Relevance of hydrological variables in water-saving efficiency of domestic rainwater tanks: multivariate statistical analysis

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Research papers

Relevance of hydrological variables in water-saving efficiency of

domestic rainwater tanks: Multivariate statistical analysis

Leonardo Rosa Andrade

a

, Adelena Gonçalves Maia

b,⇑

, Paulo Sérgio Lucio

c

a_{Faculdades Integradas de Cacoal (UNESC), Rua dos Esportes, 1.038, INCRA, CEP 78.976-215 Cacoal, RO, Brazil} b

Civil Engineering Department, Universidade Federal do Rio Grande do Norte (UFRN), UFRN/CT/LARHISA, Cx. Postal 1524, Campus Universitário Lagoa Nova, CEP: 59072-970 Natal, RN, Brazil

c

Atmospheric and Climate Sciences Department, UFRN, UFRN/CCET/PPGCC, Cx. Postal 1524, Campus Universitário Lagoa Nova, CEP: 59072-970 Natal, RN, Brazil

a r t i c l e i n f o

Article history:

Received 12 August 2016

Received in revised form 29 November 2016 Accepted 15 December 2016

Available online 21 December 2016 This manuscript was handled by A. Bardossy, Editor-in-Chief Keywords: Rainfall regime Hydrological variables Principal components Canonical correlation

a b s t r a c t

This research investigated the relevance of four hydrological variables in the performance of a domestic rainwater harvesting (DRWH) system. The hydrological variables investigated are average annual rainfall (P), precipitation concentration degree (PCD), antecedent dry weather period (ADWP), and ratio of dry days to rainy days (nD/nR). Principal component analyses are used to group the water-saving efficiency into a select set of variables, and the relevance of the hydrological variables in a water-saving efficiency system was studied using canonical correlation analysis. The P associated with PCD, ADWP, or nD/nR attained a better correlation with water-saving efficiency than single P. We conclude that empirical mod-els that represent a large combinations of roof-surface areas, rainwater-tank sizes, water demands, and rainfall regimes should also consider a variable for precipitation temporal variability, and treat it as an independent variable.

1. Introduction

Population growth in cities has a significant impact on the quantitative and qualitative availability of fresh water resources, requiring new approaches to water management in urban areas (Palla et al., 2011). In some regions of the world, authorities are adopting alternatives to meet the growing demand for fresh water, including the use of rainwater, water reuse, and desalination.

The use of alternative water sources is very important for managing water resources, and the use of rainwater is one measure that has been adopted for water conservation, not only for domes-tic use but also for industrial use (Khastagir and Jayasuriya, 2010; Palla et al., 2012). Rainwater has been used to supplement other water supplies in several parts of the world where the conven-tional water supply system does not satisfactorily meet the needs of the population (Liaw and Tsai, 2004). In Australia, government officials offer incentives and subsidies to promote the installation of rainwater utilization systems (Imteaz et al., 2012; Rahman et al., 2012). In Brazil, public funding supported the installation of more than 580,000 rainwater tanks in rural areas throughout the country.

Currently, the performance of domestic rainwater harvesting (DRWH) systems is evaluated by reservoir water balance using long-term rainfall time series (Ghisi et al., 2006; Ghisi et al., 2007; Imteaz et al., 2012; Rahman et al., 2012; Mehrabadi et al., 2013). These studies provide DRWH performance indices for differ-ent rainwater tank sizes, roof-surface areas, and levels of demand. However, these results are applied only in specific locations that provided rainwater tank outcomes related to specific rainfall time series. Other research has sought an empirical relationship between the DRWH performance indices and some of the following variables: rainwater tank size, roof area, demand, and a hydrolog-ical variable (Eroksuz and Rahman, 2010; Rahman et al., 2012; Hajani and Rahman, 2014b). The empirical models can also use dimensionless index-like independent variables (Fewkes, 1999; Khastagir and Jayasuriya, 2010; Liaw and Chiang, 2014a). However, the equation that represents the relationship between the DRWH performance indices and the independent variables only can be applied in the region that provides the long-term time series data used for analysis.

In empirical models, the main hydrological variable used is the average annual rainfall (P), without incorporating variable repre-senting the temporal variability of precipitation for a specific region. This modeling method cannot therefore be extrapolated across regions with variations in rainfall depending on the time of year; for example, localities may have the same annual rainfall

http://dx.doi.org/10.1016/j.jhydrol.2016.12.027

E-mail addresses:[email protected] (L.R. Andrade), adelenam@gmail. com(A.G. Maia),[email protected](P.S. Lucio).

Contents lists available atScienceDirect

Journal of Hydrology

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but different temporal variability, thereby producing different rainwater tank efficiency.

According toImteaz et al. (2012), many studies have used aver-age annual rainfall data to model a DRWH system; however, in areas of high inter-annual rainfall variability, analysis that consid-ers long-term mean annual rainfall may not be useful.Imteaz et al. (2013)evaluated the results of DRWH reliability in different areas of Melbourne, Australia, and concluded that it is necessary to change the traditional design practice of considering a single annual rainfall value for rainwater-tank sizing, and the results of these studies should vary if applied in places with different rainfall intensities and patterns.

Palla et al. (2012)evaluated DRWH reliability in the different climates of Europe and studied the effect of meteorological param-eters such as antecedent dry weather period (ADWP), depth, and intensity and duration of rainfall on the performance of DRWH sys-tems. They concluded that ADWP was the most significant param-eter correlated with DRWH performance indices.

The present study investigated the relevance of four hydrologi-cal variables in the performance of DRWH systems with the intent to evaluate each hydrological variable in terms of being able to be used in empirical models. The hydrological variables investigated are average annual rainfall (P), precipitation concentration degree (PCD), antecedent dry weather period (ADWP), and ratio of dry days to rainy days (nD/nR). With the exception of P, all other vari-ables represent the temporal variability of precipitation. The results of this research: (1) give support for the inclusion of hydro-logical variables that represent the temporal variability of precipi-tation in empirical models; (2) present a methodology for analyzing the relevance of different variables in water-saving effi-ciency of a DRWH system (methodology not unpublished, but not previously applied in DRWH system analyses); and (3) introduce the variable PCD, which has never been used before in DRWH sys-tem analyses.

2. Material and methods 2.1. The behavior model

The present study was performed in 50 locations in the state of Rio Grande do Norte, Brazil (Fig. 1), all of them with 48-year data series of daily rainfall, covering the same period from 1963 to 2010. All rainfall series data were provided by Agricultural Research Corporation of Rio Grande do Norte State (Empresa de Pesquisa Agropecuária do Rio Grande do Norte, EMPARN).

The system behavior analysis was performed with the water balance simulation model using daily simulations. Yield before spillage (YBS) and yield after spillage (YAS) models were developed byJenkins et al. (1978)and indicate different rules for reservoir operations to carry out simulations. In the YBS model, after-rainfall water has been added, demand is met, and spillage is com-puted in the model. In the YAS model, demand is met after rainfall water has been added to the reservoir and spillage has occurred.

Mitchell (2007)investigated the impact of the computational time step, the computational operation rule (YAS and YBS), the ini-tial volume of the reservoir, and the length of the simulation period on the accuracy of the model. In the results of this study, the YAS model was more accurate than the YBS model, regardless of the computational time step adopted, and YAS model provided more conservative efficiency values. However, the fact that the YAS model provides a conservative estimate of system performance was pointed out by another researcher (Fewkes, 1999) as a critique of the model.Liaw and Tsai (2004)recommended the YBS model, especially when there is a combination of a small reservoir and

large demand, because in these situations the water-saving effi-ciency can be zero, preventing evaluation of the system.

In this study, the simulations were conducted with the YAS model. The YAS model is based on the following equations (Eqs.

(1) and (2)): Yt¼ min Dt Vt1þ It ; ð1Þ Vt¼ min Vt1þ It Yt C Yt : ð2Þ

where Ytis the volume that supplied the demand in the final time interval t; Vt1is the stored volume in the final time t 1; Vtis the stored volume in the final time interval t (current time); Itis the water drained from the roof to the reservoir in the time interval t; Dtis the total demand for water in the time interval t; and C is the rainwater tank capacity.

The reservoir behavior was analyzed for water-saving effi-ciency, according toFewkes (1999), in Eq.(3):

E¼ PT t¼1Yt PT t¼1Dt 100: ð3Þ

where E is the system’s water-saving efficiency to meet the demand (%); Ytis the volume that supplied the demand in any time interval t; Dtis the total water demand in the time interval t (daily demand); and T is the total time (in days) of the series. The water-saving effi-ciency is interpreted as a measure of the system’s quantitative per-formance over the long-term simulation period (Palla et al., 2012). To simulate the performance of a DRWH system, a daily water balance simulation model was built in Visual Basic language for Excel (Microsoft; Redmond, Washington, USA). For each locality, a 48-year data series of daily rainfall was used. This sort of approach with long-term rainfall data produces average water-saving efficiency. For the current research, this approach is suffi-cient, as we are investigating the relevance of a set of hydrological variables in water-saving efficiency; the use of average outcomes is adequate. It should be noted, however, that if the purpose of the research is to indicate the degree of water-saving efficiency for the system-user, it should be clarified that the results presented are the average efficiency, and because of inter-annual rainfall variabilities, is not certain to save the same amount of water every year. Another alternative is to present the efficiency of rainwater tanks under different climate conditions (i.e. dry, average, wet years), as was done inImteaz et al. (2012), Imteaz et al. (2013)

andHajani and Rahman (2014a). 2.2. Scenarios

Simulations of DRWH systems were created for combinations of hydrological conditions and system characteristics as follows: 50 rainfall regimes (with annual rainfall ranging from 477 to 1699 mm), four rainwater demand amounts (50, 100, 150, and 200 L day1), five rainwater-tank sizes (1, 5, 10, 15, and 20 m3), and four roof-surface areas (50, 100, 150, and 200 m2_{). Diverse} combinations of demand, rainwater-tank size, and roof-surface area generated 80 simulations for each of the 50 locations.

These scenarios were evaluated by a dimensionless index, pi (Eq.(4)) defined using the variables: annual rainwater demand (D), rainwater-tank capacity (C) and roof-surface area (A).

p

¼A C

D5=3 ð4Þ

The

p

results of 80 combinations of D, C, and A were divided into four groups of equal size (20 elements), classified according to the schema presented inTable 1.

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Comfort level 1 shows the best water-saving efficiency results, once it represents the combination with higher tank capacities and roof-surface areas with lower annual demand. There is no physical meaning of this dimensionless index, but it can be used to analyze the different combinations of demand, rainwater-tank capacity, and roof-surface area and allow classification of these combinations in the functioning of the facility to meet demand. We consider that the comfort level 1 represents the situations in which demand is met more easily. The comfort levels 2 and 3 are able to represent situations with intermediate difficulty in meeting the demand, and comfort level 4 represents the situations in which demand is hardly met.

2.3. Hydrological variables

The hydrological variables investigated in this research are average annual rainfall (P), precipitation concentration degree (PCD), antecedent dry weather period (ADWP), and ratio of dry days to rainy days (nD/nR). Of all of these variables, P is the only one that does not represent the temporal variability of precipita-tion. For this reason, the variations in rainfall level in these loca-tions were characterized using the PCD, ADWP, and nD/nR.

The PCD is defined as the degree to which the total annual rain-fall is distributed over 12 months.Li et al. (2011)studied the spa-tial and temporal precipitation variability in Xinjiang, China; the PCD indicated that the rainfall in Northern Xinjiang was more dis-persed within a year than that in Southern Xinjiang.Zhang et al. (2007) applied this index and analyzed the spatial and temporal variability features across different regions of North China in the rainy season. The results showed that the precipitation over the eastern region of North China were more concentrated than in the western region and that low pressure from north of the Qinghai-Tibet Plateau to the Mongolian Plateau plays an important role in PCD.

Fig. 1. Location of the municipalities studied.

Table 1

Classification of comfort level by the p values (p0: 0th percentile; p25: 25th

percentile;p50: 50th percentile;p75: 75th percentile;p100: 100th percentile).

Pintervals Comfort level

P75<P6P100 Comfort level 1

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Calculation of the PCD assumes that the monthly precipitations (rj) are vector quantities, with its origin in the Cartesian axis pole and end in polar coordinates (rj,

a

), the

a

value obtained by

Table 2.

The PCD is the relationship between the resultant vector of monthly precipitation vectors and annual precipitation and calcu-lated using Eqs.(5)–(8):

P¼Xrj; ð5Þ Rx¼Xrj sin hj; ð6Þ Ry¼Xrj cos hj; ð7Þ PCD¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rx2_{þ Ry}2 q P : ð8Þ

where j represents the month (j = 1, 2,. . ., 12) in a year; rjrepresents the average monthly rainfall in a month j; and_hjis the azimuth of the month vector j.

The annual PCD interval ranges from 0 to 1, with the maximum value obtained when the total annual rainfall is concentrated in a specific month, and the minimum value when the rainfall is evenly distributed over the months during the year.

The ADWP is an important parameter in rainwater quality and is defined as the average time value (in days) between the end of one rain event and the beginning of another; the ADWP represents the average number of dry days before a rainy day (Davis and McCuen, 2005). The ADWP was used byPalla et al. (2012)to assess the impact of hydrological characteristics on the performance of rainwater utilization systems in regions with different climates.

The nD/nRindex is defined as the average value of dry days for each rainy day in the year, where nD is the number of dry days and nRis the number of rainy days in the year.Campisano and

Modica (2012) used this index to compose a dimensionless methodology for the optimal design of DRWH in Italy.

2.4. Statistical procedures

From each locality we have 80 water-saving efficiency results, from the scenarios considered, and it was necessary to include a dimensional reduction for the purpose of analyzing the influence of hydrological variables on these results. A statistical procedure, principal component analysis (PCA), was then designed to reduce the sample space. In addition, canonical correlation analysis (CCA) was used to find the influence of each hydrological variable analyzed in the water-saving efficiency results, already reduced by PCA. Both procedures (PCA and CA) were performed using R soft-ware (R Development Core Team, 2012), and (specifically for PCA) the package developed byButts (2012).

2.4.1. Principal component analysis

PCA transforms the original variables into a new set of variables that are (1) linear combinations of the original variables, (2) uncor-related with each other, and (3) ordered according to the amount of variation in the original variables that can be accounted for by the new variables (Everitt and Hothorn, 2011). In mathematical terms, PCA involves the following steps: (1) standardization of variables X1, X2,. . ., Xp, for having mean zero and unit variance; (2) calculation of correlation matrix R; (3) determination of

eigen-valuesk1,k2,. . ., kpand the corresponding eigenvectors a1, a2,. . ., ap, through solution of Eq.(9),

jR Ikj ¼ 0 ð9Þ

where I is the identity matrix; (4) elimination of components that contribute little to the variance of the original data set; and (5) use of matrices of eigenvectors as the factors in a linear combina-tion of standardized variables for the composicombina-tion of the principal components (Noori et al., 2010). For further details of this statistical procedure, seeJohnson and Wichern (2007).

2.4.2. Canonical correlation analysis

CCA was used in the present study in order to determine the degree of association between a group of hydrological variables (Pannual, PCD, ADWP andnnDR) with a second group that represents

the standard water-saving efficiency system, which will be repre-sented by the principal components adopted. For this analysis, we determined which of the hydrological variables are more rele-vant for characterizing rainfall regimes in the analysis of a DRWH system.

The objective of CCA is to build two new sets of canonical vari-ables, U =

a

X and V =bY, which are linear combinations of the orig-inal variables X and Y, so that the correlation between U and V is maximized (Noori et al., 2010).

The random vectors X and Y represent the two groups of vari-ables to be analyzed, of dimensions px1 and qx1, where it is assumed that:

VarðXÞ ¼ A ð10Þ

VarðYÞ ¼ B ð11Þ

CovðX; YÞ ¼ C ð12Þ

CovðY; XÞ ¼ C0 _ð13Þ

The construction of the matrix is shown in(14)below:

ð14Þ

From this matrix, a matrix q q can be calculated by B1_C0_A1_C, and the eigenvalue (_{k) can be considered as presented in Eq.}(15):

ðB1_C0_A1_C_{kIÞb ¼ 0:} _ð15Þ

From the eigenvalues (k), we can find the canonical correlation of each pair of canonical variables that is equal to ffiffiffiffiki

p

. The corre-sponding eigenvectors b1, b2, . . ., br provide the variable coeffi-cients Y for the construction of the canonical variable V, being V =bY. The vectors a1, a2,. . ., ar(

a

) are calculated with Eq.(16) for the construction of the canonical variable U =

a

X:

ai¼ A1Cbi: ð16Þ

Table 2

Corresponding relationship betweenaand month.

Month January February March April May June July August September October November December

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The process is performed so that the pairs of canonical variables are sorted in descending order, considering the first major canon-ical correlation to the k-th highest canoncanon-ical correlation.

Canonical loadings reflect the variance that a particular variable shares with the group. The higher the canonical loadings, the more important the variable is to derive the canonical statistical vari-able. Canonical cross-loadings represent the correlations between an original variable of a particular group and the canonical statis-tical variable of the other group.

3. Results and discussion 3.1. Rainfall pattern

In the study area, high spatial variability in the average total rainfall and the temporal variability of the rainfall throughout the year were observed (Fig. 2). The eastern region of Rio Grande do Norte has the highest annual rainfall, and it is better distributed throughout the year. Moving from east to west, the annual rainfall decreases and becomes concentrated; the highest degree of con-centration is shown in the central region of the state. In the west-ern region of the state, there is a recovery in annual rainfall values with better temporal distribution, especially in the southwest region due to localities at higher altitude being subject to the oro-graphic effect. The wide variety of rainfall patterns is due to the dif-ferent weather systems acting in the region, with an emphasis on the intertropical convergence zone and easterly wave disturbances.

The distribution pattern of the ADWP and nD/nRare similar, since these data are used daily to establish the temporal variability of rainfall, rather than to analyze the distribution of this rainfall throughout the year, thus differing from the PCD.

3.2. System water-saving efficiency

The simulations were performed for 80 scenarios (including diverse combinations of demand, rainwater-tank size, and roof-surface area) for 50 municipalities. The water-saving efficiencies are summarized in a boxplot (Fig. 3). In only six municipalities the median values were higher than 80%. Most of the municipali-ties present median values around 50% and 70%. Equador is the only municipality with a median water-saving efficiency below 50%, since Equador has the lowest average annual rainfall (478.4 mm) among all the studied locations associated with high precipitation temporal variability. The differences in the water-saving efficiencies obtained by each municipality happened because of differences between the scenarios of demand, rainwater-tank size, and roof area.

3.3. Principal component analysis

The principal components were generated from the efficiencies of 80 cases with different combinations of roof-surface area, demand, and rainwater-tank size. The results of the first two com-ponents are shown inFig. 4. The first component (PC1) captured

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93.86% of the data variance, the second component (PC2) captured 3.97%, and the third component (PC3) captured 1.10%.

In all the 80 combinations of D, C, and A, the differences between water-saving efficiency results for each location occurred due to the difference of daily rainfall series. These differences accounts for the spatial variability of the PC’s. Analyzing the PC1 results, it is found higher values at the east, lower values in the central area, and increase of PC1 at the west region. The spatial dis-tribution pattern of the PC1 values are very similar to the pattern distribution of the annual rainfall (Fig. 2), which reflects high inter-ference by the annual rainfall values in the water-saving efficiency system and, consequently, in the PC1 values.

The spatial variability of PC2 presents low values in the west area and in the east coast, and high values in a specific area in the east. This spatial distribution pattern is not similar to the other hydrological variables (PCD, ADWP, or nD/nR), although we con-sider that the spatial distribution of PC2 is related with the rainfall temporal variability.

To choose the principal components that will be used for the canonical correlation analysis, a preliminary analysis was carried out with the first two principal components.Fig. 5 the axis ‘‘y” presents the Pearson correlation coefficient between PC1 and the

water-saving efficiency results of each of the 80 scenarios; and the axis ‘‘x” presents the Pearson correlation coefficient between PC2 and the water-saving efficiency results.

In the combination belonging to comfort levels 2, 3, 4 and in part of the combination for level 1, the correlations of water-saving efficiencies are larger with PC1 than with PC2 (clear area,

Fig. 5). Even so, for seven combination of rainwater tank size, water demand, and roof-surface area (comfort level 1), the PC2 presented a larger correlation with water-saving efficiency than PC1 (fea-tured area,Fig. 5.).

This research considered the PC’s that properly represent the results of water-saving efficiency in order to reduce the 80 water-saving efficiency results in some PC’s. Using only PC1 would not generate completely valid conclusions for comfort level 1. To solve this challenge, the first two components were used in canon-ical correlation analysis to define which of the hydrologcanon-ical vari-ables shows better correlation with water-saving efficiency in different situations of comfort levels. We highlight that PC1 is bet-ter able to represent situations with higher and inbet-termediate diffi-culty in meeting the demand (comfort level 2, 3 and 4), and PC2 best represents situations in which demand is met more easily (comfort level 1).

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3.4. Canonical correlation analysis

CCA was performed with the set Y = PC1 and PC2, and two sets of X, in analysis 1, X = (P, ADWP, nD/nR, PCD), and in analysis 2, X = (ADWP, nD/nR, PCD); seeTable 3.

In Analysis 1, the two pairs of U are relevant (significance level <0.001), but in Analysis 2, only U1 was considered, because U2 has

a high significance level (0.036). The canonical cross-loadings pro-vide a correlation between the original variables (P, ADWP, nD/nR, and PCD) and the canonical variable V, created by linear combina-tion of PC1 and PC2. Through cross-correlacombina-tion analysis, in Analysis 1, we can analyze (for the bolded values) that for the first pair of canonical variables, annual rainfall has the highest correlation with V1, followed by PCD, ADWP, and nD/nR, and for the second pair of canonical variables, PCD has the highest correlation with V2. When only the rainfall variability indices were analyzed (Analysis 2), PCD showed a higher correlation with the variable V1 (bolded value), which represents the principal components. Analysis 1 confirmed that annual rainfall is the most important hydrological variable, followed by PCD, ADWP, and nD/nR. Analysis 2 verified that PCD was the best indicator among the indices that represented rainfall temporal variability.

Additional CCAs (Tables 4–6) were performed to verify whether the use of a rainfall temporal variability index associated with annual rainfall would increase its correlation with the system’s water-saving efficiency, in different comfort situations. Higher and intermediate difficulties in meeting demand are represented by PC1; the situation of the facility to meet demand is represented by PC2.

The canonical cross-loadings provide a simple correlation between the original variable and the canonical variable Y, which represents PC1 or PC2. Thus, in all cases the annual rainfall associ-ated with another hydrological variable (PCD, ADWP or nD/nR) gen-erated a correlation increase with the PC, which can be checked by Fig. 4. Spatial distribution of first and second principal components.

Fig. 5. Relationship between the correlations of the water-saving efficiency with PC1 and PC2.

Table 3

Canonical correlation analysis of data (analyses 1 and 2).

Analysis 1 Analysis 2

X = (P, ADWP, nD/nR, PCD) X = (ADWP, nD/nR, PCD)

Canonical variables U1 U2 U1 U2

Canonical Correlation 0.984 0.946 0.974 0.367

Significance level <0.001 <0.001 <0.001 0.036

Canonical loadings Canonical cross-loadings Canonical loadings Canonical cross-loadings Canonical variables U1 U2 V1 V2 U1 U2 V1 V2 P 0.930 0.356 0.916 0.337 – – – – ADWP 0.772 0.175 0.760 0.166 0.760 – 0.741 – nD/nR 0.678 0.107 0.668 0.101 0.645 – 0.629 – PCD 0.780 0.557 0.784 0.527 0.967 – 0.943 –

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comparing the canonical correlation with the canonical cross-loadings (bolded values). Annual rainfall has the highest correla-tion with PC1, but PCD and ADWP also presented high correlacorrela-tions with this component. For PC2, the PCD has the highest correlation, and was more important than the annual rainfall in comfortable situations.

Palla et al. (2012)evaluated the reliability of DRWH systems in different climates of Europe and studied the effect of hydrological parameters including ADWP, depth, and intensity and duration of rainfall on DRWH system performance. They found that the depth, intensity, and duration of rainfall were weakly correlated with sys-tem performance, as shown by the corresponding coefficient deter-mination of the regression analysis between each of these different hydrological parameters and the water-saving efficiency and detention time (time period during which water is stored in the tank).The results indicate that the coefficient of determination (R2_{) is generally greater than 0.80 for ADWP, about 0.50 for event} rainfall duration, about 0.20 for rainfall intensity, and lower than 0.20 for rainfall depth. The simulation that supports these results was conducted by YAS model considering five main climate zones (annual precipitation of less than 300 mm to approximately 1900 mm), roof-surface area of 100 m2_{, demand fraction (annual} water demand/annual inflow) equal to one, storage capacity ranges between 0.4 and 150 m3, and storage fraction (S/Q, storage capac-ity/annual rainfall) varying from 0.01 to 1. However, the regression

analysis was performed with a range of S/Q between 0.01 and 0.1, which corresponds to a storage capacity smaller than 20 m3_._Palla

et al. (2012)concluded that ADWP, that represents the temporal rainfall variability, was the main hydrological parameter that affected the system behavior, corroborating with our results.

4. Conclusions

Our study area covers 52,811 km2_{with annual rainfall} ity ranging from 490 to 1640 mm where there is also high variabil-ity in the rainfall temporal distribution patterns due to different regional weather systems. Therefore, the conclusions reached can be generalized for other areas with significant variability in rainfall.

We concluded that empirical models that attempt to represent a large combination range of roof-surface areas, rainwater-tank sizes, levels of water demand, and rainfall variations should con-sider a variable that represents the precipitation temporal variabil-ity as an independent variable. For the study area, it is found that the most influential hydrological variables are the average annual rainfall and PCD, which should be used in empirical models. In comfortable situations in terms of meeting the demand (low demand and large rainwater-tank and roof-surface areas), the use of PCD, as an independent variable, was more relevant than the Table 4

Canonical correlation analysis of data (Analysis 3).

Analysis 3: X = (P, PCD)

Y = PC1 Y = PC2

Canonical variables ations U1 U1

Canonical correlation 0.974 0.925

Significance level <0.001 <0.001

Canonical loadings Canonical cross-loadings Canonical loadings Canonical cross-loadings

Canonical Variables U1 V1 U1 V1

P 0.953 0.928 0.327 0.303

PCD 0.785 0.764 0.601 0.556

Table 5

Analysis 4: X = (P, ADWP)

Y = PC1 Y = PC2

Canonical variables U1 U1

Canonical U1 V1 U1 V1

P 0.978 0.928 0.509 0.303

ADWP 0.793 0.753 0.327 0.194

Table 6

Analysis 5: X = (P, nD/nR)

Y = PC1 Y = PC2

Canonical variables U1 U1

Canonical U1 V1 U1 V1

P 0.983 0.928 0.647 0.303

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annual rainfall. However, these situations should consider the use of the two variables concurrently.

Rainfall temporal variability is an important factor, and its rel-evance in the analysis of the performance of a rainwater-tank sys-tem should be studied. In certain situations, where there is low variability in rainfall patterns, or even when there is substantial variability in annual rainfall but a small variation in the rainfall temporal distribution, such as in areas where the weather-forming rainfall systems act across an entire area, it may not be necessary to use temporal hydrological variables. However, this possibility should be investigated, and this work presents a methodology applicable for any region (nationally and internation-ally) to make this evaluation.

The development of models for areas with homogeneous rain-fall can be an alternative to reduce the number of hydrological variables, as implemented inLiaw and Chiang (2014a)andLiaw and Chiang (2014b). But even in these cases, prior use of temporal hydrological variables would still be necessary to determine the homogeneous areas. The use of appropriate hydrological variables for the development of DRWH-system analysis models will result in more efficient models that lead to higher system-user satisfac-tion and greater diffusion of this technology.

Acknowledgements

The authors thank the Agricultural Research Corporation of Rio Grande do Norte State (EMPARN) for providing the rainfall time series data and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for providing a Master degree scholar-ship to the first author.

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