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Intensity-duration-frequency of maximum rainfall in Mato Grosso State

Intensive rainfall is an important meteorological variable that is of technical interest in hydraulic projects. This study therefore generated Intensity-Duration-Frequency equations (IDF) for 14 weather stations in Mato Grosso State, based on pluviograph analysis. Annual maximum rainfall data regarding 10-to-1440-minute long rainfall events were collected from digitized daily pluviographs. Data adherence to the generalized extreme value distribution (GEV) was checked through the Kolmogorov-Smirnov test at a 20% significance level. Next, the maximum probable rainfall for return periods such as 2, 5, 10, 20, 30, 50 and 100 years was calculated and the IDF equations were adjusted. The performance of the IDF equations was evaluated based on mean absolute error (MAE), root mean square error (RMSE), bias, Willmott's concordance index and Nash-Sutcliffe efficiency index (ENS). Adjusting the IDF equations was only possible for rainfall durations ranging from 10 to 360 min at each station due to the low frequency of longer rainfalls. High variation was present in parameters of the IDF equation and in maximum rainfall intensity between stations. The satisfactory performance of the models, as attested to by statistical indices, allows using IDF equations adjusted for rainfall durations from 10 to 360 min, and return periods from 2 to 100 years, in the regions of the Mato Grosso weather stations.


INTRODUCTION
Intensive rainfall is one of the most important meteorological variables in climate studies, as it generates a considerable volume of water in short intervals (Pereira et al., 2017). Thus, knowledge regarding the variables characterizing maximum rainfall, as well as the correlations between rainfall intensity, duration and frequency, are of technical interest to hydraulic projects such as spillways, channel terraces, agricultural, urban and road drainage systems, among others (Cheng and Amir, 2014).
The most accepted way to characterize maximum rainfall relies on the intensity-durationfrequency equation (IDF) (Campos et al., 2014). The IDF curves are based on historical rainfall time-series data and are designed to capture the intensity and frequency of precipitation for different durations by fitting a theoretical probability distribution to annual extreme rainfall (Oriani et al., 2017;Ouarda et al., 2019). The first uses of IDF relationships date back to the 1930s (Bernard, 1932) and the first curve in Brazil was done in 1958 (Pfafstetter, 1958). Nowadays, because of its importance and ease of application, IDF curves are widely used in water management and other engineering design applications (Cheng and Amir, 2014). Therefore, the difficulty in generating intense rainfall equations in Brazil is mainly attributed to the limited availability of data about rainfall network density and time range (Silva et al., 2012). These issues are particularly concerning in the Northern and Central-Western regions of the country, as in Mato Grosso State, where pluviograph data sets comprise less than 30 years of collected data -the assessment period recommended by the World Meteorological Organization.
In addition to the lack of historical data about intense rainfall events, Mato Grosso State presents a large area that includes the Cerrado, Pantanal and Amazon Biomes, which have distinct rainfall characteristics (Marcuzzo et al., 2011). Moreover, the State has been facing deep land-use and occupation changes in the last decades, mainly because of increasing urbanization, expansion of agriculture and construction of hydroelectric power plants.
However, the State has not yet defined intense rainfall equations based on the analysis of pluviograph records for most of its counties, since previous studies focused on adjusting IDF equations to Mato Grosso State were based on daily rainfall data disaggregation processes (Fietz et al., 2010;Pizzato et al., 2012;Mossini Junior et al., 2016). Thus, the current study calibrated and evaluated the statistical performance of intensity-duration-frequency equations generated for 14 pluviograph stations located in the Cerrado, Amazon and Cerrado-Amazon-Pantanal transition biomes in Mato Grosso State.

MATERIALS AND METHODS
Mato Grosso State is located between geographic coordinates 06°00' S, 19°45' S and 50°06' W, 62°45' W; its territory covers 903,202,446 km 2 (IBGE, 2017). According to the Köppen classification, Aw (tropical savanna climate) and Cwa (tropical climate) are the predominant climates in the region; mean monthly temperature ranges from 23.00°C to 26.84°C, and total annual rainfall ranges from 1,200 to 2,000 mm (Souza et al., 2013). The region has two well-defined seasons: the rainy season, from October to April; and the dry season, from May to September.
Rainfall data were collected from pluviographs belonging to the National Hydrometeorological Network (CPRM / ANA); these pluviographs referred to 14 counties (Table 1): 4 in the Amazon Biome (Northern mesoregion), 3 in Cerrado-Amazon-Pantanal transition biomes (Southwestern mesoregion) and 7 in the Cerrado biome (Southeastern mesoregion) (IBGE, 2012;2013) (Figure 1). Since the stations did not have coincident data periods, no baseline study was adopted. In addition, only stations with at least 10 years of data were selected, however, and it was decided not to fill in data gaps in order to avoid bias in the estimation of the maximum annual rainfall.
Rainfall data recorded by rain gauges were obtained through the rainwater digitization system (HidroGraph 1.02) developed for the National Water Agency (ANA) by the Water Resource Research Group of the Agricultural Engineering Department of the University of Viçosa. Maximum annual precipitation heights recorded by each station for 10,20,30,40,50,60,120,180,240,360, 720 and 1440-minute-long rainfall events were used to build the annual dataset about extreme rainfall events.    Table 1.
The data set was tested for non-stationarity using the Mann-Kendall Test Modified as recommended in Cheng and Amir (2014) and Ouarda (2019). Then, the annual dataset for extreme rainfall events was adjusted to Generalized Extreme Value distribution (GEV), based on the maximum likelihood estimation (MLE) method applied to estimate probability distribution parameters. The adherence of the adjustments made in the series to the GEV distribution was investigated through the Kolmogorov Smirnov test, at a 20% probability level. This significance level was selected to limit the hypothesis test, since increased significance levels reduce the critical value of the statistical test (Naghettini and Pinto, 2007). After the GEV distribution parameters were adjusted, the probable maximum rainfall of each rainfall duration was estimated for return periods of 2, 5, 10, 20, 30, 50 and 100 years.
Parameters of the intensity-duration-frequency equation applied to each station were determined based on the Gauss-Newton non-linear adjustment technique (Chapra and Canale, 2006;Naghettini and Pinto, 2007), by using the maximum annual rainfall intensity recorded for return periods (RP) of 2, 5, 10, 20, 30, 50 and 100 years and rainfall durations (t) of 10,20,30,40,50,60,120,180,240,360, 720 and 1440 minutes, as shown in Equation 1: Where in: i is the maximum intensity (mm h -1 ); RP is the return period (years); t is the rain duration time (min); and K, a, b and c are the adjusted local coefficients.
Rev. Ambient. Água vol. 15 n. 1, e2373 -Taubaté 2020 Where in: Pi is the estimated intensity (mm h -1 ); Oi is the observed intensity (mm h -1 ); O is the mean of observed intensities (mm h -1 ); and N is the number of sample values. Table 2 presents the means and standard deviations of the annual series of maximum rainfall intensities corresponding to rainfall durations from 10 to 1440 minutes recorded by the stations in Mato Grosso State. Although some stations presented a nonstationarity tendency, in order to avoid inconsistencies between the evaluations of different stations and due to the fact that the data series generally only had 10 years, the data series were treated in a traditional way without considering non-stationary influences.

RESULTS AND DISCUSSION
Maximum rainfall intensities tended to be higher in the Amazonian biome (Northern region of the State), whose means varied from 102.4 mm h -1 (10-min-long rainfalls) to 0.7 mm h -1 (for 360-min-long rainfalls). Stations located in the Southwestern and Southeastern regions of the State presented similar means, which ranged from 97.8 and 98.8 mm h -1 (10 min) to 0.1 and 0.3 mm h -1 (360 min), respectively. These results match the ones found by Marcuzzo et al. (2011), who recorded higher rainfall indices in the Amazonian biome (Northern Mato Grosso State) than in other phyto physiognomies in Southern Mato Grosso State.
There was no record of 1440-min-long (or longer) rainfall events in Mato Grosso State; however, there were only 7 records of 720-min-long rainfall events in the State. Alves et al. (2013) conducted a study in Cuiabá County-MT and also found low frequency of long-term rainfall events: there were three 1440-min-long rainfall events and thirteen 720-min-long events in a 22-year dataset.
The low frequency of rainfalls longer than 360 minutes results from the predominance of convective precipitations in the State, which, according to Salio et al. (2007), overall last less than 9 hours. On the other hand, longer precipitations are linked to frontal rainfall caused by cold fronts brought from the Southern region of the country by polar anticyclones (Seluchi, 2009). However, the incidence of polar anticyclones strong enough to cause precipitation is rare and often results in low-intensity/volume rainfalls (Nimer, 1972;Seluchi, 2009). Table 3 presents the IDF equations applied to the herein-investigated locations and the respective values of the adjusted parameters "K", "a", "b" and "c". Parameters "K" and "b" showed higher coefficient of variation (CV), thus, according to Silva and Oliveira (2017) indicating no spatial dependence between stations. Parameters "a" and "c" presented average variability in the State; CV values were 33.39% and 21.08%, respectively.  The number in parentheses (below the mean) corresponds to the standard deviation (S).
High variation in parameters of the IDF equation were also reported in Bahia (Silva et al., 2002); Tocantins (Silva et al., 2003), Mato Grosso do Sul (Santos et al., 2009), Pernambuco (Silva et al., 2012), and Piauí (Campos et al., 2014) states; such variation was associated with the large number of attributes involved in the process of modeling the dynamics of environmental phenomena (Mello and Silva, 2009).
Although the parameters of the IDF equation presented high variation in most stations of Mato Grosso State, the ones located near each other in the Cerrado Biome (Southeastern Mesoregion) recorded average CVs for parameters "k" (33.94%) and "A" (36.92) , as well as low variability for "b" (36.95%) and "c" (7.66%). Table 4 presents the results of the performance analysis applied to the IDF curves generated to estimate the maximum rainfall intensity at pluviographic stations in Mato Grosso State, based on different durations and return periods. The analysis of statistical indices showed that all the equations presented satisfactory performance, as seen in R 2 values higher than 86.65% 8 Marlus Sabino et al. (Humboldt), which reached maximum value of 95.96% (Jusante Foz Peixoto de Azevedo). Mean absolute error (MAE) and root mean square error (RMSE) estimates showed good fit of the equations, which recorded 15.2 mm h -1 (MAE) and 18.9 mm h -1 (RMSE) for Pontes and Lacerda stations, as well as 9.9 mm h -1 (MAE) and 12.7 mm h -1 (RMSE) when all stations were taken into consideration. The bias index analysis indicated that all models were underestimated, except for Fazenda Taquari station, where there was a model overestimation of 0.3 mm h -1 . The Willmott concordance (d) and the Nash-Sutcliffe efficiency (Ens) indices confirmed the good fit of the models, since their values were close to 1 and they were classified as suitable, according to the criterion set by Van Liew et al. (2007). Figure 2 shows the results of the adjusted rainfall intensities at different return periods, based on IDF equations of maximum rainfall intensities in Mato Grosso State. For the probable maximum precipitation estimates to design hydro-agricultural projects, usually those obtained for rains of 10-min-duration in a 10-year return period, Alto Garça station recorded the highest intensity (183.2 mm h -1 ), while the lowest was recorded at Ponte Branca station (106.1 mm h-1). These results are on average 9.5 mm h -1 lower than the ones found by Fietz et al. (2010) for the same stations at Mato Grosso. Although, it's important to notice that the estimates presented by Fietz et al. (2010) were done by means of desegregation of daily precipitation and not by pluviograph analysis, which could lead to rainfall-intensity overestimation, as can be shown when compared to the study of Castro et al. (2011) that found by pluviogram analysis an intensity of 165.6 mm h -1 at Cuiabá, results these similar to those of this work.
The lowest 60-and 360-minute rainfall estimates were recorded for Fazenda Taquari station (35.2 and 4.4 mm h -1 , respectively). Porto do Gaúchos and Pontes e Lacerda stations recorded the highest intensity for 60-and 360-minute-duration rainfalls (67.3 and 19.1 mm h -1 , Rev. Ambient. Água vol. 15 n. 1, e2373 -Taubaté 2020 respectively). The highest rainfall intensity variations happened at rainfall durations shorter than 120 minutes due to the prevalence of convective rains; the curve smoothed after 120minute-duration rainfalls. Topography is also a key factor for rainfall intensity in Mato Grosso State (Souza et al., 2013), and as such similar rainfall estimates were found in Amazon Biome stations located at altitudes ranging from 242 to 400 m, and high variation was found in estimates for the Cerrado Biome located at Petrovina Mountain, where the altitude varied from 220 to 845 m. The orographic effect became evident when Alto Garça station was compared to Ponte Branca station. These stations were 75 km away from each other; however, they recorded rainfallestimate differences of 77.1 mm h -1 , (183.2 mm h -1 at Alto Garça and 106.1 mm h -1 at Ponte Branca) for hydro-agricultural rainfall projects, corresponding to precipitation of 10-minute duration and 10 years of return time.
The small-scale variability of rainfall intensity can sensibly increase the complexity of the hydrological response, conditioned by weather and soil indicators and topography (elevation) (Oriani et al., 2017). For Haiden e Pistotnik (2009), in mountainous terrain (similar to what occurs in the southern and central regions of Mato Grosso state), elevation differences strongly contribute to the small-scale spatial variability of precipitation. The effect is most pronounced for long accumulation periods such as monthly or annual; however, they may interfere with the intensity of local precipitation. Marlus Sabino et al.

CONCLUSIONS
The IDF equations presented satisfactory adjustments, with determination coefficients above 87%, allowing the estimation of intense rainfall with 10 to 360 minutes duration and return periods from 2 to 100 years. The equations may therefore be used as a basis for hydrological studies in the state of Mato Grosso, and may also serve as a reference for engineering projects and prevention of extreme precipitation events.
The regionalization of the IDF equations for the state is not indicated due to the high variability of the IDF curves parameters of the current stations. The high variability is still indicative of the need for more weather stations to be distributed in Mato Grosso.