Performance of multiparameter distributions in estimating rainfall extremes and deriving IDF equations in Paraná

Desempenho das distribuições multiparamétricas na estimativa de extremos de chuva e derivação das equações IDF no Paraná

ABSTRACT

Frequency analysis of rainfall involves selecting a probabilistic model to represent sample data and perform design estimates. Various probability distributions are available for this purpose, but there is no general consensus on the most appropriate one. In Brazil, distributions such as Log-Normal, Gumbel, and Generalized Extreme Value (GEV) are commonly used, while less conventional models like Kappa (KAP) and Wakeby (WAK) require further investigation. This study provides a comparative evaluation of the Gumbel, GEV, KAP, and WAK distributions for modeling extreme precipitation events in the state of Paraná, selected for their ability to represent a broad range of rainfall data types. Additionally, an alternative methodology was applied to derive Intensity-Duration-Frequency (IDF) equations. The results demonstrate that the GEV distribution exhibited performance comparable to the WAK and KAP distributions, while outperforming the Gumbel distribution. Furthermore, the shape parameter of the GEV distribution exerted a significant effect on the accuracy of IDF relationship estimations.

Keywords:
Wakeby; GEV; IDF equations

RESUMO

A análise de frequência de chuvas requer a seleção de um modelo probabilístico adequado para representar os dados amostrais e realizar as estimativas de projeto. Diversas distribuições de probabilidade estão disponíveis para essa finalidade, mas ainda não há consenso sobre qual é a mais apropriada. No Brasil, distribuições como Log-Normal, Gumbel e Generalizada de Valores Extremos (GEV) são frequentemente empregadas, enquanto modelos menos convencionais, como Kappa (KAP) e Wakeby (WAK), necessitam de mais investigação. Este estudo realiza uma avaliação comparativa das distribuições Gumbel, GEV, KAP e WAK para modelar eventos extremos de precipitação no estado do Paraná, escolhidas por sua capacidade de representar uma ampla gama de tipos de dados pluviométricos. Adicionalmente, foi aplicada uma metodologia alternativa para derivar as equações de intensidade-duração-frequência (IDF). Os resultados demonstram que a distribuição GEV apresentou desempenho comparável às distribuições WAK e KAP, enquanto superou a distribuição Gumbel. Além disso, o parâmetro de forma da distribuição GEV exerceu um efeito significativo na precisão das estimativas das relações IDF.

Palavras-chave:
Wakeby; GEV; Equações IDF

INTRODUCTION

Understanding and mitigating the risks associated with intense rainfall is vital for safeguarding human lives and property against disasters like floods (Boudrissa et al., 2017). This task necessitates a meticulous risk assessment, typically conducted through frequency analysis (Ibrahim, 2019).

A fundamental aspect of frequency analysis is selecting the most appropriate probability distribution for intense rainfall data. This decision plays a critical role in estimating rainfall values associated with various return periods, directly impacting the design of hydraulic structures (Flowers Cano & Ortiz Gómez, 2021). However, the vast array of models found in the literature may yield diverse outcomes, potentially impacting the accuracy of risk assessments. Overestimations can lead to excessive construction costs, whereas underestimations may result in material damage and loss of lives (Öztekin, 2007).

In the Brazilian context, distributions with two parameters like the Log-Normal and Gumbel models are commonly used (Beskow et al., 2015; Caldeira et al., 2015; Abreu et al., 2018). Additionally, three-parameter distributions such as the Generalized Extreme Value, Generalized Pareto, and Generalized Logistic distributions are also employed (Pansera et al., 2022; Abreu et al., 2018). However, more research is needed on distributions like Kappa (four parameters) and Wakeby (five parameters) (Blain & Meschiatti, 2014).

In this context, accurately defining the analytical form of the distribution to model rainfall data can be quite challenging. Therefore, it's essential to explore a range of probability distributions, each differing in shape and the number of parameters that define them (Naghettini, 2017).

Once the probability distribution for modeling intense rainfall is chosen, it becomes possible to create intensity-duration-frequency (IDF) equations (Gomes & Vargas Júnior, 2018). The IDF equations can be constructed using the disaggregation approach (Aragão et al., 2013). An essential aspect of developing IDF equations is defining their mathematical structure (Pansera et al., 2020). Back (2020) recently proposed a promising alternative approach for estimating IDF equations, but further studies are needed to validate this method.

This study investigated the performance of multiparametric distributions in analyzing the frequency of intense rainfall in Paraná state, Brazil. The selected distributions included Gumbel, Generalized Extreme Value, Kappa, and Wakeby, chosen to cover a broad spectrum of shapes and parameters. Additionally, IDF equations were estimated using an alternative approach proposed by Back (2020).

MATERIAL AND METHODS

Rainfall data

For this study, historical precipitation data from Paraná state (Brazil) were obtained from the National Water and Sanitation Agency (ANA) database via the HidroWeb system. Station selection was based primarily on the length of available historical records. It's important to note that using the Wakeby distribution in small samples should be avoided due to potential overfitting, as recommended by Rahman et al. (2014). Stations with a minimum of 40 years of records and less than 5% data gaps were chosen. In total, 65 stations were selected, as depicted in Figure 1.

Figure 1
Spatial distribution of the rainfall monitoring stations used in the study.

The 65 stations chosen for this study, shown in Figure 1, provide the data for the distribution of annual daily maximum rainfall presented in Figure 2. It is evident that the 100 to 200 mm precipitation range predominates across the stations. At 20 stations, annual daily maximums between 200 and 300 mm were observed, while three stations recorded rainfall exceeding 300 mm. The highest values, approaching 400 mm, were recorded at two stations.

Figure 2
Distribution of annual daily maximum precipitation for the 65 studied stations.

Distributions

The distributions used in this study vary in complexity based on the number of parameters required to define them. First, the Gumbel distribution (GUM) (Naghettini, 2017) uses two parameters: location (u) and scale (α). The Generalized Extreme Value distribution (GEV) (Naghettini, 2017) adds a shape parameter (k) to these. The Kappa distribution (KAP) (Murshed et al., 2014) includes the location, scale, and two shape parameters (u, α, k, h). Lastly, the Wakeby distribution (WAK) (Busababodhin et al., 2016) is the most complex, with three shape parameters (β, γ, δ) in addition to the location and scale parameters. This sequence illustrates how these distributions become more complex and require additional parameters to accurately describe them, as detailed in Equations 1-4.

$x_{GUM}\left(F\right)=u-\alpha ln\left(-ln\left(F\right)\right)$ (1)

$x_{GEV}\left(F\right)=u+\alpha /k\left[1-{\left(-lnF\right)}^k\right]$ (2)

$x_{KAP}\left(F\right)=$ $u+\alpha /k\left[1-{\left\{\left(1-F^h\right)/h\right\}}^k\right]$ (3)

$x_{WAK}\left(F\right)=$ $u+\alpha /\beta \left[1-{\left(1-F\right)}^{\beta }\right]-\gamma /\delta \left[1-{\left(1-F\right)}^{-\delta }\right]$ (4)

Where 𝑥 represents the quantile function, 𝐹 denotes the cumulative probability of non-exceedance (1 - 1/T), and 𝑇 denotes the return period.

The distribution parameters were estimated following the metaheuristic methodology proposed by Hassanzadeh et al. (2011). Metaheuristic techniques enable a more efficient exploration of the search space, facilitating the identification of distribution parameters that optimally fit the sample data (Gomes & Vargas Júnior, 2018). Furthermore, metaheuristics provide significant advantages over conventional methods by improving the probability of achieving optimal solutions. In contrast, increasing the number of parameters in traditional methods not only increases computational complexity but also hinders convergence toward effective solutions (Hassanzadeh et al., 2011; Murshed et al., 2014; Busababodhin et al., 2016). This approach aims to minimize the objective function to achieve an optimal fit, as defined by Equation 5.

$FOB\left(\theta \right)=\frac{{{\displaystyle \sum }}_{i=1}^n{\left(O_i-Q_i\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(O_i-\overline{O}\right)}^2}$ (5)

Where θ represents the parameter vector characterizing the quantile function, $O$ denotes the observed values, $\overline{O}$ represents the mean of the observed values, Q are obtained from the quantile functions (Equations 1-4) using the Weibull plotting position, and n is the total number of observations.

To minimize the objective function, the study employed the differential evolution metaheuristic, implemented using the R package “DEoptim” (Mullen et al., 2011). Further procedural details can be found in Gomes & Vargas Júnior (2018).

Performance evaluation

For evaluating the overall goodness of fit of the distributions, the root mean square error (RMSE) was utilized:

$RMSE=\sqrt{\frac{1}{n-m}{\displaystyle \sum }_{i=1}^n{\left(O_i-Q_i\right)}^2}$ (6)

Where m is the number of parameters of the distribution.

The Modified Prediction Absolute Error (MPAE) was employed to evaluate the fit of the distributions above the 50th percentile, following equation 7.

$MPAE=\frac{2}{n}{\displaystyle \sum }_{i=n/2}^n\left|O_i-Q_i\right|$ (7)

This metric provides an assessment of distribution fitting quality in upper percentile regions, important for decision-making accuracy (Murshed et al., 2014).

Furthermore, the conformity of the distributions to the sample data was assessed using the Kolmogorov-Smirnov (KS) test (Naghettini, 2017), with a significance level set at 5%.

Intense rainfall equations

The disaggregation and derivation of Intensity-Duration-Frequency (IDF) relationships were carried out using the model proposed by Back (2020), as outlined in Equation 8:

$i=\left(\frac{60}{27.9327+3.8346t^{0.7924}}\right)\left(aln\left(T\right)+b\right)$ (8)

Where i represents rainfall intensity in mm/h, t denotes rainfall duration in minutes, and a and b are constants derived from logarithmic regression of quantiles and the return period.

RESULTS AND DISCUSSIONS

Performance evaluation

The GUM, GEV, KAP, and WAK distributions were initially fitted to the sampled data from the 65 selected rainfall stations using the metaheuristic approach. The KS test indicated that one fit was rejected for the GUM, GEV, and KAP distributions, while six stations showed rejection for the WAK distribution. Subsequently, performance metrics (MPAE and RSME) were computed and are presented in Figure 3.

Figure 3
Boxplots of MPAE and RMSE values for the 65 stations studied.

Upon analyzing Figure 3, it is evident that the RMSE metric predominantly indicates values below approximately 10 mm across most stations, while the MPAE metric suggests values below approximately 7.5 mm. Additionally, both the GUM and WAK distributions exhibit a broader range in the boxplot compared to the GEV and KAP distributions for both metrics. This pattern, combined with the six rejections observed in the KS test for the WAK distribution, suggests that the GEV and KAP distributions demonstrated superior performance compared to the WAK and GUM distributions.

Another notable point is the outlier stations that are highlighted in Figure 3. According to the RMSE metric, it is observed that two stations exhibited values above approximately 25 mm for the GUM distribution, whereas only one station showed this for the other distributions. Similarly, based on the MPAE metric, two stations displayed values above 20 mm for the GUM distribution, whereas for the GEV, KAP, and WAK distributions, only one station had a value exceeding 15 mm.

For a more thorough assessment, the two most discrepant stations identified in Figure 2 were compared with the two stations that exhibited the lowest values of the studied metrics, as depicted in Figure 4.

Figure 4
Comparison between stations showing the poorest performance (a, b) and those with the best performance (c, d).

From the analysis of Figure 4, it is clear that the two stations with the poorest performance (a, b) recorded extreme precipitation values exceeding 400 mm. On the other hand, stations with better fit did not have precipitation values above 200 mm. This highlights how extreme values can affect the accuracy of distribution fitting with metaheuristic methods.

Another assessment conducted involved comparing the estimated quantiles by the Gumbel distribution with the other distributions, and the GEV distribution with the KAP and WAK distributions, as depicted in Figure 5.

Figure 5
Pearson correlation of estimated quantiles by Gumbel (a) and GEV (b).

In Figure 5, it is evident that the Gumbel distribution exhibits strong correlation with the other distributions for quantiles up to a 20-year return period. The correlation diminishes to moderate levels between 20 and 100 years, and becomes notably low for periods exceeding 100 years. Conversely, the GEV distribution shows correlations consistently above 0,95 across all return periods with the WAK and KAP distributions, demonstrating particularly robust correlation with the KAP distribution.

The GUM distribution, with two parameters, exhibited inferior performance compared to the other distributions. The KAP (four-parameter) and GEV (three-parameter) distributions demonstrated similar performance, indicating that both are suitable. The similarity between the KAP and GEV distributions can be attributed to GEV being one of the possible forms of the KAP distribution (Murshed et al., 2014). Although Figure 5 reveals similarities among the WAK, KAP, and GEV distributions, it is important to note that six rejections occurred for the WAK distribution in the KS test.

Consequently, the GEV distribution presents notable advantages for metaheuristic parameter estimation due to its more straightforward parameter structure compared to the KAP and WAK distributions. This simplicity enhances the efficiency of convergence and lowers the risk of local optima. Martins & Stedinger (2000) highlighted that conventional methods may experience convergence difficulties, potentially resulting in implausible parameter estimates. However, such convergence issues were not observed in this study. Therefore, selecting an appropriate estimation method for the GEV distribution is essential for obtaining a reliable fit to sample data, as conventional techniques may occasionally result in the rejection of the model fit (Marques et al., 2014).

On the other hand, the KAP and WAK distributions present challenges due to the intricate nature of their quantile functions. The complexity associated with these distributions, which varies with different parameter combinations, can lead to suboptimal results, local minima, or difficulties in finding feasible solutions (Hassanzadeh et al., 2011; Öztekin, 2011; Murshed et al., 2014; Busababodhin et al., 2016; Papukdee et al., 2022). Despite these complexities, this study successfully achieved feasible solutions for all the stations analyzed.

This study highlights the general preference for three-parameter distributions over two-parameter models, indicating that increasing the number of parameters beyond three does not necessarily improve the accuracy of replicating sample statistics. Supporting this observation, Papalexiou & Koutsoyiannis (2013) demonstrated that the GEV distribution is the most suitable model for annual maximum daily rainfall data from over 15,000 global records, outperforming the GUM distribution. Similarly, Beskow et al. (2015) assessed the performance of GUM, GEV, and KAP distributions in modeling extreme precipitation events in Rio Grande do Sul. Their findings confirmed that while all three distributions were viable, the KAP distribution was the most appropriate, with the GEV serving as a viable alternative. This preference for KAP is attributed to the fact that the GEV is a special case of the KAP distribution. Furthermore, Blain & Meschiatti (2014) found that the GEV distribution surpassed the WAK in characterizing precipitation data spanning over 100 years in Campinas (São Paulo). Similarly, Abreu et al. (2018) concluded that both the GEV and GUM distributions were effective in representing extreme precipitation events in the Rio Sapucaí basin, Minas Gerais.

Intense rainfall equations

The GEV distribution exhibited superior performance relative to the GUM and was comparable to the KAP and WAK distributions. Consequently, it was chosen for deriving the IDF relationships using equantion 8 for all stations not rejected by the KS test. This selection facilitates a clearer evaluation of how the distribution's shape influences IDF relationship derivation. If the KAP and WAK distributions were selected, the analysis would become more complex due to the numerous potential special cases introduced by their two and three shape parameters, respectively.

Quantiles for return periods ranging from 2 to 1000 years were estimated using the GEV distribution and categorized into two groups: 2-100 years and 2-1000 years. The parameters of Equation 8 were then determined for each group. Figure 6 presents the coefficient of determination (R²) for Equation 8 relative to the shape parameter of the GEV distribution.

Figure 6
GEV Shape Parameter vs Coefficient of Determination (R²).

The analysis of Figure 6 reveals distinct trends based on the return period considered. For return periods ranging from 2 to 100 years, R² values consistently exceed 0,96 across various shape parameter values, indicating robust model performance. However, when extending the analysis to return periods of 2 to 1000 years, the tail behavior of the GEV distribution becomes influential. Particularly noteworthy is the observation that for shape parameter values below -0,2, there is a notable decline in R². This phenomenon is attributed to the GEV distribution's transformation towards a Fréchet-like distribution with heavier tails for negative shape parameters $k<-\mathrm{0,2}$ (Naghettini, 2017). Consequently, the linearization proposed by Equation 8 begins to exhibit reduced effectiveness under these conditions.

Finally, Table 1 presents the estimated parameters for the IDF equations derived using Equation 8 at 64 pluviometric stations where the GEV distribution was not rejected by the Kolmogorov test. The coefficients are provided for return periods ranging from 2 to 100 years.

Table 1
Estimated parameters for intensity-duration-frequency (IDF) equations at 64 pluviometric stations using the GEV distribution for return periods ranging from 2 to 100 years.

No.	Code	City	Latitude (º)	Longitude (º)	Altitude (m)	a	b	R2
1	2349033	Tomazina	-23.77	-49.95	483	23.1218	61.7362	0.9992
2	2350000	Jataizinho	-23.26	-50.98	350	14.0475	71.0003	0.9905
3	2350002	Andirá	-23.09	-50.29	375	43.4682	33.4674	0.9815
4	2350017	Cambará	-23.00	-50.03	450	15.7825	74.4587	0.9931
5	2350062	Figueira	-23.85	-50.39	526	19.6379	68.5392	0.9967
6	2352000	Rondon	-23.32	-52.67	250	24.1548	62.9710	0.9999
7	2353001	Santa Isabel do Ivaí	-23.01	-53.19	400	55.7495	30.2281	0.9820
8	2353002	Mariluz	-23.92	-53.13	320	30.8192	67.9060	0.9989
9	2449000	Adrianópolis	-24.66	-49.00	180	17.0266	56.6369	0.9747
11	2449007	Doutor Ulysses	-24.75	-49.33	400	16.9695	60.8865	0.9747
12	2450002	Tibagi	-24.51	-50.40	720	21.6236	59.1327	0.9987
13	2450005	Telêmaco Borba	-24.28	-50.63	650	24.6665	53.1858	0.9959
14	2450006	Ortigueira	-24.05	-50.71	550	31.9551	55.1298	0.9973
15	2450009	Tibagi	-24.68	-50.48	750	24.3756	55.0735	0.9997
16	2450010	Castro	-24.78	-50.00	1009	23.4987	52.0539	0.9989
17	2451003	Cândido de Abreu	-24.83	-51.15	550	17.3626	73.4535	0.9971
18	2451013	Pitanga	-24.75	-51.77	860	26.9211	66.4739	0.9998
19	2451014	Ivaiporã	-24.25	-51.65	360	25.3915	76.1506	0.9980
20	2451015	Manoel Ribas	-24.52	-51.67	972	31.2444	70.9732	0.9997
21	2451017	Grandes Rios	-24.34	-51.42	600	35.2896	49.4124	0.9901
22	2452009	Ubiratã	-24.53	-52.98	509	18.4177	77.2020	0.9984
23	2452011	Campina da Lagoa	-24.60	-52.80	618	18.7622	84.1112	0.9915
24	2452014	Mamborê	-24.28	-52.52	702	25.6531	77.5002	0.9991
25	2452015	Roncador	-24.60	-52.27	730	11.5146	88.3032	0.9747
26	2453013	Iguatu	-24.63	-53.10	730	47.5969	39.3491	0.9689
27	2454000	Marechal Cândido Rondon	-24.49	-54.32	225	38.9665	59.6286	0.9868
28	2454001	Guaíra	-24.07	-54.25	218	27.5568	68.6234	0.9992
29	2454003	Entre Rios do Oeste	-24.69	-54.23	245	35.1080	53.0579	0.9867
30	2548000	Morretes	-25.47	-48.83	8	52.9417	32.1844	0.9777
31	2548002	Morretes	-25.43	-48.94	680	40.0608	82.0634	0.9995
32	2548003	Antonina	-25.23	-48.75	80	20.8966	93.6646	0.9899
33	2549000	Lapa	-25.93	-49.78	750	29.5911	50.4394	0.9954
34	2549001	Porto Amazonas	-25.55	-49.88	793	15.0258	69.6499	0.9943
35	2549003	Quitandinha	-25.95	-49.39	810	18.5754	64.6766	0.9963
36	2549004	Piraquara	-25.45	-49.07	900	27.6774	56.5059	0.9998
37	2549006	Curitiba	-25.43	-49.27	924	17.4660	65.9200	0.9947
38	2550000	Prudentópolis	-25.24	-50.96	690	23.0110	64.5645	0.9999
39	2550001	São Mateus do Sul	-25.88	-50.39	760	42.9311	30.1680	0.9744
40	2550003	Ponta Grossa	-25.13	-50.15	790	34.1382	48.1222	0.9936
41	2550005	Mallet	-25.95	-50.68	750	41.6320	26.3246	0.9611
42	2550025	Teixeira Soares	-25.45	-50.58	893	25.7007	64.7877	0.9995
43	2551000	Guarapuava	-25.35	-51.48	950	36.5451	49.3133	0.9894
44	2551001	Inácio Martins	-25.69	-51.20	960	43.3518	36.6477	0.9729
45	2551004	Guarapuava	-25.64	-51.97	740	36.8069	54.1753	0.9926
46	2552000	Quedas do Iguaçu	-25.45	-52.90	550	29.9638	88.6630	0.9994
47	2552001	São Jorge do Ivaí	-25.77	-52.93	390	17.3812	95.1730	0.9806
48	2552006	Guaraniaçú	-25.08	-52.88	920	33.8256	69.8353	0.9968
49	2553004	Cruzeiro do Iguaçu	-25.57	-53.13	450	39.8647	58.4323	0.9913
50	2553009	Céu Azul	-25.13	-53.85	610	23.6512	99.6874	0.9914
51	2553012	Ampére	-25.92	-53.48	330	50.9019	56.3552	0.9870
52	2554001	Foz do Iguaçu	-25.63	-54.48	131	24.5806	86.3385	0.9973
53	2554002	Foz do Iguaçu	-25.68	-54.43	152	23.3782	90.9022	0.9885
54	2554006	São Miguel do Iguaçu	-25.35	-54.24	309	30.8985	73.5306	0.9997
55	2649006	Rio Negro	-26.10	-49.80	770	39.5071	24.6699	0.9604
56	2651000	União da Vitória	-26.23	-51.08	736	55.2893	27.5348	0.9809
57	2651003	General Carneiro	-26.39	-51.27	800	14.7167	80.2421	0.9838
58	2651004	Porto Vitória	-26.17	-51.22	790	41.4496	41.3969	0.9774
59	2651005	União da Vitória	-26.03	-51.14	840	27.7710	60.1390	0.9986
60	2651013	Bituruna	-26.19	-51.30	802	28.6120	63.1174	0.9983
61	2651015	Bituruna	-26.17	-51.55	900	21.7623	71.9757	0.9998
62	2651016	União da Vitória	-26.05	-51.20	800	37.0609	44.6728	0.9829
63	2652009	Bom Sucesso do Sul	-26.05	-52.80	550	23.3549	88.8334	0.9940
64	2652010	Palmas	-26.48	-52.00	1060	26.7814	69.7195	0.9989
65	2652011	Mariópolis	-26.35	-52.57	850	21.5005	76.7212	0.9999

CONCLUSIONS

This study evaluated the performance of the GUM, GEV, KAP, and WAK distributions across 65 pluviometric stations in Paraná and explored IDF relationships using the alternative model by Back (2020). The key findings are:

1. The GEV distribution (three parameters) performed similarly to the WAK (five parameters) and KAP (four parameters) distributions, while outperforming the GUM distribution (two parameters).

2. The GEV distribution is preferred for modeling intense rainfall in Paraná due to its superior performance over the GUM and comparable performance to WAK and KAP distributions. It also requires less computational effort for parameter estimation, making it more user-friendly for practical applications.

3. The tail behavior of the GEV distribution significantly influenced the accuracy of the IDF relationship estimates for return periods exceeding 100 years when using the alternative model proposed by Back (2020).

Additionally, a basic quality control procedure was applied to the rainfall data to eliminate obvious inconsistencies or errors. However, advanced consistency analyses and homogenization were not performed. Therefore, while the results provide valuable insights, more thorough quality control procedures in future studies may lead to refinements in the findings.

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Edited by

• Editor-in-Chief: Adilson Pinheiro
  Associated Editor: Carlos Henrique Ribeiro Lima

Publication Dates

• Publication in this collection
  17 Feb 2025
• Date of issue
  2025

History

• Received
  10 July 2024
• Reviewed
  29 July 2024
• Accepted
  15 Nov 2024