Fuzzified effect of ENSO and macrocirculation patterns on precipitation: an Arizona case study

INTERNATIONAL JOURNAL OF CLIMATOLOGY

Int. J. Climatol. 19: 1411–1426 (1999)

FUZZIFIED EFFECT OF ENSO AND MACROCIRCULATIONPATTERNS ON PRECIPITATION: AN ARIZONA CASE STUDY

AGNES GALAMBOSIa,*, LUCIEN DUCKSTEINa,†, ERTUNGA C. O8 ZELKANb and ISTVAN BOGARDIc

a The Department of Systems and Industrial Engineering, Uni6ersity of Arizona, Tucson, AZ 85721-0020, USAb i2 Technologies, 909 E. Las Colinas Bl6d., 16th Floor, Ir6ing, TX 75039, USA

c The Ci6il Engineering Department, Uni6ersity of Nebraska, Lincoln, NE 68588, USA

Recei6ed 26 June 1998Re6ised 1 February 1999

Accepted 6 February 1999

ABSTRACT

A fuzzy rule-based model (FRBM) is developed to analyse local monthly precipitation events conditioned onmacrocirculation patterns and El Nino–Southern Oscillation (ENSO). A case study in Arizona is presented toillustrate the methodology. The inputs of the FRBM are those Southern Oscillation Index (SOI) values which havehigh absolute lag correlation with monthly Arizona precipitation and the frequencies of all circulation patterns (CPs)in a given month; the output of the model is an estimate of local monthly precipitation. After analysing the basicproperties of the precipitation events, fuzzy rules are constructed, and then the results are interpreted and comparedwith those of a multivariate linear regression model. Using two goodness-of-fit criteria, first, the root mean squarederror (RMSE) and then the correlation between the model results and the observed values, the FRBM is found toperform better than the multiple linear regression model for the Arizona case investigated. The results show that theFRBM can provide a good basis for future work to downscale general circulation model results to study localprecipitation under climate change. The results of using only SOI lags or CP frequencies as inputs, which are alsopresented here, clearly show how much the results are improved using both inputs jointly instead of only one.Copyright © 1999 Royal Meteorological Society.

KEY WORDS: fuzzy rule-based model; multivariate linear regression model; El Nino–Southern Oscillation (ENSO); SouthernOscillation Index (SOI); macrocirculation patterns (CPs); Arizona precipitation

1. INTRODUCTION

The purpose of this paper is to analyse the effect of El Nino–Southern Oscillation (ENSO) events andlarge-scale circulation patterns (CPs) on local precipitation using first a fuzzy rule-based model (FRBM),and then a multiple linear regression model (MLRM). Since several studies (cited below) have shown thata linkage exists between not only CPs and local precipitation, but also ENSO and local precipitation, theuse of the joint information from the CPs and ENSO (represented by the Southern Oscillation Index,SOI) is expected to provide better results than using just the CPs or the SOI as inputs to downscaleprecipitation. The present paper provides a basis for the further climate change studies by specifying theconnection between the precipitation events and their two possible causes as mentioned above.

The atmospheric CPs play an important role in climate research. For example, recently, Woodhouse(1997) applied principal component analysis for studying winter atmospheric circulation patterns in theUS Sonoran desert, and concluded that two types of circulation mechanisms explain up to 63% of thevariation in numbers of rainy days here: an ENSO and Pacific North American mechanism, and asouth-western low mechanism, and they are likely to be dependent. Moreover, Schubert and Henderson-Sellers (1997) downscaled local daily temperature extreme from CPs in Australia, Cavazos (1997) used an

* Correspondence to: The Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721-0020, USA.Tel.: 1-817-543-1560; fax: 1-817-543-1560; e-mail: [email protected]† ENGREF, 19 avenue du Maine, 75732 Paris Cedex 15, France.

CCC 0899–8418/99/131411–16$17.50Copyright © 1999 Royal Meteorological Society

A. GALAMBOSI ET AL.1412

artificial neural network model for downscaling large-scale circulation to local winter rainfall inMexico, O8 zelkan et al. (1996) established a relationship between monthly CPs and precipitation byusing fuzzy logic and regression approaches. Here, a similar but modified approach is used, asin O8 zelkan et al. (1996). The present paper is an extension of the research presented in Galambosi etal. (1998) for building the fuzzy rules, providing methodological steps adapted to the problem onhand.

Although the phenomenon called El Nino was initially recognised as a local climate anomaly of theeastern equatorial Pacific region, today it is also associated with the Southern Oscillation, affectingclimate events worldwide through teleconnection patterns: droughts in Indonesia, India, Australia;floods in southern Brazil, Peru, Ecuador; epidemics in South America and the flow of the river Nilein Africa—just to name a few of them. To cite a few recent examples, Hoerling et al. (1997) studiedEl Nino, La Nina and the nonlinearity of their teleconnections, Chang (1997) looked at the ENSOextreme climate events and their impacts on Asian deltas, Quinn (1992) analysed Southern Oscillation-related climatic activity from Nile River flood data, Lanzante (1996) has studied the tropical seasurface temperature (SST) lag relationships, Chen et al. (1996) analysed the interannual variation ofatmospheric circulation associated with ENSO. Further details on the physical aspects and societalimpacts of ENSO may also be found in Glantz (1996).

The linkage between ENSO and precipitation events is also at the centre of interest. Recently,Seleshi and Demaree (1995) studied the Ethiopian rainfall variability and its linkage with SOI, Nichol-son and Kim (1997) linked ENSO to African rainfall, Waylen et al. (1996b) reported a study relatingCosta Rica annual precipitation and the Southern Oscillation, Cullather et al. (1996) investigated theinterannual variations of Antarctic precipitation related to ENSO, Kane (1997a) explored the relation-ships of ENSO with rainfall in Australia, Suppiah (1997) considered the connection between extremesof the Southern Oscillation and Sri Lanka rainfall, and Kane (1997b) compared ENSO, Pacific SSTand rainfall in various regions. The Inter-American rainfall and tropical Atlantic SST and Pacificvariability have also been investigated (Enfield, 1996), as well as ENSO and Northern Plains precipita-tion and temperature anomalies (Bunkers et al., 1996), or ENSO and US drought (Piechota andDracup, 1996).

Models of the relationship between ENSO and climate variables also cover a wide range. Theyinclude a vector time-domain approach to model Florida precipitation by Chu et al. (1995) rotationalempirical orthogonal function analysis for relationships between semi-arid southern Africa rainfall and700 hPa geopotential heights by Shinoda and Kawamura (1996), Monte Carlo simulations for ENSO-related precipitation and temperature across the Northern Plains (Bunkers et al., 1996), lag relation-ships to predict global rainfall probability from SOI (Stone et al., 1996), lag cross-correlation studiesfor Costa Rica monthly precipitation interannual variability (Waylen et al., 1996a), and waveletanalysis of summer rainfall in China and India using SOI (Hu and Nitta, 1996). Several stochasticmodels of daily precipitation conditioned on CP types have also been developed, they include, forexample, Hay (1991), Bardossy and Plate (1992), Bogardi et al. (1993) and Bartholy et al. (1995).

The study presented here provides another method, FRBM, to couple the input pair (ENSO andCPs) to produce the output data set, namely, the local monthly precipitation. The results of theFRBM are also compared to those of a MLRM. Since the FRBM is capable of handling even weaklycorrelated and/or insufficient input data, it is expected to outperform the results of the MLRM. Asensitivity analysis is also performed using four different thresholds to define precipitation events(including zero threshold which corresponds to the entire database).

This paper is organised as follows: after describing the data, the precipitation events in Arizona aredefined and analysed. The description and construction of the FRBM and the MLRM then followsfor both the training set and the validation set of data. Finally, the results of the FRBM and theMLRM are analysed, interpreted and compared on the basis of the root mean squared error (RMSE)and the correlation between the model results and observed values. Results of using only SOI lags oronly CP frequencies as inputs are also presented.

Copyright © 1999 Royal Meteorological Society Int. J. Climatol. 19: 1411–1426 (1999)

ENSO AND ARIZONIAN PRECIPITATION 1413

2. DATA ANALYSIS

This section is devoted to the description and preliminary analysis of the input and output variables ofthe FRBM and MLRM to establish a physical basis for the models to investigate the relationship betweenthese input and output variables.

2.1. Data

The input variables of the FRBM and MLRM consist of clustered pressure height data and ENSOdata. Forty-two years of daily observed 500 hPa level height data are used to describe the atmosphericCPs defined at 35 grid points on a diamond grid over the south-western US (Figure 1). The results of anautomated, nonhierarchical method, namely, principal component analysis coupled with K-means cluster-ing technique are used as an input to both models. Twenty principal components are retained to explain97–98% of the variance. There are six, seven, seven and eight types for winter (January–March), spring(April–June), summer (July–September) and fall (October–December), respectively. The number of CPschosen each season was a compromise between the increasing number of clusters and the decreasing sum

Figure 1. Diamond grid over the western US for the definition of CPs



Figure 2. Arizona precipitation stations selected for this study

of inner distances among them. Another criteria was to retain the same number of types as Bartholy andDuckstein (1994) for comparison purposes. The description of the K-means clustering algorithm can befound in many statistical books and application papers as well, e.g. MacQueen (1967) and Matyasovszkyet al. (1993). Further details on the results over this area, such as the description and hydrometeorologicalinterpretation of these types are found in Galambosi et al. (1996). A related study using a fuzzyclassification scheme can also be found in O8 zelkan et al. (1998).

In order to quantitatively describe the ENSO phenomenon, it is very common to use some indiceswhich are functions of climatological variables. Here, the so-called SOI is selected, which is defined as thepressure difference between Tahiti and Darwin (Clarke and Li, 1995):

SOI=p(Tahiti)−p(Darwin)Although an SOI historical time series of about 130 years is available, the only period utilised here is onewhich also has both CP and Arizona precipitation data available, that is 1949–1989. Since the SOI is ona monthly rather than on a daily basis, the model either has to be restricted to a monthly resolution, ora way must be found to distribute the monthly SOI data into smaller temporal scales. Here, the model isanalysed on a monthly basis as in Galambosi et al. (1998).

The output variable of the models is local precipitation at Arizona stations. Daily precipitation datacan be selected from about 300 Arizona stations with daily observations. The data stations have beenselected on the basis of data availability, data homogeneity and representativity of Arizona subclimates asdescribed in Bartholy and Duckstein (1994). Figure 2 shows the eight Arizona stations selected for thisstudy.

Here the ‘threshold cut method’ is chosen for defining the severity of a precipitation event: precipitationevents are defined on the basis of a threshold value. The higher the threshold, the more severe theprecipitation event. Since the resulting set of precipitation events depends on the threshold value, asensitivity analysis will be performed by changing the threshold values. Note that here only severe highprecipitation events are discussed, since drought (lack of precipitation) should be treated in a differentway. For drought, the precipitation is zero, so other indices have to be used to differentiate the severityof the events: e.g. Pesti et al. (1996), Pongracz et al. (1998).



Unfortunately, the database is not sufficient to try too many thresholds higher than 0.4, because as thethreshold increases there is less and less data to train and validate the model. For example, for somestations, setting the threshold value at 0.6 results in only 3 years (i.e. three data points) out of 37 yearsbeing above this threshold. In other words, lack of data determines the threshold levels used.

2.2. Results of a preliminary data analysis

An example of two stations (Grand Canyon, Betatakin) and 2 months (August, December) have beenselected to illustrate the preliminary data analysis. For sensitivity analysis, four threshold values have beenchosen to distinguish the severity of the events, namely, 0.0, 0.2, 0.4, 0.6 on a normalised scale from 0 to1, where 1 represents the highest rainfall depth on record for the given month (for example, for Februaryit is 6.7876 inches=172.41 mm, at Roosevelt, in 1980). Note that choosing 0.0 as a threshold means thatthe model is applied to the complete database.

Figure 3 shows the box plot of the monthly distribution of precipitation at the selected stations ofGrand Canyon and Betatakin. In general, the highest precipitation values occur in July and August(summer monsoon) in Betatakin. Note that the very outstanding value of close to 250 mm in Betatakinin October is indeed a value in the database: in October of 1972 the measurement was 9.63 inches (244.6mm) precipitation, and the values for other stations during the same month were all between 3.22 inches(81.88 mm) and 9.87 inches (250.65 mm)). At the Grand Canyon station the distribution exhibits asomewhat greater variability, which means that the difference between the wet and dry months is a littlestronger than in Betatakin. High precipitation occurs not only in July and August, but also in December,January and March.

Figure 3. The distribution of precipitation in Grand Canyon and Betatakin



Figure 4. Lag correlations between SOI and local precipitation from 0 to 12 months, threshold 0.0

Figure 4 shows the lag correlations between SOI and local precipitation from 0 to 12 months lag, atthreshold 0.0, in August and December, for both stations. The term ‘lag k ’, k=0, . . . , 12 means thatprecipitation in a given month was correlated against SOI which occurred k months before. The littleblack circles on the figure show those lags which have been selected as inputs to the models in the givenmonth and station for the given threshold. Table I shows the correlation threshold values with theircorresponding significance level (Draper and Smith, 1981). From the table it can be seen that the SOI lagcorrelations chosen as inputs to the FRBM and MLRM are at least 80% significant. For example, inAugust, in Betatakin, lags 0, 1, 2, 3 and 4 yield high correlation values, whereas in December, in GrandCanyon those lags are 2, 5, 9 and 10.

Figure 5 shows an example of the precipitation versus SOI index for lag 0, in August and for lag 2 inDecember, at both stations. The lag 2 case in December clearly shows that there is a good relationshipbetween local precipitation and El Nino events at both stations since high values of precipitation areusually associated with negative values of SOI. This was also indicated by the corresponding high valuesof correlation in Figure 4. In August, the relationship is not very well defined for Grand Canyon stationfor lag 0 as can be seen from both Figures 4 and 5. This shows that other factors than SOI must playsome causal role for precipitation in August, as it can be expected when convective rainfall is the mainoccurrence.

Figure 6 shows several histograms of the CPs, the examination of which provides an understanding fora given month, station and threshold as to the type of CP that may cause precipitation. From Figure 6it is clear that the histograms are a function of the thresholds, meaning that it is not the same CPs thatcause the ‘normal’ and the more severe precipitation events, as also discussed in Galambosi et al. (1996)and Shresta et al. (1996). For example, for threshold 0.0 (all events) in August, CP 2 has the highestfrequency, whereas for threshold 0.6, at both stations, CP 7 occurred most often, meaning that it is a

Table I. Correlation values with their corresponding significance levels

Significance level (%) Correlation threshold value

95 0.310.2690

85 0.2280 0.19



Figure 5. Precipitation versus SOI, Grand Canyon and Betatakin

severe precipitation-causing type. Furthermore, the histograms for the two selected stations and the twoselected months appear to be different, reflecting the temporal and spatial climatic differences. In anycase, since every type of CP seems to generate some rainfall, the input into the FRBM includes thefrequency of all the types.

Figure 6. Histograms of circulation patterns. The horizontal axes are on a nominal scale



Figure 7. An example of a rule (thick lines)

3. METHODOLOGY

3.1. Fuzzy rule-based model

In this section, the fuzzy framework used in this paper is summarised. The methodology used here issimilar to the one in O8 zelkan et al. (1996) and Galambosi et al. (1998). However, in addition to thepreliminary results of Galambosi et al. (1998), the present paper provides precise methodological stepsadapted to the problem on hand, presenting additional examples to illustrate the methodology. Further-more, models using only SOI or CPs as inputs are interpreted besides the model that uses both kinds ofinput jointly.

In general, a fuzzy rule consists of a set of explanatory variables, called premises, given in the form offuzzy numbers which are described by membership functions. The consequence or output of each rule isalso in the form of a fuzzy number. A membership function shows how much each observed data pointbelongs to the set. In this work, triangular fuzzy numbers (piecewise linear functions) are used for thecalculations as shown in Figure 7.

A typical rule is written as:

IF SOI–lag1 is Low AND SOI–lag2 is Medium AND SOI–lag3 is Low

AND CP1–frequency is Medium AND CP2–frequency is High AND. . .

AND CP6–frequency is High THEN Precipitation is Extreme High (1)

In contrast to ordinary (crisp) rules, fuzzy rules allow partial and simultaneous fulfillment of a subset ofthe rules. This means that instead of the usual case when a rule is applied or is not applied, a partialapplicability of more than one rule is also possible.



The following steps summarise the FRBM methodology used here.

3.1.1. Step 0: Split sampling. Divide the input–output database into two parts: the first part is used forcalibration or training, that is to establish the relationship, and the second part, for validation, namely,to test the predictive performance of the model.

3.1.2. Step 1: Di6ide input–output space into fuzzy regions. The domain of each premise is partitionedinto several fuzzy numbers with overlapping supports. Here, triangular fuzzy numbers are used for themodel. As was mentioned above, the inputs of the FRBM are the SOI values which have high absolutelag correlation with monthly Arizona precipitation and the frequencies of all CPs. For example, the rangeof the SOI index is divided into Low, Medium and High fuzzy sets denoting El Nino, normal and LaNina years, respectively. Similarly, the ranges of each CP frequency and of precipitation amount are alsodivided into overlapping sets as sketched in Figure 7. From a modelling point of view, fuzzy rules can beconstructed with nonsymmetrical, nonlinear functions. But as also shown in the literature (Wang andKim, 1995), the output of a FRBM is not very sensitive to the shape of the fuzzy numbers.

3.1.3. Step 2: Generate fuzzy rules from gi6en data. First, the membership values in the fuzzy sets orregions are calculated for a given data point. Second, each element of the given data vector is assignedto the fuzzy set with maximum membership function. The input–output data thus yields a rule. Forexample, as shown in Figure 7, the first data point of the first SOI lag variable, S1(1) belongs to the Lowset with membership value of 0.8 and to the Medium set with membership value of 0.2, thus it is assignedto the Low set (thick line). Similarly, S2(1) belongs to the Medium set with membership value 0.9. andto the High set with 0.1 value, thus it is assigned to the Medium set. Continuing in a similar way for theother premises and consequences. the membership values are 0.9 (Low), 0.87 (Medium), 0.75 (High), 0.88(Medium), 0.94 (Low), 0.91 (High), 0.75 (High), and for the precipitation, 0.78 (Extreme High). Thus therule in Equation (1) is constructed. If this rule has not been previously encountered, it is added to the rulebank, which will be used later for prediction purposes. This procedure is repeated for each input–outputdata vector.

3.1.4. Step 3: Rule fulfilment. Once the fuzzy partitions have been determined, the degree of fulfilment(DOF) of any rule can be calculated using the AND logical operator (Bardossy and Disse, 1993):

ni(t)= 5N

n=1

mXi,n(xn(t)) (2)

where xn(t) denotes the input data at time t, and mXi,n(xn(t)) is the membership function for

the explanatory variable Xi,n. For the example, the degree of fulfillment becomes n5(1)=0.8*0.9*0.9*0.87*0.75*0.88*0.94*0.91*0.75=0.24.

3.1.5. Step 4: Assign a weight to each rule. To show which portion of the calibration data is explainedby a given rule, one can weigh each rule by:

ji= %T

t=1

ni(t)mYi(y(t)) (3)

where mYi(y(t)) is the membership function of the precipitation (consequence of a rule) at each

observation point t=1, . . . , T. Thus ji is the weight associated with rule i. At this step, an eliminationof the rules can be carried out if desired by setting a positive threshold. This way those rules whichcontribute to the solution more significantly than others could be identified. Thus, leaving out rules willsimplify the rule-based system, and in case of large set of data, it will enable the decision maker to avoid‘noise-causing’ rules which explain only a small percent of variation. Note that in the present study j*=0has been selected so that all rules which can be constructed from the observed data are retained.

3.1.6. Step 5: Calibration using the rule-based system. After constructing the rules, the response of thefuzzy rules to a given input vector (a1(t), . . . , aN(t)) can be computed using the training set. For thispurpose, the DOF of each rule is calculated using Equation (2).



3.1.7. Step 6: Defuzzification. The response is now expressed as a real number:

y(t)=Si ni(t)jib i

0

Si ni(t)ji

(4)

where y(t) is the estimated value of y(t), ji is given by Equation (3), and b i0 is the mode of the

membership function of the consequence Yi. Note that, here as shown in Figure 7, except for the extremelow and extreme high values of precipitation, intermediate fuzzy values are represented using symmetricalfuzzy numbers for which the mode and mean are the same.

3.1.8. Step 7: Adaptation of the model. The model developed here can be made adaptive if after eachprediction, steps 2–4 of the above described algorithm are repeated as new data are observed. This way,if the rule explained by the new observation is different from the rules in the rule bank, a new rule islearned by the model and the rule weights ji are updated.

3.1.9. Step 8: Validation of the model. The output for each data point in the second part of the database(validation set) is estimated by using the rules of the system constructed in the calibration phase.

3.1.10. Step 9: Optimisation. In general, FRBM can be optimised to minimize the error between modelprediction and observed data (Jang, 1992; Nomura et al., 1992)

u*=arg minu

y(t � u)−y(t) p (5)

where . p denotes the Lp-norm. and u is the parameter set defining the shape of the fuzzy numbers,defining the input and output variables. Equation (5) usually results in nonlinear optimisation problems(O8 zelkan, 1997), and as shown in Bardossy and Duckstein (1995), least squares estimators can be foundunder certain assumptions. Here, except for SOI data (where low, medium and high fuzzy numbers areused to denote the physically accepted El Nino, normal and La Nina years), optimisation is reduced tothe identification of the optimal number of fuzzy intervals to describe the input and output variables. Thisapproach works for the case of symmetrical, fully overlapping triangular fuzzy numbers with equal basewidth as shown in Figure 7, where u in Equation (5) would reduce to the number of fuzzy numbers usedto describe the input and output variables. Note that the asymmetry of extreme low and high fuzzy valuesdoes not contradict with the approach. Also using the L2 norm (RMSE) as the distance measure to beminimised, the optimisation problem can be expressed formally as:

l n*=arg minln

� %T

t=1

y(t � ln)−y(t)2�1/2

. (6)

where l n* shows the optimal number of triangular fuzzy numbers to describe variable n. Therefore, FRBMis optimised by a ‘neighborhood search’ algorithm in terms of the RMSE such that the number of fuzzyregions for each input and output of Step 1 is modified, and the one which yields the minimum value ofRMSE of the model is used.

3.2. Multi6ariate linear regression

In the case of a statistical regression it is supposed that the true relationship between the CPfrequencies, SOI values and precipitation events is linear:

y(t)= %T

t=1

bnxn(t)+o(t) (7)

where o(t) is a random error. Adding higher order terms into the MLRM might improve the fit of themodel since it is ‘converging’ into a polynomial line fitting in the calibration phase but this will providevery little predictability for forecasting. The other problem is that there are not enough data forcalibrating a multiple linear regression with terms in xi

2 or xixj, where xi is the ith independent variable.For testing the model goodness of fit statistically, it must satisfy a set of hypotheses often violated in

practice, such as no correlation between the input variables and homoscedasticity. The model parameters



are usually estimated by the least squares method. Further details on regression can be found for examplein Chatterjee and Price (1991). As for the FRBM, split sampling is also used when applying this model,so that there is a basis for comparing the results of the two models.

4. APPLICATION AND RESULTS

As mentioned before, a split sampling approach has been used to calibrate and validate the models. Thenumber of years chosen for validation is up to 8 years, depending on the record length. In other words,the higher the threshold, the smaller the validation period. For example, 8 years are used for all validationof 0.0 threshold cases, whereas only 4 years are used for validation with threshold 0.2, December, GrandCanyon case.

The frequency distribution of the CPs and the SOI values, with highest absolute value lag correlationshave been selected as inputs to both the FRBM and the MLRM. Thus the number of inputs changes foreach month; for example, for Betatakin precipitation in August, the number of inputs is 12 for the 0.0threshold case, since five values have been selected for the SOI lags (0, 1, 2, 3 and 4), and the number ofCPs in summer is 7. Furthermore, as stated before, the number of fuzzy partitions for CPs andprecipitation has been optimised so as to minimise the RMSE. Finally, FRBM and MLRM results arecompared, based on the RMSE and correlation of the observed and model values in both calibration andvalidation periods.

Figure 8. Comparison of FRBM and MLRM results with observations, Grand Canyon, December. Threshold 0.2, sample sizes:calibration 15 years; validation 4 years



Figure 9. Values of RMSE and correlation, August, Grand Canyon, for thresholds 0.0, 0.2 and 0.4. Inputs: SOI lags and CPs.Sample sizes: Threshold=0.0, Calib=31, Valid=8; Threshold=0.2, Calib=18, Valid=8; Threshold=0.4, Calib=7, Valid=4

Figure 8 shows a comparison of the model results with the observations for both FRBM and MLRM.An example of Grand Canyon in December, threshold 0.2 is selected to illustrate the results. From theexample shown here it is clear that the FRBM gives better fit for both calibration and validation periods.

The overall comparison of the two methods is shown more precisely in Figures 9 and 10, for a typicalexample for three threshold values in August. From these figures it can be seen that the FRBM is superiorto the MLRM in every case when the CPs and the SOI data have been used jointly as inputs,

Figure 10. Values of RMSE and correlation, August, Betatakin, for thresholds 0.0, 0.2, 0.4. Inputs: SOI lags and CPs. Sample sizes:Threshold=0.0, Calib=31, Valid=8; Threshold=0.2, Calib=18, Valid=8; Threshold=0.4, Calib=5, Valid=6



Figure 11. Values of RMSE and correlation, Grand Canyon, December, for thresholds 0.0, 0.2, 0.4. Inputs: CPs only

independently of location or threshold. Using high threshold values, the MLRM usually gets worse withboth indicators, which may be explained by sample sizes becoming smaller thus fewer data points are usedfor the regression which then becomes more unstable. Sometimes there are fewer data points than thenumber of input variables, leaving no ‘degrees of freedom’ for a valid regression analysis. A MLRM with,say, ten independent variables requires lots of data. On the other hand, even when data are not sufficientfor MLRM, FRBM performs well, yielding robust results.

As another extension of Galambosi et al. (1998), precipitation outputs first using only CP frequenciesas inputs (Figure 11) and then only SOI lags as inputs (Figure 12) have been estimated. The number of

Figure 12. Values of RMSE and correlation, Grand Canyon, December, for thresholds 0.0, 0.2, 0.4. Inputs: SOI lags only



validation years for threshold 0.0, 0.2 and 0.4 are 8, 4 and 2 years, respectively, for both cases. Althoughthe results for threshold 0.4 probably mean that 2 years for validation might not be sufficient, the resultsin general clearly show the necessity of using both inputs to improve the prediction of the model. It shouldbe noted that El Nino appears to produce a strong hydrometeorological effect in the south-western USabout 40% of the time and very little effect the rest of the time (Stakhiv, 1997).

5. DISCUSSION AND CONCLUSIONS

This study examines the possible effects of El Nino and CPs on Arizona precipitation. The methodologyuses the frequencies of atmospheric CPs of the 500 hPa level and SOI of various time lags as inputs inorder to predict local precipitation events. Unlike some previous downscaling methods, not only CPs areused here for the prediction of a local hydroclimatological variable, but the effects of ENSO are alsotaken into consideration by using various lagged SOIs as inputs into the model. Since El Nino plays animportant role in the local weather events via teleconnections, better results are obtained here than by justusing the CPs. On the other hand, because El Nino has a strong effect about 40% of the time, a stochasticmodel seems to be difficult to apply in the absence of another factor indicating when such an effectoccurs. This model might also have the potential to be used for precipitation forecasting, but then a lagrelation might need to be added to the CP data, as it has been shown in O8 zelkan et al. (1996).

The concluding points, based on the results of all eight stations in Arizona, may be stated as follows:

� The FRBM appears to yield good predictions for the given area for both the normal and the severeprecipitation cases.

� In terms of RMSE and correlation between observed and modeled values, FRBM is usually superiorto MLRM.

� The FRBM is capable of giving good results even when there are insufficient or correlated data forMLRM.

� The FRBM gives more robust results in both the calibration and the validation phases than theMLRM.

� The better performance of the FRBM is partly due to its capability to represent nonlinearity better.� The prediction of the model is improved using the CP and SOI data as joint inputs to the models.

Future work might include the expansion of the model to more ENSO variables (such as SST). Sincethe FRBM works well in establishing the relationship between the pair ENSO–CP types and localprecipitation, further studies of severe precipitation events are possible for downscaling under climatechange by using scenarios such as 2×CO2 GCM outputs, although due to the large number of possibleinputs, the observed data might not be sufficient to produce rules for all future situations.

ACKNOWLEDGEMENTS

The authors wish to thank Judit Bartholy and Istvan Matyasovszky for their help.

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Fuzzified effect of ENSO and macrocirculation patterns on precipitation: an Arizona case study