# Modeling Spatiotemporal Rainfall Variability in Paraíba, Brazil

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{2}in northeastern Brazil between the 6° and 8° parallels of south latitude and the 34° and 39° meridians of west longitude. The state of Paraíba is situated in a tropical region, and it is divided into the following four mesoregions: Zona da Mata, Agreste, Borborema, and Sertão.

#### 2.1. Spatiotemporal Model

#### 2.2. Components of the Trend

#### 2.3. Spatiotemporal Variogram

#### 2.4. Generalized Product-Sum Model

#### 2.5. Spatiotemporal Kriging

#### 2.6. Selection and Performance of the Model

## 3. Results

#### 3.1. Exploratory Analysis

#### 3.2. Regression Analysis for the Trend Component

^{2}) for the monthly total rainfall for the 1994–2014 period are presented in Table 2.

^{2}estimated by the model shows that this trend model explained 29% of the variation in the total monthly precipitation. This low R

^{2}strongly indicates that the trend component was unable to explain the spatiotemporal variability in the precipitation. Moreover, it was challenging to provide a precise estimate for climatic variables, especially precipitation, because the spatial and temporal distributions of such variables exhibited large variations, causing the trend model to present a low R

^{2}[15]. It is worth noting that the methodology proposed in this paper to estimate the precipitation accounts not only for the trend but also for the dependence of spatiotemporal data. Then, the residues produced by this trend component were modeled.

#### 3.3. Spatiotemporal Variogram of the Residuals

#### 3.4. Selection and Validation of the Model

^{2}value of 0.85, which indicates that 85% of the total precipitation variability could be explained using spatiotemporal kriging. Additionally, a visual analysis of the differences between these values presented in Figure 7a shows that the differences were distributed symmetrically with a frequency near zero. In other words, the scatter plot of the observed versus the estimated values during the 1994–2014 period, as well as the histogram of the differences between these values (Figure 7b), showed that the predictions were not significantly biased.

#### 3.5. Spatiotemporal Kriging

## 4. Discussion

^{2}values [15,32]. However, there is a lack of meteorological variables for the study region. For the state of Paraíba, only eight weather stations provided recorded data, and it was impossible to use a few of those stations in this work because their data could negatively affect the predictions in the model. For future studies, the results obtained in this study could be compared with those derived from the application of data obtained using simulations. Another important topic addressed in this research is the ability to explain the variability in the data beyond just the trend component; this analytical ability was achieved by adjusting the covariance functions for the residuals obtained by adjusting the trend because they had space-time dependence [14,21].

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Monthly rainfall of the 269 stations in the period of 1994–2014. Missing data is shown as white tones.

**Figure 2.**Spatial distribution of 269 rainfall stations sampled (black dots) in the State of Paraíba, divided into four mesoregions. The larger points are the 54 rainfall stations used to adjust the trend and the variogram.

**Figure 3.**Boxplot for monthly rainfall for the period 1994–2014 in the mesoregions of Zona da Mata, Agreste, Borborema, and Sertão.

**Figure 5.**Sample spatiotemporal variogram (

**a**) and the adjusted product-sum model (

**b**) obtained from the residuals of the multiple linear regression.

**Figure 7.**Comparison between the observed and estimated values (

**a**) and the histogram of the difference between these values (

**b**).

**Table 1.**Mean values, standard deviation (SD), and coefficient of variation (CV) of rainfall (mm) for the period 1994–2014 in the mesoregions of Zona da Mata, Agreste, Borborema, and Sertão.

Months | Zona da Mata | Agreste | Borborema | Sertão | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | SD | CV (%) | Mean | SD | CV (%) | Mean | SD | CV (%) | Mean | SD | CV (%) | |

January | 86.6 | 79.8 | 92 | 62.9 | 83.8 | 133 | 61.5 | 85.9 | 140 | 124.7 | 116.4 | 93 |

February | 99.4 | 79.1 | 80 | 66.2 | 62.8 | 95 | 64.9 | 61.6 | 95 | 133.2 | 87.7 | 66 |

March | 121.1 | 85.2 | 70 | 85.8 | 62.3 | 73 | 106.6 | 86.5 | 81 | 187.6 | 115.1 | 61 |

April | 173.0 | 111.7 | 65 | 95.5 | 70.1 | 73 | 84.2 | 76.1 | 90 | 156.1 | 106.9 | 69 |

May | 191.1 | 121.1 | 63 | 100.2 | 77.2 | 77 | 67.8 | 66.7 | 98 | 92.9 | 76.8 | 83 |

June | 273.3 | 146.2 | 54 | 131.0 | 80.2 | 61 | 44.9 | 41.5 | 93 | 34.0 | 36.5 | 107 |

July | 190.3 | 114.8 | 60 | 106.0 | 76.2 | 72 | 26.3 | 26.3 | 100 | 17.1 | 19.6 | 114 |

August | 116.3 | 67.2 | 58 | 64.8 | 48.7 | 75 | 14.4 | 18.7 | 130 | 6.8 | 14.3 | 210 |

September | 59.9 | 68.9 | 115 | 29.5 | 41.7 | 141 | 4.6 | 9.5 | 207 | 2.1 | 7.7 | 359 |

October | 20.8 | 19.2 | 92 | 9.8 | 13.3 | 135 | 7.8 | 23.6 | 302 | 13.0 | 30.3 | 233 |

November | 17.7 | 18.1 | 103 | 12.0 | 18.9 | 158 | 6.1 | 16.0 | 260 | 13.5 | 25.8 | 192 |

December | 31.7 | 30.8 | 97 | 20.6 | 23.8 | 116 | 20.0 | 29.7 | 149 | 41.5 | 45.6 | 110 |

**Table 2.**Estimation of the parameters of the model described in the Equation (2) of the rainfall in the region analyzed from 1994 to 2014.

Variable | Estimation | Standard Error | t-Value | p-Value | R^{2} |
---|---|---|---|---|---|

Intercept | 323.193 | 124.843 | 2.589 | <0.01 | 0.290 |

cosx | −19.981 | 0.854 | −23.392 | <0.01 | |

senx | 52.554 | 0.854 | 61.525 | <0.01 | |

Latitude | 0.061 | 0.013 | 4.884 | <0.01 | |

Longitude | −2.308 | 0.079 | −29.330 | <0.01 | |

Longitude^{2} | 0.002 | 0.001 | 30.082 | <0.01 |

**Table 3.**Estimations of parameters (sill, range, and nugget) of the generalized product-sum variogram model fitted for the residuals and root mean square error (RMSE) and Nash–Sutcliffe efficiency (NSE). The parameter k involves the global sill.

Model | Sill | Range | Nugget | k | RMSE | NSE | |
---|---|---|---|---|---|---|---|

Space | Gaussian | 7.648 | 181 km | 1.713 | 23.480 | 10.207 | 0.903 |

Time | Exponential | 26.683 | 43 days | 8.446 | |||

Space | Gaussian | 4.140 | 178 km | 0.930 | 25.129 | 10.155 | 0.905 |

Time | Spherical | 45.686 | 143 days | 24.184 | |||

Space | Gaussian | 8.591 | 180 km | 1.933 | 19.861 | 10.150 | 0.905 |

Time | Gaussian | 27.963 | 71 days | 17.075 | |||

Space | Exponential | 6.889 | 192 km | 0.616 | 23.486 | 10.646 | 0.899 |

Time | Gaussian | 29.500 | 71 days | 18.056 | |||

Space | Exponential | 4.380 | 192 km | 0.400 | 23.614 | 10.667 | 0.900 |

Time | Spherical | 46.065 | 147 days | 24.232 | |||

Space | Exponential | 6.280 | 195 km | 0.574 | 24.823 | 10.686 | 0.897 |

Time | Exponential | 30.775 | 45 days | 10.401 | |||

Space | Spherical | 1.804 | 388 km | 0.237 | 15.857 | 10.341 | 0.901 |

Time | Exponential | 162.584 | 42 days | 49.179 | |||

Space | Spherical | 5.967 | 391 km | 0.758 | 9.251 | 10.288 | 0.904 |

Time | Gaussian | 85.384 | 70 days | 51.693 | |||

Space | Spherical | 3.047 | 392 km | 0.385 | 23.750 | 10.294 | 0.904 |

Time | Spherical | 65.444 | 143 days | 34.775 |

**Table 4.**Mean values of RMSE of rainfall (mm) for 1994–2014 in the mesoregions of Zona da Mata, Agreste, Borborema, and Sertão.

Month | Zona da Mata | Agreste | Borborema | Sertão |
---|---|---|---|---|

January | 30.4 | 32.9 | 30.4 | 30.9 |

February | 31.5 | 30.2 | 29.2 | 31.0 |

March | 30.9 | 30.0 | 32.1 | 29.7 |

April | 30.5 | 31.9 | 32.9 | 31.4 |

May | 31.6 | 35.1 | 32.1 | 30.6 |

June | 31.2 | 32.6 | 35.8 | 33.7 |

July | 33.3 | 31.9 | 32.7 | 33.4 |

August | 33.4 | 33.4 | 31.5 | 31.4 |

September | 28.4 | 31.2 | 31.5 | 31.3 |

October | 34.5 | 33.4 | 33.1 | 33.3 |

November | 33.6 | 34.6 | 33.7 | 33.0 |

December | 33.1 | 35.2 | 33.4 | 34.2 |

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**MDPI and ACS Style**

Medeiros, E.S.d.; Lima, R.R.d.; Olinda, R.A.d.; Santos, C.A.C.d.
Modeling Spatiotemporal Rainfall Variability in Paraíba, Brazil. *Water* **2019**, *11*, 1843.
https://doi.org/10.3390/w11091843

**AMA Style**

Medeiros ESd, Lima RRd, Olinda RAd, Santos CACd.
Modeling Spatiotemporal Rainfall Variability in Paraíba, Brazil. *Water*. 2019; 11(9):1843.
https://doi.org/10.3390/w11091843

**Chicago/Turabian Style**

Medeiros, Elias Silva de, Renato Ribeiro de Lima, Ricardo Alves de Olinda, and Carlos Antonio Costa dos Santos.
2019. "Modeling Spatiotemporal Rainfall Variability in Paraíba, Brazil" *Water* 11, no. 9: 1843.
https://doi.org/10.3390/w11091843