Research Article Signal Feature Extraction and

Research ArticleSignal Feature Extraction and QuantitativeEvaluation of Metal Magnetic Memory Testing for Oil WellCasing Based on Data Preprocessing Technique

Zhilin Liu1 Lutao Liu2 and Jun Zhang3

1 College of Automation Harbin Engineering University Harbin Heilongjiang 150001 China2 College of Information and Telecommunication Harbin Engineering University Harbin Heilongjiang 150001 China3 School of Electrical and Information Engineering Jiangsu University Zhenjiang Jiangsu 212013 China

Correspondence should be addressed to Lutao Liu liulutaomsncom

Received 17 April 2014 Revised 4 June 2014 Accepted 4 June 2014 Published 23 June 2014

Academic Editor Hamid Reza Karimi

Copyright copy 2014 Zhilin Liu 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

Metal magnetic memory (MMM) technique is an effective method to achieve the detection of stress concentration (SC) zone foroil well casing It can provide an early diagnosis of microdamages for preventive protection MMM is a natural space domain signalwhich is weak and vulnerable to noise interference So it is difficult to achieve effective feature extraction ofMMM signal especiallyunder the hostile subsurface environment of high temperature high pressure high humidity and multiple interfering sources Inthis paper a method of median filter preprocessing based on data preprocessing technique is proposed to eliminate the outlierspoint of MMM And based on wavelet transform (WT) the adaptive wavelet denoising method and data smoothing arithmeticare applied in testing the system of MMM By using data preprocessing technique the data are reserved and the noises of thesignal are reduced Therefore the correct localization of SC zone can be achieved In the meantime characteristic parameters innew diagnostic approach are put forward to ensure the reliable determination of casing danger level through least squares supportvector machine (LS-SVM) and nonlinear quantitative mapping relationship The effectiveness and feasibility of this method areverified through experiments

1 Introduction

Caused by the factors of erosion geology and engineeringwell casing damage leads to huge economic losses in oilfield every year because of the long-term nonuniform loadon downhole casing which results in severe local stressconcentration and bending deformation and breaking [1ndash3] Therefore it is one of the difficulties in nondestructivetesting to predict the abnormal stress concentration of oilwell casing in order to prevent casing damage Stress isregarded as one of the major factors affecting ferromagneticbehavior along with magnetic field and temperature Effectof stress or strain on magnetization is called the Villarieffect inverse magnetostrictive effect or piezomagnetism Ingeneral it is simply referred to as magnetomechanical effectSince an applied stress can alter the domain structure andhave a substantial effect on the low-field magnetic properties

such as remanence and permeability recently these effectsare mostly found in practical applications of magnetic non-destructive testing actuators and magnetic sensors As aresult the effects have been paid considerable attention inthe literature [1ndash4] Nevertheless the coupling effect betweenmechanical and magnetic properties is so complicated tostunt the development of these properties in nondestructivetesting application

Traditional nondestructive testing such as radiographictesting ultrasonic testing penetrant testing magnetic parti-cle testing and eddy current testing is effective in detectingexisting cracks which means that the detection of crack inthe bud can not be achieved in time before crack developsseverely However MMM technique is a new method ofnondestructive testing which can accomplish early diagnosisand detection of defects With the testing characteristic ofstress concentration it is capable of detecting the most

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 902304 9 pageshttpdxdoiorg1011552014902304

2 Abstract and Applied Analysis

dangerous damage in advance It has been widely appliedin the field of electric power railway and petrochemicalindustry and has been proved effective Despite of all theseMMM is found to be a natural space domain signal and havethe same order of magnitude as the earth magnetic fieldwhich means it is of random nonsmooth signal as well as lowsignal-to-noise ratio and is vulnerable to noise interference[4] Meanwhile quantitative evaluation is affected becauseit is difficult to correctly extract the features of MMMsignal under the hostile subsurface environment such ashigh temperature high pressure and great noise Anotherdifficulty for feature extraction is that underground casing iswrapped in thermal-protective coating of a large dampingwhich results in producing weaker signal or even no signalfrom the stress concentration zone where cracks appear

Data preprocessing and data driven is the first step indiagnosis for casting by usingMMMThe reliability usabilityand integrality of measurement data can affect the accuracyefficiency and effectiveness of model reconstruction directlyThe concept of data preprocessing and data driven is oftenused in computer science But because of the exceedingprogress of computer techniques massive process data can beobtained by the intelligentized industry so data preprocess-ing and data driven have been taking up a lot of attentions inengineering science Rapid improvement of database capacitymake people use data more effectively and fulfill morefunctions Data means information so the so-called datapreprocessing and data driven are drawing information fromdata and use information to realize different objects To drawinformation from data statistical techniques are the chiefmethod and applications based on multivariate statisticaltechniques become the main part of data preprocessing anddata-driven area At present there have been many papersabout the application in different fields of industry based ondata-driven algorithm and techniques [5ndash8] Based on thebasic data-driven methods for process monitoring and faultdiagnosis a comparison study in [9] is provided Authorsin [9] illustrated the efficiencies of data-driven methods dis-cussed in their paper through the application of an industrialbenchmark of Tennessee Eastman (TE) process

The most important mathematical tools in statisticaltechniques filtering algorithm and wavelet analysis arecommonly used in data preprocessing and data-driven fieldIn the process of data acquisition redundancy and mea-surement noise will be introduced inevitably These errorpoints can bring great impacts on the model reconstructionand analysis of data feature In order to extract the featureof data better data filtering must be applied to make theerrors removed Authors in [10] proposed an intelligent datafiltering method based on artificial neural networks to detectbearing defects of induction motors In [11] a robust 119867

infin

filtering problem is investigated for a class of complex net-work systems which has stochastic packet dropouts and timedelays combined with disturbance inputs Authors in [12]deal with the design problem of minimum entropy 119867

infinfilter

in terms of linearmatrix inequality (LMI) approach for linearcontinuous-time systems with a state-space model subjectto parameter uncertainty that belongs to a given convexbounded polyhedral domain In [13] a filtering algorithm for

maneuvering target tracking is presented based on smoothingspline fitting

The wavelet analysis theory is gradually developing tobecome one of important technologies in the dynamic mea-surement signal process field by its advantage of multires-olution and multidimensional analysis on time frequencyAuthors in [14] propose control strategy for the energymanagement which is based on the combination of wavelettransform and neural network arithmetic In the controlstrategy proposed wavelet is in charge of decomposing andthen reconfiguring the power difference between generatedpower and consumed power by loads In [15] authors addressan application of wavelet networks in identification andcontrol design for a class of structures equipped with a typeof semiactive actuators The wavelet analysis theories andmethods are developing and are far frommaturationWaveletanalysis and its application have great potentialities in manyapplied fields of natural science and its application in MMMtest and estimating stress concentration zone is increasing

In this paper according to the foregoing reference amethod of multisource information processing andmultifea-ture quantitative evaluation is described to solve the problemsofMMMsignal processing of underground casing First of allmedian filter preprocessing is adopted to eliminate the out-liers point of MMM Secondly based on wavelet transform(WT) the estimation ismade by adaptive threshold ofwavelettransform coefficients with various scaled space signalsthrough optimal soft threshold denoising which restrains thenoise signal to extract gradient and zero-crossing point fea-tures to achieve correct localization of SC zone Thirdly newdiagnostic approach of combined characteristic parametersis put forward to ensure the reliable determination of casingdanger level through LS-SVM and nonlinear quantitativemapping relationship The effectiveness and feasibility of thismethod are verified through experiments

2 Mechanism of Metal MagneticMemory Testing

The studies of modern material science and ferromagnetichave proved that if iron artifacts are influenced by its workingload under geomagnetic environment their interior structurewill showmagnetostrictive magnetic domain orientation andirreversible reorientationTheoretically relationship betweenthe magnetic field leakage of (119867

119901) and the changes of

mechanical stress (Δ120590) of ferromagnetic artifacts under testis as follows [2 3]

119867119901=

120582119867

1205830

Δ120590 (1)

where 120582119867 is an irreversible component in magnetoelastic

effect and it is a function that depends on mechanical stressas well as the intensity and temperature of the externalmagnetic field 120583

0= 4120587 times 10

minus7 is the permeability ofvacuum Metal magnetic memory theory has proved thatthe maximum variation of scattered magnetic leakage field119867119901occurs in stress concentration and deformation zone

that is the tangential component of magnetic leakage 119867119901(119909)

Abstract and Applied Analysis 3

Hp

Hp

x

N S

y

Stress concentrationline

Figure 1 Principle diagram of MMM testing

shows the maximum value while normal component of themagnetic leakage 119867

119901(119910) is shown to be zero (Figure 1)

This irreversibility of magnetic state will remain after theelimination of working load which makes it possible toaccomplish the accurate diagnosis of component defects and(or) stress concentration zone through the determinationof normal component 119867

119901(119910) in the magnetic leakage of

scatteredmagnetic leakage field In [16 17] the absolute valueofmaximumgradient value inmagnetic variation is taken as adiagnostic parameter to estimate stress concentration level (apatent belongs to Energodiagnostika Co Ltd of Russia) thatis 119870 = |d119867

119901(119910)d119909| is taken as a measurement indicator

The features of zero-crossing point and gradient value aretwo key characteristic parameters inMMMtesting techniqueExperts from Energodiagnostika Co Ltd have proposedsupplementary rules at the MMM application conference inAnshan in 2004 which is mainly about the comprehensivelocalization of stress concentration zone through maximumgradient area and zero-crossing point area [1ndash4] Though itis easy to locate stress concentration zone according to thesmooth MMM curves like the ones in Figure 1 the actualcurves collected are obviously much more complicated thanwhat is in Figure 1 because noise variation is often includedwhen differential derivative technique is employed whichmakes it hard to extract the accurate gradient value of signalsaltation In addition subsurface environment is extremelyhostile and has the characteristics of high temperature highpressure high humidity and great noise underground casingis wrapped in thermal-protective coating of a large dampingwhich results in worse signal-to-noise ratio of weak MMMsignal or even totally overwhelmed by noise Therefore it isdifficult to extract gradient characteristic value of MMM aswell as achieve quantitative evaluation of danger level whichproves the necessity of MMM signal processing

3 Signal Analysis of MMM Testing for OilWell Casing

While variable characteristics of MMM signal in tensile testshave been introduced in most of the present research papers[18 19] few extrusion tests have been involved The mainfunction of oil well casing is pressure-bearing In this text a

Figure 2 Stress test facility

250

200

150

100

50

0

minus50

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)

Figure 3 MMM signal of casing before stress test

ground test was carried out in the simulation well of Daqingoilfield During the test short oil well casings with the lengthof 1m were truncated respectively from a 11-meter longoil well casing (specimen is about 14mm wall thickness is76mm) and were labeled as specimen 1 and specimen 2The two specimens were placed respectively in the middleof hydraulic platform of a NYL-300 compression testingmachine and stress tests were carried out in the middle partof the specimens in order to eliminate the effect of end faceThe nominal working strengths were 20 40 and 240 kNStress application was done every other 20 kN and lastedfor 10 minutes each time After each stress application thespecimen was removed and was vertically placed at a fixedposition The experimental facility is shown in Figure 2

Before stress tests there was no stress concentration zonein oil well casing see the MMM curve in Figure 3 Thereis no obvious peak-peak singular feature of local gradientand signal When the working strength is equal to or largerthan 40 kNs pressure the curve showed significant changethat is obvious peak-peak value in Figure 4 and maximumgradient area after Fourier analysis of the curve See Figure 5This proves that MMM signal energy is mainly concentratedin low-frequency area From Figure 6 it can be seen thatin several areas the gradient values are above 8 or close to8 According to the criterion of Russian patent this meansthere is dangerous stress concentration zone (oil well casing


80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)

Figure 4 MMM signal of casing after stress test

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250

H(A

m)

Frequency (Hz)

Figure 5 MMM signal frequency

is close to hazardous situation when gradient values areabove 8) However after the experiment conclusion has beendrawn that maximum stress concentration level only appearsintensively in the intermediate region which means thereshould be low stress concentration level in other areas andthe utilization will not be affected Therefore it is difficultto determine danger level depending only on differentialderivative [20] It is necessary to conduct MMM signalprocessing especially under more complicated undergroundenvironment

4 MMM Signal Processing and FeatureExtraction for Oil Well Casing

There is a great influence of underground interference andnoise on MMM data and the main source of noise is foundto be the high-frequency noise of measurement noise andthe interfering signal from probe vibration Since MMMsignal is random signal strictly speaking it lacks stability

14

12

10

8

6

4

2

0

0 100 200 300 400 500 600 700 800 900 1000

K(A

mm

m)

L (mm)

Figure 6 Gradient value variation of MMM signal

Though Fourier analysis can achieve a general analysis onthe spectrum features of MMM signal it does not possesslocal analysis feature of time domain and frequency domain[20] Therefore wavelet analysis is adopted in this section toprocess MMM signal

41 Data Smoothing Since MMM signal is weak spatialdomain signal with low frequency [21 22] it is necessary tofirstly accomplish the smoothing of data collected in order toremove possible interference signal and meaningless isolatedoutliers point To ensure high fidelity of signal amplitude aswell as real time of the testing system without producing newquantization parameters median smooth filter is adoptedConsider

119910 (119898) = Median [119909 (119898) 119909 (119898 minus 1) 119909 (119898 minus 2)] (2)

where 119910(119898) is the output 119909(119898) is the input signal sequencein spatial domain and Median is median function

42Wavelet Analysis Sincewavelet transformation (MT)hasa goodmultiresolution time-frequency analysis featurewhichis capable of reducing noise as well as retaining the edge ithas become the key method of the extraction ofMMM signalsingularity [23ndash26] For MMM signal 119891(119905) which containsnoise the model in wavelet domain is

119910 (119905) = 119891 (119905) + 120575 sdot 119899 (119905) (3)

where 119899(119905) represents Gaussian white noise and 120590119899indicates

noise intensity 119864[119899(119905)] shows the mathematical expectationof random variable thus

119864 [119899 (119905)] = 0 119864 [119899 (119906) 119899 (V)] = 1205902

119899 119906 = V

0 119906 = V(4)


Define 119882119904(119899(119905)) as the wavelet transform value of 119899(119905) To

some degree it is also a random variable of 119905 Under waveletdecomposition scale 119904 there is

1003816100381610038161003816119882119904(119899(119905))1003816100381610038161003816

2

= ∬

+infin

minusinfin

119899 (119906) 119899 (V) 120595 (119883 minus 119906)120595 (119883 minus V) 119889119906 119889V

(5)

Its mathematical expectation is

119864 (1003816100381610038161003816119882119904(119899(119905))

1003816100381610038161003816

2

)

= ∬

+infin

minusinfin

1205902

119899120575 (119906 minus V) 120595 (119883 minus 119906)120595 (119883 minus V) 119889119906 119889V

=

10038171003817100381710038171205951003817100381710038171003817

2

1205902

119899

119904

(6)

This indicates that the average density of white noisemodulusmaximum is inverse proportion to the scale that is thegreater the scale is the sparser themodulusmaximumwill beWhileWTof noise on different scales is highly irrelevantWTof MMM singular signal often has a strong correlation thatis local modulus maximums on adjacent scales almost sharethe same position and the same sign which is the theoreticalfoundation of MMM signal processing

43 Wavelet-Based Adaptive Threshold Denoising The prin-ciple of wavelet-based denoising is to accomplish a waveletanalysis onmeasurement signal mixed with noise and to sep-arate them according to the different characteristics betweensignal and noise under WT The wavelet coefficients whichbelong to noise are set to zero wavelet reconstruction iscarried out on the left to get useful signalWavelet coefficientsobtained fromhard thresholdmethod are discrete [22] whichoften result in oscillation effect for signal reconstruction Softthreshold function can achieve smooth denoising of waveletcoefficients in low scales However in high scales it will causea decline of signal-to-noise ratio Therefore it is necessary tocontract wavelet coefficients in low scales as well as protectwavelet coefficients in high scales in order to restrain theoscillation

The combination of soft and hard threshold methods isadopted that means that under the premise of maximumdenoising [23] errors are reduced to the greatest extent withthe purpose of achieving optimal denoising Consider

sgn (119882

119904119896) (

10038161003816100381610038161198821199041198961003816100381610038161003816 minus 119886 sdot 120590)

10038161003816100381610038161198821199041198961003816100381610038161003816 ge 120590

01003816100381610038161003816119882119904119896

1003816100381610038161003816 lt 120590

0 le 119886 le 1

(7)

When 119886 = 0 it is hard thresholdmethod when 119886 = 1 it is softthreshold method 119882

119904119896represents wavelet coefficient To get

optimal estimation of the signal the value of 119886 is determined

by minimummean square error criterion Optimal thresholdvalue 119886 are [23]

119886 = 119864 [|119899|] sdot119875+minus 119875minus

119875++ 119875minus

119875+= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

119875minus= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

(8)

For MMM signal which has undergone filtering preprocess-ing the noise is mainly Gaussian white noise thus 119864[|119899|] asymp

06744 and 119886 = 06744Under regular denoising method universal threshold

value is 120590 = 120590119899radic2 log119873 119873 represents signal length

According to

120590119904= 21minus(120590119904120590119899)sdot119904

sdot 120590 (9)

it can be seen that actually noise threshold will decline alongwith the increase of scales So it is not inadvisable to choosethe fixed threshold value If the threshold value is too largeuseful signal is prone to be filtered out which will have animpact on signal-to-noise ratio So adaptive threshold valueis chosen by (9) which changes according to the scales andstandard deviation Noise standard deviation is estimated asfollows

120590119899=

Median (100381610038161003816100381610038161199081119895

10038161003816100381610038161003816)

06745 (10)

Since WT is linear transformation both signal and noisewavelet coefficients accord Gaussian distribution and signalstandard deviation of each wavelet scale is estimated in termsof approximate maximum probability

1205902

119904= max(0

1

1198992

119899

sum

119895=1

1199082

119904119905minus 1205902

119899) (11)

Therefore new wavelet coefficients are obtained and signalreconstruction is achieved through inverse transforms

Signal denoising procedures are as follows

(1) wavelet coefficients are obtained through six-layerdecomposition according to Db10 wavelet motherfunction

(2) based on (9) the first four-layer threshold valuesare determined and optimal threshold value (7) isemployed to process wavelet coefficients to get newwavelet coefficients

(3) signal is reconstructed through inverse transforms

Data fromFigure 4 is applied to adaptive threshold denoisingsee Figures 7 and 8 It is obvious that gradient valuemaximumof oil well casing in the middle area is 8 and the stressconcentration level is the highest which is in accordancewithexperimental results and should be well focused on Gradientvalues in other areas are less than 4 the stress concentrationzone is relatively smallTherefore gradient values after signaldenoising are more accessible to qualitative and quantitative


80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)

Figure 7 MMM signal after adaptive denoising

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

8

7

6

5

4

3

2

1

0

K(A

mm

m)

Figure 8 MMM gradient value after adaptive denoising

evaluation on service life of oil well casing for influence ofnoise is eliminated

After the denoising reconstruction of MMM signal col-lected feature extraction should be accomplished Patentedtechnology raised by Energodiagnostika Co Ltd of Russiais to determine stress concentration zone through gradientvalue maximum and zero-crossing point Gradient valueis considered as characteristic parameter which has beenpointed out in existing researches that one single parameteroften causes misjudgment in real In this section peak-peakvalue and gradient value are chosen to be characteristicparameters as a whole with the purpose of ensuring accuratedetermination of stress concentration zone

Signal peak-peak value PP0 is as follows peak-peak valueis taken as characteristic parameter which eliminates theimpact of signal baseline and enhances the reliability ofdetection because it is an important distribution featureof signal detection When peak-peak value is calculatedmaximum and minimum values of signal are firstly sought

and then absolute value (range) of the difference betweenthese two adjacent extremums is obtained which is easy toachieve through computer Consider

PP0 = max [119891 (119898)] minus min [119891 (119898)] (12)

wheremax[119891(119898)] andmin[119891(119898)] represent a pair of adjacentextremums During the experiment peak-peak value is thekey index

119885 = [119879PP0]119879 refers to the feature vector 119879 is gradientvalueWhen both of the two characteristic component valuesare larger than the corresponding threshold value thereis the real dangerous area of stress concentration zone Ifgradient value is larger than threshold value while peak-peak value is relatively small there is not necessarily a stressconcentration zone but possibly an uncertain area created bynoise Accurate determination needs to be done with the helpof other nondestructive testing methods Qualitative analysisis only a way to determine stress concentration zone of oilwell casing it is unable to satisfy the need of the quantitativeevaluation on danger level which will be in the next section

5 Quantitative Evaluation on Danger Level ofOil Well Casing

Mechanism model which reflects MMM effect is the quan-titative basis of nondestructive testing Present mechanismstudies are mainly based on theories such as common effectof stress and external magnetic field stress magnetization ofenergy maximum principle and stress permeability effectCombined with test data MMMmechanism model is estab-lished from various angles however complete and rigoroustheoretical system has not been formed yet So the scopeof applications is limited In the meantime local elasticand plastic deformation of ferromagnetic material has beennoticed during the experiment and nonlinear variation ofnormal magnetic flux leakage happened on the surface oflocal variation area Therefore feature parameter and stressconcentration of MMM signal are of unknownmathematicalrelation which requires a solution of quantitative evaluationproblem through nonlinear modeling technology Gradientvalue and peak-peak value as combined vector are taken todescribe signal features and there are two input ends Outputends of quantitative model are also two According to thesafe requirement of real engineering oil well casing ldquo00rdquorepresents the fact that elastic deformation is relatively smallwhich has no influence on using ldquo01rdquo shows a relatively largeelastic deformation which requires regular inspection ldquo10rdquomeans that critical plastic changing area protective measuresshould be carried out ldquo11rdquo indicates that severe deformationof oil well casing has happened and they must be replaced

Since 119879PP0 rarr 119884 obtained from experiment is smallsample quantitative classification of danger level throughsmall sample study method is adopted in this section andnonlinear mapping relationship is established through LS-SVM LS-SVM is a machine learning algorithm based onstatistical learning theory Learning according to the principleof structural risk minimization it translates optimizationproblem into a problem of convex quadratic programming


which ensure that the extreme solution is the globally optimalsolution [27]

A training sample set is taken into consideration whichincludes 119873 data point 119909

119896 119910119896 119896 = 1 2 119873 119909

119896isin 119877119899

which represents the input sample and 119910119896

isin 119877 shows theoutput sample Regression model is as follows

119910 (119909) = 119908119879

120593 (119909) + 119887 (13)

In the above formula nonlinearmapping120593(119909)map input datato high-dimension feature space which makes the nonlinearregression problem in the original space to translate intolinear regression problem in feature space And 119908 isin 119877

119899119887 isin 119877

Thedefinition of objective function in regressionLS-SVMis

min119908119890

119869 (119908 119890) =1

2119908119879

119908 +119862

2

119873

sum

119896=1

1198902

119896

119910119896= 119908119879

120593 (119909119896) + 119887 + 119890

119896 119896 = 1 2 119873

(14)

where 119896 = 1 2 119873 119862 represents the constant whichis called penalty factor and it is used to balance modelcomplexity and fitting precision and 119890

119896refers to the error

term Constrained optimization problem is translated intounconstrained optimization problem and Lagrange multi-plier 120572

119896is introduced the corresponding Lagrange function

is

119871 (119908 119887 119890 120572)

= 119869 (119908 119890) minus

119873

sum

119896=1

120572119896119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896

(15)

Under Karush-Kuhn-Tucker condition [16] we can get

120597

120597119908119871 = 0 997888rarr 119908 =

119873

sum

119894=1

120572119896120593 (119909119896)

120597

120597119887119871 = 0 997888rarr

119873

sum

119894=1

120572119896= 0

120597

120597119890119896

119871 = 0 997888rarr 120572119896= 119862119890119896

120597

120597120572119896

119871 = 0 997888rarr 119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0

(16)

[0 11198791 Ω + Cminus1I] [

119887

120572] = [

0

Y] (17)

where Y = [1199101 1199102 119910

119873]119879 1 = [1 1 1]

119879 120572 =

[1205721 1205722 120572

119873]119879 and Ω(119896 119897) = 120593(119909

119896)119879

120593(119909119897) = 119870(119909

119896 119909119897)

119896 119897 = 1 2 119873 According to formula of (15) and (16) theregression model in which LS-SVM is applied to parameterestimation is

119910 =

119873

sum

119894=1

120572119896119870(119909 119909

119896) + 119887 (18)

In the above formula 120572119896 119887 are the solution to formula (16)

Table 1 Quantitative evaluation on danger level for oil well casing

Serial number Gradient value(Ammm)

Peak-peakvalue(Am)

Outputclassification

1 4 130 002 5 150 003 8 210 014 9 260 015 9 230 016 14 300 107 10 230 108 12 250 109 18 280 1110 14 270 1111 7 200 0112 10 286 10

Figure 9 Probe device of underground MMM testing

Table 1 is the results of danger level classification for oilwell casing in which the first ten groups are training samplesand the last two are testing samples Therefore quantitativeevaluation of danger level for oil well casing is achievedthrough support vector machine

Notation 1 Accurate classification requires a large quantity oftraining samples Samples given here are related to the modelnumber of MMM device (has an influence on the sensitivityof characteristic parameter) and material and model numberof oil well casing Experimental facility of underground test inDaqing Oilfield (located in Heilongjiang Province in China)is shown in Figures 9 and 10 and research results are the sameas ground ones no repeat will be done here The casing afterMMM testing is shown in Figure 11 As the casing cube isnondestructive the MMM testing technique is suitable forpractical application

6 Conclusion

The experiment indicates that MMM is capable of achievingan effective prediction on underground stress concentration


Figure 10 Sketch of scan test

Figure 11 The casing after MMM testing

zone for oil well casing Due to the complicated undergroundenvironment and various interferences digital processingtechnology is introduced to MMM analysis software in orderto enhance signal-to-noise ratio as well as eliminate high-frequency noise In addition smooth data fromfilter process-ing achieve accurate feature extraction of MMM signal Atthe same time new combined feature vector is put forwardwhich is used to closely approach the nonlinear relationshipof magnetic flux leakage signal and danger level throughSVM Therefore the stress concentration level for oil wellcasing could be predicted in a timely and reliable mannerAnd it should also be noted that MMM is influenced bytemperature greatly With the oil well casing drilling intothe ground geothermal temperature will inevitably have animpact on the MMM signals In the future work effortswill be made to design the robust hardware to reduce thetemperature drift It is also a challenge to use advanced data-driven technology to deal with the complex MMM signalsand remove the disturbance effect of geothermal temperature

Conflict of Interests

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

Acknowledgments

This work is partially supported by National Natural ScienceFoundation of China (51379044) Fundamental ResearchFunds for the Central Universities (HEUCFX41304) and

Heilongjiang Province Natural Science Foundation Projects(F200916)

References

[1] A A Doubov ldquoDiagnostics of equipment and constructionsstrength with usage of magnetic memoryrdquo Inspection Diagnos-tics no 6 pp 19ndash29 2001

[2] A A Dubov and K Sergey ldquoThe metal magnetic memorymethod application for online monitoring of damage develop-ment in steel pipes and welded joints specimensrdquoWelding in theWorld vol 57 no 1 pp 123ndash136 2013

[3] A A Doubov ldquoExpress method of quality control of a spotresistance welding with usage of metal magnetic memoryrdquoWelding in the World vol 46 no 6 pp 317ndash320 2002

[4] A A Dubov ldquoDevelopment of a metal magnetic memorymethodrdquo Chemical and Petroleum Engineering vol 47 no 11pp 837ndash839 2012

[5] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[6] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[7] S Yin S X Ding X Xie and H Luo ldquoA review on basic data-driven approaches for industrial process monitoringrdquo IEEETransactions on Industrial Electronics vol 61 no 11 pp 6418ndash6428 2014

[8] S Yin H Luo and S X Ding ldquoReal-time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[9] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

[10] J Zarei M A Tajeddini and H R Karimi ldquoVibration analysisfor bearing fault detection and classification using an intelligentfilterrdquoMechatronics vol 24 no 2 pp 151ndash157 2014

[11] J Zhang M Lyu H R Karimi P Guo and Y Bo ldquoRobust119867infin

filtering for a class of complex networks with stochasticpacket dropouts and time delaysrdquo The Scientific World Journalvol 2014 Article ID 560234 11 pages 2014

[12] J Zhang H R Karimi Z Zheng M Lyu and Y Bo ldquoHinfin

filter design with minimum entropy for continuous-time linearsystemsrdquo Mathematical Problems in Engineering vol 2013Article ID 579137 9 pages 2013

[13] Y Liu J Suo H R Karimi and X Liu ldquoA filtering algorithm formaneuvering target tracking based on smoothing spline fittingrdquoAbstract and Applied Analysis vol 2014 Article ID 127643 6pages 2014

[14] Y D Song Q Cao X Du and H R Karimi ldquoControl strategybased on wavelet transform and neural network for hybridpower systemrdquo Journal of AppliedMathematics vol 2013 ArticleID 375840 8 pages 2013

[15] H R Karimi M Zapateiro and N Luo ldquoApplication ofadaptive wavelet networks for vibration control of base isolatedstructuresrdquo International Journal of Wavelets Multiresolutionand Information Processing vol 8 no 5 pp 773ndash791 2010


[16] M Roskosz ldquoMetal magnetic memory testing of welded jointsof ferritic and austenitic steelsrdquo NDT amp E International vol 44no 3 pp 305ndash310 2011

[17] T Yan J Zhang G Feng and J Chen ldquoInspection of wet steamgenerator tubes based on metal magnetic memory methodrdquoProcedia Engineering vol 15 pp 1140ndash1144 2011

[18] Z D Wang K Yao B Deng and K Q Ding ldquoTheoreticalstudies of metal magnetic memory technique on magnetic fluxleakage signalsrdquo NDT amp E International vol 43 no 4 pp 354ndash359 2010

[19] Z D Wang K Yao B Deng and K Q Ding ldquoQuantitativestudy of metal magnetic memory signal versus local stressconcentrationrdquo NDT amp E International vol 43 no 6 pp 513ndash518 2010

[20] K Yao B Deng and Z D Wang ldquoNumerical studies tosignal characteristics with the metal magnetic memory-effectin plastically deformed samplesrdquo NDT amp E International vol47 pp 7ndash17 2012

[21] J Leng Y Liu G Zhou and Y Gao ldquoMetal magnetic memorysignal response to plastic deformation of low carbon steelrdquoNDTamp E International vol 55 pp 42ndash46 2013

[22] X Hai-Yan W Wen-Jiang W Ri-Xing X Min-Qiang and LXue-Feng ldquoStress distribution testing of 50MW turbine frac-ture blade with metal magnetic memory methodrdquo Proceedingsof the Chinese Society of Electrical Engineering vol 26 no 4 pp72ndash76 2006

[23] H Y Kang Y Lu and W Y Shuzi ldquoSome algorithms fornondestructive testing of wire ropesłsignal pre-processing andcharacter extractionrdquoNonde Structive Testing vol 22 no 11 pp483ndash488 2000

[24] J Zhang B Wang and B Ji ldquoSignal processing for metalmagnetic memory testing of borehole casing based on wavelettransformrdquoActa Petrol Ei Sinica vol 27 no 2 pp 137ndash140 2006

[25] Z Xiaoyong and Y Yinzhong ldquoMulti-fault diagnosis methodon Mallat pyramidal algorithm wavelet analysisrdquo Control andDecision vol 19 no 5 pp 592ndash594 2004

[26] R Manojit V Kumar B D Kulkarni J Sanderson M Rhodesand M Vander Stappen ldquoSimple denoising algorithm usingwavelet transformrdquo American Institute of Chemical EngineersJournal vol 45 no 11 pp 2461ndash2466 1999

[27] J A K Suykens J De Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 1 pp85ndash105 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


dangerous damage in advance It has been widely appliedin the field of electric power railway and petrochemicalindustry and has been proved effective Despite of all theseMMM is found to be a natural space domain signal and havethe same order of magnitude as the earth magnetic fieldwhich means it is of random nonsmooth signal as well as lowsignal-to-noise ratio and is vulnerable to noise interference[4] Meanwhile quantitative evaluation is affected becauseit is difficult to correctly extract the features of MMMsignal under the hostile subsurface environment such ashigh temperature high pressure and great noise Anotherdifficulty for feature extraction is that underground casing iswrapped in thermal-protective coating of a large dampingwhich results in producing weaker signal or even no signalfrom the stress concentration zone where cracks appear

Data preprocessing and data driven is the first step indiagnosis for casting by usingMMMThe reliability usabilityand integrality of measurement data can affect the accuracyefficiency and effectiveness of model reconstruction directlyThe concept of data preprocessing and data driven is oftenused in computer science But because of the exceedingprogress of computer techniques massive process data can beobtained by the intelligentized industry so data preprocess-ing and data driven have been taking up a lot of attentions inengineering science Rapid improvement of database capacitymake people use data more effectively and fulfill morefunctions Data means information so the so-called datapreprocessing and data driven are drawing information fromdata and use information to realize different objects To drawinformation from data statistical techniques are the chiefmethod and applications based on multivariate statisticaltechniques become the main part of data preprocessing anddata-driven area At present there have been many papersabout the application in different fields of industry based ondata-driven algorithm and techniques [5ndash8] Based on thebasic data-driven methods for process monitoring and faultdiagnosis a comparison study in [9] is provided Authorsin [9] illustrated the efficiencies of data-driven methods dis-cussed in their paper through the application of an industrialbenchmark of Tennessee Eastman (TE) process

The most important mathematical tools in statisticaltechniques filtering algorithm and wavelet analysis arecommonly used in data preprocessing and data-driven fieldIn the process of data acquisition redundancy and mea-surement noise will be introduced inevitably These errorpoints can bring great impacts on the model reconstructionand analysis of data feature In order to extract the featureof data better data filtering must be applied to make theerrors removed Authors in [10] proposed an intelligent datafiltering method based on artificial neural networks to detectbearing defects of induction motors In [11] a robust 119867

infin

filtering problem is investigated for a class of complex net-work systems which has stochastic packet dropouts and timedelays combined with disturbance inputs Authors in [12]deal with the design problem of minimum entropy 119867

infinfilter

in terms of linearmatrix inequality (LMI) approach for linearcontinuous-time systems with a state-space model subjectto parameter uncertainty that belongs to a given convexbounded polyhedral domain In [13] a filtering algorithm for

maneuvering target tracking is presented based on smoothingspline fitting

The wavelet analysis theory is gradually developing tobecome one of important technologies in the dynamic mea-surement signal process field by its advantage of multires-olution and multidimensional analysis on time frequencyAuthors in [14] propose control strategy for the energymanagement which is based on the combination of wavelettransform and neural network arithmetic In the controlstrategy proposed wavelet is in charge of decomposing andthen reconfiguring the power difference between generatedpower and consumed power by loads In [15] authors addressan application of wavelet networks in identification andcontrol design for a class of structures equipped with a typeof semiactive actuators The wavelet analysis theories andmethods are developing and are far frommaturationWaveletanalysis and its application have great potentialities in manyapplied fields of natural science and its application in MMMtest and estimating stress concentration zone is increasing

In this paper according to the foregoing reference amethod of multisource information processing andmultifea-ture quantitative evaluation is described to solve the problemsofMMMsignal processing of underground casing First of allmedian filter preprocessing is adopted to eliminate the out-liers point of MMM Secondly based on wavelet transform(WT) the estimation ismade by adaptive threshold ofwavelettransform coefficients with various scaled space signalsthrough optimal soft threshold denoising which restrains thenoise signal to extract gradient and zero-crossing point fea-tures to achieve correct localization of SC zone Thirdly newdiagnostic approach of combined characteristic parametersis put forward to ensure the reliable determination of casingdanger level through LS-SVM and nonlinear quantitativemapping relationship The effectiveness and feasibility of thismethod are verified through experiments

2 Mechanism of Metal MagneticMemory Testing

The studies of modern material science and ferromagnetichave proved that if iron artifacts are influenced by its workingload under geomagnetic environment their interior structurewill showmagnetostrictive magnetic domain orientation andirreversible reorientationTheoretically relationship betweenthe magnetic field leakage of (119867

119901) and the changes of

mechanical stress (Δ120590) of ferromagnetic artifacts under testis as follows [2 3]

119867119901=

120582119867

1205830

Δ120590 (1)

where 120582119867 is an irreversible component in magnetoelastic

effect and it is a function that depends on mechanical stressas well as the intensity and temperature of the externalmagnetic field 120583

0= 4120587 times 10

minus7 is the permeability ofvacuum Metal magnetic memory theory has proved thatthe maximum variation of scattered magnetic leakage field119867119901occurs in stress concentration and deformation zone

that is the tangential component of magnetic leakage 119867119901(119909)


Hp

Hp

x

N S

y













250

200

150

100

50

0

minus50

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)





80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250

H(A

m)

Frequency (Hz)





14

12

10

8

6

4

2

0

0 100 200 300 400 500 600 700 800 900 1000

K(A

mm

m)

L (mm)







119910 (119905) = 119891 (119905) + 120575 sdot 119899 (119905) (3)



119864 [119899 (119905)] = 0 119864 [119899 (119906) 119899 (V)] = 1205902

119899 119906 = V

0 119906 = V(4)




1003816100381610038161003816119882119904(119899(119905))1003816100381610038161003816

2

= ∬

+infin

minusinfin

119899 (119906) 119899 (V) 120595 (119883 minus 119906)120595 (119883 minus V) 119889119906 119889V

(5)


119864 (1003816100381610038161003816119882119904(119899(119905))

1003816100381610038161003816

2

)

= ∬

+infin

minusinfin

1205902


=

10038171003817100381710038171205951003817100381710038171003817

2

1205902

119899

119904

(6)




sgn (119882

119904119896) (

10038161003816100381610038161198821199041198961003816100381610038161003816 minus 119886 sdot 120590)

10038161003816100381610038161198821199041198961003816100381610038161003816 ge 120590

01003816100381610038161003816119882119904119896

1003816100381610038161003816 lt 120590

0 le 119886 le 1

(7)





119886 = 119864 [|119899|] sdot119875+minus 119875minus

119875++ 119875minus

119875+= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

119875minus= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

(8)




According to

120590119904= 21minus(120590119904120590119899)sdot119904

sdot 120590 (9)


120590119899=

Median (100381610038161003816100381610038161199081119895

10038161003816100381610038161003816)

06745 (10)


1205902

119904= max(0

1

1198992

119899

sum

119895=1

1199082

119904119905minus 1205902

119899) (11)








80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


0 100 200 300 400 500 600 700 800 900 1000

L (mm)

8

7

6

5

4

3

2

1

0

K(A

mm

m)






PP0 = max [119891 (119898)] minus min [119891 (119898)] (12)









119896 119910119896 119896 = 1 2 119873 119909

119896isin 119877119899



119910 (119909) = 119908119879

120593 (119909) + 119887 (13)


119899119887 isin 119877


min119908119890

119869 (119908 119890) =1

2119908119879

119908 +119862

2

119873

sum

119896=1

1198902

119896

119910119896= 119908119879

120593 (119909119896) + 119887 + 119890

119896 119896 = 1 2 119873

(14)





is

119871 (119908 119887 119890 120572)

= 119869 (119908 119890) minus

119873

sum

119896=1

120572119896119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896

(15)


120597

120597119908119871 = 0 997888rarr 119908 =

119873

sum

119894=1

120572119896120593 (119909119896)

120597

120597119887119871 = 0 997888rarr

119873

sum

119894=1

120572119896= 0

120597

120597119890119896

119871 = 0 997888rarr 120572119896= 119862119890119896

120597

120597120572119896

119871 = 0 997888rarr 119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0

(16)

[0 11198791 Ω + Cminus1I] [

119887

120572] = [

0

Y] (17)

where Y = [1199101 1199102 119910

119873]119879 1 = [1 1 1]

119879 120572 =

[1205721 1205722 120572

119873]119879 and Ω(119896 119897) = 120593(119909

119896)119879

120593(119909119897) = 119870(119909

119896 119909119897)


119910 =

119873

sum

119894=1

120572119896119870(119909 119909

119896) + 119887 (18)




Peak-peakvalue(Am)


1 4 130 002 5 150 003 8 210 014 9 260 015 9 230 016 14 300 107 10 230 108 12 250 109 18 280 1110 14 270 1111 7 200 0112 10 286 10




6 Conclusion








Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Hp

Hp

x

N S

y













250

200

150

100

50

0

minus50

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)





80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250

H(A

m)

Frequency (Hz)





14

12

10

8

6

4

2

0

0 100 200 300 400 500 600 700 800 900 1000

K(A

mm

m)

L (mm)







119910 (119905) = 119891 (119905) + 120575 sdot 119899 (119905) (3)



119864 [119899 (119905)] = 0 119864 [119899 (119906) 119899 (V)] = 1205902

119899 119906 = V

0 119906 = V(4)




1003816100381610038161003816119882119904(119899(119905))1003816100381610038161003816

2

= ∬

+infin

minusinfin

119899 (119906) 119899 (V) 120595 (119883 minus 119906)120595 (119883 minus V) 119889119906 119889V

(5)


119864 (1003816100381610038161003816119882119904(119899(119905))

1003816100381610038161003816

2

)

= ∬

+infin

minusinfin

1205902


=

10038171003817100381710038171205951003817100381710038171003817

2

1205902

119899

119904

(6)




sgn (119882

119904119896) (

10038161003816100381610038161198821199041198961003816100381610038161003816 minus 119886 sdot 120590)

10038161003816100381610038161198821199041198961003816100381610038161003816 ge 120590

01003816100381610038161003816119882119904119896

1003816100381610038161003816 lt 120590

0 le 119886 le 1

(7)





119886 = 119864 [|119899|] sdot119875+minus 119875minus

119875++ 119875minus

119875+= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

119875minus= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

(8)




According to

120590119904= 21minus(120590119904120590119899)sdot119904

sdot 120590 (9)


120590119899=

Median (100381610038161003816100381610038161199081119895

10038161003816100381610038161003816)

06745 (10)


1205902

119904= max(0

1

1198992

119899

sum

119895=1

1199082

119904119905minus 1205902

119899) (11)








80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


0 100 200 300 400 500 600 700 800 900 1000

L (mm)

8

7

6

5

4

3

2

1

0

K(A

mm

m)






PP0 = max [119891 (119898)] minus min [119891 (119898)] (12)









119896 119910119896 119896 = 1 2 119873 119909

119896isin 119877119899



119910 (119909) = 119908119879

120593 (119909) + 119887 (13)


119899119887 isin 119877


min119908119890

119869 (119908 119890) =1

2119908119879

119908 +119862

2

119873

sum

119896=1

1198902

119896

119910119896= 119908119879

120593 (119909119896) + 119887 + 119890

119896 119896 = 1 2 119873

(14)





is

119871 (119908 119887 119890 120572)

= 119869 (119908 119890) minus

119873

sum

119896=1

120572119896119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896

(15)


120597

120597119908119871 = 0 997888rarr 119908 =

119873

sum

119894=1

120572119896120593 (119909119896)

120597

120597119887119871 = 0 997888rarr

119873

sum

119894=1

120572119896= 0

120597

120597119890119896

119871 = 0 997888rarr 120572119896= 119862119890119896

120597

120597120572119896

119871 = 0 997888rarr 119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0

(16)

[0 11198791 Ω + Cminus1I] [

119887

120572] = [

0

Y] (17)

where Y = [1199101 1199102 119910

119873]119879 1 = [1 1 1]

119879 120572 =

[1205721 1205722 120572

119873]119879 and Ω(119896 119897) = 120593(119909

119896)119879

120593(119909119897) = 119870(119909

119896 119909119897)


119910 =

119873

sum

119894=1

120572119896119870(119909 119909

119896) + 119887 (18)




Peak-peakvalue(Am)


1 4 130 002 5 150 003 8 210 014 9 260 015 9 230 016 14 300 107 10 230 108 12 250 109 18 280 1110 14 270 1111 7 200 0112 10 286 10




6 Conclusion








Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250

H(A

m)

Frequency (Hz)





14

12

10

8

6

4

2

0

0 100 200 300 400 500 600 700 800 900 1000

K(A

mm

m)

L (mm)







119910 (119905) = 119891 (119905) + 120575 sdot 119899 (119905) (3)



119864 [119899 (119905)] = 0 119864 [119899 (119906) 119899 (V)] = 1205902

119899 119906 = V

0 119906 = V(4)




1003816100381610038161003816119882119904(119899(119905))1003816100381610038161003816

2

= ∬

+infin

minusinfin

119899 (119906) 119899 (V) 120595 (119883 minus 119906)120595 (119883 minus V) 119889119906 119889V

(5)


119864 (1003816100381610038161003816119882119904(119899(119905))

1003816100381610038161003816

2

)

= ∬

+infin

minusinfin

1205902


=

10038171003817100381710038171205951003817100381710038171003817

2

1205902

119899

119904

(6)




sgn (119882

119904119896) (

10038161003816100381610038161198821199041198961003816100381610038161003816 minus 119886 sdot 120590)

10038161003816100381610038161198821199041198961003816100381610038161003816 ge 120590

01003816100381610038161003816119882119904119896

1003816100381610038161003816 lt 120590

0 le 119886 le 1

(7)





119886 = 119864 [|119899|] sdot119875+minus 119875minus

119875++ 119875minus

119875+= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

119875minus= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

(8)




According to

120590119904= 21minus(120590119904120590119899)sdot119904

sdot 120590 (9)


120590119899=

Median (100381610038161003816100381610038161199081119895

10038161003816100381610038161003816)

06745 (10)


1205902

119904= max(0

1

1198992

119899

sum

119895=1

1199082

119904119905minus 1205902

119899) (11)








80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


0 100 200 300 400 500 600 700 800 900 1000

L (mm)

8

7

6

5

4

3

2

1

0

K(A

mm

m)






PP0 = max [119891 (119898)] minus min [119891 (119898)] (12)









119896 119910119896 119896 = 1 2 119873 119909

119896isin 119877119899



119910 (119909) = 119908119879

120593 (119909) + 119887 (13)


119899119887 isin 119877


min119908119890

119869 (119908 119890) =1

2119908119879

119908 +119862

2

119873

sum

119896=1

1198902

119896

119910119896= 119908119879

120593 (119909119896) + 119887 + 119890

119896 119896 = 1 2 119873

(14)





is

119871 (119908 119887 119890 120572)

= 119869 (119908 119890) minus

119873

sum

119896=1

120572119896119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896

(15)


120597

120597119908119871 = 0 997888rarr 119908 =

119873

sum

119894=1

120572119896120593 (119909119896)

120597

120597119887119871 = 0 997888rarr

119873

sum

119894=1

120572119896= 0

120597

120597119890119896

119871 = 0 997888rarr 120572119896= 119862119890119896

120597

120597120572119896

119871 = 0 997888rarr 119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0

(16)

[0 11198791 Ω + Cminus1I] [

119887

120572] = [

0

Y] (17)

where Y = [1199101 1199102 119910

119873]119879 1 = [1 1 1]

119879 120572 =

[1205721 1205722 120572

119873]119879 and Ω(119896 119897) = 120593(119909

119896)119879

120593(119909119897) = 119870(119909

119896 119909119897)


119910 =

119873

sum

119894=1

120572119896119870(119909 119909

119896) + 119887 (18)




Peak-peakvalue(Am)


1 4 130 002 5 150 003 8 210 014 9 260 015 9 230 016 14 300 107 10 230 108 12 250 109 18 280 1110 14 270 1111 7 200 0112 10 286 10




6 Conclusion








Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1003816100381610038161003816119882119904(119899(119905))1003816100381610038161003816

2

= ∬

+infin

minusinfin

119899 (119906) 119899 (V) 120595 (119883 minus 119906)120595 (119883 minus V) 119889119906 119889V

(5)


119864 (1003816100381610038161003816119882119904(119899(119905))

1003816100381610038161003816

2

)

= ∬

+infin

minusinfin

1205902


=

10038171003817100381710038171205951003817100381710038171003817

2

1205902

119899

119904

(6)




sgn (119882

119904119896) (

10038161003816100381610038161198821199041198961003816100381610038161003816 minus 119886 sdot 120590)

10038161003816100381610038161198821199041198961003816100381610038161003816 ge 120590

01003816100381610038161003816119882119904119896

1003816100381610038161003816 lt 120590

0 le 119886 le 1

(7)





119886 = 119864 [|119899|] sdot119875+minus 119875minus

119875++ 119875minus

119875+= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

119875minus= Pr sgn (119910) = sgn (119899)

10038161003816100381610038161199101003816100381610038161003816 gt 120590

(8)




According to

120590119904= 21minus(120590119904120590119899)sdot119904

sdot 120590 (9)


120590119899=

Median (100381610038161003816100381610038161199081119895

10038161003816100381610038161003816)

06745 (10)


1205902

119904= max(0

1

1198992

119899

sum

119895=1

1199082

119904119905minus 1205902

119899) (11)








80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


0 100 200 300 400 500 600 700 800 900 1000

L (mm)

8

7

6

5

4

3

2

1

0

K(A

mm

m)






PP0 = max [119891 (119898)] minus min [119891 (119898)] (12)









119896 119910119896 119896 = 1 2 119873 119909

119896isin 119877119899



119910 (119909) = 119908119879

120593 (119909) + 119887 (13)


119899119887 isin 119877


min119908119890

119869 (119908 119890) =1

2119908119879

119908 +119862

2

119873

sum

119896=1

1198902

119896

119910119896= 119908119879

120593 (119909119896) + 119887 + 119890

119896 119896 = 1 2 119873

(14)





is

119871 (119908 119887 119890 120572)

= 119869 (119908 119890) minus

119873

sum

119896=1

120572119896119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896

(15)


120597

120597119908119871 = 0 997888rarr 119908 =

119873

sum

119894=1

120572119896120593 (119909119896)

120597

120597119887119871 = 0 997888rarr

119873

sum

119894=1

120572119896= 0

120597

120597119890119896

119871 = 0 997888rarr 120572119896= 119862119890119896

120597

120597120572119896

119871 = 0 997888rarr 119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0

(16)

[0 11198791 Ω + Cminus1I] [

119887

120572] = [

0

Y] (17)

where Y = [1199101 1199102 119910

119873]119879 1 = [1 1 1]

119879 120572 =

[1205721 1205722 120572

119873]119879 and Ω(119896 119897) = 120593(119909

119896)119879

120593(119909119897) = 119870(119909

119896 119909119897)


119910 =

119873

sum

119894=1

120572119896119870(119909 119909

119896) + 119887 (18)




Peak-peakvalue(Am)


1 4 130 002 5 150 003 8 210 014 9 260 015 9 230 016 14 300 107 10 230 108 12 250 109 18 280 1110 14 270 1111 7 200 0112 10 286 10




6 Conclusion








Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

0 100 200 300 400 500 600 700 800 900 1000

L (mm)

H(A

m)


0 100 200 300 400 500 600 700 800 900 1000

L (mm)

8

7

6

5

4

3

2

1

0

K(A

mm

m)






PP0 = max [119891 (119898)] minus min [119891 (119898)] (12)









119896 119910119896 119896 = 1 2 119873 119909

119896isin 119877119899



119910 (119909) = 119908119879

120593 (119909) + 119887 (13)


119899119887 isin 119877


min119908119890

119869 (119908 119890) =1

2119908119879

119908 +119862

2

119873

sum

119896=1

1198902

119896

119910119896= 119908119879

120593 (119909119896) + 119887 + 119890

119896 119896 = 1 2 119873

(14)





is

119871 (119908 119887 119890 120572)

= 119869 (119908 119890) minus

119873

sum

119896=1

120572119896119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896

(15)


120597

120597119908119871 = 0 997888rarr 119908 =

119873

sum

119894=1

120572119896120593 (119909119896)

120597

120597119887119871 = 0 997888rarr

119873

sum

119894=1

120572119896= 0

120597

120597119890119896

119871 = 0 997888rarr 120572119896= 119862119890119896

120597

120597120572119896

119871 = 0 997888rarr 119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0

(16)

[0 11198791 Ω + Cminus1I] [

119887

120572] = [

0

Y] (17)

where Y = [1199101 1199102 119910

119873]119879 1 = [1 1 1]

119879 120572 =

[1205721 1205722 120572

119873]119879 and Ω(119896 119897) = 120593(119909

119896)119879

120593(119909119897) = 119870(119909

119896 119909119897)


119910 =

119873

sum

119894=1

120572119896119870(119909 119909

119896) + 119887 (18)




Peak-peakvalue(Am)


1 4 130 002 5 150 003 8 210 014 9 260 015 9 230 016 14 300 107 10 230 108 12 250 109 18 280 1110 14 270 1111 7 200 0112 10 286 10




6 Conclusion








Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119896 119910119896 119896 = 1 2 119873 119909

119896isin 119877119899



119910 (119909) = 119908119879

120593 (119909) + 119887 (13)


119899119887 isin 119877


min119908119890

119869 (119908 119890) =1

2119908119879

119908 +119862

2

119873

sum

119896=1

1198902

119896

119910119896= 119908119879

120593 (119909119896) + 119887 + 119890

119896 119896 = 1 2 119873

(14)





is

119871 (119908 119887 119890 120572)

= 119869 (119908 119890) minus

119873

sum

119896=1

120572119896119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896

(15)


120597

120597119908119871 = 0 997888rarr 119908 =

119873

sum

119894=1

120572119896120593 (119909119896)

120597

120597119887119871 = 0 997888rarr

119873

sum

119894=1

120572119896= 0

120597

120597119890119896

119871 = 0 997888rarr 120572119896= 119862119890119896

120597

120597120572119896

119871 = 0 997888rarr 119908119879

120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0

(16)

[0 11198791 Ω + Cminus1I] [

119887

120572] = [

0

Y] (17)

where Y = [1199101 1199102 119910

119873]119879 1 = [1 1 1]

119879 120572 =

[1205721 1205722 120572

119873]119879 and Ω(119896 119897) = 120593(119909

119896)119879

120593(119909119897) = 119870(119909

119896 119909119897)


119910 =

119873

sum

119894=1

120572119896119870(119909 119909

119896) + 119887 (18)




Peak-peakvalue(Am)


1 4 130 002 5 150 003 8 210 014 9 260 015 9 230 016 14 300 107 10 230 108 12 250 109 18 280 1110 14 270 1111 7 200 0112 10 286 10




6 Conclusion








Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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