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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
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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.
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Climate divisions
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30
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1
2
3
4
5 6
78
9
10
(b)
-106 -104 -102 -100 -98 -96 -94
Climate map
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28
30
32
34
36
Sub-tropic humid
Sub-tropicsemi-humid
Semi-aridArid
Continental (a)
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)
Copyright 2010 Royal Meteorological Society Int. J. Climatol.
31: 20212032 (2011)
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2024 M. OZGER et al.
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
050
100150
200250
300350
4000 2 4 6 8 10
Power (%)
(1)(2)(3)
(4)
(5)
(a) (b)
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
Copyright 2010 Royal Meteorological Society Int. J. Climatol.
31: 20212032 (2011)
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ESTIMATING PALMER DROUGHT SEVERITY INDEX 2025
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.
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2026 M. OZGER et al.
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
R2 Correlationcoefficient
R2 Correlationcoefficient
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
CD-3 NINO 3.4,PDO;Pcp,Temp 0.296 0.543 0.171 0.472 0.674 0.822
0.369 0.715NINO 3.4;Pcp,Temp 0.227 0.475 0.109 0.453 0.657 0.811
0.339 0.707
PDO;Pcp,Temp 0.254 0.503 0.176 0.472 0.648 0.805 0.367
0.728Pcp,Temp 0.220 0.468 0.063 0.409 0.669 0.819 0.346 0.681
NINO 3.4,PDO 0.064 0.252 0.030 0.297 0.145 0.383 0.015 0.258CD-4
NINO 3.4,PDO;Pcp,Temp 0.322 0.567 0.228 0.505 0.698 0.836 0.379
0.703
NINO 3.4;Pcp,Temp 0.270 0.519 0.159 0.470 0.682 0.826 0.441
0.726PDO;Pcp,Temp 0.281 0.530 0.226 0.505 0.656 0.810 0.496
0.766
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
CD-5 NINO 3.4,PDO;Pcp,Temp 0.291 0.539 0.285 0.550 0.645 0.815
0.523 0.740NINO 3.4;Pcp,Temp 0.264 0.513 0.261 0.512 0.619 0.798
0.601 0.790
PDO;Pcp,Temp 0.278 0.526 0.282 0.549 0.606 0.790 0.596
0.779Pcp,Temp 0.258 0.507 0.267 0.520 0.588 0.778 0.614 0.796
NINO 3.4,PDO 0.065 0.253 0.069 0.293 0.129 0.387 0.018 0.105CD-6
NINO 3.4,PDO;Pcp,Temp 0.320 0.565 0.301 0.614 0.664 0.815 0.573
0.760
NINO 3.4;Pcp,Temp 0.249 0.499 0.314 0.568 0.642 0.801 0.557
0.750PDO;Pcp,Temp 0.313 0.558 0.284 0.592 0.644 0.803 0.603
0.782
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
CD-7 NINO 3.4,PDO;Pcp,Temp 0.286 0.534 0.341 0.590 0.698 0.836
0.492 0.727NINO 3.4;Pcp,Temp 0.238 0.487 0.277 0.528 0.680 0.825
0.568 0.763
PDO;Pcp,Temp 0.281 0.529 0.341 0.588 0.678 0.824 0.538
0.752Pcp,Temp 0.229 0.477 0.247 0.501 0.668 0.817 0.561 0.765
NINO 3.4,PDO 0.087 0.293 0.113 0.342 0.228 0.477 0.026 0.365CD-8
NINO 3.4,PDO;Pcp,Temp 0.314 0.559 0.330 0.580 0.652 0.810 0.530
0.764
NINO 3.4;Pcp,Temp 0.275 0.524 0.258 0.543 0.649 0.808 0.529
0.761PDO;Pcp,Temp 0.309 0.555 0.357 0.606 0.639 0.802 0.537
0.775
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
CD-9 NINO 3.4,PDO;Pcp,Temp 0.253 0.502 0.261 0.514 0.676 0.824
0.622 0.797NINO 3.4;Pcp,Temp 0.225 0.474 0.249 0.519 0.658 0.814
0.630 0.806
PDO;Pcp,Temp 0.246 0.495 0.275 0.538 0.662 0.816 0.582
0.786Pcp,Temp 0.201 0.447 0.210 0.480 0.656 0.812 0.610 0.800
NINO 3.4,PDO 0.002 0.013 0.006 0.009 0.140 0.377 0.081
0.298CD-10 NINO 3.4,PDO;Pcp,Temp 0.280 0.528 0.367 0.632 0.544
0.756 0.636 0.812
NINO 3.4;Pcp,Temp 0.189 0.434 0.205 0.451 0.524 0.742 0.583
0.778PDO;Pcp,Temp 0.251 0.500 0.318 0.590 0.527 0.745 0.617
0.799
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.
Copyright 2010 Royal Meteorological Society Int. J. Climatol.
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Figure 4. Average wavelet spectra of precipitation and PDSI time
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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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2028 M. OZGER et al.
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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
Copyright 2010 Royal Meteorological Society Int. J. Climatol.
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ESTIMATING PALMER DROUGHT SEVERITY INDEX 2029
(a) (b)
(c) (d)
(e) (f)
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
Copyright 2010 Royal Meteorological Society Int. J. Climatol.
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2030 M. OZGER et al.
(a)
(b) (c)
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
Copyright 2010 Royal Meteorological Society Int. J. Climatol.
31: 20212032 (2011)
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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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