Using satellite-based precipitation for developing intense rainfall equations for ungauged areas of the Southern Bahia Mesoregion

Estimativas de precipitação por satélite para definição de equações de chuvas intensas em áreas não monitoradas da Mesorregião Sul Baiano

ABSTRACT

Satellite-based precipitation products can be an important alternative for developing IDF equations in ungauged regions, in search for more flood-resilient hydraulic systems. This study evaluated precipitation estimates from the product CHIRPS for developing IDF equations in the Southern Bahia Mesoregion. The statistical metrics used were Percent Bias (PBIAS), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and the Nash-Sutcliffe Efficiency coefficient (NSE). The results indicate that CHIRPS does not perform satisfactorily in its original state, but it can provide reasonable IDF equations if the data bias is corrected. Based on a linear regression model, a bias correction method was proposed, which can be applied in ungauged areas of the studied region, showing better performance than IDF equations provided by the software Pluvio 2.1 (RMSE 45.27% lower; MAE 46.31% lower).

Keywords:
IDF curves; CHIRPS; Bias correction

RESUMO

Produtos de precipitação por satélite podem ser uma alternativa importante para a determinação de equações IDF em regiões não monitoradas, em busca de sistemas hidráulicos mais resilientes aos impactos das inundações. Este trabalho avaliou as estimativas de precipitação do produto CHIRPS para a determinação de equações IDF na Mesorregião Sul Baiano. As métricas estatísticas utilizadas foram o Bias Percentual (PBIAS), a Raíz do Erro Quadrático Médio (RMSE), o Erro Absoluto Médio (MAE), e o coeficiente de eficiência de Nash-Sutcliffe (NSE). Os resultados indicam que o CHIRPS não apresenta desempenho satisfatório em seu estado original, mas pode fornecer equações IDF razoáveis se o viés dos dados for corrigido. A partir de um modelo de regressão linear, foi proposto um método de correção de viés que pode ser aplicado em áreas não monitoradas da região estudada, apresentando desempenho melhor que equações IDF fornecidas pelo Pluvio 2.1 (RMSE 45,27% menor; MAE 46,31% menor).

Palavras-chave:
Curvas IDF; CHIRPS; Correção de viés

INTRODUCTION

Flood-induced disasters are the most frequent worldwide and can cause severe impacts, such as loss of human lives, considerable structural damage, and significant economic losses (Dinis et al., 2021; Kasi et al., 2020; Parvin et al., 2016; Ramos Filho et al., 2022). Over the past decades, nearly 20,000 people have lost their lives worldwide, and approximately 58 million have been affected by severe floods (Kumar et al., 2021; International Federation of Red Cross, 2010; World Bank, 2018). Intense rainfall in urban environments can be a strong indicator for the occurrence of floods and landslides, especially in high-risk areas such as riverbanks and slopes (Valverde et al., 2018; Yu et al., 2022). The investigation of such extreme events has gained prominence in the international scientific community due to the socioeconomic and environmental damage that excessive rainfall causes in various regions of the planet (Ferreira et al., 2017).

To minimize the effects of intense rainfall, drainage and flood control systems are constructed in urban environments. These are essential structures for directing runoff and preventing or mitigating the impacts of floods (Binesh et al., 2019). For this, an accurate estimate of the design floods is necessary while designing hydrological/hydraulic structures (Venkatesh et al., 2022). This estimation can be obtained from a time series of observed flow data or from a rainfall-runoff model based on design storm (Dorneles et al., 2019; Lima Neto et al., 2021; Suzuki et al., 2022). Generally, the design of these structures requires the study of the three variables that characterize the precipitation of a given region: intensity, duration, and frequency (IDF) (Dorneles et al., 2019). Understanding these variables allows for the determination of an IDF equation, which can also be expressed in the form of IDF curves, the primary tool for representing intense rainfall in a specific location (Noor et al., 2018, 2021; Sabino et al., 2020).

In many countries, including Brazil, the determination of IDF equations remains challenging due to the insufficiency of precipitation records, especially for sub-daily durations, primarily due to the low density of the monitoring network, large number of missing days and the short observation period available (Abreu et al., 2022; Aragão et al., 2024; Barbosa et al., 2022; Costa et al., 2024; Freitas et al., 2020; Nunes et al., 2021; Soares et al., 2016). According to Ombadi et al. (2018), the availability of long-term rainfall records with adequate spatial distribution is limited and unable to reflect the temporal variation and spatial heterogeneity of precipitation. Moreover, the availability of sub-daily precipitation data is low, with daily monitoring being more common, which requires the use of rainfall disaggregation methods that may not be suitable for the region (Abreu et al., 2022).

To increase data availability, satellite-based precipitation products (Beck et al., 2019; Chen et al., 2008; Funk et al., 2015; Huffman et al., 2020; Nguyen et al., 2020) have become an important alternative (Llauca et al., 2021), mainly due to their wide spatial coverage and refined spatiotemporal resolution (Min et al., 2020). However, these products should be used with caution, as it is necessary to verify the accuracy of their estimates. Several studies indicate that satellite precipitation products exhibit both random and systematic errors, especially in detecting extreme precipitation values (Ombadi et al., 2018; Prakash et al., 2016; Sadeghi et al., 2021). Despite the wide spatial coverage, satellite data are also mostly available in daily temporal resolution. Some products offer data at smaller temporal scales, but they show high inaccuracies in the estimates (Freitas et al., 2020; Ramos Filho et al., 2022; Trang et al., 2020). However, although ground-based data is still the most reliable source of precipitation data, satellite-based precipitation monitoring is a promising alternative.

In this context, the present work aims to contribute to the discussion and understanding of the relationships between rainfall intensity, duration, and frequency, considering satellite-based precipitation data as an important alternative for designing reliable hydraulic structures in ungauged areas, in search of more resilient systems to extreme flood conditions.

MATERIAL AND METHODS

Figure 1 briefly presents the methodological flowchart of the research, identifying the activities carried out, the methods used, and the data sources required. Details about each step of the methodology are described in the following sections.

Figure 1
Flowchart briefly describing the research method.

Study area

The state of Bahia, located in the Northeast of Brazil, is divided into seven mesoregions according to the 1989/90 regionalization by IBGE: Extremo Oeste Baiano, Vale São-Franciscano da Bahia, Centro Norte Baiano, Nordeste Baiano, Metropolitana de Salvador, Centro Sul Baiano, and Sul Baiano. The Sul Baiano (or Southern Bahia) mesoregion (Figure 2) has historically faced the consequences of intense rainfall events, which have significant impacts on the region and have become increasingly frequent. In recent years, the state of Bahia has experienced extreme episodes, occurring in 2021, 2022, and 2023, which resulted in multiple flooding events, fatalities, displaced families, and state of emergency in several municipalities in the southern part of the state. In 2021, 24 people died and 629,000 were affected throughout Bahia (G1 Bahia, 2022).

Figure 2
Location of the Southern Bahia Mesoregion.

This mesoregion encompasses 70 municipalities, it has an approximate area of 54,685.580 km², and an estimated population of 2,100,238 people (Instituto Brasileiro de Geografia e Estatística, 2024). According to Köppen's climate classification for Brazil (Alvares et al., 2013), the predominant climate type in the area is tropical rainforest (Af), with annual precipitation between 1900 and 2000 mm. This climate type is characterized by the absence of a dry season, with precipitation exceeding 60 mm in the driest month and the highest volume of rainfall between March and August (Ferraz et al., 2020). In addition to the Af climate, which covers the coastal area to the east, the region has two other climatic domains varying from east to west: the tropical monsoon climate (Am) in the central portion and the tropical sub-humid climate with summer rains and dry periods in the winter (Aw) in the western portion (Engelbrecht et al., 2019).

Ground-based rain gauge data

The daily precipitation data was collected from 31 rain gauge stations located in the study area (Figure 2), which is available on the National Water and Sanitation Agency’s (ANA) database called Hidroweb. Stations with the most data availability and in accordance with the temporal coverage of the satellite-based product were selected. Only stations with less than 10% of missing days were considered, and corrected data was prioritized over raw data, when available. According to the temporal availability of CHIRPS, the analysis was carried out with data from 1982 to 2022 (41 years).

Satellite-based precipitation data

One of the satellite-based precipitation products available and widely used for climate research lately is CHIRPS (Climate Hazards Group InfraRed Precipitation with Stations), which is a data collection developed by the United States Geological Survey (USGS) and the Climate Hazards Group at the University of California, Santa Barbara (UCSB). This product has been developed based on global CCD (Cold Cloud Duration) rainfall estimates calibrated by the TMPA 3B42 v7 (Tropical Rainfall Measuring Mission Multi-Satellite Precipitation Analysis version 7). It uses ground-based data for bias correction and provides daily precipitation data at 0.05º×0.05º and 0.25º×0.25º resolutions with temporal coverage from 1981 until present (Funk et al., 2015).

CHIRPS was selected for this study due to its extensive temporal coverage, high spatial resolution and, most importantly, its good performance compared to other products (Mianabadi, 2023), including in tropical regions of South America (Costa et al., 2019; López-Bermeo et al., 2022; Paredes-Trejo et al., 2017; Xavier et al., 2021).

CHIRPS data is available on the UCSB website (University of California, 2024) in NetCDF, GeoTiff, and Esri BIL formats. To extract point data corresponding to the location of the rain gauge stations available in the ANA database, R code was used and processed in RStudio.

Bias correction of the satellite-based data

Previous studies indicate that satellite precipitation products exhibit both random and systematic errors, especially in detecting extreme precipitation (Ombadi et al., 2018; Prakash et al., 2016; Sadeghi et al., 2021). Therefore, the bias in extreme precipitation values must be corrected before applying them for the development of IDF curves. In this study, the daily Annual Maximum Precipitation (AMP) series method was used to define extreme precipitation values, as proposed by Venkatesh et al. (2022) and also adopted by Mianabadi (2023). For this purpose, the AMP series were extracted from both field data and satellite data. The temporal extent of the series is 41 years, from 1982 to 2022. Then, the daily AMP series were ordered from the highest value to the lowest. The bias correction factor (f_bc), as proposed by Mianabadi (2023), is defined as the ratio between the AMP measurement at station j ($AM{P}_{St\left(i,j\right)}$) and the satellite-based AMP at station j ($AM{P}_{Sa\left(i,j\right)}$), as follows:

$f_{bc}=\frac{AM{P}_{St\left(i,j\right)}}{AM{P}_{Sa\left(i,j\right)}}$ (1)

In this equation, i indicates a given station. After calculating the correction factor for each event, the average factor is multiplied by the original AMP series, thus obtaining the bias-corrected AMP series.

This methodology applied for bias correction requires the use of ground-based precipitation data series. However, due to the low density and poor distribution of the rain gauge stations in Brazil, specifically in the State of Bahia, these data are not available in many locations. In many places, even where rain gauge stations exist, the historical data series contain a large number of gaps, mainly due to a lack of adequate maintenance. Therefore, this study sought to relate the calculated bias correction factors with some physical and climatic parameters of the stations used, aiming to propose a bias correction method that does not rely on ground-based data.

After determining the f_bc for all stations, an effort was made to verify the relationship between this factor and the following physical and climatic parameters of the stations used: geographical location (latitude, longitude), elevation, mean annual maximum daily precipitation (AMP), and mean total annual precipitation (PRCPTOT). This relationship was assessed by using the coefficient of determination (R²). Ombadi et al. (2018) observed a relationship between the bias correction factor and elevation, indicated by a Pearson correlation coefficient of 0.54. According to the authors, this relationship suggests that, in general, the studied satellite product tends to have a greater error, particularly underestimation, in higher elevation regions. This is due to the fact that estimating warm orographic rainfall over high-altitude regions is challenging for infrared image-based algorithms used in satellite products (Dinku et al., 2008).

After verifying these relationships, it was possible to determine an equation to calculate a bias correction factor (f_bc) that does not rely on ground-based data, by using multiple linear regression. This provides a simple bias correction method for CHIRPS data in ungauged areas of the Southern Bahia Mesoregion. Based on the relationships obtained, simple linear regression models were tested with each of the variables individually. Additionally, two multiple linear regression models were tested with the variables that showed the strongest relationships. In total, five linear regression models were tested: Reg1, Reg2, Reg3, Reg4, and Reg5. The coefficient of determination (R²) and the Standard Error (SE) were used to evaluate the models. SE can be defined as follows:

$SE=\sqrt{\frac{\sum {\left(y_i-{\widehat{y}}_i\right)}^2}{n-p}}$ (2)

In this equation, $y_i$ indicates the observed value of the dependent variable. ${\widehat{y}}_i$ is the predicted value of the dependent variable from the regression equation. n is the number of observations in the dataset, and n - p indicates the degrees of freedom for the regression, where p is the number the of estimated parameters (e.g., for a simple linear regression there are two parameters: slope and intercept, so p = 2).

Due to the lack of IDF curves for many locations in the state of Bahia, it has become common practice for designers to use IDFs obtained from the Pluvio 2.1 software to estimate design flows in their hydraulic projects. Pluvio 2.1 was developed by the Water Resources Research Group (GPRH) at the Federal University of Viçosa (UFV) and was made available in 2006. This application uses an interpolation method to determine IDF equations for any coordinate entered by the user, based on locations where the IDF equation is known. Most (19) of the equations used as references in the software for the state of Bahia were determined by Silva et al. (2002), based on historical pluviographic series with data up to 1999. This method of obtaining IDF equations has become a widely used alternative in ungauged areas of Brazil. However, for the safe design of hydraulic structures, it is important to use updated IDF equations, and the ones available in the software may be outdated. Studies indicate the impact of ongoing climate changes on hydrological records in many parts of the world (Obeysekera & Salas, 2014; Sun et al., 2021; Tabari, 2021; Wasko et al., 2021), including Brazil (Nunes et al., 2021; Ribeiro et al., 2019; Xavier et al., 2020), violating the assumption of stationarity of the historical series and emphasizing the need for updating IDF curves. Therefore, the results obtained from the IDFs developed from CHIRPS data corrected by the proposed method were also compared to those obtained from IDFs provided by Pluvio, to assess their performance relative to current practices.

Developing IDF curves for daily AMP

The IDF curves were developed on the Genetic Algorithm Methodology for IDF (GAM-IDF), which is an application developed by the Research Group in Hydrology and Hydrological Modeling in Watersheds at the Federal University of Pelotas (UFPel) (Cunha et al., 2019; Vargas et al., 2019). Firstly, the algorithm checks for the presence of a trend in the data series by applying the non-parametric Mann-Kendall test at a 5% significance level. The stations that presented a trend in their series were disregarded.

The next procedure is extracting the AMP series and, for this, the algorithm considers valid years those with no more than 31 missing days. After obtaining the AMP series, the algorithm tests twelve probability density functions (PDFs) and selects the one with the best fit, namely Exponential, Gamma, Generalized Extreme Values (GEV), Generalized Logistic, Generalized Normal, Generalized Pareto, Gumbel, Kappa, Pearson type III, Wakeby, Log-Normal 3 parameters and Weibull (Hosking, 2024). The parameters of all PDFs are estimated using the L-moments technique (Hosking & Wallis, 1997). The Anderson-Darling goodness-of-fit test is employed, at a 5% significance level, as it is a robust test for evaluating the fit of PDFs to AMP series (Beskow et al., 2015).

The return periods considered for the development of the IDF curves were 2, 5, 10, 20, 50, and 100 years. The next step was the disaggregation of the daily AMP associated with each return period. GAM-IDF uses the “different durations ratio method” to estimate precipitation on a sub-daily scale. The disaggregation constants from Companhia Ambiental do Estado de São Paulo (1986) were used, as they are commonly employed in Brazil for the disaggregation of daily rainfall (Caldeira et al., 2015) in the absence of local disaggregation coefficients. The disaggregated rainfall depths (h), associated with different return periods and durations, are converted into rainfall intensity (mm.h-1).

After all the processing, the parameters of the IDF equations were obtained, and the intensities (i) were calculated for different return periods (2, 5, 10, 15, 20, and 25 years) considering a rainfall duration time (d) of 10 minutes. This choice was made considering the application of the IDF curves for micro drainage designs.

Evaluation metrics

In search for a robust method to evaluate the accuracy of satellite product estimates compared to ground-based data, several studies have employed a combination of at least four statistical metrics (Costa et al., 2019, 2024; López-Bermeo et al., 2022; Mianabadi, 2023; Ramos Filho et al., 2022; Silva et al., 2020; Venkatesh et al., 2022; Xavier et al., 2021). Therefore, the performance of the IDF curves estimated by CHIRPS was evaluated by Percent Bias (PBIAS), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and the Nash-Sutcliffe Efficiency Coefficient (NSE), as follows:

$PBIAS=100\frac{{{\displaystyle \sum }}_{i=1}^n\left(S{a}_i-S{t}_i\right)}{{{\displaystyle \sum }}_{i=1}^n\left(S{t}_i\right)}$ (3)

$RMSE={\left[\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(S{a}_i-S{t}_i\right)}^2\right]}^{1/2}$ (4)

$MAE=\frac{1}{N}{{\displaystyle \sum }}_{i=1}^n\left|S{a}_i-S{t}_i\right|$ (5)

$NSE=1-\frac{{{\displaystyle \sum }}_{i=1}^n{\left(S{a}_i-S{t}_i\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(S{t}_i-\overline{S{t}_i}\right)}^2}$ (6)

In these equations, Sa and St are the rainfall intensity acquired from the satellite-based product and stations, respectively. i is the representative of each return period (2, 5, 10, 15, 20 and 25 years) and n is the number of different return periods considered (n=6). The comparison was conducted for each station separately and for all return periods of the daily AMP.

According to Silva et al. (2020), the PBIAS measures the accuracy of a model by comparing the mean of satellite-estimated data with the mean of observed data. It evaluates the extent to which the model tends to overestimate or underestimate actual values. An ideal PBIAS value is zero, indicating alignment between the estimated and observed data. Positive values suggest that the model tends to overestimate, while negative values indicate underestimation.

The RMSE assesses the average magnitude of errors between CHIRPS data and ground-based data (Bayissa et al., 2017; Costa et al., 2019). Meanwhile, the MAE is a statistical analysis less influenced by outliers. This metric represents the cumulative sum of errors and provides a robust estimate to indicate the ability of the estimated data to replicate observed values (Costa et al., 2019). Finally, the NSE, which ranges from –infinity to 1, helps evaluate whether there is an overall underestimation of the sample and whether the fit between the data series is satisfactory.

To obtain parameters for interpreting the results, the classifications recommended by Anjinho et al. (2021) and Rauf & Ghumman (2018) for the PBIAS and NSE indicators were also considered, as shown in Table 1, where PBIAS is classified in four classes and NSE is classified in five classes.

Table 1
Classification for PBIAS and NSE.

Classes	PBIAS	NSE
Very good	< 5	0.75 - 1.00
Good	5 - 10	0.65 - 0.75
Satisfactory	10 - 15	0.50 - 0.65
Acceptable	-	0.40 - 0.50
Unsatisfactory	≥ 15	≤ 0.40

• Source: Adapted from Rauf & Ghumman (2018); Anjinho et al. (2021).

RESULTS AND DISCUSSION

Bias correction

To correct the bias in the AMP series from CHIRPS, the bias correction factor (f_bc) was calculated for each station. The calculated f_bc ranged from 0.360 to 1.388, considering all stations. Values lower than 1 indicate that CHIRPS overestimated the maximum precipitation at the given location, while values greater than 1 indicate that the rainfall was underestimated by the product.

Figure 3 presents the relationship between f_bc and the other analyzed parameters in scatter plots, along with their respective trend lines and R² values. Longitude showed the strongest relationship, with an R² of 0.802, followed by latitude (R² = 0.5715), PRCPTOT (R² = 0.4804), while elevation showed the weakest relationship, with an R² of 0.198. The AMP parameter showed a very weak relationship with f_bc (R² = 0.057) and was therefore excluded from the analyses.

Figure 3
Relationship between the bias correction factor (f_bc) and longitude (a), latitude (b), PRCPTOT (c) and elevation (d).

In Figure 4, the relationship between the f_bc and the geographical location of the stations (latitude, longitude) can be clearly seen. In the dark green zone, the f_bc values are greater than 1, meaning that the CHIRPS estimates are being adjusted upward. This indicates that in this zone, CHIRPS tends to underestimate maximum precipitation. On the other hand, as we move toward the red zone, the f_bc values are lower than 1, meaning that the CHIRPS estimates are adjusted downward, indicating that maximum precipitation is overestimated by the product in this zone.

Figure 4
Interpolation of the f_bc across the Southern Bahia Mesoregion.

Considering the results showed in Figure 3, elevation was also disregarded due to its weaker relationship (R² = 0.198). Thus, three simple linear regression models were tested, involving longitude (Reg1), latitude (Reg2) and PRCPTOT (Reg3), individually. In addition, two multiple linear regression models were tested: the first involving longitude and latitude (Reg4), and the second involving longitude, latitude, and PRCPTOT (Reg5), as these variables showed the strongest relationship with the calculated f_bc. The details of each of the five models and the respective statistical indicators of the regression are presented in Table 2.

Table 2
Linear regression models tested for bias correction.

Model	Variables	Type	R2	SE
Reg1	Longitude	Simple	0.795	0.1255
Reg2	Latitude	Simple	0.557	0.1638
Reg3	PRCPTOT	Simple	0.462	0.1868
Reg4	Longitude and Latitude	Multiple	0.926	0.1030
Reg5	Longitude, Latitude and PRCPTOT	Multiple	0.928	0.1076

• SE = Standard Error.

Longitude is the variable with the strongest relationship to f_bc, however, when this variable is combined with latitude and PRCPTOT, the regression model results are better than when used alone. The models that performed best were the multiple regression models Reg4 and Reg5, with R² values of 0.926 and 0.928, respectively. However, the f_bc calculated by Reg5 had an overall standard error (SE) of 0.1076, slightly higher than Reg4's SE of 0.1030, when compared to the original f_bc.

Table 3 presents the statistical indicators of the estimated parameters involved in each regression. Overall, Reg5 showed a higher Standard Error than Reg4 across all parameters. Furthermore, in Reg5, the PRCPTOT variable did not show statistical significance at the 5% level to explain the response variable (f_bc) in this model, as the P-value for this variable was 0.325. All other models had parameters with statistical significance, that is, P-values under 5%.

Table 3
Statistics of the coefficients estimated for each regression model.

Name	Intersection		Longitude		Latitude		PRCPTOT
	Standard error	P-value	Standard error	P-value	Standard error	P-value	Standard error	P-value
Reg 1	2.129	2.483E-11	0.0537	1.033E-11
Reg 2	0.325	0.00127			0.0206	8.753E-07
Reg 3	0.143	7.481E-12					0.000112	1.547E-05
Reg 4	2.255	1.140E-08	0.0608	3.863E-08	0.0158	0.00279
Reg 5	2.957	9.796E-06	0.0758	1.074E-05	0.0158	0.00320	8.491E-05	0.325

Therefore, the multiple linear regression model involving longitude e latitude (Reg4) was the one that best represented the response variable (f_bc), resulting in the following equation:

$f_{bc\left(reg\right)}=-0.45404\times long-0.05185\times lat-17.954$ (7)

The confidence interval was calculated based on the Standard Error found for the Reg4 estimates, and it can be affirmed, with 95% confidence that, when using Reg4 to determine the bias correction factor in the study area, the associated error will be between -0.4% and 21% of the expected value for the f_bc.

Figure 5 shows the interpolation of the Standard Error (SE) for the estimates from the chosen model (Reg4) across the study area. This figure allows for the visualization of the zones where the model performed better and, most importantly, the zones where the errors were higher and, consequently, where the use of this model may not be recommended.

Figure 5
Interpolation of the Standard Error (SE) from f_{bc (reg)} compared to the original f_bc.

Developing IDF curves

The series were tested for the presence of trends by applying the non-parametric Mann-Kendall test at a 5% significance level and, of the thirty-one stations tested, four ground-based data series showed a trend, while only one of the satellite-based series did. Therefore, the series that presented a trend were disregarded and only twenty-six of the initial thirty-one stations were used in the analysis.

For the ground stations, the probability density function (PDF) that best fit the data from each station varied significantly, with a total of seven out of twelve functions tested being used for the probabilistic modelling on GAM-IDF. The most common PDFs were Generalized Logistic (7), Kappa (6), GEV (6), and Weibull (5). As for the CHIRPS data, the PDFs varied even more, with a total of eight out of the twelve functions being used. However, Kappa (11) and Generalized Logistic (9) were much more recurrent than the other functions. Notably, although the Gumbel distribution is widely used in studies of extreme events, it was selected in only two stations. However, the fact that a given distribution was not selected does not imply that it did not fit the data series, but rather that another distribution provided a better fit.

In most cases, the PDF chosen for the ground data was not the same as the one chosen for the satellite data in the corresponding station. The same PDF was selected in only 5 out of 26 stations. However, each equation was modelled using the PDF that provided the best fit, aiming to minimize the errors, with no need to match the PDF used in the corresponding station.

Performance of the IDF curves derived from CHIRPS

Based on the estimated parameters for each IDF, the intensities for a 10-minute duration rainfall for different return periods (2 to 25 years) were calculated. Using the intensities obtained, the performance statistical indicators for CHIRPS in comparison to the ground-based data (reference) were calculated, and the results are presented in Table 4. This table shows CHIRPS’ performance in its original state (Orig.), after the traditional method for bias correction with the application of f_bc (BC.1), and after the proposed method with the application of f_{bc (reg)} (BC.2).

Table 4
Performance indicators for IDF curves from CHIRPS.

N	Code	PBIAS			RMSE (mm/h)			MAE (mm/h)			NSE
		Orig.	BC.1	BC.2	Orig.	BC.1	BC.2	Orig.	BC.1	BC.2	Orig.	BC.1	BC.2
1	1439023	31.48%	-4.26%	-8.97%	41.22	5.91	12.04	41.02	5.54	11.69	-4.02	0.90	0.57
2	1439044	-11.74%	-11.91%	-20.79%	14.79	14.98	24.87	13.39	13.59	23.72	0.62	0.61	-0.07
3	1439089	23.60%	-1.71%	-2.25%	26.43	1.91	2.48	25.90	1.88	2.46	0.10	1.00	0.99
4	1539002	59.53%	2.47%	10.51%	69.72	2.87	12.27	69.10	2.87	12.20	-9.08	0.98	0.69
5	1539006	14.58%	3.46%	-6.09%	18.34	5.72	7.54	17.29	4.91	7.22	0.14	0.92	0.85
6	1539010	57.04%	-8.65%	9.73%	61.92	13.67	12.70	61.91	11.54	10.56	-2.60	0.82	0.85
7	1539014	19.32%	-5.74%	-5.31%	21.22	6.46	5.99	21.16	6.29	5.82	0.23	0.93	0.94
8	1639000	-0.37%	-2.72%	-14.18%	3.42	5.22	19.64	2.98	4.47	18.79	0.97	0.93	0.06
9	1739001	28.00%	-7.81%	-3.69%	46.19	12.56	6.53	44.62	12.45	5.87	-0.46	0.89	0.97
10	1739005	14.96%	6.16%	1.73%	20.47	9.80	5.40	19.01	8.47	4.85	-0.82	0.58	0.87
11	1739010	4.72%	-5.88%	-19.80%	7.29	9.06	30.54	7.25	9.03	30.40	0.93	0.89	-0.21
12	1740001	-17.99%	1.26%	6.97%	24.33	6.86	12.43	24.24	6.10	10.68	-0.73	0.86	0.55
13	1740005	-11.38%	0.16%	7.67%	14.43	2.75	9.41	13.74	2.36	9.26	0.46	0.98	0.77
14	1740006	-13.26%	-2.60%	-5.41%	16.52	3.79	6.97	16.11	3.27	6.57	0.21	0.96	0.86
15	1740008	-26.71%	-5.70%	-8.61%	34.38	9.68	12.75	33.12	8.38	10.99	-2.16	0.75	0.56
16	1839000	-1.40%	-3.99%	12.21%	6.52	7.95	20.93	5.45	6.34	17.84	0.96	0.94	0.57
17	1840012	-22.57%	-1.29%	-5.83%	35.28	9.91	12.29	35.12	8.46	9.64	-0.15	0.91	0.86
18	1539016	-21.50%	-14.08%	-30.45%	28.70	20.05	39.50	26.57	17.40	37.64	-0.82	0.11	-2.44
19	1439002	47.48%	1.51%	21.17%	43.45	2.00	19.52	42.59	1.70	18.99	-0.02	1.00	0.79
20	1539008	-24.15%	-15.62%	-34.00%	31.49	21.29	43.63	29.69	19.21	41.81	-0.45	0.34	-1.78
21	1339000	47.64%	-3.84%	0.63%	55.06	4.47	0.73	54.47	4.39	0.72	-4.45	0.96	1.00
22	1339012	57.78%	-0.81%	-13.50%	87.52	3.91	21.45	86.76	3.48	20.27	-5.08	0.99	0.63
23	1339027	69.21%	-7.46%	15.52%	79.24	8.75	17.69	78.60	8.48	17.63	-10.03	0.87	0.45
24	1339038	98.86%	7.25%	-4.27%	156.53	11.58	6.69	154.42	11.32	6.67	-18.12	0.90	0.97
25	1339044	116.51%	-21.93%	11.73%	160.62	35.14	19.16	160.00	30.11	16.12	-17.99	0.09	0.73
26	1439006	23.13%	-19.70%	-12.75%	27.57	26.16	18.96	26.70	22.74	16.26	0.06	0.16	0.56
Average		21.64%	-4.75%	-3.77%	43.56	10.09	15.46	42.74	9.03	14.41	-2.78	0.78	0.45

Considering the results for CHIRPS in its original state, PBIAS was positive for some stations and negative for others, with values reaching up to 116.51%, at station 1339044. RMSE and MAE show average errors of 43.56 mm/h and 42.74 mm/h, respectively, while NSE has an average value of -2.78. These results reveal that, overall, CHIRPS in its original state does not provide satisfactory performance for developing IDF curves in the Southern Bahia Mesoregion. These results are consistent with previous studies regarding the errors presented by satellite products in estimating extreme precipitation (Ombadi et al., 2018; Prakash et al., 2016; Sadeghi et al., 2021). Therefore, it was found necessary to correct the bias of CHIRPS estimations before developing IDF curves.

The results for the bias corrected CHIRPS (BC.1) (Table 4) show that, of the 26 stations analyzed, only 4 showed decline (red color) in statistical indicators after applying the f_bc, while the other 22 showed improvement (green color). Regarding PBIAS, the value of 116.51% for station 1339044 dropped to -21.93% after bias correction. The average RMSE decreased from 43.56 mm/h to 10.09 mm/h, and the average MAE decreased from 42.74 mm/h to 9.03 mm/h. The average NSE also showed a significant improvement, rising from -2.78 to 0.78. Despite the overall improvement in the indicators, some stations still show considerable errors even after bias correction, such as stations 1439044, 1539016, 1539008, 1339044, and 143900, all of which have a negative PBIAS greater than 10%.

Considering the results after the application of the proposed bias correction method (BC.2) (Table 4), only 6 stations showed a decline in statistical indicators, while the others showed improvement, but with overall performance inferior to the previous bias correction method. Regarding PBIAS, 12 stations had a bias greater than 10%, either positive or negative. The average values of the RMSE and E indicators also increased compared to the previous bias correction, being 15.46 mm/h and 14.41 mm/h, respectively. The average NSE coefficient also worsened, dropping to 0.45.

PBIAS and NSE results obtained in Table 4 were classified according to Table 1 to facilitate the interpretation of the results (Table 5).

Table 5
Classification of the results for PBIAS and NSE.

Classes	PBIAS			NSE
	Orig.	BC.1	BC.2	Orig.	BC.1	BC.2
Very good	3	13	5	3	19	12
Good	0	8	9	0	1	2
Satisfactory	5	2	6	1	2	6
Acceptable	-	-	-	1	0	1
Unsatisfactory	18	3	6	21	4	5

Considering the classification adopted for PBIAS, in most stations (18) original CHIRPS showed unsatisfactory results, 5 were satisfactory, and only 3 were very good. Regarding the classification based on NSE, 21 stations had unsatisfactory performance, while 1 station was acceptable, 1 was satisfactory, and 3 were very good.

The NSE indicator (Table 5) reveals that the performance of the original data estimated by CHIRPS was unsatisfactory in 21 of the 26 stations analyzed. However, for the BC.1, this number dropped to 4, while 19 stations showed very good performance. This demonstrates that CHIRPS data in their original state do not perform sufficiently for application in developing IDF curves in the Southern Bahia Mesoregion; however, they can be used after bias correction.

For BC.2, the performance was inferior compared to BC.1, but there was still a significant improvement over the original data, with unsatisfactory performance in only 5 stations and very good performance in 12. Thus, in the absence of ground-based data, bias correction can be performed, with some limitations, using only geographic coordinates (latitude and longitude) in decimal degrees as parameters, as shown in Equation 6.

Aiming to compare the performance of the IDFs obtained after the BC.2 with one of the common practices for obtaining IDFs for ungauged areas in the state of Bahia, the results were compared with the IDFs obtained by Pluvio 2.1 (Table 6). In general, BC.2 data showed better performance than Pluvio data for all indicators, performing better in 20 stations and worse in only 6.

Table 6
Comparison between the performance obtained by IDFs from Pluvio and IDFs from BC.2 CHIRPS.

N	Code	PBIAS		RMSE (mm/h)		MAE (mm/h)		NSE
		Pluvio	BC.2	Pluvio	BC.2	Pluvio	BC.2	Pluvio	BC.2
1	1439023	10.63%	-8.97%	16.71	12.04	14.29	11.69	0.17	0.57
2	1439044	25.30%	-20.79%	30.49	24.87	28.87	23.72	-0.61	-0.07
3	1439089	30.27%	-2.25%	34.37	2.48	33.22	2.46	-0.52	0.99
4	1539002	23.10%	10.51%	28.93	12.27	26.82	12.20	-0.74	0.69
5	1539006	4.11%	-6.09%	6.75	7.54	5.80	7.22	0.88	0.85
6	1539010	13.80%	9.73%	15.44	12.70	14.97	10.56	0.78	0.85
7	1539014	12.74%	-5.31%	16.12	5.99	13.96	5.82	0.56	0.94
8	1639000	-6.83%	-14.18%	9.07	19.64	9.06	18.79	0.80	0.06
9	1739001	-13.91%	-3.69%	22.21	6.53	22.18	5.87	0.66	0.97
10	1739005	7.98%	1.73%	13.41	5.40	11.57	4.85	0.22	0.87
11	1739010	-10.55%	-19.80%	18.32	30.54	16.20	30.40	0.56	-0.21
12	1740001	18.04%	6.97%	26.72	12.43	24.31	10.68	-1.08	0.55
13	1740005	25.26%	7.67%	32.32	9.41	30.50	9.26	-1.71	0.77
14	1740006	24.65%	-5.41%	32.09	6.97	29.96	6.57	-1.98	0.86
15	1740008	21.98%	-8.61%	28.88	12.75	27.25	10.99	-1.23	0.56
16	1839000	45.66%	12.21%	67.25	20.93	66.72	17.84	-3.48	0.57
17	1840012	68.04%	-5.83%	108.93	12.29	105.84	9.64	-10.01	0.86
18	1539016	15.67%	-30.45%	20.73	39.50	19.37	37.64	0.05	-2.44
19	1439002	59.41%	21.17%	54.92	19.52	53.29	18.99	-0.63	0.79
20	1539008	16.28%	-34.00%	20.17	43.63	20.01	41.81	0.40	-1.78
21	1339000	2.04%	0.63%	3.29	0.73	2.81	0.72	0.98	1.00
22	1339012	-22.52%	-13.50%	35.20	21.45	33.81	20.27	0.02	0.63
23	1339027	2.74%	15.52%	4.13	17.69	3.55	17.63	0.97	0.45
24	1339038	-25.02%	-4.27%	39.39	6.69	39.08	6.67	-0.21	0.97
25	1339044	-15.29%	11.73%	26.75	19.16	22.65	16.12	0.47	0.73
26	1439006	18.85%	-12.75%	21.81	18.96	21.76	16.26	0.41	0.56
Average		13.56%	-3.77%	28.25	15.46	26.84	14.41	-0.55	0.45

Table 7 summarizes PBIAS and NSE results for the BC.2 CHIRPS data in comparison with the IDFs provided by Pluvio.

Table 7
Classification of the results for PBIAS and NSE from Table 6.

Classes	PBIAS		NSE
	Pluvio	BC.2	Pluvio	BC.2
Very good	3	5	5	12
Good	2	9	1	2
Satisfactory	5	6	2	6
Acceptable	-	-	3	1
Unsatisfactory	16	6	15	5

Considering the NSE indicator, the BC.2 data showed a very good result in 12 stations and unsatisfactory results in only 5, whereas the Pluvio data showed a very good result in only 5 stations and unsatisfactory results in 15. This analysis indicates that using CHIRPS data after applying the proposed bias correction factor (f_{bc (reg)}) could be more recommended than using IDFs provided by Pluvio in the study area.

CONCLUSIONS

In this study, satellite-based rainfall data derived from the product CHIRS was evaluated to determine its ability to develop IDF curves in the Southern Bahia Mesoregion, which primarily lead to the following conclusions:

• Initial results indicated that the product, in its original state, does not provide satisfactory performance for developing IDF curves in the study area (PBIAS 21.54%; RMSE 43.56 mm/h; MAE 42.74 mm/h; NSE -2.78). However, the results looked much more promising when the bias of CHIRPS estimates were corrected (PBIAS -4.75%; RMSE 10.09 mm/h; MAE 9.03 mm/h; NSE 0.78);

• This study proposed a bias correction method based on a linear regression model, and results (PBIAS -3.77%; RMSE 15.46 mm/h; MAE 14.41 mm/h; NSE 0.45) showed that, in the absence of ground-based data, bias correction can be performed for the study area, with some limitations, using only geographic coordinates (latitude and longitude);

• The performance was compared with IDF curves provided by Pluvio (PBIAS 13.56%; RMSE 28.25 mm/h; MAE 26.84 mm/h; NSE -0.55), which is a common software used by engineers/designers, and results indicated that using CHIRPS data after applying the proposed bias correction factor (f_{bc (reg)}) could be more recommended than using IDFs interpolated by Pluvio in the study area.

Further analysis is being conducted to consider the trends found in some of the time series and develop IDF curves considering non-stationary conditions, which is important for adaptation to the new and future climatic scenario. Future studies might also test different satellite-based precipitation products and compare the results. In addition, improving IDF curves contributes to more reliable hydraulic structures design, but it is important to acknowledge that this is only part of the solution. In order to improve flood risk management and achieve more resilient systems to extreme events, other aspects must be considered such as sustainable urban drainage solutions, advancing in weather forecasting and flash flood warning systems, as well as improve community risk awareness.

ACKNOWLEDGEMENTS

The authors thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for the funding that enabled the development of this work, as part of the project approved by PEPEEC - Emergency Program for Prevention and Response to Disasters Related to Climate Emergencies, Extreme Events, and Environmental Accidents (PDPG-EC20222163351P).

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Edited by

• Editor-in-Chief:
  Adilson Pinheiro

• Associated Editor:
  Carlos Henrique Ribeiro Lima

Publication Dates

• Publication in this collection
  21 Apr 2025
• Date of issue
  2025

History

• Received
  12 Sept 2024
• Reviewed
  04 Dec 2024
• Accepted
  19 Feb 2025