Probability distribution of heavy rainfall and determination of IDF in the city of Caruaru - PE

Mendes, Kevin Matheus Correia; Oliveira, Aline Lima de; Alcântara, Lucas Ravellys Pyrrho de; Alves, Adriana Thays Araújo; Santos Neto, Severino Martins dos; Coutinho, Artur Paiva; Montenegro, Suzana Maria Gico Lima; Soares, José Moura; Antonino, Antonio Celso Dantas

doi:10.4136/ambi-agua.2555

Kevin Matheus Correia Mendes

Centro Acadêmico do Agreste. Núcleo de Tecnologia. Universidade Federal de Pernambuco (UFPE), Avenida Marielle Franco, s/n, Km 59, CEP: 55014-900, Caruaru, PE, Brazil. E-mail: kevintjmendes@outlook.com, arthur.coutinho@yahoo.com.br, dscgeo@gmail.com

http://orcid.org/0000-0003-4107-7760

Aline Lima de Oliveira

Departamento de Energia Nuclear. Universidade Federal de Pernambuco (UFPE), Avenida Luiz Freire, n° 1000, CEP: 50740-545, Recife, Pernambuco, Brazil. E-mail: alinelimadeoliveira.alo@gmail.com, ravellyspyrrho@gmail.com, martinsdsn@gmail.com, acdantonino@gmail.com

http://orcid.org/0000-0002-4607-3825

Lucas Ravellys Pyrrho de Alcântara ^**Corresponding author: Lucas Ravellys Pyrrho de Alcântara, e-mail: ravellyspyrrho@gmail.com

Departamento de Energia Nuclear. Universidade Federal de Pernambuco (UFPE), Avenida Luiz Freire, n° 1000, CEP: 50740-545, Recife, Pernambuco, Brazil. E-mail: alinelimadeoliveira.alo@gmail.com, ravellyspyrrho@gmail.com, martinsdsn@gmail.com, acdantonino@gmail.com

http://orcid.org/0000-0002-7259-8510

Adriana Thays Araújo Alves

Centro de Estudos Superiores de Bacabal. Departamento de Ciências Exatas e Naturais. Universidade Estadual do Maranhão (UEMA), Rua Teixeira de Freitas, n° 2492-2706, CEP: 65700-000, Bacabal, MA, Brazil. E-mail: adrianathaaraujo@gmail.com

http://orcid.org/0000-0002-1878-9076

Severino Martins dos Santos Neto

Departamento de Energia Nuclear. Universidade Federal de Pernambuco (UFPE), Avenida Luiz Freire, n° 1000, CEP: 50740-545, Recife, Pernambuco, Brazil. E-mail: alinelimadeoliveira.alo@gmail.com, ravellyspyrrho@gmail.com, martinsdsn@gmail.com, acdantonino@gmail.com

http://orcid.org/0000-0003-2940-2129

Artur Paiva Coutinho

Centro Acadêmico do Agreste. Núcleo de Tecnologia. Universidade Federal de Pernambuco (UFPE), Avenida Marielle Franco, s/n, Km 59, CEP: 55014-900, Caruaru, PE, Brazil. E-mail: kevintjmendes@outlook.com, arthur.coutinho@yahoo.com.br, dscgeo@gmail.com

http://orcid.org/0000-0002-0644-0037

Suzana Maria Gico Lima Montenegro

Centro de Tecnologia e Geociências. Universidade Federal de Pernambuco (UFPE), Avenida da Arquitetura, s/n, CEP: 50740-550, Recife, PE, Brazil. E-mail: suzanam.ufpe@gmail.com

http://orcid.org/0000-0002-2520-5761

José Moura Soares

Centro Acadêmico do Agreste. Núcleo de Tecnologia. Universidade Federal de Pernambuco (UFPE), Avenida Marielle Franco, s/n, Km 59, CEP: 55014-900, Caruaru, PE, Brazil. E-mail: kevintjmendes@outlook.com, arthur.coutinho@yahoo.com.br, dscgeo@gmail.com

http://orcid.org/0000-0002-6118-0709

Antonio Celso Dantas Antonino

Departamento de Energia Nuclear. Universidade Federal de Pernambuco (UFPE), Avenida Luiz Freire, n° 1000, CEP: 50740-545, Recife, Pernambuco, Brazil. E-mail: alinelimadeoliveira.alo@gmail.com, ravellyspyrrho@gmail.com, martinsdsn@gmail.com, acdantonino@gmail.com

http://orcid.org/0000-0002-4120-9404

^*Corresponding author: Lucas Ravellys Pyrrho de Alcântara, e-mail: ravellyspyrrho@gmail.com

Year	Annual maximum precipitation (mm)	Year	Annual maximum precipitation (mm)
1979	47.3	1999	39
1980	47.3	2000	55.4
1981	58.3	2001	82.6
1982	72	2002	56.4
1983	76	2003	27.1
1984	92	2004	62.9
1985	186.8	2005	89.7
1986	59	2006	48.9
1987	50	2007	36.6
1988	80.8	2008	50
1989	48.6	2009	50
1990	46.4	2010	134.2
1991	68.8	2011	75.2
1992	30.4	2012	49.4
1993	61.9	2013	40.7
1994	42.7	2014	21.3
1995	49.8	2015	20.9
1996	26.6	2016	26.2
1997	40.5	2017	150.6
1998	36.1	2018	36.3

Distribution	Probability Density Function		Description of parameters
GUM	$f_{x} (x) = \frac{ⅇ^{- ⅇ^{\frac{- x + α}{β}} + \frac{- x + α}{β}}}{β}$	(2)	α - location parameter β - scale parameter
W2P	$f_{x} (x) = \frac{ⅇ^{- {(\frac{x}{β})}^{α}} α {(\frac{x}{β})}^{- 1 + α}}{β}$	(3)	α - shape parameter β - scale parameter
LN2P	$f_{x} (x) = \frac{ⅇ^{- \frac{{(- α + L o g [x])}^{2}}{2 β^{2}}}}{\sqrt{2 π} x β}$	(4)	α - first sample moment β - second sample moment
GAM2P	$f_{x} (x) = \frac{ⅇ^{- \frac{x}{β}} x^{- 1 + α} β^{- α}}{Γ (α)}$	(5)	α - shape parameter β - scale parameter
PAR2P	${f_{x} (x) = x}^{- 1 - β} α^{β} β$	(6)	α - minimum value parameter β - shape parameter
FRE2P	$f_{x} (x) = \frac{ⅇ^{- {(\frac{x}{β})}^{- α}} α {(\frac{x}{β})}^{- 1 - α}}{β}$	(7)	α - shape parameter β - scale parameter
GEV	$f_{x} (x) = \frac{ⅇ^{- {(1 + \frac{(x - α) κ}{β})}^{\frac{- 1}{κ}}} {(1 + \frac{(x - α) κ}{β})}^{- 1 - \frac{1}{κ}}}{β}$	(8)	α - location parameter β - scale parameter κ- shape parameter
W3P	$f_{x} (x) = \frac{ⅇ^{- {(\frac{x - κ}{β})}^{α}} α {(\frac{x - κ}{β})}^{- 1 + α}}{β}$	(9)	α - shape parameter β - scale parameter κ- location parameter
FRE3P	$f_{x} (x) = \frac{ⅇ^{- {(\frac{x - κ}{β})}^{- α}} α {(\frac{x - κ}{β})}^{- 1 - α}}{β}$	(10)	α - shape parameter β - scale parameter κ- location parameter
PAR3P	$f_{x} (x) = \frac{β {(\frac{x + α - κ}{α})}^{- 1 - β}}{α}$	(11)	α - minimum value parameter β - shape parameter κ- location parameter

Distribution	Parameters of random distributions
	MM			ML
	α	β	κ	α	β	κ
GUM	46.01	20.84	-	44.26	26.17	-
W2P	1.91	67.35	-	1.83	66.81	-
LN2P	3.96	0.48	-	3.95	0.53	-
GAM2P	4.21	14.10	-	3.05	19.47	-
PAR2P	20.90	1.09	-	39.78	3.03	-
FRE2P	2.30	41.63	-	3.32	45.68	-
GEV	43.71	18.76	0.21	43.93	22.16	0.11
W3P	41.20	1.19	20.52	1.00	33.56	25.80
FRE3P	4.76	89.21	-45.49	3.02	13.57	18.52
PAR3P	92675.41	2409.95	20.90	171358.42	4473.06	21.19

	Distribution	χ2	Distribution	AD	Distribution	KS	Distribution	R²
MM	FRE2P	0.97	GEV	0.76	GEV	0.97	GEV	0.988
	GUM	0.80	LN2P	0.73	LN2P	0.93	LN2P	0.985
	GEV	0.71	GUM	0.37	GUM	0.88	GUM	0.973
	LN2P	0.56	GAM2P	0.32	W3P	0.75	W3P	0.968
	GAM2P	0.56	W2P	0.17	GAM2P	0.71	GAM2P	0.966
	PAR3P	0.27	PAR3P	0.08	W2P	0.68	FRE2P	0.966
	W3P	0.21	FRE2P	0.03*	FRE2P	0.41	W2P	0.951
	W2P	0.02*	W3P	0.01*	PAR3P	0.27	PAR3P	0.937
	PAR2P	0.00*	PAR2P	0.00*	PAR2P	0.01*	PAR2P	0.861
	FRE3P	0.00*	FRE3P	0.00*	FRE3P	0.00*	FRE3P	0.000
ML	LN2P	0.95	FRE3P	0.93	FRE3P	0.99	LN2P	0.992
	GEV	0.89	GEV	0.93	GEV	0.99	GUM	0.991
	FRE3P	0.89	LN2P	0.85	LN2P	0.95	FRE3P	0.990
	W3P	0.38	GUM	0.70	GUM	0.79	GEV	0.990
	GUM	0.27	FRE2P	0.63	W3P	0.75	GAM2P	0.984
	GAM2P	0.24	W3P	0.56	FRE2P	0.75	W3P	0.977
	FRE2P	0.21	GAM2P	0.49	GAM2P	0.71	FRE2P	0.973
	PAR3P	0.21	W2P	0.18	W2P	0.63	W2P	0.956
	W2P	0.08	PAR3P	0.08	PAR3P	0.24	PAR3P	0.934
	PAR2P	0.00*	PAR2P	0.00*	PAR2P	0.00*	PAR2P	0.739

	Distribution	Parameters IDF					Statistics				Performance
	Distribution	a	b	c	d	R²	RMSE	R	D	C	Performance
MM	GUM	783.43	0.22	11.83	0.77	0.999	2.29	0.999	1.000	0.999	Excellent
	W2P	856.73	0.18	11.83	0.77	0.999	3.28	0.999	0.999	0.998	Excellent
	LN2P	719.09	0.25	11.83	0.77	0.999	2.48	0.999	1.000	0.999	Excellent
	GAM2P	803.41	0.21	11.83	0.77	0.999	2.71	0.999	0.999	0.998	Excellent
	PAR2P	499.04	0.33	11.83	0.77	1.000	8.67	1.000	1.000	1.000	Excellent
	FRE2P	553.14	0.31	11.83	0.77	1.000	2.37	1.000	1.000	1.000	Excellent
	GEV	704.50	0.25	11.83	0.77	0.999	2.10	0.999	1.000	0.999	Excellent
	W3P	725.22	0.25	11.83	0.77	0.999	3.35	0.999	0.999	0.998	Excellent
	FRE3P	344.15	0.23	11.83	0.77	1.000	2.10	1.000	1.000	1.000	Excellent
	PAR3P	746.70	0.27	11.83	0.77	0.999	3.97	0.999	0.999	0.998	Excellent
MS	GUM	738.83	0.19	11.83	0.77	0.999	2.97	0.999	0.999	0.998	Excellent
	W2P	860.32	0.17	11.83	0.77	0.999	3.41	0.999	1.000	0.999	Excellent
	LN2P	719.23	0.23	11.83	0.77	0.999	2.77	0.999	0.999	0.999	Excellent
	GAM2P	792.07	0.19	11.83	0.77	1.000	3.29	1.000	1.000	1.000	Excellent
	PAR2P	262.16	0.92	11.83	0.77	1.000	1.41	1.000	1.000	1.000	Excellent
	FRE2P	500.78	0.44	11.83	0.77	1.000	1.51	1.000	1.000	0.999	Excellent
	GEV	627.55	0.29	11.83	0.77	0.999	2.53	0.999	0.999	0.998	Excellent
	W3P	769.38	0.22	11.83	0.77	1.000	3.35	1.000	1.000	0.999	Excellent
	FRE3P	627.55	0.29	11.83	0.77	0.999	0.74	0.999	0.999	0.998	Excellent
	PAR3P	746.01	0.27	11.83	0.77	0.999	3.85	0.999	1.000	0.999	Excellent

f_{x} (x) = \frac{ⅇ^{- {(1 + \frac{(x - α) κ}{β})}^{\frac{- 1}{κ}}} {(1 + \frac{(x - α) κ}{β})}^{- 1 - \frac{1}{κ}}}{β}

f_{x} (x) = \frac{ⅇ^{- {(\frac{x - κ}{β})}^{α}} α {(\frac{x - κ}{β})}^{- 1 + α}}{β}

f_{x} (x) = \frac{ⅇ^{- {(\frac{x - κ}{β})}^{- α}} α {(\frac{x - κ}{β})}^{- 1 - α}}{β}

D = 1 - [\frac{\sum {(F o b s - F c a l c)}^{2}}{\sum {(|F c a l c - \bar{F} o b s| - |F o b s - \bar{F} o b s|)}^{2}}]

[1] ^*Corresponding author: Lucas Ravellys Pyrrho de Alcântara, e-mail: ravellyspyrrho@gmail.com

Brasil

Brasil

Probability distribution of heavy rainfall and determination of IDF in the city of Caruaru - PE

Distribuição de probabilidade de chuvas intensas e determinação de IDF no município de Caruaru - PE

Abstract