Research ArticleIdentifying and Evaluating Chaotic Behavior inHydro-Meteorological Processes

Soojun Kim1 Yonsoo Kim2 Jongso Lee2 and Hung Soo Kim2

1Columbia Water Center Columbia University New York NY 10027 USA2Department of Civil Engineering Inha University Incheon 402-751 Republic of Korea

Correspondence should be addressed to Hung Soo Kim sookiminhaackr

Received 20 November 2014 Accepted 7 April 2015

Academic Editor Ismail Gultepe

Copyright copy 2015 Soojun Kim et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of this study is to identify and evaluate chaotic behavior in hydro-meteorological processes This study poses the twohypotheses to identify chaotic behavior of the processes First assume that the input data is the significant factor to provide chaoticcharacteristics to output data Second assume that the system itself is the significant factor to provide chaotic characteristics tooutput data For solving this issue hydro-meteorological time series such as precipitation air temperature discharge and storagevolume were collected in the Great Salt Lake and Bear River Basin USA The time series in the period of approximately one yearwere extracted from the original series using the wavelet transform The generated time series from summation of sine functionswere fitted to each series and used for investigating the hypothesesThen artificial neural networks had been built for modeling thereservoir system and the correlation dimension was analyzed for the evaluation of chaotic behavior between inputs and outputsFrom the results we found that the chaotic characteristic of the storage volume which is output is likely a byproduct of the chaoticbehavior of the reservoir system itself rather than that of the input data

1 Introduction

Hydrologic phenomena arise as a result of interactionsbetween climate inputs and landscape characteristics thatoccur over a wide range of space and time scales Due tothe tremendous heterogeneities in climatic inputs and land-scape properties such phenomena may be highly variableand ldquocomplexrdquo at all scales [1] The nonlinear behavior ofhydrologic systems had been known for a long time [2 3]Therainfall-runoff process is nonlinear almost regardless of thebasin area land uses rainfall intensity and other influencingfactors which are changing in a highly nonlinear fashion andso are the outputs often in unknown ways [1]

To study the nonlinear characteristics of natural phe-nomena many statisticians and scientists have suggestedthe chaos theory which analyze and forecast the nonlinearphenomena of the natural system Lorenz [4] suggested thestrange attractor in a simple model of convection roll inthe atmosphere Packard et al [5] suggested the method ofdelays and Takens [6] proved the method of delays usingdifferential topology Grassberger and Procaccia [7] and

Farmer et al [8] demonstrated the estimation of chaoticcharacterization using correlation dimension Wolf et al [9]calculated the largest Lyapunov exponent using the Benettinrsquosmethod Fraser and Swinney [10] suggested a method forthe estimation of time delay using the mutual informationGilmore [11] introduced the topological method for chaoscharacterization especially useful for small data sets Farmerand Sidorowich [12] forecasted the chaotic time series usingthe local linear approximation Also Casdagli [13] forecastedthe chaotic time series using the radial basis functionsand Casdagli and Weigend [14] modeled and forecastedthe chaotic time series using DVS (deterministic versusstochastic) algorithm Kim et al [15 16] suggested a newmethod for the estimation of delay parameters in chaosanalysis Falanga and Petrosino [17] estimated the complexityof the system by the degrees of freedom necessary to describethe asymptotic dynamics in a reconstructed phase space Themechanism of stochastic resonance which is a nonlinearphenomenon has been applied in the field of the physics ofatmosphere since it was introduced by Benzi et al [18 19] andNicolis [20]

Hindawi Publishing CorporationAdvances in MeteorologyVolume 2015 Article ID 195940 12 pageshttpdxdoiorg1011552015195940

2 Advances in Meteorology

Many hydrologists have been also analyzed hydrologicphenomena using nonlinear deterministic chaos to inter-pret the nonlinear characteristic of the hydrologic systemRodriguez-Iturbe et al [21] found the chaotic characteris-tics in rainfall data recorded with the time interval of 15seconds using the correlation dimension and the Lyapunovexponent Wilcox et al [22] tested the chaotic behaviorof daily snowmelt runoff data by correlation dimensionSangoyomi et al [23 24] used theGreat Salt Lake volume datarecordedwith the time interval of 15 days for searching for thechaotic characteristics Jeong and Rao [25] used 13 tree ringseries to determine their chaos characteristics Rodriguez-Iturbe et al [26] investigated the nonlinear dynamics ofsoil moisture using a soil moisture balance equation Kimet al [27] searched strange attractor in wastewater flowusing the C-C method Ahn and Kim [28] showed that thenonlinear stochastic model is more valid for the SOI timeseries analysis and modeling than linear stochastic analogby the BDS statistic Kim et al [29] assessed nonlineardeterministic characteristics in hydrologic time series likerainfall stream flow and reservoir volume series Sivakumaret al [30] examined the utility of nonlinear dynamic conceptsfor analysis of rainfall variability across Western AustraliaKim et al [31] assessed the applicability chaotic dynamics andfiltering techniques in radar rainfall

Even though Salas et al [32] investigated how hydrologicprocess (eg precipitation) which is low-dimensional chaoticis changed by its transformations such as aggregation andsampling mostly the single hydrologic time series havebeen analyzed for investigating its chaotic and nonlineardynamic characteristics Therefore the aim of this study isto identify chaotic behavior for the components in hydro-meteorological processes such as air temperature precipita-tion discharge and lake storage volume series The compo-nents contribute to hydro-meteorological system as inputsand outputs For this the main question is given that whatis the significant factor to provide chaotic characteristicsto output data We pose the following two hypotheses (1)Assume that the input data is the significant factor to providechaotic characteristics to output data (2) Assume that thesystem itself is the significant factor to provide chaoticcharacteristics to output data

This paper is organized to solve the issue as follows InSection 2 we give brief overview of the methodology toestimate the correlation dimension which can detect chaoticcharacteristics of data series In Section 3 we also give briefoverview of the wavelet transform to extract the data ofthe representative period from the original time series andartificial neural networks (ANN) for modeling the hydro-meteorological system In Section 4 we apply methods foridentifying chaotic behavior of the data series and discuss theresults Finally in Section 5 we summarize the findings andconclusions

2 Estimation of Correlation Dimension

21 Phase Space Reconstruction Phase space is a usefultool for representing the evolution of a system in time

It is essentially a graph or a coordinate diagram whosecoordinates represent the variables necessary to completelydescribe the state of the system at any moment (in otherwords the variables that enter the mathematical formulationof the system) The trajectories of the phase space diagramdescribe the evolution of the system from some initial statewhich is assumed to be known and hence represent thehistory of the system [5] The ldquoregion of attractionrdquo of thesetrajectories in the phase space provides at least importantqualitative information on the ldquoextent of complexityrdquo of thesystem which can subsequently be verified quantitativelyusing methods based on for example the concept of dimen-sionality

For a dynamic system with known partial differentialequations (PDEs) the system can be studied by discretizingthe PDEs and the set of variables at all grid points constitutesa phase space One difficulty in constructing the phase spacefor such a system is that the (initial) values of many of thevariables may not be known However a time series of asingle variable of the system may be available which mayallow the attractor (a geometric object that characterizes thelong-term behavior of a system in the phase space) to bereconstructed The idea behind such a reconstruction is thata (nonlinear) system is characterized by self-interaction sothat a time series of a single variable can carry the informationabout the dynamics of the entire multivariable system Manymethods are available for phase space reconstruction from anavailable time series Among these themethod of delays (eg[6]) is the most widely used one According to this methodgiven a single-variable series 119883

119894 where 119894 = 1 2 119873 a

multidimensional phase space can be reconstructed as

119884119895= (119883119895 119883119895+120591

119883119895+2120591

119883119895+(119898minus1)120591

) (1)

where 119895 = 1 2 119873 minus (119898 minus 1)120591 119898 is the dimensionof the vector 119884

119895 called embedding dimension and 120591 is an

appropriate delay time (an integermultiple of sampling time)A correct phase space reconstruction in a dimension 119898

generally allows interpretation of the system dynamics (if thevariable chosen to represent the system is appropriate) in theform of an 119898-dimensional map 119891

119879 given by

119884119895+119879

= 119891119879(119884119895) (2)

where119884119895and119884

119895+119879are vectors of dimension119898 describing the

state of the system at times 119895 (current state) and 119895 + 119879 (futurestate) respectively

22 Correlation Integral and Correlation Dimension Thedimension of a time series is in a way a representation of thenumber of variables dominantly governing the underlyingsystem dynamics Correlation dimension is a measure ofthe extent to which the presence of a data point affectsthe position of the other points lying on the attractor inthe phase space The correlation dimension method usesthe correlation integral (or function) for determining thedimension of the attractor and hence for distinguishingbetween low-dimensional chaos and high-dimensional sys-tem The concept of the correlation integral is that a time

Advances in Meteorology 3

series arising from deterministic dynamics will have a limitednumber of degrees of freedom equal to the smallest numberof first-order differential equations that capture the dominantfeatures of the dynamics Thus when one constructs phasespaces of increasing dimension a point will be reached wherethe dimension equals the number of degrees of freedombeyond which increasing the phase space dimension willnot have any significant effect on correlation dimensionMany algorithms have been formulated for the estimation ofthe correlation dimension Among these the Grassberger-Procaccia algorithm [7] has been the most popular Thealgorithm uses the concept of phase space reconstruction forrepresenting the dynamics of the system from an availablesingle-variable time series as presented in (1) For an m-dimensional phase space the correlation integral or function119862(119903) is given by

119862 (119903) = lim119873rarrinfin

2

119873 (119873 minus 1)sum

119894119895 (1le119894lt119895le119873)

119867(119903 minus10038161003816100381610038161003816119884119894minus 119884119895

10038161003816100381610038161003816) (3)

where 119867 is the Heaviside step function with 119867(119906) = 1 for119906 gt 0 and 119867(119906) = 1 for 119906 le 0 where 119906 = 119903 minus 119884

119894minus 119884119895 119903

is the vector norm (radius of sphere) centered on 119884119894or 119884119895 If

the time series is characterized by an attractor then 119862(119903) and119903 are related according to

119862 (119903)119903rarr0119873rarrinfin asymp 120572119903] (4)

where 120572 is a constant and ] is the correlation exponent orthe slope of the Log119862(119903) versus Log 119903 plot The slope isgenerally estimated by a least square fit of a straight line overa certain range of 119903 (scaling regime) or through estimation oflocal slopes between 119903 values The distinction between low-dimensional (perhaps determinism) and high-dimensional(perhaps stochasticity) can be made using the ] versus 119898

plot If ] saturates after a certain 119898 and the saturation valueis low then the system is generally considered to exhibitlow-dimensional and possibly deterministic dynamics Thesaturation value of ] is defined as the correlation dimension(1198632) of the attractor and the nearest integer above thisvalue is generally an indication of the number of variablesdominantly governing the dynamics On the other hand if119898 increases without bound with increase in 119898 the systemunder investigation is generally considered to exhibit high-dimensional and possibly stochastic behavior

3 Wavelet Transform and ArtificialNeural Networks

31 Wavelet Transform According to Fourier theory a signalcan be expressed as the sum of a possibly infinite series ofsine and cosines referred to as a Fourier expansion [33]However a Fourier expansion has only frequency resolutionand not time resolution that is no amplitude modulationof the signal at a given frequency is considered Moving-window Fourier transforms have been used to address thisissue but this method is sensitive to the choice of windowwidth Alternatively the wavelet transform [34 35] enablesthe identification of frequency components as well as their

variation in time The continuous wavelet transform of adiscrete sequence 119909119899 is defined by the convolution of 119909

119899with

a scaled and translated wavelet function 120595

119882119899 (119904) =

119873minus1

sum

1198991015840=0

1199091198991015840120595 lowast [

(1198991015840minus 119899) 120575119905

119904] (5)

where (lowast) indicates the complex conjugate 119899 is the localizedtime index 119904 = 0 is the scale parameter and119873 is the numberof points in the time series In this study we use the Morletwavelet function defined as 120595(120578) = 120587

minus14119890119894120596012057811989012057822 where 120596

0

is a frequency and 120578 is a nondimensional ldquotimerdquo parameter Byvarying the wavelet scales and translating along the localizedtime index 119899 one can construct a picture that shows boththe amplitude of any features versus the scale and how thisamplitude varies with time A vertical slice through a waveletplot is a measure of the local spectrum The time-averagedwavelet spectrum over all the local wavelet spectra gives theglobal wavelet spectrum

1198822

119905(119904) =

1

119879

119879minus1

sum

119905=0

1003816100381610038161003816119882119905 (119904)1003816100381610038161003816

2 (6)

A more detailed presentation for wavelet transform anal-ysis is referred to read Torrence and Compo [35]

32 Artificial Neural Networks ANN is a model of neuro-transmission by a neuron which is a nerve cell in the humanbrain ANN is an empirical pattern search technique thatenables the consideration of a nonlinear relationship betweeninput variables and output variables ANN is used in variousareas because of its unique applicability [36 37]This includesthe field of climate science where its applicability is proven[38 39]

Many studies suggest the ANN technique which is anonlinear model of the data series and ANN is better thanother techniques by way of systematic evaluation of varioustechniques [40 41] Therefore this study also applies ANNwhich is judged to have superior applicability in the simula-tion of nonlinear characteristics of the hydro-meteorologicalsystem

4 Applications and Results

41 Study Area and Data Series Used The Bear River Basinlocated in northeastern Utah southeastern Idaho and south-westernWyoming comprises 7500 squaremiles ofmountainand valley lands including 2700 in Idaho 3300 in Utah and1500 in Wyoming The Bear River crosses state boundariesfive times and is the largest stream in the western hemispherethat does not empty into the ocean It ranges in elevationfrom over 1278 to 3868 feet and is unique in that it isentirely enclosed by mountains thus forming a huge basinwith no external drainage outlets (httpwwwgreatsaltlake-infoorgBackgroundBearRiver) The Bear River is the larg-est tributary to the Great Salt Lake (see Figure 1)

4 Advances in Meteorology

Great Salt Lake

Bear River Basin

Weather Station

N

Discharge StationWater Level StationBear River

DEM

(ft)

High 386823

Low 127839

0 15 30 60

(km)

Figure 1 Study area

The data was collected from weather gauging station(USC00424856 NOAA) stream flow gauging station (num-ber 10126000 USGS) and lake water level gauging station(number 10010000 USGS) for the period of 1903 to 1995Themonthly rainfall shows its statistics of average 2585mm andstandard deviation 2164mm the monthly mean tempera-ture shows average 57∘C and standard deviation 88∘C themonthly mean runoff shows average 15336 ft3s and standarddeviation 13312 ft3s the monthly mean storage in the lakeshows average 1494 times 105 ft3 and standard deviation 398 times

105 ft3 and these time series plots are shown in Figure 2

42 Extraction of a Representative Time Series by WaveletTransform All hydrological measurements are to someextent contaminated by noise And the noise limits theperformance of many techniques of identification modelingprediction and control of deterministic systems [42] Inde-pendent component analysis (ICA) as a popular method isable to extract periodic signals from noise or nonlinear mix-ture [43 44] It has been applied in the fields of meteorology[45] oceanography [46] volcanology [47 48] and remotesensing [49] This study however uses wavelet transformfor extracting the representative periodic components whichaffect the data series because ICA often leads to local mini-mum solution and the suitable source signals are not isolated[50] Moreover the order of the independent components(ICs) is difficult to be determined in comparisonwith wavelettransform

Wavelet power spectrum that estimated the waveletmother function using the Morlet function is shown inFigure 3 (left) and the extent of spectrum in each periodfor time series can be identified In this figure a solid half-circle line shows the edge of the cone of influence (COI) effectthat can be caused by the discontinuity of the beginning andend of data series In particular the upper part of the solidline is statistically significant (a 95 confidence interval) andthe lower part is excluded from interpretation Parts withhigh-density spectrum are observed in some periods within aconfidence interval Global wavelet power spectrum (GWP)

in Formula (6) which represents the average value accordingto the length of each period provides more effective infor-mation about spectrum Figure 3 (center) shows the resultof GWP about spectrum Considering that the right partof a solid line is statistically significant on a basis of a 95confidence level the periodic characteristics of the time seriescould be classified into one band The band shows a strongspectrum of the period of approximately 1 year The periodextracted from the wavelet spectrum is shown in the right ofFigure 3

43 Analysis of the Time Series Using Attractor andCorrelation Dimension

431 Attractor Analysis The attractor obtained by (1) candescribe the characteristics of a time series To obtain theattractor using (1) the index lag 119905 and embedding dimension119898 must be chosen appropriately The autocorrelation func-tion (ACF) is expected to provide a reasonable measure ofthe transition from redundance to irrelevance as a functionof delay The decorrelation time which is equal to the lag(delay time 120591) at which the ACF first attains the value zerois considered Otherwise 120591 should be chosen as the localminimum of ACF whichever occurs first [51 52] When theACF decays exponentially we select 120591 at which theACF dropsto zero [53] at lag time 4months in all of seriesTherefore thedelay times of the systems can be obtained fromACFs and theattractors are drawn in Figure 4 for each time series

For the attractor analysis this study uses the extractedtime series by the wavelet transform The attractors of thetime series have a circle with a boundary If the attractor inthe phase-space exhibits clearly within a very well definedboundary suggest that the dynamics are simple and thesystem is potentially low dimension Every time series havehas a shape with a boundary which looks like a chaotic seriesParticularly air temperature shows a verywell defined bound-ary and it is potentially low-dimensional series Precipitationhowever shows relatively high complex and irregular and itis a potentially high-dimensional system than the other dataseries

432 Correlation Dimension Analysis Figure 5 shows therelationship between the correlation dimension 1198632 andthe embedding dimensions 119898 from 1 to 15 for each timeseries The correlation dimension seems to increase with theembedding dimension up to a certain point and saturatebeyond that point Such a saturation of the correlationdimension is an indication of the existence of deterministicdynamicsThe saturation values of the correlation dimensionfor the series are showing 392 141 302 and 265 in Fig-ures 5(a)ndash5(d) The low correlation dimensions suggest thepresence of low-dimensional chaotic nature of the underlyingsystem dynamics As the nearest integer above the correlationdimension value generally provides the number of dominantvariables influencing the dynamics of the underlying systemthe correlation dimensions for the series indicate that thetime series of precipitation air temperature discharge andstorage volume are dominantly governed by four two four

Advances in Meteorology 5

1910 1920 1930 1940 1950 1960 1970 1980 19900

500

1000

1500

2000

2500

Time (year)

Mon

thly

pre

cipi

tatio

n (m

m)

(a) Precipitation

1910 1920 1930 1940 1950 1960 1970 1980 1990

0

10

20

30

Time (year)

minus20

minus10

Mon

thly

mea

n te

mpe

ratu

re (∘

C)

(b) Air temperature

1910 1920 1930 1940 1950 1960 1970 1980 19900

2000

4000

6000

8000

Time (year)

Mon

thly

mea

n di

scha

rge (

ft3s

)

(c) Discharge

1910 1920 1930 1940 1950 1960 1970 1980 199050

100

150

200

250

300

Time (year)M

onth

ly m

ean

vol

(times105

ft3)

(d) Storage volume

Figure 2 Monthly time series plots for the period of 1903ndash1995

and three variables respectively Here we can find the truththat the time series have different chaos characteristics evenif they are collected from a hydro-meteorological system

44 Correlation Dimension Analysis Using Synthetically Gen-erated Series Precipitation and air temperature from themeteorological system are considered as input time series ofthe runoff system On the same principle the output serieswhich is a discharge at the runoff system or the basin outletoccurred by input series of precipitation and air temperaturefrom the meteorological system can be the input data of thereservoir system Here the methodology is suggested to solvethe two hypotheses as follows We composed the input datasets which have an arbitrary correlation dimension and buildup ANNs as a nonlinear model for modeling the reservoirsystem The modeling results from the input data sets will bethe criterion of the hypotheses The first hypothesis will bereasonable if the system responses sensitively depending onthe arbitrary input data sets whereas the second hypothesiswill be reasonable if the system does not response sensitivelydepending on the input data sets

441 Correlation Dimensions of Generated Input Series to theReservoir System Theattractors in each time series (shown inFigure 4) have limit cycle regime which is the characteristicsof a periodic system Each time series as a periodic functioncan be written as an infinite sum of sine and cosine termsFourier [54] realized this first so that this infinite sum iscalled a Fourier series

The input data sets are composed of the nine sets using thethree sine functions in the each hydro-meteorological timeseries Here the sine function which is useful for applicationto a periodic time series data ismade using the fitting toolboxof MATLAB Therefore each time series is composed of thethree cases of case (a) case (b) and case (c) as shown inTable 1 Case (a) is composed of the sum of few sine functionsand case (c) is composed of the sum of lots of sine functionsrelatively Case (b) is between case (a) and case (c) In case ofprecipitation and discharge the functions are set to have atleast three sine functions because the series are dominantlygoverned by four variables from the results of the correlationdimension analysis in Section 432 It is found that thefitting results have a good applicability with the correlationcoefficient (CC) in precipitation 054ndash065 air temperature098ndash099 and discharge 088ndash092 for 1116 months (1903ndash1995) The results of the correlation dimension analysis ineach case are shown in Figure 6 The saturated correlationdimensions in each series are (a) 254 (b) 326 and (c) 405 inprecipitation (a) 102 (b) 184 and (c) 252 in air temperatureand (a) 248 (b) 313 and (c) 38 Case (c) is composed ofmany sine functions which showed the highest correlationdimension whereas case (a) shows the lowest correlationdimension in each time series

442 ANN Modeling and Correlation Dimension Analysisof Hydro-Meteorological System In order to build up theANN model this study sets precipitation air temperatureand discharge as the input layer and storage volume as theoutput layer As seen in Figure 7 a multilayered ANNmodel

6 Advances in Meteorology

Table 1 Fitting functions in each case of each time series

Fitting function CC

Precipitation(a) = 151 sdot sin(053 sdot 119905 minus 050) + 155 sdot sin(054 sdot 119905 minus 194) + 1548 sdot sin(054 sdot 119905 + 102) 054(b) = 150 sdot sin(053 sdot 119905 minus 053) + 1881 sdot sin(054 sdot 119905 minus 202) + 1882 sdot sin(054 sdot 119905 + 097) + 123 sdot sin(055 sdot 119905 minus 061) 060(c) = 150 sdot sin(053 sdot 119905 minus 053) + 1949 sdot sin(054 sdot 119905 minus 202) + 195 sdot sin(054 sdot 119905 + 098) + 123 sdot sin(055 sdot 119905 minus 060) +

1131 sdot sin(050 sdot 119905 + 063)065

Airtemperature

(a) = 1615 sdot sin(052 sdot 119905 minus 221) 098(b) = 1617 sdot sin(052 sdot 119905 minus 222) + 038 sdot sin(051 sdot 119905 + 046) 099(c) = 1614 sdot sin(052 sdot 119905 minus 220) + 037 sdot sin(051 sdot 119905 + 044) + 033 sdot sin(053 sdot 119905 minus 516) 099

Discharge(a) = 768 sdot sin(052 sdot 119905 minus 059) + 1202 sdot sin(053 sdot 119905 minus 261) + 1111 sdot sin(053 sdot 119905 + 019) 088(b) = 766 sdot sin(052 sdot 119905 minus 057) + 1286 sdot sin(053 sdot 119905 minus 269) + 12 sdot sin(053 sdot 119905 + 015) + 108 sdot sin(055 sdot 119905 minus 243) 089(c) = 3723 sdot sin(052 sdot 119905 minus 168) + 3161 sdot sin(052 sdot 119905 + 116) + 072 sdot sin(053 sdot 119905 minus 167) + 098 sdot sin(055 sdot 119905 minus 251) +

120 sdot sin(051 sdot 119905 minus 177)092

1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

sminus10

minus201910 1930 1950 1970 1990

Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 6

Rel variance

(a) Precipitation

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 5

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(b) Air temperature

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 4E + 142E + 7

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(c) Discharge

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 14

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(d) Storage volume

Figure 3 Extraction of the representative time series using the wavelet transform (left the wavelet power spectrum center the global waveletpower spectrum and right the extracted time series about the period of approximately 1 year)

consisting of one input layer two hidden layers and oneoutput layer has been built

Monthly data series from 1903 to 1970 (800 months)has been used for the learning period 316 months fromthe learning period (1970ndash1995) are set up as verification

periods and the applicability of the constructed ANN modelis reviewed by comparing it to observed storage volume asa target data series (see Table 2) For the composition of theprediction data we again compose the three input data likecases (A) (B) and (C) Case (A) is integrated from each

Advances in Meteorology 7

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(a) Precipitation

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(b) Air temperature

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(c) Discharge

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(d) Storage volume

Figure 4 Attractors in each time series

case (a) in each time series Case (B) and case (C) are alsointegrated in the same way as case (A) Data series of 1116months (1903ndash1995) in each case (A) (B) and (C) has beenused for the prediction period

First of all according to the model verification measures(see Figure 8) such as the coefficient of correlation (CC0986) and root mean squared error (RMSE 0061) ANN isfitted very well and found its good applicability

The storage volume series of a reservoir system is esti-mated using the ANN model after setting case (A) case (b)and case (c) as the input data And then the correlationdimension analysis is performed for the estimated storagevolume in each case The results show that 255 in case (A)integrated the low-dimensional cases (a) 281 in case (B)integrated the middle-dimensional cases (b) and 289 in case(C) integrated the high-dimensional cases (c) as shown inFigure 9

45 Summary and Discussions In this study we posedthe two hypotheses to identify chaotic behavior in hydro-meteorological processes For solving this issuewe composedthe input data sets like cases (A) (B) and (C) and appliedthem to ANN model on the reservoir system of the GreatSalt Lake The criterion of the hypotheses is the sensitivityof chaotic behavior in the system In other words the firsthypothesis is reasonable if chaotic behavior in the system issensitive depending on chaotic characteristics of the inputdata otherwise the second hypothesis is reasonable Theresults of the correlation dimension analysis on every caseanalyzed in this study were summarized in Table 3

As shown in Table 3 the correlation dimensions are 255in case (A) obtained from integrating the low dimensions(precipitation 254 air temperature 102 and discharge 248)and 281 in case (B) from integrating the middle dimensions(precipitation 326 air temperature 184 and discharge 331)

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

2 Advances in Meteorology

Many hydrologists have been also analyzed hydrologicphenomena using nonlinear deterministic chaos to inter-pret the nonlinear characteristic of the hydrologic systemRodriguez-Iturbe et al [21] found the chaotic characteris-tics in rainfall data recorded with the time interval of 15seconds using the correlation dimension and the Lyapunovexponent Wilcox et al [22] tested the chaotic behaviorof daily snowmelt runoff data by correlation dimensionSangoyomi et al [23 24] used theGreat Salt Lake volume datarecordedwith the time interval of 15 days for searching for thechaotic characteristics Jeong and Rao [25] used 13 tree ringseries to determine their chaos characteristics Rodriguez-Iturbe et al [26] investigated the nonlinear dynamics ofsoil moisture using a soil moisture balance equation Kimet al [27] searched strange attractor in wastewater flowusing the C-C method Ahn and Kim [28] showed that thenonlinear stochastic model is more valid for the SOI timeseries analysis and modeling than linear stochastic analogby the BDS statistic Kim et al [29] assessed nonlineardeterministic characteristics in hydrologic time series likerainfall stream flow and reservoir volume series Sivakumaret al [30] examined the utility of nonlinear dynamic conceptsfor analysis of rainfall variability across Western AustraliaKim et al [31] assessed the applicability chaotic dynamics andfiltering techniques in radar rainfall

Even though Salas et al [32] investigated how hydrologicprocess (eg precipitation) which is low-dimensional chaoticis changed by its transformations such as aggregation andsampling mostly the single hydrologic time series havebeen analyzed for investigating its chaotic and nonlineardynamic characteristics Therefore the aim of this study isto identify chaotic behavior for the components in hydro-meteorological processes such as air temperature precipita-tion discharge and lake storage volume series The compo-nents contribute to hydro-meteorological system as inputsand outputs For this the main question is given that whatis the significant factor to provide chaotic characteristicsto output data We pose the following two hypotheses (1)Assume that the input data is the significant factor to providechaotic characteristics to output data (2) Assume that thesystem itself is the significant factor to provide chaoticcharacteristics to output data

This paper is organized to solve the issue as follows InSection 2 we give brief overview of the methodology toestimate the correlation dimension which can detect chaoticcharacteristics of data series In Section 3 we also give briefoverview of the wavelet transform to extract the data ofthe representative period from the original time series andartificial neural networks (ANN) for modeling the hydro-meteorological system In Section 4 we apply methods foridentifying chaotic behavior of the data series and discuss theresults Finally in Section 5 we summarize the findings andconclusions

2 Estimation of Correlation Dimension

21 Phase Space Reconstruction Phase space is a usefultool for representing the evolution of a system in time

It is essentially a graph or a coordinate diagram whosecoordinates represent the variables necessary to completelydescribe the state of the system at any moment (in otherwords the variables that enter the mathematical formulationof the system) The trajectories of the phase space diagramdescribe the evolution of the system from some initial statewhich is assumed to be known and hence represent thehistory of the system [5] The ldquoregion of attractionrdquo of thesetrajectories in the phase space provides at least importantqualitative information on the ldquoextent of complexityrdquo of thesystem which can subsequently be verified quantitativelyusing methods based on for example the concept of dimen-sionality

For a dynamic system with known partial differentialequations (PDEs) the system can be studied by discretizingthe PDEs and the set of variables at all grid points constitutesa phase space One difficulty in constructing the phase spacefor such a system is that the (initial) values of many of thevariables may not be known However a time series of asingle variable of the system may be available which mayallow the attractor (a geometric object that characterizes thelong-term behavior of a system in the phase space) to bereconstructed The idea behind such a reconstruction is thata (nonlinear) system is characterized by self-interaction sothat a time series of a single variable can carry the informationabout the dynamics of the entire multivariable system Manymethods are available for phase space reconstruction from anavailable time series Among these themethod of delays (eg[6]) is the most widely used one According to this methodgiven a single-variable series 119883

119894 where 119894 = 1 2 119873 a

multidimensional phase space can be reconstructed as

119884119895= (119883119895 119883119895+120591

119883119895+2120591

119883119895+(119898minus1)120591

) (1)

where 119895 = 1 2 119873 minus (119898 minus 1)120591 119898 is the dimensionof the vector 119884

119895 called embedding dimension and 120591 is an

appropriate delay time (an integermultiple of sampling time)A correct phase space reconstruction in a dimension 119898

generally allows interpretation of the system dynamics (if thevariable chosen to represent the system is appropriate) in theform of an 119898-dimensional map 119891

119879 given by

119884119895+119879

= 119891119879(119884119895) (2)

where119884119895and119884

119895+119879are vectors of dimension119898 describing the

state of the system at times 119895 (current state) and 119895 + 119879 (futurestate) respectively

22 Correlation Integral and Correlation Dimension Thedimension of a time series is in a way a representation of thenumber of variables dominantly governing the underlyingsystem dynamics Correlation dimension is a measure ofthe extent to which the presence of a data point affectsthe position of the other points lying on the attractor inthe phase space The correlation dimension method usesthe correlation integral (or function) for determining thedimension of the attractor and hence for distinguishingbetween low-dimensional chaos and high-dimensional sys-tem The concept of the correlation integral is that a time

Advances in Meteorology 3

series arising from deterministic dynamics will have a limitednumber of degrees of freedom equal to the smallest numberof first-order differential equations that capture the dominantfeatures of the dynamics Thus when one constructs phasespaces of increasing dimension a point will be reached wherethe dimension equals the number of degrees of freedombeyond which increasing the phase space dimension willnot have any significant effect on correlation dimensionMany algorithms have been formulated for the estimation ofthe correlation dimension Among these the Grassberger-Procaccia algorithm [7] has been the most popular Thealgorithm uses the concept of phase space reconstruction forrepresenting the dynamics of the system from an availablesingle-variable time series as presented in (1) For an m-dimensional phase space the correlation integral or function119862(119903) is given by

119862 (119903) = lim119873rarrinfin

2

119873 (119873 minus 1)sum

119894119895 (1le119894lt119895le119873)

119867(119903 minus10038161003816100381610038161003816119884119894minus 119884119895

10038161003816100381610038161003816) (3)

where 119867 is the Heaviside step function with 119867(119906) = 1 for119906 gt 0 and 119867(119906) = 1 for 119906 le 0 where 119906 = 119903 minus 119884

119894minus 119884119895 119903

is the vector norm (radius of sphere) centered on 119884119894or 119884119895 If

the time series is characterized by an attractor then 119862(119903) and119903 are related according to

119862 (119903)119903rarr0119873rarrinfin asymp 120572119903] (4)

where 120572 is a constant and ] is the correlation exponent orthe slope of the Log119862(119903) versus Log 119903 plot The slope isgenerally estimated by a least square fit of a straight line overa certain range of 119903 (scaling regime) or through estimation oflocal slopes between 119903 values The distinction between low-dimensional (perhaps determinism) and high-dimensional(perhaps stochasticity) can be made using the ] versus 119898

plot If ] saturates after a certain 119898 and the saturation valueis low then the system is generally considered to exhibitlow-dimensional and possibly deterministic dynamics Thesaturation value of ] is defined as the correlation dimension(1198632) of the attractor and the nearest integer above thisvalue is generally an indication of the number of variablesdominantly governing the dynamics On the other hand if119898 increases without bound with increase in 119898 the systemunder investigation is generally considered to exhibit high-dimensional and possibly stochastic behavior

3 Wavelet Transform and ArtificialNeural Networks

31 Wavelet Transform According to Fourier theory a signalcan be expressed as the sum of a possibly infinite series ofsine and cosines referred to as a Fourier expansion [33]However a Fourier expansion has only frequency resolutionand not time resolution that is no amplitude modulationof the signal at a given frequency is considered Moving-window Fourier transforms have been used to address thisissue but this method is sensitive to the choice of windowwidth Alternatively the wavelet transform [34 35] enablesthe identification of frequency components as well as their

variation in time The continuous wavelet transform of adiscrete sequence 119909119899 is defined by the convolution of 119909

119899with

a scaled and translated wavelet function 120595

119882119899 (119904) =

119873minus1

sum

1198991015840=0

1199091198991015840120595 lowast [

(1198991015840minus 119899) 120575119905

119904] (5)

where (lowast) indicates the complex conjugate 119899 is the localizedtime index 119904 = 0 is the scale parameter and119873 is the numberof points in the time series In this study we use the Morletwavelet function defined as 120595(120578) = 120587

minus14119890119894120596012057811989012057822 where 120596

0

is a frequency and 120578 is a nondimensional ldquotimerdquo parameter Byvarying the wavelet scales and translating along the localizedtime index 119899 one can construct a picture that shows boththe amplitude of any features versus the scale and how thisamplitude varies with time A vertical slice through a waveletplot is a measure of the local spectrum The time-averagedwavelet spectrum over all the local wavelet spectra gives theglobal wavelet spectrum

1198822

119905(119904) =

1

119879

119879minus1

sum

119905=0

1003816100381610038161003816119882119905 (119904)1003816100381610038161003816

2 (6)

A more detailed presentation for wavelet transform anal-ysis is referred to read Torrence and Compo [35]

32 Artificial Neural Networks ANN is a model of neuro-transmission by a neuron which is a nerve cell in the humanbrain ANN is an empirical pattern search technique thatenables the consideration of a nonlinear relationship betweeninput variables and output variables ANN is used in variousareas because of its unique applicability [36 37]This includesthe field of climate science where its applicability is proven[38 39]

Many studies suggest the ANN technique which is anonlinear model of the data series and ANN is better thanother techniques by way of systematic evaluation of varioustechniques [40 41] Therefore this study also applies ANNwhich is judged to have superior applicability in the simula-tion of nonlinear characteristics of the hydro-meteorologicalsystem

4 Applications and Results

41 Study Area and Data Series Used The Bear River Basinlocated in northeastern Utah southeastern Idaho and south-westernWyoming comprises 7500 squaremiles ofmountainand valley lands including 2700 in Idaho 3300 in Utah and1500 in Wyoming The Bear River crosses state boundariesfive times and is the largest stream in the western hemispherethat does not empty into the ocean It ranges in elevationfrom over 1278 to 3868 feet and is unique in that it isentirely enclosed by mountains thus forming a huge basinwith no external drainage outlets (httpwwwgreatsaltlake-infoorgBackgroundBearRiver) The Bear River is the larg-est tributary to the Great Salt Lake (see Figure 1)

4 Advances in Meteorology

Great Salt Lake

Bear River Basin

Weather Station

N

Discharge StationWater Level StationBear River

DEM

(ft)

High 386823

Low 127839

0 15 30 60

(km)

Figure 1 Study area

The data was collected from weather gauging station(USC00424856 NOAA) stream flow gauging station (num-ber 10126000 USGS) and lake water level gauging station(number 10010000 USGS) for the period of 1903 to 1995Themonthly rainfall shows its statistics of average 2585mm andstandard deviation 2164mm the monthly mean tempera-ture shows average 57∘C and standard deviation 88∘C themonthly mean runoff shows average 15336 ft3s and standarddeviation 13312 ft3s the monthly mean storage in the lakeshows average 1494 times 105 ft3 and standard deviation 398 times

105 ft3 and these time series plots are shown in Figure 2

42 Extraction of a Representative Time Series by WaveletTransform All hydrological measurements are to someextent contaminated by noise And the noise limits theperformance of many techniques of identification modelingprediction and control of deterministic systems [42] Inde-pendent component analysis (ICA) as a popular method isable to extract periodic signals from noise or nonlinear mix-ture [43 44] It has been applied in the fields of meteorology[45] oceanography [46] volcanology [47 48] and remotesensing [49] This study however uses wavelet transformfor extracting the representative periodic components whichaffect the data series because ICA often leads to local mini-mum solution and the suitable source signals are not isolated[50] Moreover the order of the independent components(ICs) is difficult to be determined in comparisonwith wavelettransform

Wavelet power spectrum that estimated the waveletmother function using the Morlet function is shown inFigure 3 (left) and the extent of spectrum in each periodfor time series can be identified In this figure a solid half-circle line shows the edge of the cone of influence (COI) effectthat can be caused by the discontinuity of the beginning andend of data series In particular the upper part of the solidline is statistically significant (a 95 confidence interval) andthe lower part is excluded from interpretation Parts withhigh-density spectrum are observed in some periods within aconfidence interval Global wavelet power spectrum (GWP)

in Formula (6) which represents the average value accordingto the length of each period provides more effective infor-mation about spectrum Figure 3 (center) shows the resultof GWP about spectrum Considering that the right partof a solid line is statistically significant on a basis of a 95confidence level the periodic characteristics of the time seriescould be classified into one band The band shows a strongspectrum of the period of approximately 1 year The periodextracted from the wavelet spectrum is shown in the right ofFigure 3

43 Analysis of the Time Series Using Attractor andCorrelation Dimension

431 Attractor Analysis The attractor obtained by (1) candescribe the characteristics of a time series To obtain theattractor using (1) the index lag 119905 and embedding dimension119898 must be chosen appropriately The autocorrelation func-tion (ACF) is expected to provide a reasonable measure ofthe transition from redundance to irrelevance as a functionof delay The decorrelation time which is equal to the lag(delay time 120591) at which the ACF first attains the value zerois considered Otherwise 120591 should be chosen as the localminimum of ACF whichever occurs first [51 52] When theACF decays exponentially we select 120591 at which theACF dropsto zero [53] at lag time 4months in all of seriesTherefore thedelay times of the systems can be obtained fromACFs and theattractors are drawn in Figure 4 for each time series

For the attractor analysis this study uses the extractedtime series by the wavelet transform The attractors of thetime series have a circle with a boundary If the attractor inthe phase-space exhibits clearly within a very well definedboundary suggest that the dynamics are simple and thesystem is potentially low dimension Every time series havehas a shape with a boundary which looks like a chaotic seriesParticularly air temperature shows a verywell defined bound-ary and it is potentially low-dimensional series Precipitationhowever shows relatively high complex and irregular and itis a potentially high-dimensional system than the other dataseries

432 Correlation Dimension Analysis Figure 5 shows therelationship between the correlation dimension 1198632 andthe embedding dimensions 119898 from 1 to 15 for each timeseries The correlation dimension seems to increase with theembedding dimension up to a certain point and saturatebeyond that point Such a saturation of the correlationdimension is an indication of the existence of deterministicdynamicsThe saturation values of the correlation dimensionfor the series are showing 392 141 302 and 265 in Fig-ures 5(a)ndash5(d) The low correlation dimensions suggest thepresence of low-dimensional chaotic nature of the underlyingsystem dynamics As the nearest integer above the correlationdimension value generally provides the number of dominantvariables influencing the dynamics of the underlying systemthe correlation dimensions for the series indicate that thetime series of precipitation air temperature discharge andstorage volume are dominantly governed by four two four

Advances in Meteorology 5

1910 1920 1930 1940 1950 1960 1970 1980 19900

500

1000

1500

2000

2500

Time (year)

Mon

thly

pre

cipi

tatio

n (m

m)

(a) Precipitation

1910 1920 1930 1940 1950 1960 1970 1980 1990

0

10

20

30

Time (year)

minus20

minus10

Mon

thly

mea

n te

mpe

ratu

re (∘

C)

(b) Air temperature

1910 1920 1930 1940 1950 1960 1970 1980 19900

2000

4000

6000

8000

Time (year)

Mon

thly

mea

n di

scha

rge (

ft3s

)

(c) Discharge

1910 1920 1930 1940 1950 1960 1970 1980 199050

100

150

200

250

300

Time (year)M

onth

ly m

ean

vol

(times105

ft3)

(d) Storage volume

Figure 2 Monthly time series plots for the period of 1903ndash1995

and three variables respectively Here we can find the truththat the time series have different chaos characteristics evenif they are collected from a hydro-meteorological system

44 Correlation Dimension Analysis Using Synthetically Gen-erated Series Precipitation and air temperature from themeteorological system are considered as input time series ofthe runoff system On the same principle the output serieswhich is a discharge at the runoff system or the basin outletoccurred by input series of precipitation and air temperaturefrom the meteorological system can be the input data of thereservoir system Here the methodology is suggested to solvethe two hypotheses as follows We composed the input datasets which have an arbitrary correlation dimension and buildup ANNs as a nonlinear model for modeling the reservoirsystem The modeling results from the input data sets will bethe criterion of the hypotheses The first hypothesis will bereasonable if the system responses sensitively depending onthe arbitrary input data sets whereas the second hypothesiswill be reasonable if the system does not response sensitivelydepending on the input data sets

441 Correlation Dimensions of Generated Input Series to theReservoir System Theattractors in each time series (shown inFigure 4) have limit cycle regime which is the characteristicsof a periodic system Each time series as a periodic functioncan be written as an infinite sum of sine and cosine termsFourier [54] realized this first so that this infinite sum iscalled a Fourier series

The input data sets are composed of the nine sets using thethree sine functions in the each hydro-meteorological timeseries Here the sine function which is useful for applicationto a periodic time series data ismade using the fitting toolboxof MATLAB Therefore each time series is composed of thethree cases of case (a) case (b) and case (c) as shown inTable 1 Case (a) is composed of the sum of few sine functionsand case (c) is composed of the sum of lots of sine functionsrelatively Case (b) is between case (a) and case (c) In case ofprecipitation and discharge the functions are set to have atleast three sine functions because the series are dominantlygoverned by four variables from the results of the correlationdimension analysis in Section 432 It is found that thefitting results have a good applicability with the correlationcoefficient (CC) in precipitation 054ndash065 air temperature098ndash099 and discharge 088ndash092 for 1116 months (1903ndash1995) The results of the correlation dimension analysis ineach case are shown in Figure 6 The saturated correlationdimensions in each series are (a) 254 (b) 326 and (c) 405 inprecipitation (a) 102 (b) 184 and (c) 252 in air temperatureand (a) 248 (b) 313 and (c) 38 Case (c) is composed ofmany sine functions which showed the highest correlationdimension whereas case (a) shows the lowest correlationdimension in each time series

442 ANN Modeling and Correlation Dimension Analysisof Hydro-Meteorological System In order to build up theANN model this study sets precipitation air temperatureand discharge as the input layer and storage volume as theoutput layer As seen in Figure 7 a multilayered ANNmodel

6 Advances in Meteorology

Table 1 Fitting functions in each case of each time series

Fitting function CC

Precipitation(a) = 151 sdot sin(053 sdot 119905 minus 050) + 155 sdot sin(054 sdot 119905 minus 194) + 1548 sdot sin(054 sdot 119905 + 102) 054(b) = 150 sdot sin(053 sdot 119905 minus 053) + 1881 sdot sin(054 sdot 119905 minus 202) + 1882 sdot sin(054 sdot 119905 + 097) + 123 sdot sin(055 sdot 119905 minus 061) 060(c) = 150 sdot sin(053 sdot 119905 minus 053) + 1949 sdot sin(054 sdot 119905 minus 202) + 195 sdot sin(054 sdot 119905 + 098) + 123 sdot sin(055 sdot 119905 minus 060) +

1131 sdot sin(050 sdot 119905 + 063)065

Airtemperature

(a) = 1615 sdot sin(052 sdot 119905 minus 221) 098(b) = 1617 sdot sin(052 sdot 119905 minus 222) + 038 sdot sin(051 sdot 119905 + 046) 099(c) = 1614 sdot sin(052 sdot 119905 minus 220) + 037 sdot sin(051 sdot 119905 + 044) + 033 sdot sin(053 sdot 119905 minus 516) 099

Discharge(a) = 768 sdot sin(052 sdot 119905 minus 059) + 1202 sdot sin(053 sdot 119905 minus 261) + 1111 sdot sin(053 sdot 119905 + 019) 088(b) = 766 sdot sin(052 sdot 119905 minus 057) + 1286 sdot sin(053 sdot 119905 minus 269) + 12 sdot sin(053 sdot 119905 + 015) + 108 sdot sin(055 sdot 119905 minus 243) 089(c) = 3723 sdot sin(052 sdot 119905 minus 168) + 3161 sdot sin(052 sdot 119905 + 116) + 072 sdot sin(053 sdot 119905 minus 167) + 098 sdot sin(055 sdot 119905 minus 251) +

120 sdot sin(051 sdot 119905 minus 177)092

1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

sminus10

minus201910 1930 1950 1970 1990

Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 6

Rel variance

(a) Precipitation

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 5

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(b) Air temperature

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 4E + 142E + 7

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(c) Discharge

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 14

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(d) Storage volume

Figure 3 Extraction of the representative time series using the wavelet transform (left the wavelet power spectrum center the global waveletpower spectrum and right the extracted time series about the period of approximately 1 year)

consisting of one input layer two hidden layers and oneoutput layer has been built

Monthly data series from 1903 to 1970 (800 months)has been used for the learning period 316 months fromthe learning period (1970ndash1995) are set up as verification

periods and the applicability of the constructed ANN modelis reviewed by comparing it to observed storage volume asa target data series (see Table 2) For the composition of theprediction data we again compose the three input data likecases (A) (B) and (C) Case (A) is integrated from each

Advances in Meteorology 7

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(a) Precipitation

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(b) Air temperature

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(c) Discharge

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(d) Storage volume

Figure 4 Attractors in each time series

case (a) in each time series Case (B) and case (C) are alsointegrated in the same way as case (A) Data series of 1116months (1903ndash1995) in each case (A) (B) and (C) has beenused for the prediction period

First of all according to the model verification measures(see Figure 8) such as the coefficient of correlation (CC0986) and root mean squared error (RMSE 0061) ANN isfitted very well and found its good applicability

The storage volume series of a reservoir system is esti-mated using the ANN model after setting case (A) case (b)and case (c) as the input data And then the correlationdimension analysis is performed for the estimated storagevolume in each case The results show that 255 in case (A)integrated the low-dimensional cases (a) 281 in case (B)integrated the middle-dimensional cases (b) and 289 in case(C) integrated the high-dimensional cases (c) as shown inFigure 9

45 Summary and Discussions In this study we posedthe two hypotheses to identify chaotic behavior in hydro-meteorological processes For solving this issuewe composedthe input data sets like cases (A) (B) and (C) and appliedthem to ANN model on the reservoir system of the GreatSalt Lake The criterion of the hypotheses is the sensitivityof chaotic behavior in the system In other words the firsthypothesis is reasonable if chaotic behavior in the system issensitive depending on chaotic characteristics of the inputdata otherwise the second hypothesis is reasonable Theresults of the correlation dimension analysis on every caseanalyzed in this study were summarized in Table 3

As shown in Table 3 the correlation dimensions are 255in case (A) obtained from integrating the low dimensions(precipitation 254 air temperature 102 and discharge 248)and 281 in case (B) from integrating the middle dimensions(precipitation 326 air temperature 184 and discharge 331)

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 3

series arising from deterministic dynamics will have a limitednumber of degrees of freedom equal to the smallest numberof first-order differential equations that capture the dominantfeatures of the dynamics Thus when one constructs phasespaces of increasing dimension a point will be reached wherethe dimension equals the number of degrees of freedombeyond which increasing the phase space dimension willnot have any significant effect on correlation dimensionMany algorithms have been formulated for the estimation ofthe correlation dimension Among these the Grassberger-Procaccia algorithm [7] has been the most popular Thealgorithm uses the concept of phase space reconstruction forrepresenting the dynamics of the system from an availablesingle-variable time series as presented in (1) For an m-dimensional phase space the correlation integral or function119862(119903) is given by

119862 (119903) = lim119873rarrinfin

2

119873 (119873 minus 1)sum

119894119895 (1le119894lt119895le119873)

119867(119903 minus10038161003816100381610038161003816119884119894minus 119884119895

10038161003816100381610038161003816) (3)

where 119867 is the Heaviside step function with 119867(119906) = 1 for119906 gt 0 and 119867(119906) = 1 for 119906 le 0 where 119906 = 119903 minus 119884

119894minus 119884119895 119903

is the vector norm (radius of sphere) centered on 119884119894or 119884119895 If

the time series is characterized by an attractor then 119862(119903) and119903 are related according to

119862 (119903)119903rarr0119873rarrinfin asymp 120572119903] (4)

where 120572 is a constant and ] is the correlation exponent orthe slope of the Log119862(119903) versus Log 119903 plot The slope isgenerally estimated by a least square fit of a straight line overa certain range of 119903 (scaling regime) or through estimation oflocal slopes between 119903 values The distinction between low-dimensional (perhaps determinism) and high-dimensional(perhaps stochasticity) can be made using the ] versus 119898

plot If ] saturates after a certain 119898 and the saturation valueis low then the system is generally considered to exhibitlow-dimensional and possibly deterministic dynamics Thesaturation value of ] is defined as the correlation dimension(1198632) of the attractor and the nearest integer above thisvalue is generally an indication of the number of variablesdominantly governing the dynamics On the other hand if119898 increases without bound with increase in 119898 the systemunder investigation is generally considered to exhibit high-dimensional and possibly stochastic behavior

3 Wavelet Transform and ArtificialNeural Networks

31 Wavelet Transform According to Fourier theory a signalcan be expressed as the sum of a possibly infinite series ofsine and cosines referred to as a Fourier expansion [33]However a Fourier expansion has only frequency resolutionand not time resolution that is no amplitude modulationof the signal at a given frequency is considered Moving-window Fourier transforms have been used to address thisissue but this method is sensitive to the choice of windowwidth Alternatively the wavelet transform [34 35] enablesthe identification of frequency components as well as their

variation in time The continuous wavelet transform of adiscrete sequence 119909119899 is defined by the convolution of 119909

119899with

a scaled and translated wavelet function 120595

119882119899 (119904) =

119873minus1

sum

1198991015840=0

1199091198991015840120595 lowast [

(1198991015840minus 119899) 120575119905

119904] (5)

where (lowast) indicates the complex conjugate 119899 is the localizedtime index 119904 = 0 is the scale parameter and119873 is the numberof points in the time series In this study we use the Morletwavelet function defined as 120595(120578) = 120587

minus14119890119894120596012057811989012057822 where 120596

0

is a frequency and 120578 is a nondimensional ldquotimerdquo parameter Byvarying the wavelet scales and translating along the localizedtime index 119899 one can construct a picture that shows boththe amplitude of any features versus the scale and how thisamplitude varies with time A vertical slice through a waveletplot is a measure of the local spectrum The time-averagedwavelet spectrum over all the local wavelet spectra gives theglobal wavelet spectrum

1198822

119905(119904) =

1

119879

119879minus1

sum

119905=0

1003816100381610038161003816119882119905 (119904)1003816100381610038161003816

2 (6)

A more detailed presentation for wavelet transform anal-ysis is referred to read Torrence and Compo [35]

32 Artificial Neural Networks ANN is a model of neuro-transmission by a neuron which is a nerve cell in the humanbrain ANN is an empirical pattern search technique thatenables the consideration of a nonlinear relationship betweeninput variables and output variables ANN is used in variousareas because of its unique applicability [36 37]This includesthe field of climate science where its applicability is proven[38 39]

Many studies suggest the ANN technique which is anonlinear model of the data series and ANN is better thanother techniques by way of systematic evaluation of varioustechniques [40 41] Therefore this study also applies ANNwhich is judged to have superior applicability in the simula-tion of nonlinear characteristics of the hydro-meteorologicalsystem

4 Applications and Results

41 Study Area and Data Series Used The Bear River Basinlocated in northeastern Utah southeastern Idaho and south-westernWyoming comprises 7500 squaremiles ofmountainand valley lands including 2700 in Idaho 3300 in Utah and1500 in Wyoming The Bear River crosses state boundariesfive times and is the largest stream in the western hemispherethat does not empty into the ocean It ranges in elevationfrom over 1278 to 3868 feet and is unique in that it isentirely enclosed by mountains thus forming a huge basinwith no external drainage outlets (httpwwwgreatsaltlake-infoorgBackgroundBearRiver) The Bear River is the larg-est tributary to the Great Salt Lake (see Figure 1)

4 Advances in Meteorology

Great Salt Lake

Bear River Basin

Weather Station

N

Discharge StationWater Level StationBear River

DEM

(ft)

High 386823

Low 127839

0 15 30 60

(km)

Figure 1 Study area

The data was collected from weather gauging station(USC00424856 NOAA) stream flow gauging station (num-ber 10126000 USGS) and lake water level gauging station(number 10010000 USGS) for the period of 1903 to 1995Themonthly rainfall shows its statistics of average 2585mm andstandard deviation 2164mm the monthly mean tempera-ture shows average 57∘C and standard deviation 88∘C themonthly mean runoff shows average 15336 ft3s and standarddeviation 13312 ft3s the monthly mean storage in the lakeshows average 1494 times 105 ft3 and standard deviation 398 times

105 ft3 and these time series plots are shown in Figure 2

42 Extraction of a Representative Time Series by WaveletTransform All hydrological measurements are to someextent contaminated by noise And the noise limits theperformance of many techniques of identification modelingprediction and control of deterministic systems [42] Inde-pendent component analysis (ICA) as a popular method isable to extract periodic signals from noise or nonlinear mix-ture [43 44] It has been applied in the fields of meteorology[45] oceanography [46] volcanology [47 48] and remotesensing [49] This study however uses wavelet transformfor extracting the representative periodic components whichaffect the data series because ICA often leads to local mini-mum solution and the suitable source signals are not isolated[50] Moreover the order of the independent components(ICs) is difficult to be determined in comparisonwith wavelettransform

Wavelet power spectrum that estimated the waveletmother function using the Morlet function is shown inFigure 3 (left) and the extent of spectrum in each periodfor time series can be identified In this figure a solid half-circle line shows the edge of the cone of influence (COI) effectthat can be caused by the discontinuity of the beginning andend of data series In particular the upper part of the solidline is statistically significant (a 95 confidence interval) andthe lower part is excluded from interpretation Parts withhigh-density spectrum are observed in some periods within aconfidence interval Global wavelet power spectrum (GWP)

in Formula (6) which represents the average value accordingto the length of each period provides more effective infor-mation about spectrum Figure 3 (center) shows the resultof GWP about spectrum Considering that the right partof a solid line is statistically significant on a basis of a 95confidence level the periodic characteristics of the time seriescould be classified into one band The band shows a strongspectrum of the period of approximately 1 year The periodextracted from the wavelet spectrum is shown in the right ofFigure 3

43 Analysis of the Time Series Using Attractor andCorrelation Dimension

431 Attractor Analysis The attractor obtained by (1) candescribe the characteristics of a time series To obtain theattractor using (1) the index lag 119905 and embedding dimension119898 must be chosen appropriately The autocorrelation func-tion (ACF) is expected to provide a reasonable measure ofthe transition from redundance to irrelevance as a functionof delay The decorrelation time which is equal to the lag(delay time 120591) at which the ACF first attains the value zerois considered Otherwise 120591 should be chosen as the localminimum of ACF whichever occurs first [51 52] When theACF decays exponentially we select 120591 at which theACF dropsto zero [53] at lag time 4months in all of seriesTherefore thedelay times of the systems can be obtained fromACFs and theattractors are drawn in Figure 4 for each time series

For the attractor analysis this study uses the extractedtime series by the wavelet transform The attractors of thetime series have a circle with a boundary If the attractor inthe phase-space exhibits clearly within a very well definedboundary suggest that the dynamics are simple and thesystem is potentially low dimension Every time series havehas a shape with a boundary which looks like a chaotic seriesParticularly air temperature shows a verywell defined bound-ary and it is potentially low-dimensional series Precipitationhowever shows relatively high complex and irregular and itis a potentially high-dimensional system than the other dataseries

432 Correlation Dimension Analysis Figure 5 shows therelationship between the correlation dimension 1198632 andthe embedding dimensions 119898 from 1 to 15 for each timeseries The correlation dimension seems to increase with theembedding dimension up to a certain point and saturatebeyond that point Such a saturation of the correlationdimension is an indication of the existence of deterministicdynamicsThe saturation values of the correlation dimensionfor the series are showing 392 141 302 and 265 in Fig-ures 5(a)ndash5(d) The low correlation dimensions suggest thepresence of low-dimensional chaotic nature of the underlyingsystem dynamics As the nearest integer above the correlationdimension value generally provides the number of dominantvariables influencing the dynamics of the underlying systemthe correlation dimensions for the series indicate that thetime series of precipitation air temperature discharge andstorage volume are dominantly governed by four two four

Advances in Meteorology 5

1910 1920 1930 1940 1950 1960 1970 1980 19900

500

1000

1500

2000

2500

Time (year)

Mon

thly

pre

cipi

tatio

n (m

m)

(a) Precipitation

1910 1920 1930 1940 1950 1960 1970 1980 1990

0

10

20

30

Time (year)

minus20

minus10

Mon

thly

mea

n te

mpe

ratu

re (∘

C)

(b) Air temperature

1910 1920 1930 1940 1950 1960 1970 1980 19900

2000

4000

6000

8000

Time (year)

Mon

thly

mea

n di

scha

rge (

ft3s

)

(c) Discharge

1910 1920 1930 1940 1950 1960 1970 1980 199050

100

150

200

250

300

Time (year)M

onth

ly m

ean

vol

(times105

ft3)

(d) Storage volume

Figure 2 Monthly time series plots for the period of 1903ndash1995

and three variables respectively Here we can find the truththat the time series have different chaos characteristics evenif they are collected from a hydro-meteorological system

44 Correlation Dimension Analysis Using Synthetically Gen-erated Series Precipitation and air temperature from themeteorological system are considered as input time series ofthe runoff system On the same principle the output serieswhich is a discharge at the runoff system or the basin outletoccurred by input series of precipitation and air temperaturefrom the meteorological system can be the input data of thereservoir system Here the methodology is suggested to solvethe two hypotheses as follows We composed the input datasets which have an arbitrary correlation dimension and buildup ANNs as a nonlinear model for modeling the reservoirsystem The modeling results from the input data sets will bethe criterion of the hypotheses The first hypothesis will bereasonable if the system responses sensitively depending onthe arbitrary input data sets whereas the second hypothesiswill be reasonable if the system does not response sensitivelydepending on the input data sets

441 Correlation Dimensions of Generated Input Series to theReservoir System Theattractors in each time series (shown inFigure 4) have limit cycle regime which is the characteristicsof a periodic system Each time series as a periodic functioncan be written as an infinite sum of sine and cosine termsFourier [54] realized this first so that this infinite sum iscalled a Fourier series

The input data sets are composed of the nine sets using thethree sine functions in the each hydro-meteorological timeseries Here the sine function which is useful for applicationto a periodic time series data ismade using the fitting toolboxof MATLAB Therefore each time series is composed of thethree cases of case (a) case (b) and case (c) as shown inTable 1 Case (a) is composed of the sum of few sine functionsand case (c) is composed of the sum of lots of sine functionsrelatively Case (b) is between case (a) and case (c) In case ofprecipitation and discharge the functions are set to have atleast three sine functions because the series are dominantlygoverned by four variables from the results of the correlationdimension analysis in Section 432 It is found that thefitting results have a good applicability with the correlationcoefficient (CC) in precipitation 054ndash065 air temperature098ndash099 and discharge 088ndash092 for 1116 months (1903ndash1995) The results of the correlation dimension analysis ineach case are shown in Figure 6 The saturated correlationdimensions in each series are (a) 254 (b) 326 and (c) 405 inprecipitation (a) 102 (b) 184 and (c) 252 in air temperatureand (a) 248 (b) 313 and (c) 38 Case (c) is composed ofmany sine functions which showed the highest correlationdimension whereas case (a) shows the lowest correlationdimension in each time series

442 ANN Modeling and Correlation Dimension Analysisof Hydro-Meteorological System In order to build up theANN model this study sets precipitation air temperatureand discharge as the input layer and storage volume as theoutput layer As seen in Figure 7 a multilayered ANNmodel

6 Advances in Meteorology

Table 1 Fitting functions in each case of each time series

Fitting function CC

Precipitation(a) = 151 sdot sin(053 sdot 119905 minus 050) + 155 sdot sin(054 sdot 119905 minus 194) + 1548 sdot sin(054 sdot 119905 + 102) 054(b) = 150 sdot sin(053 sdot 119905 minus 053) + 1881 sdot sin(054 sdot 119905 minus 202) + 1882 sdot sin(054 sdot 119905 + 097) + 123 sdot sin(055 sdot 119905 minus 061) 060(c) = 150 sdot sin(053 sdot 119905 minus 053) + 1949 sdot sin(054 sdot 119905 minus 202) + 195 sdot sin(054 sdot 119905 + 098) + 123 sdot sin(055 sdot 119905 minus 060) +

1131 sdot sin(050 sdot 119905 + 063)065

Airtemperature

(a) = 1615 sdot sin(052 sdot 119905 minus 221) 098(b) = 1617 sdot sin(052 sdot 119905 minus 222) + 038 sdot sin(051 sdot 119905 + 046) 099(c) = 1614 sdot sin(052 sdot 119905 minus 220) + 037 sdot sin(051 sdot 119905 + 044) + 033 sdot sin(053 sdot 119905 minus 516) 099

Discharge(a) = 768 sdot sin(052 sdot 119905 minus 059) + 1202 sdot sin(053 sdot 119905 minus 261) + 1111 sdot sin(053 sdot 119905 + 019) 088(b) = 766 sdot sin(052 sdot 119905 minus 057) + 1286 sdot sin(053 sdot 119905 minus 269) + 12 sdot sin(053 sdot 119905 + 015) + 108 sdot sin(055 sdot 119905 minus 243) 089(c) = 3723 sdot sin(052 sdot 119905 minus 168) + 3161 sdot sin(052 sdot 119905 + 116) + 072 sdot sin(053 sdot 119905 minus 167) + 098 sdot sin(055 sdot 119905 minus 251) +

120 sdot sin(051 sdot 119905 minus 177)092

1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

sminus10

minus201910 1930 1950 1970 1990

Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 6

Rel variance

(a) Precipitation

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 5

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(b) Air temperature

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 4E + 142E + 7

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(c) Discharge

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 14

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(d) Storage volume

Figure 3 Extraction of the representative time series using the wavelet transform (left the wavelet power spectrum center the global waveletpower spectrum and right the extracted time series about the period of approximately 1 year)

consisting of one input layer two hidden layers and oneoutput layer has been built

Monthly data series from 1903 to 1970 (800 months)has been used for the learning period 316 months fromthe learning period (1970ndash1995) are set up as verification

periods and the applicability of the constructed ANN modelis reviewed by comparing it to observed storage volume asa target data series (see Table 2) For the composition of theprediction data we again compose the three input data likecases (A) (B) and (C) Case (A) is integrated from each

Advances in Meteorology 7

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(a) Precipitation

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(b) Air temperature

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(c) Discharge

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(d) Storage volume

Figure 4 Attractors in each time series

case (a) in each time series Case (B) and case (C) are alsointegrated in the same way as case (A) Data series of 1116months (1903ndash1995) in each case (A) (B) and (C) has beenused for the prediction period

First of all according to the model verification measures(see Figure 8) such as the coefficient of correlation (CC0986) and root mean squared error (RMSE 0061) ANN isfitted very well and found its good applicability

The storage volume series of a reservoir system is esti-mated using the ANN model after setting case (A) case (b)and case (c) as the input data And then the correlationdimension analysis is performed for the estimated storagevolume in each case The results show that 255 in case (A)integrated the low-dimensional cases (a) 281 in case (B)integrated the middle-dimensional cases (b) and 289 in case(C) integrated the high-dimensional cases (c) as shown inFigure 9

45 Summary and Discussions In this study we posedthe two hypotheses to identify chaotic behavior in hydro-meteorological processes For solving this issuewe composedthe input data sets like cases (A) (B) and (C) and appliedthem to ANN model on the reservoir system of the GreatSalt Lake The criterion of the hypotheses is the sensitivityof chaotic behavior in the system In other words the firsthypothesis is reasonable if chaotic behavior in the system issensitive depending on chaotic characteristics of the inputdata otherwise the second hypothesis is reasonable Theresults of the correlation dimension analysis on every caseanalyzed in this study were summarized in Table 3

As shown in Table 3 the correlation dimensions are 255in case (A) obtained from integrating the low dimensions(precipitation 254 air temperature 102 and discharge 248)and 281 in case (B) from integrating the middle dimensions(precipitation 326 air temperature 184 and discharge 331)

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

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4 Advances in Meteorology

Great Salt Lake

Bear River Basin

Weather Station

N

Discharge StationWater Level StationBear River

DEM

(ft)

High 386823

Low 127839

0 15 30 60

(km)

Figure 1 Study area

The data was collected from weather gauging station(USC00424856 NOAA) stream flow gauging station (num-ber 10126000 USGS) and lake water level gauging station(number 10010000 USGS) for the period of 1903 to 1995Themonthly rainfall shows its statistics of average 2585mm andstandard deviation 2164mm the monthly mean tempera-ture shows average 57∘C and standard deviation 88∘C themonthly mean runoff shows average 15336 ft3s and standarddeviation 13312 ft3s the monthly mean storage in the lakeshows average 1494 times 105 ft3 and standard deviation 398 times

105 ft3 and these time series plots are shown in Figure 2

42 Extraction of a Representative Time Series by WaveletTransform All hydrological measurements are to someextent contaminated by noise And the noise limits theperformance of many techniques of identification modelingprediction and control of deterministic systems [42] Inde-pendent component analysis (ICA) as a popular method isable to extract periodic signals from noise or nonlinear mix-ture [43 44] It has been applied in the fields of meteorology[45] oceanography [46] volcanology [47 48] and remotesensing [49] This study however uses wavelet transformfor extracting the representative periodic components whichaffect the data series because ICA often leads to local mini-mum solution and the suitable source signals are not isolated[50] Moreover the order of the independent components(ICs) is difficult to be determined in comparisonwith wavelettransform

Wavelet power spectrum that estimated the waveletmother function using the Morlet function is shown inFigure 3 (left) and the extent of spectrum in each periodfor time series can be identified In this figure a solid half-circle line shows the edge of the cone of influence (COI) effectthat can be caused by the discontinuity of the beginning andend of data series In particular the upper part of the solidline is statistically significant (a 95 confidence interval) andthe lower part is excluded from interpretation Parts withhigh-density spectrum are observed in some periods within aconfidence interval Global wavelet power spectrum (GWP)

in Formula (6) which represents the average value accordingto the length of each period provides more effective infor-mation about spectrum Figure 3 (center) shows the resultof GWP about spectrum Considering that the right partof a solid line is statistically significant on a basis of a 95confidence level the periodic characteristics of the time seriescould be classified into one band The band shows a strongspectrum of the period of approximately 1 year The periodextracted from the wavelet spectrum is shown in the right ofFigure 3

43 Analysis of the Time Series Using Attractor andCorrelation Dimension

431 Attractor Analysis The attractor obtained by (1) candescribe the characteristics of a time series To obtain theattractor using (1) the index lag 119905 and embedding dimension119898 must be chosen appropriately The autocorrelation func-tion (ACF) is expected to provide a reasonable measure ofthe transition from redundance to irrelevance as a functionof delay The decorrelation time which is equal to the lag(delay time 120591) at which the ACF first attains the value zerois considered Otherwise 120591 should be chosen as the localminimum of ACF whichever occurs first [51 52] When theACF decays exponentially we select 120591 at which theACF dropsto zero [53] at lag time 4months in all of seriesTherefore thedelay times of the systems can be obtained fromACFs and theattractors are drawn in Figure 4 for each time series

For the attractor analysis this study uses the extractedtime series by the wavelet transform The attractors of thetime series have a circle with a boundary If the attractor inthe phase-space exhibits clearly within a very well definedboundary suggest that the dynamics are simple and thesystem is potentially low dimension Every time series havehas a shape with a boundary which looks like a chaotic seriesParticularly air temperature shows a verywell defined bound-ary and it is potentially low-dimensional series Precipitationhowever shows relatively high complex and irregular and itis a potentially high-dimensional system than the other dataseries

432 Correlation Dimension Analysis Figure 5 shows therelationship between the correlation dimension 1198632 andthe embedding dimensions 119898 from 1 to 15 for each timeseries The correlation dimension seems to increase with theembedding dimension up to a certain point and saturatebeyond that point Such a saturation of the correlationdimension is an indication of the existence of deterministicdynamicsThe saturation values of the correlation dimensionfor the series are showing 392 141 302 and 265 in Fig-ures 5(a)ndash5(d) The low correlation dimensions suggest thepresence of low-dimensional chaotic nature of the underlyingsystem dynamics As the nearest integer above the correlationdimension value generally provides the number of dominantvariables influencing the dynamics of the underlying systemthe correlation dimensions for the series indicate that thetime series of precipitation air temperature discharge andstorage volume are dominantly governed by four two four

Advances in Meteorology 5

1910 1920 1930 1940 1950 1960 1970 1980 19900

500

1000

1500

2000

2500

Time (year)

Mon

thly

pre

cipi

tatio

n (m

m)

(a) Precipitation

1910 1920 1930 1940 1950 1960 1970 1980 1990

0

10

20

30

Time (year)

minus20

minus10

Mon

thly

mea

n te

mpe

ratu

re (∘

C)

(b) Air temperature

1910 1920 1930 1940 1950 1960 1970 1980 19900

2000

4000

6000

8000

Time (year)

Mon

thly

mea

n di

scha

rge (

ft3s

)

(c) Discharge

1910 1920 1930 1940 1950 1960 1970 1980 199050

100

150

200

250

300

Time (year)M

onth

ly m

ean

vol

(times105

ft3)

(d) Storage volume

Figure 2 Monthly time series plots for the period of 1903ndash1995

and three variables respectively Here we can find the truththat the time series have different chaos characteristics evenif they are collected from a hydro-meteorological system

44 Correlation Dimension Analysis Using Synthetically Gen-erated Series Precipitation and air temperature from themeteorological system are considered as input time series ofthe runoff system On the same principle the output serieswhich is a discharge at the runoff system or the basin outletoccurred by input series of precipitation and air temperaturefrom the meteorological system can be the input data of thereservoir system Here the methodology is suggested to solvethe two hypotheses as follows We composed the input datasets which have an arbitrary correlation dimension and buildup ANNs as a nonlinear model for modeling the reservoirsystem The modeling results from the input data sets will bethe criterion of the hypotheses The first hypothesis will bereasonable if the system responses sensitively depending onthe arbitrary input data sets whereas the second hypothesiswill be reasonable if the system does not response sensitivelydepending on the input data sets

441 Correlation Dimensions of Generated Input Series to theReservoir System Theattractors in each time series (shown inFigure 4) have limit cycle regime which is the characteristicsof a periodic system Each time series as a periodic functioncan be written as an infinite sum of sine and cosine termsFourier [54] realized this first so that this infinite sum iscalled a Fourier series

The input data sets are composed of the nine sets using thethree sine functions in the each hydro-meteorological timeseries Here the sine function which is useful for applicationto a periodic time series data ismade using the fitting toolboxof MATLAB Therefore each time series is composed of thethree cases of case (a) case (b) and case (c) as shown inTable 1 Case (a) is composed of the sum of few sine functionsand case (c) is composed of the sum of lots of sine functionsrelatively Case (b) is between case (a) and case (c) In case ofprecipitation and discharge the functions are set to have atleast three sine functions because the series are dominantlygoverned by four variables from the results of the correlationdimension analysis in Section 432 It is found that thefitting results have a good applicability with the correlationcoefficient (CC) in precipitation 054ndash065 air temperature098ndash099 and discharge 088ndash092 for 1116 months (1903ndash1995) The results of the correlation dimension analysis ineach case are shown in Figure 6 The saturated correlationdimensions in each series are (a) 254 (b) 326 and (c) 405 inprecipitation (a) 102 (b) 184 and (c) 252 in air temperatureand (a) 248 (b) 313 and (c) 38 Case (c) is composed ofmany sine functions which showed the highest correlationdimension whereas case (a) shows the lowest correlationdimension in each time series

442 ANN Modeling and Correlation Dimension Analysisof Hydro-Meteorological System In order to build up theANN model this study sets precipitation air temperatureand discharge as the input layer and storage volume as theoutput layer As seen in Figure 7 a multilayered ANNmodel

6 Advances in Meteorology

Table 1 Fitting functions in each case of each time series

Fitting function CC

Precipitation(a) = 151 sdot sin(053 sdot 119905 minus 050) + 155 sdot sin(054 sdot 119905 minus 194) + 1548 sdot sin(054 sdot 119905 + 102) 054(b) = 150 sdot sin(053 sdot 119905 minus 053) + 1881 sdot sin(054 sdot 119905 minus 202) + 1882 sdot sin(054 sdot 119905 + 097) + 123 sdot sin(055 sdot 119905 minus 061) 060(c) = 150 sdot sin(053 sdot 119905 minus 053) + 1949 sdot sin(054 sdot 119905 minus 202) + 195 sdot sin(054 sdot 119905 + 098) + 123 sdot sin(055 sdot 119905 minus 060) +

1131 sdot sin(050 sdot 119905 + 063)065

Airtemperature

(a) = 1615 sdot sin(052 sdot 119905 minus 221) 098(b) = 1617 sdot sin(052 sdot 119905 minus 222) + 038 sdot sin(051 sdot 119905 + 046) 099(c) = 1614 sdot sin(052 sdot 119905 minus 220) + 037 sdot sin(051 sdot 119905 + 044) + 033 sdot sin(053 sdot 119905 minus 516) 099

Discharge(a) = 768 sdot sin(052 sdot 119905 minus 059) + 1202 sdot sin(053 sdot 119905 minus 261) + 1111 sdot sin(053 sdot 119905 + 019) 088(b) = 766 sdot sin(052 sdot 119905 minus 057) + 1286 sdot sin(053 sdot 119905 minus 269) + 12 sdot sin(053 sdot 119905 + 015) + 108 sdot sin(055 sdot 119905 minus 243) 089(c) = 3723 sdot sin(052 sdot 119905 minus 168) + 3161 sdot sin(052 sdot 119905 + 116) + 072 sdot sin(053 sdot 119905 minus 167) + 098 sdot sin(055 sdot 119905 minus 251) +

120 sdot sin(051 sdot 119905 minus 177)092

1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

sminus10

minus201910 1930 1950 1970 1990

Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 6

Rel variance

(a) Precipitation

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 5

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(b) Air temperature

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 4E + 142E + 7

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(c) Discharge

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 14

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(d) Storage volume

Figure 3 Extraction of the representative time series using the wavelet transform (left the wavelet power spectrum center the global waveletpower spectrum and right the extracted time series about the period of approximately 1 year)

consisting of one input layer two hidden layers and oneoutput layer has been built

Monthly data series from 1903 to 1970 (800 months)has been used for the learning period 316 months fromthe learning period (1970ndash1995) are set up as verification

periods and the applicability of the constructed ANN modelis reviewed by comparing it to observed storage volume asa target data series (see Table 2) For the composition of theprediction data we again compose the three input data likecases (A) (B) and (C) Case (A) is integrated from each

Advances in Meteorology 7

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(a) Precipitation

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(b) Air temperature

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(c) Discharge

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(d) Storage volume

Figure 4 Attractors in each time series

case (a) in each time series Case (B) and case (C) are alsointegrated in the same way as case (A) Data series of 1116months (1903ndash1995) in each case (A) (B) and (C) has beenused for the prediction period

First of all according to the model verification measures(see Figure 8) such as the coefficient of correlation (CC0986) and root mean squared error (RMSE 0061) ANN isfitted very well and found its good applicability

The storage volume series of a reservoir system is esti-mated using the ANN model after setting case (A) case (b)and case (c) as the input data And then the correlationdimension analysis is performed for the estimated storagevolume in each case The results show that 255 in case (A)integrated the low-dimensional cases (a) 281 in case (B)integrated the middle-dimensional cases (b) and 289 in case(C) integrated the high-dimensional cases (c) as shown inFigure 9

45 Summary and Discussions In this study we posedthe two hypotheses to identify chaotic behavior in hydro-meteorological processes For solving this issuewe composedthe input data sets like cases (A) (B) and (C) and appliedthem to ANN model on the reservoir system of the GreatSalt Lake The criterion of the hypotheses is the sensitivityof chaotic behavior in the system In other words the firsthypothesis is reasonable if chaotic behavior in the system issensitive depending on chaotic characteristics of the inputdata otherwise the second hypothesis is reasonable Theresults of the correlation dimension analysis on every caseanalyzed in this study were summarized in Table 3

As shown in Table 3 the correlation dimensions are 255in case (A) obtained from integrating the low dimensions(precipitation 254 air temperature 102 and discharge 248)and 281 in case (B) from integrating the middle dimensions(precipitation 326 air temperature 184 and discharge 331)

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 5

1910 1920 1930 1940 1950 1960 1970 1980 19900

500

1000

1500

2000

2500

Time (year)

Mon

thly

pre

cipi

tatio

n (m

m)

(a) Precipitation

1910 1920 1930 1940 1950 1960 1970 1980 1990

0

10

20

30

Time (year)

minus20

minus10

Mon

thly

mea

n te

mpe

ratu

re (∘

C)

(b) Air temperature

1910 1920 1930 1940 1950 1960 1970 1980 19900

2000

4000

6000

8000

Time (year)

Mon

thly

mea

n di

scha

rge (

ft3s

)

(c) Discharge

1910 1920 1930 1940 1950 1960 1970 1980 199050

100

150

200

250

300

Time (year)M

onth

ly m

ean

vol

(times105

ft3)

(d) Storage volume

Figure 2 Monthly time series plots for the period of 1903ndash1995

and three variables respectively Here we can find the truththat the time series have different chaos characteristics evenif they are collected from a hydro-meteorological system

44 Correlation Dimension Analysis Using Synthetically Gen-erated Series Precipitation and air temperature from themeteorological system are considered as input time series ofthe runoff system On the same principle the output serieswhich is a discharge at the runoff system or the basin outletoccurred by input series of precipitation and air temperaturefrom the meteorological system can be the input data of thereservoir system Here the methodology is suggested to solvethe two hypotheses as follows We composed the input datasets which have an arbitrary correlation dimension and buildup ANNs as a nonlinear model for modeling the reservoirsystem The modeling results from the input data sets will bethe criterion of the hypotheses The first hypothesis will bereasonable if the system responses sensitively depending onthe arbitrary input data sets whereas the second hypothesiswill be reasonable if the system does not response sensitivelydepending on the input data sets

441 Correlation Dimensions of Generated Input Series to theReservoir System Theattractors in each time series (shown inFigure 4) have limit cycle regime which is the characteristicsof a periodic system Each time series as a periodic functioncan be written as an infinite sum of sine and cosine termsFourier [54] realized this first so that this infinite sum iscalled a Fourier series

The input data sets are composed of the nine sets using thethree sine functions in the each hydro-meteorological timeseries Here the sine function which is useful for applicationto a periodic time series data ismade using the fitting toolboxof MATLAB Therefore each time series is composed of thethree cases of case (a) case (b) and case (c) as shown inTable 1 Case (a) is composed of the sum of few sine functionsand case (c) is composed of the sum of lots of sine functionsrelatively Case (b) is between case (a) and case (c) In case ofprecipitation and discharge the functions are set to have atleast three sine functions because the series are dominantlygoverned by four variables from the results of the correlationdimension analysis in Section 432 It is found that thefitting results have a good applicability with the correlationcoefficient (CC) in precipitation 054ndash065 air temperature098ndash099 and discharge 088ndash092 for 1116 months (1903ndash1995) The results of the correlation dimension analysis ineach case are shown in Figure 6 The saturated correlationdimensions in each series are (a) 254 (b) 326 and (c) 405 inprecipitation (a) 102 (b) 184 and (c) 252 in air temperatureand (a) 248 (b) 313 and (c) 38 Case (c) is composed ofmany sine functions which showed the highest correlationdimension whereas case (a) shows the lowest correlationdimension in each time series

442 ANN Modeling and Correlation Dimension Analysisof Hydro-Meteorological System In order to build up theANN model this study sets precipitation air temperatureand discharge as the input layer and storage volume as theoutput layer As seen in Figure 7 a multilayered ANNmodel

6 Advances in Meteorology

Table 1 Fitting functions in each case of each time series

Fitting function CC

Precipitation(a) = 151 sdot sin(053 sdot 119905 minus 050) + 155 sdot sin(054 sdot 119905 minus 194) + 1548 sdot sin(054 sdot 119905 + 102) 054(b) = 150 sdot sin(053 sdot 119905 minus 053) + 1881 sdot sin(054 sdot 119905 minus 202) + 1882 sdot sin(054 sdot 119905 + 097) + 123 sdot sin(055 sdot 119905 minus 061) 060(c) = 150 sdot sin(053 sdot 119905 minus 053) + 1949 sdot sin(054 sdot 119905 minus 202) + 195 sdot sin(054 sdot 119905 + 098) + 123 sdot sin(055 sdot 119905 minus 060) +

1131 sdot sin(050 sdot 119905 + 063)065

Airtemperature

(a) = 1615 sdot sin(052 sdot 119905 minus 221) 098(b) = 1617 sdot sin(052 sdot 119905 minus 222) + 038 sdot sin(051 sdot 119905 + 046) 099(c) = 1614 sdot sin(052 sdot 119905 minus 220) + 037 sdot sin(051 sdot 119905 + 044) + 033 sdot sin(053 sdot 119905 minus 516) 099

Discharge(a) = 768 sdot sin(052 sdot 119905 minus 059) + 1202 sdot sin(053 sdot 119905 minus 261) + 1111 sdot sin(053 sdot 119905 + 019) 088(b) = 766 sdot sin(052 sdot 119905 minus 057) + 1286 sdot sin(053 sdot 119905 minus 269) + 12 sdot sin(053 sdot 119905 + 015) + 108 sdot sin(055 sdot 119905 minus 243) 089(c) = 3723 sdot sin(052 sdot 119905 minus 168) + 3161 sdot sin(052 sdot 119905 + 116) + 072 sdot sin(053 sdot 119905 minus 167) + 098 sdot sin(055 sdot 119905 minus 251) +

120 sdot sin(051 sdot 119905 minus 177)092

1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

sminus10

minus201910 1930 1950 1970 1990

Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 6

Rel variance

(a) Precipitation

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 5

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(b) Air temperature

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 4E + 142E + 7

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(c) Discharge

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 14

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(d) Storage volume

Figure 3 Extraction of the representative time series using the wavelet transform (left the wavelet power spectrum center the global waveletpower spectrum and right the extracted time series about the period of approximately 1 year)

consisting of one input layer two hidden layers and oneoutput layer has been built

Monthly data series from 1903 to 1970 (800 months)has been used for the learning period 316 months fromthe learning period (1970ndash1995) are set up as verification

periods and the applicability of the constructed ANN modelis reviewed by comparing it to observed storage volume asa target data series (see Table 2) For the composition of theprediction data we again compose the three input data likecases (A) (B) and (C) Case (A) is integrated from each

Advances in Meteorology 7

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(a) Precipitation

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(b) Air temperature

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(c) Discharge

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(d) Storage volume

Figure 4 Attractors in each time series

case (a) in each time series Case (B) and case (C) are alsointegrated in the same way as case (A) Data series of 1116months (1903ndash1995) in each case (A) (B) and (C) has beenused for the prediction period

First of all according to the model verification measures(see Figure 8) such as the coefficient of correlation (CC0986) and root mean squared error (RMSE 0061) ANN isfitted very well and found its good applicability

The storage volume series of a reservoir system is esti-mated using the ANN model after setting case (A) case (b)and case (c) as the input data And then the correlationdimension analysis is performed for the estimated storagevolume in each case The results show that 255 in case (A)integrated the low-dimensional cases (a) 281 in case (B)integrated the middle-dimensional cases (b) and 289 in case(C) integrated the high-dimensional cases (c) as shown inFigure 9

45 Summary and Discussions In this study we posedthe two hypotheses to identify chaotic behavior in hydro-meteorological processes For solving this issuewe composedthe input data sets like cases (A) (B) and (C) and appliedthem to ANN model on the reservoir system of the GreatSalt Lake The criterion of the hypotheses is the sensitivityof chaotic behavior in the system In other words the firsthypothesis is reasonable if chaotic behavior in the system issensitive depending on chaotic characteristics of the inputdata otherwise the second hypothesis is reasonable Theresults of the correlation dimension analysis on every caseanalyzed in this study were summarized in Table 3

As shown in Table 3 the correlation dimensions are 255in case (A) obtained from integrating the low dimensions(precipitation 254 air temperature 102 and discharge 248)and 281 in case (B) from integrating the middle dimensions(precipitation 326 air temperature 184 and discharge 331)

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

6 Advances in Meteorology

Table 1 Fitting functions in each case of each time series

Fitting function CC

Precipitation(a) = 151 sdot sin(053 sdot 119905 minus 050) + 155 sdot sin(054 sdot 119905 minus 194) + 1548 sdot sin(054 sdot 119905 + 102) 054(b) = 150 sdot sin(053 sdot 119905 minus 053) + 1881 sdot sin(054 sdot 119905 minus 202) + 1882 sdot sin(054 sdot 119905 + 097) + 123 sdot sin(055 sdot 119905 minus 061) 060(c) = 150 sdot sin(053 sdot 119905 minus 053) + 1949 sdot sin(054 sdot 119905 minus 202) + 195 sdot sin(054 sdot 119905 + 098) + 123 sdot sin(055 sdot 119905 minus 060) +

1131 sdot sin(050 sdot 119905 + 063)065

Airtemperature

(a) = 1615 sdot sin(052 sdot 119905 minus 221) 098(b) = 1617 sdot sin(052 sdot 119905 minus 222) + 038 sdot sin(051 sdot 119905 + 046) 099(c) = 1614 sdot sin(052 sdot 119905 minus 220) + 037 sdot sin(051 sdot 119905 + 044) + 033 sdot sin(053 sdot 119905 minus 516) 099

Discharge(a) = 768 sdot sin(052 sdot 119905 minus 059) + 1202 sdot sin(053 sdot 119905 minus 261) + 1111 sdot sin(053 sdot 119905 + 019) 088(b) = 766 sdot sin(052 sdot 119905 minus 057) + 1286 sdot sin(053 sdot 119905 minus 269) + 12 sdot sin(053 sdot 119905 + 015) + 108 sdot sin(055 sdot 119905 minus 243) 089(c) = 3723 sdot sin(052 sdot 119905 minus 168) + 3161 sdot sin(052 sdot 119905 + 116) + 072 sdot sin(053 sdot 119905 minus 167) + 098 sdot sin(055 sdot 119905 minus 251) +

120 sdot sin(051 sdot 119905 minus 177)092

1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

sminus10

minus201910 1930 1950 1970 1990

Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 6

Rel variance

(a) Precipitation

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 5

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(b) Air temperature

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 4E + 142E + 7

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(c) Discharge

1910 1930 1950 1970 1990Time (year)

025051248Pe

riod

(yea

rs)

0 1E + 14

Rel variance1910 1920 1930 1940 1950 1960 1970 1980 1990

01020

Time (year)

Nor

mal

ized

serie

s

minus10

minus20

(d) Storage volume

Figure 3 Extraction of the representative time series using the wavelet transform (left the wavelet power spectrum center the global waveletpower spectrum and right the extracted time series about the period of approximately 1 year)

consisting of one input layer two hidden layers and oneoutput layer has been built

Monthly data series from 1903 to 1970 (800 months)has been used for the learning period 316 months fromthe learning period (1970ndash1995) are set up as verification

periods and the applicability of the constructed ANN modelis reviewed by comparing it to observed storage volume asa target data series (see Table 2) For the composition of theprediction data we again compose the three input data likecases (A) (B) and (C) Case (A) is integrated from each

Advances in Meteorology 7

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(a) Precipitation

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(b) Air temperature

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(c) Discharge

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(d) Storage volume

Figure 4 Attractors in each time series

case (a) in each time series Case (B) and case (C) are alsointegrated in the same way as case (A) Data series of 1116months (1903ndash1995) in each case (A) (B) and (C) has beenused for the prediction period

First of all according to the model verification measures(see Figure 8) such as the coefficient of correlation (CC0986) and root mean squared error (RMSE 0061) ANN isfitted very well and found its good applicability

The storage volume series of a reservoir system is esti-mated using the ANN model after setting case (A) case (b)and case (c) as the input data And then the correlationdimension analysis is performed for the estimated storagevolume in each case The results show that 255 in case (A)integrated the low-dimensional cases (a) 281 in case (B)integrated the middle-dimensional cases (b) and 289 in case(C) integrated the high-dimensional cases (c) as shown inFigure 9

45 Summary and Discussions In this study we posedthe two hypotheses to identify chaotic behavior in hydro-meteorological processes For solving this issuewe composedthe input data sets like cases (A) (B) and (C) and appliedthem to ANN model on the reservoir system of the GreatSalt Lake The criterion of the hypotheses is the sensitivityof chaotic behavior in the system In other words the firsthypothesis is reasonable if chaotic behavior in the system issensitive depending on chaotic characteristics of the inputdata otherwise the second hypothesis is reasonable Theresults of the correlation dimension analysis on every caseanalyzed in this study were summarized in Table 3

As shown in Table 3 the correlation dimensions are 255in case (A) obtained from integrating the low dimensions(precipitation 254 air temperature 102 and discharge 248)and 281 in case (B) from integrating the middle dimensions(precipitation 326 air temperature 184 and discharge 331)

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 7

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(a) Precipitation

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(b) Air temperature

0

5

10

15

20

0 10 20x(i+120591)

minus20 minus10

x(i)minus5

minus10

minus15

minus20

(c) Discharge

0

5

10

15

20

0 10 20x(i)

x(i+120591)

minus20 minus10

minus5

minus10

minus15

minus20

(d) Storage volume

Figure 4 Attractors in each time series

case (a) in each time series Case (B) and case (C) are alsointegrated in the same way as case (A) Data series of 1116months (1903ndash1995) in each case (A) (B) and (C) has beenused for the prediction period

First of all according to the model verification measures(see Figure 8) such as the coefficient of correlation (CC0986) and root mean squared error (RMSE 0061) ANN isfitted very well and found its good applicability

The storage volume series of a reservoir system is esti-mated using the ANN model after setting case (A) case (b)and case (c) as the input data And then the correlationdimension analysis is performed for the estimated storagevolume in each case The results show that 255 in case (A)integrated the low-dimensional cases (a) 281 in case (B)integrated the middle-dimensional cases (b) and 289 in case(C) integrated the high-dimensional cases (c) as shown inFigure 9

45 Summary and Discussions In this study we posedthe two hypotheses to identify chaotic behavior in hydro-meteorological processes For solving this issuewe composedthe input data sets like cases (A) (B) and (C) and appliedthem to ANN model on the reservoir system of the GreatSalt Lake The criterion of the hypotheses is the sensitivityof chaotic behavior in the system In other words the firsthypothesis is reasonable if chaotic behavior in the system issensitive depending on chaotic characteristics of the inputdata otherwise the second hypothesis is reasonable Theresults of the correlation dimension analysis on every caseanalyzed in this study were summarized in Table 3

As shown in Table 3 the correlation dimensions are 255in case (A) obtained from integrating the low dimensions(precipitation 254 air temperature 102 and discharge 248)and 281 in case (B) from integrating the middle dimensions(precipitation 326 air temperature 184 and discharge 331)

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

8 Advances in Meteorology

Table 2 Input data of ANN

Test Prediction Time scale

DataPrecipitation

Air temperatureDischarge

Storage volume

Case (A)Case (B)Case (C)

MonthlyLearning (calibration) period 1903sim1970 (68 years)

1ndash800 (800 months) mdash

Verification period 1970sim1995 (26 years)801ndash1116 (316 months) mdash

Prediction period 1ndash1116 (1116 months)

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 392

(a) Precipitation

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 141

(b) Air temperature

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 302

(c) Discharge

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

D2 = 265

(d) Storage volume

Figure 5 The estimated correlation dimension for each time series

and 289 in case (C) from integrating the highest dimensions(precipitation 405 air temperature 252 and discharge 380)The input data did not impact significantly on chaotic char-acteristics of the storage volume as the output even thoughthere was a little difference of the dimension 034 between

case (A) and case (B) Therefore the chaotic characteristic ofthe storage volume output in the Great Salt Lake ismost likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data However this chaoticbehavior will depend on each hydro-meteorological system

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 9

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

(i) Precipitation (ii) Air temperature

(iii) Discharge

D2 = 254

D2 = 326

D2 = 405

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 102

D2 = 184

D2 = 252

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(a)(b)(c)

D2 = 248

D2 = 313

D2 = 38

Figure 6 Correlation dimensions of generated time series

Table 3 Summary of correlation dimension in each case and timeseries

Correlation dimension (D2)Precipitation Air temperature Discharge GSL volume

Case (A) (a) 254 (a) 102 (a) 248 255Case (B) (b) 326 (b) 184 (b) 313 281Case (C) (c) 405 (c) 252 (c) 380 289Initial data 392 141 302 265

For example small hydro-meteorological systemswill be verysensitive and the chaotic characteristic will be also sensitivedepending on the input data

5 Conclusions

This study tried to identify and evaluate chaotic behaviorin hydro-meteorological processes For solving the issuesuggested in this study the two hypotheses were posed Firstassume that the input data is the significant factor to providechaotic characteristics to output data Second assume thatthe system itself is the significant factor to provide chaoticcharacteristics to output dataThe hydro-meteorological timeseries such as precipitation air temperature discharge andstorage volumewere collected in the Great Salt Lake and BearRiver Basin and the time series in the period of approximatelyone year were extracted from the original time series usingthe wavelet transform The results of the correlation dimen-sion analysis showed precipitation 392 air temperature 141

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

10 Advances in Meteorology

Airtemperature

Precipitation

Discharge

Storage volume

FHL-7 SHL-7

Input layer Hidden layer Output layer

FHL-2 SHL-2

FHL-1 SHL-1

Figure 7 Building up ANNmodel

800 850 900 950 1000 1050 1100 11500

02

04

06

08

1

Time (month)

Nor

mal

ized

GSL

vol

ume

ObservationSimulation

CC = 0986 RMSE = 0061

Figure 8 Verification result of ANN model for 316 months (1970ndash1995)

discharge 302 and storage volume 265 in each time seriesThe input data sets by the summation of sine functions werecomposed and applied them to the artificial neural networksfor modeling the reservoir system depending on the datasets and integrated the high middle and low dimensionsFinally the correlation dimension was analyzed to evaluatechaotic behavior of storage volume which is the final outputwith inputs of precipitation air temperature and dischargein the hydro-meteorological system The results showed thatthe chaotic characteristic of the storage volume is most likelya byproduct of the chaotic behavior of the reservoir systemitself rather than that of the input data We expect thatthe methodology and procedure suggested in this study willprovide a clue to understand chaotic behavior in hydro-meteorological processes

5 10 150

1

2

3

4

5

Embedding dimension

Cor

relat

ion

dim

ensio

n

(A)(B)(C)

D2 = 255

D2 = 281D2 = 289

Figure 9 Correlation dimension results in each case (A) (B) and(C)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Research Foun-dation of Korea (NRF) and grant funded by the KoreanGovernment (MEST no 2011-0028564) Also this work wassupported by INHA UNIVERSITY Research Grant

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 11

References

[1] B Sivakumar and V P Singh ldquoHydrologic system complexityand nonlinear dynamic concepts for a catchment classificationframeworkrdquoHydrology and Earth System Sciences vol 16 no 11pp 4119ndash4131 2012

[2] S L S Jacoby ldquoAmathematical model for nonlinear hydrologicsystemsrdquo Journal of Geophysical Research vol 71 no 20 pp4811ndash4824 1966

[3] J C I Dooge ldquoAnew approach to nonlinear problems in surfacewater hydrology hydrologic systems with uniform nonlinear-ityrdquo The International Association of Hydrological Sciences vol76 pp 409ndash413 1967

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980

[6] F Takens ldquoDetecting strange attractors in turbulencerdquo inDyna-mical Systems andTurbulence DA Rand andDAYoung Edsvol 898 of Lecture Notes in Mathematics pp 336ndash381 SpringerBerlin Germany 1981

[7] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[8] J D Farmer E Ott and J A Yorke ldquoThe dimension of chaoticattractorsrdquo Physica D Nonlinear Phenomena vol 7 no 1ndash3 pp153ndash180 1983

[9] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[10] A M Fraser and H L Swinney ldquoIndependent coordinates forstrange attractors frommutual informationrdquo Physical Review Avol 33 no 2 pp 1134ndash1140 1986

[11] C G Gilmore ldquoA new test for chaosrdquo Journal of EconomicBehavior and Organization vol 22 no 2 pp 209ndash237 1993

[12] J D Farmer and J J Sidorowich ldquoPredicting chaotic timeseriesrdquo Physical Review Letters vol 59 no 8 pp 845ndash848 1987

[13] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D Nonlinear Phenomena vol 35 no 3 pp 335ndash3561989

[14] M Casdagli and A Weigend ldquoExploring the continuumbetween deterministic and stochastic modelingrdquo in Forecastingthe Future and Understanding the Past A Weigend and S F IGershenfeld Eds vol 15 of Studies in the Sciences of Complexityp 993 Addison-Wesley 1994

[15] H S Kim R Eykholt and J D Salas ldquoDelay time window andplateau onset of the correlation dimension for small data setsrdquoPhysical Review E vol 58 no 5 pp 5676ndash5682 1998

[16] H S Kim R Eykholt and J D Salas ldquoNonlinear dynamicsdelay times and embedding windowsrdquo Physica D NonlinearPhenomena vol 127 no 1-2 pp 48ndash60 1999

[17] M Falanga and S Petrosino ldquoInferences on the source of long-period seismicity at Campi Flegrei from polarization analysisand reconstruction of the asymptotic dynamicsrdquo Bulletin ofVolcanology vol 74 no 6 pp 1537ndash1551 2012

[18] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[19] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climatic changerdquo Tellus vol 34 pp 10ndash16 1982

[20] C Nicolis ldquoStochastic aspects of climatic transitionsmdashresponseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[21] I Rodriguez-Iturbe B Febres De Power M B Sharifi and K PGeorgakakos ldquoChaos in rainfallrdquoWater Resources Research vol25 no 7 pp 1667ndash1675 1989

[22] B P Wilcox M S Seyfried and T H Matison ldquoSearchingfor chaotic dynamics in snowmelt runoffrdquo Water ResourcesResearch vol 27 no 6 pp 1005ndash1010 1991

[23] T Sangoyomi Climatic variability and dynamics of Great SaltLake hydrology [PhD thesis] Utah State University LoganUtah USA 1993

[24] T B Sangoyomi U Lall and H D I Abarbanel ldquoNonlineardynamics of the Great Salt Lake dimension estimationrdquoWaterResources Research vol 32 no 1 pp 149ndash159 1996

[25] G D Jeong and A R Rao ldquoChaos characteristics of tree ringseriesrdquo Journal of Hydrology vol 182 no 1ndash4 pp 239ndash257 1996

[26] I Rodriguez-Iturbe D Entekhabi and R L Bras ldquoNonlineardynamics of soil moisture at climate scales 1 Stochastic analy-sisrdquoWater Resources Research vol 27 no 8 pp 1899ndash1906 1991

[27] H S Kim Y N Yoon J H Kim and J H Kim ldquoSearching forstrange attractor in wastewater flowrdquo Stochastic EnvironmentalResearch and Risk Assessment vol 15 no 5 pp 399ndash413 2001

[28] J H Ahn andH S Kim ldquoNonlinear modeling of elninoSouth-ern osciilation indexrdquo Journal of Hydrologic Engineering vol 10no 1 pp 8ndash15 2005

[29] H S Kim K H Lee M S Kyoung B Sivakumar and E T LeeldquoMeasuring nonlinear dependence in hydrologic time seriesrdquoStochastic Environmental Research and Risk Assessment vol 23no 7 pp 907ndash916 2009

[30] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 no 1 pp 17ndash272014

[31] S Kim H Noh N Kang et al ldquoNoise reduction analysis ofradar rainfall using chaotic dynamics and filtering techniquesrdquoAdvances in Meteorology vol 2014 Article ID 517571 10 pages2014

[32] J D Salas H S Kim R Eykholt P Burlando and T R GreenldquoAggregation and sampling in deterministic chaos implicationsfor chaos identification in hydrological processesrdquo NonlinearProcesses in Geophysics vol 12 no 4 pp 557ndash567 2005

[33] C Karamperidou V Engel U Lall E Stabenau and T JSmith III ldquoImplications of multi-scale sea level and climatevariability for coastal resources a case study for south Floridaand Everglades National Park USArdquo Regional EnvironmentalChange vol 13 no 1 pp 91ndash100 2013

[34] C Chui An Introduction to Wavelets Wavelet Analysis and ItsApplication vol 1 Elsevier New York NY USA 1992

[35] C Torrence and G P Compo ldquoA Practical Guide to WaveletAnalysisrdquo Bulletin of the American Meteorological Society vol79 no 1 pp 61ndash78 1998

[36] C Bishop Neural Networks for Pattern Recognition ClarendonPress Oxford UK 2000

[37] P Picton Neural Networks Palgrave Basingstoke UK 2ndedition 2000

[38] W Hsieh Machine Learning Methods in the EnvironmentalSciences Cambridge University Press Cambridge UK 2009

[39] S Haupt A Pasini and CMarzbanArtificial IntelligenceMeth-ods in the Environmental Sciences Springer Berlin Germany2009

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

12 Advances in Meteorology

[40] R J Kuligowski and A P Barros ldquoLocalized precipitation fore-casts from anumerical weather predictionmodel using artificialneural networksrdquo Weather and Forecasting vol 13 no 4 pp1194ndash1204 1998

[41] Yuval and W W Hsieh ldquoAn adaptive nonlinear MOS schemefor precipitation forecasts using neural networksrdquo WeatherForecasting vol 18 no 2 pp 303ndash310 2003

[42] B Sivakumar K-K Phoon S-Y Liong and C-Y Liaw ldquoAsystematic approach to noise reduction in chaotic hydrologicaltime seriesrdquo Journal of Hydrology vol 219 no 3-4 pp 103ndash1351999

[43] A Hyvarinen J Karhunen and E Oja Independent ComponentAnalysis John Wiley amp Sons New York NY USA 2001

[44] A Hyvarinen and P Pajunen ldquoNonlinear independent com-ponent analysis existence and uniqueness resultsrdquo NeuralNetworks vol 12 no 3 pp 429ndash439 1999

[45] J Basak A Sudarshan D Trivedi and M S SanthanamldquoWeather data mining using independent component analysisrdquoThe Journal of Machine Learning Research vol 5 pp 239ndash253200304

[46] P Capuano E De Lauro S De Martino and M FalangaldquoWater-level oscillations in the Adriatic Sea as coherent self-oscillations inferred by independent component analysisrdquoProgress in Oceanography vol 91 no 4 pp 447ndash460 2011

[47] A Ciaramella E De Lauro S De Martino B Di Lieto MFalanga and R Tagliaferri ldquoCharacterization of Strombolianevents by using independent component analysisrdquo NonlinearProcesses in Geophysics vol 11 no 4 pp 453ndash461 2004

[48] E de Lauro S deMartinoM Falanga andM Palo ldquoDecompo-sition of high-frequency seismic wavefield of the Strombolian-like explosions at Erebus volcano by independent componentanalysisrdquo Geophysical Journal International vol 177 no 3 pp1399ndash1406 2009

[49] M S Karoui Y Deville S Hosseini A Ouamri and DDucrot ldquoImprovement of remote sensing multispectral imageclassification by using Independent Component Analysisrdquo inProceedings of the 1st Workshop on Hyperspectral Image andSignal Processing Evolution in Remote Sensing (WHISPERS rsquo09)pp 1ndash4 IEEE August 2009

[50] L Chen andC Lu ldquoAn improved independent component anal-ysis algorithmbased on artificial immune systemrdquo InternationalJournal ofMachine Learning andComputing vol 3 no 1 pp 93ndash97 2013

[51] J Holzfuss and G Mayer-Kress ldquoAn approach to error-estimation in the application of dimension algorithmsrdquo inDimensions and Entropies in Chaotic Systems G Mayer-KressEd vol 32 pp 114ndash122 Springer New York NY USA 1986

[52] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function E Basar Ed pp 135ndash152Springer Berlin Germany 1990

[53] A A Tsonis and J B Elsner ldquoThe weather attractor over veryshort time scalesrdquo Nature vol 333 no 6173 pp 545ndash547 1988

[54] J FourierTheorie Analytique de la Chaleur Firmin Didot Pereet Fils Paris France 1822 (French)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in