Estimating Palmer Drought Severity Index using a wavelet.pdf

INTERNATIONAL JOURNAL OF CLIMATOLOGYInt. J. Climatol. 31: 20212032 (2011)Published online 13 September 2010 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/joc.2215

Estimating Palmer Drought Severity Index using a waveletfuzzy logic model based on meteorological variables

Mehmet Ozger,a,b,c* Ashok K. Mishraa,b and Vijay P. Singha,ba Department of Biological and Agricultural Engineering, Texas A&M University, College Station, TX 77843-2117, USA

b Department of Civil and Environmental Engineering, Texas A&M University, College Station, TX 77843-2117, USAc Hydraulics Division, Istanbul Technical University, Maslak 34469, Istanbul, Turkey

ABSTRACT: The Palmer Drought Severity Index (PDSI) is widely used to characterize droughts. The PDSI is basedon the water balance equation over an area of concern. Calculating PDSI requires data on precipitation, temperature,soil moisture, and the previous PDSI value. While precipitation and temperature time series data are easily available formost locations, it is not always the case with soil moisture due to the lack of soil-moisture monitoring networks. Thisstudy developed a wavelet fuzzy logic model (WFL) to overcome the problem. The proposed model employs commonlyavailable precipitation, temperature, and large-scale climate indices as predictors and PDSI as a predictand. The WFLmodel is applied to ten climate divisions in Texas and its performance is compared with conventional fuzzy logic (FL)model performance. It is shown that the WFL model outperforms the FL model. The variation of WFL model performancealong with the average wavelet spectra of precipitation time series is evaluated. Results show that the WFL model iscapable of predicting PDSI. Copyright 2010 Royal Meteorological Society

KEY WORDS Palmer Drought Severity Index; continuous wavelet transform; fuzzy logic; average wavelet spectra

Received 27 November 2009; Revised 12 June 2010; Accepted 29 July 2010

1. Introduction

Evaluation of droughts is important for water resourcesplanning and management. There are several indicesthat are used to characterize drought properties. Amongthem, the most used are standardized precipitation index(SPI), deciles, Palmer drought severity index (PDSI), andderivatives of PDSI. A drought index should representsome basic characteristics to describe droughts satisfacto-rily. The three main characteristics that must be includedin the definition of a drought index are duration, magni-tude, and severity (Mishra et al., 2007; Mishra and Singh,2009). The index should also include information on theonset and termination of a drought event. It should havean ability to distinguish a drought from aridity. PDSI,which is widely used in drought studies, involves all thesecharacteristics to define a drought.

Since its first formulation by Palmer (1965), therehave been several studies on PDSI (Szinell et al., 1998;Heim, 2002; Ntale and Gan, 2003). Temperature andprecipitation are the most important two inputs usedin the calculation of PDSI. Guttmann (1991) examinedthe sensitivity of PDSI to departures from averagetemperature and precipitation conditions. It was foundthat the effects of precipitation anomalies were greaterthan the effects of temperature anomalies. Hu and Willson

* Correspondence to: Mehmet Ozger, Department of Biological andAgricultural Engineering, Texas A&M University, College Station, TX77843-2117, USA. E-mail: [email protected]

(2000) investigated the temperature and precipitationeffects on the PDSI. They showed that the PDSI can beequally affected by temperature and precipitation, whenboth have similar magnitudes of anomalies. However,estimating soil moisture from drought indices can bea practical approach. Sims et al. (2002) studied thepossible estimation of soil moisture from PDSI andSPI. Rao and Padmanabhan (1984) investigated thestochastic nature of yearly and monthly PDSI, andcharacterized those using stochastic models to forecastand simulate the PDSI series. Lohani and Loganathan(1997) used PDSI to characterize the stochastic behaviourof droughts.

Sometimes fuzzy logic (FL) is preferred when linkinginputs to outputs in a nonlinear manner. Pesti et al. (1996)modelled the relationship between drought characteristicsand general circulation patterns (CP) using FL. Pongraczet al. (1999) applied fuzzy rule-based modelling for theprediction of regional droughts using two forcing inputs,ENSO and large scale atmospheric CPs in a typical GreatPlains state, Nebraska. These FL models are applicablefor only short-term drought forecasting.

Cutore et al. (2009) developed an artificial neuralnetwork model to forecast Palmer Hydrological DroughtIndex (PHDI) up to a 4-month lead time by consideringpersistence and some climate indices. Although theyobtained high R2 values (around 0.90) for 1 monthahead forecasting, which is the consequence of high autocorrelation coefficient at lag-1, the R2 values decreasedto around 0.4 for 4 months ahead forecasts. Kim and

Copyright 2010 Royal Meteorological Society

2022 M. OZGER et al.

Valdes (2003) proposed a conjunction model, which isthe combination of discrete wavelet transform and neuralnetwork method to forecast the PDSI values up to a 12-month lead time.

PDSI is the most used index to assess the severityof droughts. Temperature, precipitation, soil moisture,and the previous PDSI value are required for calculationof PDSI. Information on soil moisture includes poten-tial evapotranspiration, recharge, loss, and runoff. Also,incoming extraterrestrial solar radiation, relative humid-ity, mean monthly minimum temperature, and meanmonthly maximum temperature are used to calculatepotential evapotranspiration. Although temperature andprecipitation records are widespread, other data requiredto calculate PDSI may not exist for certain locations.

Although several studies have been conducted to pre-dict PDSI, the prediction of PDSI from simultaneousconsideration of temperature, precipitation, and some cli-mate indices has not been pursued. Generally persistence(lagged values) has been taken as a predictor variable toincrease the capability of prediction models. However,past knowledge of PDSI may not exist when the areaswhose drought properties have not been investigated pre-viously, are considered. The advantage of excluding theprevious PDSI value from predictor variables is to makeindependent predictions of PDSI. In this way, one canproduce the PDSI values in the absence of soil-waterbalance variables and the past knowledge of PDSI.

The objective of this study is to employ FL andWFL models for predicting PDSI from predictor vari-ables, which are temperature, precipitation, and climateindices such as NINO 3.4 (is an index that representsthe sea surface temperature anomalies in eastern tropicalPacific), PDO (Pacific decadal oscillation). The purposeof the study and methodologies are to address specificquestions: (1) Is it possible to simulate PDSI series with-out using soil-moisture data, if so, with what accuracy?(2) Can the simulated PDSI series be improved using cli-mate indices? (3) The strength of wavelet fuzzy modelin simulating the PDSI series? (4) How can the possi-ble effects of temperature and precipitation on droughtsbe interpreted through wavelet analysis and spectral bandseparation?

2. Palmer Drought Index

The PDSI is widely used in drought evaluation studies.The method is based on the soil-water balance equa-tion. The climate coefficients are computed as a propor-tion between averages of actual versus potential valuesfor each of 12 months. Palmer (1965) defined climati-cally appropriate for existing conditions (CAFEC), whichshows the actual situation in the area of concern. Theamount of precipitation required for CAFEC can be com-puted from climate coefficients. Subsequently, the waterdeficiency for each month is indicated by the difference,d , between actual (P ) and CAFEC precipitation (P ) as

follows:

d = P P = P (PE + PR + PRO + PL) (1)where = ET/PE = R/PR = RO/PRO = L/PLfor 12 months. The terms are actual evapotranspiration(ET) and potential evapotranspiration (PE); recharge (R)and potential recharge (PR); runoff (RO) and potentialrunoff (PRO); net loss (L) and potential loss (PL). APalmer Moisture Anomaly Index (PMAI), Z, for an ithmonth is then defined as follows:

Zi = Kidi (2)Palmer (1965) discovered that K , the weighting factor,

varied inversely with D, the mean of the absolute valuesof di . An empirical relationship was suggested as follows:

Ki = 17.67K i/[12

i=1 DiKi

](3)

Ki depends on the average water supply and demand,

expressed as:

K i = 1.5 log10[(Mi + 2.8)/Di] (4)M i = (PE + R + RO)

/(P + L) (5)

where PE is the potential evapotranspiration, R is therecharge, RO is the runoff, P is the precipitation, and Lis the loss. The PDSI is now given by

PDSIi = 0.897PDSIi1 + 13Zi (6)

3. DataThere are five distinct climate zones in Texas show-ing the variation from arid to sub-tropic humid zones(Figure 1(a)). Texas is divided into ten climate divisionsby the National Climatic Data Center (Figure 1(b)). Eachclimate division exhibits its own specific characteristics,such as vegetation, temperature, humidity, rainfall, andseasonal weather. Representative data are calculated foreach division by taking the stations which are withinthe borders of that division and then averaging over allstations. Precipitation, temperature, drought indices, andother variables are reported using these divisions.

PDSI indicates the severity of a wet or dry spelland is reported monthly. PDSI, which is a standard-ized index, is used in the assessment of meteorologicaldroughts. It is also considered a hydrological droughtindicator due to its relation to evapotranspiration andsoil moisture. It is capable of representing the spatialcontent of droughts. While negative values stand fordry spells, wet spells are represented by positive val-ues. The PDSI data on a 20 latitude 30 longitudegrid were obtained from a nearest neighbour griddingprocedure of Cook et al. (1999). PDSI, precipitationand temperature time series for each climate division

Copyright 2010 Royal Meteorological Society Int. J. Climatol. 31: 20212032 (2011)

ESTIMATING PALMER DROUGHT SEVERITY INDEX 2023

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Figure 1. (a) Climate zones and (b) climate divisions for Texas.

obtained from the National Climate Data Center forthe period 19002007 can be found at NOAA website(http://www7.ncdc.noaa.gov/CDO/CDODivisionalSelect.jsp#).

NINO 3.4 and PDO were used as variables forlarge-scale climate indices. Time-series data for NINO3.4 region are available every month from 1856 to2007 (http://iridl.ldeo.columbia.edu/SOURCES/.Indices/.nino/.EXTENDED/).

The PDO Index is defined as the leading principalcomponent of the North Pacific monthly sea-surfacetemperature variability. The monthly data covering theperiod 19002007 is downloaded from the website(http://jisao.washington.edu/pdo/PDO.latest).

4. Methodology

4.1. Fuzzy logicFL modelling is based on the fuzzy set theory which wasintroduced by Zadeh (1965). These days many applica-tions of the FL theory are seen in all areas of engineering.FL can be used to relate multiple inputs with outputand has the ability to establish nonlinear relationships.This relationship is achieved by a fuzzy inference system.There are mainly two types of inference systems whichare Mamdani and Takagi-Sugeno (TS). While the Mam-dani type inference relies on both linguistic and numericaldata, the TS inference system works only with numeri-cal data. The TS approach has an advantage of using dataefficiently in the training procedure and makes it possibleto incorporate a suitable training algorithm, e.g. ANFIS(Adaptive neural network fuzzy inference system).

Fuzzy rules and fuzzy sets are the main elements ofthe FL modelling. On one hand, fuzzy rules providethe connection between predictors and predictand andon the other hand, fuzzy sets produce weights forthose rules. Fuzzy rules are in the form of IFTHENstatements. While the part between IF and THEN iscalled antecedent, the consequent part is found afterTHEN. Here, the antecedent part consists of precipitation,

temperature, and climate indices. The PDSI values aretaken in the consequent part. A typical fuzzy rule can bewritten for this case as follows:

Ri : IF Precip is in F1 and Temp is in S1THEN PDSI = pi1 Precip + pi2 Temp + pi0

Rk : IF Precip is in F2 and Temp is in S2THEN PDSI = pk1 Precip + pk2 Temp + pk0

where F and S are the membership functions forthe precipitation and temperature variables that includeantecedent part parameters, and ps are the consequentpart parameters.

The fuzzy inference system consists of four steps:(1) The predictor variables are fuzzified by assigningmembership functions to each variable. The type (Gaus-sian, triangular, etc.) and the number of the membershipfunctions are determined by the user. (2) The fuzzy rulebase is constructed based on the previous step. The rulebase consists of rules which are combinations of member-ship functions of predictor variables. For instance, if thereare two variables with three membership functions in theantecedent part, there would be 3 3 = 9 rules totally inthe rule base. (3) The implication step incorporates out-comes of the antecedent part to the consequent part andaggregate the consequent part of all rules. (4) Since theaggregated results appear in the form of fuzzy sets, it isrequired to find a one-crisp value by using defuzzifica-tion as a final step. The following equations are used toobtain the final outcome of a fuzzy inference system:

i(x) = exp[(

x cibi

)2](7)

IF input 1 = n and input 2 = m THENoutput is zi = pi1n + pi2m + pi0 (8)

wi = 1 2 (9)



Final output =

Ni=1

wizi

Ni=1

wi

(10)

where b and c are the antecedent part parameters; psare the consequent part parameters; is the membershipfunction; and w is the weighting of the each rule. In thisstudy, the ANFIS technique was employed to determinethe antecedent and consequent part parameters. Detailsof this technique can be found in Jang (1993).

4.2. Continuous wavelet transformThe continuous wavelet transform (CWT) is used todecompose a signal into wavelets, small waves thatgrow and decay over a small distance, whereas theFourier transform decomposes a signal into an infinitenumber of sine and cosine terms losing most time-localization information. A continuous wavelet trans-form of a signal produces coefficients at a given scale.Comparison between Fourier analysis and wavelet anal-ysis is given by Kumar and Foufoula-Georgiou (1997)who presented only the basics regarding wavelet analy-sis. CWTs basis functions are scaled and shifted ver-sions of the time-localized mother wavelet. A Morletwavelet is one of the many wavelet functions whichhas a zero mean and is localized in both frequency andtime. Since the Morlet wavelet provides a good balancebetween time and frequency localizations, it is preferredfor application and can be represented as (Torrence andCompo, 1998; Torrence and Webster, 1999; Grinstedet al., 2004):

() = 1/4ei0.52 (11)

where is the dimensionless frequency, and is thedimensionless time parameter. The wavelet is stretchedin time (t) by varying its scale (s), so that = s/t . When

using wavelets for feature extraction purposes, the Morletwavelet (with = 6) is a good choice, since it satisfiesthe admissibility condition (Farge, 1992; Torrence andCompo, 1998).

For a given wavelet 0(), it was assumed that Xj isa time series of length N (Xj, i = 1, . . . , N ) with equaltime spacing t . The continuous wavelet transform of adiscrete sequence Xj is defined as a convolution of Xjwith the scaled and translated wavelet 0():

WnX(s) =

Nj=1

Xj[(j n)t

s

](12)

where asterisk indicates the complex conjugate. CWTdecomposes a time series into time-frequency space,enabling the identification of both the dominant modesof variability and how those modes vary with time.

4.3. Wavelet fuzzy logicGeophysical time series include different patterns, such asperiodicity, trend, noise which are the results of differentmechanisms affecting the process. Filtering such patternshelps understand the behaviour of time series. One oflatest techniques used for filtering time series in timeand scale domains is the wavelet transform. There is atendency to filter the data before its use, especially in pre-dicting problems. Several researchers (Kim and Valdes,2003; Webster and Hoyos, 2004; Partal and Kisi, 2007;Nourani et al., 2009) have proposed that it is better tomake predictions after decomposing both predictors andpredictand into several bands. Wavelet transform makesit possible to separate time series into its subseries. Here,the important question is how the significant bands canbe selected. For this purpose, Webster and Hoyos (2004)proposed the use of average wavelet spectra obtainedfrom continuous wavelet transform of a variable of con-cern. The significant spectral bands can be selected, basedon the average wavelet spectra which show the varia-tion of power with scales. A sample band selection forPDSI is shown in Figure 2 along with its wavelet power

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Figure 2. (a) Continuous wavelet map of PDSI series for climate division 7 and its corresponding (b) average wavelet spectra over the period19002007. There are five significant bands detected from average spectra which are 264 months. This

figure is available in colour online at wileyonlinelibrary.com/journal/joc



map for climate division 7. There has not been a ruleestablished to separate the bands so far. The importantcriterion for the separation of bands is to detect the bandsthat have significant power compared to others. Otherbands are separated according to their average waveletpower, respectively. For instance, in this study, band 3(66111 months) shows peak power and can be distin-guished from others. The Morlet wavelet was employedfor the continuous wavelet transform. It can be seen fromFigure 2 that it is possible to separate the original timeseries into five different significant bands. These are 264 months. Thus, atthe end, we have five different sub-series each of whichcarries specific information about the process. However,each predictor time series is separated into five differentsubseries using the same spectral bands as of predic-tand.

Subsequently, it is required to relate each band ofpredictors to the corresponding band of predictand witha statistical scheme. Here, we used a fuzzy logic modelto establish a connection between predictors and thepredictand band. Five fuzzy models would be needed tomake predictions. Finally, all those five predicted bandsof the predictand variable are reconstructed to obtain thefinal series. A schematic diagram of the overall procedureis shown in Figure 3.

5. Results and discussion

To predict PDSI from meteorological variables andclimate indices, FL and WFL models were applied. Theresults were obtained for ten different climate divisionsin Texas. Five scenarios, each of them included differentpredictand combinations (Table I), were employed to seehow combinations affect the accuracy of models.

Figure 3. Flowchart of the methodology. This figure is available incolour online at wileyonlinelibrary.com/journal/joc

5.1. Wavelet band separationThe selection of bands which carry significant poweris important for the model setup. The separation ofbands was made by considering the average waveletspectra of the PDSI series for each climate division.We obtained different groups of spectral bands accordingto the average wavelet spectra of PDSIs shown inFigure 4. The significant bands detected from the averagewavelet spectra of PDSI are presented in Table II. Thepredictors were separated into their bands according tothose intervals, identically.

The PDSI average wavelet spectra consist of severalpeaks each of which represents a significant power atthe corresponding period. It is apparent from Table IIthat the PDSI time series for all climate divisions canbe separated into five significant bands. While the firstband shows noisy data, the fourth and fifth bands standfor low-frequency variation of PDSI. The higher poweris observed at around 60120-month period in climatedivisions 7 and 8, which are located in the south-central Texas. In panhandle (climate divisions 1 and 2), ahigher power occurs for 60240 months which showsthe importance of mediumrange droughts. However,low-frequency variation is remarkable in the arid zone(climate division 5).

All the bands carry specific information related to theoriginal time series. It can be said that the bands arerectified from the effects of processes involved in thegeneration of time series and represent only one feature ofthe concerned series. For instance, the higher level band(e.g. >200 months) contains only information on long-time cycles of the concerned variable and excludes otherproperties such as noisy data, trends. However, short-timecycles (e.g.


Table I. Results of FL and WFL modelling of PDSI for each ten climate divisions.

CDs Inputs FL WFL

Train Test Train Test

R2 Correlationcoefficient




CD-1 NINO 3.4,PDO;Pcp,Temp 0.266 0.515 0.175 0.533 0.7873 0.8882 0.4202 0.6876NINO 3.4;Pcp,Temp 0.243 0.492 0.062 0.473 0.7537 0.8705 0.5485 0.7525

PDO;Pcp,Temp 0.531 0.816 0.162 0.715 0.7131 0.845 0.5175 0.7198Pcp,Temp 0.237 0.486 0.082 0.484 0.6615 0.8138 0.6035 0.7774

NINO 3.4,PDO 0.036 0.186 0.023 0.334 0.091 0.303 0.050 0.155CD-2 NINO 3.4,PDO;Pcp,Temp 0.226 0.474 0.221 0.502 0.697 0.836 0.334 0.639

NINO 3.4;Pcp,Temp 0.192 0.437 0.098 0.419 0.7917 0.89 0.4163 0.6854PDO;Pcp,Temp 0.215 0.462 0.255 0.534 0.752 0.8676 0.3921 0.6489

Pcp,Temp 0.204 0.451 0.138 0.457 0.6452 0.8067 0.4893 0.7149NINO 3.4,PDO 0.080 0.282 0.106 0.095 0.115 0.342 0.101 0.334





Pcp,Temp 0.269 0.518 0.178 0.485 0.648 0.805 0.491 0.764NINO 3.4,PDO 0.047 0.213 0.074 0.071 0.111 0.336 0.127 0.088





Pcp,Temp 0.244 0.493 0.272 0.528 0.631 0.794 0.560 0.755NINO 3.4,PDO 0.116 0.340 0.004 0.332 0.170 0.412 0.017 0.358





Pcp,Temp 0.258 0.507 0.223 0.514 0.647 0.807 0.516 0.765NINO 3.4,PDO 0.064 0.250 0.049 0.268 0.135 0.372 0.097 0.192





Pcp,Temp 0.184 0.429 0.192 0.437 0.506 0.731 0.589 0.784NINO 3.4,PDO 0.081 0.282 0.162 0.468 0.086 0.337 0.139 0.421

CC, correlation coefficient.



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Figure 4. Average wavelet spectra of precipitation and PDSI time series. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

in most of the cases. This significant improvement in themodel accuracy makes it possible to use these models inpractical applications.

The increase in the R2 values by the WFL modelcan be related to its setup. The main idea behind theWFL method is based on the wavelet banding explainedabove. Since WFL uses information at various spectral

bands separately, it can capture and model the databehaviour (e.g. periodicity, noise) easily compared tothe simple FL model. WFL models consist of a certainnumber of FL models which is equal to the numberof separated bands from an original time series. Forinstance, assume that five different spectral bands aredetected from the average wavelet spectra of PDSI.



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Figure 4. (Continued).

Subsequently, predictors are separated into the same fivebands. Thus, there are five different series all of whichcontain significant spectral power. Five different fuzzymodels were established and then the results of them werereconstructed to obtain a single series. Figure 5 showssignificant spectral bands and predicted bands of PDSIfrom the identical bands of NINO 3.4, precipitation, andtemperature. Observed and predicted time series of PDSIfor climate division 7 along with the scatter diagrams

of observed and predicted PDSI in the validation period(19702007) for WFL and FL models are shown inFigure 6.

5.3. WFL model capability in climate divisionsConsidering climate divisions, the WFL model resultswere evaluated throughout Texas. The WFL model per-formance shows variability from one division to the other,as given in Table I, for all climate divisions in terms



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Figure 5. (ae)Time series of the five observed and predicted wavelet bands for climate division 7 PDSIs, and (f) final reconstructed and observedPDSI time series. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

of the R2 values and correlation coefficients. The tablepresents the results of both FL and WFL models fortraining (calibration) and testing (validation) periods. Inthe evaluation, the R2 value for the testing period wastaken as an indicator of model performance. It is evidentfrom the table that WFL outperforms the FL model inall cases. Precipitation and temperature time series alongwith climate indices (NINO 3.4 and PDO) as predictorswere combined variously to determine the best combi-nation. No significant effect of climate indices was seenthat increased the WFL model capabilities. It is apparentfrom Table I that the capabilities of the models disappearcompletely when the predictor combinations constitutedwithout using precipitation and temperature. Precipitationand temperature are the driven factors in the predictionof PDSI. To understand the model capability across cli-mate divisions, the average wavelet spectra of PDSI alongwith the predictor variables were considered. The aver-age wavelet spectra of NINO 3.4 and PDO are depictedin Figure 7. Different energy patterns in their spectra canbe seen from the figure. While NINO 3.4 has significant

Table II. Significant bands selected from average waveletspectra of PDSIs for ten climate divisions.

Climatedivisions

PDSI band separation (months)

1 2 3 4 5

1 2222 1873 2224 2645 2646 2647 2648 2649 18710 187

power at the 6070-month band, PDO exhibits a signifi-cant power at around 6070 and 300340-month bands.The wavelet spectra of the temperature time series for all



(a)

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Figure 6. (a) Observed and predicted time series of PDSI for climate division 7. FL and WFL models were employed to predict PDSI fromNINO 3.4, precipitation, and temperature. Scatter diagrams of observed and predicted PDSI in validation period (19712006) for (b) wavelet

fuzzy logic model (WFL), and (c) fuzzy logic (FL) model. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

climate divisions show nearly the same pattern (only twoare shown, Figure 8). The significant energy is presentat 12-month (1 year) band which shows the annual cycleof temperature variation. However, the average waveletspectra of precipitation changes from division to division.The wavelet spectra of precipitation along with the cor-responding PDSI time series are shown in Figure 4. It isseen from the figure that annual cycles (significant powerat 12-month band) in precipitation are dominant for cli-mate divisions 1, 2, 5, 6, 9, and 10. For the other climatedivisions (3, 4, 7, and 8), it is observed that low-frequency

bands are significant which indicate the presence of dif-ferent precipitation generating mechanisms.

Since the wavelet spectra of precipitation and theWFL model results show different patterns throughoutthe climate divisions, a possible relation between thesespectra and the WFL model performance scores (R2) canbe expected. Investigation of average wavelet plots alongwith R2 values reveals that the WFL model performsbetter in the climate divisions where the annual cycleof precipitation is dominant. However, the accuracyof WFL model reduces in the places where multiple

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Figure 7. Average wavelet spectra for (a) NINO 3.4 and (b) PDO index. This figure is available in colour online at wileyonlinelibrary.com/journal/joc



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Figure 8. Average wavelet spectra of temperature time series for (a) climate division 5 located in arid zone and (b) climate division 8located in sub-tropic humid zone. Same pattern of variation is seen for other climate divisions. This figure is available in colour online at

wileyonlinelibrary.com/journal/joc

peaks of significant power are seen in their spectra. Thereason behind this result can be related to significantpower bands other than 12-month band which makethe prediction issue more complex. Multiple powerpeaks in the wavelet spectra of precipitation show thepresence of various frequency regime combinations in thetime series. This kind of nonlinear interactions betweenseveral physical processes leads to more disorderedPDSI series which finally makes the prediction of PDSIdifficult.

6. Conclusions

Prediction of PDSI is achieved from precipitation, tem-perature, and large scale climate indices by using WFL,which is a relatively new methodology. This method isapplied to ten climate regions in Texas to model PDSI.The model results are compared to FL approach. Thefollowing conclusions can be drawn from this study:

1. The WFL model predicts PDSI satisfactorily from pre-cipitation and temperature. This enables to determinePDSI in the absence of soil moisture information andother parameters required for the calculation of PDSI.Inversely, it is possible to estimate soil moisture fromthe predicted PDSI values.

2. A significant improvement is obtained over the FLmodel in the prediction of PDSI by using WFL whichis capable of modelling more complex systems.

3. The effect of large-scale climate indices on the predic-tion of PDSI is not important. While in some climatedivisions they improve the WFL model performanceslightly, in general their impact on prediction is minor.Precipitation and temperature are the main predictorsfor PDSI.

4. The evaluation of average wavelet spectra of predictorvariables reveals that only precipitation time seriesexhibits different spectral patterns throughout climatedivisions. Temperature time series shows nearly thesame pattern which is a significant power at 12 monthsfor all climate divisions.

5. The behaviour of precipitation has a significant impacton the PDSI time series which eventually affects theperformance of the WFL model. It is found that WFLperforms better in the climate divisions where theaverage wavelet spectra of precipitation show singlepeak energy around 12-month cycle which indicatesthe regular annual cycle. These regular precipitationevents also put the PDSI time series in order.

A future work can be the estimation of soil moisturefrom PDSI. The predicted PDSI values from precipita-tion and temperature can be used to obtain soil moisturefor the places where a network of soil-moisture measure-ments does not exist.

AcknowledgementsThis work was financially supported by the United StatesGeological Survey (USGS, Project ID: 2009TX334G)and Texas Water Resources Institute (TWRI) throughthe project Hydrological Drought Characterization forTexas under Climate Change, with Implications for WaterResources Planning and Management. The authors arethankful to the reviewers for their insightful commentswhich helped improve the quality of the manuscript.

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