Regionalization of precipitation with determination of homogeneous regions via fuzzy c-means

Gomes, Evanice Pinheiro; Blanco, Claudio José Cavalcante; Pessoa, Francisco Carlos Lira

doi:10.1590/2318-0331.231820180079

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Scientific/Technical Article • RBRH 23 • 2018 • https://doi.org/10.1590/2318-0331.231820180079 copy

Regionalization of precipitation with determination of homogeneous regions via fuzzy c-means

Regionalização de precipitação com determinação de regiões homogêneas via agrupamento fuzzy c-means

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Knowledge about precipitation is indispensable for hydrological and climatic studies because precipitation subsidizes projects related to water supply, sanitation, drainage, flood and erosion control, reservoirs, agricultural production, hydroelectric facilities, and waterway transportation and other projects. In this context, methodologies are used to estimate precipitation in unmonitored locations. Thus, the objectives of this work are to i) identify homogeneous regions of precipitation in the Tocantins-Araguaia Hydrographic Region (TAHR) via the fuzzy c-means method, ii) regionalize and estimate the probability of occurrence of monthly and annual average precipitation using probability distribution models, and iii) regionalize and estimate the precipitation height using multiple regression models. Three homogeneous regions of precipitation were identified, and the results of the performance indices from the regional models of probability distribution were satisfactory for estimating average monthly and annual precipitation. The results of the regional multiple regression models showed that the annual mean precipitation was satisfactorily estimated. For the average monthly precipitation, the estimates of multiple regression models were only satisfactory when the months used were distributed in the dry and rainy seasons. Therefore, our results show that the methodology developed can be used to estimate precipitation in unmonitored locations in the TAHR.

Keywords:
PBM index; Probability distribution models; Multiple regression models; Tocantins-Araguaia Hydrographic Region

ID	Station	Code	P (mm)	Alt (m)	Lat	Lon
E1	São José da Serra	01555005	1614	797	-15.8	-55.3
E2	Rio das Mortes	01554005	1690	551	-15.3	-54.2
E3	Alô Brasil	01251000	1658	339	-12.2	-51.7
E4	Santo Antônio Leverger	01250001	1581	205	-12.3	-51
E5	Barreira do Campo	00950001	1417	195	-9.2	-50.2
E6	Fazenda Caiçara	00650001	1730	95	-6.8	-50.5
E7	Faz. Estrela Norte	00350000	1931	22	-3.9	-50.5
E8	Acampamento IBDF	00151001	2012	11	-1.8	-51.4
E9	Maracacuera Florestal	00251000	2620	20	-2.2	-51.2
E10	Cachoeira Tracambeua	00349001	2382	50	-3.5	-49.2
E11	Abaetetuba	00148010	2584	13	-1.7	-48.9
E12	Vigia	00048006	2843	15	-0.9	-48.1
E13	Faz. Maringa	00348001	1933	20	-3.2	-48.1
E14	Tomé-Açu	00248003	2553	45	-2.4	-48.1
E15	Abreulandia	00949000	2117	240	-9.6	-49.2
E16	Almas	01147000	1524	427	-11.6	-47.2
E17	Alto Araguaia	01753000	1681	659	-17.3	-53.2
E18	Ananas	00648001	1562	191	-6.4	-48.1
E19	Araguacema	00849002	2048	203	-8.8	-49.6
E20	Araguatins	00548000	1552	122	-5.6	-48.1
E21	Arapoema	00749000	1867	215	-7.7	-49.1
E22	Aruanã	01451000	1537	200	-14.9	-51.1
E23	Bandeirantes	01350000	1456	276	-13.7	-50.8
E24	Bom Jardim de Goiás	01652000	1650	402	-16.2	-52.2
E25	Britânia	01551000	1417	297	-15.2	-51.2
E26	Cachoeira GO	01650000	1514	766	-16.7	-50.6
E27	Caiaponia	01651000	1632	713	-16.9	-51.8
E28	Campinaçu	01348000	2441	683	-13.8	-48.6
E29	Cavalcante	1347000	1850	821	-13.8	-47.5
E30	Colinas do Sul	01448000	1573	530	-14.2	-48.1
E31	Colinas TO	00848000	1801	229	-8.1	-48.5
E32	Colonha	01248001	1420	264	-12.4	-48.7
E33	Contagem	01547010	1570	1242	-15.7	-47.9
E34	Córrego do ouro	01650001	1544	569	-16.3	-50.6
E35	Dianópolis	01146000	1449	679	-11.6	-46.8
E36	Dois Irmãos Tocantins	00949001	2029	264	-9.3	-49.1
E37	Entroncamento S M	01349003	1635	345	-13.1	-49.2
E38	Estrela do Norte	01349000	1751	467	-13.9	-49.1
E39	Fátima	01048000	1897	352	-10.8	-48.9
E40	Faz Primavera	00748002	1816	257	-7.6	-48.4
E41	Faz São Bernardo	01752002	1674	750	-17.7	-52.8
E42	Faz. Lobeira	01148000	1556	243	-11.5	-48.3
E43	Faz. Santa sé	01547001	1684	573	-15.2	-47.2
E44	Flores GO	01447001	1144	200	-14.5	-47
E45	Goiantins	00747001	1572	185	-7.7	-47.3
E46	Israelândia	01650002	1597	406	-16.3	-50.9
E47	Itaberaí	01649007	1828	726	-16	-49.8
E48	Itacaja	00847001	1845	250	-8.4	-47.8
E49	Itapirapua	01550000	1589	343	-15.8	-50.6
E50	Itapuranga	01549002	1645	646	-15.8	-50.6
E51	Jeroaquara	01550001	1780	400	-15.4	-50.5
E52	Lagoa da Flexa	01450000	1436	200	-14.3	-50.7
E53	Mimoso	01548001	1308	687	-15.1	-48.2
E54	Miracema Tocantins	00948000	1707	210	-9.6	-48.4
E55	Monte Carlos GO	01551001	1543	400	-15.6	-51.4
E56	Mozarlandia	01450001	1654	400	-14.7	-50.6
E57	Muricilandia	00748003	1671	393	-7.2	-48.5
E58	Niquelândia	01448001	1704	568	-14.5	-48
E59	Nova América	01549004	1606	800	-15	-49.9
E60	Novo Acordo	01047001	1598	300	-10	-47.7
E61	Novo Planalto	01349001	1588	286	-13.2	-49.5
E62	Paraíso do TO	01048001	2281	390	-10.2	-48.9
E63	Perez	01551002	1499	299	-15.9	-51.9
E64	Pilar de Goiás	01449000	1948	765	-14.8	-49.6
E65	Pindorama do Tocantins	01147002	1615	444	-11.1	-47.6
E66	Piranhas	01651002	1583	356	-16.4	-51.8
E67	Piraquê	00648002	1761	184	-6.7	-48.5
E68	Ponte Paranã	01347001	1245	363	-13.4	-47.1
E69	Porto Gilandia	01047002	1656	220	-10.8	-47.8
E70	Porto Real	00948001	1599	200	-9.3	-47.9
E71	Porto Uruaçu	01449001	1468	572	-14.6	-49.1
E72	Rio Pintado	01350001	1444	200	-13.5	-50.2
E73	Sama	01348001	1411	375	-13.5	-48.2
E74	Santa fé	01551003	1615	400	-15.8	-51.1
E75	Santa Terezinha GO	01449002	1505	400	-14.4	-49.7
E76	São Ferreira	01651003	1673	361	-16.3	-51.5
E77	São João Aliança	01447002	1499	1009	-14.7	-47.5
E78	Tesouro	01653000	1715	389	-16.1	-53.5
E79	Torixoreu	01652002	1406	307	-16.2	-52.6
E80	Travessão	01550002	1517	450	-15.4	-50.7
E81	Tupiratins	00848003	1740	192	-8.4	-48.1
E82	Xambioá	00648000	1695	148	-6.4	-48.5
E83	Xavantina	01452000	1526	263	-14.7	-52.4
E84	Tucuruí ^* * Target Stations;	01449000	2422	40	-3.8	-49.7
E85	Cametá*	01147002	2590	24	-2.2	-49.5
E86	Belém*	01651002	2943	10	-1.5	-48.5
E87	Trecho Médio*	00648002	1555	232	-14.1	-51.7
E88	Gurupi*	01347001	1497	353	-11.7	-49.1
E89	Formosa* do Araguaia	01047002	1708	247	-11.8	-49.5
E90	Faz. Marajá*	00948001	1498	666	-15.6	-48.6
E91	Pirenópolis*	01449001	1687	740	-15.9	-49
E92	Faz. Babilônia*	01350001	1632	699	-17.4	-53.1

Coefficients
F - Test F of Significance. If it is < 5% the model is useful.
$R^{2} = \frac{S Q R e g}{S Q T} = \frac{{[\hat{β}]}^{T} {[Y]}^{T} [Y] - n . {\hat{Y}}^{2}}{{[Y]}^{T} [Y] - n . \hat{Y} ²}$	Determination coefficient⁽¹⁾
$R_{a}^{2} = 1 - \frac{n - 1}{n - p - 1} (1 - R^{2})$	Adjusted coefficient of determination⁽²⁾
$E = (\frac{p_{o} - p_{e}}{p_{o}}) * 100$	Percentage relative error⁽³⁾
$ε = N^{- 1} {[\sum_{i = 1}^{N} {(\frac{p_{o -} p_{e}}{p_{o}})}^{2}]}^{1 / 2}$	Mean relative root square error⁽⁴⁾
$N A S H = 1 - \frac{\sum {(p_{o} - p_{e})}^{2}}{\sum {(p_{o} - \bar{X})}^{2}}$	Nash coefficient⁽⁵⁾
$R M S E = \sqrt{\frac{1}{n} * \sum {(p_{o} - p_{e})}^{2}}$	Root mean Ssquared error⁽⁶⁾

H. R.	Result of the chi-square
H. R.	Normal	Log-Normal	Gamma	Weibull
I	0.79	0.54	3.57	2.93
II	0.46	0.36	4.39	3.95
III	0.09	0.04	5.18	3.08

Region	Target Stations	X ²
Region	Target Stations	Log-Normal
I	Faz. Marajá	1.70
	Pirenópolis	0.82
	Faz. Babilônia	1.12
II	Trecho Médio	2.95
	Gurupi	0.76
	Formosa do Araguaia	1.80
III	Tucuruí	2.37
	Cametá	0.59
	Belém	2.96

	Normal			Log Normal			Gamma			Weibull
	HR I	HR II	HR III	HR I	HR II	HR IIII	HR I	HR II	HR III	HR I	HR II	HR III
Jan	0.89	0.85	0.083	0.52	0.52	0.22	0.72	0.56	0.17	2.32	2.16	0.005
Feb	3.85	2.17	0.016	3.1	2.43	0.06	3.31	2.36	0.04	6.7*	1.97	1.23
Mar	1.06	1.29	0.05	1.61	1.14	0.23	1.02	1.21	0.14	1.31	1.92	0.09
Apr	10.11 ^* * Inappropriate Values;	3.01	6.5*	6.15*	2.76	8.07*	5.67	2.48	7.7*	5.74	3.86	5.94
May	15.0*	1.68	0.37	4.64	2.54	5.58	1.16	2.14	1.9	1.63	1.3	1.52
June	46.58*	3.41	0.196	2.23	1.84	8.26*	2.14	1.83	1.76	1.44	1.53	1.56
July	49.58*	7.59*	0.38	2.52	7.94*	12.5*	2.02	6.9*	3.22	1.35	6.1*	3.04
Aug	12.15*	0.75	3.9	1.61	1.41	0.42	5.11	0.89	0.43	6.5*	0.58	0.47
Sept	7.75*	2.91	2.53	2.11	2.05	1.43	5.19	2.03	1.62	8.1*	2.57	2.08
Oct	2.88	0.91	0.16	5.18	1.03	0.75	2.52	0.96	0.31	3.86	0.88	0.12
Nov	4.86	0.37	0.12	15.4*	0.36	1.17	4.09	0.37	0.55	4.33	0.65	0.3
Dec	1.09	1.07	0.13	3.19	1.2	1.20	2.41	1.1	0.73	0.13	1.57	0.3

HR	Station	Months
HR	Station	Jan	Feb	Mar	Apr	May	June	July	Aug	Sept	Oct	Nov	Dec
I	Faz. Marajá	0.07	0.2	1.46	0.63	0.52	0.53	2.71	1.64	0.14	1.13	2.71	0.06
	Pirenópolis	3.8	2.84	0.03	1.36	0.96	0.46	2.21	1.04	0.19	0.77	0.81	0.5
	Faz. Babilônia	0.57	0.71	0.83	1.1	2.41	0.54	0.29	3.6	0.95	0.07	0.78	0.65
II	Trecho Médio	2.19	0.26	2.31	3.21	1.74	0.03	2.12	3.52	1.63	0.11	2.8	2.25
	Gurupi	0.28	1.17	2.64	0.16	0.59	0.57	0.26	2.52	0.3	0.31	0.61	1.23
	Formosa do Araguaia	3.19	0.92	0.98	3.28	2.39	0.09	3.12	0.75	3.4	3.36	2.58	2.36
III	Tucuruí	3.69	1.52	2.23	3.61	4.38	3.59	1.35	1.99	0.44	1.08	3.38	2.65
	Cametá	1.13	1.12	0.79	1.07	2.14	3.35	0.19	1.52	0.43	1.21	0.23	2
	Belém	0.21	1.66	1.63	1.67	1.86	3.39	1.97	4.58	2.40	0.29	0.17	0.32

Homogeneous Region I
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.46	0.32	6.09	0.138	0.46	0.00017
Potential	0.42	0.28	5.61	0.138	0.42	0.00017
Exponential	0.45	0.30	5.61	0.139	0.45	0.00017
Logarithm	0.42	0.28	6.08	0.137	0.42	0.00017
Homogeneous Region II
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.41	0.26	3.89	0.58	0.41	0.00075
Potential	0.39	0.29	3.81	0.58	0.40	0.00076
Exponential	0.40	0.28	3.90	0.58	0.40	0.00075
Logarithm	0.39	0.22	3.80	0.58	0.39	0.00076
Homogeneous Region III
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.74	0.61	4.73	0.64	0.74	0.00071
Potential	0.67	0.51	5.75	0.63	0.68	0.00064
Exponential	0.72	0.58	4.51	0.65	0.75	0.00072
Logarithm	0.70	0.55	5.64	0.63	0.70	0.00046

Homogeneous Region I
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.38	0.23	6.00	0.11	0.38	0.0011
Potential	0.32	0.18	6.30	0.11	0.32	0.00104
Exponential	0.29	0.25	6.10	0.11	0.26	0.00109
Logarithm	0.21	0.16	6.20	0.11	0.21	0.00105
Homogeneous Region II
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.37	0.340	4.70	0.44	0.37	0.00341
Potential	0.17	0.027	4.80	0.44	0.17	0.00338
Exponential	0.18	0.037	4.70	0.45	0.18	0.0034
Logarithm	0.17	0.024	4.80	0.44	0.17	0.00339
Homogeneous Region III
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.73	0.44	4.60	0.60	0.996	0.00568
Potential	0.66	0.49	4.61	0.62	0.997	0.006
Exponential	0.64	0.41	4.63	0.60	0.996	0.00562
Logarithm	0.69	0.53	4.64	0.62	0.997	0.00427

Homogeneous Region I
Models	R²	R²_a	E(%)	ε(%) (%)	Nash	RMSE
Linear	0.80	0.78	9.65	0.03	0.8	0.00391
Potential	0.80	0.78	9.22	0.04	0.79	0.00387
Exponential	0.80	0.79	9.57	0.04	0.81	0.00403
Logarithm	0.80	0.78	9.58	0.03	0.80	0.00391
Homogeneous Region II
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.60	0.53	11.09	0.17	0.6	0.00953
Potential	0.62	0.61	10.94	0.17	0.61	0.00948
Exponential	0.60	0.53	10.85	0.18	0.60	0.00965
Logarithm	0.59	0.52	11.21	0.16	0.59	0.00935
Homogeneous Region III
Models	R²	R²_a	E(%)	ε(%)	Nash	RMSE
Linear	0.85	0.77	15.73	0.16	0.85	0.00527
Potential	0.89	0.87	14.08	0.27	0.80	0.00457
Exponential	0.87	0.82	13.78	0.23	0.87	0.00561
Logarithm	0.85	0.77	15.11	0.21	0.85	0.0036

Regression models for the estimation of annual mean precipitation totals
HR	Linear
I	$P = 272.3 + 0.27 * H + 31 * l a + 33.4 * l o$
II	$P = 1475 - 0.15 * H + 21.4 * l a + 0.38 * l o$
III	$P = 100063 - 1.5 * H + 97 * l a + 149 * l o$
Regression models for the estimation of monthly precipitation for the rainy season
HR	Linear
I	$P = - 20.14 + 0.11 * H + 5.13 * l a + 5.6 * l o$
II	$P = 363.56 - 0.014 * H + 1.7 * l a + 1.86 * l o$
III	$P = 1224 + 0.02 * H - 0.6 * l a + 19.12 * l o$
Regression models for the estimation of the average monthly precipitation for the dry season
HR	Potential
I	$P = - 12.4 + H^{0.089} . l a^{- 0.23} + l o^{1.24}$
II	$P = - 8.14 + H^{0.32} . l a^{0.44} + l o^{2.19}$
III	$P = 11.67 + H^{- 0.11} . l a^{- 0.37} + l o^{- 1.59}$

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[1] E-mails: gomesevanice@ufpa.br (EPG), blanco@ufpa.br (CJCB), fclpessoa@ufpa.br (FCLP)