A coupled system based on Differential Evolution for the determination of Rainfall intensity equations

Gomes, Guilherme José Cunha; Vargas Júnior, Eurípedes do Amaral

doi:10.1590/2318-0331.231820170165

Guilherme José Cunha Gomes

Departamento de Estradas de Rodagem do Espírito Santo, Vitória, ES, Brasil

Eurípedes do Amaral Vargas Júnior

Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, Brasil

E-mails: guilhermejcg@yahoo.com.br (GJCG), vargas@puc-rio.br (EAVJ)

Authors contributions Guilherme José Cunha Gomes: Bibliographic research, data collection and treatment, computer code development, analysis of the results and elaboration of the text.

Eurípedes do Amaral Vargas Júnior: Guidance of the study, contribution in the definition and interpretation of the methodology, discussion of the results and revision of the text.

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Figure 1
General overview of the different elements of the system developed in MATLAB. The central column details the different stages of the program, whereas the left and right columns summarize the input and output data of the optimization processes, respectively. The consecutive stages of proposed modeling allow the user to obtain the rainfall intensity equation without dealing with parametrization.

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Figure 2
(a) Adjustment of the Gumbel distribution to the annual maximum discharges (

Q_{m a x}

) published by Tucci (1993) TUCCI, C. E. M. Hidrologia ciência e aplicação . Porto Alegre: ABRH, 1993. 943 p. ; (b) Comparison between the proposed DE method and the DREAM software. The dark and light-gray areas represent the confidence intervals due to the uncertainty of the parameters, and the total (parameters + model).

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Figure 3
Posterior distribution (plot (a) and plot (d)) and parameter correlation (plot (b) and plot (c)) after simulation with the DREAM algorithm.

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Figure 4
Adjustment of the Gumbel distribution to the annual maximum precipitations (

P_{m a x}

). The solid line represents the adjustment performed in this study, and the points in circles denote the result obtained by Penner and Lima (2016) PENNER, G. C.; LIMA, M. P. Comparação entre métodos de determinação da equação de chuvas intensas para a cidade de Ribeirão Preto. Geociências, v. 35, n. 4, p. 542-559, 2016. .

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Figure 5
Convergence test (optional) for visualizing the evolution of the objective function (RMSE) along the

N

generations for the Gumbel model (solid line) and for the IDF model (dotted line).

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Figure 6
Toolbox IDF results: (a) The solid lines represent the IDF curves obtained in this study, whereas the dotted lines are those published by Penner and Lima (2016) PENNER, G. C.; LIMA, M. P. Comparação entre métodos de determinação da equação de chuvas intensas para a cidade de Ribeirão Preto. Geociências, v. 35, n. 4, p. 542-559, 2016. concerning three different return periods; (b) Comparison between the calibration data, obtained via rainfall disaggregation (circles), and the model obtained through the optimization of the rainfall equation; (c) Comparison between the proposed DE method and the DREAM software. The dark and light-gray areas represent the confidence intervals due to the uncertainty of the parameters, and the total (parameters + model).

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Figure 7
Marginal posterior distribution histograms of the parameters

a, b, c

and

d

of the IDF model obtained with the DREAM software.

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Figure 2
(a) Adjustment of the Gumbel distribution to the annual maximum discharges (

Q_{m a x}

) published by Tucci (1993) TUCCI, C. E. M. Hidrologia ciência e aplicação . Porto Alegre: ABRH, 1993. 943 p. ; (b) Comparison between the proposed DE method and the DREAM software. The dark and light-gray areas represent the confidence intervals due to the uncertainty of the parameters, and the total (parameters + model).

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Figure 3
Posterior distribution (plot (a) and plot (d)) and parameter correlation (plot (b) and plot (c)) after simulation with the DREAM algorithm.

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Figure 4
Adjustment of the Gumbel distribution to the annual maximum precipitations (

P_{m a x}

). The solid line represents the adjustment performed in this study, and the points in circles denote the result obtained by Penner and Lima (2016) PENNER, G. C.; LIMA, M. P. Comparação entre métodos de determinação da equação de chuvas intensas para a cidade de Ribeirão Preto. Geociências, v. 35, n. 4, p. 542-559, 2016. .

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Figure 5
Convergence test (optional) for visualizing the evolution of the objective function (RMSE) along the

N

generations for the Gumbel model (solid line) and for the IDF model (dotted line).

Thumbnail

Figure 6
Toolbox IDF results: (a) The solid lines represent the IDF curves obtained in this study, whereas the dotted lines are those published by Penner and Lima (2016) PENNER, G. C.; LIMA, M. P. Comparação entre métodos de determinação da equação de chuvas intensas para a cidade de Ribeirão Preto. Geociências, v. 35, n. 4, p. 542-559, 2016. concerning three different return periods; (b) Comparison between the calibration data, obtained via rainfall disaggregation (circles), and the model obtained through the optimization of the rainfall equation; (c) Comparison between the proposed DE method and the DREAM software. The dark and light-gray areas represent the confidence intervals due to the uncertainty of the parameters, and the total (parameters + model).

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Figure 7
Marginal posterior distribution histograms of the parameters

a, b, c

and

d

of the IDF model obtained with the DREAM software.

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Figure 8
Adjustment of the Gumbel distribution to the annual maximum precipitations (

P_{m a x}

) in Marataízes/ES.

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Figure 9
Comparison between the calibration data, obtained via rainfall disaggregation (points in circles), and the model obtained through the optimization of the rainfall equation (IDF curve) for Marataízes/ES.

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Table 1
Annual maximum discharges in the river Mãe Luzia ( Tucci, 1993 TUCCI, C. E. M. Hidrologia ciência e aplicação . Porto Alegre: ABRH, 1993. 943 p. ).

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Table 2
Values of the adjustment parameters of the Differential Evolution algorithms employed for the optimization of the Gumbel model.

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Table 3
Coefficient of determination (

R^{2}

) and values of

α

and

β

( ) for the simulations presented in .

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Table 4
Constants of the model of disaggregation of daily precipitations into hourly ones (DNOS).

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Table 5
Values of the adjustment parameters of the Differential Evolution algorithm employed for the optimization of the IDF curves.

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Table 6
Intervals of prior uncertainty of the adjustment parameters for the IDF curves’ model.

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Table 7
Values of

a, b, c

and

d

( ) regarding the simulations performed in this study with Differential Evolution (DE), with Markov chain Monte Carlo simulations (DREAM), as well as regarding the data presented by Penner and Lima (2016) PENNER, G. C.; LIMA, M. P. Comparação entre métodos de determinação da equação de chuvas intensas para a cidade de Ribeirão Preto. Geociências, v. 35, n. 4, p. 542-559, 2016. . The obtained coefficients of determination (

R^{2}

) are listed in the last line.

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Table 8
Annual maximum daily precipitations during the period from 1947 to 2016, in Marataízes/ES.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Year	$Q$	Year	$Q$	Year	$Q$
Year	(m³/s)	Year	(m³/s)	Year	(m³/s)
1943	92.5	1955	261.0	1967	376.0
1944	228.6	1956	248.0	1976	252.0
1945	72.5	1957	256.5	1977	280.0
1946	180.0	1958	380.0	1978	470.0
1947	180.0	1959	221.4	1979	773.0
1948	350.0	1960	370.0	1980	880.0
1949	116.8	1961	360.0	1981	545.0
1950	216.0	1962	340.0	1982	117.0
1951	330.0	1963	480.0	1983	265.0
1952	241.2	1964	290.0	1984	121.0
1953	300.0	1965	216.0	1985	315.0
1954	380.0	1966	390.0

Parameter	Value
Number of generations	200
Population ( $N$ )	40
Offspring diversity ( $F_{DE}$ )	1
Crossover rate	0.4
Minimum parameter value	0.1
Maximum parameter value	${\bar{P}}_{max}$ ou ${\bar{Q}}_{max}$

Simulation	$R^{2}$	$RMSE$ (m³/s)	$α$	$β$
DE	0.934	43.52	143.1	233.9
DREAM	0.934	43.52	143.1	233.8
Tucci	0.920	45.49	132.3	234.7

Ratio	Constant	Disaggregation
24 h/1 d	1.14	$\times$ (24 h, $T_{r}$ )
12 h/24 h	0.85	$\times$ (12 h, $T_{r}$ )
12 h/ 24 h	0.82	$\times$ (10 h, $T_{r}$ )
8 h/24 h	0.78	$\times$ (8 h, $T_{r}$ )
6 h/24 h	0.72	$\times$ (6 h, $T_{r}$ )
1 h/24 h	0.42	$\times$ (1 h, $T_{r}$ )
30 min/ 1 h	0.74	$\times$ (30 min, $T_{r}$ )
25 min/ 30 min	0.91	$\times$ (25 min, $T_{r}$ )
20 min/ 30 min	0.81	$\times$ (20 min, $T_{r}$ )
15 min/ 30 min	0.70	$\times$ (15 min, $T_{r}$ )
10 min/ 30 min	0.54	$\times$ (10 min, $T_{r}$ )
5 min/ 30 min	0.34	$\times$ (5 min, $T_{r}$ )

Parameter	Value
Number of generations	250
Population ( $N$ )	200
Offspring diversity ( $F_{DE}$ )	1
Crossover rate	0.4

Range	$a$	$b$	$c$	$d$
Minimum	500	0.01	1	0.1
Maximum	5000	1	100	2

Parameter	DE	DREAM	Penner and Lima (2016) PENNER, G. C.; LIMA, M. P. Comparação entre métodos de determinação da equação de chuvas intensas para a cidade de Ribeirão Preto. Geociências, v. 35, n. 4, p. 542-559, 2016.
$a$	848.98	723.42	953.20
$b$	0.18	0.19	0.18
$c$	10.22	9.97	12
$d$	0.76	0.73	0.76
$R^{2}$	0.996	0.997	0.993

Year	$P_{max}$	Year	$P_{max}$	Year	$P_{max}$
Year	(mm)	Year	(mm)	Year	(mm)
1947	51.0	1967	55.0	1988	46.4
1948	60.4	1968	125.0	1989	60.0
1949	54.6	1969	42.0	1990	32.7
1950	68.8	1970	44.6	1991	38.2
1951	69.6	1971	80.6	1992	37.5
1952	86.8	1972	81.2	1993	50.8
1953	54.6	1973	122.0	1994	88.9
1954	58.8	1974	41.0	1995	67.7
1955	45.4	1975	64.0	1996	74.5
1956	56.6	1976	75.0	1997	88.4
1957	54.8	1977	96.2	1998	51.4
1958	100.6	1978	135.2	1999	66.8
1959	72.0	1979	147.6	2000	135.9
1960	120.4	1980	95.7	2004	70.2
1961	62.6	1981	98.3	2005	86.7
1962	56.0	1982	114.2	2006	70.8
1963	62.4	1983	127.3	2007	61.3
1964	84.0	1984	83.5	2015	38.8
1965	63.0	1985	67.0	2016	56.0
1966	95.8	1986	57.0

[1] E-mails: guilhermejcg@yahoo.com.br (GJCG), vargas@puc-rio.br (EAVJ)

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A coupled system based on Differential Evolution for the determination of Rainfall intensity equations

Um sistema acoplado baseado em Evolução Diferencial para determinação de equações de chuvas intensas

ABSTRACT