Multivariate statistical analysis to support the minimum streamflow regionalization

Estatísticas multivariadas como suporte à regionalização de vazões mínimas¹

ABSTRACT

This study aimed to develop a methodology based on multivariate statistical analysis of principal components and cluster analysis, in order to identify the most representative variables in studies of minimum streamflow regionalization, and to optimize the identification of the hydrologically homogeneous regions for the Doce river basin. Ten variables were used, referring to the river basin climatic and morphometric characteristics. These variables were individualized for each of the 61 gauging stations. Three dependent variables that are indicative of minimum streamflow (Q_7,10, Q₉₀ and Q₉₅). And seven independent variables that concern to climatic and morphometric characteristics of the basin (total annual rainfall – P_a; total semiannual rainfall of the dry and of the rainy season – P_ss and P_sc; watershed drainage area – A_d; length of the main river – L_p; total length of the rivers – L_t; and average watershed slope – S_L). The results of the principal component analysis pointed out that the variable S_L was the least representative for the study, and so it was discarded. The most representative independent variables were A_d and P_sc. The best divisions of hydrologically homogeneous regions for the three studied flow characteristics were obtained using the Mahalanobis similarity matrix and the complete linkage clustering method. The cluster analysis enabled the identification of four hydrologically homogeneous regions in the Doce river basin.

principal component analysis; cluster analysis; homogeneous regions

RESUMO

Este trabalho teve por objetivo propor uma metodologia baseada em análises estatísticas multivariadas de componentes principais e de agrupamento, com o intuito de identificar as variáveis mais representativas em estudos de regionalização de vazões mínimas e otimizar a obtenção das regiões hidrologicamente homogêneas para a bacia hidrográfica do Rio Doce. Foram utilizadas 10 variáveis referentes às características climáticas e morfométricas da bacia. Essas variáveis foram individualizadas para as 61 estações fluviométricas adotadas, sendo três variáveis dependentes e indicativas das vazões mínimas (Q_7,10, Q₉₀ e Q₉₅) e sete independentes (precipitação total anual – P_a; precipitação total do semestre seco – P_ss e chuvoso –P_sc; área de drenagem da bacia –A_d; comprimento do rio principal –L_p; comprimento total dos cursos d'água da bacia – L_t; e declividade média da bacia –S_L). O resultado da análise de componentes principais apontou a variável independente S_L como a menos representativa e foi excluída do estudo. As variáveis independentes mais representativas foram A_d e P_sc. As melhores divisões de regiões hidrologicamente homogêneas para as três vazões características estudadas foram obtidas utilizando-se conjuntamente da matriz de similaridade de Mahalanobis e do método de agrupamento do vizinho mais distante. A análise de agrupamento possibilitou a identificação de quatro regiões hidrologicamente homogêneas na bacia hidrográfica do Rio Doce.

análise de componentes principais; análise de agrupamento; regiões homogêneas

INTRODUCTION

The hydrological performance of a river basin depends on its geomorphological characteristics, climate aspects and the type of plant coverage. Thereby, several physical and biotic variables of a basin play an important role in the hydrological cycle processes.

River basins with large drainage areas can present different hydrological performances in distinct parts. For that reason, the identification of hydrological homogeneity of a determined region is one of the first goals to be reached in order to have a correct water resources management.

Hydrological generalization is defined as any process of transferring information from one region of identified hydrological performance to other places, usually without observations. This transfer can be directly reported to data series, or to certain relevant statistical parameters such as average, variance, maximum and minimum events, or even equations and parameters related to these statistical parameters.

MISHRA & COULIBALY (2009) describe the importance of having reliable information in a watershed scope due to its innumerous practical uses: hydrology, agronomy, climatology, hydrogeology, management and planning of water resources, decision processes for implementation of public policies and industrial plants installations.

In this context, multivariate statistical analyses can expressively assist the conduction of hydrological generalization studies, reducing the processing time of database and increasing the reliability of obtained results. In an international level, this affirmation can be ascertained through the development of several studies aiming hydrological generalization (ASSANI et al., 2011; MWALE et al., 2011;SAMUEL et al., 2011; ENGELAND & HISDAL, 2009; CASTIGLIONI et al., 2009).

The principal component analysis aims to review the correlations between studied variables, to summarize a large set of variables in a smaller one and in an equivalent purpose, to evaluate the importance of each variable and to promote the elimination of the ones that contribute little, in terms of variation, in the group of evaluated individuals (WILKS, 2006).

Over the past years, this technique has been used in a variety of domains, including medicine (SILVA et al., 2010), remote sensing (JESUS & EPIPHANIO, 2010), farming (ARRUDA et al., 2012; ARRUDA et al., 2011; YAMAKI et al., 2009), chemistry (BELLOMARINO et al., 2010; FARO JR. et al., 2010), analysis of soil (ISLABÃO et al., 2013), environmental studies (GUEDES et al., 2012; REID & SPENCER, 2009) among others.

Clustering multivariate statistical analysis is a data exploratory tool that aims to classify homogeneous groups (WILKS, 2006). It has been employed in several areas of knowledge, for instance, genetics (CARVALHO et al., 2009), management (COUTO JR & GALDI, 2012), health (RESENDES et al., 2010) and environmental engineering (HATVANI et al., 2011). In hydrology, cluster analysis is a technique often used to define classes or to group stations into homogeneous climate regions, i.e. regionalization.

In light of the above, this study aimed at developing a methodology based on multivariate statistical analyses of principal component and cluster analysis, targeting the identification of the most representative variable in studies of minimum streamflow regionalization, as well as to optimize the obtainment of hydrological homogeneous regions for the Doce river basin.

MATERIAL AND METHODS

Region of study

The Doce river basin is located in the Southeast region of Brazil, between the parallels 17°45' and 21°15' S and the meridians 39°30' and 43°45' W, with average altitude of 578 m. It presents drainage area of 83.400 km², approximately, of which 86% belong to the state of Minas Gerais and 14% to the state of Espírito Santo. The population in the basin is of approximately 3.1 million habitants, with 70% living in urban areas. The Doce river basin in inserted, in 98% of its area, in the Brazilian biome called Atlantic Rainforest and the rest belongs to the Cerrado biome. The leading developed economic activities are mining, metallurgy, forestry and farming (CBH-Doce, 2010).

Database and Applications

The study was conducted using data from 61 gauging stations belonging to hydro meteorological network of the National Water Agency – NWA. The employed series consisted of daily data of flow corresponding to the base period from 1976 to 2005 (Table 1). It is highlighted that the use of data up to the year 2005 was limited due to the fact that, in the beginning of the study, this was the most recent year with consisted data provided by NWA.

TABLE 1
Gauging stations selected for a minimum streamflow regionalization.

Code	Lat (o)	Long (o)	Code	Lat (o)	Long (o)	Code	Lat (o)	Long (o)
56028000	-20.6906	-43.2994	56640000	-19.9711	-43.4567	56940002	-19.0694	-41.5369
56055000	-20.8475	-43.2419	56659998	-19.7656	-43.0328	56948005	-19.3431	-41.2461
56065000	-20.9117	-43.0972	56696000	-19.5244	-42.6550	56960005	-20.1825	-41.9611
56075000	-20.6703	-43.0881	56719998	-19.3278	-42.3975	56976000	-19.7431	-41.7853
56085000	-20.7236	-42.9231	56750000	-19.0144	-43.4461	56978000	-19.6825	-41.8372
56090000	-20.7139	-42.9992	56765000	-18.9661	-43.2431	56983000	-20.1081	-41.7283
56110005	-20.3889	-42.9039	56775000	-19.2322	-43.0200	56988500	-19.7989	-41.7061
56240000	-20.3769	-43.1919	56787000	-19.3319	-43.0703	56989001	-19.8111	-41.4378
56335001	-20.3625	-43.1439	56800000	-18.8947	-43.0825	56989400	-19.5947	-41.4581
56337000	-20.2856	-43.0989	56825000	-19.1881	-42.4228	56990000	-19.4925	-41.1617
56385000	-20.6836	-42.6725	56845000	-18.8947	-42.7125	56990990	-20.0786	-41.1214
56415000	-20.2261	-42.6500	56846000	-18.9517	-42.3592	56991500	-19.9011	-41.0581
56425000	-19.9947	-42.6742	56850000	-18.8831	-41.9503	56992000	-19.5236	-41.0142
56460000	-20.2772	-42.3256	56860000	-18.3611	-42.6050	56993002	-19.7800	-40.8511
56484998	-20.1036	-42.4400	56870000	-18.2000	-42.4547	56993551	-19.5586	-40.7325
56500000	-20.2989	-42.4781	56880000	-18.2500	-42.0667	56994500	-19.5308	-40.6231
56510000	-20.0975	-42.4592	56891900	-18.5747	-42.9178	56995500	-19.4228	-40.6864
56520000	-19.9989	-42.3486	56900000	-18.2392	-41.7311	56997000	-19.0578	-40.5164
56539000	-19.7769	-42.4764	56915500	-18.4594	-41.8075	56998000	-19.4075	-40.0639
56570000	-19.7086	-42.4456	56920000	-18.9711	-41.6392
56610000	-19.9317	-43.1731	56935000	-19.3736	-42.1050

• The multivariate statistical analyses were conducted with the Statistica® 7.0 software and the multiple regression equations were obtained with the aid of SisCORV 1.0.3 software (SOUSA et al. 2008) developed by the Research Group of Water Resources, linked to the Department of Agricultural Engineering of the Viçosa Federal University.

The vector base of elevation (contour lines and elevated points) and of hydrography of the hydrographic region obtained with the Brazilian Institute of Geography and Statistics– IBGE at a scale of 1:250.000.

The generation of hydrographically conditioned digital elevation methods (HCDEM), the automatic achievement of morphometric variables, of average precipitations and the spatialization of results were conducted with the aid of the ArcGIS® 10.0 software, as a geoprocessing tool of vectors and spatial representation of data.

In the present study, 10 variables were considered, as three were dependent variables to be regionalized (minimum average flow of seven consecutive days and recurrence period of ten years - Q_7,10; and the minimum streamflow associated with permanence in time of 90% -Q₉₀ and 95% -Q₉₅) and seven independent variables (total annual rainfall – P_a, total semiannual rainfall of the dry season–P_ss and rainy season –P_sc, in mm; watershed drainage area –A_d,in km²; length of the main river – L_p,in km; total length of the rivers –L_t, in km; and average watershed slope –S_L, in %).

Principal component analysis

Based on the principal components analysis, the original set of observed independent variables (P_a, P_ss, P_sc, A_d, L_p, L_t and S_L) was transformed into a new set of variables, named principal components, meeting the following criteria (JOLLIFE, 2002):

1. Considering that Y_i is a principal component of data matrix, it will be a linear combination among the seven independent variables regarded;

2. The sum of the coefficients square aij is equal to 1;

3. each principal component has its own coefficients;

4. the components are not correlated, which means that they are independent of one another;

5. among all the components, Y1 presents the greatest variance, Y2 the second greatest and so forth;

6. the sum of variances of each principal component (Yi) is equal to the sum of variances of each variable (Xj).

As R is a symmetric correlation matrix, of dimensions p x p, from which the eigenvalues (λi) and the eigenvectors (ai) are extracted, the solution was achieved by solving the system (eq. 1):

In which,

λ_i – are the root characteristics (or eigenvalues) of R matrix. There are p eigenvalues corresponding to the variances of each one of the p principal components;

I – is the identity matrix of p x p dimension;

a_i – eigenvector or characteristic vector or ap x 1 matrix, containing the p coefficients for each eigenvalue λ_i corresponding to Y_iprincipal component,

Φ – is a zero vector, of p x 1 dimension.

One of the most common problems found in the application of multivariate statistical models is that these depend on the unities and scales in which the variables were measured. The data of variables were standardized through the [eq. (2)], eliminating the dependence of unities and scales in which the variables were presented:

in which,

Z_ij – standardized variable;

σ(X_j) – standard deviation, and

X̅ – average of j-th original variable.

The importance of each principal component was evaluated through the existing correlation with each X_j variable, which is (eq. 3):

The following criteria were used to select the principal components in this study:

• accumulated percentage of original data total variance greater or equal to 75% (JOLLIFE, 2002); and

• eigenvalues greater or equal to the average of eigenvalues (RENCHER, 2002).

Cluster analysis

The cluster process was based on two steps: firstly, the estimation of Mahalanobis similarity between the 61 grouped gauging stations and secondly the adoption of a cluster technique between the single linkage method and complete linkage method, to form the groups.

When Euclidean distance is used for cluster analysis, all variables must be considered not correlated among themselves, although this presumption is usually ignored. In order to avoid this common problem in studies of hydrological generalization, a matrix of similarities was constituted with Mahalanobis distance. In practice, Mahalanobis distance is summarized in the application of the Euclidean distance (eq. 4) to the standardized matrix of data.

In which,

Z_ij and Z_zj – observations of i-th and z-th gauging stations (i = 1, 2, …, n and z = 1, 2, …, n), with reference to j-th variable or absolute frequency in each studied class (j = 1, 2, …, p).

The single linkage method consists, initially, of a distance matrix (dissimilarity) between gauging stations (individuals). The two most similar individuals were identified (by smaller distance between them) and were reunited in the initial group. In sequence, the distance from the first group in relation to the other individuals was calculated.

The distance between a group and an individual was provided through the expression (eq. 5):

Which means, the distance between the group constituted of the individualsa and b and the individualc was provided by the smallest element from the set of distances of the pairs ac and bc.

From the identification of the smallest distances between the constituted group and the neighboring individuals, a new matrix of dissimilarity was developed of smaller dimensions compared to the initial group and the most similar individuals and/or groups were identified. They were then either incorporated into the initial group or arranged into a second group, depending on whether or not the smallest distance of the new matrix of dissimilarity had been visualized between two other individuals.

In the subsequent stages, increasingly smaller dissimilarity matrices were employed, completing the grouping of all individuals in a single group and composing a dendrogram or tree.

The complete linkage clustering method presents a procedure similar to single linkage method, with one important difference: in each stage, the distance was given by the one that enabled the greatest distance between two individuals and/or groups.

The distance between a group and an individual was provided by the expression: (eq. 6):

which means, the distance between the group constituted by the individualsa and b and the individualc was provided by the greatest element of the distance between sets of pairs of ac andbc.

The construction of dissimilarity matrices, of smaller dimensions than the initial, followed the same procedure described in the single linkage method. The only difference was the creation of groups through maximum distances (complete linkage) rather than through minimum distances (dingle linkage).

The definition of the number of homogeneous regions of flow characteristics was carried out using the criterion of inertia between jumps, in which the first visible discontinuity of the graphic is defined as ‘cut-off' (MELO JÚNIOR et al., 2006; RENCHER, 2002; WILKS, 2006).

Multiple regression analysis

The regression models used to create regionalization equations for each hydrologically homogeneous region were linear, potential, exponential, logarithmic and reciprocal.

The models resulting from the application of multiple regression considered in the hydrologically homogeneous regions provided by the cluster analysis, were selected through the following observations:

• representative equation of the studied event;

• lower number of independent variables according to the relative significance provided by the principal components analysis;

• greater values of adjusted determination coefficient;

• lower values of factorial standard-error;

• significant results by the F test; continuity of flows; and

• Convenience of geographic spatialization of the obtained equations.

In order to verify the adjustment of the adopted regression models to the data, an adjusted determination coefficient (r²a> 0.70), a standard error of estimate lower than 0.5 (EP < 0.5) and a significance level of 5% by F test, were used.

RESULTS AND DISCUSSION

Principal components analysis

Based on the seven independent variables used (P_a, P_ss, P_sc, A_d, L_p, L_t e S_L) for each one of the 61 gauging stations adopted, analysis of the principal components was conducted. The total variance existing in the set of analyzed multivariate data was equal to the number of analyzed variables after data samples were standardized with average and variance equal to 0 and 1, respectively.

In Table 2, the correlation matrix between the standardized independent variables is displayed. In order to evaluate the importance of each variable and promote the elimination of the ones that contribute little in terms of variation, the principal components for the studied variables were identified in the group of individuals evaluated in the regionalization analysis of flows (Table 3).

TABLE 2
Correlation matrix R between the independent variables considered.

Variables	Pa	Pss	Psc	Ad	Lp	Lt	SL
Pa	1.00	0.65	0.99	-0.04	-0.02	-0.04	-0.06
Pss		1.00	0.57	-0.08	-0.07	-0.08	-0.18
Psc			1.00	-0.02	0.01	-0.02	-0.06
Ad				1.00	0.94	1.00	-0.10
Lp					1.00	0.94	-0.11
Lt						1.00	-0.10
SL							1.00

• Caption: P_a – average total annual rainfall; P_ss – average total semiannual rainfall of the dry season; P_sc – average total semiannual rainfall of the rainy season; A_d - watershed drainage area; L_p - length of the main river; L_t - total length of the rivers and S_L - average watershed slope.

Table 3
Principal components (CP) of studied variables.

CP	Variance	% Var.	% acum.	Coefficients of standardized variables
	eigenvalue			Z1 (Pa)	Z2 (Pss)	Z3 (Psc)	Z4 (Ad)	Z5 (Lp)	Z6 (Lt)	Z7 (SL)
Y1	2.9628	42.33%	42.33%	0.1216	0.1310	0.1035	-0.5679	-0.5544	-0.5679	0.0700
Y2	2.4911	35.59%	77.92%	0.6025	0.4858	0.5889	0.1074	0.1207	0.1072	-0.1286
Y3	0.9950	14.21%	92.13%	0.1430	-0.1534	0.1628	0.0519	0.0427	0.0497	0.9605
Y4	0.4733	6.76%	98.89%	0.2541	-0.8447	0.4032	-0.0463	-0.0139	-0.0362	-0.2361
Y5	0.0755	1.08%	99.97%	0.0222	-0.0123	0.0068	0.3960	-0.8220	0.4082	-0.0124
Y6	0.0020	0.03%	99.996%	-0.7317	0.0976	0.6726	0.0369	-0.0144	-0.0330	0.0108
Y7	0.0003	0.004%	100.000%	0.0379	-0.0108	-0.0307	0.7092	-0.0066	-0.7032	-0.0039

According to HELENA et al. (2000), correlation coefficients superior to 0.5 express a strong relationship between evaluated variables. Table 2 demonstrates that climate variables P_a and P_sc are strongly correlated to one another and variable P_ss is moderately correlated to variables P_a and P_sc. Morphometric variables A_d, L_p and L_t are highly correlated to one another, however the morphometric variable S_L presents weak correlation to the remaining analyzed variables (R<0.5), which indicates that it should possibly be excluded from this study.

Based on data presented on Table 3, only the two first components (Y₁ and Y₂) were considered, as they simultaneously met two adopted criteria of selection (the accumulated variance explaining a value greater or equal to 75% of the total data variation and eigenvalues greater or equal to 1). The other components were not considered, which together explained 22.08% of the total variation. Table 4 presents the correlations, or load factors, between the seven standardized variables and the two first principal components.

TABLE 4
Load factors between the standardized variables (VP) and the principal components (CP) and variance (λi) of each principal component (i = 1, 2).

X	VP	CP
		Y1	Y2
Pa	Z1	0.209226	0.950899
Pss	Z2	0.225520	0.766821
Psc	Z3	0.178086	0.929543
Ad	Z4	-0.977547	0.169484
Lp	Z5	-0.954338	0.190567
Lt	Z6	-0.977535	0.169216
SL	Z7	0.120541	-0.202970
(%) λi		42.33	35.59

It is observed on Table 4 that the standardized variables Z₄, Z₅ and Z₆ present greater correlations with the first principal component (Y₁), while the variables Z₁, Z₂ and Z₃ indicate greater correlations with the second principal component (Y₂). The variable Z₇ can be discarded from the study as it contributes little to the group of evaluated individuals in terms of variation, confirming the result obtained by the analysis of correlation matrix R.

The average watershed slope (S_L) variable presents insignificant representativeness in relation to the performance of studied flow characteristics, as it defines a uniform surface of all drainage areas, which does not physically represent the natural process of river channel runoff. For this reason, the exclusion of the variable S_L from the selected set was expected.

The monitoring of water resources involves a great number of variables and the quantitative reduction of unnecessary information directly leads to savings in time and resources. MISHRA & COULIBALY (2009) demonstrated in their study, the importance of having reliable variables in engineering studies in a watershed. CASTIGLIONI et al. (2009) also used physiographic variables when trying to identify hydrologically homogeneous regions.

Physically, the principal component Y₁ represents the most representative morphometric variables and the principal component Y₂represents the average rainfalls in drainage areas upwind of each gaging station. ASSANI et al. (2011) achieved great results using the technique of principal components analysis in river basins in Canada.

According to WILKS (2006), the obtained results showed that the use of principal components tool for the regionalization of flows, even in a preliminary way, is fundamental to the elimination of little expressive variables, thus increasing the spatial reliability of hydrologically homogeneous regions.

Cluster analysis

After disregarding the variable S_L, from the results achieved in the principal components analysis, the homogeneous regions for the three flows were obtained separately, based on standardized variables that presented greater correlations with the two first principal components (A_d, L_t, L_p, P_a, P_sc e P_ss) from the distance matrix of Mahalanobis.

The closest neighbor method presented irregular clusters for the three studied flow characteristics and was discarded. MELO JÚNIOR et al. (2006) found a similar situation and also disregarded the clusters obtained for the nearest neighbor method.

The complete linkage method presented easy interpretation of results and equal number of clusters for the three evaluated flows. For this method, the cut-off can be identified as the approximate distance of 19% of dissimilarity in a dendrogram, in which four groups are formed with homogeneous characteristics of flow for all the considered flows. In order to illustrate the achieved result, in Figures 1 and 2, the graphics of dissimilarity distance vs clustering steps and dendrogram for the variable Q_7,10 are each presented.

FIGURE 1
Dissimilarities distance vs. cluster steps to Q7,10 from the furthest neighbor method.

FIGURE 2
Dendrogram for Q7,10 showing the clustering steps from the furthest neighbor method.

By analyzing Figure 1, the first discontinuity is observed between the clustering steps 56 and 57. From this result, four hydrologically homogeneous regions were identified for the three analyzed flow characteristics, which followed the same performance through the application of clustering method (Figure 2).

MELO JÚNIOR et al. (2006) obtained good results with the complete linkage clustering method in studies of precipitation.

Homogeneous regions

Through the complete linkage clustering method, four regions with homogeneous characteristics of flow for the Doce river basin were obtained, as described:

• Region I – composed of stations with smaller flows and drainage areas. Spatially comprised of headwater regions and small tributaries. Seventeen (17) gauging stations comprise this region for all studied flows with drainage areas varying from 166 to 970 km².

• Region II – intermediate region between regions I and III, which were composed of 12 gauging stations with drainage areas varying from 757 km² to 1.396 km².

• Region III – intermediate region between regions II and IV. Spatially constituting of the main tributaries of the greater flow rivers of the basin and comprised of 13 gauging stations with drainage areas varying from 1,200 to 3,055 km².

• Region IV – comprised of stations with greater flows and drainage areas. Spatially constituted of the key channel of the Doce River and its main tributaries: Piracicaba, Santo Antônio, Suaçuí and Manhuaçu. Nineteen gauging stations comprise this homogeneous region with drainage areas varying from 2,578 to 81,940 km².

Figure 3 presents the spatial configuration of the four hydrologically homogeneous regions for the flows Q_7,10, Q₉₀ and Q₉₅, that presented recurring hydrological performance. For the delimitation of homogeneous regions, the influence areas of gauging stations that comprise them were extended up to the outflow region in the largest river downstream, in accordance with the process described by MARQUES et al. (2009).

FIGURE 3
Hydrologically homogeneous regions for minimum streamflow obtained for the Doce river basin.

In analyzing Figure 3, it is noticed that the homogeneous region of greatest spatial scope is region I (headwater regions and smaller drainage areas), followed by region IV (channel of main river and main tributaries), region III and region II.

It is highlighted that drainage areas inferior to 160km² and superior to 82,000km² were included in the hydrological regions I and IV, respectively. However, it is important to emphasize that the major parts of the Doce river basin do not allow for adequate monitoring (drainage areas smaller than 160km²), thus requiring adoption of other criteria for result projections in these regions.

RIBEIRO et al. (2005) worked with minimum streamflow of reference (Q_7,10, Q₉₀ and Q₉₅) and obtained seven hydrologically homogeneous regions for the Doce River basin.MARQUES et al. (2009) investigated the same watershed and employed minimum streamflow of reference (Q_7,10, Q₉₀ and Q₉₅) in quarterly periods, obtaining hydrologically homogeneous regions.

It is highlighted that the regions the mentioned authors considered are subdivisions and/ or spatial junctions of hydrologically homogeneous regions found through the application of methodology presented in this study, based on unities of existing water resources management and sub-basins of the Doce River basin.

It is essential to distinguish that the methodology proposed in this study is based on multivariate statistical analysis and the obtained results point to the general hydrological performance of the Doce River basin.

The result obtained by the proposed methodology was complemented by the multiple regression analysis between the dependent variables (minimum streamflow characteristics) and the independent variables (climate and morphometric variables), to obtain regional equations for the four hydrologically homogeneous regions.

Multiple regression analysis

Considering the hydrologically homogeneous regions obtained through the application of the proposed scientific approach, for the investigated flow characteristics, the equations of multiple regression of linear, potential, exponential, logarithmic and reciprocate types were adjusted. Table 5 presents for each homogeneous region, the regression equations that adjusted best to the variables Q_7,10, Q₉₀ and Q₉₅.

TABLE 5
Regression models that adjusted best to the minimum and average flow characteristics and the obtained adjustments.

^3-1_d²_sc(**) Equations valid to the interval of independent variables of the gauging stations that constitute the hydrologically homogeneous region.²

In order to meet the selection criteria of regression equations, it was necessary to exclude three gauging stations for region I (56570000, 56935000, 56993002) and one gauging station for region 4 (56880000).

By analyzing Table 5, it can be observed:

• The regression model that adjusted best to the flow data was the potential. The same performance for the regional equations was achieved by RIBEIRO et al. (2005) andMARQUES et al. (2009) for the Doce River basin;

• The most important independent variable for the study was drainage area (A_d) followed by average semiannual rainfall in rainy season (P_sc);

• The regional equations presented for the four hydrologically homogeneous regions, defined by the methodology proposed in this study, showed determination coefficients higher than 0.70, standard errors of estimate lower than 0.5 and significance levels of 5% by the F test.

The results achieved through multiple regression analysis were considered satisfactory, validating the scientific methodology presented in this study.

From previous knowledge of the region, the use of spatial analysis tools and the experience of an hydrologist, multivariate statistical analyses of both principal components and of clustering can contribute to the subdivision of hydrologically homogeneous regions, thus enabling more consistent decision-making, from a more reliable database (eliminating variables that contribute little to the study) of obtained clusters (verification of statistical performance of flow characteristics from the dendrogram).

CONCLUSIONS

Principal components analysis presented satisfactory results for excluding little representative variables in the identification of hydrologically homogeneous regions.

The first two principal components, Y₁ eandY₂, were responsible for 77.92% of data total variation.

The Mahalanobis similarity matrix and the complete linkage clustering method demonstrated great results in the identification of hydrologically homogeneous regions for all studied flows.

Four hydrologically homogeneous regions were obtained for all studied minimum flow characteristics.

The regionalization equations obtained through multiple regression analysis for the minimum flow characteristics were considered satisfactory, validating the scientific methodology presented in this study.

The methodology proposed for identification of the number of homogeneous regions showed great results, enabling the elimination of subjectivity in the identification of hydrologically homogeneous regions.

ACKLOWDEGMENTS

The authors want to thank the Research Support Foundation of Minas Gerais state (FAPEMIG), the Coordination for the Improvement of Higher Education Personnel (CAPES), the National Council for Scientific and Technological Development (CNPq) and the Federal University of Viçosa (UFV) for the funding this study.

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• 1
  Parte da Tese de Doutorado desenvolvida pelo primeiro autor. Projeto financiado pela FAPEMIG, CAPES e CNPq

Publication Dates

• Publication in this collection
  Sep-Oct 2015

History

• Received
  22 Apr 2013
• Accepted
  7 Feb 2015