S.S. Ahmad1and S.P. Simonovic2
1 Delcan Corporation, Ottawa, ON, Canada
2 Department of Civil and Environmental Engineering, The University of Western Ontario, London, ON, Canada
Correspondence
Dr. Shohan S. Ahmad, Delcan Corporation, Ottawa, ON, Canada K1J 7T2
Email: s.ahmad@delcan.com
DOI:10.1111/j.1753-318X.2011.01090.x
Key words
Flooding; fuzzy set; reliability; risk; space; three- dimensional fuzzy set; time; uncertainty.
Abstract
This paper presents a methodology for assessing spatial and temporal uncertainty associated with flood risk management. Traditional modelling approaches focus on either temporal or spatial variability, but not both. There is a need to understand the dynamic characteristics of flood risk and its spatial variability.
The traditional two-dimensional (2D) fuzzy set, with one dimension for the universe of discourse and the other dimension for its membership degree, is not sufficient to handle both, spatial and temporal variation of flood risk. The theoretical foundation of this study is based on the development of a three- dimensional (3D) fuzzy set that includes flood risk variability in space and time.
The proposed methodology extends the acceptance level of partial flood damage concept to a 3D representation and allows capturing change of decision makers’
preferences in time and space. The main objective of the paper is to present an original methodology for flood risk management that is capable of (a) addressing uncertainty caused by spatial and temporal variability and ambiguity; (b) integrating objective and subjective risks; and (c) assisting flood management decision making based on a more detailed understanding of temporal and spatial variability of risk. Presented methodology is illustrated using the Red River flood of 1997 (Manitoba, Canada) as a case study.
Introduction
Flood management under uncertainty
Natural hazards are unavoidable. Among natural hazards, floods are the most frequent, widespread, and with the most severe consequences. Floods destroy lives and cause damage to land, property, and infrastructure. To ensure the sustain- ability of a community, flood risk identification should be one of the most important steps in the design, planning, and management of flood protection measures. Ganoulis (1994) argues that engineering risk and reliability analyses provide a general methodology for the quantification of uncertainty, and as a result, should be used to assess the safety of an engineering system. Risk assessment is an important com- ponent of sustainable flood management, and it is gaining more attention with population growth and climate change.
There is a scientific consensus that climate change is causing (i) higher average temperature; (ii) sea-level rise; (iii) change in precipitation patterns; and (iv) change in frequency and severity of extreme hydrological conditions – floods and droughts. Population growth and urbanization affect the land use and increase the pressure on flood-prone areas.
There are two general types of measures for management of floods: (a) structural measures; and (b) nonstructural measures (Simonovic, 1999 among others). The most com- mon structural measures used today are (i) levees or flood walls; (ii) diversion structures; (iii) channel modifications;
and (iv) flood control reservoirs. In spite of the application of many structural measures, the flood damage is still on the rise. Therefore, the structural measures are being combined with the nonstructural measures, such as flood zoning, flood warning, waterproofing, and flood insurance. Levy and Hall (2005) introduced the concept of ‘living with flood’, which requires a high public awareness of actual flood risks. The quantification of all uncertainties and the representation of flood risk in time and space are becoming elements of contemporary flood risk management, which may help to reduce both human and material damage caused by floods.
Various uncertainties present in flood risk management originate from (1) natural variability and (2) knowledge uncertainty (Simonovic, 1997; National Research Council, 2000). Natural variability is further divided into (i) tempor- al variability, (ii) spatial variability, and (iii) individual heterogeneity. Spatial and temporal variability of the main hydrologic variables (precipitation, flow, evaporation, etc.)
are of the high significance in flood risk management.
Knowledge uncertainty reflects our limited ability to repre- sent real world phenomena with mathematical models. The uncertainties in (i) model formulation, (ii) parameter estimation, and (iii) use of models in decision making have to be considered in flood risk management.
A major difficulty in flood risk management relates to an inadequate distinction between three fundamental concepts of probability and risk (Slovic, 2000): (i)objective risk(real, physical), Ro and objective probability, po, which is the property of real physical systems; (ii)subjective risk,Rsand subjective probability,ps. Probability is defined here as the degree of belief in a statement.Rsandpsare not properties of the physical systems under consideration (but may be some function ofRoandpo); and (iii)perceived risk,Rp, which is related to an individual’s feeling of fear in the face of an undesirable event and is not a property of the physical systems. Perceived risk may be a function ofRo,po,Rs, and ps. It is very common that many characteristics of subjective risk are believed to be valid factors for analysing objective risk (Simonovic, 2002).
Probabilistic flood risk analysis effectively deals with objective risks arising from the uncertainty associated with available data on, for example, precipitation, stream flow, water quality, etc. Data insufficiency and inaccuracy represent major problems in probabilistic flood risk analysis. A lack of physical data (river cross-sections, discharge estimation, flow velocity measurements, precipitation, soil data, etc.) can result in flood prediction error (Nirupama and Simonovic, 2002). In addition to the physical information, flood risk management requires economic data: depth–damage rela- tionship; and value of the exposed infrastructure (National Research Council, 2000). Uncertainty associated with eco- nomic information also plays an important role in planning, design, and management of flood protection measures and operational strategies for flood management infrastructure.
Presently, very limited attention is being given to the assessment of subjective and perceived flood risks. They are the consequence of value systems, experience, and other social factors. Akter and Simonovic (2005) dealt with some social aspects of flood risk management. In conclusion, even if risk is properly quantified and the costs of risk are generated, communicating and establishing acceptable levels of risk remain difficult, particularly as they depend on public perceptions of tolerable risk.
The probabilistic (stochastic) risk management approach has been extensively used in practice. The expected annual flood loss analysis (Hydrologic Engineering Center, 1989) is used to address the hydrologic (flood-frequency analysis), hydraulic (rating-curve development), and economic un- certainties (stage-damage analysis) in flood risk manage- ment. The main deficiency of this approach is ignorance of subjective types of flood risk.
There are several approximate methods that can deal with the main deficiencies of probabilistic flood risk manage- ment. For example, in some cases it is useful to use the normal representation of nonnormal distributions as a practical alternative based on the central limit theory (Tung and Yen, 2005). However, the data requirements for estimat- ing the first two moments of the assumed normal distribu- tion are still very high. Another approach to avoid the problem of data insufficiency is the use of the subjective judgement of the decision maker to estimate the probability distribution of a random event, i.e. subjective probability (Vick, 2002). The third approach is the integration of judgement with the observed information using Baye’s theory (Ang and Tang, 1984). The problem with Bayesian reliability analysis is in the selection of prior distribution.
The choice of subjective probability distribution, in these two approaches, presents difficulties in the translation of the prior knowledge into meaningful probability distribution, especially for multiparameter problems (Press, 2003).
Therefore, accuracy of the derived distributions is strongly dependent on the realistic estimation of the decision- maker’s judgement (El-Baroudy and Simonovic, 2004).
The practice of flood risk management relies on the use of probabilistic approach. However, this approach fails to address the issue of subjective and perceived risks. We use the concept of risk to increase our understanding of various uncertainties and to develop our capacity to cope with the negative impacts of floods. The meaning of failure and risk is different for different people. Slovic (2000) points out that the probabilistic information processing by people does not take into account the proper probabilistic principles when judging the likelihood of a certain event. However, subjec- tive probability has merit in quantifying engineering judge- ment about the likelihood of the occurrence of an uncertain event, the existence of unknown conditions, or the con- fidence in the truth of a proposition (Vick, 2002).
This work presents an innovative framework for (a) integrating different perspectives of flood risk; (b) perform- ing flood risk assessments; and (c) developing flood risk management strategies for an entire river basin. Progress in flood risk prevention and flood disaster mitigation is based on the harm directly experienced by the people. This directly affects investments in flood risk prevention and mitigation measures, as well as the development of legisla- tion, standardization, and governmental regulations and control. The framework developed in this research provides support for the broad range of flood-related decision- making processes.
Fuzzy approach to flood risk management Fuzzy set theory was developed to address people’s judge- mental beliefs or subjective uncertainty, which result from a
lack of knowledge. In comparison with the probability theory, it has a certain degree of freedom with respect to aggregation operators, types of fuzzy sets (membership functions), etc., which allows for its adaptability to different contexts (Zimmermann, 1996). The application of the fuzzy set approach in the field of water resource management has grown over the last two decades (Simonovic, 2009, chapter 6). The fuzzy set approach is widely used in water resources multiobjective decision making under conditions of uncer- tainty (Kacprzyk and Nurmi, 1998; Bender and Simonovic, 2000; Despic and Simonovic, 2000; Borsuk et al., 2001;
Prodanovic and Simonovic, 2002; Simonovic and Nirupa- ma, 2005 among others). Pioneering applications of fuzzy set theory to spatial analyses can be found in Guesgen (2005); Shiet al. (2005); and Verstraeteet al. (2005) among others.
Among the more recent applications of fuzzy set theory to flood management is the work of Akter and Simonovic (2005). Their work is applied to flood management in the Red River Basin, Manitoba, Canada, and allowed for discrete ranking of flood management alternatives using multiple objectives and input from a large number of stakeholders.
They used fuzzy set and fuzzy logic techniques to success- fully represent the imprecise and vague information and to obtain preferences of a large number of stakeholders where subjective uncertainty plays an important role. Work pre- sented in this paper builds on the work of Akter and Simonovic (2005).
Ahmad and Simonovic (2007) developed a methodology to address the spatial variability of flood risk using fuzzy set theory. They integrated GIS technology with fuzzy flood risk estimation to develop spatial representation of flood risk.
The existing fuzzy set approaches to flood risk manage- ment, however, are not capable of addressing both the spatial and temporal variability of flood risk.
Research objectives
The main objectives of the research presented here are (a) to provide the methodology for flood risk assessment while taking into consideration the spatial and temporal variabil- ity of various objective and subjective uncertainties in flood management and (b) to provide a methodology to present the flood risk as a function of time and location in the floodplain. This research uses three fuzzy perfor- mance indices (El-Baroudy and Simonovic, 2004): (1) combined reliability–vulnerability index, (2) robustness index, and (3) resiliency index for spatial and temporal reliability analysis of riverine floods. This work extends the methodology for spatial flood risk analysis using the fuzzy set approach developed by Ahmad and Simonovic (2007).
The methodology is not limited by the shape of the membership function in any way. The shape of the member-
ship function that best represents the flood damage should be selected on the basis of available damage information and stakeholder’s knowledge of the system. Despic and Simono- vic (2000) provide a review of methods for developing an appropriate membership function for flooding that com- bines available data, expert opinion, and stakeholder preferences. In this work, triangular and trapezoidal membership functions are subjectively selected and used to illustrate the methodology and to test the sensitivity of the fuzzy performance indices to the shape of a membership function.
Paper organization
The following section of the paper describes the theoretical background for the calculation of fuzzy performance indices and their ability to capture the spatial and temporal varia- bility of various sources of uncertainty in flood management.
The next section presents an application of this methodology to the case study of the Red River flood (Manitoba, Canada) of 1997. The paper ends with conclusions.
A new methodology for spatial and temporal analysis of flood risk
The new methodology presented here uses three fuzzy performance indices for flood risk analysis (El-Baroudy and Simonovic, 2004): (i) a combined reliability–vulnerability index, (ii) a robustness index, and (iii) a resiliency index.
The calculation of reliability indices depends on the exact definition of the unsatisfactory state of a system. Engineer- ing reliability analysis uses load and resistance to define the state of a system. The failure state occurs when resistance falls below the load. In flood management, for example, load can be represented by the floodwater level and the degree of resistance by the elevation of the embankment used to protect an area from flooding. The margin of safety (the difference between load and resistance)M(D) = 0.0, or safety factor (the ratio between load and resistance) Y(D) = 1.0, shown in Figure 1 by the dashed horizontal line, is used to define the partial system failure. The area below full line in
Region of no flood damage
Region of complete flood damage
Region of partial flood damage
Time M(D)<0.0
or Θ(D)<1.0 M(D)=0.0 or Θ(D)=1.0
Flood damage -state
Figure 1 Different perception of failure.
Figure 1 is the area of complete failure. The area above the dashed line is the area of complete safety. That leaves the area between these two lines as the area of partial system failure.
Accuracy in measurement of floodwater levels may vary significantly, as well as the accuracy of the embankment.
Therefore, the prediction of failure state is subject to uncertainty. The methodology developed in this work is based on the concept of partial failure first introduced by El- Baroudy and Simonovic (2004). Partial failure is defined by the introduction of subjective levels of partial flood damage.
The boundary of the acceptance level of partial failure region is ambiguous and varies with time and location, and also from one stakeholder to the other, depending on their perception of risk.
The main input for the methodology presented in this paper includes: (a) the system state membership func- tion; and (b) the predefined acceptance level of partial failure membership function. In flood risk management, for example, the system state membership function can be used to represent uncertainty in floodwater elevation, while the acceptance level of partial failure membership function can be used to define the region of partial failure – the region between the complete failure (i.e. when floodwater levels result in complete failure of the flood protection embankment and a complete inundation of the protected area) and the acceptable failure level (i.e. flood- water level exceeds the height of the flood protection embankment without causing complete failure, therefore resulting in a partial inundation of the protected area).
Because the fuzzy sets are capable of representing the notion of imprecision better than the ordinary sets, the fuzzy set theory has been used in this work to describe various aspects of risk.
Ahmad and Simonovic (2007) introduced fuzzy flood risk management approach with explicit consideration of change in fuzzy risk of flooding with location in the flood plain (spatial variability). This work is expanding their approach to address the change in fuzzy risk of flooding with time (temporal variability).
Definition of the acceptance level of partial flood damage
The shape of the acceptance level of partial flood damage membership function changes with time and location. The spatial and temporal variability of the acceptance level of partial flood damage is addressed in this work by introdu- cing time as a dimension to the two dimensional (2D) representation of the acceptance level of partial flood damage (Figure 2) (Ahmad and Simonovic, 2007). The acceptance level of partial flood damage is now represented as a fuzzy membership function,M~ijðDijÞbased on the flood
damage valueDijin time and space (Figure 3).
M~ijðDijÞ ¼ 1 jkðDijÞ 0
ifDij < Dij1
ifDij2 ½Dij1;Dij2 ifDij > Dij2
8>
><
>>
:
9>
>=
>>
;
ð1Þ
whereDijis the flood damage fori-th time step at locationj M~ijðDijÞ is the fuzzy membership function of margin of safety fori-th time step at locationj;Dij1andDij2are lower and upper bounds of the acceptance level of partial flood damage fori-th time step at locationj;jk(Dij) are functional relationships representing the subjective view of the partial risk for i-th time step at location j; k( = 1, 2, 3) is the indicator of possible type of the acceptance level of partial flood damage membership function; k= 1 denotes the conservative acceptance level of partial flood damage mem- bership function fori-th time step wherei2[i0, i1] corre- sponds to the rising limb of the stage/discharge hydrograph;
k= 2 denotes the neutral acceptance level of partial flood damage membership function fori-th time step wherei2[i1, i2] corresponds to the peak part of the stage/discharge hydrograph; and k= 3 denotes risky acceptance level of
D
D Flood damage, D
1
0
Region of no flood damage
Region of partial
flood damage Region of complete flood damage Fuzzy membership function of
acceptance level of partial flood damage ,M~ (D)
Membership value, µ (D )
Figure 2Two-dimensional fuzzy representation of acceptance level of partial flood damage.
Membership level, μ
Flood damage, D
Time step 0.2
0.4 0.6 0.8 1
0 μ
i i
i 0
Risky (k =3) Neutral (k =2) Conservative (k =1)
k =1 k =2 k =3
i
Figure 3Three-dimensional fuzzy representation of acceptance level of partial flood damage in time and space.
partial flood damage membership function for i-th time step wherei2[i2,i3] corresponds to the recession limb of the stage/discharge hydrograph.
The shape of the membership functionM~ijðDijÞis based on the decision maker’s perception of flood risk and damage levels and the shape of stage and/or discharge hydrograph. To reflect the subjectivity of the decision maker, different time steps of the hydrograph are assigned to different shapes: the rising limb with a conservative acceptance level; the crest with a neutral acceptance level; and the recession limb with a risky acceptance level of partial flood damage membership functions. The perception of flood risk and damage levels also varies with the location in the floodplain that may be under various land-use patterns, i.e. residential, agricultural, etc. Areas near the river that are highly prone to significant flood damage, compared with the areas further away from the river, may be designated with different shapes of mem- bership functionM~ijðDijÞthat will be based on the decision maker’s perception of flood risk and damage level. In this way, the methodology is capable of developing the shape of the membership function M~ijðDijÞ according to variations with time and location. This can be carried out by assigning the value to the lower bound,Dij1and/or the upper bound, Dij2of the acceptance level of partial flood damage.
If the value of flood damage exceedsDij2, then the region is exposed to complete damage (Figure 3). In this case, the membership functionM~ijðDijÞvalue is equal to zero. If the value of flood damage is belowDij2but exceedsDij1, then the region is exposed to partial flood damage. The membership function, M~ijðDijÞ of the acceptance level of partial flood damage attains its maximum value of one if the value of flood damage is belowDij1.
The reliability measure (LRij) of the acceptance level of partial flood damage calculated fori-th time step at location j is calculated according to El-Baroudy and Simonovic (2004), which is as follows:
LRij¼Dij1Dij2
Dij2Dij1
whereDij2 > Dij1 ð2Þ Using an almost crisp definition of the acceptance level of partial failure region by selecting close values forDij1and Dij2will result in a very highLRijvalue. The subjectivity of decision makers will be captured by the selection ofDij1and Dij2 and by the selection of the shape of the membership functionM~ijðDijÞ.
Fuzzy flood damage in time and space
A fuzzy approach to flood risk assessment generally invol- ves a 2D fuzzy set. Liet al. (2007) define the 2D fuzzy set having one dimension (1D) for the universe of discourse of variable and the other dimension for its membership degree.
The 2D fuzzy set consists of a 1D fuzzy membership
function. The 2D fuzzy set is appropriate for represent- ing either spatial or temporal variability of flood damage, but not both of them. A new approach is proposed here that will address spatial and temporal uncertainty in flood management together. Representation of fuzzy flood damage in time and space is considered in the following four steps.
2D fuzzy set of flood damage in time
Determining the variation of flood damage with time is performed by considering the uncertainty related to chan- ging flow. Uncertainty related to properties of spatial variability is not considered in determining flood damage in time. Agricultural damage is determined as a function of flood recession date. Flood damage in residential areas is determined using a depth–damage relationship. Uncertainty in floodwater level is the result of our inability to accurately measure, calculate, or estimate the flow value. Because a probabilistic approach usually fails to address factors of uncertainty related to human error, subjectivity, and the lack of historical records and data, a fuzzy set approach is used.
Uncertainty of flood damage in time is described in this study using a 2D fuzzy set with 1D representing the value of flood damage for i-th time step,Di and the other for its membership degree (Figure 4). The definition of this 2D fuzzy set of flood damage in time is given by
A¼ fðDi;S~iðDiÞj8Di2Dg;
0S~iðDiÞ 1 ð3Þ
whereAdenotes the 2D fuzzy set of flood damage in time;
Di is the flood damage for i-th time step in the universe of discourseD; andS~iðDiÞdenotes the membership degree of the 2D fuzzy set.
In the case of a triangular membership function, the fuzzy flood damage in time functionS~iðDiÞcan be defined fori-th
Membership value, µ (D )
Flood damage in time, D 1
D 0
D Fuzzy
membership function,
) D ( S
D Figure 4 Fuzzy flood damage in time.
time step as
S~iðDiÞ ¼
0 ifDi < DiMin
DiDiMin
DiMeanDiMin
ifDi2 ½DiMin;DiMean DiMaxDi
DiMaxDiMean
ifDi2 ½DiMean;DiMax 0 ifDi > DiMax
8>
>>
>>
><
>>
>>
>>
:
9>
>>
>>
>=
>>
>>
>>
; ð4Þ
whereS~iðDiÞis the flood damage in time membership function fori-th time step;DiMeanis the modal value of flood damage in time fori-th time step; andDiMin,DiMaxare the lower and the upper bounds of flood damage in time fori-th time step.
2D fuzzy set of flood damage in space
Flood damage in space is determined by considering only the uncertainty related to properties that change with the location in the floodplain. Uncertainty resulting from the change in properties with time is not considered in deter- mining flood damage in space. Agricultural damage de- pends, for example, on the spatial distribution of different types of crops. Methodology developed in this work con- siders average crop damage as a property of location (space).
In the case of residential areas, infrastructure/property damage and depth–damage relationships are also considered to be location dependent. These space-dependent properties are subjected to uncertainty due to lack of data, human error, etc. Therefore, the fuzzy set approach is used to capture that uncertainty. Variability of uncertainty of flood damage in space is described in this study using a 2D fuzzy set with 1D representing the value of flood damage at location j, Dj, and the other for its membership degree (Figure 5). The definition of this 2D fuzzy set of flood damage in space is given by
B¼ fðDj;S~jðDjÞj8Dj 2Dg;
0S~jðDjÞ 1 ð5Þ
whereBdenotes 2D fuzzy set of flood damage in space;Djis the flood damage in space at locationjin the universe of discourseD; andS~jðDjÞ denotes the membership degree of the 2D fuzzy set.
In the case of a triangular membership function shape, the fuzzy flood damage in space function S~jðDjÞ can be defined for locationjas
S~jðDjÞ ¼
0 ifDj < DjMin
DjDjMin
DjMeanDjMin ifDj2 ½DjMin;DjMean DjMaxDj
DjMaxDjMean
ifDj2 ½DjMean;DjMax 0 ifDj > DjMax
8>
>>
>>
>>
<
>>
>>
>>
>:
9>
>>
>>
>>
=
>>
>>
>>
>; ð6Þ
where S~jðDjÞ is the flood damage in space membership function for locationj; DjMeanis the modal value of flood damage for locationj; andDjMin,DjMaxare the lower and the upper bounds of flood damage for locationj.
3D fuzzy set of flood damage in time and space The 2D fuzzy sets: (i) fuzzy flood damage in time and (ii) fuzzy flood damage in space developed in this study are used to capture various sources of uncertainty in time and space, respectively. However, neither (i) nor (ii) is capable of representing the combined uncertainty of the flood risk that is both spatial and temporal in nature. Therefore, a new three-dimensional (3D) fuzzy set is developed in this paper to fully address the spatial and temporal uncertainty and subjectivity associated with the flood damage. Li et al.
(2007) define a 3D fuzzy set as a fuzzy set with an extra dimension for universe of discourse of variable. Similarly, a 3D fuzzy set is developed in this paper (Figure 6) with the first dimension used for the flood damage in timeDi, the second dimension for the flood damage in spaceDj, and the third dimension for the membership degree. This 3D fuzzy
Membership value, µ (D )
Flood damage in space, D 1
D 0
D Fuzzy
membership function,
) D ( S~
D Figure 5 Fuzzy flood damage in space.
Membership level, μ
Flood damage in space, D
Flood damage in time, D 0.2
0.4 0.6 0.8 1
0
D
D D D
D
D
Figure 6Three-dimensional joint fuzzy flood damage membership function in time and space.
set consists of a 2D fuzzy membership function. The joint membership function of the flood damage in time and space provides a visualization of the uncertainties in flood damage at any membership level. This 3D fuzzy set represents uncertainty in flood damage at every spatial location and at every time step. The definition of the 3D fuzzy set is given as follows:
V¼ fðDi;DjÞ;mVðDi;DjÞj8Di2D;Dj 2Dg;
0mVðDi;DjÞ 1 ð7Þ
whereVdenotes 3D fuzzy set of flood damage in time and space;Diis the flood damage in time fori-th time step in the universe of discourseD;Djis the flood damage in space at location j in the universe of discourse D; and mVðDi;DjÞ denotes membership degree of the 3D fuzzy set.
Dimension reduction
The 3D fuzzy set is developed in order to express the combined spatial and temporal variability of uncertainty related to flood management. However, the complexity of the 3D spatial and temporal components of the fuzzy set may cause difficulty in the risk analysis process. The reliability assessment, in this study (described later in ‘Fuzzy flood compatibility’) is based on the comparative analysis of two membership functions: (a) flood damage membership function in time and space; and (b) the predefined accep- tance level of partial flood damage membership function in time and space. Fori-th time step, the predefined partial level flood damage results in a 2D fuzzy set. In order to compare with 2D fuzzy set of the predefined acceptance level of partial flood damage, the 3D fuzzy set of flood damage in time and space needs to be represented by a 2D fuzzy set.
In practice, the 3D fuzzy set used to calculate flood damage in time and space can be approximated by performing a dimension reduction operation, which consists of con- structing a 2D fuzzy set for flood damage at a particular location at a particular point in time. The dimension reduction operation uses the centroid operation in the following equation to determine the centre of gravity (Figure 7) of the fuzzy flood damage in time:
DiG ¼ RDiMax
DiMin DiS~iðDiÞdD RDiMax
DiMin S~iðDiÞdD
" #
ð8Þ where DiG denotes centre of gravity of the fuzzy flood damage in time.
From the 3D fuzzy set, a new 2D fuzzy set is generated by calculating fuzzy flood damage in space at the centre of gravity of fuzzy flood damage in time,DiG (Figure 8). The new 2D fuzzy set is an approximate representation of the 3D fuzzy set of flood damage in time and space. This new 2D fuzzy set generates a trapezoidal flood damage membership
function (Figure 9) defined in the following equation, which represents flood damage uncertainty in time and space.
S~ijðDijÞ ¼
0 ifDij < DijMin
DijDijMin
DijMode1DijMin
ifDij2 ½DijMin;DijMode1
m ifDij2 ½DijMode1;DijMode2
DijMaxDij
DijMaxDijMode2 ifDij2 ½DijMode2;DijMax
0 ifDij > DijMax
8>
>>
>>
>>
>>
><
>>
>>
>>
>>
>>
:
9>
>>
>>
>>
>>
>=
>>
>>
>>
>>
>>
; ð9Þ whereS~ijðDijÞis the flood damage membership function for i-th time step at location j; DijMode1 and DijMode2 are the modal values fori-th time step at locationj; andDijMinand
DijMaxare the lower and the upper bounds of flood damage
fori-th time step at locationj.
Total flood damage
The methodology presented here proposes a simplified equation to capture the dynamic characteristics of flood risk
Membership value, µ (D )
Center of gravity of fuzzy flood damage in time, D
Flood damage in time, D 1
D 0
D Fuzzy
membership function,
) D ( S~
D D
Figure 7 Centre of gravity of the fuzzy flood damage in time.
Membership level, μ
Flood damage in space, D
Flood damage in time, D 0.2
0.4 0.6 0.8 1
0
D
D D D
D
D D
μ
Figure 8 Fuzzy flood damage membership function in space atDiG.
and its spatial variability as follows:
DT¼
Z T
i¼0
Z b x¼a
Z o
y¼u
Dðx;y;iÞdxdy
" #
di ð10Þ
where DT denotes the total flood damage; x denotes the xcoordinate of the centre of a grid cell at locationj,x[a,b];
y denotes the y coordinate of the centre of a grid cell at locationj,y[u,o];idenotes the time step,i[0,T]; and (D (x,y,i) denotes a function of flood damage with respect to space (xcoordinate,ycoordinate) and time (i).
This simplified representation of flood risk has the potential for increasing our understanding of its dynamic character in time and space.
Fuzzy flood compatibility
The basis for reliability assessment in this study is the comparative analysis of two membership functions: (a) flood damage membership function in time and space shown in Eqn (9); and (b) the predefined acceptance level of partial flood damage membership function in time and space shown in Eqn (1). The purpose of the comparative analysis is to capture the extent to which the two fuzzy sets match (Figure 10). According to Zimmermann (1996) and Simonovic (2009), the extent of overlap between the two membership functions, represented as a fraction of the total area of the flood damage membership function, illustrates more clearly the fuzzy compliance between the flood da- mage membership function and the acceptance level of partial flood damage membership function than does the fuzzy possibility and the fuzzy necessity measures.
The compliance of two fuzzy membership functions can be mathematically presented using the fuzzy compatibility measure (CM) (El-Baroudy and Simonovic, 2004):
CMij¼OAij
Aij
ð11Þ
where CMij is the fuzzy compatibility for i-th time step at location j; OAij is the overlap area for i-th time step at location j; and Aij is the total area under the flood damage membership function for i-th time step at locationj.
Verma and Knezevic (1996) state that an overlap in the area of high significance (area with high membership values) is preferable to an overlap in a low-significance area. Thus, the fuzzy compliance takes into account the weighted area approach, which modifies Eqn (11) into
CMij¼WOAij
WAij
ð12Þ where CMij is the compatibility measure for i-th time step at location j; WOAij is the weighted overlap area between the flood damage membership function and the acceptance level of partial flood damage membership func- tion fori-th time step at locationj; andWAijis the weighted area of the flood damage membership fori-th time step at locationj.
The calculation of the fuzzy compatibility measure is presented for the flood damage membership function S~ijðDijÞ, as shown in Figure 11. At any particular a-cut of width da, the corresponding left and right values of the
Figure 10 Overlap area between flood damage membership function and acceptance level of partial flood damage membership function in time and space.
Flood damage, D dα
D D D D D
D 1
0
Flood damage membership functionS(D )
Single segment of flood damage membership function )
D ( S~D)+d ( S~
µ Membership value
Figure 11 Weighted area calculation for the flood damage membership function in time and space.
Membership value, µ (D )
Flood damage in time and space, D 1
D 0
D Fuzzy
membership function,
) D ( S~ µ
D D
Figure 9 Fuzzy flood damage membership function in time and space.
universe of discourse are
Dijl1¼DijMinþ ðDijMaxDijMinÞS~ijaðDijl1Þ Dijl2¼DijMinþ ðDijMaxDijMinÞS~ijaðDijl2Þ Dijr1¼DijMinþ ðDijMaxDijMinÞS~ijaðDijr1Þ Dijr2¼DijMinþ ðDijMaxDijMinÞS~ijaðDijr2Þ
ð13Þ
whereS~ijaðDijÞis thea-cut of fuzzy flood damage member- ship function fori-th time step at locationj;Dijl1is the first left (lower) flood damage value fori-th time step at location j;Dijl2is the second left (upper) flood damage value fori-th time step at locationj;Dijr1is the first right (lower) flood damage value fori-th time step at locationj; andDijr2is the second right (upper) flood damage value fori-th time step at locationj.
The area of the incrementala-cut for i-th time step at locationjis calculated as follows:
dAij¼ðDijr1Dijl1Þ þ ðDijr2Dijl2Þ
2 da ð14Þ
The weight,wof this incremental area is the average value of the membership function
w¼S~ijaðDijl1Þ þS~ijaðDijl2Þ 2
or
w¼S~ijaðDijr1Þ þS~ijaðDijr2Þ 2
ð15Þ
As a result, the weighted area fori-th time step at location jequals
dAwij¼ ðDijr1Dijl1Þ þ ðDijr2Dijl2Þ
2 da
S~ijaðDijl1Þ þS~ijaðDijl2Þ 2
ð16Þ Integration of Eqn (16) over the entire domain ofa-cut values, i.e. from 0 tom, results in the weighted area of the flood damage membership function,WAijfori-th time step at locationj
WAij¼ Z
dAwij
¼ Z m
a¼0
ðDijr1Dijl1Þ þ ðDijr2Dijl2Þ 2da
S~ijaðDijl1Þ þS~ijaðDijl2Þ 2
da ð17Þ Similar calculations apply to the overlap area (Figure 10) between the flood damage membership function and the predefined acceptance level of partial flood damage mem- bership function. The weighted area of the overlap fori-th
time step at locationj,WOAij, is calculated for determining the fuzzy compliance measure.
Fuzzy risk indices -- combined
reliability--vulnerability index in time and space Fuzzy reliability and fuzzy compatibility of two input membership functions are used in the mathematical deriva- tion of the fuzzy combined reliability–vulnerability index in time and space. ‘Reliability and vulnerability are used to provide a complete description of system performance in the case of failure and to determine the magnitude of the failure event’ (Simonovic, 2009, p. 202). Fuzzy combined reliability–vulnerability index in time and space for flood risk assessment is calculated using the following:
REij¼f2K
CMij1;CMij2;. . .;CMijf
LRijMax
f2KLRij1;LRij2;. . .;LRijf ð18Þ where REij is the fuzzy combined reliability–vulnerability index for i-th time step at location j; LRijMaxis the fuzzy reliability of partial flood damage level corresponding to the maximum compatibility value fori-th time step at location j;LRijfis the fuzzy reliability of thef-th partial flood damage level for i-th time step at location j; CMijf is the fuzzy compatibility of thef-th of partial flood damage level for i-th time step at locationj; andK(f =1,2, . . . ,K) is the total number of the defined levels of partial flood damage.
A flow chart in Figure 12 shows the process adopted for the calculation of the fuzzy combined reliability–vulnerabil- ity index in time and space. This computation is implemen- ted in the GIS environment that assists in spatial analysis.
Computation of the fuzzy combined reliability–vulnerabil- ity index starts with the first i-th time step at location j.
Flood damage is determined fori-th time step at locationj.
The next step deals with the generation of a 2D fuzzy set of flood damage in time and a 2D fuzzy set of flood damage in space. To describe the overall spatial and temporal uncer- tainty, a 3D fuzzy set of flood damage in time and space is generated fori-th time step at locationj. Then, the dimen- sion reduction operation using Eqn (8) calculates the centre of gravityDiG of the fuzzy flood damage in time in order to compress the 3D fuzzy set into a 2D fuzzy set, thereby representing fuzzy flood damage in space forDiG. This new 2D fuzzy set generates a trapezoidal flood damage member- ship function that represents uncertainty in both time and space. The process then proceeds to the next j-th cell and follows the steps described above to generate fuzzy flood damage in time and space fori-th time at nextj-th grid cell.
Once all the grid cells have been taken into consideration, the programme generates fuzzy flood damage map fori-th time step. The lower bound, the modal value, and the upper
bound of the fuzzy flood damage in time and space are stored as GIS layers. The next step deals with the generation of the acceptance level of partial flood damage and the computation of the weighted overlap area to determine the compliance level (Figure 10). In order to illustrate the range of stakeholders’ preferences, three shapes of the acceptance level of partial flood damage membership are introduced in
this study that should in practice be obtained from the stakeholders. These levels are selected to capture the follow- ing: (i) conservative risk attitude (in our case, applied to the rising limb of the hydrograph); (ii) neutral risk attitude (applied to the crest of the hydrograph); and (iii) risky attitude (applied to the recession limb of the hydrograph).
The shape of the acceptance level of partial flood damage
Figure 12 Flow chart of fuzzy combined reliability–vulnerability index in space and time.
membership functions also differs with the land-use type. In our case, two land-use types are considered: residential and agricultural. Thus, the acceptance level of partial flood damage membership functions are developed to capture full variability of damage in space and time. The range of stakeholders’ preferences is considered with f number of different partial level of flood damage membership func- tions. The computation of the weighted overlap area forf membership functions determines the compliance level.
Then, the fuzzy compatibility is calculated using Eqn (12).
The acceptance levels of partial flood damage result in raster GIS maps with corresponding fuzzy compatibility values.
Next, Eqn (18) is used to develop a single map containing the fuzzy combined reliability–vulnerability index for the i-th time step.
Once the calculation is finished for thei-th time step, the process proceeds to the next time step. The process follows the same approach described above to generate a fuzzy combined reliability–vulnerability index for every grid cell over the time horizon of calculation. Once the calculations for all the time steps are carried out, the Eqn (10) is used to calculate total flood damage in space and time.
Fuzzy risk indices -- robustness index in time and space
The adaptability of the system to the change from one to another acceptance level of partial flood damage is repre- sented in space and time. Two maps containing compat- ibility measure values are used as inputs for the following calculation:
ROij¼ 1 CMij1CMij2
ð19Þ whereROijis the fuzzy robustness index fori-th time step at location j; CMij1 is the compatibility measure for the acceptance level of partial flood damage fori-th time step at locationj; andCMij2is the compatibility measure after the change in the acceptance level of partial flood damage for i-th time step at locationj.
The fuzzy robustness index in time and space is calculated as the inverse of the difference in compatibility values between the two acceptance levels of partial flood damage for each location and over the computation time horizon.
This inverse relation implies that the higher the change in compatibility, the lower the fuzzy robustness value.
Fuzzy risk indices -- resiliency index in time and space
The resiliency index measures the ability of a system to recover from the failure state. A resilient community is able
to recover quickly from a flood disaster. After a disaster, postflood recovery involves restoring all systems to normal or near normal condition. As a measure of the ability to recover, the time necessary to recover from flood is deter- mined on the basis of water drainage, damage assessment, provision of assistance to flood victims, time for rebuilding or repairing, and return to normal life (Morris–Oswald and Simonovic, 1997, 1999).
Failures of an engineering systems (in our case, a flood protection systems) may be of different nature, and for each type of failure, the system might have a different recovery time. The time required to recover from the failure state can be represented as a fuzzy set in order to account for the uncertainty (imprecision) in its determina- tion. Based on local factors (such as land-use type, available resources to help flood victims, etc.), an appropriate shape membership function is derived for every location and at every time step. From a series of fuzzy membership func- tions developed for various types of failure and for different locations, the maximum recovery time is chosen to repre- sent the system’s recovery time (Kaufmann and Gupta, 1985):
T~ijðaÞ ¼
k¼K½tij11ðaÞ;tij12ðaÞ;. . .;tij1kðaÞ;
k¼K½tij21ðaÞ;tij22ðaÞ;. . .;tij2kðaÞ
ð20Þ whereT~ijðaÞis the system fuzzy maximum recovery time at a-cut for i-th time step at location j; tij1kðaÞ is the lower bound of thek-th recovery time ata-cut fori-th time step at location j; tij2kðaÞis the upper bound of thek-th recovery time ata-cut fori-th time step at locationj; andKis the total number of failure events.
The extent of flood damage to structures in residential and agricultural areas is a key factor in assessing the time required to recover from the flood damage. In most cases, high recovery cost corresponds to longer recovery time, and vice versa. Based upon this assumption, for illustration of the methodology a recovery time versus flood damage relationship is generated in this research for assessing the recovery time for residential and ring-diked communities at the postflood stage. For agricultural areas, flood recovery time is assessed based on the flood recession date, i.e. the date when floodwater has completely receded. In this research, the recession dates are obtained from 2D hydro- dynamic modelling. For illustration purposes, a triangular fuzzy membership function is assigned in space and time to represent uncertainty in flood recovery time. The triangular shape of the membership function conveys the notion that the values for minimum and maximum recovery time
(tijMin and tijMax) are concentrated around the modal
value of the recovery timetijMean, expressed mathematically
as follows (Figure 13):
T~ijðtijÞ ¼
0 iftij < tijMin
tijtijMin
tijMeantijMin iftij2 ½tijMin;tijMean tijMaxtij
tijMaxtijMean
iftij2 ½tijMean;tijMax
0 iftij > tijMax
8>
>>
>>
>>
<
>>
>>
>>
>:
9>
>>
>>
>>
=
>>
>>
>>
>; ð21Þ
where T~ijðtijÞ is the membership function of the flood recovery time fori-th time step at locationj;tijMeanis the modal value of the flood recovery time fori-th time step at locationj; andtijMin,tijMaxare the lower and upper bounds of the flood recovery time fori-th time step at locationj.
The inverse of the centre of gravity of the recovery time is used to represent the resiliency in time and space. The fuzzy resiliency index in time and space is calculated as follows:
RSij¼ CGij
1
¼ RtijMax
tijMin tijT~ijðtijÞdt RtijMax
tijMin
T~ijðtijÞdt 2
4
3 5
1
ð22Þ whereRSijis the spatial fuzzy resiliency index fori-th time step at locationj;CGijis the centre of gravity of the recovery time membership fori-th time step at locationj; andT~ijðtijÞ is the fuzzy recovery time membership function fori-th time step at locationj.
A case study
The Red River basin from the community of St Agathe to south of Winnipeg floodway in Manitoba, Canada, is used to illustrate the applicability of a spatio-temporal fuzzy risk analysis to flood management. The specific characteristics of the Red River basin are flat topography, frequent flooding, presence of flood control structures such as diversions, floodway, dikes, and reservoirs. Operation of major protec- tion works such as gates, floodway, and diversions provides
the necessary protection for the City of Winnipeg. Most of the communities, including St Agathe, south of the City are protected mainly by ring dikes.
2D hydrodynamic modelling
‘Flood of the Century’ data from 1997 are used in this case study. Mike21 (Danish Hydraulic Institute, 2008), a 2D hydrodynamic modelling tool, is used for modelling flows in the Red River basin. Hourly discharges at the Red River near St Agathe and hourly water levels below the Red River floodway are used as upstream and downstream boundary conditions, respectively. The results of the Mike21 are verified by comparing the extent of flooding with satellite images taken on 1 May 1997 (Figure 14), and also with the observed water levels at Red River near St Adolphe (Figure 15). The Mike21 model results were satisfactory for assessing the flooding of Red River in 1997 in the region from St Agathe to the Red River Floodway control structure (south border of the City of Winnipeg).
Input for fuzzy risk analysis of the Red River Flood of 1997
The methodology developed and presented above has been implemented in this case study for the flood risk analysis in time and space.
Agricultural damage
Agricultural damage is assessed in this study based on the delay to seed, and as a result, the reduction of the crop yields (International Joint Commission, 1997). Agricultural da- mage assessment is carried out by following these four steps.
Step one: determine the date of flood recession from flood stage hydrograph. The spatial and temporal variability of floodwater depth across the case study area is obtained from the 2D hydrodynamic modelling. Flood stage hydrographs are generated for every location in the case study area. They provide the information of flood recession date for every location. From the flood stage hydrograph, the first date of seeding and the expected yield for the crop are determined for i-th time step at location j. As floods recede, areas with higher elevation further from the river are exposed first and are ready for seeding before areas closer to the river, at lower elevations.
Step two: add additional time to flood recession date for field drying. Additional time, i.e. a drying period, is added to the date of flood recession to allow the agricultural land to completely dry before seeding. In this study, a 14-day drying time was used to account for one rainfall event during this period.
Recovery time, t 1
Membership value, µ (t )
t 0
t Center of gravity
Fuzzy recovery time membership function,
) t ( T~
t
Figure 13 Fuzzy membership function of recovery time.
Step three: determine percentage of average yield. The rela- tive yield is then determined from the graphical relation- ship (Figure 16) between the relative yield and seeding date (KGS, 2000).
Step four: assess agricultural damage. The agricultural damage is assessed using the corresponding value of the relative yield from Figure 16 (KGS, 2000) and is calculated as Dij¼100Yj
%AjPj ð23Þ
where Dij is the agricultural damage for i-th time step at locationj;Yjis the expected yield (function of seeding date) atj-th location (bushels/acre);Ajis the area (25 m5 m) of thej-th grid cell; andPjis the 3-year average crop price (US$/
bushel).
Residential damage
For flood damage in ring-dike communities, a depth–
damage function (KGS, 2000) is used to estimate the incremental damage to a town’s infrastructure. The depth–
damage function accounts for the dike closure costs, and infrastructure losses after the dikes are overtopped. Figure 17 shows the damages as a percentage of the total reported damage, primarily preemptive and postflood clean-up costs (KGS, 2000). For ring-dike communities, the total reported damage is equal to flood fighting costs. The initial damage level for the ring-diked communities, is assumed to be 5% of the total reported damage, which then rises linearly to 100%
when floodwater level reaches the top of the ring dike (Figure 17). Once the floodwater level exceeds the top level of the dike, the damage to infrastructure is shown by a vertical line at the right of the chart, which represents the
228 229 230 231 232 233 234 235 236
04-23-97 04-28-97 05-03-97 05-08-97 05-13-97 05-18-97 05-23-97 05-28-97 06-02-97 Date
Water Elevation (meter)
Observed 2D Model
Figure 15 Water elevation (m) at Red River near St Adolphe.
0 20 40 60 80 100 120
01-May 07-May 13-May 19-May 25-May 31-May 06-Jun 12-Jun
Seeding date
Percent of average Yield
Figure 16 Graphical relationship of percentage of average yield and seeding date (KGS, 2000).
Figure 14 Satellite image (left) and simulated flooded area (right) on 1 May 1997.
total potential of infrastructure damage as a percentage of the total damage reported. The maximum flood damage of ring-diked communities remains the same, even after the floodwaters have receded. Information such as the top level of the dike, level of incipient flooding, the initial flood damage level, the area within the ring-dike, total reported flood damage, and total potential infrastructure damage for each ring-dike community, is required for the damage assessment. The output of the damage analyses is repre- sented by maps showing flood damage in space and time.
Total reported damage forj-th location within a ring-dike community
The total damage Dijfor i-th time step at locationjis determined using the following logical rules: IF Water ElevationoBase Level of Incipient Flooding THEN Dij= 0
ELSEIF Base Level of Incipient Flooding Water Eleva- tion Top Level of Dike
Spatial and temporal fuzzy risk analysis of the Red River flood of 1997
The spatial and temporal fuzzy risk analysis of the Red River flood of 1997 is performed using Eqns (18), (19), and (22).
The fuzzy flood damage membership functions for agricul- tural land and residential land (including ring-diked com-
munities) are developed based on the flood damage data.
The compliance of the flood damage membership function with different acceptance level of partial flood damage membership functions is assessed for i-th time step at location j. The maximum value of compatibility is com- bined fori-th time step at locationjto determine the fuzzy combined reliability–vulnerability index in time and space.
The inverse of the difference in compatibility values between two acceptance levels of partial flood damage represents the fuzzy robustness index in time and space.
The fuzzy robustness index measures the adaptability to change in the acceptance level of partial flood damage.
The time to recover from flood damage is determined using a flood recession time and recovery time–damage relationship. Uncertainty in the value of recovery time is accounted for using a fuzzy membership function. Ability to recover from a failure state is represented by a fuzzy resiliency index in time and space.
Results and discussion
Flood damage in time and space
Agricultural and residential damage for the Red River basin is shown using a red (light to dark) colour ramp, with red representing a location with high damage and white repre- senting a location with low damage (Figure 18). Damage
analyses show that on 23 April 1997 there was no damage in the floodplain. On 26 April 1997, however, agricultural areas closer to the river were flooded and show considerable flood damage. On 3 May 1997, more agricultural land is submerged and flood damage is significantly higher.
Agricultural damage was experienced during the period from 26 April 1997 to 21 May 1997. Each section of flooded agricultural area has a unique seeding date. Because the agricultural damage is assessed based on the seeding date, at a particular location, a section of agricultural area under the water shows the same level of damage over the time period bounded by the date of submergence and the seeding date.
231 233 235 237 239 241
0% 100%
Infrastructure damage as a percent of reported damage
Water Elevation (m)
X%
Total reported damage represented 100% of pre and post flood costs
Total potential infrastructure as percent of total closure cost
Base level costs at 5%
Top level of Dike
Base level at incipient Flooding
Figure 17 Depth–damage relationship for ring-diked communities (KGS, 2000).
Gj¼Total reported damage for a communityArea of grid cellðAjÞ
Area within the ring dike ð24Þ
THEN Dij¼ðWater LevelBase Level of Incipient FloodingÞ ð1005Þ100 Gj
Top Level of DikeBase Level of Incipient Flooding þ 5 100Gj
ELSEIF Water Elevation > Top Level of Dike THEN Dij¼Total Potential Infrastructure Damage
Total Reported Damage Gj
ð25Þ
As the seeding date of 21 May 1997 approaches, much of the agricultural land shows zero damage. There is some agricul- tural land in the floodplain where floodwater remained
stagnant over a longer period. The major delay in seeding date caused the damage in these locations to be significantly higher (Figure 18).
Figure 18 Variation of damage (US$ per 625 m2) in time and space.
