Quantifying the Rainy Season in the Brazilian Northeast Through a Modification of the Relative Entropy Method

1. Introduction

The rainy season in the Brazilian Northeast (NE) depends strongly on the geographic location. For instance, the NE capital Recife which is located in the coastal area of the state of Pernambuco, experiences excess rain from March to August [1], while the city of Serra Talhada only 400 km away in the interior of the state of Pernambuco (located in the semiarid area) experiences concentrated rain in the period from January to April [2]. Several studies report that six consecutive rainy months vary across NE: between January and June in the northern part, between March and August in the eastern coastal area, and between November and April in the southern part [3,4,5]. Based on rainfall data from 148 stations covering 1972 - 2002 period, Oliveira et al. [4], identified three rainiest months across NE: March, April and May in the northern part, June, July and August in the eastern coastal area, and December, January and February in the southern part.

Northeast Brazil is one of the regions most vulnerable to climate change, worldwide, with an area of $~1.5\times 10^6{km}^2$ and $~55\times 10^6$ inhabitants. The welfare of its habitants highly depends on climate conditions that vary from recurrent and prolonged drought in the semiarid area where the rural population is primarily dependent on subsistence agriculture [6,7], to flash floods in coastal regions that can trigger massive landslides, which can cause numerous fatalities [8,9]. These extreme conditions are related to the interannual variability of large-scale oscillation systems, El Niño - Southern Oscillation (ENSO) in the equatorial Indo-Pacific Ocean, and the meridional sea surface temperature gradient (MGRAD) over the tropical Atlantic. During the positive phase of ENSO (El Niño) and negative phase of MGRAD, there is a decrease in rainfall over NE, while during the negative phase of ENSO (La Niña) and positive phase of MGRAD the rainfall in NE is above the average. In general, severe and prolonged droughts occur in El Niño years in which the direct impact of ENSO is amplified by pre-existing Atlantic sea surface temperature (SST) anomalies [10]. Intraannual variability is characterized by well defined seasonal patterns which are related to the seasonal displacement of the Atlantic intertropical convergence zone (ITCZ) [11,12]. The seasonal cycle of rainfall in NE affects energy production [13], carbon storage in tropical dry forest [14], agricultural productivity [15] and water quality [16]. Regions characterized by high rainfall seasonality tend to have a higher percentage of total annual precipitation concentrated during the short period of time, which leads to a higher probability of the occurrence of floods [17]. In order to mitigate water related hazards, that can compromise various economic sectors such as energy production and agriculture [18,19], infrastructure [20], and affect ecosystems [21] and human health [22], it is fundamental to enhance our understanding of the variability of rainfall anomalies at different temporal and spatial scales, especially in the regions with vulnerable population and higher frequency of severe drought. To this end, a systematic quantification of the rainy and dry seasons from historical data, at different geographic locations across the Brazilian NE, appears to be of utmost importance.

Currently, systematic, spatially explicit quantification of the rainy season duration and timing for the Brazilian Northeast, based on established statistical methods, is still lacking in the body of literature. Recent studies in this direction provide results based on monthly rainfall climatology, identifying six rainiest months and three months as a peak of rainy season. The results depend on the dataset used and the period considered. Rao et al. [3] studied rainfall regime in Brazil, using Climate Research Center (CPC) gauge based global daily precipitation with a horizontal resolution of 0.50 latitude/longitude and time span from 1979 to 2011. For the NE they found that the three rainiest months were FMA in the northern part, JFM in the central part, NDJ in the southern part and MJJ in the eastern coastal part. Using station data for approximately the same period, Oliveira et al. [4] found peak of the rainy season to be shifted forward one month: MAM in the northern part, DJF in the southern part and JJA in the eastern coastal part. Rainfall seasonality can be evaluated using specially designed indices, such as statistics-based precipitation seasonality index SI [23], that classify rainfall regime according to the degree of variability in monthly rainfall throughout the year. When calculated for the entire period and as average of individual years, SI provides the information of the replicability of rainfall regime at a certain location [24,25]. However, a complete description of rainfall seasonality requires the knowledge of the magnitude, duration and timing of the wet season. Feng et al. [26] introduced the Relative entropy (RE) method based on information theory, which provides a set of indicators for quantification of the magnitude, duration and timing of the wet season. This method was shown successful for regions with a well-defined wet and dry season and was used with both observational and model data [27,28].

The principal purpose of the current work is to introduce a modification of the Relative Entropy method in order to perform geographically explicit quantification of the rainy season across the Brazilian Northeast. The results of this study, in geographically explicit form, should be taken into account for agricultural planning, better planning of use of water resources for renewable energy, as well as to develop more efficient measures for the protection and mitigation of the impact from water related natural hazards (drought and floods).

2. Materials and Methods

2.1. Data

ERA5-Land is an advanced reanalysis dataset that provides high-resolution (about 9 km) land surface data over the global domain [29]. The high resolution of the dataset enhances the accuracy and detail of land surface representations, making it particularly valuable for localized studies. ERA5-Land provides a continuous record of land surface conditions from 1950 to the present with hourly time resolution. This extensive temporal coverage supports long-term studies, climate monitoring, and trend analysis. The dataset is freely accessible through the Copernicus Climate Data Store (CDS). The CDS portal provides user-friendly tools and services for data retrieval and analysis, ensuring that researchers and practitioners can easily integrate ERA5-Land data into their workflows.

In this work we used data from the ERA5-Land for the period 1980 to 2022, as this period matches most of the other works that we compare with. For each grid point at resolution of ~9km the data are integrated into monthly precipitation series, with the goal of quantifying the timing and duration of the rainy season from historical data, in a geographically explicit manner.

2.2. Relative Entropy

Relative Entropy method was introduced [26,27] to quantify the length and timing of the rainy season in the regions with a single pronounced rainy epoch, across the globe. For a given discrete probability distribution $p\equiv \left\{p_m, m=1,\dots ,N\right\}$ information entropy (measure of uncertainty) is defined as

$$H\left(p\right)=-{\sum }_{m=1}^N{p}_m{\log }_2{p}_m,$$

(1)

attaining a maximum of ${\mathrm{l}\mathrm{o}\mathrm{g}}_{2}N$ for the uniform distribution $p_m=1/N$ (maximum uncertainty), and a minimum of zero when only one of the $p_m$ values is unity, all the others being zero (no uncertainty). For two distributions $p$ and $q$ the relative entropy

$$D\left(p|q\right)=\sum _{m=1}^N{p}_m{\log }_2\left({\displaystyle \frac{p_m}{q_m}}\right)$$

(2)

measures the inefficiency of assuming $q$ when the true distribution is $p$.

For the particular case of monthly rain distribution $p\equiv \left\{p_m, m=1,\dots ,12\right\}$, assuming uniform distribution $q=1/12$ leads to relative entropy

$$D\left(p\right)={\sum }_{m=1}^{12}{p}_m{\log }_2\left(12 p_m\right).$$

(3)

Relative entropy $D\left(p\right)$ quantifies strength of seasonality and varies between 0 for a uniform rain distribution across the year, to ${\log }_2\left(12\right)=3.58$ when all the rain occurs in a single month.

The effective number of months of the rainy season duration is found as [27]

$$n\left(p\right)=12\times 2^{-D\left(p\right)},$$

(4)

and the centroid of the rainy season is found as [27]

$$C\left(p\right)=\arg \left(z\right), z=\sum _{m=1}^{12}{p}_m{e}^{-i{\displaystyle \frac{2\pi m}{12}}}.$$

(5)

The quantities $D\left(p\right)$ (strength of seasonality), $n\left(p\right)$ (duration of the rainy season, in months), and $C\left(p\right)$ (center of the rainy season), represent the principal foci of interest of the current work.

2.2. Modification of the Relative Entropy Method

To test the applicability of the Relative Entropy method for the Brazilian Northeast, a test was performed on precipitation data for Recife (Caxanga station) in the period 1911-1970 and Serra Talhada in the period 1911-1991, collected by the Superintendência de Desenvolvimento do Nordeste (SUDENE) and provided by APAC [30,31]. The original Relative Entropy definition yields values of n~9.62 and n~8.45 for Recife and Serra Talhada, respectively, in agreement with findings of Ref. [27], but in disaccord with the commonly accepted values of n=6 and n=4 months of the rainy season in the two regions [1,2]. The thresholding adopted in [26,27] between the rainy season and the dry season may be adequate for some parts of the world, but here it appears too stringent, and the duration of the rainy season is by far overestimated.

To mitigate this discrepancy between the results of the RE method and the “common sense” adopted by the scientific community (and community in general) for the Brazilian Northeast, rather than adopting for the probability $p\equiv \left\{p_m, m=1,\dots ,12\right\}$ the ratio of precipitation in month $m$ with total yearly precipitation as in [26,27], here we address the excess precipitation – the probability of precipitation above the mean, in any given month

To this end, first the z-score $z_m=({prec}_m-\overline{prec})/{\sigma }_m$ is calculated for each month $m=1,\dots ,12$, where $\overline{prec}$ is the average and ${\sigma }_m$ is the monthly precipitation standard deviation along the years of the period under study, and then the months with precipitation below the average $z_m<0$ are discarded, to find the set of months $M_{sup}$ with precipitation above the average, and finally the excess precipitation probability is defined as

$$p_m=\left\{\begin{array}{cc}{\displaystyle \frac{{prec}_m}{\sum _{k\in M_{sup}}{prec}_k}},& z_m>0\\ 0,& z_m\le 0\end{array}\right.,$$

(6)

The precipitation histograms for Recife and Serra Talhada, for SUDENE data, are shown in Figure 1,

while the excess rain probability distributions as defined in (6) for the two sites are shown in Figure 2.

Using the definition of excess precipitation probability (6) in equations (3-5) leads to values of $n~5.03$ and n~3.59 for Recife ad Serra Talhada, centered about June and March, respectively, in agreement with (albeit slightly underestimating) the widely accepted observational (phenomenological) conclusions.

The objective of this work is to implement this modification of the Relative Entropy method over the vortices across the Brazilian Northeast, and thus establish geographically explicit areas of rainfall season duration and timing on purely phenomenological grounds, without assuming any functional form for precipitation probability distributions, therefore free of any parametrization.

3. Results

The mean monthly precipitation data from the ERA5-Land project across the Brazilian NE, for the period 1980 to 2023 is shown in Figure 3, the results being quite compatible with the interpolated station precipitation data for the period 1904-1998 [13].

The mean yearly precipitation is shown in Figure 4, where the orange/yellow region corresponds to the so called “Poligono das Secas” (the Drought Polygon).

The main results of this work are presented in Figure 5, where the Strength of seasonality $D\left(p\right)$, duration of the rainy season in months $n\left(p\right)$, and the center of the rainy season $C\left(p\right)$ are displayed across the Brazilian NE. The spatial distribution of $C\left(p\right)$ coincides with the rainiest months as found in ref. [3,4] where different datasets were used. Rao et al. [3] found that the three rainiest months were FMA in the northern part, JFM in the central part, NDJ in the southern part and MJJ in the eastern coastal part, while Oliveira et al. [4] found a peak of rainy season in MAM in the northern part, DJF in the southern part and JJA in the eastern coastal part.

The results for the capitals of the nine NE states Alagoas (AL), Bahia (BA), Ceará (CE), Maranhão (MA), Paraíba (PB), Pernambuco (PE), Piauí (PI), Rio Grande do Norte (RN) and Sergipe (SE) are presented in Table 1.

Hänsel at al. [32] investigated climate variability in the capitals of North-Eastern Brazil, based on temperature and precipitation data from nine meteorological stations recorded during the period 1961-2011. They found that the rainy season occurred in different periods in northern and the south-eastern stations. In the northern region, the wettest months are January to April in Teresina and January to June at São Luís and Fortaleza. The rainy season in other six stations span from March to July, with peak in April and May in Aracaju and Salvador, and from April to July at Natal, João Pessoa, Recife, and Maceió. The values of duration of the rainy season presented in Table 1 agree with these results. For Terezinha, Recife, Maceo, Aracaju and Salvador we obtained the same duration of rainy season, for Sao Luiz and Fortaleza one month shorter, and for Natal and João Pessoa one month longer. The values for the center of the rainy season in Natal, João Pessoa, Recife and Maceo agree with the period of highest rainfall in these cities found in [32]. On the other hand, the values of the duration of the rainy season calculated using the original RE method (shown in parentheses in the one before last column of Table 1) by far overestimate these findings.

3.2. Sofware

For ease of extraction of the rainy season parameters (strength, duration and the center of the rainy season), together with the monthly precipitation series for any point or municipality average, we also provide user-friendly software with Graphical User Interface for the Windows platform as presented in Figure 6, as part of supplementary material of this paper, together with an Excel file with mean season parameters for all the 1794 municipalities of the Brazilian NE.

The software displays a map of the Brazilian NE with state (dark gray) and municipality (medium gray) borders, together with (red) points corresponding to ERA5-Land grid vertex positions. Municipality Barra in the state of Bahia (BA) is outlined in blue in Figure 6, being the current choice, that may be changed through the combo box in the lower left part of the control dialog box, or by simply clicking the left mouse button with cursor positioned anywhere inside the municipality – the point of choice is displayed as a red circle. As the cursor is moved across the map, the current cursor position and the corresponding local rain season parameters are displayed in the top part of the dialog box. The middle part of the control dialog box also permits choice of a specific point, by specifying the latitude and longitude values. Finally, the “Extract” button generates an ASCII file (displayed immediately in Notepad), where the pointwise or municipality average values are listed (together with the monthly precipitation sequence), depending on the choice of the “Municipality” or “Point” radio buttons. If the “Point” option is chosen the values are calculated applying the Inverse Distance Squared method on up to four neighboring grid points (SW, SE, NW and NE), while points close to the NE border area may have less than four neighboring grid points, and the values are derived from these.

5. Conclusions

We analyzed rainfall seasonality in the Brazilian Northeast (NE), by using the Relative Entropy (RE) method. To better match the results of other works, and commonly accepted values for the length and the center of the rainy season in this region, we introduced the modification of the original RE method (MRE), which is based on the concept of excess precipitation – the probability of precipitation above the average. By applying the MRE approach on ERA5-Land high resolution gridded dataset (~9 km) for the period 1980–2022 we obtained spatial distribution of seasonal characteristics of rainfall (strength, duration and center of rainy season) across the NE. We found that the length of the rainy season lasts 4-5 months in the northern region, 5-6 months in the central and southern regions and 6-7 months in the eastern coastal region. The center of the rainy season ranges from February to March in the northern part, January to February in the central and southern parts and April to June in the eastern coastal part. The results for the NE capitals agree well with the results obtained by classical statistical methods, while the original RE method overestimated by far the length of the rainy season. We also developed a dedicated user-friendly software that permits to obtain the parameters of rainfall seasonality, together with the monthly precipitation series, for any point on the map, or any of the 1794 municipalities (by averaging over the corresponding grid cells). The results of this study showed that the modified RE method provides reliable information about rainfall seasonality in the Brazilian NE, and should be taken into account for agricultural planning, better planning of efficient use of water resources, and to develop more efficient measures for the protection and mitigation of the impact from extreme water related conditions (drought and floods).