Department of Systems and Industrial Engineering, University of Arizona, Tucson. Istvfin Bogfirdi .... western United States by the subjective classification of Bar-.
WATER RESOURCES
RESEARCH,
VOL. 32, NO. 6, PAGES 1741-1747, JUNE 1996
A fuzzy rule-based approach to drought assessment G6za Pesti Civil EngineeringDepartment,Universityof Nebraska-Lincoln
Biijaya P. Shresthaand Lucien Duckstein Departmentof Systemsand IndustrialEngineering,Universityof Arizona, Tucson
IstvfinBogfirdi Civil EngineeringDepartment,Universityof Nebraska-Lincoln
Abstract. A methodologyfor predictingregionaldroughtsfrom atmosphericpressure patternsis presented.Drought characteristics are stronglyrelated to generalcirculation patterns(CP). CPs are determinedfrom daily atmosphericpressuredata. The link betweenlarge-scaleCPs and regionalscaledroughtsis modeledusinga fuzzy rule-based approach.A fuzzy rule-basedmodel operateson an "if" -o "then" principle,where "if" correspondsto a vector of fuzzy inputsand "then" correspondsto somefuzzy consequences. The rulesare derivedfrom a so-calledtrainingset which includesa daily time seriesof CP classesand a corresponding monthlysequenceof Palmer Drought SeverityIndices(PDSI). Split samplingof historicaldata availablefor a 35-yeartime period is usedto deriveand then to validatethe rules.Then, thesefuzzyrulesmay be appliedto predictdroughtsin termsof atmosphericcirculationpatterns.The occurrence and persistenceof CPs are expectedto vary under globalclimatechange.Thus the approachmay alsobe usefulin estimatingthe potentialimpactof climaticchange(e.g., 2 x CO2 scenario)on droughts.The methodologyis illustratedusingdroughtindexdata from New Mexico and atmosphericpressuredata over the westernUnited States. Introduction
The purpose of the paper is to present an approachto drought prediction from atmospheric circulation patterns (CP), usinga so-calledfuzzyrule-basedapproachto estimatea drought index on the basis of imprecise or fuzzy inputs. Droughtsare rather difficultto assess primarilydue to the lack of a widely accepteddroughtdefinition[Dracupet al., 1980]. The followingstatementof Yevjevich[1967, p. 4] may still be valid nowadays:"The failure to developa succinctand objective definitionof droughtsis one of the principalobstaclesto the effectiveinvestigationof these events."Drought may be defined from a meteorological,hydrological,agricultural,or socio-economicviewpoint. Many researchersand organizations have defined drought from the viewpoint of their own requirementsin termsof meteorologicphenomena.For example, Palmer [1965] and Krishnan [1979] compiled a set of drought definitionsdealing with precipitation and moisture deficit. Clearly, it is useful to categorizethe variousdrought types,but the boundariesseparatingthesecategoriesare often vague[Wilhiteand Glantz, 1985]. A droughtrecord may be a time seriesof a state variable whichcanbe usedfor quantitativecharacterization of drought events.In a very simplifiedcase,droughtrecordscan be obtained directlyfrom precipitationrecords.Consideringhydrological droughts,a partial duration seriesof flow in a river sectioncan be used as a drought record. In a more general case, time series of a drought index such as Standardized Copyright1996by the AmericanGeophysicalUnion. Paper number 96WR00271. 0043-1397/96/96WR-00271 $09.00
Anomaly Index (SAI) or Palmer's Drought SeverityIndex (PDSI) can be used as droughtrecords.Drought indicesare calculatedas functionsof hydrometeorologicvariables.Variousindiceshavebeen developedfor quantitativeassessment of wet and dry periods.Here, PDSI time serieswill be used, althoughdrought recordsobtained in any other way may be utilized with the methodology.PDSI, a monthly index, has been selectedbecauseit is a widely accepteddrought index which has been successfully applied to drought characterization in manydifferentregions[Paulsonet al., 1991;Wilhiteand Glantz, 1985]. PDSI, through a recursion,is able to consider the historyand cumulativevaluesof hydrometeorological elements and anomaliesusing the moisturebalance of the root zone, as briefly describedin Appendix A. Circulation
Patterns
Atmosphericcirculationpatterns(CP) are generallycharacterized by the spatial distributionof sea level pressureor of somelow or middle troposphericpressureheightover an area at leastaslargeasEurope.Any CP typemaypersistfor several days,and duringthistime the main featuresof weatherremain essentiallyconstant.After this, there usuallyis a rapid transition to another CP type. Daily circulationpatterns may be classifiedbasedon characteristicssuchas high or low centers, cyclonicand anticyclonic,rotation and direction of motion. Yarnal [1984] categorizedclassificationtechniquesas either subjectiveor objective.Subjectiveor manual techniquesdepend on the analyst'sinterpretation of major features and persistenceof patterns.Objectiveor automatedclassification methods,in contrast,are based on mathematicaltechniques. Someof them are hierarchical[Johnson,1967],K means[Mac-
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Queen,1967],correlationmethods[Bradleyet al., 1982;Yarnal, 1984], principalcomponentanalysis(PCA) [Kutzbach,1970], and PCA coupledwith K means [Matyasovszky et al., 1993]. Objectivetechniquesare often impreciseand unlikelyto preserve persistenceof CP types over several successivedays [Yarnal and White, 1987]. Bardossyet al. [1995] and C. E. Ozelkan et al. (Systemsand Industrial Engineeringworking paper, 1994) have used fuzzy set techniquesto classifyCPs under imprecision.Here the CP types developedfor southwesternUnited Statesby the subjectiveclassificationof Bartholyand Duckstein[1994] are used.
Relationship Between Circulation Patterns and Droughts
ASSESSMENT
Rule:
IF
(Input)
THEN (Conseq. 1 PDSI
IF• &•& A THEN IF• &A &• THEN
Rule 1:
Rule 2:
Rule m:
The important effect of atmosphericcirculationon the ocFigure 1. A set of fuzzy rules. currence,amount,and spatialdistributionof precipitationhas been demonstratedby numerousresearchers[Lamb, 1977;McCabeet at., 1989;Bardossy and Plate, 1991].Wilsonet at. [1991] developeda daily precipitationmodel usinga weather classi- Fuzzy Rule-Based Methods fication
scheme for the Pacific Northwest
United
States. Bar-
dossyand Plate [1991, 1992] and Bogardiet at. [1993] have conditionedthe modelingof precipitationoccurrenceon atmosphericcirculationpatterns.The link betweenlocalweather phenomena,includingtemperatureand precipitation,and atmosphericcirculationpatternsmakes it reasonableto relate precipitation-baseddrought indices to CPs. Hydrologic extremessuchas floodsand dry periodshave been shownto be influencedby atmosphericcirculationpatterns[Bardossy et at., 1990;Ducksteinet at., 1993;Bogardiet at., 1994]. The central question is how the drought-producingcirculation pattern typescan be identifiedand usedin droughtprediction,particularly in the estimation of the potential effect of climatic changes.Bogardiet at. [1994] estimate the effect of climate changeon regionaldroughtsusinga stochasticlinkagebetween CPs and a rainfall deficit-baseddrought index, the BhalmeMooley index. In most of the above mentioned studies,hydrometeorological variableshavebeen linked to CPsby means of a stochasticmodel in a conditionalprobabilityframework. The linkagebetweenmonthlydroughtindicesand dailyCPsis more complex.In addition to the problem of different timescales(i.e., monthlydroughtindicesare to be linked to daily CPs), further difficultiesarisefrom the fact that droughtindices are in functionalrelationshipwith a sequenceof several hydrometeorological variableswhich are also but differently related to CPs. Thus a drought index, such as the monthly PDSI, dependson a sequenceof precedingCPs.Propertiesof a CP sequence(e.g.,frequencies andarrangementof CP types) are rather difficult to account for in a stochasticmodel. Thus,
Fuzzy rule-basedmodelshave been describedin detail and illustratedthroughpracticalexamplesof Bardossyand Duckstein[1993, 1995] and Bardossyand Disse[1993], whosework providesthe basisfor the presentapproach.A fuzzyrule-based model (Figure 1) operateson an "if" -• "then" principle, where "if" correspondsto a vector of fuzzy explanatoryor input variablesand "then" correspondsto somefuzzy consequences.Both the input variablesand consequences are assumedto be in the form of specialfuzzysets,namely,triangular fuzzynumbers.Basicdefinitionsof fuzzysets[Zadeh,1965] and fuzzyarithmeticis givenby Zimmermann[1985]andKaufmann and Gupta [1991]. In AppendixB, a brief reviewof the fuzzyset definitionsare necessary to the understandingof the paper is given.
Fuzzy Rules A fuzzy rule i, with an "and" combinationof K input variablesand one consequence (Figure 1), may be formulatedas
i' if {g,,•andg,,2ßßßandgix}, then b, where {gi,•,' k = 1,...,
(1)
K} and bi are triangularfuzzy
numbers (seeAppendix B)withmembership functions {m•,,k: k -- 1, ..., K} andmr,l, respectively. Unlikeordinary rules, which can either be appliedor not be appliedin a givensituation, fuzzy rules allow partial and simultaneousfulfillment and applicabilityof rules.Applicationof sucha fuzzyrule may be illustrated
as follows.
Let a = {a •, ..., as:} be a vectorof observedor calculated for eight CP types and 2-day sequencesthe probabilitiesof parametersnonfuzzy(or crisp)values;the fuzzyresponsebi is PDSI shouldbe conditionedon at least64 statevariables(i.e., to be determinedusingthe abovefuzzyrule. Figure 2 showsan eight for the occurrencestimes eight for the arrangementsof illustrativeexamplewith two rules and two input variables, CPs).As differentsetof CP typesare definedfor eachseason, where the observationvector is {13, 5}. The grade of fulfillonly one-fourthof the availablePDSI time seriescan be used mentof a•, in gi,•,is definedasthe membership functionvalue to estimate these conditional probabilities.Here, only 105 m•,,k (a•,).Thedegree offulfillment vi of theabove ruler is PDSI data (i.e., three-monthlydata for each seasonover 35 then definedasthe minimumamongthe fulfillmentdegreesof years)are availablefor calculatingprobabilitiesconditionedon all inputsinto the rule [Bardossy and Duckstein,1995] 16 randomvariables,whichis clearlynot enough.Therefore a v, = min {m•,,•(ak)) (2) different approachis used here, namely, a fuzzy rule-based Vk methodwhichis not a substitutefor a stochasticapproachbut is a viable and easyto usetechniquewhen data are scarceand The responsebi will have a membershipfunctionreducedby Pi h} (B2) variables,sucha CPscorresponding to morethanonepressure levels or even paleohydrologicdata [Dracup and Kendall, The level setA(h) includesa set of numberswith a given minimum(h) membership(credibility). 1991]. Fuzzynumbers. A fuzzynumber• (FigureB1) is a special Conclusions may be summarizedas follows: fuzzy setwith the followingproperties:(1) it is definedon the 1. Droughts are linked to persistenceof specificatmoset of real numbers,(2) it has at leastone point t*, where its sphericcirculationpatternsat differentspatialand temporal membership functionma(t* ) -- max (ma) -- 1, (3) ma(t* ) scales;here, daily continental-scale CPs have been related to is unimodalwith an increasingand decreasingpart (quasimonthlyregional-scaledroughts. 2. The fuzzy rulesderivedfrom historicaldata of CPs and convexityassumption). Triangular fuzzy numbers. The fuzzynumber• r is called droughtindicesover a 20-yearperiod maybe usedto predict triangular and denoted(a •, a2, a3)r if its membershipfuncthe frequencydistributionof droughts. tion is a triangle,whoseequationmay be written as 3. The method is easyto implement,requiringrelatively simplecodingand little computertime.
Appendix A: Palmer's [1965] Drought Severity Index
PDSI is calculatedusingthe actual,potential(i.e., physically attainable)and ClimaticallyAppropriatefor ExistingConditions(CAFEC) valuesof precipitation(P), evapotranspiration (ET), runoff(RO), recharge(R), andmoistureloss(L). Any locationcan be characterizedby the ratiosof the averagesof the actualand potentialvalues.Theseare a,/3, % and/• for the evapotranspiration, recharge,runoff,andsoilmoisture,respectively.For eachmonththe CAFEC valuefor precipitationcan be calculated
•y(0
1l............. '
h
as 0
P = aPET +/3PR + yPRO- 8PL
(A•)
wherePET, PR, PRO, and PL are the potentialvaluesfor ET,
t*
Figure B1. Fuzzynumber.
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ma(t) - 0
ma(t) m•(t) =
t-al a2-
a1
t -a3 a2-
a3
ET AL.:
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Engineering RiskAssessment, editedby G. Ganoulis,NA TO ASI Ser.
t < a•, t -->a3
a• < t
