Frequency analysis of minimum flows

Granemann, Adelita Ramaiana Bennemann; Mine, Miriam Rita Moro; Kaviski, Eloy

doi:10.1590/2318-0331.0318170080

Adelita Ramaiana Bennemann Granemann

Universidade Federal do Paraná, Curitiba, PR, Brasil

Miriam Rita Moro Mine

Universidade Federal do Paraná, Curitiba, PR, Brasil

Eloy Kaviski

Universidade Federal do Paraná, Curitiba, PR, Brasil

E-mails: adelitarbg@hotmail.com (ARBG), mrmine.dhs@ufpr.br (MRMM), eloy.dhs@ufpr.br (EK)

Adelita Ramaiana Bennemann Granemann: Contributed in the conception of the work, literature review, data acquisition, treatment and analysis, map elaboration, discussion of results, writing, formatting and review of the article.

Miriam Rita Moro Mine: Contributed to the conception of the work, literature review, data analysis, discussion of results, writing, formatting and review of the article.

Eloy Kaviski: Contributed to literature review, data treatment and analysis, discussion of results, writing, formatting and article revision.

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Figure 1
Iguaçu river basin and fluviometric stations of the study.

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Figure 2
(a) PDF and (b) CDF to Porto Amazonas for 7-day minimum moving average.

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Figure 3
(a) PDF and (b) CDF to Rio da Várzea dos Lima for 7-day minimum moving average.

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Figure 4
Gamma distribution normalized in bilogarithmic scale for different values of

V_{X}

(a) with the approximation of the power function (b) for the Porto Amazonas station.

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Figure 5
Gamma distribution normalized in bilogarithmic scale for different values of

V_{X}

(a) with the approximation of the power function (b) for the União da Vitoria station.

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Figure 6
Lognormal distribution normalized in bilogarithmic scale for different values of

V_{X}

(a) with the approximation of the power function (b) for the São Mateus do Sul station.

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Figure 7
Lognormal distribution normalized in bilogarithmic scale for different values of

V_{X}

(a) with the approximation of the power function (b) for the Ponte do Vitorino station.

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Figure 8
Moment diagram (

γ

x

V_{X}

) with the normalized daily mean flows and the theoretical curve of the Weibull and Lognormal distributions.

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Figure 9
Dependence of the k parameter of the Weibull distribution, the series of normalized annual minimum flows, the coefficient of variation of the Gamma and Lognormal distributions.

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Figure 10
Dependence of the parameter b of the Weibull distribution, the series of normalized annual minimum flows, the coefficient of variation of the Gamma and Lognormal distributions, for

v = 10

.

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Figure 11
Dependence of the parameter b of the Weibull distribution, the series of normalized annual minimum flows, the coefficient of variation of the Gamma and Lognormal distributions, for

v = 20

.

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Figure 12
Dependence of the parameter b of the Weibull distribution, the series of normalized annual minimum flows, the coefficient of variation of the Gamma and Lognormal distributions, for

v = 1

.

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Figure 13
Dependence of the μ parameter of the Weibull distribution, with the normalized annual minimum flows, of the coefficient of variation of the Gamma and Lognormal distributions.

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Figure 14
Dependence of the σ parameter of the Weibull distribution, with the normalized annual minimum flows, of the coefficient of variation of the Gamma and Lognormal distributions.

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Table 1
Values of

a_{k}

,

a_{c}

and

p_{C}

for different probability levels α.

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Table 2
Minimum moving average of 7 days flow and return period of 10 years in m³/s.

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Table 3
Minimum annual flows for return period of 10 years in m³/s.

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Table 4
Adhesion tests KS and

χ^{2}

for minimum moving average daily flow of 7 days.

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Table 5
Adhesion tests KS and

χ^{2}

for minimum flow.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

$α$	$a_{k}$	$a_{c}$	$p_{c}$
0.001	0.3195	59.8159	0.9350
0.002	0.3366	36.4476	0.8875
0.005	0.3651	18.5459	0.8184
0.010	0.3929	10.9772	0.7604
0.020	0.4291	6.4447	0.6963
0.050	0.4984	3.1774	0.5995
0.100	0.5812	1.8807	0.5140
0.200	0.7231	1.1498	0.4131
0.500	1.2543	0.6991	0.2382

Station	Empirical	LN2	W2
PA	5.40	5.98	5.07
UV	66.26	66.55	53.85
SC	10.41	11.28	9.94
SB	7.17	7.34	6.40
AV	16.33	14.97	14.54
SM	14.80	15.10	12.50
PV	1.37	1.37	1.33
RN	13.33	13.53	11.66
VL	2.79	2.58	2.21
JS	2.18	2.38	1.83
FM	0.86	0.90	0.71

Station	Empirical	LN2	W2
PA	4.29	5.00	4.07
UV	61.94	62.54	51.39
SC	9.51	10.39	9.28
SB	6.66	6.93	6.02
AV	14.17	13.32	13.52
SM	13.40	14.17	11.57
PV	1.16	0.80	1.11
RN	12.40	12.44	10.60
VL	2.54	2.38	1.99
JS	1.65	1.82	1.45
FM	0.66	0.76	0.63

Station	Lognormal Distribution						Weibull Distribution
	$χ^{2}$			KS			$χ^{2}$			KS
	*D_1calc*	*D_1tab*	Hypothesis H_o	*D_2calc*	*D_2tab*	Hypothesis H_o	*D_1calc.*	*D_1tab*	Hypothesis H_o	*D_2calc.*	*D_2tab*	Hypothesis H_o
PA	2.20	7.81	Accepted	0.41	0.23	Rejected	5.15	7.81	Accepted	0.36	0.23	Rejected
UV	6.01	9.49	Accepted	0.19	0.23	Accepted	13.18	9.49	Rejected	0.21	0.23	Accepted
SC	2.35	7.81	Accepted	0.39	0.23	Rejected	3.29	7.81	Accepted	0.33	0.23	Rejected
SB	3.65	9.49	Accepted	0.29	0.23	Rejected	6.76	9.49	Accepted	0.27	0.23	Rejected
AV	2.12	9.49	Accepted	0.38	0.23	Rejected	3.90	9.49	Accepted	0.31	0.23	Rejected
SM	2.93	7.81	Accepted	0.40	0.23	Rejected	10.19	7.81	Rejected	0.36	0.23	Rejected
PV	5.45	7.81	Accepted	0.46	0.23	Rejected	3.99	7.81	Accepted	0.39	0.23	Rejected
RN	3.68	7.81	Accepted	0.23	0.23	Accepted	10.10	7.81	Rejected	0.17	0.23	Accepted
VL	5.38	9.49	Accepted	0.36	0.23	Rejected	22.14	9.49	Rejected	0.31	0.23	Rejected
JS	2.48	7.81	Accepted	0.34	0.23	Rejected	6.90	7.81	Accepted	0.31	0.23	Rejected
FM	5.15	7.81	Accepted	0.31	0.23	Rejected	7.34	7.81	Accepted	0.29	0.23	Rejected

Station	Lognormal Distribution						Weibull Distribution
	$χ^{2}$			KS			$χ^{2}$			KS
	*D_1calc*	*D_1tab*	Hypothesis H_o	*D_2calc*	*D_2tab*	Hypothesis H_o	*D_1calc.*	*D_1tab*	Hypothesis H_o	*D_2calc.*	*D_2tab*	Hypothesis H_o
PA	0.61	7.81	Accepted	0.40	0.23	Rejected	2.50	7.81	Accepted	0.36	0.23	Rejected
UV	4.24	5.99	Accepted	0.22	0.23	Accepted	7.62	5.99	Rejected	0.23	0.23	Accepted
SC	1.10	7.81	Accepted	0.44	0.23	Rejected	3.16	7.81	Accepted	0.38	0.23	Rejected
SB	3.63	9.49	Accepted	0.29	0.23	Rejected	8.69	9.49	Accepted	0.26	0.23	Rejected
AV	4.15	9.49	Accepted	0.41	0.23	Rejected	3.39	9.49	Accepted	0.34	0.23	Rejected
SM	4.69	7.81	Accepted	0.39	0.23	Rejected	13.41	7.81	Rejected	0.36	0.23	Rejected
PV	8.86	7.81	Rejected	0.50	0.23	Rejected	0.58	7.81	Accepted	0.47	0.23	Rejected
RN	4.98	7.81	Accepted	0.24	0.23	Rejected	17.03	7.81	Rejected	0.18	0.23	Accepted
VL	7.24	9.49	Accepted	0.40	0.23	Rejected	22.88	9.49	Rejected	0.34	0.23	Rejected
JS	3.02	7.81	Accepted	0.30	0.23	Rejected	7.29	7.81	Accepted	0.25	0.23	Rejected
FM	2.98	7.81	Accepted	0.31	0.23	Rejected	4.31	7.81	Accepted	0.27	0.23	Rejected

f_{X} (x) = \frac{1}{x \sqrt{2 π \ln (1 + V_{X}^{2})}} e x p {- \frac{1}{2} \frac{{(\ln (x) + \frac{1}{2} \ln (1 + V_{X}^{2}))}^{2}}{\ln (1 + V_{X}^{2})}}; 0 < x < \infty

[1] E-mails: adelitarbg@hotmail.com (ARBG), mrmine.dhs@ufpr.br (MRMM), eloy.dhs@ufpr.br (EK)

Brasil

Brasil

Frequency analysis of minimum flows

Análise de frequência de vazões mínimas

ABSTRACT