Water quality analysis in rivers with non-parametric probability distributions and fuzzy inference systems: Application to the Cauca River, Colombia

Environment International 52 (2013) 17–28

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Water quality analysis in rivers with non-parametric probability distributions and fuzzy inference systems: Application to the Cauca River, Colombia William Ocampo-Duque a,⁎, Carolina Osorio a, Christian Piamba a, Marta Schuhmacher b, José L. Domingo c a b c

Faculty of Engineering, Pontificia Universidad Javeriana, Cll. 18 #118-250, Cali, Colombia Department of Chemical Engineering, Universitat Rovira i Virgili, Av. Países Catalans 26, 43007 Tarragona, Spain Laboratory of Toxicology and Environmental Health, School of Medicine, IISPV, Universitat Rovira i Virgili, Sant Llorens 21, 43201 Reus, Spain

a r t i c l e

i n f o

Article history: Received 20 April 2012 Accepted 16 November 2012 Available online 23 December 2012 Keywords: Water quality Non-parametric density estimators Uncertainty Fuzzy inference systems Monte Carlo simulation Cauca River (Colombia)

a b s t r a c t The integration of water quality monitoring variables is essential in environmental decision making. Nowadays, advanced techniques to manage subjectivity, imprecision, uncertainty, vagueness, and variability are required in such complex evaluation process. We here propose a probabilistic fuzzy hybrid model to assess river water quality. Fuzzy logic reasoning has been used to compute a water quality integrative index. By applying a Monte Carlo technique, based on non-parametric probability distributions, the randomness of model inputs was estimated. Annual histograms of nine water quality variables were built with monitoring data systematically collected in the Colombian Cauca River, and probability density estimations using the kernel smoothing method were applied to fit data. Several years were assessed, and river sectors upstream and downstream the city of Santiago de Cali, a big city with basic wastewater treatment and high industrial activity, were analyzed. The probabilistic fuzzy water quality index was able to explain the reduction in water quality, as the river receives a larger number of agriculture, domestic, and industrial effluents. The results of the hybrid model were compared to traditional water quality indexes. The main advantage of the proposed method is that it considers flexible boundaries between the linguistic qualifiers used to define the water status, being the belongingness of water quality to the diverse output fuzzy sets or classes provided with percentiles and histograms, which allows classify better the real water condition. The results of this study show that fuzzy inference systems integrated to stochastic non-parametric techniques may be used as complementary tools in water quality indexing methodologies. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Despite the huge numeric datasets collected nowadays, it is well known that the assessment of water quality still relies heavily upon subjective judgments and interpretation. Linguistic computations should be considered together with numerical scoring systems to give appropriate water quality classifications (Ocampo-Duque et al., 2006). There is no doubt that the introduction of intelligent linguistic operations to analyze databases is producing self-interpretable water quality indicators for a better assessment. Moreover, to simplify and improve the understanding and the interpretation of water quality, methodologies for integration, aggregation, and fusion of data must be developed (Sadiq and Tesfamariam, 2007). Data aggregation is not simply a problem of calculations; rather it is a problem of judgment. Therefore, it deals not only with uncertainty or variability related to random phenomena, but also with the subjective uncertainty related to linguistic, subjective, vague and imprecise concepts faced in decision-making processes. Consequently, Fuzzy Logic and

⁎ Corresponding author. Tel.: +57 2 321 8200. E-mail address: [email protected] (W. Ocampo-Duque). 0160-4120/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.envint.2012.11.007

Monte Carlo based methods are highly recommended in water quality management since they are appropriate tools to deal with all diverse types of uncertainties (Chowdhury et al., 2009; Darbra et al., 2008). Fuzzy inference systems (FIS) have recently attracted the attention of environmental scientists as suitable platforms to evaluate multiple criteria related to water quality, and other environmental conditions (Marchini et al., 2009). A common application of FIS has been the integration of water quality variables to design suitable integrative systems, which are successfully compared to traditional indexing techniques. Water quality is a vague term that cannot be easily described using crisp data or limited indicators. Instead, water quality should be considered as a fuzzy term appropriately estimated with linguistic computations (Mahapatra et al., 2011). The amount of linguistic “if-then” rules, as well as the number of indicators considered, seems to be definitive for a robust and reliable evaluation (Lermontov et al., 2009). In a previous study, we developed a structured fuzzy hierarchy to interconnect various partial inference engines intended to define water quality (Ocampo-Duque et al., 2006). Here, the FIS contained an analytical hierarchy process to deal with the relative weight of the variables involved in the evaluation process. Adaptive and cooperative neuro-FIS models have also been implemented to provide water

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quality management solutions (Ocampo-Duque et al., 2007, 2012). An integrated risk assessment methodology, based on the weight of evidence approach, which implemented a FIS in order to hierarchically aggregate a set of biological indicators following the precepts of the Water Framework Directive, was recently described (Gottardo et al., 2011). Also, Bayesian networks and probabilistic neural networks have been recently used to train a water quality index supported in FIS (Nikoo et al., 2011). Probabilistic approaches are commonly applied in environmental analysis and modeling to control uncertainty propagation. Parameter uncertainty is a major aspect of the model-based estimation of the risk of human exposure to pollutants. The Monte Carlo method is extensively applied despite it relies heavily on a statistical representation of available information. The probability distributions of each variable are defined according to the Bayesian theory (Ramaswami et al., 2005). For instance, in human health risk assessment some variables are usually managed as probability density functions (PDF) (Legay et al., 2011; Mari et al., 2009). Probabilistic Monte Carlo computations are powerful tools for water quality modeling (Cardona et al., 2011; Misha 2011). However, their use in water quality indexing systems is scarce. A new probabilistic water quality index intended for use in the production of drinking water is described by Beamonte-Cordoba et al. (2010). In this approach, each water quality variable is considered random with normal distribution. Likewise, classical water quality indexes available worldwide could be computed with Monte Carlo methods, assuming probability distributions, or estimating them from monitoring data to provide a most comprehensive evaluation. Recently, fuzzy-probabilistic methods have emerged to deal with complex problems related to water management (Chen et al., 2010; Zhang K. et al., 2009; Zhang X. et al., 2009). Hybrid methods allow address model parameter uncertainty in situations where available information is not sufficient to identify statistically representative distributions. Therefore, they assign fuzzy numbers when the amount of data is short, or when the information about the confidence intervals of variables and parameters is unknown (Baudrit et al., 2007; Kentel and Aral, 2005). For example, Faybishenko (2010) showed a recent application of combining probability and possibility theory for simulating a soil water balance. Moreover, fuzzy-stochastic hybrid methods are currently used to solve optimization and management issues associated to water pollution (Guo et al., 2010; Rehana and Mujumdar, 2009; Zhang K. et al., 2009; Zhang X. et al., 2009). In order to preserve the origin of uncertainties, some methods partitioning the total variance in risk analysis have been developed (Kumar et al., 2009). Likewise, current methodologies are handling both random uncertainty and epistemic uncertainty, because they can combine the fuzzy set theory and Monte Carlo simulations (Li and Zhang, 2010; Li et al., 2007). The method proposed in the present study is somehow inspired in the formal concept of fuzzy randomness, which was first introduced to structural analysis in civil engineering (Möller and Beer, 2004; Möller et al., 2002). The idea behind such concept is that stochastic as well as non-stochastic uncertainty is treated on the basis of the super-ordinated uncertainty model fuzzy randomness. This new uncertainty model contains the special cases of real valued random variables and fuzzy variables, and permits to take into account both uncertainty characteristics, simultaneously. Hybrid stochastic fuzzy model was also applied for in-flight gas turbine engine diagnostics, where the random fluctuations of performance parameters were modeled with PDF while the complex functional relationships were dealt with Neural Networks with FIS structure, commonly called ANFIS (Ghiocel and Altmann, 2001). In the present study, the objective was to model variables with two layers of analysis for uncertainty estimation, one inner layer of FIS using fuzzy membership functions and rules, and one outer layer using Monte Carlo simulation. Randomness in water quality input variables was dealt with probability theory. Then, decision about the water quality status was made by integration of these variables with the help of a FIS. In that way, we introduce a combined stochastic fuzzy model to assess water quality in rivers. The

purpose of this research was to manage both the random nature of input variables and the linguistic subjectivity present in the water quality indexing process. A case study, with information from a Colombian River, was selected to explain the application of the proposed method and its benefits. The results are here reported. Comparison with common indexes is also discussed. Consequently, the simulation outputs involved both kinds of uncertainty: fuzzy and probabilistic. 2. Methods 2.1. Case study: the Cauca River The Cauca River is one of the most important water resources in Colombia. It has a length of 1350 km, with a basin area of approximately 63300 km 2. It goes across the country from south to north through nine departments and a number of cities and towns without appropriate wastewater treatment plants (WWTP). In fact, there are municipalities without any kind of treatment of their sewage. In the Department of Valle del Cauca there is a notable deterioration of water quality in the river, especially when it receives discharges from the City of Santiago de Cali. In this zone, a number of big river releases from domestic, agricultural, and industrial activities are present. The City of Santiago de Cali, with more than two million inhabitants and several companies located at Yumbo Industrial Park, is the main source of river pollution. After crossing these areas, the organic loads are as high as to diminish dissolved oxygen levels below 1 mg/L, compromising the ecosystems living downstream and producing a clear reduction in its ecological status. Although the environmental concerns about water pollution in the river are commonly expressed by people and expert scientists, little actions to recover the river to its original good ecological status, are undertaken. For the current assessment, a water quality monitoring database including nineteen sampling sites was used. Data were provided by the regional environmental protection agency, called the CVC Corporation (www.cvc.gov.co). Data from ten years, considering four sampling campaigns per year, were used. Fig. 1 shows the sampling points where the data were collected: SP1 (Antes Suarez), SP2 (Antes Ovejas), SP3 (Antes Timba), SP4 (Paso de La Balsa), SP5 (Paso de La Bolsa), SP6 (Hormiguero), SP7 (Antes Navarro), SP8 (Juanchito), SP9 (Paso del Comercio), SP10 (Yumbo — Puerto Isaacs), SP11 (Paso de la Torre), SP12 (Vijes), SP13 (Yotoco), SP14 (Mediacanoa), SP15 (Puente Río frío), SP16 (Puente Guayabal), SP17 (Puente la Victoria), SP18 (Puente Anacaro), SP19 (Puente La Virginia) (CVC Corporation, 2004). 2.2. Water quality analysis and traditional indexes According to the objectives of this study, the Cauca River was divided into three river sections: Section I (SP1 to SP6), Section II (SP7 to SP14), and Section III (SP15 to SP19). Thereby, the division includes a relative less impacted area, an area highly impacted because of the discharges from the city of Santiago de Cali and its industrial parks, and an area where these impacts should be reduced due to natural attenuation. Table 1 displays the main statistics of water quality variables used in this study. These were: dissolved oxygen (DO), fecal coliforms (FC), biochemical oxygen demand (BOD5), temperature (T), phosphates (PO4), nitrates (NO3), turbidity (TUR), total solids (TS), and hydrogen potential (pH). Three years are displayed equally time spaced. Sampling campaigns included monitoring data in field (pH, DO, T), and laboratory measurements of composite samples. The sampling campaigns were carried out during the same day in all sites. The 19 sites are monitored in 4 periods: February–March, May–June, July–August, and October– November, seeking stable hydrological conditions which are complex in tropical regions. Traditional water quality indexes are designed to integrate water quality variables or indicators to provide a class or score about physicochemical and biological water quality status. It is intended that

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19

RISARALDA SP19 SP18

CHOCO QUINDIO

SP17 SP16

SP15

SP14 SP13

CAUCA RIVER SP12 SP11 SP10

COLOMBIA

SP9 SP8

VALLE DEL CAUCA

SP7 SP6

SP5 SP3

SP4

SP2

0

16.000

32.000

Meters 64.000

SP1 Fig. 1. Map of the studied area: the Cauca River in the Valle Department (Colombia).

they are useful in environmental decision making. A commonly referred water quality index was developed by the National Sanitation Foundation of United States (NSF_WQI) (Brown et al., 1970). It was defined for any use of water by simply determining the specifications required by that use. This index included various physical, chemical and biological characteristics. For each variable, the index included a quality-value function that expressed the equivalence between the variable and its quality level. The strongly subjective character of the equivalence functions is a problem with that index (Beamonte-Cordoba et al., 2010). The NSF_WQI is computed with Eq. (1), NSF WQI ¼ ∑i¼1 wi Q i N

variable i. At local level, in the Cauca river basin, the CVC Corporation also uses the ICAUCA index to evaluate the water status (Torres et al., 2010). This index is computed according to Eq. (2), N

wi

ICAICA ¼ ∏i¼1 Ii

ð2Þ

where Ii is a special function defined for the variable i to transform the real value to a normalized quality number. The functions to calculate both indexes may be consulted in (CVC Corporation, 2004). 2.3. Fuzzy inference systems

ð1Þ

where wi is the weight of the variable, usually defined by experts, N is the number of variables, and Qi is the quality value function of the

It has been recently shown that linguistic computations used in fuzzy inference systems (FIS) are superior to algebraic common expressions for water quality indexing evaluation (Lermontov et al.,

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Table 1 Basic statistics of water quality variables involved in the study. Indicator, abbr., units

Fecal coliforms, FC, CFU/100 mL

Biochemical oxygen demand, BOD5, mg/L

Temperature, T, °C

Phosphates, PO4, mg/L

Nitrates, NO3, mg/L

Turbidity, TUR, NTU

Total solids, TS, mg/L

Hydrogen potential, pH, (−)

2002 2006 2010 2002 2006 2010 2002 2006 2010 2002 2006 2010 2002 2006 2010 2002 2006 2010 2002 2006 2010 2002 2006 2010 2002 2006 2010

Section I

Section II

Section III

X


s

N

Min

Max

X


s

N

Min

Max

X


s

N

Min

Max

76.09 73.08 72.16 1.51E + 05 1.12E + 04 1.05E + 05 1.55 1.87 8.51 20.4 21.3 22.4 0.062 0.034 0.069 0.30 0.42 0.84 30.8 110.8 79.1 131.33 181.25 163.29 6.93 6.88 7.15

20.71 14.87 18.65 5.13E + 05 2.52E + 04 2.61E + 05 1.12 0.76 3.31 2.8 1.9 0.8 0.008 0.010 0.016 0.20 0.02 0.89 20.0 117.5 95.1 63.25 108.37 145.42 0.47 0.69 0.30

24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24

17.11 37.99 22.65 0.00E + 00 2.30E + 01 7.30E + 02 0.30 1.09 5.33 15.0 18.0 20.9 0.060 0.021 0.064 0.11 0.40 0.11 3.0 9.0 2.0 68.00 59.00 58.00 5.76 5.30 6.45

83.26 94.42 94.68 2.40E + 06 1.10E + 05 9.30E + 05 5.30 4.02 16.00 24.2 28.8 24.3 0.099 0.050 0.125 1.05 0.44 2.57 75.0 349.0 344.0 310.00 396.00 721.00 7.98 7.62 7.65

27.90 47.95 35.70 5.97E + 07 2.72E + 05 5.35E + 06 5.28 3.96 19.68 23.8 21.1 24.4 0.076 0.099 0.089 0.26 0.57 0.98 67.3 143.1 131.1 172.94 270.09 233.25 6.98 6.82 7.05

25.45 26.16 23.96 1.01E + 08 4.96E + 05 1.62E + 07 2.92 1.51 6.99 2.0 1.2 1.3 0.039 0.047 0.025 0.25 0.10 1.10 53.4 125.4 135.6 51.33 134.88 121.32 0.19 0.32 0.19

32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32

2.76 7.47 7.02 2.40E + 04 7.50E + 03 9.10E + 04 1.30 1.75 9.82 20.0 18.0 22.2 0.060 0.031 0.064 0.04 0.45 0.11 30.0 18.0 17.0 68.00 129.00 116.00 6.58 5.65 6.68

77.34 85.32 75.30 2.40E + 08 2.40E + 06 9.30E + 07 13.80 7.52 36.90 27.0 23.0 27.1 0.216 0.241 0.157 1.53 0.69 3.29 300.0 404.0 670.0 302.00 811.00 621.00 7.27 7.39 7.32

34.64 32.00 37.22 1.82E + 05 1.79E + 04 3.15E + 05 2.79 3.44 20.66 25.2 24.0 25.3 0.065 0.083 0.084 0.43 0.57 1.39 61.2 201.3 107.4 203.55 338.35 256.71 6.90 7.04 7.36

8.84 7.51 15.68 5.30E + 05 2.98E + 04 7.16E + 05 0.83 1.08 24.20 0.9 1.4 1.5 0.015 0.026 0.045 0.20 0.38 1.40 34.5 231.6 75.4 80.53 156.07 139.37 0.31 0.36 0.33

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

20.50 18.20 8.81 2.40E + 03 2.40E + 02 2.30E + 03 1.20 2.05 6.73 22.8 21.5 22.6 0.060 0.053 0.064 0.07 0.40 0.11 29.0 21.0 23.0 136.00 191.00 0.08 6.22 6.40 6.72

55.98 45.90 69.32 2.40E + 06 1.10E + 05 2.40E + 06 4.30 5.77 121.00 26.7 26.3 27.5 0.123 0.142 0.271 0.78 2.01 4.28 185.0 892.0 265.0 406.00 901.00 551.00 7.48 7.54 7.90

Note: X
is the median, s is the standard deviation, N is the number of data, Min is the minimum, Max is the maximum. Abr. is the abbreviation of the water quality variable.

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Dissolved oxygen, DO, % Sat.

Year

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2009; Ocampo-Duque et al., 2006). Water quality assessment is a subjective task that must be carried out with tools able to manage such subjectivity and imprecision. Here, linguistic operations in a FIS frame are proposed to compute water quality by integrating parameters within an inference engine. Thus, a methodology to design a water quality index is proposed. It could be adapted to diverse purposes with different number of inputs. In this sense, a FIS is a mapping process from given water quality inputs to desired water quality index. The FIS involves three important parts: membership functions, fuzzy set operations, and inference rules. The Fuzzy Logic toolbox of MATLAB (R2010) was used to build and compute the FIS. A FIS was parameterized to assess water quality considering nine input indicators (Table 1), using the same indicators that those included in the well-known NSF_WQI and the ICAUCA. The FIS output is a fuzzy water quality (FWQ) index. Table 2 summarizes the parameters of the membership functions. Five fuzzy sets were defined for input variables: “Very low”, “low”, “medium”, “high”, and “extreme”. In turn, the output water quality was defined according to five fuzzy sets (qualifiers): “poor”, “bad”, “regular”, “good”, and “excellent”. Gaussian functions were used at low, medium, high, bad, regular and good fuzzy sets, having the following expressions:
 −ðx−cÞ μ ðx; s; cÞ ¼ exp 2 2s

ð3Þ

where s and c are the parameters shown in Table 2, x is the value of the input, and μ is the belongingness (or membership) of the input to the respective fuzzy set, which is a number between 0 and 1, meaning none and total membership, respectively. The parameter c represents the center of the function in the abscissa where the membership value is 1, and the parameter s defines the width of the function. It is important to point out that in fuzzy logic reasoning an x value may belong to more than one fuzzy set. Z-shape functions were used in very low and poor fuzzy sets, having the following equations to represent them: 9 8 1; x≤a > > > > x−a2 > > > > a þ b > > > > ; a≤x≤ = < 1−2 b−a 2
2 μ ðx; a; bÞ ¼ x−b a þ b > > > ≤x≤b > ; > > 2 > > > > b−a 2 > > ; : 0; x≥b

ð4Þ

where a and b are the parameters displayed in Table 2. These parameters locate the extremes of the sloped portion of the curve. Finally, S-shape

21

functions were used in extreme and excellent fuzzy sets, having the following equations to represent them: 8 >
0; 2 > > > x−d > > ; < 2 e−d
2 μ ðx; a; bÞ ¼ > x−d > > 1−2 ; > > e−d > : 1;

9 x≤d > > > d þ e> > > d≤x≤ = 2 > dþe > ≤x≤e > > > 2 > ; x≥e

ð5Þ

where d, and e are the parameters shown in Table 2. These parameters locate the extremes of the sloped portion of the curve. The design and selection of membership functions from intervals of the input variables is a very subjective task. The main questions arise from the number of fuzzy sets used to divide the ranges of the variables, and the own shape of these sets. A division in five fuzzy sets seems appropriate. However, the number of rules may considerably increase, especially if rules with more than one antecedent are desired. In continuous variables the number of fuzzy sets to represent any range could be selected from three to seven, being five a reasonable number. The shape of the functions selected above was considered because of the low number of parameters required. Notwithstanding, other functions could be also used, perhaps requiring more than two parameters. The Colombian Decree 1594/1984 and Resolution 2115/2007, the Spanish Decree 927/ 1988, the boundaries taken in the Lermontov fuzzy water quality index (Lermontov et al., 2009) and the limits set by our previous study (Ocampo-Duque et al., 2006) were used to define the ranges from very low to extreme that water quality variables could take. Then, the division in five qualifiers was given trying to equally divide the universe of discourse with appropriate fuzzy intersection between sets. The inference engine is where the linguistic computations are executed. It was created considering two kinds of rules: rules with only one antecedent, and rules with two antecedents or water quality variables. Forty five (45) rules were written with one antecedent and one consequent (9 water quality variables per 5 fuzzy sets or options, from very low to extreme). Nine hundred (900) rules were written with two antecedents and one consequent. All the likely combinations without repetitions were considered ((81 − 9)/2 = 36 pair combinations and 25 options). In each rule, the most conservative output was considered, and the importance of the rule was defined according to the importance of the variables involved. Rules with DO, pH, BOD5 and FC received a weight of 1.0. Rules with NO3, PO4 and T, received a weight of 0.75. Finally, rules with TUR and TS received a weight of 0.5. More complex rules with three or more antecedents could be created,

Table 2 Parameters of the fuzzy inference system. Indicator*

Units

Membership function parameters “Very Low” a

“Low” b

Z-shape DO FC BOD5 T PO4 NO3 TUR TS pH FWQ

% Sat. CFU/100 mL mg/L ° C mg/L mg/L NTU mg/L – –

0.0 58.9 0.0 15.1 0.0 0.0 3.0 25.6 5.0 “Poor” a Z-shape 0.0

s

“Medium” c

Gaussian 27.8 272.0 2.2 19.9 0.15 3.8 30.7 230.4 6.5 b 38.9

15.0 143.3 1.2 2.6 0.07 1.6 15.0 80.0 0.5 “Bad” s Gaussian 10.5

s

“High” c

Gaussian 31.3 337.5 1.5 18.9 0.14 3.2 33.5 150.6 6.4 c 35.5

15.0 143.3 1.2 2.6 0.07 1.6 15.0 80.0 0.5 “Regular” s Gaussian 11.6

s

“Extreme” c

Gaussian 58.2 675.0 3.5 23.0 0.25 6.1 70.7 300.0 7.5 c 60.0

15.0 143.3 1.2 2.6 0.07 1.6 15.0 80.0 0.5 “Good” s Gaussian 9.4

d

e

S-shape 84.1 1013.0 5.2 27.2 0.4 9.5 107.4 450.0 8.5 c 81.4

70.0 1078.0 5.0 25.1 0.3 7.2 88.7 395.0 8.0 “Excellent” d S-shape 68.2

110.0 1284.0 6.9 30.0 0.5 12.0 136.8 642.0 9.5 e 100.0

*DO: dissolved oxygen, FC: fecal coliforms, BOD5: biochemical oxygen demand, T: temperature, PO4: phosphates, NO3: nitrates, TUR: turbidity, TS: total solids, FWQ: Fuzzy water quality index. a, b, s, c, d, and e, are the parameters to build the membership functions according to Eqs. (3)–(5).

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Fig. 2. Conceptual integration of non-parametric Monte Carlo modeling with a Fuzzy Inference System.

although the improvements are not significant. Rules and ranges were tested with several environmental experts from the CVC Corporation and Academia. Some examples of rules are shown:

are defined on the universe X, for a given element x belonging to X, the following operations can be carried out: Intersection; AND :

If “fecal coliform” is very low then “water quality” is excellent, If “dissolved oxygen” is high then “water quality” is good, If “phosphate” is medium then “water quality” is regular, If “nitrate” is high then “water quality” is bad, If “BOD5” is very high then “water quality” is poor, If “fecal coliform” is very low and “dissolved oxygen” is very high then “water quality” is excellent. Computations with words within the inference engine followed standard fuzzy set operations. These are: union (OR), intersection (AND) and additive complement (NOT). If two fuzzy sets A and B

μ A∩B ðxÞ ¼ minðμ A ðxÞ; μ B ðxÞÞ

ð6Þ

Union; OR : μ A∪B ðxÞ ¼ maxðμ A ðxÞ; μ B ðxÞÞ

ð7Þ

Additive complement; NOT : μ A ðxÞ ¼ 1−μ A ðxÞ:

ð8Þ

Vector inputs are fuzzified to enter to the inference engine using the membership functions. When there are two antecedents, fuzzy logic operations are applied to give a degree of support for these rules. In rules with one antecedent, their degree of support is the degree of membership. The degree of support for the entire rule is used to shape the output fuzzy set. The consequent of a fuzzy rule assigns an entire fuzzy set to the

Fig. 3. Propagation of uncertainty when a probabilistic variable is introduced to a fuzzy inference system.

W. Ocampo-Duque et al. / Environment International 52 (2013) 17–28

output. This fuzzy set is represented by a membership function, which is chosen to indicate the qualities of the consequent. If the antecedent is only partially true (i.e., μ b 1), the output fuzzy set is truncated at this value. This procedure is called the minimum implication method. Since decisions are based on the testing of all the rules in the system, these must be aggregated to make a final decision. Therefore, output fuzzy sets of each rule are aggregated to a single output fuzzy set that may have a complex geometry. The aggregation procedure used here was the maximum method, which is the union of all truncated output fuzzy sets (Mathworks, 2012). The final step was defuzzification to provide a numerical water quality score. A convenient way to give FIS outputs is also by means of the linguistic fuzzy sets with their respective membership degrees. In the current study, the centroid method was used for defuzzification. It delivers a numeric score to water quality, so

FWQ ¼

∫μ ðzÞ⋅zdz ∫μ ðzÞdz

ð9Þ

where FWQ is a fuzzy water quality index which is a score between 0 a 100, and z is the independent variable of the output fuzzy set in each rule. Fuzzy water quality indexes have recently been proposed (Lermontov et al., 2009; Ocampo-Duque et al., 2006). 2.4. Monte Carlo simulation of FIS When the fuzzy water quality index is stochastically computed with Monte Carlo method, a stochastic fuzzy water quality index is obtained. The stochastic model used in this study is described below. Obviously, the building of a FIS for water quality analysis is extremely subjective. The number of input variables should be considerably higher than nine, since the number of physicochemical, microbiological, and biological variables measured nowadays in rigorous water protection agencies may be greater than hundreds. The creation of appropriate fuzzy rules is an important issue for increasing the preciseness of the simulation. A considerable number of fuzzy rules may make more accurate the decision from the inference engine. However, if the number of input variables increases, the number of rules would also increase exponentially to thousands or millions, which would make extremely more complex the model requiring powerful computation to deliver a single score under stochastic conditions. Moreover, the number and form of the rules, as well as the shape of the ranges of the membership functions, are also subjective complex decisions, which could be designed for specific, regional and/or local requirements. Therefore, we here propose a convenient method to build a FIS for water quality evaluation purposes

23

rather than a standard index for use anywhere. Because of the high random uncertainty in water quality variables, due to experimental measurement, human errors, and propagation of error due to the methods used to measure the water quality variable, we propose treating the FIS inputs as stochastic. The conceptual model is depicted in Fig. 2. The algorithm for Monte Carlo simulation assumes each computation with the FIS as deterministic. A vector with water quality variables is randomly selected according to its probability distribution over the domain. Then the corresponding water quality score to that vector is computed with the FIS. The computation is carried out a consistent number of times to cover the entire range of likely inputs, and to build a well-defined histogram of the water quality scores. Random numbers were generated with the inverse transform method. The quantity of random numbers was set at 10 000 in all cases. Fig. 3 outlines the propagation of uncertainty when a probabilistic variable is introduced to a FIS. A, and C, are fuzzy sets. Arrows point out the information flow. Suppose a measured water quality variable X, continuous, positive, and random, with probability density function, f(X) ~ PDF, as shown in Fig. 3, to be introduced to the fuzzy system. Let X
, Q1, and Q3 be the median, the 25th and 75th percentiles of the data, respectively. When X is introduced to the fuzzy system, the probabilistic or random uncertainty is transformed into fuzzy uncertainty. First, X
is fuzzified to take the membership value μA(X
). μA( X
) is the degree of membership of X
to the set A. Then, μA( X
) is transformed to μC(y) according to the rule: If X is A then y is C:

ð10Þ

Such transformation is computed according to the implication method in fuzzy reasoning. In the case of the Figure, the reasoning leads to the horizontal projection line from left to right, or μA( X
) = μC(y), as shown in the Fig. 3. The shape and size of the output fuzzy set is defined by the μC(y) value where the output set is truncated. The area of the output fuzzy set is shown in dark gray. Observe that ΔU is the uncertainty in the height of the output fuzzy set after fuzzification of the random variable X when the interquartile range (IQR) is computed. The area of the output fuzzy set in every rule is required in the centroid method to provide the final output single water quality score. The centroid computes the center of area under the curve resulting after aggregation of all fuzzy sets within the inference engine. Therefore, the uncertainty in the area of the fuzzy set do affects the water quality score. The Monte Carlo method allows computing the final effect over the propagation of uncertainty when dealing with a random input variable in a FIS to provide a final defuzzified water quality score, which also leaves the system with an empirical probability

Fig. 4. Examples of optimized fitting of non-parametric versus parametric distributions of two input variables. (Data of 2009, Section II).

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Fig. 5. Box-and-Whisker plots for assessed water quality with the stochastic fuzzy water quality index (SFWQI) for different years and the three river sections. Reported values are the medians.

density function. Thus, the shapes of the output fuzzy sets vary with each run as a random input is chosen. Propagation of uncertainty is somewhat expressed in this context as the transformation of probabilistic uncertainty into fuzzy uncertainty through the every membership function and rule evaluation. Such propagation is graphically represented as the uncertainty in the area of the output fuzzy set (ΔU) when the random input takes a number between Q1 and Q3. To compute such uncertainty, deterministic computations of the FWQ index are performed depending on the probability of water quality inputs randomly chosen within the statistic range of the water quality variables. Therefore, two layers of uncertainty may clearly be identified. The fuzzy uncertainty is self-contained in the FWQ number as long as probabilistic uncertainty is observed through the output FWQ histogram.

parameters of the assumed distribution from the data. This is the most common way to apply the PDF in environmental uncertainty analysis, with multiple tools available. The main disadvantage of the parametric approach is the lack of flexibility. Each parametric family of distributions imposes restrictions on the shapes that f(x) may have. For example, the density function of the normal distribution is symmetrical and bell-shaped, and therefore, it is unsuitable for representing skewed densities or bimodal densities, which may appear in real water quality datasets. The idea of the non-parametric approach is to avoid restrictive assumptions about the form of f(x), and to estimate it directly from the water quality monitoring data (Qin et al., 2011). It could be especially useful if data are limited. A well-known non-parametric estimator of the PDF is the histogram, when classes are properly well defined. Likewise, the kernel density estimation method is a widely used method for density estimation. The most attractive feature of non-parametric kernel density estimation is that it directly makes use of sample data without a need of estimating characteristic parameters in a theoretical distribution. In other words, there is no error caused by assumption of a theoretical distribution for data and by mismatch between estimated parameters and actual behaviors of water quality indicators. Let X1, X2,…, Xn denote n water quality variable samples. The real probability density function f(x) of a water quality variable can be estimated by the following density function:
 n 1 X x−X i f^n ðxÞ ¼ K nh i¼1 h

where h is the bandwidth, K is called the kernel function and n is the sample size. Gaussian functions are commonly selected as kernel functions:

2.5. Non-parametric kernel density estimator The use of probability distributions to assess water quality, when the integration of variables is required, may provide a better estimation, since the outputs of the fuzzy water quality index will also have probability density rather than point estimation. Consequently, stochastic fuzzy water quality indexes are estimated. Thus, the final classification will be more realistic. The probability distribution of a continuousvalued random variable X is conventionally described in terms of its probability density function (PDF), from which probabilities associated with X can be determined using the relationship b

P ðabXbbÞ ¼ ∫ f ðxÞdx:

ð11Þ

a

The parametric approach for estimating f(x) is to assume some parametric family of probability distributions, and then to estimate the

ð12Þ

K

!
 x−X i 1 ðx−X i Þ2 ¼ pffiffiffiffiffiffi exp − : h 2h2 2π

ð13Þ

The determination of the bandwidth h is crucial for accurate estimation of water quality variable distributions. There are many ways to estimate an optimal bandwidth (hopt). An approximation, known as the Silverman's rule (Silverman, 1998) has been proposed:
hopt ¼ σ

 4 1=5 3n

 where σ ¼ min s;

ð14Þ IQR 20:6745

 n 2 1 , s2 ¼ n−1 ∑i¼1 ðxi −x
Þ , and IQR is the

interquartile range of the data. Therefore, parametric or non-parametric PDF should be estimated for annual data sets to each input water quality

Table 3 Classification of the water quality according to the membership degree of the fuzzy sets. Year

2002

2006

2008

2009

2010

Section

I II III I II III I II III I II III I II III

Lower Quartile (0.25)

Median

Upper Quartile (0.75)

Bad

Regular

Good

Bad

Regular

Good

Bad

Regular

Good

0.344 0.408 0.444 0.336 0.431 0.459 0.322 0.379 0.395 0.348 0.364 0.419 0.379 0.376 0.534

0.741 0.678 0.642 0.750 0.654 0.628 0.766 0.707 0.691 0.737 0.722 0.667 0.707 0.709 0.553

0.005 0.003 0.002 0.005 0.003 0.002 0.006 0.004 0.035 0.005 0.004 0.003 0.004 0.004 0.001

0.310 0.355 0.406 0.302 0.382 0.418 0.292 0.347 0.358 0.326 0.331 0.377 0.343 0.315 0.413

0.777 0.730 0.679 0.786 0.703 0.667 0.797 0.739 0.727 0.760 0.755 0.708 0.743 0.745 0.672

0.006 0.005 0.003 0.007 0.004 0.003 0.007 0.005 0.005 0.006 0.005 0.004 0.005 0.005 0.006

0.266 0.317 0.360 0.268 0.333 0.369 0.268 0.314 0.323 0.301 0.377 0.339 0.314 0.313 0.354

0.823 0.769 0.725 0.822 0.753 0.716 0.822 0.773 0.764 0.787 0.708 0.747 0.773 0.775 0.732

0.009 0.006 0.004 0.009 0.005 0.004 0.009 0.088 0.006 0.007 0.004 0.005 0.006 0.006 0.005

W. Ocampo-Duque et al. / Environment International 52 (2013) 17–28

25

2010: Water Quality

600

700 600

500

800

500

400

600

Frequency

1000

Frequency

Frequency

Water Quality

Water Quality

1200

400 300

400

300

200 200

200

100

100

0

0 47.25

49.50

51.75

54.00

56.25

58.50

60.75

0

63.00

48

49

50

2010 Section I

51

52

53

54

45.6

46.8

48.0

2010 Section II

49.2

50.4

51.6

52.8

54.0

2010 Section III

2009: Water Quality

Water Quality

Water Quality

700

500 900

600

800

400

700

400 300

Frequency

Frequency

Frequency

500 300

200

600 500 400 300

200

200

100 100

100

0 48.8

49.6

50.4

51.2

52.0

52.8

53.6

0 46.2

54.4

47.3

48.4

2009 Section I

49.5

50.6

51.7

52.8

0 44.8

53.9

46.2

47.6

2009 Section II

49.0

50.4

51.8

53.2

2009 Section III

2008: Water Quality

Water Quality

Water Quality 800

600 400

700 600

400 300

300

Frequency

Frequency

Frequency

500

200

500 400 300

200 200

100 100

100 0

0 49.6

50.4

51.2

52.0

52.8

2008 Section I

53.6

54.4

0 47.7

48.6

49.5

50.4

51.3

52.2

53.1

54.0

2008 Section II

45.6

46.8

48.0

49.2

50.4

51.6

52.8

54.0

2008 Section III

Fig. 6. Non parametric distributions of the stochastic fuzzy water quality index in the Cauca River for some selected years.

variable prior to the FIS calculations. Fig. 4 depicts two examples of distribution fittings carried out to estimate the best probability distributions better representing the variables BOD5 and total solids, for 2009. Non-parametric versus parametric distributions are compared, corresponding the best fit to the kernel method in both cases. The Kolmogorov–Smirnov (K–S) test was used to assess the goodness-of-fit of the entire input PDF variables. It is a nonparametric test for the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution. The most attractive characteristic of K–S test is that it is applicable for any continuous variable distribution, and any sample size. A smaller statistic represents the better goodness-of-fit between assumed theoretical distribution, and actual variable samples. Therefore, the statistics can be used to rank the performances of all the water quality variable distributions including the proposed non-parametric kernel distribution estimation (Qin et al., 2011). The K–S test evaluates if a sample comes from a continuous distribution with specified parameters, against the alternative that it does not come from that distribution. The test rejects the null hypothesis at the 5% significance level (p b 0.05). All statistic calculations were performed with the statistics toolbox of MATLAB (R2010). The best goodness of fit for 84% of data was obtained for non-parametric kernel density estimators. The remaining 16% of data presented K–S statistics for parametric fittings similar to that for non-parametric estimators. That was a good reason to choose the

nonparametric method to build all the probability distributions. The ksdensity (Kernel smoothing density) algorithm was used to compute the probability density estimates of the input variables (Mathworks, 2012). The estimate is based on a normal kernel function, using a window parameter (bandwidth) that is a function of the number of points (Eq. (14)). The density is evaluated at equally spaced points that cover the range of the data. The ksdensity algorithm makes no assumptions about the mechanism producing the data or the form of the underlying distribution. Therefore, no parameter estimates are made. In other words, it produces a nonparametric density estimate that adapts itself to the data. Likewise, the ProbDistUnivKernel object, which represents a nonparametric probability distribution based on a normal kernel smoothing function, was used to deal with all the PDFs (Mathworks, 2012). 3. Results and discussion The water condition for the Cauca River when crossing the Valle Department in Colombia has been here assessed with a water quality index built on a FIS. Input data have been provided by the CVC Corporation, a regional environmental protection agency. We assessed various years using stochastic modeling with non- parametric kernel estimators of inputs. Hence, integration between fuzzy and stochastic models was carried out to manage both random and linguistic

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Table 4 Comparison of the fuzzy water quality index versus other indexes after Monte Carlo simulations (medians are provided). Membership values in linguistic scores, computed with fuzzy modeling, are provided. Index

Stochastic NSF_WQI

Stochastic ICAUCA

Stochastic FWQ

Year

Section I

Section II

Section III

Numeric score

Linguistic score

Numeric score

Linguistic score

Numeric score

Linguistic score

2002 2006 2008 2009 2010 2002 2006 2008 2009 2010 2002

63 56 58 61 56 42.65 63.74 63.03 74.96 40.49 51.58

48 45 50 49 40 27.49 54.95 28.80 30.64 26.70 50.59

51.81

2008

51.98

2009

51.18

2010

50.85

Bad Bad Bad Bad Bad Bad Good Bad Bad Bad 0.35 bad 0.73 regular 0.38 bad 0.70 regular 0.34 bad 0.73 Regular 0.33 bad 0.75 regular 0.32 bad 0.75 regular

49 47 46 49 41 30.09 44.03 42.54 55.26 26.77 49.74

2006

Regular Regular Regular Regular Regular Regular Good Good Good Regular 0.31 bad 0.77 regular 0.30 bad 0.78 regular 0.29 bad 0.79 regular 0.32 bad 0.76 regular 0.34 bad 0.74 regular

Bad Bad Bad Bad Bad Bad Regular Regular Good Bad 0.40 bad 0.67 regular 0.41 bad 0.66 regular 0.35 bad 0.72 regular 0.37 bad 0.70 regular 0.41 bad 0.67 regular

uncertainty in the analysis. Consequently, a stochastic fuzzy water quality index was developed. Fig. 5 shows the behavior of the stochastic FWQ index for five years and the three river sections assessed using a Box-and-Whiskers plot. Most of the time, the stochastic FWQ index in river Section I was higher than that in river Section II, and in river Section III, respectively. Therefore, it is noted with the index that water quality decreases downstream. Generally, there is a bigger dispersion in river Section III, as it is shown by their box heights. Moreover, it is noticeable the symmetry of the median, as well as the close distance between estimated boundary values. Such numeric scores agree well with expert and non-expert opinions, since water quality control is minimal in such area, and water pollution is considerably increased downstream. Despite the limited monitored variables considered in the current study, and included in the index, the FIS model adequately describes the observed condition. The low water quality scores should be inferred from a brief inspection to Table 1. Low dissolved oxygen concentrations, specifically in Sections II and III, were observed with very low saturation percentages. Also, high fecal coliform concentrations were available. Moderate (medium to high) concentration of total solids and turbidity are also common features in the area. BOD5 was high in Section III with observed increase in time, since in 2010, the values were considerably higher than those in 2006 and 2002. With the aid of the fuzzy stochastic analysis, it is possible to map fuzzy random input parameter into fuzzy random responses. The stochastic fuzzy behavior of the FWQ index and some of its advantages are shown in Table 3. Here, the membership functions (μ) to the diverse output fuzzy sets are calculated. As mentioned above, the membership to each fuzzy set is a number between zero (0) and one (1), meaning none and total membership, respectively. Partial memberships are also possible, which is one of the advantages of fuzzy logic for environmental decision making. It must be remarked that the sum of specific set membership values could be higher than 1. The membership degrees may be stated as possibility values to not confuse them with probabilistic computations. Table 3 presents the calculated membership values to the sets bad, regular and good. Such score was zero in all years, reaching to poor and excellent water quality fuzzy sets. In all cases, the belongingness estimation to the good water quality fuzzy sets was really low. The fuzzy sets with the higher membership values were the ones to regular water quality classification during all years, and through the three river sections with median variations between 0.797 and 0.667. Likewise, the membership value to bad water quality classification during all years and through the three river sections had

50.00 50.78 51.05 50.89

49.47 50.54 50.10 49.45

median variations between 0.413 and 0.292, which is a considerable belongingness. Moreover, it is noticeable that the membership values to the regular water quality fuzzy sets decreased downstream while they increased for bad water quality fuzzy sets. This result may be associated to the higher number of domestic, agricultural and industrial loads available downstream. In Table 3 can be seen that the membership degrees for bad water quality are higher in the lower quartile than in the median, which is due to the lower quartile of the index is the worst condition. For example, in 2002 and Section I, the membership degrees to the bad fuzzy set were 0.344, 0.310, and 0.266 for the lower quartile, median and upper quartiles, respectively. Output histograms are necessary in indexing computations since with point estimates the results could be limited. Fig. 6 shows the frequency histograms (or the nonparametric density estimations) for the stochastic FWQ index of some assessed years through the three sections. They point out the randomness of water quality integration outputs when random inputs are provided to the FIS. So, the output spread is easily observed in such figures. As it can be noted, diverse shapes are possible. Some histograms showed peak shapes with relative symmetrical variability. Likewise, some histograms with wide peaks (2008, Sections II and III) were calculated. The biggest dispersion was generally observed for river Section III. Moreover, in some cases appeared bimodal histograms, especially in Section III, although the closeness between peaks was enough to get appropriate classification and unambiguous outputs. In order to validate the performance of the stochastic fuzzy water quality index, similar stochastic computations were carried out for the indexes: NSF_WQI and ICAUCA. Results of the medians are given in Table 4. From the NSF_WQI calculations, it can be observed that they always provided a consistent output, classifying as regular the river Section I, and bad the river Sections II and III, for all the assessed years. In this case, numeric scores ranged between 63 and 41, with a spread of 22. The assessment with the ICAUCA index was less strict, delivering good water quality classification in some cases. ICAUCA index outputs were between 74.93 and 26.70, with a range of 48.26. The outputs from stochastic FWQ index were similar to the other indexes. The dispersion of the stochastic FWQ index results was lower than the other indexes, being within a maximum of 51.98 and a minimum of 49.45, with a range of 2.53. Although the numeric score of the defuzzified stochastic FWQ index is important, the advantage of the hybrid probabilistic fuzzy index over the others is that the last one provides a classification with more information related to the

W. Ocampo-Duque et al. / Environment International 52 (2013) 17–28

belongingness to diverse possible classifications. In all cases, the stochastic FWQ index outputs have classified water quality in the studied area as “partially bad and partially regular”. Lower possibility has the water quality to belong to the good class (in Table 3, observe that μ b 0.01). As above stated, the belongingness to bad water quality sets increased downstream from the river Section I to Section III. It agrees with the results from the NSF_WQI. Consequently, the belongingness to the regular class decreases downstream. Water quality indexes based on fuzzy systems have been recently proposed in scientific literature with relative success. The fuzzy frame clearly improves the conceptual design of the indexes, because they are computed with expert rules and sets to provide final numerical/ linguistic scores which include a convenient treatment of linguistic uncertainty and subjectivity. However, the computation of water quality index scores is clearly deterministic even within the fuzzy method. A vector of water quality variables is given to the FIS, and a unique water quality score is obtained. The challenge now is how to deal with computation in non-deterministic real world scenarios. Water quality variables collected in rivers are essentially stochastic, and density probability functions may easily be computed. Then, the key question is how to perform computations of water quality indexes when sufficient data have been collected, and the statistics are dependable. Currently, the easiest way to deal with stochastic computations is through Monte Carlo methods. Moreover, fuzzy alpha-cuts to deal with uncertainty in inputs could also be considered (Kumar et al., 2009). In this paper, we used Monte Carlo simulation to calculate the fuzzy index to analyze historic and geographic trends in water quality. The method was powerful because provided better water quality classification, and we observed graphically the consistency in fuzzy classification. However, the use of combined probabilistic and fuzzy methods is still under development, and a generalized theory of uncertainty is required (Zadeh, 2005). Moreover, mathematical foundations about propagation of probabilistic uncertainty through fuzzy systems may also require further research. Finally, we found that the method was powerful not only by providing consistent histograms of defuzzified water quality scores but also delivering the membership values to more than one water quality class. The value of the membership function of the output fuzzy sets resulted highly sensitive to input conditions. With this tool, the decision makers may be able to relax the boundaries between two or more likely water quality classes. Moreover, a consistent classification in water quality after stochastic simulations was observed which showed that the fuzzy index was stable in providing appropriate classification. Finally, the use of fuzzy systems avoids using crisp values to water quality classification which is the most important fact in applying this methodology. With the Monte Carlo and FIS approach, the strongly subjective character of the equivalence functions of traditional water quality indexes is avoided, and the assessment is closer to human reasoning, becoming the technique very useful under many similar environmental assessment problems. 4. Conclusion We have implemented stochastic simulation to a fuzzy water quality index in order to improve the water quality assessment provided with deterministic indexes. The hybrid stochastic fuzzy method combined the benefits of Mont Carlo simulations with the advantages of fuzzy inference. The proposed method updated the design of indexing techniques to integrate water quality variables available to date. Nonparametric kernel density estimators resulted appropriate tools to build empirical probability density functions from raw data since normal and other parametric distributions did not fit well the real data, especially when number of data was limited. The Monte Carlo simulation improved the results from point estimate of fuzzy water quality indexes since the dispersion of the final indexes was estimated. The water quality classification preserved the linguistic uncertainty of the subjective index and the randomness from real measurements. The main advantage of the

27

proposed method is that membership to two or more classes is also possible which gives to decision makers a better conceptual assessment. When the developed method was applied to the Cauca River, the results for several years showed that water quality was possibly “regular” with a membership degree of approximately 0.7, and possibly “bad” with a membership degree of approximately 0.4. The index also predicted that water quality decreased downstream. The results have complex origins, since the river is plainly affected by the presence of towns and cities without adequate treatment for wastewater. We observed that the environmental impact was not reduced downstream. Intense sugarcane agriculture and some industrial plants could also be responsible of surface water pollution. An intensive environmental protection program from regional and national government is suggested if ecosystem restoration and biodiversity conservation is desired in the area. Acknowledgments The authors thank the Agencia Española de Cooperación Internacional para el Desarrollo (AECID) for financial support (Projects D/026977/09, and D/031370/10). We also thank the CVC Corporation for providing water quality monitoring data. References Baudrit C, Guyonnet D, Dubois D. Joint propagation of variability and imprecision in assessing the risk of groundwater contamination. J Contam Hydrol 2007;93:72–84. Beamonte-Cordoba E, Casino Martinez A, Veres-Ferrer E. Water quality indicators: comparison of a probabilistic index and a general quality index. The case of the Confederación Hidrográfica del Júcar (Spain). Ecol Indic 2010;10:1049–54. Brown RM, McClelland NI, Deininger RA, Tozer RG. A water quality index: do we dare? Water Sew Works 1970;117:339–43. Cardona CM, Martin C, Salterain A, Castro A, San Martín D, Ayesa E. CALHIDRA 3.0 — new software application for river water quality prediction based on RWQM1. Environ Model Softw 2011;26:973–9. Chen Z, Zhao L, Lee K. Environmental risk assessment of offshore produced water discharges using a hybrid fuzzy-stochastic modeling approach. Environ Modell Softw 2010;25:782–92. Chowdhury S, Champagne P, McLellan PJ. Uncertainty characterization approaches for risk assessment of DBPs in drinking water: a review. J Environ Manage 2009;90: 1680–91. CVC Corporation. Estudio de la calidad del agua del río cauca y sus principales tributarios mediante la aplicación de índices de calidad y contaminación. Project Report 0168, Oct 2004. Available at: http://190.97.204.39/cvc/Mosaic/dpdf2/ Volumen10/1-ECARCpag1-158.pdf2004. (Accessed 1/9/2012). Darbra RM, Eljarrat E, Barcelo D. How to measure uncertainties in environmental risk assessment. Trends Anal Chem 2008;27:377–85. Faybishenko B. Fuzzy-probabilistic calculations of water-balance uncertainty. Stoch Environ Res Risk A 2010;24:939–52. Ghiocel DM, Altmann J. Hybrid stochastic-neuro-fuzzy model-based system for in-flight gas turbine engine diagnostics. In: Pusey HC, Pusey SC, Hobbs WR, editors. New frontiers in integrated diagnostics and prognosticsProceedings of the 55th meeting of the Society for Machinery Failure Prevention Technology, Virginia; 2001. Gottardo S, Semenzin E, Giove S, Zabeo A, Critto A, de Zwart D, et al. Integrated risk assessment for WFD ecological status classification applied to Llobregat river basin (Spain). Part I—fuzzy approach to aggregate biological indicators. Sci Total Environ 2011;409:4701–12. Guo P, Huang GH, Zhu H, Wang XL. A two-stage programming approach for water resources management under randomness and fuzziness. Environ Modell Softw 2010;25:1573–81. Kentel E, Aral M. 2D Monte Carlo and Monte Carlo-fuzzy health risk assessment. Stoch Environ Res Risk A 2005;19:86–96. Kumar V, Mari M, Schuhmacher M, Domingo JL. Partitioning total variance in risk assessment: application to a municipal solid waste incinerator. Environ Modell Softw 2009;24:247–61. Legay C, Rodriguez MJ, Sadiq R, Sérodes JB, Levallois P, Proulx F. Spatial variations of human health risk associated with exposure to chlorination by-products occurring in drinking water. J Environ Manage 2011;92:892–901. Lermontov A, Yokoyama L, Lermontov M, Soares-Machado MA. River quality analysis using fuzzy water quality index: Ribeira do Iguape river watershed, Brazil. Ecol Indic 2009;9:1188–97. Li H, Zhang K. Development of a fuzzy-stochastic nonlinear model to incorporate aleatoric and epistemic uncertainty. J Contam Hydrol 2010;111:1-12. Li J, Huang GH, Zeng G, Maqsood I, Huang Y. An integrated fuzzy-stochastic modeling approach for risk assessment of groundwater contamination. J Environ Manage 2007;82:173–88. Mahapatra SS, Nanda SK, Panigrahy BK. A cascaded fuzzy inference system for Indian River water quality prediction. Adv Eng Softw 2011;42:787–96. Marchini A, Facchinetti T, Mistri M. F-IND: a framework to design fuzzy indices of environmental conditions. Ecol Indic 2009;9:485–96.

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