Proceedings of the Annual Conference of the Prognostics and Health Management Society 2010

603 Pages·2011·49.29 MB·English

www.phmsociety.org
  ISBN
 -­‐
 978-­‐1-­‐936263-­‐03-­‐5

Proceedings of
The Annual Conference of the
Prognostics and Health Management
Society 2011

PHM’11
ISBN - 978-­‐1-­‐936263-­‐03-­‐5
Montreal, Quebec Canada
September 25 – 29, 2011
Edited
 by:

José
 R.
 Celaya,

Sankalita
 Saha,
 and

Abhinav
 Saxena
AnnualConferenceofthePrognosticsandHealthManagementSociety,2011
ii
AnnualConferenceofthePrognosticsandHealthManagementSociety,2011
Table of Contents
Full Papers
ABayesianProbabilisticApproachtoImprovedHealthManagementofSteamGeneratorTubes
KaushikChatterjeeandMohammadModarres 1
A Combined Anomaly Detection and Failure Prognosis Approach for Estimation of Remaining Useful Life in Energy
StorageDevices
MarcosE.Orchard,LiangTang,GeorgeJ.Vachtsevanos 8
AMobileRobotTestbedforPrognostics-EnabledAutonomousDecisionMaking
EdwardBalaban,SriramNarasimhan,MatthewDaigle,JoséR.Celaya,IndranilRoychoudhury,BhaskarSaha,Sankalita
Saha,KaiGoebel 15
AModel-basedPrognosticsMethodologyforElectrolyticCapacitorsBasedonElectricalOverstressAcceleratedAging
JoséR.Celaya,ChetanKulkarni,GautamBiswas,SankalitaSaha,KaiGoebel 31
AStructuralHealthMonitoringSoftwareToolforOptimization,DiagnosticsandPrognostics
SethS.Kessler,EricB.Flynn,ChristopherT.Dunn,MichaelD.Todd 40
AStudyontheparameterestimationforcrackgrowthpredictionundervariableamplitudeloading
SangHyuckLeem,DawnAn,SanghoKo,Joo-HoChoi 48
ATestbedforReal-TimeAutonomousVehiclePHMandContingencyManagementApplications
LiangTang,EricHettler,BinZhang,JonathanDeCastro 56
AdaptiveLoad-AllocationforPrognosis-BasedRiskManagement
BrianBole,LiangTang,KaiGoebel,GeorgeVachtsevanos 67
AnAdaptiveParticleFiltering-basedFrameworkforReal-timeFaultDiagnosisandFailurePrognosisofEnvironmental
ControlSystems
IoannisA.Raptis,GeorgeJ.Vachtsevanos 77
E2GK-pro: AnEvidentialEvolvingMultimodelingApproachforSystemsBehaviorPrediction
LisaSerir,EmmanuelRamasso,NoureddineZerhouni 85
BayesianfatiguedamageandreliabilityanalysisusingLaplaceapproximationandinversereliabilitymethod
XuefeiGuan,JingjingHe,RatneshwarJha,YongmingLiu 94
BayesianSoftwareHealthManagementForAircraftGuidance,Navigation,andControl
JohannSchumann,TimmyMbaya,OleMengshoel 104
CommercializationofPrognosticsSystemsLeveragingCommercialOff-The-ShelfInstrumentation,Analysis,andData
BaseTechnologies
PrestonJohnson 114
ComparisonofFaultDetectionTechniquesforanOceanTurbine
MustaphaMjit,Pierre-PhilippeJ.Beaujean,DavidJ.Vendittis 123
ComparisonofParallelandSingleNeuralNetworksinHeartArrhythmiaDetectionbyUsingECGSignalAnalysis
EnsiehSadatHosseiniRooteh,YouminZhang,ZhigangTian 134
ConditionBasedMaintenanceOptimizationforMulti-componentSystemsCostMinimization
ZhigangTian,YouminZhang,JialinCheng 143
CostComparisonofMaintenancePolicies
LeMinhDuc,TanCherMing 149
DecisionandFusionforDiagnosticsofMechanicalComponents
RenataKlein,EduardRudyk,EyalMasad 159
DefectsourcelocationofanaturaldefectonahighspeedrollingelementbearingwithAcousticEmission
BEftekharnejad,A.Addali,DMba 168
DerivingBayesianClassifiersfromFlightDatatoEnhanceAircraftDiagnosisModels
DanielL.C.Mack,GautamBiswas,XenofonD.Koutsoukos,DinkarMylaraswamy,GeorgeD.Hadden 175
iii
AnnualConferenceofthePrognosticsandHealthManagementSociety,2011
DesignforFaultAnalysisUsingMulti-partite,Multi-attributeBetweennessCentralityMeasures
Tsai-ChingLu,YiluZhang,DavidL.Allen,MutasimA.Salman 190
DistributedDamageEstimationforPrognosticsbasedonStructuralModelDecomposition
MatthewDaigle,AnibalBregon,IndranilRoychoudhury 198
ExperimentalPolymerBearingHealthEstimationandTestStandBenchmarkingforWaveEnergyConverters
MichaelT.Koopmans,StephenMeicke,IremY.Tumer,RobertPaasch 209
ExperimentswithNeuralNetworksasPrognosticsEnginesforPatientPhysiologicalSystemHealthManagement
PeterK.Ghavami,KailashKapur 222
ExploringtheModelDesignSpaceforBatteryHealthManagement
BhaskarSaha,PatrickQuach,KaiGoebel 231
FaultDiagnosisinAutomotiveAlternatorSystemUtilizingAdaptiveThresholdMethod
AliHashemi,PierluigiPisu 239
Fault-TolerantTrajectoryTrackingControlofaQuadrotorHelicopterUsingGain-ScheduledPIDandModelReference
AdaptiveControl
ImanSadeghzadeh,AnkitMehta,YouminZhang,Camille-AlainRabbath 247
FeatureSelectionandCategorizationtoDesignReliableFaultDetectionSystems
H.Senoussi,B.Chebel-Morello,M.Denaï,N.Zerhouni 257
From measurements collection to remaining useful life estimation: defining a diagnostic-prognostic frame for optimal
maintenanceschedulingofchokevalvesundergoingerosion
GiulioGola,BentH.Nystad 267
GearHealthThresholdSettingBasedOnaProbabilityofFalseAlarm
EricBechhoefer,DavidHe,PaulaDempsey 275
GearboxVibrationSourceSeparationbyIntegrationofTimeSynchronousAveragedSignals
GuicaiZhang,JoshuaIsom 282
HealthMonitoringofanAuxiliaryPowerUnitUsingaClassificationTree
WlamirO.L.Vianna,JoaoP.P.Gomes,RobertoK.H.Galvao,TakashiYoneyama,JacksonP.Matsuura 293
IdentificationofCorrelatedDamageParametersunderNoiseandBiasUsingBayesianInference
DawnAn,Joo-HoChoi,NamH.Kim 300
IntegratingProbabilisticReasoningandStatisticalQualityControlTechniquesforFaultDiagnosisinHybridDomains
BrianRicks,CraigHarrison,OleMengshoel 310
InvestigatingtheEffectofDamageProgressionModelChoiceonPrognosticsPerformance
MatthewDaigle,IndranilRoychoudhury,SriramNarasimhan,SankalitaSaha,BhaskarSaha,KaiGoebel 323
Investigationontheopportunitytointroduceprognostictechniquesinrailwaysaxlesmaintenance
MattiaVismara 334
Lithium-ionBatteryStateofHealthEstimationUsingAh-VCharacterization
DanielLe,XidongTang 361
Model-BasedPrognosticsUnderNon-stationaryOperatingConditions
MatejGašperin,PavleBoškoski,DaniJuricˇic´ 368
ModelingwavepropagationinSandwichCompositePlatesforStructuralHealthMonitoring
V.N.Smelyanskiy,V.Hafiychuk,D.G.Luchinsky,J.Miller,C.Banks,R.Tyson 375
OnlineEstimationofLithium-IonBatteryState-of-ChargeandCapacitywithaMultiscaleFilteringTechnique
ChaoHu,ByengD.Youn,JaesikChung,TaejinKim 385
Optimizationoffatiguemaintenancestrategiesbasedonprognosisresults
YibingXiang,YongmingLiu 398
PhysicsBasedPrognosticHealthManagementforThermalBarrierCoatingSystem
AmarKumar,BhavayaSaxena,AlkaSrivastava,AlokGoel 409
PhysicsbasedPrognosticsofSolderJointsinAvionics
AvisekhBanerjee,AshokK.Koul,AmarKumar,NishithGoel 419
iv
AnnualConferenceofthePrognosticsandHealthManagementSociety,2011
Pointprocessesforbearingfaultdetectionundernon-stationaryoperatingconditions
PavleBoškoski,DaniJuricˇic´ 427
PowerCurveAnalyticforWindTurbinePerformanceMonitoringandPrognostics
OnderUluyol,GirijaParthasarathy,WendyFoslien,KyusungKim 435
PrognosticsofPowerMOSFETsunderThermalStressAcceleratedAgingusingData-DrivenandModel-BasedMethod-
ologies
JoséR.Celaya,AbhinavSaxena,SankalitaSaha,KaiGoebel 443
StructuralIntegrityAssessmentUsingIn-SituAcousticEmissionMonitoring
MasoudRabiei,MohammadModarres,PaulHoffman 453
StudyonMEMSboard-levelpackagereliabilityunderhigh-Gimpact
JiuzhengCui,BoSun,QiangFeng,ShengKuiZeng 463
SymbolicDynamicsandAnalysisofTimeSeriesDataforDiagnosticsofadc-dcForwardConverter
GregoryBower,JeffreyMayer,KarlReichard 469
UsingtheValidatedFMEAtoUpdateTroubleShootingManuals: aCaseStudyofAPUTSMRevision
ChunshengYang,SylvainLétourneau,MarvinZaluski 479
Utilizing Dynamic Fuel Pressure Sensor For Detecting Bearing Spalling and Gear Pump Failure Modes in Cummins
PressureTime(PT)Pumps
J.ScottPflumm,JeffC.Banks 490
Poster Papers
ADiscussionofthePrognosticsandHealthManagementAspectsofEmbeddedConditionMonitoringSystems
RogerI.Grosvenor,PaulW.Prickett 502
***(no license)A new method of bearing fault diagnostics in complex rotating machines using multi-sensor mixtured
hiddenMarkovmodels
Z.S.Chen,Y.M.Yang,Z.Hu,Z.X.Ge 510
APrognosticHealthManagementBasedFrameworkforFault-TolerantControl
DouglasW.Brown,GeorgeJ.Vachtsevanos 516
DamageIdentificationinFrameStructures,UsingDamageIndex,BasedonH2-Norm
MahdiSaffari,RaminSedaghati,IonStiharu 527
EnhancedMultivariateBasedApproachforSHMUsingHilbertTransform
RafikHajrya,NazihMechbal,MichelVergé 532
FaultDetectioninNonGaussianProblemsUsingStatisticalAnalysisandVariableSelection
JoãoP.P.Gomes,BrunoP.Leão,RobertoK.H.Galvão,TakashiYoneyama 540
Fleet-widehealthManagementArchitecture
***(nolicense,othertextthere)MaximeMonnin,AlexandreVoisin,Jean-BaptisteLéger,BenoitIung 547
Improvingdata-drivenprognosticsbyassessingpredictabilityoffeatures
KamranJaved,RafaelGouriveau,RyadZemouri,NoureddineZerhouni 555
IntegratedRobustFaultDetection,DiagnosisandReconfigurableControlSystemwithActuatorSaturation
JinhuaFan,YouminZhang,ZhiqiangZheng 561
MultipleFaultDiagnosticStrategyforRedundantSystem
YangPeng,QiuJing,LiuGuanjun,LvKehong 572
OnlineAbnormalityDiagnosisforreal-timeImplementationonTurbofanEnginesandTestCells
JéroˇmeLacaille,ValerioGerez 579
ProficyAdvancedAnalytics: aCaseStudyforRealWorldPHMApplicationinEnergy
SubratNanda,XiaohuiHu 588
Author Index 595
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AnnualConferenceofthePrognosticsandHealthManagementSociety,2011
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AnnualConferenceofthePrognostics a ndHealthManagementSociety,2011
A Bayesian Probabilistic Approach to Improved Health
Management of Steam Generator Tubes
Kaushik Chatterjee and Mohammad Modarres
Center for Risk and Reliability, University of Maryland, College Park, MD, 20742, USA
[email protected]
[email protected]
ABSTRACT between 1975 and 2000. One such incident occurred in the
North Anna power station in 1987 when the plant reached
Steam generator tube integrity is critical for the safety and
its 100% capacity (US Nuclear Regulatory Commission,
operability of pressurized water reactors. Any degradation
1988). The cause of tube rupture was found to be fatigue,
and rupture of tubes can have catastrophic consequences,
caused by combination of alternating stresses resulting from
e.g., release of radioactivity into the atmosphere. Given the
flow-induced tube vibration and flaws resulting from
risk significance of steam generator tube ruptures, it is
denting of tubes at support plates.
necessary to periodically inspect the tubes using
nondestructive evaluation methods to detect and Given the risk significance of SGTRs, it is absolutely
characterize unknown existing defects. To make accurate necessary to periodically inspect the tubes using
estimates of defect size and density, it is essential that nondestructive evaluation methods in order to detect and
detection uncertainty and measurement errors associated quantify the severity of unknown existing defects.1 All
with nondestructive evaluation methods are characterized nondestructive evaluation methods have detection
properly and accounted for in the evaluation. In this paper uncertainty and measurement errors associated with them
we propose a Bayesian approach that updates prior that are a result of test equipment complexity, defect
knowledge of defect size and density with nondestructive attributes, as well as human error. These uncertainties and
evaluation data, accounting for detection uncertainty and errors need to be characterized properly and accounted for
measurement errors. An example application of the while estimating the size and density of defects.
proposed approach is then demonstrated for estimating
A defect of a given size might be detected only a certain
defect size and density in steam generator tubes using eddy
percentage of the time (out of total attempts during
current evaluation data. The proposed Bayesian probabilistic
nondestructive testing) depending on factors such as, noise
approach helps improve health management of steam
level, test probe sensitivity, test equipment repeatability and
generator tubes, thereby enhancing the overall safety and
human error. Hence, a defect has an associated probability
operability of pressurized water reactors.
of detection, which can be defined as the probability the
inspection will detect the defect of true size, , and is
denoted by POD (Kurtz, Heasler, & Anderso(cid:1)n, 1992).
1. INTRODUCTION The data from w(cid:2)h(cid:1)i(cid:3)ch POD curves are generated can be
categorized into two types: qualitative data, i.e., hit/miss;
Pressurized water reactors (PWR) use heat produced from
and quantitative data, i.e., signal response amplitude
nuclear fission in the reactor core to generate electricity. In
( ), where is signal response. The hit/miss data type
the process of generating electricity, steam generators (SG)
is(cid:1)(cid:4) (cid:6)b(cid:7)a.s(cid:1)ed on a (cid:1)b(cid:4)i nary process, i.e., whether a defect is
play an important role by keeping the reactor core at a safe
detected or not detected. The POD for this data type is
temperature and acting as the primary barrier between
calculated as the ratio of the number of successful detection
radioactive and non-radioactive sides of a nuclear power
over the total number of inspections performed for a
plant. Since SG tubes play such an important role, any
particular defect size, and is called the averaged POD.
degradation and rupture in the tubes can be catastrophic
Hit/miss data are obtained from test equipments such as
(Chatterjee & Modarres, 2011). According to the US
Sonic IR, and are very subjective in nature depending on
Nuclear Regulatory Commission (2010), there have been 10
operator experience (Li & Meeker, 2008), which induces
steam generator tube rupture (SGTR) occurrences in the US
uncertainty in the values of the POD. A logistic function is

Chatterjee, K. et al. This is an open-access article distributed under the 1 In this paper defect may indicate a crack, flaw, pit, or any other
terms of the Creative Commons Attribution 3.0 United States License, degradation in a structural component. Size may refer to either through-
which permits unrestricted use, distribution, and reproduction in any wall depth or surface length of a defect, unless specified. Density refers to
medium, provided the original author and source are credited. number of defects observed per unit volume.

1
[paper1]
AAnnnnuuaall CCoonnffeerreennccee ooff tthhee PPrrooggnnoossttiiccss aanndd HHeeaalltthh MMaannaaggeemmeenntt SSoocciieettyy,, 22001111
found to best-fit hit/miss data for modeling POD (Jenson, or random variation in measured values (measurement
Mahaut, Calmon & Poidevin, 2010). uncertainty).
The other type of POD data is more continuous in nature In the past, there have been efforts to model defect severity
and is a measure of the amplitude of signal response in structural components considering nondestructive
recorded by the nondestructive test equipment, e.g., evaluation uncertainties. Celeux, Persoz, Wandji, and Perrot
ultrasonic or eddy current. In the signal response data-based (1999) describe a method to model defects in PWR vessels
POD estimation method, the most important parameters are considering the POD and random error in measurements.
the inspection threshold (noise level) and the decision Yuan, Mao, and Pandey (2009) followed the idea of Celeux
threshold. The inspection threshold is chosen to account for et al. (1999), to propose a probabilistic model for pitting
the noise indications by test equipment, and responses corrosion in SG tubes considering the POD and random
above this threshold are considered for detection/non- error of the eddy current measurements. However, both
detection decisions. Decision threshold is often based on Celeux et al. (1999) and Yuan et al. (2009) did not consider
previous field inspections and knowledge of the noise the effect of systematic error or bias in measured defect
distribution, laboratory experience, and operator experience. sizes. Also, the POD has not been adjusted for measurement
The POD curve for signal response data type is modeled errors in their models. Further, they did not consider
using a cumulative log-normal distribution function uncertainties in the values of the POD, which can affect the
(Department of Defense, 1999; Jenson, et al., 2010), by defect severity estimates considerably.
determining the cumulative probability of responses (defect
This paper addresses some of the shortcomings of existing
signals) greater than the decision threshold. The selection of
literature and develops a Bayesian probabilistic approach for
decision threshold also determines the probability of false
modeling defect severity (size and density) in structural
call (or false positive).2 Hence, there is lot of uncertainty
components considering the detection uncertainty (i.e., POD
associated with the values chosen for inspection and
and associated uncertainty) and measurement errors (and
decision threshold, which lead to uncertainties in the values
associated uncertainty) associated with nondestructive
of the POD. In some cases, the signal response data is also
evaluation methods. The paper then presents example
converted into hit/miss data (Jenson et al., 2010) by using
application of the proposed approach for estimating defect
the decision threshold and averaged POD values are
severity in SG tubes using eddy current evaluation data.
estimated, which are then fitted into a logistic function.

The precision and accuracy of nondestructive test
equipment, and also the techniques used to analyze and
2. PROPOSED BAYESIAN APPROACH
process the test results can contribute to measurement
errors. For example, large volume of sensor data (such as The proposed Bayesian approach updates prior knowledge
ultrasound or digital images) are filtered, smoothed, of defect size and density with nondestructive evaluation
reduced, and censored into another form by subjectively data, considering the POD, measurement errors (systematic
accounting for only certain features of the data. Also, often and random), and associated uncertainties, to infer the
measurement models are used to convert the form of a posterior distributions of defect size and density. The
measured or observed data into the corresponding value of combined effect of POD, measurement errors, and
the reality of interest (i.e., defect size). Uncertainties associated uncertainties on measured defect sizes is captured
associated with data processing, model selection and human by a likelihood function. In this section, models for
error can contribute to measurement errors. Measurement measurement errors and POD function will be first defined;
error is defined as the difference between the measured and then the defect severity models will be presented, followed
the true value of a defect size. There are two components of by the likelihood functions and Bayesian inference
measurement error: systematic (bias) error and random equations.
(stochastic) error (Jaech, 1964; Hofmann, 2005). Systematic
The analysis of measurement error is based on assessing the
error or bias is a consistent and permanent deflection in the
deviation of the measured defect size from the actual or true
same direction from the true value (Hofmann, 2005).
defect size, as shown in Eq. (1):
Systematic error (bias) may indicate overestimation

(positive bias) or underestimation (negative bias). In most

nondestructive measurements, small defects are oversized
(cid:12)
and large defects are undersized (Kurtz et al., 1992; Wang (cid:9)(cid:10) (cid:11)(cid:1) (cid:13)(cid:1) (cid:2)1(cid:3)
where, is the measurement error, is measured and is
& Meeker, 2005). Random error arises due to the scattering
(cid:12)
the tru(cid:9)e(cid:10) defect size. Generally (cid:1)a linear regress(cid:1)ion
2 A nondestructive test equipment response interpreted as having detected a relationship of the form shown in Eq. (2) is used to model
flaw but associated with no known flaw at the inspection location measurement error (Kurtz et al., 1992; Jaech, 1964).
(Department of Defense, 1999).

(cid:12)
(cid:1) (cid:11)(cid:15)(cid:1)(cid:16)(cid:17)(cid:16)(cid:18)(cid:2)0,(cid:21)(cid:22)(cid:3) (cid:2)2(cid:3)
2 2
[pape r1]
AAnnnnuuaall CCoonnffeerreennccee ooff tthhee PPrrooggnnoossttiiccss aanndd HHeeaalltthh MMaannaaggeemmeenntt SSoocciieettyy,, 22001111
where, and are regression coefficients obtained through The marginal POD independent of random variables, and
a regres(cid:15)sion an(cid:17)alysis of , and is the random error , can be expressed as shown in Eq. (8), w@here,
(cid:12)
in measurement (scatterin(cid:1)g o(cid:6)f(cid:7) .th(cid:1)e data)(cid:18), which is assumed to (cid:21)ABC represents the PDF of random variable, .
follow a normal distribution with mean zero and standard (cid:15)(cid:2)(cid:21)ABC(cid:3) (cid:21)ABC
deviation (function of defect size). The regression

coefficients(cid:21) (cid:22)( ) are jointly measure of systematic error
=>?(cid:2)(cid:1)(cid:3)(cid:11) 5 5=>?(cid:2)(cid:1)|@,(cid:21)ABC(cid:3)H(cid:2)@(cid:3)(cid:15)(cid:2)(cid:21)ABC(cid:3);@;(cid:21)ABC (cid:2)8(cid:3)
or bias in me(cid:15)a s&ur(cid:17)ements. Distributions of bias parameters +JKLI
represent epistemic uncertainty in the chosen measurement
The likelihood function for detecting defect of true size, ,
error model. From Eqs. (1) and (2), the measurement error
given that the defect is detected , can then b(cid:1)e
can be expressed as:
expressed as shown in Eq. (9) (Celeux(cid:2)? et(cid:11) al1., (cid:3)1999):

(cid:29)(cid:30)(cid:22)(cid:31) (cid:22)!"#(cid:10) $%%#%
2(cid:2)(cid:1)|4(cid:3)O=>?(cid:2)(cid:1)(cid:3)
(cid:9)(cid:10) (cid:11)(cid:2)(cid:25)(cid:15)(cid:26)(cid:26)(cid:13)(cid:26)(cid:27)1(cid:3)(cid:26)(cid:1)(cid:26)(cid:26)(cid:16)(cid:28)(cid:17)(cid:16) (cid:18)(cid:25)(cid:2)(cid:26)0(cid:27),(cid:21)(cid:26)(cid:22)(cid:28)(cid:3) (cid:2)3(cid:3) N(cid:2)(cid:1)|? (cid:11)1(cid:3)(cid:11) (cid:2)9(cid:3)
Measurement error can then be expressed as a function of ="(cid:2)4(cid:3)
where, is the marginal POD that is a function of
measured defect size using Eqs. (1) and (3) as:
defect s=iz"e(cid:2) 4d(cid:3)istribution parameters only (independent of
defect size), and can be expressed as:
(cid:29)(cid:30)(cid:22)(cid:31) (cid:22)!"#(cid:10) $%%#%
(cid:25)(cid:15)(cid:26)(cid:26)(cid:13)(cid:26)(cid:26)1(cid:27)(cid:26)(cid:26)(cid:12)(cid:26)(cid:26)(cid:28)(cid:17) (cid:18)(cid:25)(cid:2)(cid:26)0(cid:27),(cid:21)(cid:26)(cid:22)(cid:28)(cid:3) S
(cid:9)(cid:10) (cid:11)' ((cid:1) (cid:16) (cid:16) (cid:2)4(cid:3)
(cid:15) (cid:15) (cid:15)
The probability density function (PDF) of the measurement ="(cid:2)4(cid:3)(cid:11)Pr(cid:2)? (cid:11)1(cid:3)(cid:11)5=>?(cid:2)(cid:1)(cid:3)2(cid:2)(cid:1)|4(cid:3);(cid:1) (cid:2)10(cid:3)
error can then be defined using a normal distribution with T
During nondestructive measurements true defect sizes are
mean as the bias, , standard deviation as that of random
unknown, while the only known quantities are the measured
error, , and meas*ur(cid:22)ement error as random variable.
+, defect sizes and number of detections. The likelihood
(cid:10) function of true defect sizes corresponding to measurements
consisting of exact defect sizes (using Eq. 9) considering
(cid:21)(cid:22) (cid:12)
-(cid:2)(cid:9)(cid:10)(cid:3)(cid:11)./*(cid:22), 0 (cid:2)5(cid:3) measurement Uer$rors can be represented as:
(cid:15)
Assume that true defect size, , is treated as random
variable with the PDF, , w(cid:1)here is the vector of the !Z(cid:12)
ePrDroFr pcaarna tmheente rbse. eDxepfreecst2s es(cid:2)id(cid:1)z |ea4 sP (cid:3)sDhoFw cno nins4i dEeqr.i n(6g) .m easurement N (cid:2) (cid:1) $ V(cid:22)WE|4 (cid:3)(cid:11)X="(cid:2)14(cid:3) Y ! Z (cid:12) [ (cid:30) \ ] 9 5: = >?(cid:2)(cid:1)(cid:30)(cid:12)(cid:13)(cid:9) (cid:10)(cid:3)26(cid:2)(cid:1)(cid:30)(cid:12)(cid:13)(cid:9) (cid:10)(cid:3)|48-(cid:2)(cid:9)(cid:10) (cid:3) ; ((cid:9)1(cid:10)1)
Nondestructive measurements are in most cases interval or
left censored, in which case the likelihood function of true
(cid:12) defect sizes corresponding to measurements consisting of
2(cid:2)(cid:1)|4(cid:3)(cid:11) 526(cid:2)(cid:1) (cid:13)(cid:9)(cid:10)(cid:3)748-(cid:2)(cid:9)(cid:10)(cid:3);(cid:9)(cid:10) (cid:2)6 (cid:3)
defects within the interval (or in a left censored
9: (cid:12)
Uin(cid:30)t!eEr,^val) (Cook, Duckwor_tEhF, Kaiser, Meeker & Stephenson,
All the defects in a structure are not detected during
2003), can be expressed as shown in Eq. (12).
nondestructive testing. The detection of a defect depends on

its size and is represented by the POD curve. The POD of a
di ne fEeqc.t (o7f) :s ize, (cid:1), can be represented by a function as shown N^(cid:2)(cid:1)(cid:30)!E|4(cid:3)(cid:11) `="1(cid:2)4(cid:3)5(cid:22)(cid:22)a(cid:12) ba(cid:12)c95:=>?(cid:2)(cid:1) (cid:12)(cid:13)(cid:9)(cid:10)(cid:3)26(cid:2)(cid:1) (cid:12)(cid:13)(cid:9)(cid:10)(cid:3)|48- (cid:2)(cid:9)(cid:10)(cid:3);(cid:9)(cid:10); (cid:1) (cid:12) d(!1e(cid:12)f2g,a)
Therefore, the likelihood function of true defect sizes
= >?(cid:2)(cid:1)|@,(cid:21)ABC(cid:3)(cid:11)D(cid:2)(cid:1),@,(cid:1)EF(cid:3)(cid:16)(cid:18)ABC(cid:2)0,(cid:21)ABC(cid:3) (cid:2)7(cid:3) corresponding to total measurements consisting of defect
where, is the POD function, is the detection size intervals each with certain number of defects ((cid:15) in
thresholDd(cid:2),(cid:1) ,@ i,s(cid:1) EvFe(cid:3)ctor of parameters of (cid:1)thEFe POD function, interval), and exact defect sizes can then be exUp(cid:30)(cid:12)!reE,s^s ed
and is@ the random error, which represents uncertainty a_EsF s hown in Eq. (1U3$(cid:12)).
in th(cid:18)eA BCPOD data and is assumed to follow a normal
distribution with mean zero and standard deviation
(function of true defect size). The POD function is sele(cid:21)cAtBeCd N(cid:2)(cid:1)|4(cid:3)(cid:11)∏(cid:10)^ \]iAj](cid:2)k(cid:3)l(cid:22)(cid:22)a(cid:12)a(cid:12)b cl9:=>?(cid:2)(cid:1) (cid:12)(cid:13)(cid:9)(cid:10)(cid:3)26(cid:2)(cid:1) (cid:12)(cid:13)(cid:9)(cid:10)(cid:3)|48 -(cid:2)(cid:9)(cid:10)(cid:3);(cid:9)(cid:10); (cid:1) (cid:12) m ! e(cid:12)fg,a
based on the type of data, e.g., hit/miss or signal response as
discussed in Section 1. Joint distribution of the parameters 1 Un(cid:12) (cid:12) (cid:12) (13)
of the POD function, , represents the epistemic OX=;(cid:2)4(cid:3)YUn(cid:12)∏o(cid:11)1l(cid:9)(cid:15)=>?(cid:2)(cid:1)o(cid:13)(cid:9)(cid:15)(cid:3)26(cid:2)(cid:1)o(cid:13)(cid:9)(cid:15)(cid:3)|48-(cid:2)(cid:9)(cid:15)(cid:3);(cid:9)(cid:15)
uncertainty associated withH (cid:2)th@e(cid:3) choice of the POD function.
3 3
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The posterior defect size distribution parameters can then be where, and are parameters of gamma distribution.

estimated using Bayesian inference as: Then thye] posteryiozr distribution of defect density can be
expressed as shown in Eq. (19).

N(cid:2)?(cid:1)q(cid:1)|4(cid:3)pT(cid:2)4(cid:3)
p ] (cid:2)4|?(cid:1)q(cid:1)(cid:3)(cid:11)lkN(cid:2)?(cid:1)q(cid:1)|4(cid:3)pT(cid:2)4(cid:3);4 (cid:2)14(cid:3) p](cid:2)u(cid:3)(cid:11)-(cid:1)(cid:15)(cid:15)(cid:1)(cid:2)u|t(cid:16)y],U(cid:16)yz(cid:3) (cid:2)19(cid:3)
A MATLAB routine was developed to implement this entire
where, is posterior distribution of defect size
Bayesian approach for estimating defect severity in
parametper]s(cid:2) 4|a?n(cid:1)d q(cid:1)(cid:3) is prior distribution of the
structural components. The proposed Bayesian approach
parameters. The popsTt(cid:2)e4rio(cid:3)r defect size parameters obtained
considers systematic (bias) and random error in
from Bayesian inference can then be used to estimate the
nondestructive measurements; suitably adjusts measurement
corresponding marginal POD values (Eq. 10).
errors in POD; considers uncertainty in POD values;
The likelihood of observing number incorporates prior knowledge of defect size and density;
of defects given actualU (cid:12)n (cid:2)u(cid:11)mUb$(cid:12)er(cid:16) o∑f(cid:10)^ \d]eUfe(cid:30)(cid:12)!cEt,s^ (cid:3)can be provides a framework for updating probability distributions
expressed by a biUnomial function (detection process is of defect model parameters when new data become
binary, i.e., either detection or no detection), as shown by available; and is applicable to exact, interval, and censored
Eq. (15): measurements.

(cid:12) U !(cid:12) !s!(cid:12) 3. APPLICATION OF PROPOSED BAYESIAN
N(cid:2)U |U(cid:3)(cid:11)/ (cid:12)0X="(cid:2)4(cid:3)Y X1(cid:13)="(cid:2)4(cid:3)Y (cid:2)15(cid:3) APPROACH TO EDDY CURRENT DATA
U
where, is the marginal POD value corresponding to
An example application of the proposed Bayesian approach
posterio=r" (cid:2)d4ef(cid:3)ect size parameters. In Eq. (15), the actual
is presented in this section for estimating flaw severity in
number of defects, , is unknown whereas and are
(cid:12) SG tubes using eddy current measurements of flaw sizes
known. The actual nUumber of defects can bUe estima=t"e(cid:2)d4 u(cid:3)sing
(through-wall depth). In this section, we first model POD
Bayesian inference as shown in Eq. (16):
and measurement error for eddy current evaluation using

available data from literature, and then use the proposed
(cid:12) Bayesian approach to estimate the posterior distributions of
p](cid:2)U|U(cid:12)(cid:3)(cid:11)∑N!(cid:2)NU(cid:2)U|(cid:12)U|(cid:3)Up(cid:3)Tp(cid:2)TU(cid:2)(cid:3)U(cid:3) (cid:2)16(cid:3) flaw size and density.
where, is posterior distribution of actual number The eddy current measurement error is assessed in this
of defecpts] (cid:2)gUiv|Ue(cid:12)n(cid:3) the observation, , and is the prior paper by a Bayesian regression approach (Azarkhail &
distribution of number of defects.U T(cid:12)he prpioTr(cid:2) Udi(cid:3)stribution of Modarres, 2007) in light of available data from literature
number of defects can be estimated from a Poisson function, (Kurtz, Clark, Bradley, Bowen, Doctor, Ferris & Simonen,
which gives the likelihood of observing total number of 1990). The regression result is illustrated by Figure 1 with
defects in a volume , given prior defect dUensity as shown the 50% regression line representing the bias corresponding
in Eq. (17). Here tPoisson distribution is useud because to mean values of the parameters and of Eq. (3). The
defects are assumed to occur with the same average 95% uncertainty bounds of Figur(cid:15)e 1 co(cid:17)rresponds to the
intensity and independent of each other. random error with a constant standard deviation, . The
parameters and obtained through Bayesian reg(cid:21)ression
were then (cid:15)u,s(cid:17)ed in(cid:21) Eq. (5) to estimate the PDF of
! measurement error as a function of measured flaw size.
svw(cid:2)ut(cid:3)
pT(cid:2)U(cid:3)(cid:11)n (cid:2)17(cid:3)
U! In order to derive the POD model, it was assumed in this
The posterior distribution of actual number of defects (Eq.
paper that eddy current signal response data were converted
16) can then be used to obtain the posterior defect density.
into equivalent hit/miss. The POD curve can then be
The standard conjugate prior employed for Poisson
expressed by a logistic function of the form as shown in Eq.
distribution likelihood (Eq. 17) is a two-parameter gamma
(20) (Yuan et al., 2009):
distribution (Simonen, Doctor, Schuster, & Heasler, 2003),

in which case the posterior has the same functional form as
the gamma distribution. Assume that prior distribution of ]}$b~c~(cid:127)
defect density is: =>?(cid:2)(cid:1)|{],{z,(cid:1)EF(cid:3)(cid:11)|1(cid:13)]}$~c(cid:2),b~(cid:127)b,g€(cid:3)(cid:16)(cid:18)ABC(cid:2)0,(cid:21)ABC(cid:3) 2(cid:129)‚ (cid:1)ƒ(cid:1)EF…
0 (cid:129)qDn‚ „ (o2(cid:7)n0 )

where, is flaw size, is threshold size for detection,

pT (cid:2) u (cid:3)(cid:11)-(cid:1)(cid:15)(cid:15)(cid:1)(cid:2)u|y],yz(cid:3) (cid:2)18(cid:3) raanndd o{mz (cid:1) aererr olro,g iswtihci cfhu(cid:1) nEiFcst ioans spuamraemd etteor s,f oalnlodw (cid:18) AaB Cn iosr mt{ha]el
distribution with mean zero and standard deviation . A
(cid:21)ABC
4 4
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