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Environmental Modelling & Software 60 (2014) 31e44

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Environmental Modelling & Software

journal homepage: www.elsevier .com/locate/envsoft

Identifying low impact development strategies for flood mitigationusing a fuzzy-probabilistic approach

J. Yazdi, S.A.A. Salehi Neyshabouri*

Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran

a r t i c l e i n f o

Article history:Received 3 April 2014Received in revised form31 May 2014Accepted 2 June 2014Available online 27 June 2014

Keywords:UncertaintyLow impact developmentCopulaFuzzyNSGA-IIANNOptimizationFlood risk

* Corresponding author.E-mail addresses: j.yazdi@modares.ac.ir (J. Y

(S.A.A. Salehi Neyshabouri).

http://dx.doi.org/10.1016/j.envsoft.2014.06.0041364-8152/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

Low impact development (LID) includes strategies and practices that are designed to control surfacerunoff at its sources in a sustainable way. The performance of these strategies has been frequentlyaddressed through curve number approach. This approach however subjects to a great deal of un-certainties owing to uncertain nature of curve numbers and temporal/spatial variability of flood events.This paper represents a novel methodology to deal with both inherent flood uncertainties and epistemicuncertainties identifying optimal LID strategies for flood mitigation. The proposed methodology in-tegrates a great variety of mathematical tools including copula functions, MCS method, hydrological andhydraulic models, NSGA-II algorithm as well as ANN and fuzzy set theory. The obtained results from acase study clearly demonstrate that the proposed methodology not only presents cost-effective mea-sures, but also can simultaneously handle both inherent and epistemic uncertainties in flood riskmanagement.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Population growth and urbanization have altered the streamflow regimes associated with frequent flood events. This has led toincrease runoff volumes, flood damages and stream-bank erosions.Low impact developments (LIDs) and best management practices(BMPs) are a set of mitigation measures that can be considered forreducing the hydrologic consequences of urbanization (Karamouzand Nazif, 2013). In contrast of traditional procedures like dikesand floodwalls which try to evacuate excess waters as soon aspossible and do not take into account cumulative damages of floodsover long periods of times, LID or BMP measures seek a sustainableway which enable better management of runoff (in terms of bothquality and quantity) at source by preserving natural drainagepatterns and emulating the natural hydrological cycle.

LIDs includes a wide variety of measures such as runoff reuse,soak ways, infiltration trenches, swales, green roofs, land usechange, permeable paving, wetlands, etc. For each case study, theforms of LIDs are determined by local conditions.

Application of LIDs for mitigating the hydrologic consequenceshas been reported in different research works. USEPA (2000)

azdi), salehi@modares.ac.ir

represented a literature review of low impact developments andtheir effectiveness for controlling stormwater volume and reducingpollutant loadings into receiving waters. Perez-Pedini et al. (2005)and Zhang (2009) developed and applied Simulation-optimizationmethodologies to allocate infiltration-based LID technologies basedon the reduction of peak flow at the watershed outlet. Damodaramet al. (2010) demonstrated that a combination of LID and ponds ismore efficient compared to LID or ponds alone. Damodaram andZechman (2013) also developed a simulationeoptimizationapproach for optimal allocation of permeable parking lots, rooftopsand detention ponds for a few design storms. Oraei Zare et al.(2012) represented multi-objective optimization methods toselect BMP measures for urban storm management consideringboth quality and quantity aspects of controlling surface runoff.Among different types of LIDs, land use change strategies are ofgreat importance for flood mitigation due to their compatibilitieswith environmental conditions. More attention however has beenpaid to find out optimal land allocation from the soil conservationand agricultural-benefit viewpoint (see Onal et al., 1998; Amir andFisher, 1999; Randhir et al., 2001; Mohseni Saravi et al., 2003,Ducourtieux et al., 2005; Gabriel et al., 2006; Luo and You, 2007;Sadeghi et al., 2009). Yazdi et al. (2013a) represented a methodol-ogy to find out the most suitable land use allocation for flood riskmitigation. The proposed methodology integrated a hydraulicmodel and a multi-objective optimization model for considering

Fig. 1. A schematic view of a watershed and scattered rain gauges.

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e4432

the interaction of various land use change actions on flood miti-gation and finding out the best land use change strategies. Theirwork however did not involve the inherent and epistemicuncertainties.

Despite various researchworks on selecting LID strategies, thereare still some challenges in design of LID and BMP measures con-cerning the significant flood uncertainties. Most of the previousworks have represented LID strategies for one or a few designstorms, separately. From decision making viewpoint, the questionis which storm design should be considered to select the beststrategies. Since floods or storms are naturally stochastic, instead ofa scenario based approach, a risk based approach seems moreappropriate to represent optimal practices considering flood un-certainties. Inherent uncertainties of floods in watershed scaleinclude both temporal and spatial uncertainties. Temporal un-certainties are normally considered in flood risk management byflood frequency analysis of maximum annual time series and rep-resenting as the probability density functions (PDFs) of rainfalls ordischarges which these PDFs can subsequently be used in MonteCarlo Simulation (MCS). Spatial uncertainties however, cannoteasily be handled through MCS task, due to dependence structureof flood rainfalls in different parts of watersheds. Researches haveshown that the assumption of independence can have a significanteffect on magnitudes of generated surface runoffs and may leads toerroneous results (e.g. see AghaKouchak et al., 2010; Golian et al.,2011). Nevertheless, these spatial uncertainties have not beenconsidered so far in optimization of cost-effective designs offloodplain systems, likely due to the difficulty and complexity ofgenerating joint probability distributions for rainfall variables. Inthis paper, spatial variability of rainfalls along their dependencestructures are addressed through joint cumulative distributionfunctions (CDFs) derived by copula functions within a Monte Carlo(MC) framework. A major advantage of copula method for gener-ating CDFs is that marginal distributions of individual variables canbe of any form and the variables can be correlated (Favre et al.,2004). This represents a significant advantage compared to con-ventional multivariate analysis as many variables from hydrologicalphenomena cannot be described using the same type of probabilitydistributions (Fu and Butler, 2013).

Besides the inherent uncertainties, epistemic uncertainties,stemmed from incomplete knowledge or imprecise informationabout a physical system, also play an important role in flood riskmanagement. In optimal design of flood mitigation measures, theperformance of LID strategies is often considered through updatingthe curve numbers (CNs). CN modifications however are subject toa great deal of epistemic uncertainties, arising from the lack ofknowledge and existing ambiguity in the curve number of practicesand land use changes in the future plans. Less attention howeverhas been paid to these parameter uncertainties in representingcost-effective LIDs. To cover this gap, fuzzy approach is addressedthroughout this paper which is able to characterize curve numberuncertainties into the system design.

Generally, in flood risk management, sparse efforts have beenspent to develop methodologies that can propagate both types ofuncertainties simultaneously through themodeling process. Fu andKapelan (2011) employed a random set based method to bridge thegap between probability and fuzzy set for sewer flooding estima-tion. Their method however is not able to handle spatial un-certainties of floods while these uncertainties are of greatimportance in flood risk assessment when studies are carried out inwatershed scale.

The work described here represents an innovative methodologywhich integrates inherent and epistemic uncertainties in a unifiedframework for identifying the optimal flood risk managementstrategies, offering a flexible facility to handle both types of

uncertainty. Multivariate joint distribution of observed extremerainfalls is constructed for considering temporal and spatial flooduncertainties within a MC procedure. On the other hand, epistemicuncertainties of curve numbers are characterized through the fuzzymembership functions and a-cut concept in order to propagatecurve number uncertainties into the flood damage fuzzy mem-bership functions. Both approaches are aggregated with NSGA-IIoptimization model to represent optimal land use strategies asLID measures. Details of the methodology are represented incontinued sections.

Hereafter throughout this paper, the terms “LID strategies”, and“land use change strategies” are used interchangeably.

2. The analysis of extreme rainfall data

Flood rainfalls/discharges are naturally uncertain i.e. theirvalues constantly fluctuate with random patterns. These un-certainties are a property of the system and cannot be reduced dueto their inherent nature. Henceforth, they are so called as inherentuncertainties. Inherent uncertainties are traditionally representedby probability theory in which the value of uncertain variables isdescribed by probability density functions (PDFs). The probabilisticapproach has been extensively used in practice for consideringflood uncertainties on optimal design of hydro-systems (e.g. Afsharand Marino, 1990; Tung and Bao, 1990; Kort and Booij, 2007;Karamouz et al., 2008; Afshar et al., 2009; Qi and Altinakar, 2011;Delelegn et al., 2011; Sun et al., 2011; Yazdi and SalehiNeyshabouri, 2012). The methodology represented by US ArmyCorp of Engineers (1996) is commonly used to address the hydro-logic (flood-frequency analysis), hydraulic (rating-curve develop-ment), and economic uncertainties (stage-damage analysis) inflood risk management, particularly for floodplain systems. Despiteextensive application, there are still some challenges when thisprobabilistic approach is used on awatershed scale for selecting thebest flood risk mitigation measures which have not been reportedyet through other research works. Flood magnitudes are not thesame at different parts of the watershed (Fig. 1) and this means inaddition to temporal uncertainties, floods have considerable vari-ations on spatial scales.

Thus, a multi-site sampling approach is required for rainfallgeneration from the PDFs of extreme rainfalls at monitoring raingauges of the watershed. Multi-variate sampling however is a

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J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e44 33

crucial task since samples cannot be generated independentlyowing to the existence of well correlations among flood rainfallsin adjacent sub-basins. This paper, for the first time, implementsstatistical copula functions to address aforementioned problem.Copulas provide the possibility of combining univariate distribu-tions of stochastic variables to construct a joint probability dis-tribution while the correlations among variables are preserved.Henceforth, sampling from the joint PDF assures the existence ofdependence among generated floods of adjacent regions. Copulasare first introduced by Sklar (1959) in statistics and recently havegained great attention in hydrology for describing the dependencestructure among different hydrological variables. Some examplesare: modeling the intensity-duration of rainfall events (DeMichele and Salvadori, 2003), hydrological frequency analysis(Favre et al., 2004), analysis of drought severity and duration(Shiau, 2006), the interpolation of ground water quality parame-ters (Bardossy, 2006), multivariate analysis of water deficit andannual evapotranspiration (Nazemi and Elshorbagy, 2012), andassessing the reliability of flood warning systems (Yazdi et al.,2013b).

Based on the Sklar theorem, for two variables x and y, withmarginal distributions as GX and FY , the joint distribution func-tion, Hðx; yÞ can be represented as below:

Hðx; yÞ ¼ CðFXðxÞ;GYðyÞÞ ¼ Cðu; vÞ (1)

where C is a copula function and u and v are uniformly distributedrandom variables in [0,1]. Copula function C can be uniquelydetermined when u and v are continuous functions. Some of thefamous copula functions are presented in Table 1. To select thebest family of copulas for a problem, the maximum likelihoodmethod (Zhang and Singh, 2007), Inference Function for Margins(IFM) method (Joe, 1997) and Canonical Maximum Likelihood(CML) method (Genest et al., 1995) can be employed. More detailsabout the concepts and theory of copulas can be found in (Joe,1997).

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2.1. Generating multi-variate synthetic rainfall events

One of the primary applications of copulas is in simulation andMonte Carlo studies. For generating a sample ðx; yÞ from thespecified joint distribution, the inverse distribution functionmethod is suggested. Based on the Sklar theorem, the conditionaldistribution function for V given U ¼ u, which we denote cuðvÞ, is:

cuðvÞ¼P½V�vjU¼u�¼ limDu/0

CðuþDu;vÞ�Cðu;vÞDu

¼vCðu;vÞvu

(2)

Now, the following steps are used for generating a pair sampleðx;yÞ of random variables ðX;YÞ with a joint distribution functionH, where H is H¼CðU;VÞ:

1. Generate two independent uniform ð0;1Þ variates u and t;2. Set v ¼ cð�1Þ

u ðtÞ where v ¼ cð�1Þu denotes a quasi-inverse of cu.

3. For the pair ðu; vÞ, Set x ¼ F�1ðuÞ and y ¼ G�1ðvÞ. The desiredpair is ðx; yÞ.

It is worth mentioning in the first step, the Latin HypercubeSampling (LHS)method (McKay et al., 1979) is used to improve thecomputational efficiency of sampling approach.

Table

1Differentco

pula

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type

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bos

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gaard

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3. Hydrologic and hydraulic modeling

In this paper, MC method is used to map generated rainfalls asthe input variables to the associated flood damages as the system

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e4434

output. MC method may be the only procedure that can estimatethe complete probability distribution of the model output for caseswith highly non-linear and or complex system relationship(Melching, 1995). In the studied problem here, there are highlynon-linear relationships among variables for converting the floodrainfalls to the hydrograph discharges, water elevations and dam-ages. Using the MCS method provides an efficient and reliablemethod to estimate probability distribution of flood damages fromdistribution of flood rainfalls as input variables. For calculatingflood damages, a couple of calibrated hydrological and hydraulicmodels as well as the damage-elevation curves of properties onflood prone areas are commonly required. Furthermore, in a prac-tical procedure on the scale of watershed, a significant number ofmulti-site rainfall samples should be generated. For each sample set(q1,q2,…,qm), synthetic rainfall hyetographs can easily be con-structed and introduced to the hydrological model. If a land usechange plan is considered, the sub-watershed curve numbers(~c1;~c2; :::;~cn) should also be updated in hydrological model. Thenrunning the hydrological model gives the runoff hydrographs ofdifferent sub-basins which are introduced to hydrodynamic modelas the boundary conditions. Consequently, hydrodynamic model isperformed to calculate water depth of different properties in flood-prone areas and finally the flood damages are estimated throughdamage-elevation curves (Fig. 2). It is noticeable that in this studyonly physical vulnerabilities are considered in flood damage anal-ysis using non-dimensional damage-elevation curves of floodplainproperties. Study of other flood vulnerabilities is usually carried outusing the index approach. To obtain more details about differentapproaches of flood vulnerability assessment, interested readerscan refer to the work of Balica et al. (2013).

Since a significant number of random sampling from joint PDF ofstochastic rainfall variables and curve numbers are needed tocalculate expected flood damages, simulation process becomes toocomputationally expensive when it is implemented in the optimi-zation model. To avoid this problem here, a multi-layer feed-for-ward artificial neural network as a surrogate model (also calledMeta model) is trained and after verification is replaced withsimulator engine to alleviate large computational budgets. Toobtain reliable responses from the surrogate model, a wide varietyof rainfall samples ranging from high frequent rainfalls to veryextreme rainfalls with low frequencies are provided for training thesurrogate model using the complex simulator engine. This im-proves the fidelity of surrogatemodel responses for extreme eventsgenerated and risk estimations. The weights and biases of thenetwork are obtained with back propagation Lev-enbergeMarquardt algorithm. After tuning the weights and veri-fication, trained network can be used for flood damage predicting.

Fig. 2. Schematic view of simulato

4. Risk based optimal LID strategies

There are different sources of uncertainties in flood risk man-agement including hydrological and hydraulic uncertainties,model uncertainties, parameter uncertainties as well as economicuncertainties and uncertainties related to measurement errors andsmall samples of random phenomena. Flood management un-certainties can generally be classified into inherent and epistemicuncertainties (Hall and Solomatine, 2008). The inherent or alea-tory uncertainty represents the randomness and variabilityobserved in nature (both in space and time), whereas epistemic orknowledge uncertainty refers to the state of knowledge of aphysical system and our ability to measure and model. Here twoimportant sources of uncertainties i.e. hydrological uncertainties,referred to as inherent uncertainties, and curve number parameteruncertainties, referred to as epistemic uncertainties, are consid-ered on flood risk mitigation. For this purpose, a probabilisticapproach is utilized to deal with spatial and temporal hydrologicaluncertainties while a fuzzy approach is implemented to involveknowledge uncertainties in conjunction with curve number offuture land use change strategies, both approaches embeddedwithin a Monte Carlo (MC) procedure. The proposed approach forhandling the inherent uncertainties was thoroughly discussed insection 2. Quantifying the relevant epistemic uncertainties andtheir propagation into the modeling are discussed in the followingsection.

4.1. Epistemic uncertainties

It is argued that the main deficiency of probabilistic approachfor flood risk management is ignorance of subjective types of floodrisk arising from knowledge uncertainties (Ahmad and Simonovic,2011). Nevertheless, less considered uncertainties but not lessimportant are epistemic uncertainties (knowledge uncertainties)and thus, they need more attention to be taken into account. Anappropriate tool to handle this kind of uncertainties is fuzzy settheory. Fuzzy set theory was developed specifically to deal withuncertainties that are not statistical in nature (Zadeh, 1965). Awiderange of real-world problems which are involved linguistic de-scriptions or are based on expert knowledge may effectively bedealt with fuzzy sets (Yen and Langari, 1998). Fuzzy set theory hasbeen successfully applied in the field of water resources manage-ments over the last decades. Some applications of fuzzy set theoryon simulationeoptimization methodologies can be enumerated as:Esogbue et al. (1992), Chang and Chen (1996), Mujumdar andSubbarao Vemula (2004), Fu (2008), Labadie and Wan (2010), Fuand Kapelan (2011) and Teegavarapu et al. (2013).

r engine to calculate damages.

Fig. 3. a) Input hydrographs with uncertainty, b) Propagating the input fuzzy membership functions to the output membership function.

Fig. 4. a-cut of a fuzzy number.

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e44 35

Among the more recent application of fuzzy set theory to floodmanagement is the work of Ahmad and Simonovic (2007). Theydeveloped a methodology to address the spatial variability of floodrisk using fuzzy set. Their work however was not capable to addressboth spatial and temporal variability of flood risk. They extendedthe fuzzy methodology for assessing both spatial and temporalvariability and also ambiguity in flood risk analysis (Ahmad andSimonovic, 2011). Nevertheless, the new proposed approach doesnot deal to flood risk management underlying flood mitigationmeasures.

The methodology described here considers the epistemic un-certainties of curve numbers associated to future land use changestrategies in an optimization framework to find out the best stra-tegies for flood risk management. For each land use change strat-egy, the curve number of the region is expected to change, but thenew CN is uncertain. Henceforth, the new CNs can be assumed asfuzzy numbers instead of crisp values based on expert knowledge.Selection of the appropriate shape of fuzzy numbers is highlydependent on the stochastic nature of the parameter. Therefore, theshape of membership function for curve number is subjectivelyselected by the expert's opinion based on existing informationabout the curve number of considered land uses. For the study areaand for suggested land covers, by consulting with a team of expertsand studying the table guides of curve numbers for different landuses, triangular shapes are recognized suitable for the curvenumbers. Subjective judgment and the role of expert knowledge forselecting the membership functions of input fuzzy variables arealso reported in other researches (e.g. see Ahmad and Simonovic,2011) and this is one of the advantages of fuzzy attitude thanprobabilistic approach in which subjectivity can be imported intothe modeling. Although, triangular shapes are used in this study,but the proposed methodology is general enough and other shapescan also be implemented in this methodology.

As shown in Fig. 3, fuzzy CNs lead to uncertain flood hydro-graphs for a certain rainfall, in particular fuzzy peak discharge fordifferent sub-basins. Consequently, uncertain hydrographs cause touncertain water surfaces in flood plains and eventually fuzzy

damage (instead of a crisp value of damage) that should be mini-mized in an optimization algorithm. Identifying fuzzy membershipfunction of damage through propagating the input CN fuzzynumbers is a difficult task owing to nonlinear and complicate re-lationships among different components and variables (the solu-tion is discussed in continue).

In the most of engineering problems, it is necessary to convertfuzzy numbers (in this case fuzzy damages) to the crip numbers forinterpretation of the results. There are two approaches for thispurpose: first, using the usual defuzzification methods and thesecond, using the concept of a-cut. In this research the secondapproach was used to convert fuzzy damage results to the crispvalues.

In general, the concept of a-cut is used to convert a fuzzy set to aclassic set. Assume the fuzzy set ~A and the a number (0 � a � 1).The a-cut of the fuzzy set ~A is shown by Aa and defined as (Fig. 4):

Aa ¼nxjm~AðxÞ � a

o(3)

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e4436

where m~AðxÞ is a membership function taking values from [0, 1],specifying to what degree x belongs to ~A. The Aa set is a classic set.Indeed, each member x2Aa is also a member of ~A, which itsmembership function is greater or equal a.

Based on the a-cut concept, an uncertain variable representedby a fuzzy number can be transformed into crisp sets. Therefore, theCN interval at a level can be calculated using the two extremevalues of the a-cut interval of CN. CN intervals can then propagateto damage interval:

½Dmin;Dmax�a¼a* ¼ f�½CN1min;CN1max�a¼a* ;

½CN2min;CN2max�a¼a* ; ::::� (4)

where CNimin and CNimax are extreme values of CNi at a* level, fstands for the nonlinear hydrological and hydraulic equations anddamage-elevation curves for calculating damages. Dmin and Dmax

are also extreme values of damage at a* level.It is worth mentioning the concept of a-cut is general and is not

related to special shape of membership function. When the shapesof fuzzy membership functions are not left-right forms such astriangular forms, e.g. when there is a uniform fuzzy membershipfunction, two extreme values of all a-cut intervals correspond to thelower and upper bounds of the uniform fuzzy membership func-tion. It's true because when uniform membership functions areselected for CN variables by an engineer or an expert, it meansbased on his/her knowledge and information on hand, he does nothave any preference among the CN values on the variability range,i.e. the same possibility exists for all CN values inside the supportand therefore the extreme values of different a-cuts are the same.This rule is also valid for one-sided uniform shapes of membershipfunctions. In this case, there are different a-cuts for decreasing sideof the function while for the uniform side, for all a-cuts, theextreme value of that side is the same and equal to the boundaryvalue of the support.

Although some analytical methods have been represented toobtain output variable interval from the given input variable in-tervals (e.g. see Moore,1979; Neumaier, 1990; Hansen,1992; Revelliand Ridolfi, 2002), these methods are not applicable here since theinput intervals have different variability ranges and the function f istoo complicated and nonlinear. A sampling approach from theinput intervals is used here based on the Latin Hypercube Sampling(LHS) technique to obtain the output interval. LHS assures wellcovering the range of all input intervals in sampling while a sig-nificant number of sampling and running a simulator engine foreach sample set gives the corresponding damages. Damage intervalis then considered as two extreme values of calculated damages.

The a-cuts of damages and consequently benefits allow thedecision makers to impose their risk acceptance level through se-lection of the appropriate value for a-cut of curve numbers. Se-lection of 1 for a-cut (i.e. 100% risk acceptance) leads to acompletely crisp space and full ignorance of the uncertaintiesoriginated from uncertain curve numbers in the benefit to cost (B/C) analysis. A quite conservative decision maker with zero riskacceptance level may select zero value for a, leading to a very highrange of expected damage/benefits and consequently B/C ratios.The cumulative effects of curve number uncertainties in the valueof estimated damages underlying different LID strategies mayresult in very diverse possible benefit to cost ratios. It is veryimportant for the decision makers to have a notion on the possibleminimum and maximum expected benefits or B/C ratios. The pro-posed fuzzy approach provides the decisionmaker with an efficienttool to analyze the expected damages and B/C ratios along theirvariability ranges for different LID alternatives and therefore, toprovide more insights to select final LID plan.

4.2. Optimization model

Decision making on flood management issues like most of theother water resources problems involves a wide variety of stake-holders and parties with many interests which usually leads toconflict objectives. Inevitably, this requires the use of multi-objective approaches and tools to deal the conflicts that arise(Barreto Cordero, 2012). Multi objective optimization models give aset of optimal solutions providing a possibility of negotiating bydifferent stakeholders to select the final strategies among differentcriteria. Here initial investment cost and expected flood damagesare considered as two separate conflicting objectives. The generalformulations of the objective functions are as follows:

Min F1 ¼ AC�x�¼

qð1þ qÞtð1þ qÞt � 1

!Xni¼1

CostðxiÞ ¼ q$Xni¼1

CostðxiÞ

(5)

Min F2 ¼ eD ¼Z∞0

~D�x�f�~D�$d~D

¼Z∞0

G�~c1;~c2; :::;~cp

�F½fðq1; q2; :::; qmÞ�$d~D

(6)

where AC is the annual cost, x is the vector of decision variablesreferred to as a set of land use change strategies, t and q stand forthe life time of plans and discount rate, respectively. As shown inEq. (5), capital recovery factor would be equal to discount ratewhen the life time of projects is unlimited (Oscounejad, 1996). eDrefers to as the expected fuzzy annual flood damage in the whole ofwatershed; ~D represents fuzzy damage of a flood with a givenseverity which is a function of fuzzy curve numbers (Gð~c1;~c2; :::;~cpÞ)related to different land uses. f ð~DÞ represents the probability dis-tribution of fuzzy damages. Identifying f ð~DÞ is a crucial taskrequiring multivariate sampling from joint probability distribution(fðq1; q2; :::; qmÞ) of stochastic variables (q1,q2,…,qm) and applicationof hydrological and hydraulic models as well as damage-elevationcurves embedded in a Monte Carlo (MC) framework. F½:::� func-tion in Eq. (6) represents this computationally expensive process toconvert multiple extreme rainfall samples to correspondingdamages.

NSGA-II algorithm (proposed by Deb et al., 2000) is chosen tosolve the aforementioned optimization problem owing to itspopularity and successfully applications in different water re-sources problems. Decision variables are coded in a discrete space.Each variable refers to a land use change for a sub-basin in thewatershed which can take the values of 0 or other integer values. Ifzero value is taken, it means there is no land use changewhile othervalues refer to different land use change strategies for the sub-basins. Number of decision variables is equal to the number ofsub-basins.

4.3. Proposed algorithm

The main steps of the proposed algorithm are illustrated inFig. 5. First, the initial population is randomly generated. Eachchromosome is coded as different land use change options and totalcost corresponding to the chromosome is computed by the sum ofthe costs of options in that chromosome. Afterwards, multi-siterainfalls are generated by multivariate sampling from the bestfitted copula function and consequently, the damage a-cut intervalis calculated for the current chromosome and generated rainfalls.The value of a-cut can be interpreted as the degree of uncertainty

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e44 37

that is considered in calculating output variable. Here the value of0.05 is typically selected whichmeans 0.95 percent of uncertaintiesin curve numbers is considered in damage interval calculation.

Owing to stochastic nature of flood rainfalls, a significantnumber of multiple sampling should be generated for each chro-mosome. Henceforth, the current damage interval is saved in anarchiving cache and multi-site sampling is frequently carried outand damage intervals are calculated each time. The expecteddamage interval (as the second objective function) is then identi-fied by retrieving the calculated intervals from the archiving cacheand averaging them.

Computations are repeated for all chromosomes in the currentpopulation. Then, based on NSGA-II algorithm operators, theoffspring chromosomes are generated and mixed with currentpopulation. After sorting all chromosomes based on the non-domination and crowding distance, the best chromosomes areselected as the new generation. This evolutionary search processcontinues until a certain convergence criterion is met.

Fig. 5. The main steps of p

5. Case study

The proposed algorithm is tested and verified for Kanwatershedin central part of Iran as a real world case study. Kan basin, with216 km2 area, is a mountainous area with steep rivers and theability of creating debris floods. Fig. 6 shows Kan's sub-basins withmain rivers and land uses. This basin has had several devastatingfloods in last decades caused considerable flood damages and lossof lives (WRI, 2011). This area is a recreational region near thecapital, Tehran city and has a considerable mobile populationvisiting the basin especially in the weekends. More description ofthe watershed can be found in the work of Yazdi et al. (2013a).

5.1. Land use change strategies for Kan watershed

Land use change LIDs include methods to create, enhance andmaintain vegetation to reduce run-off and provide flood mitigationin the watershed. These measures change the land uses and affect

roposed methodology.

Table 2The suggested LIDs and the cost of each measure for different sub-basins.

Sub-basin Seeding Brush-grassmixture

Orchards-grasscombination

Area (ha) Cost ($) Area (ha) Cost ($) Area (ha) Cost ($)

SB1 271 1938 e e 591 10,343SB2 e e e e 346 6055SB3 135 965 e e 933 16,328SB4 249 1780 e e 278 4865SB5 662 4733 e e 1494 26,145SB6 50 358 e e 161 2818SB7 50 358 104 16,224 15 263SB8 442 3160 104 16,224 1137 19,898SB9 52 372 89 13,884 381 6668SB10 72 515 355 55,380 159 2783

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e4438

flood characteristics (the shapes of hydrographs). Kan watershedwas considered as a pilot project for “Integrated Flood Manage-ment” studies in Iran (WRI, 2011). Based on these studies, feasibilityof different land use change LIDs was assessed and some measuresfor each sub-basin were suggested. Table 2 shows the suggestedLIDs and the cost of each measure for different sub-basins.

These measures usually affect upon flood characteristics bychanging the peak and shape of flood hydrographs in different sub-basins. In fact, changes in land uses reduce the averaged curvenumber (CN) of sub-basin and consequently increase its retentioncoefficient. The new CNs of sub-basins corresponding to LIDs areestimated using the table guides of CNs (e.g. see Maidment, 1992),the map of land uses and the map of hydrologic soil group in GISenvironment.

Maidment (1992) represented a complete list of curve numbersfor a wide range of land uses in different conditions of soil moisture

Fig. 6. Study area including sub-basins, land uses and main rivers (Yazdi et al., 2013a).

Table 4Residual criteria for the results of model calibrations, where n is the number ofobservations, yi and byi are simulated and observed data.

Criteria Formula Qp

(m3/s)Time topeak(h)

Floodingvolume(MCM)

Maximumwaterdepth (m)

Mean RelativeAbsolute Error(MRAE)

1nPn

i¼1jyi�byi jbyi

0.04 0.04 0.09 0.01

Mean AbsoluteError (MAE)

1nPn

i¼1jyi � byij 0.312 0.8 0.04 0.02

AbsoluteMaximumError (AME)

maxjyi � byij 0.86 1 0.14 0.06

Root Mean SquareError (RMSE)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nPn

i¼1ðyi � byiÞ2q0.42 0.89 0.065 0.03

NasheSutcliffecoefficient (E)

1�Pn

i¼1ðbyi�yiÞ2Pn

i¼1ðbyi�byiÞ2

0.999 0.997 0.991 1.000

Table 5Residual criteria for the results of model validation.

Criteria Qp (m3/s) Time topeak (h)

Flooding volume(MCM)

Maximumwater depth (m)

MRAE 0.04 0.02 0.06 0.01MAE 1.17 0.67 0.068 0.02AME 2.1 1 0.09 0.02RMSE 1.35 0.82 0.071 0.02E 0.989 1.000 0.996 1.000

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e44 39

and land-use change quality. This reference has presented thecorresponding curve number of different land uses in three statesof good, average and poor conditions that can be considered as therange of uncertainty for CNs. We used these ranges to construct theprimary fuzzy membership functions of CNs. They were modifiedafter consulting with a team of experts.

In summary, based on abovementioned information, three CNnumbers were allocated for each pixel of sub-basins (CN in good,average and poor conditions). The new three CNs of each sub-basinwere obtained by averaging the whole pixel's CNs of sub-basin inGIS environment and considered as triangular fuzzy membershipfunction of CN for that sub-basin. The initial CN and the new fuzzyCNs of sub-basins are represented in Table 3 where the support offuzzy numbers is defined by the interval [a c] and the kernel is b.

The new CNs can be replaced in the calibrated hydrologicalmodel to obtain the modified flood hydrographs and modifiedhydrographs can be used as the boundary conditions of hydrody-namic model to simulate LID impacts on flood characteristics.

As shown in Fig. 6 there are three rain gauges and three hy-drometric gauges on the watershed to record flood events contin-uously in the study area. The HEC-HMS and MIKE11 models wereserved as the hydrological and hydraulic models and were cali-brated based on the rainfall records and observed hydrographs inthe hydrometric gauges. Details of model calibrations and damage-elevation curves of the study area have been represented in otherreferences (e.g. see Yazdi et al., 2013a).

6. Model evaluations

In order to use environmental models effectively for manage-ment and decision-making, it is vital to establish an appropriatelevel of confidence in their performance (Bennett et al., 2013).Accordingly, available observed data of recorded flood events in thestudy areawas used to validate simulationmodels and several well-known benchmark problems were selected from the literature toevaluate the performance of optimization model which are illus-trated in the following sections.

6.1. Simulation models

Simulation models include hydrological and hydraulic modelswhich are applied for rainfall-runoff modeling and flood routing,respectively. The performance of hydrological model was evaluated

Table 3Membership function of new CNs after LID strategies.

Sub-basin Initial CN Seeding O

a b c a

SB1 77.95 77.26 77.39 77.52 7SB2 78.50 e e e 7SB3 78.50 78.19 78.24 78.29 7SB4 79.50 78.70 78.84 78.98 7SB5 78.53 76.81 77.05 77.30 7SB6 80.00 79.33 79.45 79.57 7SB7 85.18 84.74 84.82 84.91 8SB8 83.19 82.07 82.31 82.55 8SB9 84.00 83.44 83.51 83.57 8SB10 80.46 79.20 79.34 79.48 7

Sub-basin Brush Seedingebrush

a b c a b c

SB7 84.32 84.47 84.62 83.88 84.11 84.35SB8 82.54 82.64 82.74 81.43 81.76 82.09SB9 82.88 83.02 83.17 82.32 82.53 82.74SB10 75.63 76.32 77.01 74.37 75.20 76.03

based on the accuracy of flood peak discharges (Qp), time to peakand flood volumes estimated and hydraulic model was assessedaccording to comparison of the calculated and observed floodingwater depths since damage analysis is based on these estimations.For this purpose, observed data sets including rainfall and dis-charges inmonitoring gauges of the study area (Fig. 6) were dividedinto two independent subsets, namely the calibration and valida-tion subsets. The calibration sets were used to adjust the parame-ters of hydrological and hydraulic models and the validation setswere used to verify the calibrated models, before applying into theoptimization process. The results of calibration and validationbased on some residual metrics are shown in Tables 4 and 5,respectively. All metrics have an ideal value equal to zero except thelast metric which the ideal value is one. According to the residualerrors, it is obvious that both models have suitable performance in

rchards Seeding- orchards

b c a b c

6.40 76.85 77.30 75.71 76.29 76.876.60 77.32 78.04 e e e

6.49 77.02 77.55 76.17 76.76 77.348.94 79.10 79.25 78.15 78.44 78.747.16 77.71 78.25 75.43 76.23 77.027.70 78.35 78.99 77.03 77.80 78.564.39 84.93 85.48 83.94 84.58 85.211.26 81.84 82.42 80.15 80.96 81.781.47 82.20 82.93 80.91 81.71 82.508.87 79.11 79.34 77.62 77.99 78.36

Brusheorchards Seedingebrusheorchards

a b c a b c

83.53 84.22 84.92 83.08 83.87 84.6580.62 81.29 81.97 79.50 80.41 81.3280.34 81.23 82.11 79.79 80.73 81.6874.04 74.97 75.89 72.78 73.85 74.91

Table 6Multi-objective evolutionary algorithm (MOEA) suite test functions.

MOP Definition Constraints

FON Fonseca (1995) F ¼ ðf1ðxÞ; f2ðxÞÞ; where

f1ð x!Þ ¼ 1� exp

�Pn

i¼1

�xi � 1ffiffiffi

np�2!;

f2ð x!Þ ¼ 1� exp

�Pn

i¼1

�xi þ 1ffiffiffi

np�2!

�4 � xi � 4; i ¼ 1;2; 3

KUR Kursawe (1991) F ¼ ðf1ðxÞ; f2ðxÞÞ; where

f1ð x!Þ ¼Pn�1i¼1 ð�10eð�0:2Þ*

ffiffiffiffiffiffiffiffiffiffiffiffiffix2i þx2iþ1

pÞ;

f2ð x!Þ ¼Pni¼1ðjxija þ 5 sinðxiÞbÞ

�5 � xi � 5; i ¼ 1;2; 3a ¼ 0:8;b ¼ 3

ZDT2 Zitzler et al. (2000) f1 x1ð Þ ¼ x1;

g x2; :::; xnð Þ ¼ 1þ 9n� 1

Xni¼2

xi;

f2 x2; :::; xnð Þ ¼ g 1� f1=ð gÞ2� �

0 � xi � 1; i ¼ 1;2; :::; 30q ¼ 4;a ¼ 2

ZDT3 Zitzler et al. (2000) f1 x1ð Þ ¼ x1;

g x2; :::; xnð Þ ¼ 1þ 9n� 1

Xni¼2

xi;

f2 x2; :::; xnð Þ ¼ g 1�ffiffiffiffiffiffiffiffiffif1=g

q� f1=ð g

� �sin 10pf1ð ÞÞ

0 � xi � 1; i ¼ 1;2; :::; 30

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e4440

flood modeling and thus can be applied for the next steps of theproposed methodology.

In order to decrease computational time of simulator engine(hydrological and hydraulic models), a two-layer artificial neuralnetwork was also trained and used for calculating flood damages.Input variables were considered as rainfalls of the rain gauges andcurve numbers of sub-basins in the watershed and the outputvariable was considered as flood damage. The ANN training datawas provided by performing the “simulator engine” (Fig. 2) for asignificant number of randomly generated rainfalls and curvenumbers (2000 data sets). After cross validation, the best archi-tecture was obtained as an ANN with 16 neurons in the hiddenlayer and one neuron in the output layer. The maximum and

Fig. 7. Pareto fronts produced by NSGA-II for the a) F

average of the absolute errors for test datawas obtained as 0.07 and0.003 respectively.

6.2. Optimization model

In multi-objective evolutionary computations, researchers haveused many different benchmark problems with known sets ofPareto-optimal solutions. Here, several well-known test functionswere taken from the specialized literature to evaluate the results ofNSGA-II algorithm. The considered test functions are FON (Fonseca,1995), KUR (Kursawe, 1991), ZDT1 and ZDT2 (Zitzler et al., 2000)multi-objective test problems. The objective functions and theirconstraints for these benchmarks are represented in Table 6. Fig. 7

ON, b) KUR, c) ZDT2 and d) ZDT3 test functions.

Fig. 9. Optimal Pareto front solutions in the final population.

Fig. 8. Joint PDF and CDF of extreme rainfall records in P1 (SB1) and P2 (SB3) rain gauges.

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e44 41

shows the graphical results produced by NSGA-II algorithm for testfunctions chosen. The global Pareto front of the problems is shownas a continuous line. As can be observed, NSGA-II algorithm givesthe true Pareto in comparison with global optimal solutions.

7. Results and discussion

Statistical analysis of extreme rainfall records for three raingauges of the watershed showed that data correlation of thirdgauge (in sub-basin SB7, Fig. 6) with that of two other gauges is low.So, random sampling from PDF of this gauge was carried outindependently. For this gauge, among different well known prob-ability distributions, log-normal distribution was selected as thebest distribution based on the goodness of fit criteria. For two othergauges, a joint distribution copula function was fitted. Best mar-ginal distributions for two rain gauge data obtained as the “Gen.Extreme Value” distribution function and the best copula function(among seven different copulas represented in Table 1) was knownas “Ali-Mikhail-Haq” copula based on the likelihood criterion withparameter q ¼ 2:308. The fitted copula is illustrated in Fig. 8.

A sensitivity analysis was carried out to achieve the number ofadequate multiple samples from the joint probability distributionin MC experiments. The number of 200 samples was deemedadequate for MCS in optimization framework.

LID technologies in the study area include a combination ofmeasures: Seeding, Brush, and Orchard in different sub-basins.

NSGA-II algorithm for the study area was set using a populationsize of 50 individuals and performed for a maximum 100 genera-tions where the search algorithm is converged. To keep the di-versity in the population, assuming an initial population size, 3 to 5times of chromosome length is a good selection. The crossover andmutation probabilities were set to 0.9 and 1/n, respectively where nis the population size.

The results obtained indicated that the model is convergedbefore attaining the final generation. Investment costs and

Table 7Optimal combination of land use change LIDs for flood risk mitigation in the watershed after clustering the Pareto front solutions.

Design No. Cost($/year)

EAD($/year)

EABa

($/year)B/Cb SB1 SB2 SB3 SB4 SB5 SB6 SB7 SB8 SB9 SB10

(1) 2397 899,032 32,769 13.7 Seeding e e SeedingOrchard

Seeding e e e e e

(2) 9892 865,585 66,216 6.7 SeedingOrchard

e SeedingOrchard

SeedingOrchard

Seeding SeedingOrchard

SeedingOrchard

Seeding SeedingOrchard

e

(3) 15,090 858,431 73,370 4.9 SeedingOrchard

Orchard SeedingOrchard

SeedingOrchard

Seeding SeedingOrchard

SeedingOrchard

SeedingOrchard

Orchard SeedingOrchard

(4) 19,362 847,288 84,513 4.4 SeedingOrchard

Orchard SeedingOrchard

SeedingOrchard

SeedingOrchard

SeedingOrchard

SeedingOrchard

SeedingOrchard

SeedingOrchard

Seeding

(5) 25,704 839,802 91,999 3.6 SeedingOrchard

Orchard SeedingOrchard

SeedingOrchard

SeedingOrchard

SeedingOrchard

SeedingOrchard Bush

SeedingOrchard Bush

SeedingOrchard

SeedingOrchard

a EAB ¼ Expected Annual Benefit (EAB(combination) ¼ EAD(No option) � EAD(combination)).b B/C ¼ benefit to cost ratio.

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e4442

expected flood damages (EADs) for the final generation (Paretofront) are shown in Fig. 9 for two a-cut levels of 1 and 0.05. a-cutlevel of 1 shows complete ignoring the epistemic uncertaintieswhile a-cut level of 0.05 means 0.95 percentage of uncertaintyranges (arising from uncertain CNs) is considered in damagecalculation (0.95 percent reliability). As shown in Fig. 9, the range ofvariability for the expected damages increases as the investmentcost level rises up. This originates from this fact that more expen-sive plans include more land use change strategies and involvingmore uncertain curve numbers for damage calculation, leading tohigher range of uncertainty for expected damages or benefits. Eachcost level represents an LID plan (including different land usechange strategies) with its variability ranges.

Selection of the most practical solution among the optimalPareto set of solutions is a challenging issue. When the number ofoptimal solutions is high, different clustering algorithms may beused to decrease the number of solutions into the smaller wellrepresentative solutions for further analysis. Multi-criteria decisionmaking (MCDM) techniques can then help to identify desiredportfolio among the remaining options or alternatives throughanalyzing multiple criteria by which the strengths andweaknesses of various adaptation options could be evaluated.Malekmohammadi et al. (2009) has already followed such anapproach for ranking the optimal solutions obtained by NSGA-II foroperation of a cascade system of reservoirs. Among others, Zagonariand Rossi (2013) also used MCDM approach for scoring the com-binations of floodmitigation and recovery options. MCDM providesan opportunity for further incorporation of decision makers' pref-erences in choosing the final plan. Nevertheless, developing anMCDM approach for decision making analysis is beyond the scopesof this study and may be considered in future researches. Here, five

Fig. 10. The variability range of expected benefits for selected optimal land use changestrategies.

plans with different investment costs are selected from the Paretofront solutions based on k-means clustering method for moreconsideration.

Table 7 shows the chosen optimal plans and their cost-benefitanalysis. As can be seen, the proposed approach has identifiedthe best land use change strategies at different sub-basins. Theconcentration of land use “Bush” is very low in the optimal com-binations due to its low performance upon flood damage reductionand the option “Seeding” is the best measure, emerged in the mostof optimal combinations. All optimum combinations have benefitto cost ratio greater than 1 and are economically justifiable. High B/C ratio for land use change LIDs is one of the advantages of thesestrategies for flood risk mitigation in comparison with structuralmeasures.

Fig. 10 illustrates the variability range of expected benefitsarising from the uncertain CNs for the selected plans (representedin Table 7). The expected benefits are variable in the range of 31, 32,34, 36 and 34% for plans 1, 2, 3, 4 and 5, respectively. This infor-mation provides valuable insights to decision makers about therobustness of different optimal designs and the degree of reliabilityof the estimated benefits. For selecting the final optimum plan ofland uses including orchard, brush, and seeding, decision makerscan select one of the optimal Pareto front solutions based on theirdecision criteria such as investment level, degree of reliability andsocial constraints.

The effects of plan No. 5 (high LID plan in Table 7) on reducingthe peak of typical floods with return periods of 5yr and 25yr inoutlet of the watershed are shown in Fig. 11 and 12, respectively. Ascan be seen in Fig. 11, optimal allocation of land uses reduces theflood peak at the range of 18e35% for plan No. 5 while for floodwith return period of 25 years; this ratio was obtained as 12e25%(Fig. 12). As the results are shown, it is obvious that LIDs are more

Fig. 11. Flood hydrographs in outlet of the watershed (return period: 5 year).

Fig. 12. Flood hydrographs in outlet of the watershed (return period: 25year).

J. Yazdi, S.A.A. Salehi Neyshabouri / Environmental Modelling & Software 60 (2014) 31e44 43

effective for floods with lower magnitudes and higher frequencies.Because of the significant effects of low and frequent floods on theamount of stream-bank erosion, land use change LIDs are alsoimportant from the viewpoint of soil conservation.

8. Concluding remarks

This paper represents a fuzzy probabilistic approach to mitigateflood risk sustainably onwatershed scale handling randomness andfuzziness simultaneously for the land use change plans. Multivar-iate joint distribution of observed extreme rainfalls was con-structed via copula functions considering temporal and spatialinherent flood uncertainties within a Monte Carlo framework.Epistemic uncertainties of curve numbers were also characterizedthrough fuzzy membership functions and a-cut concept to propa-gate curve number intervals to those of flood damages. Both ap-proaches in a unified framework was successfully employed inconjunction with NSGA-II optimization model to determine cost-effective land use change (LID) strategies including orchard, brushand seeding measures in different part of Kan watershed. Obtainedresults represents the optimal location of land use change strate-gies, their B/C ratio, expected damages and benefits as well as thevariability range of the estimated parameters for optimal solutions.More expensive optimal plans have higher expected benefits withlarger variability range, but lower B/C ratios. This informationprovides valuable insights to decision makers about the robustnessof different optimal plans and the degree of reliability of the esti-mated benefits or damages.

While the land use change strategies efficiently mitigatestream-bank erosion and the risk of low floods with high fre-quency, their combinations with structural measures like deten-tion dams is a better strategy for mitigating wider variety offloods with low and high frequencies. Although the proposedapproach here was focused on land use change strategies, themethodology is general enough and thus it can be also applied todetermine best combination of soft and hard flood mitigationstrategies.

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