Research Article Markov Chain Models for the Stochastic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 108386 13 pageshttpdxdoiorg1011552013108386

Research ArticleMarkov Chain Models for the Stochastic Modeling ofPitting Corrosion

A Valor1 F Caleyo1 L Alfonso2 J C Velaacutezquez3 and J M Hallen1

1 Departamento de Ingenierıa Metalurgica IPN-ESIQIE UPALM sn Edificio 7 Zacatenco 07738 Mexico DF Mexico2 Universidad Autonoma de la Ciudad de Mexico 09790 Mexico DF Mexico3 Departamento de Ingenierıa Quımica Industrial ESIQIE-IPN UPALM Edificio 7 Zacatenco 07738 Mexico DF Mexico

Correspondence should be addressed to A Valor almavalorgmailcom

Received 1 February 2013 Revised 19 April 2013 Accepted 3 May 2013

Academic Editor Wuquan Li

Copyright copy 2013 A Valor et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The stochastic nature of pitting corrosion of metallic structures has been widely recognized It is assumed that this kind ofdeterioration retains no memory of the past so only the current state of the damage influences its future development Thischaracteristic allows pitting corrosion to be categorized as aMarkov process In this paper two differentmodels of pitting corrosiondeveloped using Markov chains are presented Firstly a continuous-time nonhomogeneous linear growth (pure birth) Markovprocess is used tomodel external pitting corrosion in underground pipelines A closed-form solution of the system of Kolmogorovrsquosforward equations is used to describe the transition probability function in a discrete pit depth space The transition probabilityfunction is identified by correlating the stochastic pit depth mean with the empirical deterministic mean In the second modelthe distribution of maximum pit depths in a pitting experiment is successfully modeled after the combination of two stochasticprocesses pit initiation and pit growth Pit generation is modeled as a nonhomogeneous Poisson process in which induction timeis simulated as the realization of aWeibull process Pit growth is simulated using a nonhomogeneousMarkov process An analyticalsolution of Kolmogorovrsquos system of equations is also found for the transition probabilities from the firstMarkov state Extreme valuestatistics is employed to find the distribution of maximum pit depths

1 Introduction

Localized corrosion specifically pitting corrosion of metalsand alloys constitutes one of the main failure mechanismsof corroding structures such as pressurized containers andpipes Pits cause failure through perforation of the compo-nent wall In other cases pits become nucleation centers forcracks [1] In the oil and gas industry pitting corrosion is amajor problem especially for transporting pipelines [2]

Pitting corrosion comprises two main processes pit ini-tiation and stable pit growth (pit repassivation is not consid-ered in this paper) It is accepted that pit initiation can be aconsequence of the breakdown of the passive layer caused byrandom fluctuations in local conditions This process takessome time usually called induction (nucleation or initiation)period [3] Passive layer breakdown followed by localizedmetal dissolution is the most commonmechanism of pittingcorrosion However pitting can also occur as the result of

the active dissolution of certain regions of the material at itssurface such as nonmetallic inclusions which are susceptibleand dissolve faster than the rest of the surface [4] in this casethe pitting induction time is typically shorter

After a pit has nucleated it can repassivate (stop growing)immediately or grow and then repassivate This process isregarded as metastable pitting If a pit is able to grow indef-initely it becomes a stable growing pit [5] Those pits thatreach the stable growth regime become part of a pit popu-lation that shows a remarkable stochastic behavior [6 7]

The time evolution of pit depth due to corrosion is oftenexpressed as a power function of time [7ndash11] 119863(119905) = 120572(119905 minus

119905ini)120573 where 119905 is the exposure time 120572 and 120573 are parameters

related to the corrosion process and 119905ini stands for pit ini-tiation time This same function has been found to governthe pitting corrosion growth in stainless and mild steels andaluminum alloys [11 12]

2 Mathematical Problems in Engineering

Pitting corrosion occurs in a wide range of metals andenvironments This fact points out to the universality of thisphenomenon and suggests that randomness is an inherentand unavoidable characteristic of this damage over time sothat stochastic models are better suited to describe pittingcorrosion than deterministic ones Localized corrosion can-not be explained without assuming stochastic points of viewdue to the large scatter in the measurable parameters such ascorrosion rate maximum pit depth and time to perforation[13] Many variables of the metal-environment system suchas alloy composition andmicrostructure and composition ofthe surrounding media and temperature are all involved inthe pitting process [4] Such complexity imposes the devel-opment of theoretical models and simulation tools for a bet-ter understanding of the outcome of the pitting corrosionprocess These tools help predict more accurately the timeevolution of pit depth in corroding structures as the key factorin structural reliability assessment

Another important characteristic of the pitting corrosionprocess that is worth noting is the time and pit-depthdependence of the corrosion rate [6 7] It has been establishedthat for a given pit the growth rate decreases with time whilefor pits with equal lifetimes the corrosion rate is larger fordeeper ones

Provan and Rodriguez [14] are amongst the first authorsto use a nonhomogenous Markov process to model pit depthgrowth In their model the authors divided the space ofpossible pit depths into discrete non overlapping statesand numerically solved the system of Kolmogorovrsquos forwardequations (1) for the transition probabilities 119901119894119895(119905) betweendamage states 119894 and 119895 However Provan and Rodriguez mod-eled pitting without taking into account the pit generationprocess and proposed an expression for the intensities 120582119894(119905)of the process that conveyed no physical meaning They didnot discuss the method used to solve the system of equationseither This has made it impossible to reproduce their results(deeper discussion on this topic can be found in [15])

119889119901119894119895 (119905)

119889119905=

120582119895minus1 (119905) 119901119894119895minus1 (119905) minus 120582119895 (119905) 119901119894119895 (119905) 119895 ge 119894 + 1

120582119894 (119905) 119901119894119894 (119905)

(1)

Other authors who made use of Markov chains to modelcorrosionwereMorrison andWorthingham [16]Making useof a continuous time birth process with linear intensity 120582they determined the reliability of high-pressure corrodingpipelines For their purpose these authors divided the spaceof the load-resistance ratio into discrete states and numeri-cally solved the Kolmogorovrsquos equations in order to find theintensities of transition between damage states Afterwardsprobability distribution function of the load-resistance ratiowas estimated and compared to the distribution obtainedfrom field measurements Worthinghamrsquos model was furtherimproved byHong [17] who used an analytical solution to thesystemofKolmogorovrsquos equations for the same homogeneouscontinuous type of Markov process and derived the process

probability transition matrix in order to evaluate the prob-ability of failure Hong also investigated the influence of pitdepth on the load-resistance ratioThemerits and limitationsof these two models are discussed in detail in [15 18]

In recent years modeling of pitting corrosion withMarkov chains has shown new advances For exampleBolzoni et al [19] have modeled the first stages of localizedcorrosion using a continuous-time three-state Markov pro-cessTheMarkov states of themetal surface are passivitymetastability and stable pit growth On the other hand Timashevand coworkers [20] formulated a model based on the useof a continuous-time discrete-state pure birth homogenousMarkov process for stochastically describing the growth ofcorrosion-caused metal loss The goal was to assess theconditional probability of pipeline failure and to optimizethe maintenance of operating pipelines In their model theintensities of the process were calculated by iteratively solvingthe proposed system of Kolmogorovrsquos forward equationsThedrawbacks of Timashevrsquosmodel are discussed in detail in [18]

In the present paper a review of the Markov modelsdeveloped by the authors in an attempt to describe pittingcorrosion is presentedThe first model intends to solve a pro-blem that is crucial in reliability-based pipeline integritymanagement the accurate estimation of future pit depthand growth rate distributions from a (single) measured orassumed pit depth distribution It has been recognized [21]that such estimation can be carried out only if oversimpli-fications are made or if additional information besides thepit depth distribution is available In the developed modelit is postulated that in the case of external pitting corrosionin underground pipelines this additional information can beattained from the available predictivemodels for pit growth asa function of the soil characteristics [8 22] A model for pitgrowth previously developed by the authors has been used toperformMonte Carlo simulations in order to predict the dis-tribution of maximum pit depths as a function of the pipelineage and the physicochemical characteristics of the soil [9]

A nonhomogenous linear growth pure birth Markovprocess with discrete states in continuous time is used tomodel external pitting corrosion in underground pipelinesThe system of forward Kolmogorovrsquos equations (1) is ana-lytically solved using the binomial closed-form solution forthe transition probabilities between Markov states in a giventime interval This Markov framework is used to predict thetime evolution of the pitting depth and rate distributionsTheanalytical solution becomes available under the assumptionthat Markov-derived stochastic mean of the pit depth distri-bution is equal to the deterministic mean of the distributionobtained through Monte Carlo simulations This assumptionis made for different exposure times and different soil classesdefined according to soil physicochemical characteristics thatare easy to measure in the field In this way the transitionprobabilities are obtained as a function of soil propertiesfor a given time point and the corrosion rate and futurepit depth distributions are predicted Real-life case studiesare presented to illustrate the proposed Markov model Themain advantage of this model resides in its capability ofcorrectly predicting the time evolution of pit depth and ratedistributions over time

Mathematical Problems in Engineering 3

One of the main goals of reliability analysis is to estimatethe risk of perforation of in-service components produced bythe deepest pit Extreme value statistics is commonly usedtogether with pit depth growth modeling to predict the riskof failure of in-service components and structures [23 24]

It is recognized [6 7] that the deepest pits grow at higherrates than the rest of the defects right from the beginning ofthe corrosion process so that the maximum pit depths arecommonly sampled from an exponential parent distributionthat constitutes the right tail of the pit depth distribution ofthe whole pit population [7] Besides that in the corrosionliterature [13 14 24 25] it is a well-established fact that theGumbel extreme value distribution [26 27] is a good fit tothe maximum pit depth distribution obtained by measuringthe maximum pit depth on several areas The cumulativefunction of the Gumbel distribution with location parameter120583 and scale parameter 120590 is

119866 (119909) = exp [minus exp (minus(119909 minus 120583

120590))] minusinfin lt 119909 lt infin (2)

It is important to underline that the previouslymentionedMarkov model has proved successful in modeling the timeevolution of the entire pit depth population but failed tocorrectly describe the evolution of pit-depth extremes Thesecond model presented in this work focuses on the simula-tion of the time evolution of extreme pit depths The modelis based on the stochastic description of pitting corrosiontaking into account pit initiation and growth A nonho-mogeneous Poisson process is used to model pit initiationThe distribution of pit nucleation times is simulated usinga Weibull process Pit depth growth is also modeled as anonhomogeneous Markov process The system of forwardKolmogorovrsquos equations (1) is solved analytically for thetransition probability from the first state to any 119895th stateduring a given time interval [28] From this solution thecumulative distribution function of pit depth for the one-pit case can be found This distribution function has anexponential character and corresponds to the parent distri-bution from which the extremes can be readily drawn Theextreme depths distribution for the119898-pit case is obtained bythe multiplication of the 119898 parent cumulative distributionfunctions of the pits populationThis stochastic model is ableto predict the time evolution of the probability distribution ofmaximum pit depths

2 Stochastic Models of Pitting Corrosion

In this section two different Markov chain models arepresented to describe pitting corrosion The first one isfocused on the description of time evolution of pit depth andrate distributions in operating underground pipelines Thesecond model deals with the description of maximum pitdepths when multiple corrosion pits are taken into accountin a laboratory (controlled) pitting corrosion experiment

In the following only the definitions relevant to the focusof the presented models are given The reader is referred to[28 29] for the formalism theory of Markov processes

In both models the possible pit depths constitute theMarkov space The material thickness (along which pits

propagate) is divided into 119873 discrete Markov states Thecorrosion damage (pit) depth at time 119905 is represented by adiscrete random variable 119863(119905) such that 119875119863(119905) = 119894 = 119901119894(119905)with 119894 = 1 2 119873 Furthermore it is postulated that theprobability that the damage that is at the 119894th state at themoment 119905 increases by one state in a very small interval oftime 120575119905 is expressed as 120582119894(119905)120575119905 + 119900(120575119905) For a continuous-time nonhomogenous linear growth Markov process withintensities 120582119894(119905) = 119894120582(119905) the probability119901119894119895(119905) that the processin state 119894 will be in state 119895 (119895 ge 119894) at some later time obeys thesystem of Kolmogorovrsquos forward equations presented in (1)In this infinitesimal transition scheme120582(119905) can be interpretedas the jump frequency between the 119894th to the (119894 + 1)th statesduring the time interval [119905 119905+120575119905]Thismeans that the numberof states transited by the corrosion pit in a short time interval[0 119905] can be written as

120588 (119905) = int

119905

0120582 (120591) 119889120591 (3)

It is not difficult to find out that the functions 120582(119905) and120588(119905) have direct physicalmeaningwhenMarkov processes areused to model pitting corrosion They are related to the pitgrowth rate and pit depth respectively

21 Markov Chain Modeling of Pitting Corrosion Depth andRate in Underground Pipelines In [28 page 304] it is shownthat for a Markov process defined by the system of (1) theconditional probability 119901119898119899(1199050 119905) = 119875119863(119905) = 119899 | 119863(1199050) = 119898

of transition from the119898th state to the 119899th state (119899 ge 119898) in theinterval (1199050 119905) can be obtained analytically and has the form

119901119898119899 (1199050 119905)

= (119899 minus 1

119899 minus 119898) 119890minus120588(119905)minus120588(1199050)119898(1 minus 119890

minus120588(119905)minus120588(1199050))119899minus119898

(4)

where 120588(119905) is defined by (3)Equation (4) shows that the increase in pit depth over

Δ119905 = 119905 minus 1199050 follows a negative binomial distributionNegBin(119903 119901) with parameters 119903 = 119898 and 119901 = 119901119904 =

119890minus120588(119905)minus120588(1199050) From the transition probability 119901119898119899(1199050 119905) it ispossible to estimate the probability distribution function119891(120592)of the pitting corrosion rate 120592 over the time interval Δ119905 whenthe pit depth is at the119898th state as

119891 (120592119898 1199050 119905) = 119901119898 (1199050) 119901119898119898+120592Δ119905 (1199050 119905) Δ119905 (5)

From the distribution function119891(120592119898 1199050 119905) it is straight-forward to derive the pitting rate probability distributionassociated with the entire pit population (all possible depths)as

119891 (120592 1199050 119905) =

119873

sum

119898=1

119891 (120592119898 1199050 119905) (6)

Until this point we have the transition probabilitiesfrom any state 119898 to the state 119899 in the interval (1199050 119905) givenby (4) The corrosion rate distribution can also be derivedthrough (6) In principle if the probability distribution of the


corrosion depth at 1199050 is known that is 119875119863(1199050) = 119898 =

119901119898(1199050) the pit depth distribution at any future moment intime can be estimated using

119901119899 (119905) =

119899

sum

119898=1

119901119898 (1199050) 119901119898119899 (1199050 119905) (7)

For the case of buried pipelines the initial probabilitydistribution of pit depths 119901119898(1199050) can be obtained if thecorrosion damage in the pipeline is monitored using in-lineinspection (ILI) being 1199050 the time of the inspection Thevalues of the probabilities 119901119898 can be estimated from the ratioof the number of corrosion pits with depths in the 119898th stateto the total number of observed pits

It is evident from (4) that the transition probabilities119901119898119899(1199050 119905) depend on the functions 120582(119905) and 120588(119905) At thispoint physically sound expression for 120588(119905) and 120582(119905) should beproposed in order to estimate the evolution of the pit depthand corrosion rate distributions For that the knowledgeabout the pitting corrosion process in soils must be used Itis postulated that the stochastic mean pit depth 119872(119905) can beassumed to be equal to the deterministic mean K(119905) of thepitting process which can be estimated from the observedevolution of the average pit depth over time as

119872(119905) = K (119905) (8)

This equality is true under certain simple assumptionsthat are explained in [29] In summary a sufficient conditionfor the equality of the stochastic and deterministic means isthat for any positive integer 119902 the structure of the processstarting from 119899 individuals is identical to that of the sum of 119902separated systems each starting from 119899

Cox andMiller [29] have shown that if the initial damagestate is 119899119894 at 119905 = 119905119894 so that119863(119905119894) = 119899119894 then the time-dependentstochastic mean119872(119905) = 119864[119863(119905)] of the linear growthMarkovprocess can be expressed as

119872(119905) = 119899119894119890120588(119905minus119905119894) (9)

If we consider that a power function is an accurate deter-ministic representation of the pit growth process one canpostulate that the deterministic mean pit depth at time 119905 is[8]

K (119905) = 120581(119905 minus 119905sd)] (10)

where 120581 and ] are the pitting proportionality and exponentparameters respectively and 119905sd is the starting time of the pit-ting corrosion process In systemswhere passivity breakdownandor inclusion dissolution are the prevalent mechanismsfor pit initiation 119905sd would represent the initiation time ofstable pit growth In the case of underground pipelines thisparameter corresponds to the total elapsed time frompipelinecommissioning to coating damage plus the time periodwhen the cathodic protection is still effective preventing orattenuating external pitting corrosion after coating damage

The increase of the deterministic mean pit depth in thetime interval Δ119905 is

ΔK (119905) = 120582 (119905)K (119905) Δ119905 (11)

where 120582 can be interpreted as the deterministic intensity(rate) of the process The pitting rate obtained by taking thetime derivative ofK(119905) (10) obeys the functional form of (11)with 120582 = ]120591 and 120591 = 119905 minus 119905sd as

119889K (119905)

119889119905=]

120591K (119905) (12)

If 120575119905 represents the stay time of the corrosion pit in thefirst state of the chain then 119899119894 = 1 during the time intervalbetween 119905sd and 119905sd + 120575119905 If 120575119905 is significantly less than thesimulation time span it is easy to show from (8)ndash(10) that thevalue of the function 120588(119905) can be approximated as

120588 (119905) = ln [120581(119905 minus 119905sd)]] (13)

From this taking into account (3) it follows that

120582 (119905) =]

119905 minus 119905sd (14)

Note that the intensity of the Markov process 120582(119905) isinversely proportional to the exposure time 120591 as it is the caseof the deterministic intensity 120582(119905) according to (11) and (12)

One can substitute the expression for 120588(119905) from (13)into (4) and to show that the probability parameter 119901119904 =

119890minus120588(119905)minus120588(1199050) can be expressed as

119901119904 = (1199050 minus 119905sd119905 minus 119905sd

)

]

119905 ge 1199050 ge 119905sd (15)

Suppose that a pit is in the state119898 at 1199050 Let119864[119899minus119898]Δ119905 bethe average damage rate in the time interval (1199050 1199050+Δ119905) It canbe shown [19] that the instantaneous pitting rate 120592 predictedby the stochastic model is

120592 (119898 1199050) =119864 [119899 minus 119898]

Δ119905997888997888997888997888997888rarrΔ119905rarr0

]119898

1199050

(16)

The stochastically predicted instantaneous damage rateagrees with the deterministic rate given by (12) This coin-cidence is a sign of the adequacy of the proposed Markovpitting-corrosion model

For the case of underground pipelines the pitting cor-rosion damage evolution can be undertaken as follows Themeasured or assumed pit depth probability distribution at 1199050is used as the initial corrosion damage distribution 119901119898(1199050)where 1199050 is the time of the pit depth measurement The valueof the probabilities 119901119898 is estimated from the ratio of thenumber of corrosion pits with depths in the 119898th state tothe total number of observed pits The transition probabilityfunction 119901119898119899(1199050 119905) can be identified from (4) and (15) if theparameters 119905sd and ] are known

The estimation of the pitting parameters is possiblethanks to the previously developed [8 9] predictive modelfor localized corrosion in buried pipelines which relates thephysicochemical conditions of the pipe and soil to parameters119905sd 120581 and ] The model is based on a multivariate nonlinearregression analysis using (10) with the field-measured maxi-mumpit depths as the dependent variable to be predicted and


Table 1 Estimated pit initiation times for underground pipelines(from [8])

Soil class 119905sd (years)Clay 30Clay loam 31Sandy clay loam 26All 29

the pipe age and soil-pipe characteristics as the independentones The experimental details and corrosion data used toproduce the model can be found elsewhere [8 9] From thismodel the pitting initiation time was estimated to have thevalues displayed in Table 1 for different soil classes In Table 1class ldquoAllrdquo corresponds to a general class that includes all thesoils found in the field survey carried out in South Mexico[8] The pitting parameters 120581 and ] in (10) were estimated aslinear combinations of soil and pipe characteristics [8] for thedifferent soil classes

Later on Caleyo et al [9] used a Monte Carlo simulationbased on (10) and on the obtained linear combinations ofenvironmental variables for the pitting parameters to predictthe time evolution of the maximum pit depth probabilitydistribution in underground pipelines The measured dis-tributions of the considered soil-pipe independent variableswere used as inputs in Monte Carlo simulations of the pittingprocess for each soil class to produce the simulated extremepit depth distribution for each soil category and exposuretime consideredThe reader is referred to [9] for details aboutthe used pipe-soil properties distributions and the MonteCarlo simulation framework The Monte Carlo producedextreme pit depth values for each soil class at different timepoints were fitted to the generalized extreme value (GEV)distribution [30]

GEV (119909) = expminus[1 + 120577 (119909 minus 120583

120590)]

minus1120577

(17)

where 120577120590 and120583 are the shape scale and location parametersof the distribution respectively

Figure 1(a) shows the time evolution of the GEV distribu-tion fitted to theMonteCarlo predicted extremepit depths fora general soil class found in southernMexico [8]The inset ofFigure 1(a) shows the evolution of the mean and variance ofsuch distributions with time These average values are usedto estimate the pitting proportionality and exponent factors(120581 and ]) associated with the typical (average) values of thepredictor variables in this soil categoryThe time evolution ofthe mean of the simulated maximum pit depth distributionswas fitted to (10) using the corresponding value of 119905sd givenin Table 1 Typical values 120581Typ and ]Typ derived following thismethod are displayed in Table 2 for the analyzed soil texturalclasses

From the estimated values of 120581 and ] for the general soilclass and the value of 119905sd shown in the last column of Table 1it is possible to obtain the function 120588(119905) (13) and 119901119904 (15) fortypical conditions of the ldquoAllrdquo soil category [8 9]

Figure 1(b) shows the time evolution of 120588(119905) and 119901119904

predicted from the parameters associated with the pitting

20

25

30

35

Pipeline age 5 years 10 years 15 years

25 years 35 years

00

05

10

15

20

25

30

Mean

Mea

n (m

m)

00

05

10

15

20

25

30

Variance

00

05

10

15

Den

sity

of p

roba

bilit

y (1

mm

)

0 1 2 3 4 5 6 7Maximum pit depth (mm)

5 10 15 20 25 30 35 40Pipeline age (years)

Varia

nce (

mm

2)

(a)

25

30

35

06

08

10

10

15

20

00

02

04

Pipeline age (years)5 10 15 20 25 30 35 40

120588(119905

)

120588(119905)

Prob

abili

ty119901119904

119901119904 1199050 = 15 years119901119904 1199050 = 5 years

Func

tion

(b)

Figure 1 (a) Evolution of the Generalized Extreme Value distri-bution fitted to the extreme pit depths predicted in a Monte Carloframework [7] for a general soil classThe time evolution of themeanvalue and variance of such distributions are shown in the inset (b)Time evolution of the functions120588(119905) (13) and119901119904 (15) calculated usingthe predicted pitting parameters 120581 ] and 119905sd The two curves of 119901119904correspond to two different values of 1199050 in (15)

process in soils Given that 120588(119905) is completely determined bythe extent of the damage [15] its value is unique for each soilclass Also the probability119901119904 is unique in each soil typeHow-ever in Figure 1(b) two different curves of 119901119904 are displayed toillustrate the dependence of 119901119904 on time They correspond totwo distinct values of 1199050 (see (15)) the time when the initialpit depth distribution is measured The value of 119901119904 at a givenmoment in time 119905 ge 1199050 increases with increasing 1199050 and


Table 2 Estimated pitting parameters (10) associated with thetypical values of the physicochemical pipe-soil variables in each soilcategory [9]

Soil class Typical pitting coefficient Typical pitting120581Typ (mmyears]Typ )a exponent ]Typ (years)

a

Clay 0178 0829Clay loam 0163 0793Sandy clayloam 0144 0734

All 0164 0780aCorrespond to parameters in (10)

decreases when the length of the interval (1199050 119905) increasesTheincrease in 119901119904 with 1199050 indicates that the mean and varianceof the pitting rate decrease as the lifetime of the pittingdamage increases which is in agreement with experimentalobservations Also the form of 120588(119905) in Figure 1(b) suggeststhat for pits with equal lifetimes the deeper the pit thesmaller the value of 119901119904 and therefore the larger themean andvariance of the pitting rate This characteristic of the devel-oped model will permit accurately predicting the corrosionrate distribution in a pitting corrosion experiment

To estimate the evolution of pit depth and pitting ratedistributions using the proposed Markov model one can use(7) and (6) respectively As has been already stated from theresults of an ILI of the pipe and the knowledge of the localsoil characteristics it would be possible to estimate the pittingcorrosion damage evolution

To illustrate the application of the proposed Markovchain model it is employed in the analysis of an 82 km longoperating pipeline used to transport sweet gas since itscommissioning in 1981 This pipeline is coal-tar coated andhas a wall thickness of 952mm (037510158401015840) It was inspected in2002 and 2007 using magnetic flux leakage (MFL) ILI Thepit depth distributions present in the pipeline were obtainedby calibrating the ILI tools using amethodology described byCaleyo et al [31]These distributions are shown in Figure 2 inthe form of histograms The first grey-shadowed histogramrepresents the depth distribution of 11987302 = 3577 pits locatedand measured by ILI in 2002 while the hatched histogramrepresents the depth distribution of 11987307 = 3851 pitsmeasured by ILI in mid-2007

In order to apply the proposed model the pipe wallthickness was divided in 01mm thick non overlappingMarkov states so that the pitting damage is representedthrough Markov chains with states ranging from 119898 = 1 to119898 = 119873 = 100 Unless otherwise specified this scheme ofdiscretization of the pipe wall thickness is used hereafter torepresent the pitting damage penetration

The empirical depth distribution observed in 2002 wasused as the initial distribution so that 1199050 = 21 years Δ119905 =

55 years and 119901119898(1199050 = 21) = 11987311989811987302 119873119898 being thenumber of pits withmeasured depth in the119898th stateThe soilcharacteristics along the pipeline were taken as those of theldquoAllrdquo (generic) soil class and the value of 119905sd was taken equalto 29 years as indicated in Table 1 From Table 2 a value of

008

009

010

006

008

010

002

003

004

005

006

007

ILI 2002

000

002

004

000

001

ILI 2007Markov-chain

1 2 3 4 5 6 7 8 9Pit depth (mm)

Den

sity

of p

roba

bilit

y (1

mm

)

00 01 02 03 04Pitting rate (mmyr)

Prob

abili

ty d

ensit

y(y

rm

m)

Markov-chainGEVD fit

func

tion

Figure 2 Histograms of the pipeline pit depth distributions mea-sured by ILI in 2002 (grey) and in 2007 (hatched) The solid line isthe Markov-estimated pit depth distribution for 2007 In the insetthe Markov-derived pitting rate distribution for the period 2002ndash2007 is presented

0780 was assigned to ] in order to obtain the predicted pitdepth distribution 119901119899 (119905 = 26 years) for 2007 The values of119901119899(119905) can be calculated using the following expression whichis deducted from (4) (7) and (15)

119901119899 (119905)

=

119899

sum

119898=1

119901119898 (1199050) (119899 minus 1

119899 minus 119898)(

1199050 minus 119905sd1199050 minus 119905sd

)

]119898

[1 minus (1199050 minus 119905sd1199050 minus 119905sd

)

]

]

119899minus119898

(18)

In Figure 2 the Markov chain-predicted pit depth distri-bution for 2007 is represented with a thick line

In order to test the degree of coincidence between theempirical distribution (for 2007) and the Markov-modelpredicted distribution the two-sample Kolmogorov Smirnov(K-S) and the two-sample Anderson-Darling (A-D) testswere used This was done by means of a Monte Carlosimulation in which 1000 samples of 50 depth values weregenerated for each one of the distributions under comparisonand then used in pairs for the tests In the case of theMarkov-predicted distribution the data samples were generated as(pseudo) random variates with probability density function(pdf) as displayed in Figure 2 and given by (18) The samplesfrom the empirical data were produced by sampling theexperimental 2007 dataset with replacement The K-S testgave an average 119875 value of 042 with only 7 of cases wherethe test failed at 5 significance By its part the A-D testproduced an average 119875 value of 039 with an 11 fractionof failed tests This is an evidence of the validity of thenull hypothesis about both samples modeled and empiricalcoming from the same distribution The good agreementbetween the empirical pit depth distribution observed in 2007


and theMarkov chain-modeled distribution also points to theaccuracy of the proposed model

The corrosion rate distribution 119891(120592 1199050 119905) associated withthe entire population of pits in the 2002ndash2007 periodwas alsoestimated using (19) whichwas derived from (4) (5) (6) and(15) Consider

119891 (120592 1199050 119905) =

119873

sum

119898=1

119901119898 (1199050) (119898 + 120592 (119905 minus 1199050) minus 1

120592 (119905 minus 1199050))(

1199050 minus 119905sd119905 minus 119905sd

)

]119898

times [1 minus (1199050 minus 119905sd119905 minus 119905sd

)

]

]

120592(119905minus1199050)

(19)

The estimated rate distribution is shown in the inset ofFigure 2 it can be fitted to a GEV distribution (17) withnegative shape parameter 120577 although it is very close to zeroThis means that the Weibull and Gumbel subfamilies [27] ofthe GEV distribution are appropriate to describe the pittingrate in the pipeline over the selected estimation period

It has to be noticed that the proposed Markov model canbe used to predict the progression of other pitting processesFor this to be done the values of the pitting exponent andstarting time must be known for the process These canbe obtained for example from the analysis of repeated in-line inspections of the pipeline from the study of corrosioncoupons or from the analysis of laboratory tests To showthis capability of the model another example is given usingthis time a different experimental data set gathered fromtwo successive MFL-ILIs The analysis involves a 28 km longpipeline used to transport natural gas since its commissioningin 1985 This pipeline is coal-tar coated made of API 5Lgrade X52 steel [32] with a diameter of 4572mm (1810158401015840) anda wall thickness of 874mm (034410158401015840) The first inspectionwas conducted in 1996 and the other in 2006 The ILIswere calibrated following procedures described elsewhere[31] Before using the ILI data the corrosion defects fromboth inspections were matched using the following criteria(i) the matched defects should agree in location and (ii) thedepth of a defect in the second inspection must be equalto or greater than its depth in the first inspection Onlythe matched defects were used in the analyses that followedto ensure that only the actual defect depth progressionwith time was considered (one-by-one if necessary) withoutincluding defects that might have nucleated in the intervalbetween the two inspections The depth distribution of thematched defects of the 1996 inspectionwas taken as the initialdepth distribution fed into the proposed Markov model Theresulting depth distribution was subsequently compared tothat of the defects observed in the 2006 inspection

From the knowledge of the mean depth values from bothinspections an empirical value of the pitting exponent ] wasestimated instead of using the value recommended in Table 2It was calculated for the specific type of soil in which thepipe under analysis was buried by using the ratio between theempirical pit depth meansK96 andK06 from the ILIs in 1996and 2006 respectively Assuming that the pit depth meancomplies with (10) and that parameters 120581 ] and 119905sd (whichdepend exclusively on the soil and pipe properties) are the

same at the times of both inspections it is possible to obtainthe pitting exponential parameter ] from

K06K96

=120581(11990506 minus 119905sd)

]

120581(11990596 minus 119905sd)] (20)

where 11990596 = 11 years and 11990506 = 21 years are the times of theinspections and 119905sd is the incubation time of the corrosionmetal losses in the pipeline The value of 119905sd was taken equalto 29 years the average pitting initiation time for a genericsoil class (Table 1)

As in the previous example the wall thickness wasdivided into119873 (01mm thick) equally spaced Markov statesThe observed defect depth was converted to Markov-stateunits and the depth distribution was given in terms of theprobability 119901119898(119905) for the depth in a state equal or less than119898 at time 119905 The proposed Markov model (18) was appliedto estimate the pit depth distribution after a time interval120575119905 = 119905 minus 1199050 of 10 years with 119905sd = 29 years 1199050 = 11 yearsand 119905 = 21 years

In Figure 3 the final (2006) pit depth distribution of theby-ILI measured and matched 179 defects in this pipelineis shown on a probability density scale in the form of ahistogram together with the pit depth distribution predictedby the Markov model for 2006 Again the Monte Carloframework described in the previous example was followedin order to apply the two-sample K-S test The sample sizewas 50 and the sampling process was repeated 1000 timesThe resulting average 119875 value was 007 which is enoughfor not rejecting the null hypothesis that both the empiricaland modeled distributions come from the same distributionFrom the results of the test and from what it is shown inFigure 3 it can be concluded that the Markov model predictsa pit depth distribution that is close to the experimentalone More important the model is also capable to correctlydescribe the form (shape) of the defect depth distribution asseen also from this figure

To further explore the capabilities of themodel in estimat-ing the time evolution of pit depths the defects in the initial(1996) inspection were grouped into 6-state (06mm) depthintervals in order to test the model efficacy in predicting thepit depth evolution by depth interval An example of thesegrouping and modeling approach is shown in the inset ofFigure 3 The chosen initial depth interval is that between9 and 15 states (09 and 15mm) In the inset a histogramof the matched defect depths in the second ILI togetherwith the distribution (solid line) predicted by the Markovmodel starting from the chosen initial depth subpopulationis depicted Here as well the model is capable of reproducingthe experimental results The same was proven true forthe rest of depth intervals in the initial (1996) pit depthpopulationTherefore it can be stated that theMarkovmodelis not only accurate in estimating the time evolution of theentire defect depth distribution but also in estimating thetime evolution of defect subpopulations that differ in depths

Additionally to check the ability of the proposed Markovmodel to predict the corrosion rate (CR) distribution anempirical CR distribution was determined using the datafrom both inspections based on the observed change in depth


0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)

Empirical Markov-chain

EmpiricalMarkov-chain

0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)

Figure 3 Histogram of the 179 matched defect depths measured in2006 together with the predicted pit depth distribution The insetshows the histogram of the observed in the second inspection defectdepths matched to 09ndash15mm depths of first inspection along withthe Markov chain-predicted distribution (solid line)

of the matched defects over the interval 120575119905 The empiricalrate distribution was then compared with the corrosion ratedistribution predicted by the model (19) This comparisoncan be observed in Figure 4 The two-sample K-S test for1000 pairs of 50-point CR samples yielded an average 119875

value of 027 with only 12 of rejections at 5 significancelevel Again the Markov model satisfactorily reproduces theempirical CR distribution

The reasons behind the foregoing results lie in the abilityof the Markov chain model to capture the influence ofboth the depth and age of the corrosion defects on thedeterministic pit depth growth together with its ability toreproduce the stochastic nature of the process also as afunction of pit depth and age The inset of Figure 4 is a proofof this last statement It shows the experimental corrosionrate mean and variance dependence on pit depth To obtainthese results pit depths were grouped into depth intervalsof ten states (1mm) and the corrosion rate values derivedfrom the comparison of the two ILIs were averaged withineach group From this inset it can be concluded that boththe corrosion rate mean and variance increase with defectdepth as was previously noted from different experimentalevidence [7 18] This result is critical to support the criteriathat a good corrosion rate model should consider both theage and depth of the to-be-evolved corrosion defects togetherwith the actual shape of the function describing the observeddependence of the corrosion defect depth with time

22 Markov Chain Modeling of Extreme Corrosion Pit DepthsThe novelty in this second stochastic model is the considera-tion of multiple pits in a given corrosion area (called couponhereafter) Pit initiation ismodeled using a nonhomogeneousPoisson process while the distribution of pit nucleation times

00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)

Corrosion rate (mmyr)

05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)

Figure 4 Comparison of the empirical corrosion rate (CR) distri-bution as determined from repeated ILIs (1996 and 2006) with theCR distribution predicted by the model The inset is a plot of theexperimental CR mean and variance against defect depth

is simulated using a Weibull process The initiation time ofeach one of the 119898 pits produced in a coupon is describedas the time to the first failure of a part of the system [33]So if pit initiation time is understood as the time to firstfailure (passive layer breakdown or inclusion dissolution) ofan individual part [33] one can assume that it follows aWeibull process with distribution

119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)

The pit initiation times 119905119896 are computed as the realizationsof a Weibull probability distribution described by (21) withboth the scale parameter 120576 and the time 119905 expressed in days

After a pit has been generated at time 119905119896 it is assumedthat it starts to grow The time evolution of pit depth 119863(119905) isthenmodeled using a nonhomogeneousMarkov processThetransition probabilities 119901119894119895 from state 119894 to state 119895 satisfy thesystem of forward Kolmogorovrsquos equations given in (1)

If it is assumed that the pitting damage (depth) is in state1 at the initial time 119905119896 (after initiation) then for the Markovprocess defined by the system of (1) the transition probabilityfrom the first state to any 119895th state during the interval (119905119896 119905)that is 1199011119895(119905) = 119875119863(119905) = 119895 | 119863(119905119896) = 1 can be foundanalytically [28] as

1199011119895 (119905) = exp [minus120588 (119905) + 120588 (119905119896)] 1 minus exp [minus120588 (119905) + 120588 (119905119896)]119895minus1

(22)

where 120588(119905) is defined by (3)


From (22) it follows that for a single pit the probabilitythat its depth is equal or less than state 119894 after a time incre-ment (119905119896 119905) is

119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)

where119873 is the total number of states in the Markov chainIt is easy to show [15] that expression (23) can be rewritten

as

119865 (119894 119905) = 1 minus 1 minus exp [minus120588 (119905) + 120588 (119905119896)]119894 (24)

Equation (24) represents the probability of finding asingle pit in a state less than or equal to state 119894 after a timeinterval (119905119896 119905) It is worth noting that in (22)ndash(24) the pitinitiation and growth processes are combined by taking intoaccount the pitting initiation time 119905119896

As it was stated before a direct physical meaning canbe given to the functions 120582(119905) and 120588(119905) which are related tothe pit growth rate and pit depth respectively Taking intoaccount that the dependence of pit depth on time is a powerfunction (10) the functional dependence of 120588 and 120582 on timewas assumed to be

120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)

where 120594 has dimensions of distance over the 120596th power oftime and 120596 is less than 10

In order to generalize from the single pit case to the 119898-pit case the probability that the maximum damage state isless than or equal than a given value for a time interval isestimated under the assumption that all the pits are describedby parent distributions 119865(119894 119905119896 119905) of the type given in (24)First let us consider the simplest situation in which it isassumed that all the pits are generated at the same time119905119896 = 119905ini In this case function (24) represents the cumulativedistribution function of a parent population of 119898 pits Tofind the extreme depth value distribution the distribution of(24) should be raised to the119898th power [27] It can be shown[15 27] that for a large number of pits (119898 rarr infin) and undera suitable variable transformation it follows that

Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)

Substituting (24) and (25) in expression (27) and aftersome transformations [15] 119866(119894 119905) converges to a Gumbelfunction given by (2)The involvedmathematical formalismsthat led to this result can be found in Appendix A of [15]

Summarizing when the pit initiation times are very short(smaller than the observation time) andor when it can beconsidered that they are the same for all the pits the distribu-tion function for maximum pit depths is obtained by raising

(24) to the power of the number of pits in the area of interestThe parameters of the model in this case are the number ofpits 119898 and the parameters of the 120588(119905) function 120594 and 120596 (seeexpression (25))

Consider now an alternative case when 119898 pits aregenerated at different times 119905119896 and expression (24) holds foreach one of them The 119898 cumulative distribution functions119865119896(119894 119905 minus 119905119896) 119896 = 1 119898 must be combined in order to esti-mate the distribution of the deepest pit under the assumptionthat pits nucleate and grow independently In such a case theprobability that the deepest pit is in a state less than or equalto state 119894 at time 119905 can be estimated using the expression

120579 (119894 119905) =

119898

prod

119896=1

1 minus [1 minus exp (minus120588 (119905) + 120588 (119905119896))]119894 (28)

It can be shown that this cumulative function also followsa Gumbel distribution for large119898 [15]

In (28) expression (25) for 120588(119905) must be substitutedtogether with the initiation times 119905119896 which are assumed tofollow a Weibull distribution (21) Therefore function 120579(119894 119905)

in (28) includes the model parameters 120576 120585 and119898 to simulatepit initiation together with parameters 120594 and 120596 to model pitgrowth through the function 120588(119905)

At this point owing to the fact that functionsΦ(119894 119905) (from(27)) and 120579(119894 119905) (from (28)) are extreme value distributions ofthe Gumbel type it can be stated that function 119865(119894 119905) (24) isthe parent distribution that lies in the domain of attractionof the Gumbel distribution for maxima [27] From (24) itcan be observed that 119865(119894 119905) is of exponential type thereforethe parent distribution for pit depth extremes is actually anexponential functionThis is in complete agreement with thefindings in [7] where it was concluded after measuring andanalyzing all the pits in corroded low carbon steel couponsthat the exponential pit depth distribution fitted to the righttail of the pit depth distribution (of the Normal type adjustedto the depths of the whole population of pits) is actuallythe extremersquos parent distribution which lies in the domainof attraction of the Gumbel extreme value distribution formaxima

Following this idea it is soundly to shift the initialMarkovstate to the depth value that constitutes the starting pointof the exponential distribution tail If we consider that thisexponential distribution starts at some depth value 119906 thenthe Markov state 119894 that appears in (24) should be changed toa new variable

1198941015840= 119894 + 119906 (29)

Because the pits that contribute to the extreme pit depthsvalues are only those whose depths exceed the threshold 119906the number119898 of pits that should be taken into account whenapplying (27) or (28) can be equated to the average numberof pits whose depths exceed the threshold value 119906 A moredetailed justification of this procedure of variable change forthe state 119894 in (24) can be found elsewhere [34]

The empirical average number 120582119890 of exceedances overthreshold 119906 per coupon (to be compared to themodel param-eter119898) can be estimated using (30) which relates 120582119890 with the


threshold value 119906 and the parameters 120590 and 120583 of the Gumbeldistribution fitted to the observed extreme pit depths Con-sider

120582119890 = exp(120583 minus 119906

120590) (30)

This expression is readily obtained from the existingrelation between the Gumbel and exponential parameters ashas been shown in [7]

To run the proposed Markov model for extremes (27)(pits initiate all at the same time) or (28) (pits initiate at dif-ferent times) is computed and fitted to a Gumbel distributionfor several times 119905 (2) The model parameters (three for theinstantaneous pit initiation 120594 120596 and 119898 and five for thegeneration of pits at different times 120576 120585 120594 120596 and 119898) areadjusted considering the experimental distributions for thedeepest pits A series of corrosion experiments is consideredeach one of them consisting in the exposition of 119899119888 couponsof a corrodible material to a corrosion environment for agiven time After exposure the depth of the deepest pitin each coupon is measured [6 7] and the distribution ofmaximum depths is fitted with a Gumbel distribution (2) Ifthe experiment is carried out for119873119905 exposure times and theobserved behavior of depth extremes is to be described withthe proposed model the model parameters are adjusted byminimizing a total error function 119864119879 defined as

119864119879 =

119873119905

sum

119897=1

(radic(120583119897119890 minus 120583119897119901)2+ radic(1205902119897119890 minus 1205902119897119901 )

2) (31)

where (120583119897119890 1205902119897119890 ) and (120583

119897119901 1205902119897119901 ) are the mean and variance values

of the 119897th experimental and estimated extreme value distribu-tions respectively

The model parameters are adjusted through minimiza-tion of the average value of the error function 119864119879 computedduring119873MC Monte Carlo simulations with119873MC = 1000

221 Application of Markov Model for Extremes to Low Car-bon Steel Corrosion ExperimentThedescribedMarkovmodelfor extreme pit depths was applied to experimental dataobtained in pitting corrosion experiments for low-carbonsteel reported byRivas et al [7]Thedetails of the experimentscan be found elsewhere [7]

In these experiments groups of 20 coupons of API-5LX52 [32] pipeline steel with 1 times 1 cm2 of exposed area wereimmersed in a corroding solution for 1 3 7 15 21 and 30days After the immersion all the pits with depths greaterthan 5 120583m were measured The maximum pit depth in eachcoupon was also recorded The Gumbel distribution (2) wasfitted to the resulting maxima data sets using the MaximumLikelihood Estimator (MLE) [27]

The Gumbel probability density functions fitted to theexperimental pit depth maxima for the six exposure timesare displayed in Figure 5 They are represented with red solidlines Table 3 shows the values of the Gumbel location (120583119864)

and scale (120590119864) parameters for the fitted distributionsIn their work Rivas et al [7] concluded that as has been

previously recognized [4 35] in low carbon steel pits initiate

008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300

VarianceModelModel EmpiricalEmpirical

000

001

002

Experimental Markov-chain

25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days

0 5 10 15 20 25 30 35Pipeline age (days)

Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)

Figure 5 Comparison of the Gumbel probability density functionsfitted to the experimental pit depth maxima with the model-predicted density functions for the six exposure times

at the site of MnS inclusions In the present experimentthe dissolution of these inclusions occurs in a matter of fewhours after immersion and then pits start propagating Theauthors also established that the deepest pits of the entirepit population are the oldest ones meaning that they are thedefects that initiate first in time [7] This conclusion leadsto the use of (27) to model the extreme pits using Markovchains Since the pit initiation time for this experiment hasbeen considered to be very short the nucleation time 119905119896 in(27) was set to zero

In order to apply the Markov model the value of thethreshold parameter 119906 of (29) should be determined from theempirical corrosion data Given that 119906 represents the depthvalue from which the exponential tail of the whole pit depthdistribution starts it should coincide with the previouslydetermined [7] threshold pit depth value 119906119875 for the tail of thewhole pit depth distribution

When applying the so-called Peak over Threshold (POT)approach in pitting corrosion experiments Rivas et al pro-posed [7] a simplified method for determining the thresholddepth value without the need of measuring the entire pitdepth population The method includes a proposition of thea priori determination of the threshold pit depth value 119906119875

from the fittedGumbel distribution to the pit depth extremesIt was suggested to take 119906119875 as a depth value for which theGumbel distribution shows a cumulative probability 119875 in therange from 000005 to 0005 This proposition was based onthe empirical fact that in the analyzed pitting corrosion exper-iments [7] the Gumbel distribution for maximum depthstarts to rise precisely at the beginning of the exponential tailof the normal distribution of the entire pit depth populationTherefore as the starting point of the Gumbel distributionit was proposed to use the 05 005 or even the 0005quantile of the experimental pit depth distributions

Following this suggestion and taking into considerationthe determined values of the location and scale from the


Table 3 Estimated location and scale parameters of the Gumbel distributions fitted to the experimental pit depth maxima and to theMarkovmodel The fourth and fifth columns correspond to the calculated pit depth threshold and average exceedances values respectively The lastcolumn displays the average 119875 value of 1000 two-sample Kolmogorov-Smirnov tests

Exp time (days) Estimated from the experimental data1199060005

a120582119890

b Estimated from the modeled distribution119875 value

120583119864 (120583m) 120590119864 (120583m) 120583119872 (120583m) 120590119872 (120583m)1 328 515 212 10 320 515 0503 470 758 3449059 5 491 702 0367 729 929 5020122 12 697 917 03115 949 1275 7036299 9 956 1194 05621 1059 1352 8167309 6 1110 1352 02930 1279 1454 9565344 9 1294 1552 053aEstimated from (32)bEstimated from (30)

0 5 10 15 20 25 3020

40

60

80

100

120

Empirical

Exposure time (days)

005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365

Figure 6 Plot of the calculated threshold depth values 1199060005 versusexposure time The solid line is a power function fitted to thedetermined threshold values

fitted Gumbel distribution it is possible to determine the119883119875quantile being 119875 = 0005 00005 or 000005 To achievethis the following equation obtained through the inversionof expression (1) is used

119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)

Thus the value of 119883119875 can be used as the threshold 119906119875

(119906119875 = 119883119875) [7] For the analyzed experiment we are fixingthe value of 119875 to 00005 Substituting this value in (32) andusing the estimated location (120583119864) and scale (120590119864) parametersfor the empirical Gumbel distributions for each one of theexposure times (Table 3) six values of 11990600005 are obtainedThese values are plotted in Figure 6 against the exposuretimes In this figure a power function fitted to the empiricalthreshold values is also displayed From this curve the valuesof 119906 to be substituted in (29) are taken Amore detailed proofof the correctness of this choice of 119906 can be found in [34]

Once the value of the threshold is determined the func-tion119865(119894 119905) of (24) is raised to the119898th power and the resultingfunction Φ(119894 119905) (27) is fitted to a Gumbel distribution (2)The fitted function is compared with the empirical Gumbeldistribution in order to adjust the model parameters Forthis particular case the parameters to take into account arethe number of pits 119898 which should approximately matchthe average experimental number of exceedances over thethreshold 119906 per coupon and the parameters 120594 and 120596 whichdefine the functionΦ(119894 119905) The model parameters are refinedby minimizing the error function of (31)

The minimization process gives an adjusted parameter119898 = 76 This value can be approximated to 8 pits Itrepresents the average number of pits per coupon whosedepths exceed the threshold value 119906 This fact can beconfirmed by comparing the adjusted value for 119898 with thevalues of the empirical exceedances 120582119890 displayed in the fifthcolumn of Table 3The exceedances for each exposure periodwere calculated by substituting the corresponding estimatedGumbel parameters 120583119864 and 120590119864 and the threshold value 11990600005(all given in Table 3) into (30) The mean value of 120582119890 (fromTable 3) equals 8 which is in good agreement with the modelprediction

The probability density functions corresponding to thedistributionsΦ(119894 119905) obtained after raising 119865(119894 119905) to the powerof the adjusted parameter 119898 = 76 are shown in Figure 5 forthe six exposure times In this figure the modeled distribu-tions can be compared with the empirical Gumbel distribu-tions One can see a good agreement between the modeledand experimental Gumbel distributions Even for this case inwhich the number of pits119898 is small (see Section 21 and (27))both the experimental and modeled distributions agree withthe Gumbel model for maxima

The Gumbel distribution (2) was also fitted to the func-tions predicted by the model using theMaximum LikelihoodEstimator (MLE) [27] In Table 3 the estimated parameters120583119872 and 120590119872 for the fitted Gumbel distributions (to theMarkov-predicted functions) are displayed together with thecorresponding parameters of the empirical Gumbel distribu-tions The differences between the empirical and modeledparameters do not exceed 8 The last column of Table 3shows the average 119875 values of the two-sample K-S testperformed on 1000 pairs of 50-depth-point samples (one for


the empirical and one for theMarkov-modeled distributions)for the six exposure periodsThe smaller119875 value among themis 029 which leads to not reject the null hypothesis that thetwo samples belong to the same distribution

The good agreement between the modeled and theempirical Gumbel distributions can also be established fromthe inset of Figure 5 where the time evolution of the Gumbelmean and variance for the observed data and for the resultof the proposed model are compared These results demon-strate the suitability of the proposed model to describe theinitiation and evolution of the maximum pit depths in apitting corrosion experiment which is of great importancein many applications such as reliability assessment and riskmanagement

The advantages of the proposed model compared withprevious Markov models (developed by other authors) canalso be established The details regarding this topic can befound in [15]

3 Concluding Remarks

Two Markov chain models have been presented to simulatepitting corrosion They have been developed and validatedusing experimental pitting corrosion data Both models areattractive due to the existence of analytical solutions of thesystem of Kolmogorovrsquos forward equations

The first model describes the time evolution of pitdepths that correspond to the general population of defectsin underground pipelines It has been developed using acontinuous-time nonhomogenous linear growth (pure birth)Markov process under the assumption that the Markov-chain-derived stochastic mean of the pitting damage equalsthe deterministic empirical mean of the defect depths Suchan assumption allows the transition probability functionto be identified only from the pitting starting time andexponent parameter Moreover this assumption requiresthat the functional form of the stochastic and deterministicinstantaneous pitting rates also agree This supports the ideathat the intensities of the transitions in theMarkov process areclosely related to the pitting damage rateThis model permitspredicting the time evolution of pit depth distribution whichis of paramount importance for reliability estimations It isalso able to capture the dependence of the pitting rate on thepit depth and lifetime This is an advantage of the Markovchain approach over deterministic and other stochasticmodels for pitting corrosion It could also be extended toinvestigate pitting corrosion in environments other than soilsfor example in laboratory experiments

The second Markov chain pitting corrosion model givesaccount for the maximum pit depths In it pitting corrosionis modeled as the combination of two independent nonho-mogeneous in time physical processes one for pit initiationand one for pit growth Both processes are combined usingwell-suited physical and statistical methodologies such asextreme value statistics to produce a unified stochasticmodelof pitting corrosion

Pit initiation is described by means of a non-homogene-ous Poisson process so that a set of multiple pit nucleation

events can be modeled as a Weibull process Pit growth ismodeled as a nonhomogeneous Markov process Given thatthe intensity of the process is related to the corrosion rateits functional form can be proposed from the results of theexperimental tests

Taking into account the experimental evidence that thepit depth parent distribution that leads to the distribution forextremes is the exponential tail of the pit depth populationdistribution the solution of theMarkov chain is shifted to thethreshold pit depth value where the exponential tail of the pitdepth distribution starts The threshold value is determinedfrom the pit depth value from which the empirical Gumbeldistribution for maximum pit depth becomes significant

Extreme value statistics has been used to show the accu-racy of the model describing the experimental observationsThe model is capable of predicting not only the correctGumbel distributions for pitting corrosion maxima in low-carbon steel but also of estimating the number of extremepits that has physical sense and thatmatches the experimentalfindings

In order to simulate the whole pitting process five modelparameters are necessary Two parameters are required tosimulate pit initiation as a Weibull process another twoparameters are required to simulate pit growth with thenonhomogeneous Markov process finally the number ofpits is necessary to combine these two processes If theassumption that all pits nucleate instantaneously holds asis the case of the presented experimental example onlythree parameters will be necessary to fit the model to theexperimental data

Given the fact that the model parameters and assump-tions do not depend on the corroded material nor on thecorrosion environment the model is suited for differentcorrosion systems The model can be easily adapted todescribe situations in which the distribution of pit initiationtimes and the functional form of the time dependence for pitgrowth differ from those considered in this work

References

[1] G S Chen K C Wan G Gao R P Wei and T H FlournoyldquoTransition from pitting to fatigue crack growthmdashmodelingof corrosion fatigue crack nucleation in a 2024-T3 aluminumalloyrdquoMaterials Science and Engineering A vol 219 no 1-2 pp126ndash132 1996

[2] M Nessin ldquoEstimating the risk of pipeline failure due to corro-sionrdquo in Uhligrsquos Corrosion Handbook W Revie Ed p 85 JohnWiley amp Sons 2nd edition 2000

[3] T Shibata and T Takeyama ldquoPitting corrosion as a stochasticprocessrdquo Nature vol 260 no 5549 pp 315ndash316 1976

[4] G S Frankel ldquoPitting corrosion of metals a review of the crit-ical factorsrdquo Journal of the Electrochemical Society vol 145 no6 pp 2186ndash2198 1998

[5] G T Burstein C Liu R M Souto and S P Vines ldquoOrigins ofpitting corrosionrdquo Corrosion Engineering Science and Technol-ogy vol 39 no 1 pp 25ndash30 2004

[6] P M Aziz ldquoApplication of the statistical theory of extreme val-ues to the analysis of maximum pit depth data for aluminumrdquoCorrosion vol 12 pp 495tndash506t 1956


[7] D Rivas F Caleyo A Valor and J M Hallen ldquoExtreme valueanalysis applied to pitting corrosion experiments in low carbonsteel comparison of block maxima and peak over thresholdapproachesrdquo Corrosion Science vol 50 no 11 pp 3193ndash32042008

[8] J C Velazquez F Caleyo A Valor and J M Hallen ldquoPredictivemodel for pitting corrosion in buried oil and gas pipelinesrdquoCorrosion vol 65 no 5 pp 332ndash342 2009

[9] F Caleyo J C Velazquez A Valor and J M Hallen ldquoProba-bility distribution of pitting corrosion depth and rate in under-ground pipelines A Monte Carlo Studyrdquo Corrosion Science vol51 no 9 pp 1925ndash1934 2009

[10] Z Szklarska-Smialowska ldquoPitting corrosion of aluminumrdquoCorrosion Science vol 41 no 9 pp 1743ndash1767 1999

[11] W Zhang andG S Frankel ldquoLocalized corrosion growth kinet-ics in AA2024 alloysrdquo Journal of the Electrochemical Society vol149 no 11 pp B510ndashB519 2002

[12] F Hunkeler and H Bohni ldquoMechanism of pit growth on alu-minum under open circuit conditionsrdquo Corrosion vol 40 no10 pp 534ndash540 1984

[13] T Shibata ldquo1996 WR Whitney Award Lecture statistical andstochastic approaches to localized corrosionrdquoCorrosion vol 52no 11 pp 813ndash830 1996

[14] J W Provan and E S Rodriguez ldquoPart I Development of aMarkov description of pitting corrosionrdquo Corrosion vol 45 no3 pp 178ndash192 1989

[15] A Valor F Caleyo L Alfonso D Rivas and J M Hallen ldquoSto-chasticmodeling of pitting corrosion a newmodel for initiationand growth of multiple corrosion pitsrdquo Corrosion Science vol49 no 2 pp 559ndash579 2007

[16] T B Morrison and R G Worthingham ldquoReliability of highpressure line pipe under external corrosionrdquo in Proceedings ofthe 11th ASME International Conference Offshore Mechanics andArctic Engineering vol 5 part B pp 401ndash408 Calgary CanadaJune 1992

[17] H P Hong ldquoInspection and maintenance planning of pipelineunder external corrosion considering generation of newdefectsrdquo Structural Safety vol 21 no 3 pp 203ndash222 1999

[18] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009

[19] F Bolzoni P Fassina G Fumagalli L Lazzari and E MazzolaldquoApplication of probabilistic models to localised corrosionstudyrdquoMetallurgia Italiana vol 98 no 6 pp 9ndash15 2006

[20] S A Timashev M G Malyukova L V Poluian and A VBushiskaya ldquoMarkov description of corrosion defects growthand its application to reliability based inspection and mainte-nance of pipelinesrdquo in Proceedings of the 7th ASME InternationalPipeline Conference (IPC rsquo08) Calgary Canada September2008 Paper IPC2008-64546

[21] T B Morrison and G Desjardin ldquoDetermination of corrosionrates from single inline inspection of a pipelinerdquo in Proceedingsof the NACE One Day Seminar Houston Tex USA December1998

[22] Y Katano K Miyata H Shimizu and T Isogai ldquoPredictivemodel for pit growth on underground pipesrdquo Corrosion vol 59no 2 pp 155ndash161 2003

[23] A K Sheikh J K Boah and D A Hansen ldquoStatistical mod-elling of pitting corrosion and pipeline reliabilityrdquo Corrosionvol 46 no 3 pp 190ndash196 1990

[24] M Kowaka Ed Introduction to Life Prediction of IndustrialPlant Materials Application of the Extreme Value StatisticalMethod for Corrosion Analysis Allerton Press New York NYUSA 1994

[25] A Valor D Rivas F Caleyo and J M Hallen ldquoDiscussionstatistical characterization of pitting corrosionmdashPart 1 Dataanalysis and part 2 Probabilistic modeling for maximum pitdepthrdquo Corrosion vol 63 no 2 pp 107ndash113 2007

[26] E J Gumbel Statistics of Extremes Columbia University PressNew York NY USA 2004

[27] E Castillo Extreme Value Theory in Engineering AcademicPress San Diego Calif USA 1988

[28] E Parzen Stochastic Processes vol 24 SIAM Philadelphia PaUSA 1999

[29] D R Cox and H D Miller The Theory of Stochastic ProcessesChapman amp HallCRC Boca Raton Fla USA 1st edition 1965

[30] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer LondonUK 2001

[31] F Caleyo L Alfonso J H Espina-Hernandez and J M HallenldquoCriteria for performance assessment and calibration of in-lineinspections of oil and gas pipelinesrdquo Measurement Science andTechnology vol 18 no 7 pp 1787ndash1799 2007

[32] Specification for Line Pipe API Specification 5L AmericanPetroleum InstituteWashington DC USA 42nd edition 2001

[33] H Ascher and H Feingold Repairable Systems ReliabilityModeling Inference Misconceptions and Their Causes vol 7Marcel Dekker New York NY USA 1984

[34] A Valor F Caleyo D Rivas and J M Hallen ldquoStochas-tic approach to pitting-corrosion-extreme modelling in low-carbon steelrdquo Corrosion Science vol 52 no 3 pp 910ndash915 2010

[35] G Wranglen ldquoPitting and sulphide inclusions in steelrdquo Corro-sion Science vol 14 no 5 pp 331ndash349 1974

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


Pitting corrosion occurs in a wide range of metals andenvironments This fact points out to the universality of thisphenomenon and suggests that randomness is an inherentand unavoidable characteristic of this damage over time sothat stochastic models are better suited to describe pittingcorrosion than deterministic ones Localized corrosion can-not be explained without assuming stochastic points of viewdue to the large scatter in the measurable parameters such ascorrosion rate maximum pit depth and time to perforation[13] Many variables of the metal-environment system suchas alloy composition andmicrostructure and composition ofthe surrounding media and temperature are all involved inthe pitting process [4] Such complexity imposes the devel-opment of theoretical models and simulation tools for a bet-ter understanding of the outcome of the pitting corrosionprocess These tools help predict more accurately the timeevolution of pit depth in corroding structures as the key factorin structural reliability assessment

Another important characteristic of the pitting corrosionprocess that is worth noting is the time and pit-depthdependence of the corrosion rate [6 7] It has been establishedthat for a given pit the growth rate decreases with time whilefor pits with equal lifetimes the corrosion rate is larger fordeeper ones

Provan and Rodriguez [14] are amongst the first authorsto use a nonhomogenous Markov process to model pit depthgrowth In their model the authors divided the space ofpossible pit depths into discrete non overlapping statesand numerically solved the system of Kolmogorovrsquos forwardequations (1) for the transition probabilities 119901119894119895(119905) betweendamage states 119894 and 119895 However Provan and Rodriguez mod-eled pitting without taking into account the pit generationprocess and proposed an expression for the intensities 120582119894(119905)of the process that conveyed no physical meaning They didnot discuss the method used to solve the system of equationseither This has made it impossible to reproduce their results(deeper discussion on this topic can be found in [15])

119889119901119894119895 (119905)

119889119905=

120582119895minus1 (119905) 119901119894119895minus1 (119905) minus 120582119895 (119905) 119901119894119895 (119905) 119895 ge 119894 + 1

120582119894 (119905) 119901119894119894 (119905)

(1)

Other authors who made use of Markov chains to modelcorrosionwereMorrison andWorthingham [16]Making useof a continuous time birth process with linear intensity 120582they determined the reliability of high-pressure corrodingpipelines For their purpose these authors divided the spaceof the load-resistance ratio into discrete states and numeri-cally solved the Kolmogorovrsquos equations in order to find theintensities of transition between damage states Afterwardsprobability distribution function of the load-resistance ratiowas estimated and compared to the distribution obtainedfrom field measurements Worthinghamrsquos model was furtherimproved byHong [17] who used an analytical solution to thesystemofKolmogorovrsquos equations for the same homogeneouscontinuous type of Markov process and derived the process

probability transition matrix in order to evaluate the prob-ability of failure Hong also investigated the influence of pitdepth on the load-resistance ratioThemerits and limitationsof these two models are discussed in detail in [15 18]

In recent years modeling of pitting corrosion withMarkov chains has shown new advances For exampleBolzoni et al [19] have modeled the first stages of localizedcorrosion using a continuous-time three-state Markov pro-cessTheMarkov states of themetal surface are passivitymetastability and stable pit growth On the other hand Timashevand coworkers [20] formulated a model based on the useof a continuous-time discrete-state pure birth homogenousMarkov process for stochastically describing the growth ofcorrosion-caused metal loss The goal was to assess theconditional probability of pipeline failure and to optimizethe maintenance of operating pipelines In their model theintensities of the process were calculated by iteratively solvingthe proposed system of Kolmogorovrsquos forward equationsThedrawbacks of Timashevrsquosmodel are discussed in detail in [18]

In the present paper a review of the Markov modelsdeveloped by the authors in an attempt to describe pittingcorrosion is presentedThe first model intends to solve a pro-blem that is crucial in reliability-based pipeline integritymanagement the accurate estimation of future pit depthand growth rate distributions from a (single) measured orassumed pit depth distribution It has been recognized [21]that such estimation can be carried out only if oversimpli-fications are made or if additional information besides thepit depth distribution is available In the developed modelit is postulated that in the case of external pitting corrosionin underground pipelines this additional information can beattained from the available predictivemodels for pit growth asa function of the soil characteristics [8 22] A model for pitgrowth previously developed by the authors has been used toperformMonte Carlo simulations in order to predict the dis-tribution of maximum pit depths as a function of the pipelineage and the physicochemical characteristics of the soil [9]

A nonhomogenous linear growth pure birth Markovprocess with discrete states in continuous time is used tomodel external pitting corrosion in underground pipelinesThe system of forward Kolmogorovrsquos equations (1) is ana-lytically solved using the binomial closed-form solution forthe transition probabilities between Markov states in a giventime interval This Markov framework is used to predict thetime evolution of the pitting depth and rate distributionsTheanalytical solution becomes available under the assumptionthat Markov-derived stochastic mean of the pit depth distri-bution is equal to the deterministic mean of the distributionobtained through Monte Carlo simulations This assumptionis made for different exposure times and different soil classesdefined according to soil physicochemical characteristics thatare easy to measure in the field In this way the transitionprobabilities are obtained as a function of soil propertiesfor a given time point and the corrosion rate and futurepit depth distributions are predicted Real-life case studiesare presented to illustrate the proposed Markov model Themain advantage of this model resides in its capability ofcorrectly predicting the time evolution of pit depth and ratedistributions over time












120588 (119905) = int

119905

0120582 (120591) 119889120591 (3)




119901119898119899 (1199050 119905)

= (119899 minus 1



(4)




119891 (120592119898 1199050 119905) = 119901119898 (1199050) 119901119898119898+120592Δ119905 (1199050 119905) Δ119905 (5)


119891 (120592 1199050 119905) =

119873

sum

119898=1

119891 (120592119898 1199050 119905) (6)





119901119899 (119905) =

119899

sum

119898=1

119901119898 (1199050) 119901119898119899 (1199050 119905) (7)



119872(119905) = K (119905) (8)



119872(119905) = 119899119894119890120588(119905minus119905119894) (9)


K (119905) = 120581(119905 minus 119905sd)] (10)



ΔK (119905) = 120582 (119905)K (119905) Δ119905 (11)


119889K (119905)

119889119905=]

120591K (119905) (12)


120588 (119905) = ln [120581(119905 minus 119905sd)]] (13)


120582 (119905) =]

119905 minus 119905sd (14)




119901119904 = (1199050 minus 119905sd119905 minus 119905sd

)

]

119905 ge 1199050 ge 119905sd (15)


120592 (119898 1199050) =119864 [119899 minus 119898]

Δ119905997888997888997888997888997888rarrΔ119905rarr0

]119898

1199050

(16)










120590)]

minus1120577

(17)






20

25

30

35


25 years 35 years

00

05

10

15

20

25

30

Mean

Mea

n (m

m)

00

05

10

15

20

25

30

Variance

00

05

10

15

Den

sity

of p

roba

bilit

y (1

mm

)



Varia

nce (

mm

2)

(a)

25

30

35

06

08

10

10

15

20

00

02

04


120588(119905

)

120588(119905)

Prob

abili

ty119901119904

119901119904 1199050 = 15 years119901119904 1199050 = 5 years

Func

tion

(b)






a









008

009

010

006

008

010

002

003

004

005

006

007

ILI 2002

000

002

004

000

001


1 2 3 4 5 6 7 8 9Pit depth (mm)

Den

sity

of p

roba

bilit

y (1

mm

)


Prob

abili

ty d

ensit

y(y

rm

m)


func

tion



119901119899 (119905)

=

119899

sum

119898=1

119901119898 (1199050) (119899 minus 1

119899 minus 119898)(

1199050 minus 119905sd1199050 minus 119905sd

)

]119898


)

]

]

119899minus119898

(18)






119891 (120592 1199050 119905) =

119873

sum

119898=1

119901119898 (1199050) (119898 + 120592 (119905 minus 1199050) minus 1

120592 (119905 minus 1199050))(

1199050 minus 119905sd119905 minus 119905sd

)

]119898


)

]

]

120592(119905minus1199050)

(19)





K06K96

=120581(11990506 minus 119905sd)

]

120581(11990596 minus 119905sd)] (20)







0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)



0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)






00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)


05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)



119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)





(22)




119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















120588 (119905) = int

119905

0120582 (120591) 119889120591 (3)




119901119898119899 (1199050 119905)

= (119899 minus 1



(4)




119891 (120592119898 1199050 119905) = 119901119898 (1199050) 119901119898119898+120592Δ119905 (1199050 119905) Δ119905 (5)


119891 (120592 1199050 119905) =

119873

sum

119898=1

119891 (120592119898 1199050 119905) (6)





119901119899 (119905) =

119899

sum

119898=1

119901119898 (1199050) 119901119898119899 (1199050 119905) (7)



119872(119905) = K (119905) (8)



119872(119905) = 119899119894119890120588(119905minus119905119894) (9)


K (119905) = 120581(119905 minus 119905sd)] (10)



ΔK (119905) = 120582 (119905)K (119905) Δ119905 (11)


119889K (119905)

119889119905=]

120591K (119905) (12)


120588 (119905) = ln [120581(119905 minus 119905sd)]] (13)


120582 (119905) =]

119905 minus 119905sd (14)




119901119904 = (1199050 minus 119905sd119905 minus 119905sd

)

]

119905 ge 1199050 ge 119905sd (15)


120592 (119898 1199050) =119864 [119899 minus 119898]

Δ119905997888997888997888997888997888rarrΔ119905rarr0

]119898

1199050

(16)










120590)]

minus1120577

(17)






20

25

30

35


25 years 35 years

00

05

10

15

20

25

30

Mean

Mea

n (m

m)

00

05

10

15

20

25

30

Variance

00

05

10

15

Den

sity

of p

roba

bilit

y (1

mm

)



Varia

nce (

mm

2)

(a)

25

30

35

06

08

10

10

15

20

00

02

04


120588(119905

)

120588(119905)

Prob

abili

ty119901119904

119901119904 1199050 = 15 years119901119904 1199050 = 5 years

Func

tion

(b)






a









008

009

010

006

008

010

002

003

004

005

006

007

ILI 2002

000

002

004

000

001


1 2 3 4 5 6 7 8 9Pit depth (mm)

Den

sity

of p

roba

bilit

y (1

mm

)


Prob

abili

ty d

ensit

y(y

rm

m)


func

tion



119901119899 (119905)

=

119899

sum

119898=1

119901119898 (1199050) (119899 minus 1

119899 minus 119898)(

1199050 minus 119905sd1199050 minus 119905sd

)

]119898


)

]

]

119899minus119898

(18)






119891 (120592 1199050 119905) =

119873

sum

119898=1

119901119898 (1199050) (119898 + 120592 (119905 minus 1199050) minus 1

120592 (119905 minus 1199050))(

1199050 minus 119905sd119905 minus 119905sd

)

]119898


)

]

]

120592(119905minus1199050)

(19)





K06K96

=120581(11990506 minus 119905sd)

]

120581(11990596 minus 119905sd)] (20)







0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)



0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)






00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)


05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)



119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)





(22)




119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901119899 (119905) =

119899

sum

119898=1

119901119898 (1199050) 119901119898119899 (1199050 119905) (7)



119872(119905) = K (119905) (8)



119872(119905) = 119899119894119890120588(119905minus119905119894) (9)


K (119905) = 120581(119905 minus 119905sd)] (10)



ΔK (119905) = 120582 (119905)K (119905) Δ119905 (11)


119889K (119905)

119889119905=]

120591K (119905) (12)


120588 (119905) = ln [120581(119905 minus 119905sd)]] (13)


120582 (119905) =]

119905 minus 119905sd (14)




119901119904 = (1199050 minus 119905sd119905 minus 119905sd

)

]

119905 ge 1199050 ge 119905sd (15)


120592 (119898 1199050) =119864 [119899 minus 119898]

Δ119905997888997888997888997888997888rarrΔ119905rarr0

]119898

1199050

(16)










120590)]

minus1120577

(17)






20

25

30

35


25 years 35 years

00

05

10

15

20

25

30

Mean

Mea

n (m

m)

00

05

10

15

20

25

30

Variance

00

05

10

15

Den

sity

of p

roba

bilit

y (1

mm

)



Varia

nce (

mm

2)

(a)

25

30

35

06

08

10

10

15

20

00

02

04


120588(119905

)

120588(119905)

Prob

abili

ty119901119904

119901119904 1199050 = 15 years119901119904 1199050 = 5 years

Func

tion

(b)






a









008

009

010

006

008

010

002

003

004

005

006

007

ILI 2002

000

002

004

000

001


1 2 3 4 5 6 7 8 9Pit depth (mm)

Den

sity

of p

roba

bilit

y (1

mm

)


Prob

abili

ty d

ensit

y(y

rm

m)


func

tion



119901119899 (119905)

=

119899

sum

119898=1

119901119898 (1199050) (119899 minus 1

119899 minus 119898)(

1199050 minus 119905sd1199050 minus 119905sd

)

]119898


)

]

]

119899minus119898

(18)






119891 (120592 1199050 119905) =

119873

sum

119898=1

119901119898 (1199050) (119898 + 120592 (119905 minus 1199050) minus 1

120592 (119905 minus 1199050))(

1199050 minus 119905sd119905 minus 119905sd

)

]119898


)

]

]

120592(119905minus1199050)

(19)





K06K96

=120581(11990506 minus 119905sd)

]

120581(11990596 minus 119905sd)] (20)







0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)



0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)






00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)


05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)



119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)





(22)




119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120590)]

minus1120577

(17)






20

25

30

35


25 years 35 years

00

05

10

15

20

25

30

Mean

Mea

n (m

m)

00

05

10

15

20

25

30

Variance

00

05

10

15

Den

sity

of p

roba

bilit

y (1

mm

)



Varia

nce (

mm

2)

(a)

25

30

35

06

08

10

10

15

20

00

02

04


120588(119905

)

120588(119905)

Prob

abili

ty119901119904

119901119904 1199050 = 15 years119901119904 1199050 = 5 years

Func

tion

(b)






a









008

009

010

006

008

010

002

003

004

005

006

007

ILI 2002

000

002

004

000

001


1 2 3 4 5 6 7 8 9Pit depth (mm)

Den

sity

of p

roba

bilit

y (1

mm

)


Prob

abili

ty d

ensit

y(y

rm

m)


func

tion



119901119899 (119905)

=

119899

sum

119898=1

119901119898 (1199050) (119899 minus 1

119899 minus 119898)(

1199050 minus 119905sd1199050 minus 119905sd

)

]119898


)

]

]

119899minus119898

(18)






119891 (120592 1199050 119905) =

119873

sum

119898=1

119901119898 (1199050) (119898 + 120592 (119905 minus 1199050) minus 1

120592 (119905 minus 1199050))(

1199050 minus 119905sd119905 minus 119905sd

)

]119898


)

]

]

120592(119905minus1199050)

(19)





K06K96

=120581(11990506 minus 119905sd)

]

120581(11990596 minus 119905sd)] (20)







0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)



0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)






00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)


05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)



119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)





(22)




119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












a









008

009

010

006

008

010

002

003

004

005

006

007

ILI 2002

000

002

004

000

001


1 2 3 4 5 6 7 8 9Pit depth (mm)

Den

sity

of p

roba

bilit

y (1

mm

)


Prob

abili

ty d

ensit

y(y

rm

m)


func

tion



119901119899 (119905)

=

119899

sum

119898=1

119901119898 (1199050) (119899 minus 1

119899 minus 119898)(

1199050 minus 119905sd1199050 minus 119905sd

)

]119898


)

]

]

119899minus119898

(18)






119891 (120592 1199050 119905) =

119873

sum

119898=1

119901119898 (1199050) (119898 + 120592 (119905 minus 1199050) minus 1

120592 (119905 minus 1199050))(

1199050 minus 119905sd119905 minus 119905sd

)

]119898


)

]

]

120592(119905minus1199050)

(19)





K06K96

=120581(11990506 minus 119905sd)

]

120581(11990596 minus 119905sd)] (20)







0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)



0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)






00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)


05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)



119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)





(22)




119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891 (120592 1199050 119905) =

119873

sum

119898=1

119901119898 (1199050) (119898 + 120592 (119905 minus 1199050) minus 1

120592 (119905 minus 1199050))(

1199050 minus 119905sd119905 minus 119905sd

)

]119898


)

]

]

120592(119905minus1199050)

(19)





K06K96

=120581(11990506 minus 119905sd)

]

120581(11990596 minus 119905sd)] (20)







0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)



0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)






00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)


05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)



119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)





(22)




119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 800

02

04

Den

sity

of p

roba

bilit

y (1

mm

)

Defect depth (mm)



0 1 2 3 4 500

02

04

06

08

10

Den

sity

of p

roba

bilit

y (1

mm

)Defect depth (mm)






00 01 02 03 04 0500

02

04

06

08

10


Den

sity

of p

roba

bilit

y (y

rm

m)


05 10 15 20 25 30 35000

004

008

012

016

020

CR mean

Mea

n (m

my

r)

Pit depth (mm)

04

08

12

16

20

CR variance

Varia

nce (

times10

minus2

mm

2y

r2)



119865 (119905) = 1 minus exp[minus( 119905120576)

120585

] (21)





(22)




119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865 (119894 119905) =

119894

sum

119895=1

1199011119895 (119905)

=119890minus120588(119905)+120588(119905119896)

1 minus 119890minus120588(119905)+120588(119905119896)

119894

sum

119895=1

(1 minus 119890minus120588(119905)+120588(119905119896))

119895 119894 = 1 119873

(23)


as




120588 (119905) = 120594(119905)120596 (25)

120582 (119905) = 120594120596(119905 minus 119905119896)120596minus1

(26)



Φ (119894 119905) = [119865 (119894 119905)]119898 119898997888rarrinfin997888997888997888997888997888rarr 119866 (119894 119905) (27)





120579 (119894 119905) =

119898

prod

119896=1







1198941015840= 119894 + 119906 (29)





120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120582119890 = exp(120583 minus 119906

120590) (30)



119864119879 =

119873119905

sum

119897=1


2) (31)

where (120583119897119890 1205902119897119890 ) and (120583









008150

Mean

400

500

003

004

005

006

007

0255075

100125

0

100

200

300


000

001

002


25 50 75 100 125 150 175 200 225 250Pit depth (120583m)

1 day

21 days30 days

15 days

7 days

3 days


Varia

nce (

120583m

2d

ay2)

Mea

n (120583

md

ay)

Prob

abili

ty d

ensit

y fu

nctio

n (1

120583m

)










a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












a120582119890



0 5 10 15 20 25 3020

40

60

80

100

120

Empirical


005

-th

resh

old

(11990600005

)

Best fit 11990600005 = 119860(119905 minus 1199050)120581

119860 = 229 1199050 = 0 120581 = 04365



119883119875 = 120583 minus 120590 ln [minus ln (119875)] (32)





















References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























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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