Brazil’s Daily Precipitation Concentration Index (CI) Using Alternative Fitting Equation and Ensemble Data

Abstract

In recent years, precipitation concentration indices have become popular, and the daily precipitation concentration index has been widely used worldwide. This index is based on the Lorenz curve fitting. Recently, some biases in the fitting process have been observed in some research. Therefore, this research’s objective consisted of testing the performance of one alternative equation for fitting the Lorenz curve through the analysis of the daily precipitation concentration in Brazil. Daily precipitation data from 735 time series were used to fit the Lorenz curve and calculate the concentration index. Therefore, the goodness of fit was evaluated to determine which equation better describes the empirical data. Results show that the mean value for the concentration index based on Equation (1) was 0.650 ± 0.079, while the mean value based on Equation (2) was 0.624 ± 0.070. The results of the fitting performance show a better fitting with Equation (2) compared to Equation (1) as indicated by R², RSS, and RMSE values, R² = 0.9959 for Equation (1) versus 0.9996 for Equation (2), RSS = 252.78 versus 22.66, and RMSE = 1.5092 versus 0.0501. Thus, Equation (2) can be considered an alternative to improve the calculation of the concentration index.

1. Introduction

Historically, precipitation has been of interest to climatologists and hydrologists due to its role in the hydrological cycle and the climatological dynamics of basins [1,2,3]. Special attention has been paid to rainfall patterns because of their impact on water availability, ecosystems, agriculture, and human societies, among other things [2,3]. Also, because spatial and temporal precipitation distribution changes can lead to significant changes in streamflow, which can cause floods or droughts, these changes have recently led to concerns due to climate change projections [1,2,3,4,5]. According to the results of studies conducted worldwide, there are increases in the frequency of intense and heavy precipitations [6] and increased disasters related to climatic conditions [7,8].

Through time, precipitation research has focused mainly on analyzing its frequency and intensity [4,6], while other research has focused on its statistical distribution. Among the first researchers to explore the statistical distribution of rainfall were Riehl [8] and Olascoaga [9], who proposed the normalized rainfall curves in the fifties. Later, based on the normalized rainfall curves, Martin-Vide [10] proposed a daily precipitation concentration index equating the normalized rainfall curve with the Lorenz curve to express the relationship between the accumulated proportion of precipitation versus the accumulated proportion of rainfall events. In their paper, Martin-Vide [10] shows two functional forms to approximate the empirical data to the Lorenz curve. However, most of the research has used the exponential equation suggested by Riehl [8] and Olascoaga [9], as shown by the research conducted in Algeria [11], Turkey [12], Spain [13], Peru [14], Argentina [15], Russia [16], Puerto Rico [17], Chile [18], and China [19,20], among others.

On the other hand, the second equation shown by Martin-Vide [10,21] and proposed by Ananthakrishnan and Soman [22] to approximate the Lorenz curve has been used only in a few research works. The curve fitting is crucial because, in the literature, it was recognized that different data could produce equal concentration values [20], which indicates that one of the fitted curves did not fit the data correctly. This result can under- or over-estimate the actual daily precipitation concentration, affecting the evaluation of possible precipitation-related risks. Recently, Núñez-González et al. [21] evaluated the performance of this equation to approximate the Lorenz curve for calculating the concentration index and compare the results with those obtained using the equation of Riehl and Olascoaga [9,10]. Their results showed similar behavior for the concentration index calculated according to both equations; however, the equation proposed by Ananthakrishnan and Soman [22] showed better results when the precipitation was more concentrated. Nevertheless, the research conducted by Núñez-González et al. [21] was carried out with the data of only forty-four meteorological stations. Therefore, testing the performance of the equation proposed by Ananthakrishnan and Soman [22] with more meteorological stations and considering different climate conditions is necessary.

Based on Brazil’s dimensions, it has a great variety of climates influenced by latitude, altitude, and sea distance [23,24]. Annual rainfall in Brazil ranges from 387 to 4000 mm [23], making it attractive to analyze the concentration of its precipitation. However, this requires precipitation information covering the entire country. To have a long-term dataset for Brazilian catchments that allows for a better understanding of the hydrological drivers related to climate, Almagro et al. [24] prepared a novel dataset with data from 1980 to 2010 for 735 catchments based on the grided dataset developed by Xavier et al. [25] and the ERA5 climate data [26], and made it freely available [24]. The concentration of precipitation in Brazil has been researched recently. On one hand, Nery et al. [27] analyzed the precipitation concentration of Southeastern Brazil using data from 120 stations.

Similarly, Siqueira and Nery [28] studied the spatial and temporal variability of precipitation concentration in Northeastern Brazil based on 82 rainfall series. On the other hand, Back et al. [29] researched to evaluate the daily concentration in Brazil. Their research covers the entire country based on the information from the 1402 rainfall series. However, the research works mentioned above consider only the first equation proposed by Martin-Vide [10]. Therefore, the objective of this research is threefold: the first part consists of an analysis of the daily precipitation concentration of Brazil using the ensemble dataset provided by Almagro et al. [24]; the second part is a test of the performance of the alternative equation proposed by Ananthakrishnan and Soman [22] in a broad context than that of Núñez-González et al. [21] and a comparison of the result with that obtained with the equation proposed by Riehl [8] and Olascoaga [9]; and finally, the third part is an analysis of the relationship between the concentration values obtained through the fitting and both equations.

2. Materials and Methods

The CABra dataset was used to calculate Brazil’s concentration index. CABra is a large-sample dataset for Brazilian catchments that includes data on climate over 30 years (1980–2010). One characteristic of this dataset is that it has less than 10% of missing data [24]. The dataset is freely available at https://thecabradataset.shinyapps.io/CABra/ (accessed on 7 May 2024). This dataset is composed of 735 time series of area-averaged precipitation. The time series were generated using the high-resolution meteorological gridded dataset (0.25° × 0.25°) [25]. They were complemented with data from ERA5 [26], the most recent version of the climate reanalysis from the European Centre for Medium-Range Weather Forecasts. ERA5 provides information in a 0.25° × 0.25° spatial resolution grid for several atmospheric variables from 1950 to the present. Figure 1 shows the location of the basin centroids used as a reference for the area-averaged precipitation. Also, this figure shows the spatial distribution of rainfall across the country according to the data of the CABra dataset.

The concentration index was calculated following the methodology proposed by Martin-Vide [10] using a 30-year time series from the CABra dataset from 1980 to 2010. To perform this, the cumulative frequencies of rainy days and precipitation were used to fit the equations proposed by Riehl and Olascoaga [9,10] (Equation (1)) and Ananthakrishnan and Soman [22] (Equation (2)).

The calculation of the cumulative frequency of rainy days and the cumulative frequency of precipitation is exemplified in

Table 1. Descriptive statistics of the concentration index were calculated using both equations by administrative regions.

Class	ni (Number)	∑ ni (Number)	pi (mm)	∑ pi (mm)	∑ ni (%)	∑ pi (%)
0.1–0.9	21	21	10.5	10.5	21.65	0.83
1.0–1.9	6	27	9	19.5	27.84	1.54
2.0–2.9	7	34	17.5	37	35.05	2.93
.	.	.	.	.	.	.
.	.	.	.	.	.	.
.	.	.	.	.	.	.
84.0–84.9	0	96	0	1178	98.87	93.23
85.0–85.9	1	97	85.5	1263.5	100.00	100.00

. To perform this, first, the data were classified into 1 mm classes (column 1). Then, each class’s rainy days and total rainfall were calculated (columns 2 and 4). Later, the cumulative sum of rainy days and total precipitation were computed (columns 3 and 5); and finally, the cumulative sums were transformed into percentual relative distributions of rainy days and precipitation (columns 6 and 7). A detailed numeric example of the calculations can be consulted in Martin-Vide’s work [10]. Once the distribution of rainy days and precipitation was obtained, Equations (1) and (2) were used to approximate the Lorenz curve, which is the basis for calculating the concentration index. Equations (1) and (2) were fitted using the statistical software R 4.2.2.

Dots indicated that table classes continue to avoid a long table.

$$y=ax{e}^{bx}$$

(1)

where a and b are regression constants.

$$y=x{e}^{[-b{\left(100-x\right)}^c]}$$

(2)

where b and c are regression constants.

After the fitting of Equations (1) and (2), the area between the equality line and the Lorenz curve was computed according to

$$A=5000-{\int }_0^{100}ax{e}^{bx}dx$$

(3)

$$A=5000-{\int }_0^{100}x{e}^{[-b{\left(100-x\right)}^c]}dx$$

(4)

Finally, the concentration index was calculated as

$$CI={\displaystyle \frac{A}{5000}}$$

(5)

To evaluate the performance of the functional forms used to approximate the Lorenz curve, three statistical measures were used: the determination coefficient (R²), the root mean square error (RMSE), and the residual sum of squares (RSS). The relationship between the concentration index obtained from the two functional forms and between coefficients was analyzed using Pearson’s correlation coefficient. Statistics and graphs were made using the R statistical package [30], and maps were produced using QGIS [31].

3. Results and Discussion

The spatial distribution of the concentration index values computed according to Equations (1) and (2) for Brazil show similar behaviors (Figure 2). Based on Equation (1), the concentration values range from 0.233 to 0.814 with a mean value of 0.650 ± 0.079, while concentration values based on Equation (2) range from 0.232 to 0.779 with a mean of 0.624 ± 0.070. The highest values were observed in the Northeast, Southeast, and South regions, while the lowest were in the North region. This behavior is opposite to that observed in the annual spatial rainfall distribution shown in Figure 1. The concentration values calculated in this research for the southeastern region are higher than those observed by Nery et al. [27].

Similarly, the values found for the northeastern part are higher than those of Siqueira and Nery [28]. Back et al. [29] found lower values at the country level than in this research. The differences can be associated with the period of records used for the calculations. Nery et al. [27] used data from 1976 to 2010, Siqueira y Nery [28] calculated the concentration index based on data from 1966 to 2014, while Back et al. [29] took into account precipitation data from 1991 to 2022. On the other hand, differences can be associated with the kind of data. Nery et al. [27], Siqueira y Nery [28], and Back et al. [29] researched using pluviometers. In contrast, this research’s precipitation data comprise time series of area-averaged precipitation obtained from a gridded dataset [24]. Nonetheless, the concentration values in this research concord with those obtained by Monjo and Martin-Vide [32], who also used a gridded dataset.

The spatial distribution of the concentration index values shows significant differences between the eastern and western parts of the country. According to both equations, critical values for the concentration index are located mainly on the east side of the country in states near the Atlantic Ocean, such as Ceará, Bahia, Minas Gerais, Espíritu Santo, and Rio Grande do Sul. Monjo and Martin-Vide have reported this spatial behavior in the past [32], and recently, Back et al. [29] also found higher concentration values over the same region.

Table 2. Descriptive statistics of the concentration index were calculated using both equations by administrative regions. The bold is evidence of high-concentration values.

State	Basins by State	Martin-Vide				Ananthakrishnan and Soman
		Min	Mean	Max	SD	Min	Mean	Max	SD
Acre	1		0.493				0.476
Alagoas	6	0.613	0.651	0.661	0.019	0.554	0.616	0.642	0.038
Amapá	4	0.486	0.494	0.504	0.009	0.479	0.486	0.493	0.007
Amazonas	15	0.233	0.385	0.484	0.073	0.232	0.381	0.48	0.073
Bahia	64	0.586	0.691	0.814	0.056	0.536	0.647	0.779	0.042
Ceará	7	0.738	0.758	0.784	0.017	0.689	0.713	0.736	0.017
Distrito Federal	4	0.562	0.57	0.583	0.009	0.549	0.558	0.569	0.008
Espírito Santo	27	0.686	0.734	0.774	0.025	0.647	0.694	0.731	0.023
Goiás	48	0.527	0.56	0.619	0.024	0.519	0.546	0.611	0.021
Maranhão	32	0.548	0.625	0.68	0.027	0.541	0.603	0.659	0.026
Mato Grosso	21	0.477	0.525	0.569	0.023	0.474	0.518	0.557	0.02
Mato Grosso do Sul	7	0.543	0.598	0.644	0.038	0.537	0.579	0.618	0.03
Minas Gerais	185	0.56	0.685	0.775	0.042	0.479	0.661	0.74	0.039
Pará	22	0.234	0.53	0.675	0.08	0.235	0.516	0.614	0.071
Paraíba	6	0.694	0.744	0.775	0.034	0.623	0.679	0.729	0.045
Paraná	57	0.548	0.683	0.715	0.023	0.545	0.653	0.679	0.02
Pernambuco	9	0.615	0.674	0.778	0.043	0.553	0.622	0.719	0.051
Piauí	8	0.59	0.662	0.735	0.051	0.576	0.641	0.7	0.045
Rio de Janeiro	37	0.623	0.661	0.725	0.029	0.603	0.636	0.695	0.026
Rio Grande do Norte	4	0.728	0.755	0.775	0.021	0.661	0.697	0.725	0.027
Rio Grande do Sul	47	0.665	0.695	0.736	0.018	0.634	0.668	0.707	0.018
Rondônia	9	0.49	0.512	0.547	0.015	0.487	0.504	0.537	0.014
Roraima	5	0.478	0.52	0.61	0.053	0.465	0.505	0.59	0.05
Santa Catarina	48	0.654	0.675	0.695	0.01	0.574	0.643	0.662	0.014
São Paulo	32	0.627	0.66	0.704	0.017	0.602	0.636	0.675	0.018
Sergipe	1		0.684				0.6
Tocantins	13	0.516	0.556	0.579	0.015	0.514	0.548	0.568	0.014

shows the descriptive statistics for the concentration index by administrative region. According to this Table, the highest values were observed in Ceará, Rio Grande do Norte, Paraiba, and Espíritu Santo, where the mean values are around 0.7, which, according to Martin-Vide [10], is considered a value for very high precipitation concentration, because this concentration value indicates that only a quarter of the rainy days concentrate around 80% of the precipitation [33]. Monjo and Martin-Vide associate the location of the highest concentration values [32] with the seasonally warm coastlines. Fortunately, the highest concentration values were observed in regions where the annual precipitation can be considered low. Similarly, Monjo and Martin-Vide [32] state that a more regular rainfall distribution, as those observed in eastern Brazil and mainly in the Amazonas, is associated with the rainfall patterns of rainforests, which have a recurrent pattern of daily storms.

The daily precipitation concentration has been used worldwide during the last few years to explore the statistical distribution of precipitation. One of the methodologies most used for this purpose is the one established by Martin-Vide [10]. This methodology is based on fitting the Lorenz curve, which describes the distribution of precipitation data versus the rainfall events. Most research based on Martin-Vide’s methodology focuses on the index’s spatial and temporal behavior, and only a few studies have investigated the performance of the equations used. In previous research, Núñez-González et al. [21] found that in cases where a high concentration of precipitation is recognized, Equation (2) shows a better performance to fit the Lorenz curve than Equation (1); however, their results were based only on data from forty-four stations.

In this sense, results obtained from 735 Brazilian stations, shown in Table 2 and Figure 2 and Figure 3a, show that all the values of the concentration index based on Equation (2) are lower than those based on Equation (1). However, they seem related, as was highlighted by Nuñez-González et al. [21]. In fact, according to the correlation analysis (Figure 3b), there is a linear relationship between the concentration index values of both equations. The slope value in this figure shows that concentration values obtained with Equation (2) are in a proportion of 0.8745 compared to those calculated with Equation (1). This value is very close to that observed by Nuñez-González et al. [21] at the annual and rainy season scale. The determination coefficient (R²) indicates that the concentration values of Equation (2) can be reasonably approximated based on the concentration values obtained with Equation (1) (R² = 0.966), which shows that both concentration values are linearly related. The result of the significance test of the model shown in Figure 3b (p-value = 2.2 × 10⁻¹⁶ according to ANOVA test) shows that the concentration values obtained according to the Martin-Vide’s equation (Equation (1)) are significant.

On the other hand, Figure 3a shows some values considered outliers in the distribution’s lower and higher tails. However, on the top of the distribution, there was only one value considered as an outlier in both cases, while in the lower tail, the following were classified as outliers: approximately 5% of the concentration values obtained using the equation by Riehl and Olascoaga [9,10], and 6% according to the equation by Ananthakrishnan and Soman [22]. These values show that concentration values lower than 0.5 are scarce, showing that Brazil’s concentration is mainly considered in medium to high concentration categories [29].

Regarding the goodness of fit of the two equations for the fit of the Lorenz curve with the CABra dataset, results using three metrics for the functions used to approximate the Lorenz curve show similar behavior (Figure 4). The average R² (Figure 4a) was 0.9959 ± 0.0050 for the Martin-Vide equation and 0.9996 ± 0.0003 for the Ananthakrishnan and Soman equation. The average RMSE (Figure 4b) was 1.5092 ± 0.5642 for the Martin-Vide equation and 0.05019 ± 0.2103 for the Ananthakrishnan and Soman equation. RSS average values (Figure 4c) were 252.78 ± 897.45 for the Martin-Vide equation and 22.66 ± 21.38 for Ananthakrishnan and Soman equation. These results are in concordance with those of Núñez-González et al. [21] and also the hierarchical ordering of the performance of the two equations used to approximate the Lorenz curve. Figure 4 also shows that more than half of the values obtained for the Martin-Vide equation are below (Figure 4a) and above (Figure 4b,c) those obtained with the Ananthakrishnan and Soman equation, which suggests that the Ananthakrishnan and Soman equation is a better option to fit the Lorenz curve, especially in those places where it is known that precipitation is not regularly distributed [21].

In their research, Ananthakrishnan and Soman [22] found that regression constants were related to the variation coefficient when they studied the normalized rainfall curves, which can be considered the Lorenz curve in the precipitation concentration index. They found that high values of the variation coefficient were related to high rainfalls. However, according to the distribution of precipitation, which is widely recognized, it is fitted to negative exponential distributions; it was decided to analyze the relationship between the regression constants of Equations (1) and (2) versus the skewness coefficient since these regression constants control the exponential growth rate of the fitted equation.

Regression constants obtained for Equation (1) are concordant with those reported in research conducted in other countries [11,12,15,16,17,18,21]. Similarly, regression constants of Equation (2) show the same behavior as those reported by Núñez-González et al. [21]. Regression constant c in Equation (2) can show the similarity between the concentration index calculated with both equations because when this constant takes values approaching 1, the concentration values obtained with both equations are more similar [21].

On the other hand, in their research, Ananthakrishnan and Soman (Ananthakrishnan and Soman 1989) stated that the constant c becomes important for large values of the coefficient of variation. However, according to Brazilian precipitation (Figure 5b,c), the constant c looks more related to the skewness coefficient (Figure 5b) than to the coefficient of variation (Figure 5d). Similarly, the constant b of Equation (1) looks related to the coefficient of variation (Figure 5c). Figure 5b shows that higher skewness coefficients are related to lower values of the c constant in Equation 2. Low values of c make the curvature of the function smoother due to a more gradual convergence than observed when values of c are high. This helps improve the goodness of fit in the time series with a high degree of precipitation concentration. Alternatively, high values of coefficient b in Equation (1) are related to high values of the coefficient of variation. The coefficient b regulates the function’s exponential growth rate, so at higher values of the coefficient of variation, the exponential growth is expected to be higher, which is related to a greater degree of precipitation concentration.

The spatial distribution of the c coefficient in the equation of Ananthakrishnan and Soman (Ananthakrishnan and Soman 1989) shows that daily precipitation concentration calculations can be improved for the warm coastlines of the northeastern, southeastern, and southern regions using Equation (2), where the highest concentration index values were found. In contrast, in the northern region, the lowest concentrations were observed, where both equations can be used equivalently, because when the c coefficient approaches the value of 1, both equations produce similar results for the concentration index [21].

On the other hand, based on the spatial distribution of the skewness coefficient of the annual precipitation in Brazil, an inverse relationship between the coefficient c and the skewness coefficient can be inferred. When the spatial distribution of this index and the skewness coefficient are compared (Figure 6), it is observed that high values of the skewness coefficient characterize the regions with low values of the coefficient c of Equation (2). Based on this, the skewness coefficient can be considered an indicator to decide the usage of either Equation (1) or Equation (2) for the Lorenz curve fitting.

4. Conclusions

The results of this research, based on an analysis of 735-time series precipitation from Brazil, confirm that asseveration is based on the result of the three metrics used to evaluate the goodness of fit of Equations (1) and (2) versus the empirical data. The improvement in the fitting is related to the role of the coefficient c of Equation (2). This coefficient can make a smoother curvature of the Lorenz curve due to their gradual convergence, which is useful when there is a high degree of precipitation concentration. Also, it was found that the index values obtained through Equations (1) and (2) are linearly related, which allows for the adjustment of the values obtained for the concentration index from Equation (1) to Equation (2) in a simple manner and with a high confidence degree. The adjustment can be considered mainly for high concentration index values because Equations (1) and (2) produce very similar results for low concentration index values. Additionally, an inverse linear relationship was found for the skewness coefficient versus the coefficient c of Equation (2), which shows that the data asymmetry determines coefficient c of Equation (2). The spatial behavior of these coefficients supports the relationship mentioned above.

Regarding the behavior of the concentration index, higher values were found than those reported by other researchers for the same zones, which showed a high degree of precipitation concentration. High concentration values are associated with regions with a high-risk potential for erosion, flooding, and torrential rains. Nevertheless, these differences can be attributed to the kind of data used. This is because the calculation dataset comprises a time series of area-averaged precipitation instead of data collected directly from pluviometers. Despite the numerical differences in the concentration values, the spatial distribution of the concentration index calculated in this research coincides with those reported for the same study area by other researchers, which proves that the CABra dataset can reproduce the behavior of precipitation distribution. The concentration values indicate the necessity to plan actions to prevent precipitation-related natural disasters, especially in the states near the coastlines.