Research ArticleNoise Reduction Analysis of Radar Rainfall UsingChaotic Dynamics and Filtering Techniques

Soojun Kim1 Huiseong Noh2 Narae Kang2 Keonhaeng Lee3 Yonsoo Kim2 Sanghun Lim3

Dong Ryul Lee3 and Hung Soo Kim2

1 Columbia Water Center Columbia University New York NY 10027 USA2Department of Civil Engineering Inha University Incheon 402-751 Republic of Korea3Water Resources Research Division Korea Institute of Civil Engineering and Building Technology (KICT)Goyang 411-712 Republic of Korea

Correspondence should be addressed to Hung Soo Kim sookiminhaackr

Received 1 July 2014 Accepted 29 July 2014 Published 20 August 2014

Academic Editor Vincenzo Levizzani

Copyright copy 2014 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 evaluate the filtering techniques which can remove the noise involved in the time series For this Logisticseries which is chaotic series and radar rainfall series are used for the evaluation of low-pass filter (LF) and Kalman filter (KF)The noise is added to Logistic series by considering noise level and the noise added series is filtered by LF and KF for the noisereduction The analysis for the evaluation of LF and KF techniques is performed by the correlation coefficient standard error theattractor and the BDS statistic from chaos theory The analysis result for Logistic series clearly showed that KF is better tool thanLF for removing the noise Also we used the radar rainfall series for evaluating the noise reduction capabilities of LF and KF Inthis case it was difficult to distinguish which filtering technique is better way for noise reduction when the typical statistics suchas correlation coefficient and standard error were used However when the attractor and the BDS statistic were used for evaluatingLF and KF we could clearly identify that KF is better than LF

1 Introduction

Recently the advances of radar rainfall estimates withhigh spatial and temporal resolution have demonstrated theprospect of improving the accuracy of rainfall inputs forthe accuracy of real time flood forecasting However theadvantage of the weather radar rainfall estimates has beenlimited by a variety of sources of uncertainty in the radarreflectivity process including random and systematic errorsThere are a lot of discussions on radar rainfall estimationerrors [1ndash8]

There are several ways of filtering a signal in one or twodimensions An example of one which is often applied is low-pass filtering an operation which removes all components ofthe power spectrumwhose frequency is higher than a chosenthreshold Having this into account several approaches havebeen proposed to reduce radar errors Panofsky and Brier[9] introduced a low-pass filter which was borrowed from

electrical engineering terminology and removed the highvariability of noise from the data and leave only the low fre-quency components In the recent years the filtering methodhas been applied to reduce noise of radar rainfall in somestudies [10ndash12] The Kalman filtering approach has the mainadvantage of providing a real time scheme to calibrate radarrainfall estimates based on rain gauge measurements Thestudies of Ahnert [13] Smith and Krajewski [14] Anagnostouand Krajewski [15] Seo et al [16] Dinku et al [17] andChumchean et al [18] Krajewski et al [19] and Wang etal [20] used Kalman filtering to predict and update themean field bias in real time There is also a methodology todecompose a multivariate signal into independent signalsnamely independent component analysis (ICA) It is efficientto decompose the complexity of the dynamics in the seismo-logical and atmospheric field [20 21]

These days much amount of radar rainfall data is beingproduced processed and used Also the radar rainfall series

Hindawi Publishing CorporationAdvances in MeteorologyVolume 2014 Article ID 517571 10 pageshttpdxdoiorg1011552014517571

2 Advances in Meteorology

is widely applied to hydrologic applications such as flash floodforecasting However radar rainfall data include noise frommany sources and there are lacks of noise reduction studieson the radar rainfall data itself Therefore this study analyzesnoise of radar rainfall using chaotic dynamics which hasnonlinear and aperiodic nature and filtering techniques forinvestigating radar rainfall characteristics

To study the nonlinear characteristics of natural phenom-enamany statisticians and scientists have suggested the chaostheory which analyze and forecast the nonlinear phenomenaof the natural system Lorenz [22] suggested the strangeattractor in a simple model of convection roll in the atmo-sphere Packard et al [23] suggested themethod of delays andTakens [24] proved the method of delays using differentialtopology Fraser and Swinney [25] suggested a method forthe estimation of time delay using the mutual informationFalanga and Petrosino [26] estimated the complexity of thesystem by the degrees of freedom necessary to describe theasymptotic dynamics in a reconstructed phase space

All hydrological measurements are to some extent con-taminated by noise And the noise limits the performanceof many techniques of identification modeling predictionand control of deterministic systems [27] Some of the mostcharacteristic examples of the effects of noise are as follows(1) self-similarity of the attractor is broken (2) a phase spacereconstruction appears as high-dimensional on small lengthscales (3) nearby trajectories diverge diffusively rather thanexponentially and (4) the prediction error is found to bebounded from below no matter in which prediction methodis used and to how many digits the data are recorded [28]

This study evaluates the noise cancellation capabilitiesof filtering techniques of low-pass filter (LF) and Kalmanfilter (KF) To do this we regenerate chaotic data series andadd noise to the series And then we perform the noisereduction analysis for the noise added chaotic series by usingtwo filtering techniques and investigate the noise cancellationcapabilities of the techniques by the attractor of the seriesand by the BDS statistic [29ndash39] The same analysis is alsoperformed for the radar rainfall in this study

2 The BDS Statistic and NoiseReduction Filters

21 Phase Space Reconstruction The first step in metricanalysis of a chaotic time series is the construction of an 119898-dimensional embedding space from the scalar time seriesThis is done using the method of delays introduced byPackard et al [23] and Takens [24] which has the advantageof distributing the noise equally among the 119898 componentsA scalar time series 119909

119894 119894 = 1 2 is embedded into 119898-

dimensional space by constructing the vectors

119894= (119909119894 119909119894+119905 119909

119894+(119898minus1)119905)

119894isin 119877119898 (1)

where 119905 is the index lag and119898 is embedding dimension bothof which must be chosen appropriately If the sampling timeis 120591119904 the delay time is 120591

119889= 119905120591119904

The reconstructed state variables 119909119894need to be indepen-

dent and the quality of reconstructed attractor depends on

the choice of the index lag 119905 If the delay time 120591119889is too small

the reconstructed attractor is compressed along the identityline and this is called redundance If 120591

119889is too large the

attractor dynamicsmay become causally disconnectedwhichis called irrelevance and which may cause the attractor toappear much more complex than it really is [40]

22 The BDS Statistic The BDS statistic is derived fromthe correlation integral and has its origins in the recentwork on deterministic nonlinear dynamics and chaos theoryGrassberger and Procaccia [41] introduced the correlationintegral as a method of measuring the fractal dimension ofdeterministic data It is measure of the frequency with whichtemporal patterns are repeated in the data The correlationintegral at embedding dimension119898 is given by

119862 (119898119873 119903) =2

119872 (119872 minus 1)sum1le119894lt119895le119872

Θ(119903 minus10038171003817100381710038171003817119894minus 119895

10038171003817100381710038171003817) 119903 gt 0

(2)

where Θ(119886) = 0 if 119886 le 0 Θ(119886) = 1 if 119886 ge 0And 119873 is the size of the data sets119872 = 119873 minus (119898 minus 1)119905 is

the number of embedded points in119898-dimensional space and sdot denotes the sup-norm 119862(119898119873 119903)measures the fractionof the pairs of points

119894 119894 = 1 2 119872 whose sup-norm

separation is not greater than 119903 If the limit of 119862(119898119873 119903 119905)as 119873 rarr infin exists for each 119903 we write the fraction of allstate vector points that are within 119903 of each other as119862(119898 119903) =lim119873rarrinfin

119862(119898119873 119903)If the data are generated by a strictly stationary stochastic

process which is absolutely regular then this limit exists Inthis case the limit is as follows

119862 (119898 119903) = intint119883

Θ(119903 minus1003817100381710038171003817 minus 119910

1003817100381710038171003817) 119889119865 () 119889119865 ( 119910) 119903 gt 0

(3)

When the process is IID and since Θ(119903 minus minus 119910) =

prod119898

119896=1Θ(119903 minus |119909

119896minus 119910119896|) (3) implies that 119862(119898 119903) = 119862119898(1 119903)

Also 119862(119898 119903) minus 119862119898(1 119903) has asymptotic normal distributionwith zero mean and variance as follows

1205902 (119898119872 119903)

4

= 119898 (119898 minus 1) 1198622(119898minus1)

(119870 minus 1198622) + 119870119898minus 1198622119898

+ 2

119898minus1

sum119894=1

[1198622119894(119870119898minus119894minus 1198622(119898minus119894)

) minus 1198981198622(119898minus119894)

(119870 minus 1198622)]

(4)

We can consistently estimate the constants 119862 by 119862(1 119903)and119870 by

119870 (119898119872 119903)

=6

119872 (119872 minus 1) (119872 minus 2)

times sum1le119894lt119895le119872

[Θ (119903 minus10038171003817100381710038171003817119894minus 119895

10038171003817100381710038171003817)Θ (119903 minus

10038171003817100381710038171003817119894minus 119895

10038171003817100381710038171003817)]

(5)

Advances in Meteorology 3

Under the IID hypothesis the BDS statistic for 119898 gt 1 isdefined as

BDS (119898119872 119903) =radic119872

120590[119862 (119898 119903) minus 119862

119898(1 119903)] (6)

which has a limiting standard normal distribution under thenull hypothesis of IID as 119872 rarr infin and obtains its criticalvalues using the standard normal distribution

Before applying the BDS statistic the first addressedissue is which region of ldquo119903rdquo yields BDS statistics that arewell approximated by the asymptotic distribution As thesample size is increased the distribution of the BDS statisticbecomes more normal So the minimal number of data mustbe provided Next the region of embedding dimension ldquo119898rdquoshould be suggested If the sample size is fixed we expect thefinite sample property to worsen as ldquo119898rdquo increases This studyfollows the recommendation of Brock et al [29] for selectingthe ranges of 119898 119903 and 119873 The embedding dimension 119898 isused in the range of 2 le 119898 le 5 Then the value of ldquo119903rdquo isselected as the half standard deviations of the data sets

23 Kalman Filter and Low-Pass Filter The Kalman filter(KF) was introduced by Kalmanrsquos famous paper describing arecursive solution to the discrete data linear filtering problem[42] The Kalman filter algorithm can be applied as anestimator of the state of a dynamic system described by thelinear difference equation

119909119896= 119860119896minus1119909119896minus1+ 119861119896minus1119906119896minus1+ 119908119896minus1 (7)

where matrix119860119896minus1

is the relationship between the state of thesystem described by the column vector 119909

119896minus1at time 119896 minus 1

and the state 119909119896at time 119896 The column form of a vector is

considered to be the normal form and a transpose into a rowvector will be denoted by 119879 this is also the notation used forthe transpose of matrices Matrix 119861

119896minus1relates the optional

control input vector 119906119896minus1

to the state and the vector 119908119896minus1

isthe process noise

The system is then measured at discrete points in timewhere the measurement vector 119911

119896is related to the true state

of the system by the equation

119911119896= 119867119896119909119896+ V119896 (8)

where vector V119896represents the measurement noise and119867

119896is

the observation matrixThey are assumed to be independent (of each other)

white and with normal probability distributions

119901 (119908) sim 119873 (0 119876) (9)

119901 (V) sim 119873 (0 119877) (10)

In practice the process noise covariance 119876 and measure-ment noise covariance 119877 matrices might change with eachtime step or measurement however here we assume they areconstant

The error covariance matrix 119875119896|119896

of the estimate 119909119896|119896

is

119875119896|119896= cov (119909

119896minus 119909119896|119896 119909119896minus 119909119896|119896) (11)

00

05

10

15

20

0 10 20 30 40 50 60 70 80 90 100N

Valu

e

minus10

minus05

xi + 10s

xi + 05s

xi + 01s

xi

Figure 1 Logistic series with the noise levels of 10 50 and 100(119873 = 1sim100)

The estimate vector 119909119896|119896

is referred to as the a posterioriestimate since it is an estimate of the system at time 119896

A posteriori state estimate 119909119896|119896

is computed as a linearcombination of a priori estimate 119909

119896|119896minus1and a weighted differ-

ence between an actual measurement 119911119896and a measurement

prediction119867119896119909119896|119896minus1

as follows

119909119896|119896= 119909119896|119896minus1

+ 119870119896(119911119896minus 119867119896119909119896|119896minus1) (12)

119870119896= 119875119896|119896minus1119867119879

119896(119867119896119875119896|119896minus1119867119879

119896+ 119877119896)minus1

(13)

The matrix 119870119896in (13) is chosen to be the gain that

minimizes the a posteriori error covariance equationKim [43] shows how to track a varying signal and at

the same time reduce the influence of measurement noiseby using a 1st order low-pass filter (LF) (an exponentiallyweighted moving average filter) described as

119909119896|119896= 120572119909119896|119896minus1

+ (1 minus 120572) 119911119896

= 120572 120572119909119896|119896minus2

+ (1 minus 120572) 119911119896minus1 + (1 minus 120572) 119911119896

= 1205722119909119896|119896minus2

+ 120572 (1 minus 120572) 119911119896minus1 + (1 minus 120572) 119911119896

(14)

where 120572 is a constant in the range of 0 lt 120572 lt 1The expression for the computation of the a posteriori

estimate 119909119896|119896

in (12) is very similar to the 1st order low-pass filter with the significant difference lying in the varyingKalman filter gain instead of the constant 120572

3 Noise Reduction Studies on Logistic andRadar Rainfall Series

31 Noise Influence on Logistic Series

311 Attractor Characteristics in Noise Level To study theeffects of noise in a time series we added Gaussian noiseto the time series Specifically it considered the noise added

4 Advances in Meteorology

00

02

04

06

08

10

12

0 02 04 06 08 1x(i)

x(i+1)

(a) Noise level = 0

00

02

04

06

08

10

12

14

0 02 04 06 08 1 12x(i)

x(i+1)

minus02

minus02

(b) Noise level = 10

00

05

10

15

20

0 04 08 12 16 2x(i)

x(i+1)

minus05

minus08 minus04

minus10

minus12

(c) Noise level = 50

00

10

20

30

0 1 2 3x(i)

x(i+1)

minus10

minus20

minus2 minus1

(d) Noise level = 100

Figure 2 The attractors with the noise level

time series 119909120576119894 to test the noise effect in the original time

series 119909119894 and define the 119909

120576119894 as

119909120576119894= 119909119894+ 120578120590120576119894 119894 = 1 2 119873 (15)

where 120578 is the noise level 120590 is a standard deviation of 119909119894 and

Gaussian noise 120576119894has119873(0 1)

May [44] emphasized that a simple nonlinear map mayhave very complicated dynamics and showed his point withLogistic map which is a discrete time analog for populationgrowth Logistic map is defined as

119909119905+1= 119903119909119905(1 minus 119909

119905) (16)

where 119903 is between 0 and 4 For small values of 119903 the system isstable andwell behaved however as the value of 119903 approaches4 it becomes chaotic We simulate Logistic map sequence of119873 = 1000 and add noise to it with the noise levels 10 50and 100 Logistic series with noise are shown in Figure 1Here 119904 is standard deviation of the sample series in Figure 1

The attractor of each Logistic series is reconstructed inphase space and the characteristics of the series can be

identified (Figure 2) For the reconstruction of the seriesusing (1) the embedding dimension 119898 = 2 and delay time120591119889= 1 are used (Figure 2) The autocorrelation function

(ACF) is expected to provide a reasonable measure of thetransition from redundance to irrelevance as a function ofdelay It is considered that the decorrelation time equals thelag (delay time 120591

119889) at which the ACF first attains the value

zero Otherwise 120591119889should be chosen as the local minimum

ofACF whichever occurs first [45 46]When theACF decaysexponentially we select 120591

119889at which the ACF drops 1119890 [47]

The original Logistic series which has one variable showsits attractor with a simple quadratic form (Figure 2(a))However as the noise level is increased the attractor isbecoming more andmore complicated form which it is high-dimensional series (Figures 2(b)ndash2(d)) For the noise level =100 especially the attractor looks like random series

312 Noise Reduction Studies of Logistic Series This sectionstudies the noise reduction of the noise added Logistic seriesusing LF and KF Noise cannot be forecasted but statisticallyestimated and the parameters of LF and KF are calibrated by

Advances in Meteorology 5

00

05

10

15

0 20 40 60 80 100t

Valu

e

(a) Noise level = 10

t

0002040608101214

0 20 40 60 80 100

Valu

e

minus02

minus04

(b) Noise level = 50

t

00030609121518

0 20 40 60 80 100

Valu

e

Raw data seriesKalman filterLow-pass filter

minus03

minus06

(c) Noise level = 100

Figure 3 Noise removed data series through Low-pass filter and Kalman filter

Table 1 Statistical characteristics of Logistic series after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error

119909119894+ 01119904 0699 0251 0994 0037

119909119894+ 05119904 0629 0273 0986 0058

119909119894+ 10119904 0501 0303 0925 0134

trial and error method The constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The results of noise reduction studies usingLF and KF are shown in Figure 3When noise level is smallernoise can be removedmore effectively by LF andKF Table 1 isshowing the statistical results for noise reduction analysis byLF and KF LF has the coefficient of correlation 050ndash070 andstandard error 030ndash025 and KF has 093ndash099 and 004ndash01Therefore KF can reduce noise more effectively than LF

The attractors for noise removed Logistic series by LFand KF are reconstructed in phase space (Figure 4) Thenoise removed series by LF show their attractors which stillhave noisy shapes (Figures 4(a)ndash4(c)) but the noise removedseries by KF show more clear attractors which describe thecharacteristics of Logistic map (Figures 4(d)ndash4(f)) Eventhough KF is more effective way for removing noise in theseries it is difficult to restore it to the original state If weinvestigate the range of the values of the series generated from

Logistic equation we can find that the values of the originalseries are in the range of 0 to 1 The values of noise removedseries by LF and KF for the series having noise level = 100are investigated and the values by LF and KF are in the rangeof minus1 to 2 (Figure 4(c)) and 0 to 1 (Figure 4(f)) respectivelyTherefore the result is showing that KF is more proper toolfor the noise reduction of the series

The BDS statistic was applied for testing for nonlinearityof each data series Not only is it useful in detecting determin-istic chaos but it also serves as a residual diagnostic If themodel (null hypothesis) is correct then the estimated resid-uals will pass the test for IID (independently and identicallydistributed) A failure to pass the test is an indication that theselected model is misspecified Here the confidence interval(CL) of 95 which is a significance level of 5 is usedfor the randomness test of a time series The original seriesnoise added series and noise removed series of Logisticmap are analyzed by the BDS statistic for their randomnessand nonlinearity And the results are shown in Table 2 The

6 Advances in Meteorology

00

02

04

06

08

10

00 02 04 06 08 10 12minus02minus02

x(i)

x(i+1)

(a) LF noise level = 10x(i+1)

00 05 10 15x(i)

minus05

00

05

10

15

minus05

(b) LF noise level = 50

x(i+1)

00

05

10

15

20

00 10 20x(i)

minus05

minus10minus10

(c) LF noise level = 100

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(d) KF noise level = 10

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(e) KF noise level = 50

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(f) KF noise level = 100

Figure 4 The noise removed attractors by LF and KF

Table 2 The BDS statistic values for data series in each case

119898 119903 119909119894

The series with noise Low-pass filter Kalman filter 95 CI119909119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904

2

05 4873 5354 333 05 2664 296 13 4460 3690 1602

[minus196 196]1 2019 1876 161 11 1519 164 13 2000 1844 72015 111 106 34 minus03 126 53 06 116 127 1592 minus203 minus197 minus40 minus18 minus102 minus13 minus05 minus181 minus149 72

3

05 6509 7208 386 09 3612 346 19 6125 5297 2567

[minus196 196]1 1826 1714 154 14 1396 153 18 1875 1777 77815 15 20 29 02 33 42 09 38 64 1492 minus176 minus169 minus34 minus11 minus92 minus11 00 minus129 minus100 66

4

05 8565 9576 404 11 4776 379 23 8071 7179 3801

[minus196 196]1 1742 1643 143 16 1343 144 21 1763 1670 74315 07 04 03 06 09 37 13 14 25 1112 minus153 minus148 minus30 minus06 minus86 minus11 04 minus105 minus81 60

5

05 11739 13264 423 14 6567 407 26 11018 10008 5577

[minus196 196]1 1684 1615 137 16 1345 137 21 1736 1628 74315 minus08 minus11 02 07 minus08 33 13 03 14 862 minus136 minus135 minus29 minus04 minus82 minus11 05 minus92 minus72 51

Advances in Meteorology 7

Table 3 Statistical characteristics of radar rainfall after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error0994 0156 0989 0231

01234567

0 500 1000 1500 2000 2500 3000

Rain

fall

(mm

)

Time (min)

(a)

01234567

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

(b)

Figure 5 The time series (a) of radar rainfall and its attractor (b)

original series is showing its nonlinearity and the series withnoise level of 100 which has bold type in Table 2 representsits randomness The noise removed series by LF which is inbold type in Table 2 is also showing its randomnessThe boldtype describes that the null hypothesis can be accepted andthe null hypotheses of the series in all other columns exceptfor two columns of the series with noise and LF with 119909

119894+10119904

cannot be accepted If we see the values of the BDS statisticin Table 2 KF has more similar BDS statistic values with thevalues of original series than LFTherefore it can be identifiedthat KF is more proper tool for noise cancellation than LF

32 Noise Influence on Radar Rainfall Series

321 Radar Rainfall Series and Its Attractor Radar rainfall isa representative hydrologic data which includes noise frommany sources This study uses the radar rainfall obtainedfrom the radar in Biseul Mountain radar (BSL radar) inGyeongbuk province Korea The radar rainfall series inGamcheon watershed especially which is produced in BSLradar is used for analyzing the series characteristics accordingto noise cancellation by LF andKF BSL radarwas constructedin 2009 and it is dual polarization radar The radar hastemporal and spatial resolutions of 25min and 250m times

250m Therefore BSL radar rainfall series of 25min-timeinterval is obtained with the data period of 6242011 0900ndash6262011 1100 (about 3000min average 173mm standarddeviation 153mm)

The ACF of radar rainfall series was exponentiallydecreased and so the delay timewas selected as 120591

119889= 825 min

(lag 119896 = 33) at which the ACF drops 1119890 (Tsonis and Elsner[47]) The time series plot and the reconstructed attractor ofradar rainfall are shown in Figure 5 Even though the ACFshowed the persistence of radar rainfall series the attractor iscomplicated and we can know that the radar rainfall is greatlyinfluenced by noise

0

2

4

6

8

10

0 500 1000 1500 2000

Raw data seriesKalman filterLow-pass filter

Rain

fall

(mm

)

Time (min)

3456

1500 1550 1600

Figure 6 The raw data and noise removed series of radar rainfall

322 Noise Reduction Studies of Radar Rainfall Series Thissection applies LF and KF for the noise reduction study ofradar rainfall series and the constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The raw data series of radar rainfall and theresults of noise reduction studies using LF and KF are shownin Figure 6 The magnified red box in Figure 6 is for heavyrainfall period Table 3 is showing the statistical results fornoise reduction analysis by LF and KF for the radar rainfallseries LF has the coefficient of correlation 0994 and standarderror 0156 and KF has 0989 and 0231 In this case thereis not much difference between LF and KF Therefore bothfiltering techniques show the similar function for removingnoise involved in radar rainfall

The attractors for noise removed radar rainfall seriesby LF and KF are reconstructed in phase space (Figure 7)The noise removed radar rainfall series by LF shows thatits attractor is more simplified shape (Figure 7(a)) than the

8 Advances in Meteorology

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

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Journal of

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OceanographyInternational Journal of

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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

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MeteorologyAdvances in

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ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

2 Advances in Meteorology

is widely applied to hydrologic applications such as flash floodforecasting However radar rainfall data include noise frommany sources and there are lacks of noise reduction studieson the radar rainfall data itself Therefore this study analyzesnoise of radar rainfall using chaotic dynamics which hasnonlinear and aperiodic nature and filtering techniques forinvestigating radar rainfall characteristics

To study the nonlinear characteristics of natural phenom-enamany statisticians and scientists have suggested the chaostheory which analyze and forecast the nonlinear phenomenaof the natural system Lorenz [22] suggested the strangeattractor in a simple model of convection roll in the atmo-sphere Packard et al [23] suggested themethod of delays andTakens [24] proved the method of delays using differentialtopology Fraser and Swinney [25] suggested a method forthe estimation of time delay using the mutual informationFalanga and Petrosino [26] estimated the complexity of thesystem by the degrees of freedom necessary to describe theasymptotic dynamics in a reconstructed phase space

All hydrological measurements are to some extent con-taminated by noise And the noise limits the performanceof many techniques of identification modeling predictionand control of deterministic systems [27] Some of the mostcharacteristic examples of the effects of noise are as follows(1) self-similarity of the attractor is broken (2) a phase spacereconstruction appears as high-dimensional on small lengthscales (3) nearby trajectories diverge diffusively rather thanexponentially and (4) the prediction error is found to bebounded from below no matter in which prediction methodis used and to how many digits the data are recorded [28]

This study evaluates the noise cancellation capabilitiesof filtering techniques of low-pass filter (LF) and Kalmanfilter (KF) To do this we regenerate chaotic data series andadd noise to the series And then we perform the noisereduction analysis for the noise added chaotic series by usingtwo filtering techniques and investigate the noise cancellationcapabilities of the techniques by the attractor of the seriesand by the BDS statistic [29ndash39] The same analysis is alsoperformed for the radar rainfall in this study

2 The BDS Statistic and NoiseReduction Filters

21 Phase Space Reconstruction The first step in metricanalysis of a chaotic time series is the construction of an 119898-dimensional embedding space from the scalar time seriesThis is done using the method of delays introduced byPackard et al [23] and Takens [24] which has the advantageof distributing the noise equally among the 119898 componentsA scalar time series 119909

119894 119894 = 1 2 is embedded into 119898-

dimensional space by constructing the vectors

119894= (119909119894 119909119894+119905 119909

119894+(119898minus1)119905)

119894isin 119877119898 (1)

where 119905 is the index lag and119898 is embedding dimension bothof which must be chosen appropriately If the sampling timeis 120591119904 the delay time is 120591

119889= 119905120591119904

The reconstructed state variables 119909119894need to be indepen-

dent and the quality of reconstructed attractor depends on

the choice of the index lag 119905 If the delay time 120591119889is too small

the reconstructed attractor is compressed along the identityline and this is called redundance If 120591

119889is too large the

attractor dynamicsmay become causally disconnectedwhichis called irrelevance and which may cause the attractor toappear much more complex than it really is [40]

22 The BDS Statistic The BDS statistic is derived fromthe correlation integral and has its origins in the recentwork on deterministic nonlinear dynamics and chaos theoryGrassberger and Procaccia [41] introduced the correlationintegral as a method of measuring the fractal dimension ofdeterministic data It is measure of the frequency with whichtemporal patterns are repeated in the data The correlationintegral at embedding dimension119898 is given by

119862 (119898119873 119903) =2

119872 (119872 minus 1)sum1le119894lt119895le119872

Θ(119903 minus10038171003817100381710038171003817119894minus 119895

10038171003817100381710038171003817) 119903 gt 0

(2)

where Θ(119886) = 0 if 119886 le 0 Θ(119886) = 1 if 119886 ge 0And 119873 is the size of the data sets119872 = 119873 minus (119898 minus 1)119905 is

the number of embedded points in119898-dimensional space and sdot denotes the sup-norm 119862(119898119873 119903)measures the fractionof the pairs of points

119894 119894 = 1 2 119872 whose sup-norm

separation is not greater than 119903 If the limit of 119862(119898119873 119903 119905)as 119873 rarr infin exists for each 119903 we write the fraction of allstate vector points that are within 119903 of each other as119862(119898 119903) =lim119873rarrinfin

119862(119898119873 119903)If the data are generated by a strictly stationary stochastic

process which is absolutely regular then this limit exists Inthis case the limit is as follows

119862 (119898 119903) = intint119883

Θ(119903 minus1003817100381710038171003817 minus 119910

1003817100381710038171003817) 119889119865 () 119889119865 ( 119910) 119903 gt 0

(3)

When the process is IID and since Θ(119903 minus minus 119910) =

prod119898

119896=1Θ(119903 minus |119909

119896minus 119910119896|) (3) implies that 119862(119898 119903) = 119862119898(1 119903)

Also 119862(119898 119903) minus 119862119898(1 119903) has asymptotic normal distributionwith zero mean and variance as follows

1205902 (119898119872 119903)

4

= 119898 (119898 minus 1) 1198622(119898minus1)

(119870 minus 1198622) + 119870119898minus 1198622119898

+ 2

119898minus1

sum119894=1

[1198622119894(119870119898minus119894minus 1198622(119898minus119894)

) minus 1198981198622(119898minus119894)

(119870 minus 1198622)]

(4)

We can consistently estimate the constants 119862 by 119862(1 119903)and119870 by

119870 (119898119872 119903)

=6

119872 (119872 minus 1) (119872 minus 2)

times sum1le119894lt119895le119872

[Θ (119903 minus10038171003817100381710038171003817119894minus 119895

10038171003817100381710038171003817)Θ (119903 minus

10038171003817100381710038171003817119894minus 119895

10038171003817100381710038171003817)]

(5)

Advances in Meteorology 3

Under the IID hypothesis the BDS statistic for 119898 gt 1 isdefined as

BDS (119898119872 119903) =radic119872

120590[119862 (119898 119903) minus 119862

119898(1 119903)] (6)

which has a limiting standard normal distribution under thenull hypothesis of IID as 119872 rarr infin and obtains its criticalvalues using the standard normal distribution

Before applying the BDS statistic the first addressedissue is which region of ldquo119903rdquo yields BDS statistics that arewell approximated by the asymptotic distribution As thesample size is increased the distribution of the BDS statisticbecomes more normal So the minimal number of data mustbe provided Next the region of embedding dimension ldquo119898rdquoshould be suggested If the sample size is fixed we expect thefinite sample property to worsen as ldquo119898rdquo increases This studyfollows the recommendation of Brock et al [29] for selectingthe ranges of 119898 119903 and 119873 The embedding dimension 119898 isused in the range of 2 le 119898 le 5 Then the value of ldquo119903rdquo isselected as the half standard deviations of the data sets

23 Kalman Filter and Low-Pass Filter The Kalman filter(KF) was introduced by Kalmanrsquos famous paper describing arecursive solution to the discrete data linear filtering problem[42] The Kalman filter algorithm can be applied as anestimator of the state of a dynamic system described by thelinear difference equation

119909119896= 119860119896minus1119909119896minus1+ 119861119896minus1119906119896minus1+ 119908119896minus1 (7)

where matrix119860119896minus1

is the relationship between the state of thesystem described by the column vector 119909

119896minus1at time 119896 minus 1

and the state 119909119896at time 119896 The column form of a vector is

considered to be the normal form and a transpose into a rowvector will be denoted by 119879 this is also the notation used forthe transpose of matrices Matrix 119861

119896minus1relates the optional

control input vector 119906119896minus1

to the state and the vector 119908119896minus1

isthe process noise

The system is then measured at discrete points in timewhere the measurement vector 119911

119896is related to the true state

of the system by the equation

119911119896= 119867119896119909119896+ V119896 (8)

where vector V119896represents the measurement noise and119867

119896is

the observation matrixThey are assumed to be independent (of each other)

white and with normal probability distributions

119901 (119908) sim 119873 (0 119876) (9)

119901 (V) sim 119873 (0 119877) (10)

In practice the process noise covariance 119876 and measure-ment noise covariance 119877 matrices might change with eachtime step or measurement however here we assume they areconstant

The error covariance matrix 119875119896|119896

of the estimate 119909119896|119896

is

119875119896|119896= cov (119909

119896minus 119909119896|119896 119909119896minus 119909119896|119896) (11)

00

05

10

15

20

0 10 20 30 40 50 60 70 80 90 100N

Valu

e

minus10

minus05

xi + 10s

xi + 05s

xi + 01s

xi

Figure 1 Logistic series with the noise levels of 10 50 and 100(119873 = 1sim100)

The estimate vector 119909119896|119896

is referred to as the a posterioriestimate since it is an estimate of the system at time 119896

A posteriori state estimate 119909119896|119896

is computed as a linearcombination of a priori estimate 119909

119896|119896minus1and a weighted differ-

ence between an actual measurement 119911119896and a measurement

prediction119867119896119909119896|119896minus1

as follows

119909119896|119896= 119909119896|119896minus1

+ 119870119896(119911119896minus 119867119896119909119896|119896minus1) (12)

119870119896= 119875119896|119896minus1119867119879

119896(119867119896119875119896|119896minus1119867119879

119896+ 119877119896)minus1

(13)

The matrix 119870119896in (13) is chosen to be the gain that

minimizes the a posteriori error covariance equationKim [43] shows how to track a varying signal and at

the same time reduce the influence of measurement noiseby using a 1st order low-pass filter (LF) (an exponentiallyweighted moving average filter) described as

119909119896|119896= 120572119909119896|119896minus1

+ (1 minus 120572) 119911119896

= 120572 120572119909119896|119896minus2

+ (1 minus 120572) 119911119896minus1 + (1 minus 120572) 119911119896

= 1205722119909119896|119896minus2

+ 120572 (1 minus 120572) 119911119896minus1 + (1 minus 120572) 119911119896

(14)

where 120572 is a constant in the range of 0 lt 120572 lt 1The expression for the computation of the a posteriori

estimate 119909119896|119896

in (12) is very similar to the 1st order low-pass filter with the significant difference lying in the varyingKalman filter gain instead of the constant 120572

3 Noise Reduction Studies on Logistic andRadar Rainfall Series

31 Noise Influence on Logistic Series

311 Attractor Characteristics in Noise Level To study theeffects of noise in a time series we added Gaussian noiseto the time series Specifically it considered the noise added

4 Advances in Meteorology

00

02

04

06

08

10

12

0 02 04 06 08 1x(i)

x(i+1)

(a) Noise level = 0

00

02

04

06

08

10

12

14

0 02 04 06 08 1 12x(i)

x(i+1)

minus02

minus02

(b) Noise level = 10

00

05

10

15

20

0 04 08 12 16 2x(i)

x(i+1)

minus05

minus08 minus04

minus10

minus12

(c) Noise level = 50

00

10

20

30

0 1 2 3x(i)

x(i+1)

minus10

minus20

minus2 minus1

(d) Noise level = 100

Figure 2 The attractors with the noise level

time series 119909120576119894 to test the noise effect in the original time

series 119909119894 and define the 119909

120576119894 as

119909120576119894= 119909119894+ 120578120590120576119894 119894 = 1 2 119873 (15)

where 120578 is the noise level 120590 is a standard deviation of 119909119894 and

Gaussian noise 120576119894has119873(0 1)

May [44] emphasized that a simple nonlinear map mayhave very complicated dynamics and showed his point withLogistic map which is a discrete time analog for populationgrowth Logistic map is defined as

119909119905+1= 119903119909119905(1 minus 119909

119905) (16)

where 119903 is between 0 and 4 For small values of 119903 the system isstable andwell behaved however as the value of 119903 approaches4 it becomes chaotic We simulate Logistic map sequence of119873 = 1000 and add noise to it with the noise levels 10 50and 100 Logistic series with noise are shown in Figure 1Here 119904 is standard deviation of the sample series in Figure 1

The attractor of each Logistic series is reconstructed inphase space and the characteristics of the series can be

identified (Figure 2) For the reconstruction of the seriesusing (1) the embedding dimension 119898 = 2 and delay time120591119889= 1 are used (Figure 2) The autocorrelation function

(ACF) is expected to provide a reasonable measure of thetransition from redundance to irrelevance as a function ofdelay It is considered that the decorrelation time equals thelag (delay time 120591

119889) at which the ACF first attains the value

zero Otherwise 120591119889should be chosen as the local minimum

ofACF whichever occurs first [45 46]When theACF decaysexponentially we select 120591

119889at which the ACF drops 1119890 [47]

The original Logistic series which has one variable showsits attractor with a simple quadratic form (Figure 2(a))However as the noise level is increased the attractor isbecoming more andmore complicated form which it is high-dimensional series (Figures 2(b)ndash2(d)) For the noise level =100 especially the attractor looks like random series

312 Noise Reduction Studies of Logistic Series This sectionstudies the noise reduction of the noise added Logistic seriesusing LF and KF Noise cannot be forecasted but statisticallyestimated and the parameters of LF and KF are calibrated by

Advances in Meteorology 5

00

05

10

15

0 20 40 60 80 100t

Valu

e

(a) Noise level = 10

t

0002040608101214

0 20 40 60 80 100

Valu

e

minus02

minus04

(b) Noise level = 50

t

00030609121518

0 20 40 60 80 100

Valu

e

Raw data seriesKalman filterLow-pass filter

minus03

minus06

(c) Noise level = 100

Figure 3 Noise removed data series through Low-pass filter and Kalman filter

Table 1 Statistical characteristics of Logistic series after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error

119909119894+ 01119904 0699 0251 0994 0037

119909119894+ 05119904 0629 0273 0986 0058

119909119894+ 10119904 0501 0303 0925 0134

trial and error method The constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The results of noise reduction studies usingLF and KF are shown in Figure 3When noise level is smallernoise can be removedmore effectively by LF andKF Table 1 isshowing the statistical results for noise reduction analysis byLF and KF LF has the coefficient of correlation 050ndash070 andstandard error 030ndash025 and KF has 093ndash099 and 004ndash01Therefore KF can reduce noise more effectively than LF

The attractors for noise removed Logistic series by LFand KF are reconstructed in phase space (Figure 4) Thenoise removed series by LF show their attractors which stillhave noisy shapes (Figures 4(a)ndash4(c)) but the noise removedseries by KF show more clear attractors which describe thecharacteristics of Logistic map (Figures 4(d)ndash4(f)) Eventhough KF is more effective way for removing noise in theseries it is difficult to restore it to the original state If weinvestigate the range of the values of the series generated from

Logistic equation we can find that the values of the originalseries are in the range of 0 to 1 The values of noise removedseries by LF and KF for the series having noise level = 100are investigated and the values by LF and KF are in the rangeof minus1 to 2 (Figure 4(c)) and 0 to 1 (Figure 4(f)) respectivelyTherefore the result is showing that KF is more proper toolfor the noise reduction of the series

The BDS statistic was applied for testing for nonlinearityof each data series Not only is it useful in detecting determin-istic chaos but it also serves as a residual diagnostic If themodel (null hypothesis) is correct then the estimated resid-uals will pass the test for IID (independently and identicallydistributed) A failure to pass the test is an indication that theselected model is misspecified Here the confidence interval(CL) of 95 which is a significance level of 5 is usedfor the randomness test of a time series The original seriesnoise added series and noise removed series of Logisticmap are analyzed by the BDS statistic for their randomnessand nonlinearity And the results are shown in Table 2 The

6 Advances in Meteorology

00

02

04

06

08

10

00 02 04 06 08 10 12minus02minus02

x(i)

x(i+1)

(a) LF noise level = 10x(i+1)

00 05 10 15x(i)

minus05

00

05

10

15

minus05

(b) LF noise level = 50

x(i+1)

00

05

10

15

20

00 10 20x(i)

minus05

minus10minus10

(c) LF noise level = 100

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(d) KF noise level = 10

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(e) KF noise level = 50

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(f) KF noise level = 100

Figure 4 The noise removed attractors by LF and KF

Table 2 The BDS statistic values for data series in each case

119898 119903 119909119894

The series with noise Low-pass filter Kalman filter 95 CI119909119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904

2

05 4873 5354 333 05 2664 296 13 4460 3690 1602

[minus196 196]1 2019 1876 161 11 1519 164 13 2000 1844 72015 111 106 34 minus03 126 53 06 116 127 1592 minus203 minus197 minus40 minus18 minus102 minus13 minus05 minus181 minus149 72

3

05 6509 7208 386 09 3612 346 19 6125 5297 2567

[minus196 196]1 1826 1714 154 14 1396 153 18 1875 1777 77815 15 20 29 02 33 42 09 38 64 1492 minus176 minus169 minus34 minus11 minus92 minus11 00 minus129 minus100 66

4

05 8565 9576 404 11 4776 379 23 8071 7179 3801

[minus196 196]1 1742 1643 143 16 1343 144 21 1763 1670 74315 07 04 03 06 09 37 13 14 25 1112 minus153 minus148 minus30 minus06 minus86 minus11 04 minus105 minus81 60

5

05 11739 13264 423 14 6567 407 26 11018 10008 5577

[minus196 196]1 1684 1615 137 16 1345 137 21 1736 1628 74315 minus08 minus11 02 07 minus08 33 13 03 14 862 minus136 minus135 minus29 minus04 minus82 minus11 05 minus92 minus72 51

Advances in Meteorology 7

Table 3 Statistical characteristics of radar rainfall after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error0994 0156 0989 0231

01234567

0 500 1000 1500 2000 2500 3000

Rain

fall

(mm

)

Time (min)

(a)

01234567

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

(b)

Figure 5 The time series (a) of radar rainfall and its attractor (b)

original series is showing its nonlinearity and the series withnoise level of 100 which has bold type in Table 2 representsits randomness The noise removed series by LF which is inbold type in Table 2 is also showing its randomnessThe boldtype describes that the null hypothesis can be accepted andthe null hypotheses of the series in all other columns exceptfor two columns of the series with noise and LF with 119909

119894+10119904

cannot be accepted If we see the values of the BDS statisticin Table 2 KF has more similar BDS statistic values with thevalues of original series than LFTherefore it can be identifiedthat KF is more proper tool for noise cancellation than LF

32 Noise Influence on Radar Rainfall Series

321 Radar Rainfall Series and Its Attractor Radar rainfall isa representative hydrologic data which includes noise frommany sources This study uses the radar rainfall obtainedfrom the radar in Biseul Mountain radar (BSL radar) inGyeongbuk province Korea The radar rainfall series inGamcheon watershed especially which is produced in BSLradar is used for analyzing the series characteristics accordingto noise cancellation by LF andKF BSL radarwas constructedin 2009 and it is dual polarization radar The radar hastemporal and spatial resolutions of 25min and 250m times

250m Therefore BSL radar rainfall series of 25min-timeinterval is obtained with the data period of 6242011 0900ndash6262011 1100 (about 3000min average 173mm standarddeviation 153mm)

The ACF of radar rainfall series was exponentiallydecreased and so the delay timewas selected as 120591

119889= 825 min

(lag 119896 = 33) at which the ACF drops 1119890 (Tsonis and Elsner[47]) The time series plot and the reconstructed attractor ofradar rainfall are shown in Figure 5 Even though the ACFshowed the persistence of radar rainfall series the attractor iscomplicated and we can know that the radar rainfall is greatlyinfluenced by noise

0

2

4

6

8

10

0 500 1000 1500 2000

Raw data seriesKalman filterLow-pass filter

Rain

fall

(mm

)

Time (min)

3456

1500 1550 1600

Figure 6 The raw data and noise removed series of radar rainfall

322 Noise Reduction Studies of Radar Rainfall Series Thissection applies LF and KF for the noise reduction study ofradar rainfall series and the constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The raw data series of radar rainfall and theresults of noise reduction studies using LF and KF are shownin Figure 6 The magnified red box in Figure 6 is for heavyrainfall period Table 3 is showing the statistical results fornoise reduction analysis by LF and KF for the radar rainfallseries LF has the coefficient of correlation 0994 and standarderror 0156 and KF has 0989 and 0231 In this case thereis not much difference between LF and KF Therefore bothfiltering techniques show the similar function for removingnoise involved in radar rainfall

The attractors for noise removed radar rainfall seriesby LF and KF are reconstructed in phase space (Figure 7)The noise removed radar rainfall series by LF shows thatits attractor is more simplified shape (Figure 7(a)) than the

8 Advances in Meteorology

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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International Journal of

Geophysics

OceanographyInternational Journal of

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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

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MeteorologyAdvances in

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Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 3

Under the IID hypothesis the BDS statistic for 119898 gt 1 isdefined as

BDS (119898119872 119903) =radic119872

120590[119862 (119898 119903) minus 119862

119898(1 119903)] (6)

which has a limiting standard normal distribution under thenull hypothesis of IID as 119872 rarr infin and obtains its criticalvalues using the standard normal distribution

Before applying the BDS statistic the first addressedissue is which region of ldquo119903rdquo yields BDS statistics that arewell approximated by the asymptotic distribution As thesample size is increased the distribution of the BDS statisticbecomes more normal So the minimal number of data mustbe provided Next the region of embedding dimension ldquo119898rdquoshould be suggested If the sample size is fixed we expect thefinite sample property to worsen as ldquo119898rdquo increases This studyfollows the recommendation of Brock et al [29] for selectingthe ranges of 119898 119903 and 119873 The embedding dimension 119898 isused in the range of 2 le 119898 le 5 Then the value of ldquo119903rdquo isselected as the half standard deviations of the data sets

23 Kalman Filter and Low-Pass Filter The Kalman filter(KF) was introduced by Kalmanrsquos famous paper describing arecursive solution to the discrete data linear filtering problem[42] The Kalman filter algorithm can be applied as anestimator of the state of a dynamic system described by thelinear difference equation

119909119896= 119860119896minus1119909119896minus1+ 119861119896minus1119906119896minus1+ 119908119896minus1 (7)

where matrix119860119896minus1

is the relationship between the state of thesystem described by the column vector 119909

119896minus1at time 119896 minus 1

and the state 119909119896at time 119896 The column form of a vector is

considered to be the normal form and a transpose into a rowvector will be denoted by 119879 this is also the notation used forthe transpose of matrices Matrix 119861

119896minus1relates the optional

control input vector 119906119896minus1

to the state and the vector 119908119896minus1

isthe process noise

The system is then measured at discrete points in timewhere the measurement vector 119911

119896is related to the true state

of the system by the equation

119911119896= 119867119896119909119896+ V119896 (8)

where vector V119896represents the measurement noise and119867

119896is

the observation matrixThey are assumed to be independent (of each other)

white and with normal probability distributions

119901 (119908) sim 119873 (0 119876) (9)

119901 (V) sim 119873 (0 119877) (10)

In practice the process noise covariance 119876 and measure-ment noise covariance 119877 matrices might change with eachtime step or measurement however here we assume they areconstant

The error covariance matrix 119875119896|119896

of the estimate 119909119896|119896

is

119875119896|119896= cov (119909

119896minus 119909119896|119896 119909119896minus 119909119896|119896) (11)

00

05

10

15

20

0 10 20 30 40 50 60 70 80 90 100N

Valu

e

minus10

minus05

xi + 10s

xi + 05s

xi + 01s

xi

Figure 1 Logistic series with the noise levels of 10 50 and 100(119873 = 1sim100)

The estimate vector 119909119896|119896

is referred to as the a posterioriestimate since it is an estimate of the system at time 119896

A posteriori state estimate 119909119896|119896

is computed as a linearcombination of a priori estimate 119909

119896|119896minus1and a weighted differ-

ence between an actual measurement 119911119896and a measurement

prediction119867119896119909119896|119896minus1

as follows

119909119896|119896= 119909119896|119896minus1

+ 119870119896(119911119896minus 119867119896119909119896|119896minus1) (12)

119870119896= 119875119896|119896minus1119867119879

119896(119867119896119875119896|119896minus1119867119879

119896+ 119877119896)minus1

(13)

The matrix 119870119896in (13) is chosen to be the gain that

minimizes the a posteriori error covariance equationKim [43] shows how to track a varying signal and at

the same time reduce the influence of measurement noiseby using a 1st order low-pass filter (LF) (an exponentiallyweighted moving average filter) described as

119909119896|119896= 120572119909119896|119896minus1

+ (1 minus 120572) 119911119896

= 120572 120572119909119896|119896minus2

+ (1 minus 120572) 119911119896minus1 + (1 minus 120572) 119911119896

= 1205722119909119896|119896minus2

+ 120572 (1 minus 120572) 119911119896minus1 + (1 minus 120572) 119911119896

(14)

where 120572 is a constant in the range of 0 lt 120572 lt 1The expression for the computation of the a posteriori

estimate 119909119896|119896

in (12) is very similar to the 1st order low-pass filter with the significant difference lying in the varyingKalman filter gain instead of the constant 120572

3 Noise Reduction Studies on Logistic andRadar Rainfall Series

31 Noise Influence on Logistic Series

311 Attractor Characteristics in Noise Level To study theeffects of noise in a time series we added Gaussian noiseto the time series Specifically it considered the noise added

4 Advances in Meteorology

00

02

04

06

08

10

12

0 02 04 06 08 1x(i)

x(i+1)

(a) Noise level = 0

00

02

04

06

08

10

12

14

0 02 04 06 08 1 12x(i)

x(i+1)

minus02

minus02

(b) Noise level = 10

00

05

10

15

20

0 04 08 12 16 2x(i)

x(i+1)

minus05

minus08 minus04

minus10

minus12

(c) Noise level = 50

00

10

20

30

0 1 2 3x(i)

x(i+1)

minus10

minus20

minus2 minus1

(d) Noise level = 100

Figure 2 The attractors with the noise level

time series 119909120576119894 to test the noise effect in the original time

series 119909119894 and define the 119909

120576119894 as

119909120576119894= 119909119894+ 120578120590120576119894 119894 = 1 2 119873 (15)

where 120578 is the noise level 120590 is a standard deviation of 119909119894 and

Gaussian noise 120576119894has119873(0 1)

May [44] emphasized that a simple nonlinear map mayhave very complicated dynamics and showed his point withLogistic map which is a discrete time analog for populationgrowth Logistic map is defined as

119909119905+1= 119903119909119905(1 minus 119909

119905) (16)

where 119903 is between 0 and 4 For small values of 119903 the system isstable andwell behaved however as the value of 119903 approaches4 it becomes chaotic We simulate Logistic map sequence of119873 = 1000 and add noise to it with the noise levels 10 50and 100 Logistic series with noise are shown in Figure 1Here 119904 is standard deviation of the sample series in Figure 1

The attractor of each Logistic series is reconstructed inphase space and the characteristics of the series can be

identified (Figure 2) For the reconstruction of the seriesusing (1) the embedding dimension 119898 = 2 and delay time120591119889= 1 are used (Figure 2) The autocorrelation function

(ACF) is expected to provide a reasonable measure of thetransition from redundance to irrelevance as a function ofdelay It is considered that the decorrelation time equals thelag (delay time 120591

119889) at which the ACF first attains the value

zero Otherwise 120591119889should be chosen as the local minimum

ofACF whichever occurs first [45 46]When theACF decaysexponentially we select 120591

119889at which the ACF drops 1119890 [47]

The original Logistic series which has one variable showsits attractor with a simple quadratic form (Figure 2(a))However as the noise level is increased the attractor isbecoming more andmore complicated form which it is high-dimensional series (Figures 2(b)ndash2(d)) For the noise level =100 especially the attractor looks like random series

312 Noise Reduction Studies of Logistic Series This sectionstudies the noise reduction of the noise added Logistic seriesusing LF and KF Noise cannot be forecasted but statisticallyestimated and the parameters of LF and KF are calibrated by

Advances in Meteorology 5

00

05

10

15

0 20 40 60 80 100t

Valu

e

(a) Noise level = 10

t

0002040608101214

0 20 40 60 80 100

Valu

e

minus02

minus04

(b) Noise level = 50

t

00030609121518

0 20 40 60 80 100

Valu

e

Raw data seriesKalman filterLow-pass filter

minus03

minus06

(c) Noise level = 100

Figure 3 Noise removed data series through Low-pass filter and Kalman filter

Table 1 Statistical characteristics of Logistic series after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error

119909119894+ 01119904 0699 0251 0994 0037

119909119894+ 05119904 0629 0273 0986 0058

119909119894+ 10119904 0501 0303 0925 0134

trial and error method The constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The results of noise reduction studies usingLF and KF are shown in Figure 3When noise level is smallernoise can be removedmore effectively by LF andKF Table 1 isshowing the statistical results for noise reduction analysis byLF and KF LF has the coefficient of correlation 050ndash070 andstandard error 030ndash025 and KF has 093ndash099 and 004ndash01Therefore KF can reduce noise more effectively than LF

The attractors for noise removed Logistic series by LFand KF are reconstructed in phase space (Figure 4) Thenoise removed series by LF show their attractors which stillhave noisy shapes (Figures 4(a)ndash4(c)) but the noise removedseries by KF show more clear attractors which describe thecharacteristics of Logistic map (Figures 4(d)ndash4(f)) Eventhough KF is more effective way for removing noise in theseries it is difficult to restore it to the original state If weinvestigate the range of the values of the series generated from

Logistic equation we can find that the values of the originalseries are in the range of 0 to 1 The values of noise removedseries by LF and KF for the series having noise level = 100are investigated and the values by LF and KF are in the rangeof minus1 to 2 (Figure 4(c)) and 0 to 1 (Figure 4(f)) respectivelyTherefore the result is showing that KF is more proper toolfor the noise reduction of the series

The BDS statistic was applied for testing for nonlinearityof each data series Not only is it useful in detecting determin-istic chaos but it also serves as a residual diagnostic If themodel (null hypothesis) is correct then the estimated resid-uals will pass the test for IID (independently and identicallydistributed) A failure to pass the test is an indication that theselected model is misspecified Here the confidence interval(CL) of 95 which is a significance level of 5 is usedfor the randomness test of a time series The original seriesnoise added series and noise removed series of Logisticmap are analyzed by the BDS statistic for their randomnessand nonlinearity And the results are shown in Table 2 The

6 Advances in Meteorology

00

02

04

06

08

10

00 02 04 06 08 10 12minus02minus02

x(i)

x(i+1)

(a) LF noise level = 10x(i+1)

00 05 10 15x(i)

minus05

00

05

10

15

minus05

(b) LF noise level = 50

x(i+1)

00

05

10

15

20

00 10 20x(i)

minus05

minus10minus10

(c) LF noise level = 100

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(d) KF noise level = 10

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(e) KF noise level = 50

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(f) KF noise level = 100

Figure 4 The noise removed attractors by LF and KF

Table 2 The BDS statistic values for data series in each case

119898 119903 119909119894

The series with noise Low-pass filter Kalman filter 95 CI119909119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904

2

05 4873 5354 333 05 2664 296 13 4460 3690 1602

[minus196 196]1 2019 1876 161 11 1519 164 13 2000 1844 72015 111 106 34 minus03 126 53 06 116 127 1592 minus203 minus197 minus40 minus18 minus102 minus13 minus05 minus181 minus149 72

3

05 6509 7208 386 09 3612 346 19 6125 5297 2567

[minus196 196]1 1826 1714 154 14 1396 153 18 1875 1777 77815 15 20 29 02 33 42 09 38 64 1492 minus176 minus169 minus34 minus11 minus92 minus11 00 minus129 minus100 66

4

05 8565 9576 404 11 4776 379 23 8071 7179 3801

[minus196 196]1 1742 1643 143 16 1343 144 21 1763 1670 74315 07 04 03 06 09 37 13 14 25 1112 minus153 minus148 minus30 minus06 minus86 minus11 04 minus105 minus81 60

5

05 11739 13264 423 14 6567 407 26 11018 10008 5577

[minus196 196]1 1684 1615 137 16 1345 137 21 1736 1628 74315 minus08 minus11 02 07 minus08 33 13 03 14 862 minus136 minus135 minus29 minus04 minus82 minus11 05 minus92 minus72 51

Advances in Meteorology 7

Table 3 Statistical characteristics of radar rainfall after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error0994 0156 0989 0231

01234567

0 500 1000 1500 2000 2500 3000

Rain

fall

(mm

)

Time (min)

(a)

01234567

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

(b)

Figure 5 The time series (a) of radar rainfall and its attractor (b)

original series is showing its nonlinearity and the series withnoise level of 100 which has bold type in Table 2 representsits randomness The noise removed series by LF which is inbold type in Table 2 is also showing its randomnessThe boldtype describes that the null hypothesis can be accepted andthe null hypotheses of the series in all other columns exceptfor two columns of the series with noise and LF with 119909

119894+10119904

cannot be accepted If we see the values of the BDS statisticin Table 2 KF has more similar BDS statistic values with thevalues of original series than LFTherefore it can be identifiedthat KF is more proper tool for noise cancellation than LF

32 Noise Influence on Radar Rainfall Series

321 Radar Rainfall Series and Its Attractor Radar rainfall isa representative hydrologic data which includes noise frommany sources This study uses the radar rainfall obtainedfrom the radar in Biseul Mountain radar (BSL radar) inGyeongbuk province Korea The radar rainfall series inGamcheon watershed especially which is produced in BSLradar is used for analyzing the series characteristics accordingto noise cancellation by LF andKF BSL radarwas constructedin 2009 and it is dual polarization radar The radar hastemporal and spatial resolutions of 25min and 250m times

250m Therefore BSL radar rainfall series of 25min-timeinterval is obtained with the data period of 6242011 0900ndash6262011 1100 (about 3000min average 173mm standarddeviation 153mm)

The ACF of radar rainfall series was exponentiallydecreased and so the delay timewas selected as 120591

119889= 825 min

(lag 119896 = 33) at which the ACF drops 1119890 (Tsonis and Elsner[47]) The time series plot and the reconstructed attractor ofradar rainfall are shown in Figure 5 Even though the ACFshowed the persistence of radar rainfall series the attractor iscomplicated and we can know that the radar rainfall is greatlyinfluenced by noise

0

2

4

6

8

10

0 500 1000 1500 2000

Raw data seriesKalman filterLow-pass filter

Rain

fall

(mm

)

Time (min)

3456

1500 1550 1600

Figure 6 The raw data and noise removed series of radar rainfall

322 Noise Reduction Studies of Radar Rainfall Series Thissection applies LF and KF for the noise reduction study ofradar rainfall series and the constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The raw data series of radar rainfall and theresults of noise reduction studies using LF and KF are shownin Figure 6 The magnified red box in Figure 6 is for heavyrainfall period Table 3 is showing the statistical results fornoise reduction analysis by LF and KF for the radar rainfallseries LF has the coefficient of correlation 0994 and standarderror 0156 and KF has 0989 and 0231 In this case thereis not much difference between LF and KF Therefore bothfiltering techniques show the similar function for removingnoise involved in radar rainfall

The attractors for noise removed radar rainfall seriesby LF and KF are reconstructed in phase space (Figure 7)The noise removed radar rainfall series by LF shows thatits attractor is more simplified shape (Figure 7(a)) than the

8 Advances in Meteorology

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

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Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

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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

4 Advances in Meteorology

00

02

04

06

08

10

12

0 02 04 06 08 1x(i)

x(i+1)

(a) Noise level = 0

00

02

04

06

08

10

12

14

0 02 04 06 08 1 12x(i)

x(i+1)

minus02

minus02

(b) Noise level = 10

00

05

10

15

20

0 04 08 12 16 2x(i)

x(i+1)

minus05

minus08 minus04

minus10

minus12

(c) Noise level = 50

00

10

20

30

0 1 2 3x(i)

x(i+1)

minus10

minus20

minus2 minus1

(d) Noise level = 100

Figure 2 The attractors with the noise level

time series 119909120576119894 to test the noise effect in the original time

series 119909119894 and define the 119909

120576119894 as

119909120576119894= 119909119894+ 120578120590120576119894 119894 = 1 2 119873 (15)

where 120578 is the noise level 120590 is a standard deviation of 119909119894 and

Gaussian noise 120576119894has119873(0 1)

May [44] emphasized that a simple nonlinear map mayhave very complicated dynamics and showed his point withLogistic map which is a discrete time analog for populationgrowth Logistic map is defined as

119909119905+1= 119903119909119905(1 minus 119909

119905) (16)

where 119903 is between 0 and 4 For small values of 119903 the system isstable andwell behaved however as the value of 119903 approaches4 it becomes chaotic We simulate Logistic map sequence of119873 = 1000 and add noise to it with the noise levels 10 50and 100 Logistic series with noise are shown in Figure 1Here 119904 is standard deviation of the sample series in Figure 1

The attractor of each Logistic series is reconstructed inphase space and the characteristics of the series can be

identified (Figure 2) For the reconstruction of the seriesusing (1) the embedding dimension 119898 = 2 and delay time120591119889= 1 are used (Figure 2) The autocorrelation function

(ACF) is expected to provide a reasonable measure of thetransition from redundance to irrelevance as a function ofdelay It is considered that the decorrelation time equals thelag (delay time 120591

119889) at which the ACF first attains the value

zero Otherwise 120591119889should be chosen as the local minimum

ofACF whichever occurs first [45 46]When theACF decaysexponentially we select 120591

119889at which the ACF drops 1119890 [47]

The original Logistic series which has one variable showsits attractor with a simple quadratic form (Figure 2(a))However as the noise level is increased the attractor isbecoming more andmore complicated form which it is high-dimensional series (Figures 2(b)ndash2(d)) For the noise level =100 especially the attractor looks like random series

312 Noise Reduction Studies of Logistic Series This sectionstudies the noise reduction of the noise added Logistic seriesusing LF and KF Noise cannot be forecasted but statisticallyestimated and the parameters of LF and KF are calibrated by

Advances in Meteorology 5

00

05

10

15

0 20 40 60 80 100t

Valu

e

(a) Noise level = 10

t

0002040608101214

0 20 40 60 80 100

Valu

e

minus02

minus04

(b) Noise level = 50

t

00030609121518

0 20 40 60 80 100

Valu

e

Raw data seriesKalman filterLow-pass filter

minus03

minus06

(c) Noise level = 100

Figure 3 Noise removed data series through Low-pass filter and Kalman filter

Table 1 Statistical characteristics of Logistic series after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error

119909119894+ 01119904 0699 0251 0994 0037

119909119894+ 05119904 0629 0273 0986 0058

119909119894+ 10119904 0501 0303 0925 0134

trial and error method The constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The results of noise reduction studies usingLF and KF are shown in Figure 3When noise level is smallernoise can be removedmore effectively by LF andKF Table 1 isshowing the statistical results for noise reduction analysis byLF and KF LF has the coefficient of correlation 050ndash070 andstandard error 030ndash025 and KF has 093ndash099 and 004ndash01Therefore KF can reduce noise more effectively than LF

The attractors for noise removed Logistic series by LFand KF are reconstructed in phase space (Figure 4) Thenoise removed series by LF show their attractors which stillhave noisy shapes (Figures 4(a)ndash4(c)) but the noise removedseries by KF show more clear attractors which describe thecharacteristics of Logistic map (Figures 4(d)ndash4(f)) Eventhough KF is more effective way for removing noise in theseries it is difficult to restore it to the original state If weinvestigate the range of the values of the series generated from

Logistic equation we can find that the values of the originalseries are in the range of 0 to 1 The values of noise removedseries by LF and KF for the series having noise level = 100are investigated and the values by LF and KF are in the rangeof minus1 to 2 (Figure 4(c)) and 0 to 1 (Figure 4(f)) respectivelyTherefore the result is showing that KF is more proper toolfor the noise reduction of the series

The BDS statistic was applied for testing for nonlinearityof each data series Not only is it useful in detecting determin-istic chaos but it also serves as a residual diagnostic If themodel (null hypothesis) is correct then the estimated resid-uals will pass the test for IID (independently and identicallydistributed) A failure to pass the test is an indication that theselected model is misspecified Here the confidence interval(CL) of 95 which is a significance level of 5 is usedfor the randomness test of a time series The original seriesnoise added series and noise removed series of Logisticmap are analyzed by the BDS statistic for their randomnessand nonlinearity And the results are shown in Table 2 The

6 Advances in Meteorology

00

02

04

06

08

10

00 02 04 06 08 10 12minus02minus02

x(i)

x(i+1)

(a) LF noise level = 10x(i+1)

00 05 10 15x(i)

minus05

00

05

10

15

minus05

(b) LF noise level = 50

x(i+1)

00

05

10

15

20

00 10 20x(i)

minus05

minus10minus10

(c) LF noise level = 100

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(d) KF noise level = 10

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(e) KF noise level = 50

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(f) KF noise level = 100

Figure 4 The noise removed attractors by LF and KF

Table 2 The BDS statistic values for data series in each case

119898 119903 119909119894

The series with noise Low-pass filter Kalman filter 95 CI119909119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904

2

05 4873 5354 333 05 2664 296 13 4460 3690 1602

[minus196 196]1 2019 1876 161 11 1519 164 13 2000 1844 72015 111 106 34 minus03 126 53 06 116 127 1592 minus203 minus197 minus40 minus18 minus102 minus13 minus05 minus181 minus149 72

3

05 6509 7208 386 09 3612 346 19 6125 5297 2567

[minus196 196]1 1826 1714 154 14 1396 153 18 1875 1777 77815 15 20 29 02 33 42 09 38 64 1492 minus176 minus169 minus34 minus11 minus92 minus11 00 minus129 minus100 66

4

05 8565 9576 404 11 4776 379 23 8071 7179 3801

[minus196 196]1 1742 1643 143 16 1343 144 21 1763 1670 74315 07 04 03 06 09 37 13 14 25 1112 minus153 minus148 minus30 minus06 minus86 minus11 04 minus105 minus81 60

5

05 11739 13264 423 14 6567 407 26 11018 10008 5577

[minus196 196]1 1684 1615 137 16 1345 137 21 1736 1628 74315 minus08 minus11 02 07 minus08 33 13 03 14 862 minus136 minus135 minus29 minus04 minus82 minus11 05 minus92 minus72 51

Advances in Meteorology 7

Table 3 Statistical characteristics of radar rainfall after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error0994 0156 0989 0231

01234567

0 500 1000 1500 2000 2500 3000

Rain

fall

(mm

)

Time (min)

(a)

01234567

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

(b)

Figure 5 The time series (a) of radar rainfall and its attractor (b)

original series is showing its nonlinearity and the series withnoise level of 100 which has bold type in Table 2 representsits randomness The noise removed series by LF which is inbold type in Table 2 is also showing its randomnessThe boldtype describes that the null hypothesis can be accepted andthe null hypotheses of the series in all other columns exceptfor two columns of the series with noise and LF with 119909

119894+10119904

cannot be accepted If we see the values of the BDS statisticin Table 2 KF has more similar BDS statistic values with thevalues of original series than LFTherefore it can be identifiedthat KF is more proper tool for noise cancellation than LF

32 Noise Influence on Radar Rainfall Series

321 Radar Rainfall Series and Its Attractor Radar rainfall isa representative hydrologic data which includes noise frommany sources This study uses the radar rainfall obtainedfrom the radar in Biseul Mountain radar (BSL radar) inGyeongbuk province Korea The radar rainfall series inGamcheon watershed especially which is produced in BSLradar is used for analyzing the series characteristics accordingto noise cancellation by LF andKF BSL radarwas constructedin 2009 and it is dual polarization radar The radar hastemporal and spatial resolutions of 25min and 250m times

250m Therefore BSL radar rainfall series of 25min-timeinterval is obtained with the data period of 6242011 0900ndash6262011 1100 (about 3000min average 173mm standarddeviation 153mm)

The ACF of radar rainfall series was exponentiallydecreased and so the delay timewas selected as 120591

119889= 825 min

(lag 119896 = 33) at which the ACF drops 1119890 (Tsonis and Elsner[47]) The time series plot and the reconstructed attractor ofradar rainfall are shown in Figure 5 Even though the ACFshowed the persistence of radar rainfall series the attractor iscomplicated and we can know that the radar rainfall is greatlyinfluenced by noise

0

2

4

6

8

10

0 500 1000 1500 2000

Raw data seriesKalman filterLow-pass filter

Rain

fall

(mm

)

Time (min)

3456

1500 1550 1600

Figure 6 The raw data and noise removed series of radar rainfall

322 Noise Reduction Studies of Radar Rainfall Series Thissection applies LF and KF for the noise reduction study ofradar rainfall series and the constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The raw data series of radar rainfall and theresults of noise reduction studies using LF and KF are shownin Figure 6 The magnified red box in Figure 6 is for heavyrainfall period Table 3 is showing the statistical results fornoise reduction analysis by LF and KF for the radar rainfallseries LF has the coefficient of correlation 0994 and standarderror 0156 and KF has 0989 and 0231 In this case thereis not much difference between LF and KF Therefore bothfiltering techniques show the similar function for removingnoise involved in radar rainfall

The attractors for noise removed radar rainfall seriesby LF and KF are reconstructed in phase space (Figure 7)The noise removed radar rainfall series by LF shows thatits attractor is more simplified shape (Figure 7(a)) than the

8 Advances in Meteorology

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

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

00

05

10

15

0 20 40 60 80 100t

Valu

e

(a) Noise level = 10

t

0002040608101214

0 20 40 60 80 100

Valu

e

minus02

minus04

(b) Noise level = 50

t

00030609121518

0 20 40 60 80 100

Valu

e

Raw data seriesKalman filterLow-pass filter

minus03

minus06

(c) Noise level = 100

Figure 3 Noise removed data series through Low-pass filter and Kalman filter

Table 1 Statistical characteristics of Logistic series after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error

119909119894+ 01119904 0699 0251 0994 0037

119909119894+ 05119904 0629 0273 0986 0058

119909119894+ 10119904 0501 0303 0925 0134

trial and error method The constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The results of noise reduction studies usingLF and KF are shown in Figure 3When noise level is smallernoise can be removedmore effectively by LF andKF Table 1 isshowing the statistical results for noise reduction analysis byLF and KF LF has the coefficient of correlation 050ndash070 andstandard error 030ndash025 and KF has 093ndash099 and 004ndash01Therefore KF can reduce noise more effectively than LF

The attractors for noise removed Logistic series by LFand KF are reconstructed in phase space (Figure 4) Thenoise removed series by LF show their attractors which stillhave noisy shapes (Figures 4(a)ndash4(c)) but the noise removedseries by KF show more clear attractors which describe thecharacteristics of Logistic map (Figures 4(d)ndash4(f)) Eventhough KF is more effective way for removing noise in theseries it is difficult to restore it to the original state If weinvestigate the range of the values of the series generated from

Logistic equation we can find that the values of the originalseries are in the range of 0 to 1 The values of noise removedseries by LF and KF for the series having noise level = 100are investigated and the values by LF and KF are in the rangeof minus1 to 2 (Figure 4(c)) and 0 to 1 (Figure 4(f)) respectivelyTherefore the result is showing that KF is more proper toolfor the noise reduction of the series

The BDS statistic was applied for testing for nonlinearityof each data series Not only is it useful in detecting determin-istic chaos but it also serves as a residual diagnostic If themodel (null hypothesis) is correct then the estimated resid-uals will pass the test for IID (independently and identicallydistributed) A failure to pass the test is an indication that theselected model is misspecified Here the confidence interval(CL) of 95 which is a significance level of 5 is usedfor the randomness test of a time series The original seriesnoise added series and noise removed series of Logisticmap are analyzed by the BDS statistic for their randomnessand nonlinearity And the results are shown in Table 2 The

6 Advances in Meteorology

00

02

04

06

08

10

00 02 04 06 08 10 12minus02minus02

x(i)

x(i+1)

(a) LF noise level = 10x(i+1)

00 05 10 15x(i)

minus05

00

05

10

15

minus05

(b) LF noise level = 50

x(i+1)

00

05

10

15

20

00 10 20x(i)

minus05

minus10minus10

(c) LF noise level = 100

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(d) KF noise level = 10

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(e) KF noise level = 50

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(f) KF noise level = 100

Figure 4 The noise removed attractors by LF and KF

Table 2 The BDS statistic values for data series in each case

119898 119903 119909119894

The series with noise Low-pass filter Kalman filter 95 CI119909119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904

2

05 4873 5354 333 05 2664 296 13 4460 3690 1602

[minus196 196]1 2019 1876 161 11 1519 164 13 2000 1844 72015 111 106 34 minus03 126 53 06 116 127 1592 minus203 minus197 minus40 minus18 minus102 minus13 minus05 minus181 minus149 72

3

05 6509 7208 386 09 3612 346 19 6125 5297 2567

[minus196 196]1 1826 1714 154 14 1396 153 18 1875 1777 77815 15 20 29 02 33 42 09 38 64 1492 minus176 minus169 minus34 minus11 minus92 minus11 00 minus129 minus100 66

4

05 8565 9576 404 11 4776 379 23 8071 7179 3801

[minus196 196]1 1742 1643 143 16 1343 144 21 1763 1670 74315 07 04 03 06 09 37 13 14 25 1112 minus153 minus148 minus30 minus06 minus86 minus11 04 minus105 minus81 60

5

05 11739 13264 423 14 6567 407 26 11018 10008 5577

[minus196 196]1 1684 1615 137 16 1345 137 21 1736 1628 74315 minus08 minus11 02 07 minus08 33 13 03 14 862 minus136 minus135 minus29 minus04 minus82 minus11 05 minus92 minus72 51

Advances in Meteorology 7

Table 3 Statistical characteristics of radar rainfall after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error0994 0156 0989 0231

01234567

0 500 1000 1500 2000 2500 3000

Rain

fall

(mm

)

Time (min)

(a)

01234567

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

(b)

Figure 5 The time series (a) of radar rainfall and its attractor (b)

original series is showing its nonlinearity and the series withnoise level of 100 which has bold type in Table 2 representsits randomness The noise removed series by LF which is inbold type in Table 2 is also showing its randomnessThe boldtype describes that the null hypothesis can be accepted andthe null hypotheses of the series in all other columns exceptfor two columns of the series with noise and LF with 119909

119894+10119904

cannot be accepted If we see the values of the BDS statisticin Table 2 KF has more similar BDS statistic values with thevalues of original series than LFTherefore it can be identifiedthat KF is more proper tool for noise cancellation than LF

32 Noise Influence on Radar Rainfall Series

321 Radar Rainfall Series and Its Attractor Radar rainfall isa representative hydrologic data which includes noise frommany sources This study uses the radar rainfall obtainedfrom the radar in Biseul Mountain radar (BSL radar) inGyeongbuk province Korea The radar rainfall series inGamcheon watershed especially which is produced in BSLradar is used for analyzing the series characteristics accordingto noise cancellation by LF andKF BSL radarwas constructedin 2009 and it is dual polarization radar The radar hastemporal and spatial resolutions of 25min and 250m times

250m Therefore BSL radar rainfall series of 25min-timeinterval is obtained with the data period of 6242011 0900ndash6262011 1100 (about 3000min average 173mm standarddeviation 153mm)

The ACF of radar rainfall series was exponentiallydecreased and so the delay timewas selected as 120591

119889= 825 min

(lag 119896 = 33) at which the ACF drops 1119890 (Tsonis and Elsner[47]) The time series plot and the reconstructed attractor ofradar rainfall are shown in Figure 5 Even though the ACFshowed the persistence of radar rainfall series the attractor iscomplicated and we can know that the radar rainfall is greatlyinfluenced by noise

0

2

4

6

8

10

0 500 1000 1500 2000

Raw data seriesKalman filterLow-pass filter

Rain

fall

(mm

)

Time (min)

3456

1500 1550 1600

Figure 6 The raw data and noise removed series of radar rainfall

322 Noise Reduction Studies of Radar Rainfall Series Thissection applies LF and KF for the noise reduction study ofradar rainfall series and the constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The raw data series of radar rainfall and theresults of noise reduction studies using LF and KF are shownin Figure 6 The magnified red box in Figure 6 is for heavyrainfall period Table 3 is showing the statistical results fornoise reduction analysis by LF and KF for the radar rainfallseries LF has the coefficient of correlation 0994 and standarderror 0156 and KF has 0989 and 0231 In this case thereis not much difference between LF and KF Therefore bothfiltering techniques show the similar function for removingnoise involved in radar rainfall

The attractors for noise removed radar rainfall seriesby LF and KF are reconstructed in phase space (Figure 7)The noise removed radar rainfall series by LF shows thatits attractor is more simplified shape (Figure 7(a)) than the

8 Advances in Meteorology

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

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

00

02

04

06

08

10

00 02 04 06 08 10 12minus02minus02

x(i)

x(i+1)

(a) LF noise level = 10x(i+1)

00 05 10 15x(i)

minus05

00

05

10

15

minus05

(b) LF noise level = 50

x(i+1)

00

05

10

15

20

00 10 20x(i)

minus05

minus10minus10

(c) LF noise level = 100

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(d) KF noise level = 10

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(e) KF noise level = 50

00

02

04

06

08

10

00 02 04 06 08 10x(i)

x(i+1)

(f) KF noise level = 100

Figure 4 The noise removed attractors by LF and KF

Table 2 The BDS statistic values for data series in each case

119898 119903 119909119894

The series with noise Low-pass filter Kalman filter 95 CI119909119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904 119909

119894+ 01119904 119909

119894+ 05119904 119909

119894+ 10119904

2

05 4873 5354 333 05 2664 296 13 4460 3690 1602

[minus196 196]1 2019 1876 161 11 1519 164 13 2000 1844 72015 111 106 34 minus03 126 53 06 116 127 1592 minus203 minus197 minus40 minus18 minus102 minus13 minus05 minus181 minus149 72

3

05 6509 7208 386 09 3612 346 19 6125 5297 2567

[minus196 196]1 1826 1714 154 14 1396 153 18 1875 1777 77815 15 20 29 02 33 42 09 38 64 1492 minus176 minus169 minus34 minus11 minus92 minus11 00 minus129 minus100 66

4

05 8565 9576 404 11 4776 379 23 8071 7179 3801

[minus196 196]1 1742 1643 143 16 1343 144 21 1763 1670 74315 07 04 03 06 09 37 13 14 25 1112 minus153 minus148 minus30 minus06 minus86 minus11 04 minus105 minus81 60

5

05 11739 13264 423 14 6567 407 26 11018 10008 5577

[minus196 196]1 1684 1615 137 16 1345 137 21 1736 1628 74315 minus08 minus11 02 07 minus08 33 13 03 14 862 minus136 minus135 minus29 minus04 minus82 minus11 05 minus92 minus72 51

Advances in Meteorology 7

Table 3 Statistical characteristics of radar rainfall after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error0994 0156 0989 0231

01234567

0 500 1000 1500 2000 2500 3000

Rain

fall

(mm

)

Time (min)

(a)

01234567

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

(b)

Figure 5 The time series (a) of radar rainfall and its attractor (b)

original series is showing its nonlinearity and the series withnoise level of 100 which has bold type in Table 2 representsits randomness The noise removed series by LF which is inbold type in Table 2 is also showing its randomnessThe boldtype describes that the null hypothesis can be accepted andthe null hypotheses of the series in all other columns exceptfor two columns of the series with noise and LF with 119909

119894+10119904

cannot be accepted If we see the values of the BDS statisticin Table 2 KF has more similar BDS statistic values with thevalues of original series than LFTherefore it can be identifiedthat KF is more proper tool for noise cancellation than LF

32 Noise Influence on Radar Rainfall Series

321 Radar Rainfall Series and Its Attractor Radar rainfall isa representative hydrologic data which includes noise frommany sources This study uses the radar rainfall obtainedfrom the radar in Biseul Mountain radar (BSL radar) inGyeongbuk province Korea The radar rainfall series inGamcheon watershed especially which is produced in BSLradar is used for analyzing the series characteristics accordingto noise cancellation by LF andKF BSL radarwas constructedin 2009 and it is dual polarization radar The radar hastemporal and spatial resolutions of 25min and 250m times

250m Therefore BSL radar rainfall series of 25min-timeinterval is obtained with the data period of 6242011 0900ndash6262011 1100 (about 3000min average 173mm standarddeviation 153mm)

The ACF of radar rainfall series was exponentiallydecreased and so the delay timewas selected as 120591

119889= 825 min

(lag 119896 = 33) at which the ACF drops 1119890 (Tsonis and Elsner[47]) The time series plot and the reconstructed attractor ofradar rainfall are shown in Figure 5 Even though the ACFshowed the persistence of radar rainfall series the attractor iscomplicated and we can know that the radar rainfall is greatlyinfluenced by noise

0

2

4

6

8

10

0 500 1000 1500 2000

Raw data seriesKalman filterLow-pass filter

Rain

fall

(mm

)

Time (min)

3456

1500 1550 1600

Figure 6 The raw data and noise removed series of radar rainfall

322 Noise Reduction Studies of Radar Rainfall Series Thissection applies LF and KF for the noise reduction study ofradar rainfall series and the constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The raw data series of radar rainfall and theresults of noise reduction studies using LF and KF are shownin Figure 6 The magnified red box in Figure 6 is for heavyrainfall period Table 3 is showing the statistical results fornoise reduction analysis by LF and KF for the radar rainfallseries LF has the coefficient of correlation 0994 and standarderror 0156 and KF has 0989 and 0231 In this case thereis not much difference between LF and KF Therefore bothfiltering techniques show the similar function for removingnoise involved in radar rainfall

The attractors for noise removed radar rainfall seriesby LF and KF are reconstructed in phase space (Figure 7)The noise removed radar rainfall series by LF shows thatits attractor is more simplified shape (Figure 7(a)) than the

8 Advances in Meteorology

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

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

Table 3 Statistical characteristics of radar rainfall after applying the filtering techniques

Low-pass filter Kalman filterCoefficient of correlation Standard error Coefficient of correlation Standard error0994 0156 0989 0231

01234567

0 500 1000 1500 2000 2500 3000

Rain

fall

(mm

)

Time (min)

(a)

01234567

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

(b)

Figure 5 The time series (a) of radar rainfall and its attractor (b)

original series is showing its nonlinearity and the series withnoise level of 100 which has bold type in Table 2 representsits randomness The noise removed series by LF which is inbold type in Table 2 is also showing its randomnessThe boldtype describes that the null hypothesis can be accepted andthe null hypotheses of the series in all other columns exceptfor two columns of the series with noise and LF with 119909

119894+10119904

cannot be accepted If we see the values of the BDS statisticin Table 2 KF has more similar BDS statistic values with thevalues of original series than LFTherefore it can be identifiedthat KF is more proper tool for noise cancellation than LF

32 Noise Influence on Radar Rainfall Series

321 Radar Rainfall Series and Its Attractor Radar rainfall isa representative hydrologic data which includes noise frommany sources This study uses the radar rainfall obtainedfrom the radar in Biseul Mountain radar (BSL radar) inGyeongbuk province Korea The radar rainfall series inGamcheon watershed especially which is produced in BSLradar is used for analyzing the series characteristics accordingto noise cancellation by LF andKF BSL radarwas constructedin 2009 and it is dual polarization radar The radar hastemporal and spatial resolutions of 25min and 250m times

250m Therefore BSL radar rainfall series of 25min-timeinterval is obtained with the data period of 6242011 0900ndash6262011 1100 (about 3000min average 173mm standarddeviation 153mm)

The ACF of radar rainfall series was exponentiallydecreased and so the delay timewas selected as 120591

119889= 825 min

(lag 119896 = 33) at which the ACF drops 1119890 (Tsonis and Elsner[47]) The time series plot and the reconstructed attractor ofradar rainfall are shown in Figure 5 Even though the ACFshowed the persistence of radar rainfall series the attractor iscomplicated and we can know that the radar rainfall is greatlyinfluenced by noise

0

2

4

6

8

10

0 500 1000 1500 2000

Raw data seriesKalman filterLow-pass filter

Rain

fall

(mm

)

Time (min)

3456

1500 1550 1600

Figure 6 The raw data and noise removed series of radar rainfall

322 Noise Reduction Studies of Radar Rainfall Series Thissection applies LF and KF for the noise reduction study ofradar rainfall series and the constant 120572 = 05 in (14) for LFis used the process noise covariance 119876 = 10 in (9) for KFis applied and the measurement noise covariance 119877 = 100in (10) is used The raw data series of radar rainfall and theresults of noise reduction studies using LF and KF are shownin Figure 6 The magnified red box in Figure 6 is for heavyrainfall period Table 3 is showing the statistical results fornoise reduction analysis by LF and KF for the radar rainfallseries LF has the coefficient of correlation 0994 and standarderror 0156 and KF has 0989 and 0231 In this case thereis not much difference between LF and KF Therefore bothfiltering techniques show the similar function for removingnoise involved in radar rainfall

The attractors for noise removed radar rainfall seriesby LF and KF are reconstructed in phase space (Figure 7)The noise removed radar rainfall series by LF shows thatits attractor is more simplified shape (Figure 7(a)) than the

8 Advances in Meteorology

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

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

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(a) Low-pass filter

0 1 2 3 4 5 6 7

x(t+120591)

x(t)

0

1

2

3

4

5

6

7

(b) Kalman filter

Figure 7 The attractors of noise removed radar rainfall series by LF and KF

Table 4 The BDS statistic values for radar rainfall and the noise removed series

119898 119903 Radar rainfall Low-pass filter Kalman filter 95 CI

2

05 1551 1681 1853

[minus196 196]1 2543 2773 314015 4459 4925 57242 8398 9434 11251

3

05 1390 1495 1670

[minus196 196]1 1739 1863 210515 2225 2388 27352 2952 3186 3703

4

05 968 1053 1161

[minus196 196]1 1056 1129 125215 1155 1225 13642 1286 1359 1525

5

05 773 829 896

[minus196 196]1 773 810 87515 768 797 8582 771 797 857

original attractor (Figure 5(b)) Also the noise removed radarrainfall series by KF shows that its attractor (Figure 7(b))is more clear shape than the attractor by LF (Figure 7(a))Therefore the attractor in which the noise of radar rainfallseries is removed by LF andKF can bemore clearly identifiedIn this case the attractor by KF is clearer than by LF

The original radar rainfall series and noise removed seriesby LF and KF are analyzed by the BDS statistic for theirrandomness and nonlinearity And the results are shownin Table 4 The original radar rainfall series is showing itsnonlinearity in the radar rainfall column of Table 4 and theseries after removing the noise by LF and KF are also showingtheir nonlinearities in columns of low-pass Filter andKalmanFilter of Table 4 If we see the BDS statistic values KF has the

largest values LF has next and the original radar rainfall hasthe smallest values This means that the noise removed radarrainfall series by KF is better than LF for noise reduction andfor describing the nonlinearity of the radar rainfall

4 Summary and Conclusions

This study investigated the filtering techniques for removingthe noise involved in Logistic series and radar rainfallThe chaotic dynamics and the BDS statistic were used foranalyzing the time series which are associated with noiseLogistic series with noise level were used for evaluatingthe filtering techniques of LF and KF The analysis for the

Advances in Meteorology 9

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

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

evaluation of LF and KF was performed by phase spacereconstruction and the BDS statistic from chaos theory Asthe noise level is increased the characteristics of Logisticseries were becoming random and this phenomenonwas alsooccurred in the attractors and the BDS statistic analysis Theapplications of LF and KF to the noise added Logistic seriesshowed that KF reduced noise more clearly involved in theLogistic series than LF

The noise in radar rainfall series was removed by LFand KF Then the attractor and the BDS statistic were usedfor evaluating the filtering techniques It was difficult todistinguish which filtering technique is better when thecorrelation coefficient and standard error were used forevaluating LF and KF However the attractor and the BDSstatistic gave us more clear answers for the determination ofthe proper filtering technique In this study we have shownthat KF is better technique than LF and chaos theory can beapplied for investigating the characteristics of the time series

Conflict of Interests

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

Acknowledgments

This study was conducted with financial support from theKorean Institute of Civil Engineering and Building Tech-nologyrsquos Strategic Research Project (Operation of Hydrolog-ical Radar and Development of Web and Mobile WarningPlatform) Also this work was supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2009-220-D00104)

References

[1] T W Harrold E J English and C A Nicholass ldquoThe accuracyof radar-derived rainfall measurements in hilly terrainrdquo Quar-terly Journal of the RoyalMeteorological Society vol 100 no 425pp 331ndash350 1974

[2] J W Wilson and E A Brandes ldquoRadar measurement ofrainfallmdasha summaryrdquo Bulletin AmericanMeteorological Societyvol 60 no 9 pp 1048ndash1058 1979

[3] M R Duncan B Austin F Fabry and G L Austin ldquoTheeffect of gauge sampling density on the accuracy of streamflowprediction for rural catchmentsrdquo Journal of Hydrology vol 142no 1-4 pp 445ndash476 1993

[4] F Fabry G L Austin and D Tees ldquoThe accuracy of rainfallestimates by radar as a function of rangerdquo Quarterly Journal ofthe Royal Meteorological Society vol 118 no 505 pp 435ndash4531992

[5] F Fabry A Bellon M R Duncan and G L Austin ldquoHighresolution rainfall measurements by radar for very small basinsthe sampling problem reexaminedrdquo Journal of Hydrology vol161 no 1ndash4 pp 415ndash428 1994

[6] M Kitchen ldquoTowards improved radar estimates of surfaceprecipitation rate at long rangerdquo Quarterly Journal of the RoyalMeteorological Society vol 123 no 537 pp 145ndash163 1997

[7] W F Krajewski and J A Smith ldquoRadar hydrology rainfallestimationrdquo Advances in Water Resources vol 25 no 8ndash12 pp1387ndash1394 2002

[8] M A Rico-Ramirez I D Cluckie G Shepherd and A PallotldquoA high-resolution radar experiment on the island of JerseyrdquoMeteorological Applications vol 14 no 2 pp 117ndash129 2007

[9] H A Panofsky and G W Brier Some Applications of Statisticsto Meteorology Pennsylvania State University University ParkPa USA 1958

[10] V Pastoriza A Nunez F MacHado PMarino F P Fontan andU C Fiebig ldquoCombining meteorological radar and network ofrain gauges data for space-time model developmentrdquo Interna-tional Journal of Satellite Communications and Networking vol29 no 1 pp 61ndash78 2011

[11] G Pegram X Llort and D Sempere-Torres ldquoRadar rainfallseparating signal and noise fields to generate meaningfulensemblesrdquoAtmospheric Research vol 100 no 2-3 pp 226ndash2362011

[12] F Fenicia L Pfister D Kavetski et al ldquoMicrowave links forrainfall estimation in an urban environment insights from anexperimental setup in Luxembourg-Cityrdquo Journal of Hydrologyvol 464ndash465 pp 69ndash78 2012

[13] P M Ahnert ldquoKalman filter estimation of radar-rainfall meanfield biasrdquo in Proceedings of the 23rd Radar Meteorology Confer-ence of American Meteorological Society pp JP33ndashJP37 1986

[14] J A Smith and W F Krajewski ldquoEstimation of the mean fieldbias of radar rainfall estimatesrdquo Journal of Applied Meteorologyvol 30 no 4 pp 397ndash412 1991

[15] E N Anagnostou and W F Krajewski ldquoCalibration of theWSR-88D precipitation processing subsystemrdquo Weather andForecasting vol 13 no 2 pp 396ndash406 1998

[16] D J Seo J P Breidenbach and E R Johnson ldquoReal-timeestimation of mean field bias in radar rainfall datardquo Journal ofHydrology vol 223 no 3-4 pp 131ndash147 1999

[17] T Dinku E N Ananostou and M Borga ldquoImproving radarbased estimation of rainfall over complex terrainrdquo AmericanMeteorological Society vol 41 no 12 pp 1163ndash1178 2002

[18] S Chumchean A Seed and A Sharma ldquoCorrecting of real-time radar rainfall bias using a Kalman filtering approachrdquoJournal of Hydrology vol 317 no 1-2 pp 123ndash137 2006

[19] W F Krajewski G Villarini and J A Smith ldquoRadar-rainfalluncertainties where are we after thirty years of effortrdquo Bulletinof the American Meteorological Society vol 91 no 1 pp 87ndash942010

[20] S Wang X Liang and Z Nan ldquoHow much improvementcan precipitation data fusion achieve with a Multiscale KalmanSmoother-based frameworkrdquoWater Resources Research vol 47no 8 Article IDW00H12 2011

[21] E De Lauro S De Martino M Falanga A Ciaramella and RTagliaferri ldquoComplexity of time series associated to dynamicalsystems inferred from independent component analysisrdquo Phys-ical Review E Statistical Nonlinear and SoftMatter Physics vol72 no 4 Article ID 046712 14 pages 2005

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

[23] 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

[24] F Takens ldquoDetecting strange attractors in turbulencerdquo inDynamical Systems and Turbulence D A Rand and L S YoungEds vol 898 of Lecture Notes in Mathematics pp 336ndash381Springer Warwick Berlin 1980

10 Advances in Meteorology

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

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

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

[26] 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

[27] B Sivakumar K Phoon S Liong and C Liaw ldquoA systematicapproach to noise reduction in chaotic hydrological time seriesrdquoJournal of Hydrology vol 219 no 3-4 pp 103ndash135 1999

[28] H Kantz and T Schreiber Nonlinear Time Series AnalysisCambridge University Press Cambridge UK 1997

[29] W A Brock D A Hsieh and B LeBaronNonlinear DynamicsChaos and Instability StatisticalTheory and Economic EvidenceMIT Press 1991

[30] W A Brock W D Dechert J A Scheinkman and B LeBaronldquoA test for independence based on the correlation dimensionrdquoEconometric Reviews vol 15 no 3 pp 197ndash235 1996

[31] 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

[32] 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

[33] 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

[34] H S Kim D S Kang and J H Kim ldquoThe BDS statisticand residual testrdquo Stochastic Environmental Research and RiskAssessment vol 17 no 1-2 pp 104ndash115 2003

[35] 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

[36] J H Ahn and H S Kim ldquoNonlinear modeling of ElninoSouthern osciilation indexrdquo Journal of Hydrologic Engineeringvol 10 no 1 pp 8ndash15 2005

[37] 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

[38] B Sivakumar F MWoldemeskel and C E Puente ldquoNonlinearanalysis of rainfall variability in Australiardquo Stochastic Environ-mental Research and Risk Assessment vol 28 pp 17ndash27 2013

[39] S Kim V P Singh Y Seo and H S Kim ldquoModeling nonlinearmonthly evapotranspiration using soft computing and datareconstruction techniquesrdquo Water Resources Management vol28 no 1 pp 185ndash206 2014

[40] M Casdagli ldquoNonlinear prediction of chaotic time seriesrdquoPhysica D vol 35 no 3 pp 335ndash356 1989

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

[42] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash461960

[43] P Kim Kalman Filters for Beginners with MATLAB ExamplesA-JIN Publishing Company 2010

[44] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[45] 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 of Springer Series in Synergetics pp 114ndash122Springer New York NY USA 1986

[46] K E Graf and T Elbert ldquoDimensional analysis of the wakingEEGrdquo in Chaos in Brain Function D Basar Ed Springer NewYork NY USA 1990

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

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