curves, one of the most widely used methodologies in rainfall-flow transformation processes (Damé et al. 2008).Intense rainfall is determined by empirical adjustments of IDF equation parameters, derived from rainfall data for each of the stations (Santos et al. 2009, Back et al. 2012).The parameters of the intense rainfall equation can be obtained via nonlinear multiple regression, based on information extracted from the series of rainfall data (Campos et al. 2014), which do not always contain duration, but are composed of daily records.However, it is necessary to know which rainfalls lasted less than 24 h, in order to adjust IDF equations (Aragão et al. 2013).One alternative is disaggregation based on proportionality factors, which makes possible to obtain rainfall duration in minutes (Cetesb 1986, Garcia et al. 2011, Aragão et al. 2013).
In the Brazilian Paraíba State, pioneering studies on intense rainfall were conducted by Pfafstetter (1957), who used rainfall records from stations located in the cities of João Pessoa and São Gonçalo.For these areas, Pfafstetter (1957) adjusted the parameters of the relationship between rainfall and return period for different durations.In other study, Souza (1972) used data from João Pessoa to develop the IDF equation for this station.Since these outdated reports were the only ones available for Paraíba, a State that depends greatly on the efficient use of water for its cropping activities, it is required to update and estimate them for other locations in this region.
Due to the significant lack of information about rainfall intensity-duration-frequency (IDF) for most locations in Paraíba, and given the importance of knowing such information for the design of irrigation projects, this study aimed at estimating the parameters of the IDF equation for rainfall stations in this State.

MATERIAL AND METHODS
The study was carried out in 2014, at the Universidade Federal do Piauí, in Bom Jesus, Piauí State, Brazil.Daily rainfall data were collected from 132 stations in the Paraíba State, obtained from the Brazilian Water Agency databank (Brasil 2012).
First, the consistency of the data series of each station was analyzed, and those with less than 16 years of observation were excluded, leaving only one per municipality, and a total of ninety stations (Figure 1).
Once the fit of distribution data to the probability model had been assessed and disaggregation was achieved at shorter times, the parameters K, a, b and c of the intensity-duration-frequency equation were established (Equation 1), according to Pfafstetter (1982): where Mi is the mean maximum rainfall, in mm h -1 ; RP the return period, in years; t the rainfall duration, in min; and K, a, b, and c the parameters adjusted based on the rainfall data for the location.
The IDF equation parameters were adjusted via nonlinear multiple regression, using the generalized reduced gradient interaction method, and goodness of fit was assessed based on the coefficient of determination (r 2 ) estimated by the Equation 2: where x is the observed values; x the observed mean values; y the estimated values; and ӯ the estimated mean values.
Goodness of fit was also evaluated using the regression equation of data observed in relation to the estimated data, considering, in this case, the angular coefficient of the straight line.
After the parameters (K, a, b and c) were adjusted, they were used to estimate the maximum rainfall intensity for the return period of 25 years, as well as duration of 5, 30, 60, 360, 720 and 1,440 min.Next, these data were regionalized for all the Paraíba State, using the kriging method and ArcGis 10 software.

RESULTS AND DISCUSSION
From the probability models that exhibited the best fit for the series of rainfall data, the highest standard error observed was 41.83.Considering the standard error values obtained, more than one of the probability models provided a good fit to a same data series, according to the return period.This indicates that the model that best fits each return period should be used to estimate the maximum rainfall in different return periods.Lyra et al. (2006) observed the goodness of fit of different probability models to estimate rains at different periods of the year, indicating that fitting models is directly linked to the temporal distribution of rainfall.This confirms the need to apply more than one probability model to analyses covering different times, providing a greater security to projects dimensioned according to these estimated rainfall values.
Of the ninety stations studied, the Log-Normal III model showed the best fit to the data series (49.3 % of the stations), followed by the Gumbel model, with 16.1 % of the stations.The models with the poorest fit were Pearson III, Log-Pearson III and Log-Normal II, with 14.1 %, 12.8 % and 7.8 % of the stations, respectively (Figure 2).Martins et al. (2011) estimated the maximum flow and rainfall using probability models, also obtaining a better fit with the Log-Normal III distribution model.These findings are similar to those reported by Silva et al. (2003a), in determining the parameters of IDF equations for the Tocantins State, Brazil.These results confirm that, in addition to the Gumbel model (Santos et al. 2009), widely recommended in literature, the Log-Normal III may also be useful to estimate rainfall Source: Cetesb (1979).

Rainfall-flow transformation interval
Coefficient 1 day to 24 h 1.14 1 day to 12 h 0.85 24 h to 10 h 0.82 24 h to 8 h 0.78 24 h to 6 h 0.72 24 h to 1 h 0.42 1 h to 30 min 0.74 1 h to 25 min 0.91 1 h to 20 min 0.81 1 h to 15 min 0.70 1 h to 10 min 0.54 1 h to 5 min 0.34  2. Adjusted values for parameters of the intensity-duration-frequency (IDF) equation 1 and respective determination coefficient (r 2 ) for each studied rainfall station in the Paraíba State, Brazil.
1 The parameters K, a, b and c are defined by Pfafstetter (1982).
in different return periods, having the lowest standard error of the estimates as a fit criterion.
The values of the fitting IDF parameters varied significantly from one station to another.The K parameter estimates ranged between 473.139 and 2095.776,respectively for the Remígio and Santa Rita stations.Moruzzi & Oliveira (2009), Souza et al. (2012) and Silva et al. (2012) also reported a similar variation for this parameter, attributing that to the interaction between K and the other IDF parameters.The estimates of the parameter "a" ranged between 0.025 and 0.386, respectively for the Sousa and Santa Rita stations; parameter "b" between 9.836 and 30.214, respectively for the Remígio and Santa Rita stations; and parameter "c" between 0.745 and 0.915, respectively for the Remígio and Santa Rita stations (Table 2).Similar results have been reported in others studies, with the variability in IDF parameters attributed primarily to the different rainfall distribution (Silva et al. 2003b, Santos et al. 2009, Campos et al. 2014).Silva et al. (2002) also found a wide variation in the estimates of the IDF equation parameters in a study carried out in the Bahia State.For the K parameter, the estimates ranged from 1121.260 to 8999.000; parameter "a" from 0.174 to 0.245; parameter "b" from 19.457 to 56.068; and parameter "c" from 0.783 to 1.119.Similarly, Santos et al. (2009) also observed a significant variation for stations assessed in the Mato Grosso do Sul State.These results confirm that the IDF of rainfall is directly linked to its spatial distribution (Silva et al. 2002, Santos. et al. 2010), corroborating the need to determine these parameters for each specific rainfall station.
In most of the equations adjusted by applying the daily rainfall disaggregation methodology, the r 2 values were above 0.99, and of the ninety stations studied, only three exhibited values below 0.99, the lowest being 0.97 at the Santa Rita station.These results show the good fit of the estimates obtained by the methodology used, in relation to the data observed.
The spatialization of maximum rainfall intensity values with the estimates of the IDF equation parameters shows that the greatest intensities, regardless of duration, occur in the coastal and highlands at the "sertão" of the Paraíba State (Figure 3).In general, the spatial distribution of maximum rainfall intensity estimated by the IDF equation corroborates other studies on rainfall pattern in the Paraíba State (Limeira et al. 2012, Silva et al. 2009).These findings demonstrate that the rainfall values estimated for different durations and return periods by IDF equations have a high correlation with the rainfall distribution observed in this region.

Figure 1 .
Figure 1.Location of ninety rainfall stations used to estimate the intensity-duration-frequency (IDF) equations for the Paraíba State.Source: Brasil (2012).

Figure 2 .
Figure 2. Frequency of the best fits to the respective probability distribution models.

CONCLUSIONS 1 .
The Log-Normal III distribution model shows a better fit to the data series, for determination of maximum rainfalls for different return periods in the Paraíba State; 2. The intense rainfall equations for most stations showed a good fit, with coefficients of determination above 0.99, supporting the methodology used in this study; 3. The intensity-duration-frequency (IDF) equation parameters show a wide variation among stations, indicating the need to determine them for each station; 4. The spatialization of the maximum rainfall intensity values with the estimates of IDF equation parameters shows that the highest intensities occur in the regions of coastal and highlands at the "sertão" of the Paraíba State.
(Sousa et al. 2009)II, Log-Normal III, Pearson III and Log-Pearson III(Santos 2010).For each station, the maximum rainfall in which the data series showed the best fit to the probability model was selected, i.e., the distribution model with the lowest standard error.All these stages were conducted using the SisCAH software(Sousa et al. 2009).Afterwards, the probability model that best fitted the data series was determined to estimate the maximum daily rainfall, and the number of occurrences of each model was counted.

Table 1 .
Coefficients used for daily rainfall disaggregation in shorter time intervals.