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Space–Time Kriging of Precipitation: Modeling the Large-Scale Variation with Model GAMLSS

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Space–Time Kriging of Precipitation: Modeling the Large-Scale Variation with Model GAMLSS water Article Space–Time Kriging of Precipitation: Modeling the Large-Scale Variation with Model GAMLSS Elias Silva de Medeiros 1,* , Renato Ribeiro de Lima 2, Ricardo Alves de Olinda 3 , Leydson G. Dantas 4 and Carlos Antonio Costa dos Santos 4,5 1 FACET, Federal University of Grande Dourados, Dourados 79.825-070, Brazil 2 Department of Statistics, Federal University of Lavras, Lavras 37.200-000, Brazil; rrlima.ufl[email protected] 3 Department of Statistics, Paraíba State University, Campina Grande-PB 58.429-500, Brazil; [email protected] 4 Academic Unity of Atmospheric Sciences, Center for Technology and Natural Resources, Federal University of Campina Grande-PB, Campina Grande 58.109-970, Brazil; [email protected] (L.G.D.); [email protected] or [email protected] (C.A.C.d.S.) 5 Daugherty Water for Food Global Institute, University of Nebraska, Lincoln, NE 68588-6203, USA * Correspondence: [email protected]; Tel.: +55-67-3410-2094 Received: 16 October 2019; Accepted: 7 November 2019; Published: 12 November 2019 Abstract: Knowing the dynamics of spatial–temporal precipitation distribution is of vital significance for the management of water resources, in highlight, in the northeast region of Brazil (NEB). Several models of large-scale precipitation variability are based on the normal distribution, not taking into consideration the excess of null observations that are prevalent in the daily or even monthly precipitation information of the region under study. This research proposes a novel way of modeling the trend component by using an inflated gamma distribution of zeros. The residuals of this regression are generally space–time dependent and have been modeled by a space–time covariance function. The findings show that the new techniques have provided reliable and precise precipitation estimates, exceeding the techniques used previously. The modeling provided estimates of precipitation in nonsampled locations and unobserved periods, thus serving as a tool to assist the government in improving water management, anticipating society’s needs and preventing water crises. Keywords: water resources; GAMLSS; geostatistics 1. Introduction An effort has emerged in several areas of knowledge, especially in the climate sciences, to analyze a phenomenon by both space and time [1–3]. It is already established that the spatial–temporal behavior of rainfall is of great importance for the water resources of a region and that it has a direct influence on human activities, such as agriculture and commerce [4]. The northeast region of Brazil (NEB) is characterized by irregular distributions of rainfall, resulting in high spatial and temporal variability of precipitation [4]. In this region, it is common to observe high rainfall indices in a particular location and no record of rain in its surroundings [5]. The state of Paraíba, located in this region, presents the same characteristics, with periods of drought in rainy seasons and behavior different from the precipitation between the mesoregions that constitute the region [6]. Consequently, Paraíba presents a problem concerning the spatiotemporal variability of rainfall. The high variability of recorded rainfall in this region may be attributed to the characteristics of the meteorological systems and their dates of installation in different regions of the state. The periodicity is related to the general behavior of sea surface temperature (SST) in the tropical Pacific region, contributing to the events of El Niño and La Niña, and most notably, by the behavior of the SST Water 2019, 11, 2368; doi:10.3390/w11112368 www.mdpi.com/journal/water Water 2019, 11, x FOR PEER REVIEW 2 of 17 Water 2019, 11, 2368 2 of 16 of the SST in the tropical Atlantic region due to its proximity to the study area. In the literature, in theseveral tropical studies Atlantic have region evaluated due to precipitation its proximity in to thethis studyregion area. using In thespatial literature, statistics several techniques studies [7,8]. haveHowever, evaluated these precipitation previous in works this region did usingnot take spatial spatial statistics and techniquestemporal dependence [7,8]. However, into these account previoussimultaneously. works did not take spatial and temporal dependence into account simultaneously. SeveralSeveral previous previous works works used used multiple multiple linear linear regression regression to model to model the deterministicthe deterministic component, component, suchsuch as the as Gaussianthe Gaussian model model [9–11 [9].– These11]. These works works assumed assume thatd that the variablethe variable response response to be to studied be studied presentedpresented a behavior a behavior that couldthat could be correctly be correctly modeled modeled by a Gaussianby a Gaussian approach. approach. However,However, recent recent studies studies have h evaluatedave evaluated non-Gaussian non-Gaussian models, models, including including generalized generalized linear linear modelsmodels (GLM), (GLM) with, with variable variable response response Gamma Gamma [12,13 [12,13]]. Once. Once the Gammathe Gamma model model is assumed, is assumed, the the variablevariable domain domain has tohas be to strictly be strictly positive, positive, and and it does it does not includenot include the valuethe value zero. zero. For example,For example, Menezes et al. (2016) [12] and Monteiro et al. (2017) [13] analyzed the concentration of NO2, which Menezes et al. (2016) [12] and Monteiro et al. (2017) [13] analyzed the concentration of NO2, which had asymmetrichad asymmetric behavior behavior with excesses with ofexcesses zero. However, of zero. whenHowever, using GLMwhen with using Gamma GLM distribution, with Gamma it is notdistribution, possible toit modelis not possible the frequency to model of zeroes the frequency in the variable of zeroes under in study.the variable To circumvent under study. this To limitation,circumvent some this models limitation, captured some the models frequency capture of zeroesd the frequency in continuous of zeroes variables. in continuous Stauffer variables. et al. (2017)Stauffer [3] analyzed et al. the(2017) spatial–temporal [3] analyzed the distribution spatial–temporal of daily precipitationdistribution atof 117daily stations precipitation located inat 117 Tirol,stations Austria, located over 42 in years. Tirol, TheseAustria, authors over 42 pointed years. toThese the high authors frequency pointed of to zero the observationshigh frequency in theof zero data,observations suggesting the in usethe ofdata, the suggesting normal zero-censored the use of the model. normal However, zero-censored only the model. trend component However, wasonly the modeled,trend notcomponent taking into was account modeled, the spatial–temporal not taking into dependence account the of precipitationspatial–temporal in this dependence region. of precipitationIn this study, wein this analyzed region. the total monthly rainfall measured in the State of Paraíba, Brazil, which presents anIn asymmetricalthis study, we behavior analyzed and the has total a high monthly incidence rainfall of zeros, measured which in is the a climate State of characteristic Paraíba, Brazil, of thewhich region presents under study. an asymmetrical Thus, to model behavior this behavior, and has a newa high approach incidence in theof regressionzeros, which adjustment is a climate is proposedcharacteristic by applying of the generalizedregion under additive study. models Thus, to for model location, this scale, behavior, and shape a new (GAMLSS) approach [14 in]. the Theseregression models have adjustment several advantages, is proposed such by applying as all distribution generalized parameters additive can models be modeled for location as covariate, scale, and functions,shape cases (GAMLSS of over-dispersion) [14]. These models and excess have of several zeroes inadvantages, the data can such be modeled, as all distribution and any distribution parameters can to thebe response modeled variable as covariate can be functions adjusted., cases of over-dispersion and excess of zeroes in the data can be modeled, and any distribution to the response variable can be adjusted. 2. Materials 2. Materials The study region was Paraíba State, Brazil (Figure1), located in the NEB, which is positioned 2 between parallelsThe study 6◦ regionand 8◦ wasS and Paraíba meridians State, 34 ◦Braziland 39 (Figure◦ W, with 1), anlocated area ofin approximatelythe NEB, which 56,500 is positioned km . Paraíbetweenba is included parallels in the 6° Tropicaland 8° SRegion, and meridi andans its area 34° isand divided 39° W into, with four an geographical area of approximately mesoregions: 56,500 Zonakm². da Mata, Paraíba Agreste, is included Borborema, in the and Tropical Sertão. Region, and its area is divided into four geographical mesoregions: Zona da Mata, Agreste, Borborema, and Sertão. Figure 1. Pluviometric stations over the four mesoregions of Paraíba State, Brazil. The information for eachFigure respective 1. Pluviometric number is presented stations over in Table the 1fo. ur mesoregions of Paraíba State, Brazil. The information for each respective number is presented in Table 1. Water 2019, 11, 2368 3 of 16 Table 1. Pluviometric stations with an identification number (ID), name of the mesoregion, name of rainfall station, latitude (Lat), longitude (Long), Altitude (Alt), and precipitation quantiles 2.5%, 50%, 97.5%. Quantiles ID Mesoregion Station Lat Long Alt (m) 2.5% 50.0% 97.5% 1 Zona da Mata Alhandra
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