Managing Information Uncertainty in Wave Height Modeling ...

Research ArticleManaging Information Uncertainty in Wave Height Modelingfor the Offshore Structural Analysis through Random Set

Keqin Yan1 Yi Zhang12 Tao Cheng1 Xianfeng Luo1 Quan Zhou1

Kun Fang2 Ankit Garg3 and Akhil Garg4

1School of Civil Engineering Hubei Polytechnic University Huangshi 435003 China2Department of Civil and Earth Resources Engineering Kyoto University Kyoto 6068501 Japan3Civil Engineering Department Indian Institute of Technology Guwahati India4Department of Mechatronics Engineering Shantou University Shantou China

Correspondence should be addressed to Yi Zhang zhang_yi87163com

Received 13 April 2017 Accepted 2 July 2017 Published 15 August 2017

Academic Editor Jurgita Antucheviciene

Copyright copy 2017 Keqin Yan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This chapter presents a reliability study for an offshore jacket structure with emphasis on the features of nonconventional modelingFirstly a random set model is formulated for modeling the random waves in an ocean site Then a jacket structure is investigatedin a pushover analysis to identify the critical wave direction and key structural elements This is based on the ultimate base shearstrengthThe selected probabilistic models are adopted for the important structural members and the wave direction is specified inthe weakest direction of the structure for a conservative safety analysisThe wave height model is processed in a P-box format whenit is used in the numerical analysis The models are applied to find the bounds of the failure probabilities for the jacket structureThe propagation of this wave model to the uncertainty in results is investigated in both an interval analysis and Monte Carlosimulation The results are compared in context of information content and numerical accuracy Further the failure probabilitybounds are compared with the conventional probabilistic approach

1 Introduction

Reliable estimation of extreme values of wave height is animportant prerequisite to the design of coastal and off-shore structures [1 2] Many estimation methods have beenadopted by the researchers and they are summarized in Muirand El-Shaarawi [3] Goda [4] Guedes Soares [5] and Zhangand Lam [6]These covered a wide range of statistical modelsin the fitting to the measured data of ocean parameterslike lognormal [7] Weibull [8] Generalized Gamma [9]and Beta [10] probability distribution models Besides thesethe Peak over Threshold (POT) method is considered tobe quite powerful in the modeling of extreme wave height[11ndash15] (Zhang and Cao 2015) But the main problem inPOT method is the choice of a suitable threshold a matterthat is currently investigated by many researchers [16] Sincethe threshold is used to determine a specific group of data

named as ldquoextremesrdquo the prediction of long term extremevalues is not reliable A generic methodology for selectingthe accurate threshold is still lackingThe review summarizesthat traditional statisticalmethodswhich rely on assumptionsare not suitable in handling such uncertainty Thus thisstudy explores nontraditional models as an alternative in themodeling of extreme wave height

The analysis and design of offshore structures includesthe consideration of waves as the decisive loads A realisticmodeling of thewave loads is particularly important to ensurea sufficiently reliable performance of these structures [17 18]Unfortunately there are large variations in the wave loadThe wave height which is a major factor in the wave load isbelieved to be time-varying and can be significantly differentunder different climate conditions [19] These variations canbe caused by a wide range of factors such as interseasonalchanges interannual changes and interdecadal changes [20]

HindawiComplexityVolume 2017 Article ID 9546272 13 pageshttpsdoiorg10115520179546272

2 Complexity

This has been noticed by many climatologists in recent times[21ndash24]The impact of these variations inmarine applicationsis also mentioned in [25]

Meanwhile it is realized that the engineering safetyassessment may always involve various types of uncertaintymodels In practical problems different forms of modelmay be needed in the same system [26] This requires newdevelopments in themodelingwhich could combine differenttypes of information One of these models is the random setmodeling which is a type of imprecise probability (Walley1990) The random set or sometimes known as P-box is anextension from the traditional probability theory allowingfor intervals or sets of probabilities [27] It can representnot only distributions with unknown parameters but alsodistributions with unknownmodes or unknown dependencyparameters The use of P-box has shown significant advan-tages in many engineering case studies investigated by theprevious researchers [28ndash30] The computational techniquesassociated with imprecise probability in structural analysishave also been developed simultaneously Ferson andDonald[31] have developed a formal probability bounds analysisthat facilitates computation Other similar methods can beseen from Berleant[32] These were shown to be quite usefulin engineering design works [33] The algorithms in thesemethods are mainly belonging to interval arithmetic orMonte Carlo simulations

The main focus of this study is to investigate the uncer-tainties in wave height modeling and its application inoffshore engineering In this study time series measureddata which is from a buoy located in the west coast of USis selected for extreme value modeling The original datais first analyzed in form of parametric models to estimatethe extremes The Peak over Threshold (POT) method isthen applied to different data set For the considerationof information uncertainty a nonstationary Poisson pointprocess is used in the characterization of the occurrencerate for the extremes The stability of the Pareto familyused in extreme value modeling is investigated from severalstatistical points of view to obtain a feasible range for thethresholdThe randomsets theory is emphasized to formulatean imprecise probability model for the extremes by usinga set of thresholds The constructed uncertainty model wasthen introduced to represent the extreme wave height Themain focus of this study is to investigate the handling of theimprecise probabilistic information and also to show thatimprecise probability can provide a framework for processingincomplete information in engineering analysis Thereforethe handling of this uncertainty model is conducted througha reliability analysis with considerations of several uncer-tainties in mechanical properties Finally the conclusions areemphasized

2 Data Used

Thedata used in this study is downloaded fromNational DataBuoyCenter (NDBC) for buoy 46029 (httpwwwndbcnoaagov accessed Nov 2010) which is located in the west ofColumbia River Mouth The exact measured location is at46∘810158403710158401015840N 124∘3010158403710158401015840W The water depth is 1353m and

0050

100150

Hs (

m)

1000 2000 3000 4000 50000(Hrs)

Figure 1 Plot of the SeptemberndashApril Hs time series of 01-02

00

50

100

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


watch circle radius is 483 km (281 yards) The recordedsignificant wave height (Hs) data dates back to 1984 and isavailable for recent times

The collected data of Hs in each year contains 8766observations which are based on hourly record It showsclearly that the winter period which is from September toApril is the roughest time throughout the yearThis dominantseason of strongest storms can cause theHs to be significantlydifferent from the other times and is considered as a differentdata set Thus the data corresponding to this period whichhas a total sample size of 5088 is chosen for the investigationFor amore accurate analysis only parts of the datawhich havehigh degree of data completeness are extracted for this studyThis corresponds to the years of 01-02 02-03 03-04 05-0606-07 and 09-10 which have percentages of missing data of279 033 094 301 214 and 110 respectivelyThe time series data can be seen from the plot in Figures1ndash6 The next section will conduct the statistical modeling ofextreme values in the wave height record

3 Application of POT in Modeling theExtreme Wave Height

Compared to the traditional extreme statistical modelsPOT does not need the data to have stringent statisticalsimilarities [34] As long as the data is stationary havinga weak dependencies structure during its reference periodPOT method is appropriate to model the extremes In thiscase since the original data is selected from a specifiedsample period the winter the statistical property of thetime series data is assumed to be stationary Therefore theuncertainties resulting from different populations could belargely eliminated

Before the POTmethod is applied to the time series datathe declustering scheme is carried out first This includesselections of appropriate threshold 119906 and time span Δ119905 toseparate the ldquoextreme eventsrdquo out from the original data (seeFigure 7) Both parameters are time dependent because of thenaturersquos variability For the selection of a suitable time spanΔ119905a wide range of values have been suggested by the researchers

Complexity 3

0050

100150

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


004080

120160

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


1000 2000 3000 4000 50000(Hrs)

004080

120

Hs (

m)


1000 2000 3000 4000 50000(Hrs)

00306090

120

Hs (

m)


Identified extremes P1 P2 P3 Pn

Δt1 Δt2 Δt3 Δtnminus1middot middot middot

middot middot middot

Δti gt ΔtDeclustering criteria

u

0

Figure 7 Illustration of declustering process in a time series data

(see [26 35ndash37]) It was suggested to choose a time span insuch an optimal waywhich is aminimumvalue to guarantee apersistent Poisson process for the extremes in the time seriesIn this study the intensity function is modeled as a constantimplying that the occurrence rate throughout the referencetime remains unchangedHere threeΔ119905 have been selected inthe testing of temporal dependencies between extremes 1 day

3 days and 10 days The threshold used in this declustering is40 The case where there is no separating zero time spanis also included to assess the effect of dependencies in themodeling While the Δ119905 is increased the number of extremeevents is reduced and thus leads to a large estimate in returnvalue This is quite the same situation when the threshold isincreasing As shown in Figure 8 by incorporating a timespan in the separating of extremes the fitting of the theo-retical Poisson model is highly improved It turns out thatthe dependencies are high when the time span is not appliedHowever the fitting turns to be poor once the time span isincreased to 10 days The reason is the shortage of data for anefficient learning of the methodology Therefore a time spanΔ119905 of 1 day or 3 days is considered as an appropriate value forthe Poisson model In this study a time span Δ119905 = 1 day isadopted

The selection of an appropriate threshold is determinedby the stability of Pareto distribution in the modeling Thecrucial aspects of generalized Pareto distribution (GPD)model are tested by increasing the threshold The change ofextreme occurrence rate for the thresholds is summarized inTable 1 It is clear that the stability of GPD model is highlymaintained within the range of threshold from 24 to 64except for some small deviation of 26 and 28 In fact thethreshold is not considered suitable when it has a value below35 As a result a lower limit of 35 for the threshold shouldbe required

Finally after inspection at these plots it can be concludedthat a range of [36 64] for the threshold is adequate for thePareto model The uncertainties still exist in the selection ofthe threshold in this region

4 Proposed Random Set and P-Box Modeling

The GPD models are found to be valid for a set of thresholdvalues The Poissonrsquos process cannot make certain judgmentto the use of a specific threshold This imprecise informationin the threshold could not be eliminated in a traditionalstatistical way as the threshold is not consistent along the timeseries data Therefore a single value in the threshold is notrecommended for an accurate consideration

In contrast to the traditional statistical method randomset could provide a good combination in the POT model toquantify the uncertainties associated with the thresholdFrom this point of view by taking the advantages of randomset a nonconventional model is proposed Different thresh-olds are considered as various information sources Each ofthese information sources is represented by a random set(I119894 119898119894)These are combined in such a way that the averagingprocedure is applied to the probability mass assignment

119898(119860) = 1119899119899

sum119894=1

119898(119860 119894) (1)

Here the feasible range for the threshold [36 64] isdivided into 14 intervals and 15 thresholds are obtained fromthese intervals Table 2 shows the focal sets for the 25-year return value obtained from each threshold and also theprobability mass assignment

4 Complexity

TheoreticalStationary




0 days 1 day

3 days 10 days

0

02

04

06

08

1

Empi

rical

CD

F

040 0602 108

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1Em

piric

al C

DF

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF

Nonstationary Nonstationary

NonstationaryNonstationary

Figure 8 Comparison of several time spans in the fitting to stationary and nonstationary Poisson model

Based on the Demspter-Shafer theory the bounds forthe cumulative probabilities can be obtained from theserandom sets (see Figure 9) The models are also calculatedfor 50- and 100-year return value and presented in the figureIt can be seen that by increasing the return period thebounds for the estimation values becomemore impreciseThebelief function in this case shows larger variations than theplausibility function and thus gives a more imprecise upperbound for the return value Figure 9 presents a comprehensiveway to represent the uncertaintieswithin theGPDmodelThesensitivity of the estimated return level value with respect tothe threshold can be easily seen from the plot This alsogives an indication whether further information is neededto reduce the uncertainties in the model formulation Itprovides enough flexibility to the engineering decisionsRandom sets approach here provides a general combinationrule which quantifies the statistical uncertainties met in the

POT model The imprecise bounds could capture the fullscope of uncertainty in the selection of the threshold Theanalysis suggests the possibility and advantages of usingthe proposed nonconventional model for the prediction ofextreme significant wave height

5 Reliability Analysis of Offshore StructuresP-Box Approach

In practical problems the random set model needs to beused in a structural numerical analysis This requires newdevelopments in themanagement of uncertainty informationfrom engineering perspectives This section presents a P-box reliability study for an offshore jacket structure withan emphasis on the features of nonconventional random setmodeling

Complexity 5

Table 1 Summary of exceedances for various thresholds in POT

Threshold Number of Exceedances01-02 02-03 03-04 05-06 06-07 09-10 Total

119906 = 24 31 18 31 28 31 31 170119906 = 27 30 17 28 28 27 28 158119906 = 30 26 17 27 28 23 30 151119906 = 33 26 16 25 30 22 30 149119906 = 36 24 14 23 25 21 27 134119906 = 39 21 14 23 27 20 26 131119906 = 42 20 13 23 26 19 21 122119906 = 45 20 16 22 27 19 19 123119906 = 48 18 15 18 27 17 17 112119906 = 51 17 12 16 23 15 16 99119906 = 54 16 12 14 21 14 18 95119906 = 57 16 10 13 19 13 16 87119906 = 60 15 10 12 16 16 13 82119906 = 63 12 8 11 13 13 16 73119906 = 66 10 8 9 11 11 11 60119906 = 70 10 7 8 11 10 8 54

Table 2 Random sets and mass assignment for different threshold

Sources Focal set Mass36 [11941615] 006738 [11901601] 00674 [11891603] 006742 [11891583] 006744 [11921563] 006746 [11891555] 006748 [11981534] 00675 [11931535] 006752 [11901532] 006754 [11831531] 006756 [11881520] 006758 [11911524] 00676 [11831525] 006762 [11761529] 006764 [11661546] 0067

51 Structure Description A realistic North Sea jacket struc-ture taken from the USFOS example model is analyzed [24]The structure is shown in Figure 10

The structure is an 8-leg jacket designed for a waterdepth of 110 meters The legs are arranged in a two by fourrectangular grid with the central pair of legs on the platformOverall the dimensions at top elevation are 27 times 54mwith launch legs twenty meters apart and the dimensions atthe bottom line are 56 times 70m Total height is 142m withhorizontal bracings at 5 levels (see Figure 11) More detailedstructural descriptions can be found in USFOS Manual [38]

52 Static Pushover Analysis The ultimate strength of thejacket structure is determined through a static pushoveranalysis inUSFOS For this example thewave and current are

considered as the only external loads and other environmen-tal factors are ignored The base shear strength is consideredas the resistance in the reliability analysis

A particular attention was paid to the effects of phase anddirection of the comingwaveHere the jacket structure is firstanalyzed with various wave directions and phases to identifythe critical response in the base shear The investigation iscarried out in all wave directions with full considerations ofthe wave phases (Figure 12) It concludes that a direction of180∘ which has the smallest ultimate base shear strengthis found to have the most critical state and is used in thefollowing analysis for a conservative consideration

Besides the modeling of variations in the wave character-istics several uncertainties associated with the key structurersquosmechanical properties are also considered Based on the ulti-mate strength analysis of the platform the diagonal membersbelow the sea level showed high degree of plasticity utilizationbefore the base shear failure of the jacket (see Figure 13)It indicates the importance of the diagonal members overthe other members in the ultimate strength of the wholestructure Thus in present study selected uncertainties asso-ciated with manufacturing and corrosion effect (reduction inthickness) are applied to the key elements of the jacketstructure The yield strength of the steel BS 968 for highstrength tubes is described with a lognormal distributionwith a coefficient of variation of 005sim008 (Baker 1969) Theuncertainty in the thickness associatedwith corrosion effect ismodeled in a normal distribution with a coefficient ofvariation of 017 This is based on an experimental study [24]Detailed summary of the data is given in Table 3

The ultimate base shear resistance is determined bythe environmental design load multiplied by the ReserveStrength Ratio (RSR) which is defined as

RSR = ultimate resistance of intact structuredesign environmental load

(2)

6 Complexity

0

01

02

03

04

05

06

07

08

09

1

10 12 14 16 18

Poss

ibili

ty

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

10 1812 14 16Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 1810Hs

25-year return value



(a)

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty12 14 16 1810

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

12 14 16 1810Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 18 2010Hs

(b)

Figure 9 Lower and upper probabilities for 25 50 and 100 year return value (a) Observed original random sets (b) The P-box plot

Complexity 7

Table 3 Summary of random variables in reliability analysis

Variable Type of distribution Mean Standard deviationYield stress 119865119910 (Nmm2) Lognormal 360 18 (cov = 008)Thickness c (mm) Normal 48 8Wave height119867 (m) P-box mdash mdash

x y

z

Figure 10 Jacket structure overview

In this example a design environmental wave load whichcorresponds to awave height of 25m and period of 16 secondsis used A constant current of 2ms is also included in thedesign of environmental load The response surface methodis utilized to approximate this response value while taking theyield strength and thickness as inputs [39] A quadratic poly-nomial with cross terms is selected in the approximation ofthe relationship between RSR and 119865119910119905

RSR = minus165 + 140119865119910 + 0238119905 minus 01631198652119910 minus 002531199052

+ 0118119865119910119905(3)

where the yield strength 119865119910 is in unit 108Nmm2 thethickness 119905 is in unit 10minus2m The adequacy of the selectedmodel can be checked by the residual plots (see Figure 14)

The response base shear can be approximated by (4)related to wave height and current in the form [40]

119878 = 1198881 (119867 + 1198883V)1198882 (4)

where V is the speed of the current and 1198881 1198882 and 1198883 areconstants As the speed of current is assumed to be fixed

X

YZ

110m

70 m

54 m 27 m

142m

56 m

Transverse directionLongitudinal direction

Conductors

Figure 11 Jacket structure framing

015

30

45

60

75

90

105

120

135

150165

180195

210

225

240

255

270

285

300

315

330345

Figure 12 Comparison of jacket structurersquos ultimate strength withvarious directional wave loads (direction angle is from 119909-axis to 119910-axis)

the factors related to the current speed can be considered asconstants The determination of these constants can beobtained by a curve fitting procedure (Figure 15) This givesthe approximated equation for the response base shear as

119878 = 004143 (119867 + 86745)20786 (5)

where 119878 is the response base shear with unit in MN 119867 isthe wave height with unit in meters The wave period istaken to be 16 seconds which is assumed consistent in the

8 Complexity

Element 355Element 352

Element 455

Element 452

Plastic utilization10

08

06

04

02

0

Figure 13 Plasticity utilization plot

analysisHowever we should realize that there aremany othermethods that could be used to predict the extreme wave load[41ndash49] For example a NewWave profile [50] will give themost probable shape of a large linear crest elevation

The quality of the curve fitting is examined by investi-gating the residuals between the real and fitted ones Theresidual plots are presented in Figure 16 It can be seen thatthe residuals can follow a very strict normal distributionThevariance is generally quite small which indicates a well fittedmodel in the equation However we should realize that theerrors contained in this formulation consist of not only theerror from (3) but also the error from (5) Both equationsapproximating the load and strength contain the errors Thediscussion in the later part of this papermust realize this errorpropagation scheme in advance

Finally the performance function can be expressed as

119866 = 63307 (minus165 + 140119865119910 + 0238119905 minus 01631198652119910minus 002531199052 + 0118119865119910119905) minus 004143 (119867+ 86745)20786

(6)

Consequently the ultimate base shear failure probabilitycan be obtained by giving 119901119891 = Pr(119866 lt 0) As bothprobabilistic model and P-box model exist in the system thefailure probability investigation may involve interval prob-ability analysis This could be viewed as a general mappingfrom the input interval119883119868 to the failure probability

119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)

where119883119868 = 119909 | 119909 isin [119909119897 119909119906]Themain difference in this caseis that the upper and lower bounding values are not preciselydetermined (P-box approach)

P-box is characterized by a mixed case which specifiesthe bounds of probability for an uncertain quantity withunderlying randomness that is not known in detail Suppose119865 and 119865 are nondecreasing functions mapping the real lineR onto [01] and 119865(119909) le 119865(119909) for all 119909 isin R Let [119865 119865]denote the set of all nondecreasing functions F from thereals into [0 1] such that 119865(119909) le 119865(119909) le 119865(119909) When thefunctions 119865 and 119865 circumscribe an imprecisely knownprobability distribution the model of [119865 119865] specified by thepair of functions is called a ldquoprobability boxrdquo or impreciseprobability (Ferson 1998) for that distribution This meansthat if [119865 119865] is a ldquoprobability boxrdquo for a random variable 119883whose distribution 119865 is unknown except that it is within theldquoprobability boxrdquo then119865(119909) is a lower boundon119865(119909)which isthe (imprecisely known) probability that the random variable119883 is smaller than 119909 Likewise 119865(119909) is an upper bound onthe same probability From a lower probability measure 119875 fora random variable 119883 one can compute upper and lowerbounds on distribution functions using the following [27]

119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)

As shown in Figure 17 the left bound119865 is an upper boundon probabilities and a lower bound on quantiles (that is the119909-values)The right bound119865 is a lower bound onprobabilitiesand an upper bound on quantiles

Therefore the results cannot be simply calculated basedon one interval analysisThe P-boxmodel for the wave heightobtained in Section 4 is in a discrete formThe calculation ofthe final results could thus be conducted by the arithmetic ofP-box structures discussed in the work of Tucker and Ferson[51] The interval analysis is carried out to find the failureprobability bound 119875119899119891119868 (119899 = 1 15) corresponding to eachof the discretized 15 intervals 119883119899119868 (119899 = 1 15) in the P-box Equal probabilities are assigned to the results 119875119899119891119868 (119899 =1 15) without the consideration of dependence withinthe system The results are then accomplished by groupingthe 15 response intervals in a stacked P-box (Figure 18)

The maximum and minimum value of the P-box whichcorrespond to 112 times 10minus11 and 277 times 10minus8 could representan envelope for the failure probability 119875119891 This is basicallya constraint in the failure probability while the wave heightprobability is specified in an imprecise form The generalaggregation formulation for the envelope in this case can beexpressed as

envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)

It should be noted that this aggregation rule may notalways be true if the mapping function is changed Alter-natively the failure probability bounds can be computed byan interval Monte Carlo method This procedure is basedon repeated calculations for a series of probability functionswithin the bounded P-box region It is quite efficient for

Complexity 9

Normal probability chart Residual versus fitted value

Histogram Residual versus observed value

011

10

50

90

99

999

Perc

enta

ge

30 35 40 4525Fitted value (RSR)

10 20 30 40 50 60 70 80 90 100 110 1201Observed value (RSR)

0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual

Figure 14 Residual plots for the adequacy check of the quadratic polynomial function

Real valueCurve fitting

200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs

Figure 15 Comparison between the calculated base shear and thecurve fitting

the P-box model which is established from some parameteruncertainties for example the mean or variance but notfeasible for the P-box model which is constructed by therandom sets However the Monte Carlo simulation can stillbe carried out by the aid of parametric model fittings tothe bounded functions Here three typical probability dis-tribution functions Gamma Lognormal and Weibull areutilized Although these distributions may be debatable theyare needed for the evaluation of the performance function bymeans of Monte Carlo simulation The failure probabilities

obtained in these parametric models are comparable Theseare summarized in Table 4 The results obtained fromMonteCarlo simulations are in good agreement with the resultscalculated from the interval analysis A general comparisonbetween these two approaches in the approximated bounds isalso illustrated in Figure 19The good agreement in the resultsfurther proves the applicability of proposed P-box approach

The random set approach proposed in this study providesa general characterization rule in quantifying the statisticaluncertainties associated with the POT model The imprecisebounds could capture the full scope of uncertainty in theextreme value modeling [52] These suggest the possibilityand advantages of using the random set model for the uncer-tainty information management in offshore engineering

6 Conclusion

In this study a nonconventional random set model withthe consideration of nonstationarity is established for theprediction of extreme wave height The random sets areintroduced to combine a set of threshold estimations whichresulted from the threshold uncertainty The applicability ofthe nonconventional model is investigated in view of theprobability bounds for the return values This is justified bythe study in an offshore structural reliability analysis which iscarried out by using various wave height models The impactof the nonstationarity in the extreme wave height distribu-tion is investigated in the failure probability bounds which

10 Complexity


Residual versus observed valueHistogram

0400 08minus08 minus04

Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100

minus04minus06 00 02 04 06minus02

Residual

Freq

uenc

y

Figure 16 Residual plots for the adequacy check of the approximation equation for base shear

1

0

F F

X

Figure 17 P-box model

are obtained through the computational procedures in thestructural analysis The key findings are as follows

(i) The P-box model possesses significant advantagesover the traditional probability It provides a conve-nient and comprehensive way to represent the non-stationary distribution model The estimated boundsfor the return level are found to have enough con-servatism compared to the estimates from traditionalprobabilistic models

(ii) The propagation of P-box model is performed in aninterval analysis and Monte Carlo simulations forthe structural analysis The result obtained in thisreliability analysis is also represented in a P-box

Maximum

Minimum

00102030405060708091

100E minus 10 100E minus 09 100E minus 06100E minus 07 100E minus 05100E minus 11 100E minus 08

Failure probability Pf(middot)

CDF

F(x

)

Figure 18 P-box for the response failure probability

format Compared with the traditional probabilisticapproach the P-box failure probability gives a moreflexible answer while the certain nonstationary effectsare revealed by the imprecise bounds This providesmore information for engineers to make decisionsespecially for an existing structure which is exposedto a changing environment

To the best of our knowledge the uncertainty informationproblem in the offshore engineering is realized throughthe random set theory Further the findings from the case

Complexity 11

Table 4 Failure probability obtained from selected models

Fitted models Minimum 119875119891 Maximum 119875119891 lowastK-S test 119901 value

Lower bound Upper bound 1 quantile in lowerbound

99 quantile in upperbound Lower bound Upper bound

Gamma Gamma 1124 times 10minus11 3840 times 10minus8 08133 04391Lognormal Lognormal 1112 times 10minus11 4351 times 10minus8 05627 02841Weibull Weibull 1070 times 10minus11 3666 times 10minus8 09828 03545lowastThe K-S test is conducted between the Monte Carlo simulated results (sample size = 100000) and the interval analysis results The hypothesis is that they arefrom different continuous distributions Significance level is at 5

0

01

02

03

04

05

06

07

08

09

1

100E minus 05100E minus 07100E minus 09100E minus 11


CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1

100E minus 09 100E minus 07 100E minus 05100E minus 11


CDF

F(x

)

(b)

00102030405060708091

100E minus 10 100E minus 09 100E minus 08 100E minus 07 100E minus 06 100E minus 05100E minus 11


CDF

F(x

)

(c)

Figure 19 Comparison of Monte Carlo simulated P-box (solid line) and interval analysis P-box (dotted line) (a) Gamma model (b)Lognormal model (c) Weibull model

analysis show its benefits for practical use in reliabilityengineeringMany other applications related to reliability anduncertainty analyses instead of only the offshore engineeringapplications can be conducted in future research

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y Zhang C W Kim and K F Tee ldquoMaintenance managementof offshore structures usingMarkov processmodelwith randomtransition probabilitiesrdquo Structure and Infrastructure Engineer-ing vol 13 no 8 pp 1068ndash1080 2017

[2] Y Zhang C Kim K F Tee and J S Lam ldquoOptimal sustain-able life cycle maintenance strategies for port infrastructuresrdquoJournal of Cleaner Production vol 142 pp 1693ndash1709 2017

12 Complexity

[3] L R Muir and A H El-Shaarawi ldquoOn the calculation ofextreme wave heights a reviewrdquo Ocean Engineering vol 13 no1 pp 93ndash118 1986

[4] Y Goda Random seas and design of maritime structures WorldScientific Toh Tuck Link Singapore 2000

[5] C Guedes Soares ldquoChapter 6 Probabilistic models of waves inthe coastal zonerdquo Elsevier Oceanography Series vol 67 no Cpp 159ndash187 2003

[6] Y Zhang and J S L Lam ldquoEstimating the economic losses ofport disruption due to extreme wind eventsrdquo Ocean amp CoastalManagement vol 116 pp 300ndash310 2015

[7] N H Jaspers ldquoStatistical distribution patterns of ocean wavesand of wave induced stresses and motions with engineeringapplicationsrdquo Transactions Society Naval Architects and MarineEngineers vol 64 pp 375ndash432 1956

[8] J A Battjes ldquoLong-term wave height distributions at sevenstations around the British Islesrdquo Deutsche HydrographischeZeitschrift vol 25 no 4 pp 179ndash189 1972

[9] M K Ochi ldquoNew approach for estimating the severest sea statefrom statistical datardquo in Proceedings of the 23rd InternationalConference on Coastal Engineering pp 512ndash525 October 1992

[10] J A Ferreira and C Guedes Soares ldquoModelling the long-term distribution of significant wave height with the Beta andGammamodelsrdquoOcean Engineering vol 26 no 8 pp 713ndash7251999

[11] S Caires andA Sterl ldquo100-year return value estimates for oceanwind speed and significant wave height from the ERA-40 datardquoJournal of Climate vol 18 no 7 pp 1032ndash1048 2005

[12] V V Kharin and FW Zwiers ldquoEstimating extremes in transientclimate change simulationsrdquo Journal of Climate vol 18 no 8 pp1156ndash1173 2005

[13] S Parey F Malek C Laurent and D Dacunha-CastelleldquoTrends and climate evolution statistical approach for very hightemperatures in Francerdquo Climatic Change vol 81 no 3-4 pp331ndash352 2007

[14] M Menendez F J Mendez I J Losada and N E GrahamldquoVariability of extreme wave heights in the northeast PacificOcean based on buoy measurementsrdquo Geophysical ResearchLetters vol 35 no 22 2008

[15] J Kysely J Picek and R Beranova ldquoEstimating extremesin climate change simulations using the peaks-over-thresholdmethod with a non-stationary thresholdrdquo Global and PlanetaryChange vol 72 no 1-2 pp 55ndash68 2010

[16] A Naess and E Haug ldquoExtreme value statistics of wind speeddata by the pot and acermethodsrdquo Journal of OffshoreMechanicsand Arctic Engineering vol 132 no 4 Article ID 041604 2010

[17] T Cheng R Hu W Xu and Y Zhang ldquoMechanical character-izations of oxidizing steel slag soil and applicationrdquo PeriodicaPolytechnica Civil Engineering 2017

[18] T Cheng K Yan J Zheng et al ldquoSemi-analytical and semi-numerical method for the single soil layer consolidation prob-lemrdquoEngineering Computations vol 34 no 3 pp 960ndash987 2017

[19] K Chatziioannou V Katsardi A Koukouselis and E Mis-takidis ldquoThe effect of nonlinear wave-structure and soil-structure interactions in the design of an offshore structurerdquoMarine Structures vol 52 pp 126ndash152 2017

[20] R Sakthivel S Selvi K Mathiyalagan and A Arunku-mar ldquoRobust reliable sampled-data 119867infin control for uncertainstochastic systems with random delayrdquo Complexity vol 21 no1 pp 42ndash58 2015

[21] F Vikeboslash T Furevik G Furnes N G Kvamstoslash andM ReistadldquoWave height variations in the North Sea and on the NorwegianContinental Shelf 1881ndash1999rdquo Continental Shelf Research vol23 no 3-4 pp 251ndash263 2003

[22] X LWang FW Zwiers and V R Swail ldquoNorth Atlantic oceanwave climate change scenarios for the twenty-first centuryrdquoJournal of Climate vol 17 no 12 pp 2368ndash2383 2004

[23] Y Zhang ldquoOn the climatic uncertainty to the environmentextremes a Singapore case and statistical approachrdquo PolishJournal of Environmental Studies vol 24 no 3 pp 1413ndash14222015

[24] Y Zhang ldquoComparing the robustness of offshore structureswith marine deteriorationsmdasha fuzzy approachrdquo Advances inStructural Engineering vol 18 no 8 pp 1159ndash1172 2015

[25] G P Harrison and A R Wallace ldquoSensitivity of wave energy toclimate changerdquo IEEE Transactions on Energy Conversion vol20 no 4 pp 870ndash877 2005

[26] Y Zhang and J S Lam ldquoEstimating economic losses of industryclusters due to port disruptionsrdquo Transportation Research PartA Policy and Practice vol 91 pp 17ndash33 2016

[27] P Walley Statistical reasoning with imprecise probabilities vol42 of Monographs on Statistics and Applied Probability Chap-man and Hall Ltd London UK 1991

[28] E Rubio J W Hall and M G Anderson ldquoUncertainty analysisin a slope hydrology and stability model using probabilistic andimprecise informationrdquo Computers and Geotechnics vol 31 no7 pp 529ndash536 2004

[29] M Oberguggenberger and W Fellin ldquoReliability boundsthrough random sets non-parametric methods and geotechni-cal applicationsrdquo Computers and Structures vol 86 no 10 pp1093ndash1101 2008

[30] J A Enszer Y Lin S Ferson G F Corliss andM A StadtherrldquoProbability bounds analysis for nonlinear dynamic processmodelsrdquo AIChE Journal vol 57 no 2 pp 404ndash422 2011

[31] S Ferson and S Donald Probability Bounds Analysis Prob-abilistic Safety Assessment and Management Springer-VerlagNew York NY USA 1998

[32] D Berleant ldquoAutomatically verified reasoning with both inter-vals and probability density functionsrdquo in Proceedings of theInternational Conference onNumerical Analysis withAUTomaticResult Verification pp 48ndash70 Lafayette La USA

[33] J M Aughenbaugh and C J J Paredis ldquoThe value of usingimprecise probabilities in engineering designrdquo Transactions ofthe ASME Journal ofMechanical Design vol 128 no 4 pp 969ndash979 2006

[34] H Juan W Jian J Xianglin X Chen and L Chen ldquoThelong-term extreme price risk measure of portfolio in inventoryfinancing an application to dynamic impawn rate intervalrdquoComplexity vol 20 no 5 pp 17ndash34 2015

[35] S Corti AGiannini S Tibaldi andFMolteni ldquoPatterns of low-frequency variability in a three-level quasi-geostrophic modelrdquoClimate Dynamics vol 13 no 12 pp 883ndash904 1997

[36] I D Morton J Bowers and G Mould ldquoEstimating returnperiod wave heights and wind speeds using a seasonal pointprocess modelrdquo Coastal Engineering vol 31 no 1-4 pp 305ndash326 1997

[37] T-C Chen and J-H Yoon ldquoInterdecadal variation of the NorthPacific wintertime blockingrdquoMonthly Weather Review vol 130no 12 pp 3136ndash3143 2002

[38] USFOS Course Manual ldquoUSFOS jacket pushoverrdquo 2001 MAR-INTEK SINTEF Group Workshop II

Complexity 13

[39] Y Zhang and J S L Lam ldquoReliability analysis of offshorestructures within a time varying environmentrdquo Stochastic Envi-ronmental Research and Risk Assessment vol 29 no 6 pp 1615ndash1636 2015

[40] J Heideman ldquoParametric response model for wavecurrentjoint probabilityrdquo 1980 API TAC 88-20

[41] J B Rozario P S Tromans P H Taylor and M EfthymiouldquoComparison of loads predicted using rsquoNewwaversquo and otherwave models with measurements on the tern structurerdquoUnder-water Technology vol 19 no 2 pp 24ndash30 1993

[42] J E Taylor and R J Adler ldquoGaussian processes kinematicformulae and Poincarersquos limitrdquo The Annals of Probability vol37 no 4 pp 1459ndash1482 2009

[43] M J Cassidy R E Taylor and G T Houlsby ldquoAnalysis of jack-up units using a Constrained NewWave methodologyrdquo AppliedOcean Research vol 23 no 4 pp 221ndash234 2001

[44] A Naess O Gaidai and S Haver ldquoEfficient estimation ofextreme response of drag-dominated offshore structures byMonte Carlo simulationrdquoOcean Engineering vol 34 no 16 pp2188ndash2197 2007

[45] A Naess O Gaidai and P S Teigen ldquoExtreme responseprediction for nonlinear floating offshore structures by MonteCarlo simulationrdquo Applied Ocean Research vol 29 no 4 pp221ndash230 2007

[46] O Gaidai and A Naess ldquoExtreme response statistics fordrag dominated offshore structuresrdquo Probabilistic EngineeringMechanics vol 23 no 2-3 pp 180ndash187 2008

[47] J J Jensen ldquoExtreme value predictions using Monte Carlo sim-ulations with artificially increased load spectrumrdquo ProbabilisticEngineering Mechanics vol 26 no 2 pp 399ndash404 2011

[48] J J Jensen A S Olsen and A E Mansour ldquoExtreme wave andwind response predictionsrdquo Ocean Engineering vol 38 no 17-18 pp 2244ndash2253 2011

[49] R Coe V Neary M Lawson Y Yu and J Weber ldquoExtremeconditions modeling workshop reportrdquo Tech Rep SNLSAND2014-16384R TP National Renewable Energy Labora-tory (NREL) Golden Colo USA 2014

[50] P S Tromans A R Anaturk and P Hagemeijer ldquoA new modelfor the kinematics of large ocean waves-application as a designwaverdquo inProceedings of the First InternationalOffshore and PolarEngineering Conference International Society of Offshore andPolar Engineers Mountain View Calif USA 1991

[51] T Tucker and S Ferson Probability Bounds Analysis in Envi-ronmental Risk Assessments Applied Biomathamatics SetauketNY USA 2003

[52] M Beer Y Zhang S T Quek and K K Phoon ldquoRelia-bility analysis with scarce information comparing alternativeapproaches in a geotechnical engineering contextrdquo StructuralSafety vol 41 pp 1ndash10 2013

Submit your manuscripts athttpswwwhindawicom

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

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

This has been noticed by many climatologists in recent times[21ndash24]The impact of these variations inmarine applicationsis also mentioned in [25]

Meanwhile it is realized that the engineering safetyassessment may always involve various types of uncertaintymodels In practical problems different forms of modelmay be needed in the same system [26] This requires newdevelopments in themodelingwhich could combine differenttypes of information One of these models is the random setmodeling which is a type of imprecise probability (Walley1990) The random set or sometimes known as P-box is anextension from the traditional probability theory allowingfor intervals or sets of probabilities [27] It can representnot only distributions with unknown parameters but alsodistributions with unknownmodes or unknown dependencyparameters The use of P-box has shown significant advan-tages in many engineering case studies investigated by theprevious researchers [28ndash30] The computational techniquesassociated with imprecise probability in structural analysishave also been developed simultaneously Ferson andDonald[31] have developed a formal probability bounds analysisthat facilitates computation Other similar methods can beseen from Berleant[32] These were shown to be quite usefulin engineering design works [33] The algorithms in thesemethods are mainly belonging to interval arithmetic orMonte Carlo simulations

The main focus of this study is to investigate the uncer-tainties in wave height modeling and its application inoffshore engineering In this study time series measureddata which is from a buoy located in the west coast of USis selected for extreme value modeling The original datais first analyzed in form of parametric models to estimatethe extremes The Peak over Threshold (POT) method isthen applied to different data set For the considerationof information uncertainty a nonstationary Poisson pointprocess is used in the characterization of the occurrencerate for the extremes The stability of the Pareto familyused in extreme value modeling is investigated from severalstatistical points of view to obtain a feasible range for thethresholdThe randomsets theory is emphasized to formulatean imprecise probability model for the extremes by usinga set of thresholds The constructed uncertainty model wasthen introduced to represent the extreme wave height Themain focus of this study is to investigate the handling of theimprecise probabilistic information and also to show thatimprecise probability can provide a framework for processingincomplete information in engineering analysis Thereforethe handling of this uncertainty model is conducted througha reliability analysis with considerations of several uncer-tainties in mechanical properties Finally the conclusions areemphasized

2 Data Used

Thedata used in this study is downloaded fromNational DataBuoyCenter (NDBC) for buoy 46029 (httpwwwndbcnoaagov accessed Nov 2010) which is located in the west ofColumbia River Mouth The exact measured location is at46∘810158403710158401015840N 124∘3010158403710158401015840W The water depth is 1353m and

0050

100150

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


00

50

100

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


watch circle radius is 483 km (281 yards) The recordedsignificant wave height (Hs) data dates back to 1984 and isavailable for recent times

The collected data of Hs in each year contains 8766observations which are based on hourly record It showsclearly that the winter period which is from September toApril is the roughest time throughout the yearThis dominantseason of strongest storms can cause theHs to be significantlydifferent from the other times and is considered as a differentdata set Thus the data corresponding to this period whichhas a total sample size of 5088 is chosen for the investigationFor amore accurate analysis only parts of the datawhich havehigh degree of data completeness are extracted for this studyThis corresponds to the years of 01-02 02-03 03-04 05-0606-07 and 09-10 which have percentages of missing data of279 033 094 301 214 and 110 respectivelyThe time series data can be seen from the plot in Figures1ndash6 The next section will conduct the statistical modeling ofextreme values in the wave height record

3 Application of POT in Modeling theExtreme Wave Height

Compared to the traditional extreme statistical modelsPOT does not need the data to have stringent statisticalsimilarities [34] As long as the data is stationary havinga weak dependencies structure during its reference periodPOT method is appropriate to model the extremes In thiscase since the original data is selected from a specifiedsample period the winter the statistical property of thetime series data is assumed to be stationary Therefore theuncertainties resulting from different populations could belargely eliminated

Before the POTmethod is applied to the time series datathe declustering scheme is carried out first This includesselections of appropriate threshold 119906 and time span Δ119905 toseparate the ldquoextreme eventsrdquo out from the original data (seeFigure 7) Both parameters are time dependent because of thenaturersquos variability For the selection of a suitable time spanΔ119905a wide range of values have been suggested by the researchers

Complexity 3

0050

100150

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


004080

120160

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


1000 2000 3000 4000 50000(Hrs)

004080

120

Hs (

m)


1000 2000 3000 4000 50000(Hrs)

00306090

120

Hs (

m)






u

0









119898(119860) = 1119899119899

sum119894=1

119898(119860 119894) (1)


4 Complexity





0 days 1 day

3 days 10 days

0

02

04

06

08

1

Empi

rical

CD

F

040 0602 108

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1Em

piric

al C

DF

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF








Complexity 5



119906 = 24 31 18 31 28 31 31 170119906 = 27 30 17 28 28 27 28 158119906 = 30 26 17 27 28 23 30 151119906 = 33 26 16 25 30 22 30 149119906 = 36 24 14 23 25 21 27 134119906 = 39 21 14 23 27 20 26 131119906 = 42 20 13 23 26 19 21 122119906 = 45 20 16 22 27 19 19 123119906 = 48 18 15 18 27 17 17 112119906 = 51 17 12 16 23 15 16 99119906 = 54 16 12 14 21 14 18 95119906 = 57 16 10 13 19 13 16 87119906 = 60 15 10 12 16 16 13 82119906 = 63 12 8 11 13 13 16 73119906 = 66 10 8 9 11 11 11 60119906 = 70 10 7 8 11 10 8 54











(2)

6 Complexity

0

01

02

03

04

05

06

07

08

09

1

10 12 14 16 18

Poss

ibili

ty

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

10 1812 14 16Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 1810Hs




(a)

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty12 14 16 1810

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

12 14 16 1810Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 18 2010Hs

(b)


Complexity 7



x y

z




+ 0118119865119910119905(3)



119878 = 1198881 (119867 + 1198883V)1198882 (4)


X

YZ

110m

70 m

54 m 27 m

142m

56 m


Conductors


015

30

45

60

75

90

105

120

135

150165

180195

210

225

240

255

270

285

300

315

330345



119878 = 004143 (119867 + 86745)20786 (5)


8 Complexity


Element 455

Element 452


08

06

04

02

0






(6)


119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)



119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)




envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)


Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3

0050

100150

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


004080

120160

Hs (

m)

1000 2000 3000 4000 50000(Hrs)


1000 2000 3000 4000 50000(Hrs)

004080

120

Hs (

m)


1000 2000 3000 4000 50000(Hrs)

00306090

120

Hs (

m)






u

0









119898(119860) = 1119899119899

sum119894=1

119898(119860 119894) (1)


4 Complexity





0 days 1 day

3 days 10 days

0

02

04

06

08

1

Empi

rical

CD

F

040 0602 108

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1Em

piric

al C

DF

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF








Complexity 5



119906 = 24 31 18 31 28 31 31 170119906 = 27 30 17 28 28 27 28 158119906 = 30 26 17 27 28 23 30 151119906 = 33 26 16 25 30 22 30 149119906 = 36 24 14 23 25 21 27 134119906 = 39 21 14 23 27 20 26 131119906 = 42 20 13 23 26 19 21 122119906 = 45 20 16 22 27 19 19 123119906 = 48 18 15 18 27 17 17 112119906 = 51 17 12 16 23 15 16 99119906 = 54 16 12 14 21 14 18 95119906 = 57 16 10 13 19 13 16 87119906 = 60 15 10 12 16 16 13 82119906 = 63 12 8 11 13 13 16 73119906 = 66 10 8 9 11 11 11 60119906 = 70 10 7 8 11 10 8 54











(2)

6 Complexity

0

01

02

03

04

05

06

07

08

09

1

10 12 14 16 18

Poss

ibili

ty

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

10 1812 14 16Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 1810Hs




(a)

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty12 14 16 1810

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

12 14 16 1810Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 18 2010Hs

(b)


Complexity 7



x y

z




+ 0118119865119910119905(3)



119878 = 1198881 (119867 + 1198883V)1198882 (4)


X

YZ

110m

70 m

54 m 27 m

142m

56 m


Conductors


015

30

45

60

75

90

105

120

135

150165

180195

210

225

240

255

270

285

300

315

330345



119878 = 004143 (119867 + 86745)20786 (5)


8 Complexity


Element 455

Element 452


08

06

04

02

0






(6)


119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)



119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)




envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)


Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity





0 days 1 day

3 days 10 days

0

02

04

06

08

1

Empi

rical

CD

F

040 0602 108

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1Em

piric

al C

DF

08 10 0402 06

Theoretical CDF

0

02

04

06

08

1

Empi

rical

CD

F

08 10 0402 06

Theoretical CDF








Complexity 5



119906 = 24 31 18 31 28 31 31 170119906 = 27 30 17 28 28 27 28 158119906 = 30 26 17 27 28 23 30 151119906 = 33 26 16 25 30 22 30 149119906 = 36 24 14 23 25 21 27 134119906 = 39 21 14 23 27 20 26 131119906 = 42 20 13 23 26 19 21 122119906 = 45 20 16 22 27 19 19 123119906 = 48 18 15 18 27 17 17 112119906 = 51 17 12 16 23 15 16 99119906 = 54 16 12 14 21 14 18 95119906 = 57 16 10 13 19 13 16 87119906 = 60 15 10 12 16 16 13 82119906 = 63 12 8 11 13 13 16 73119906 = 66 10 8 9 11 11 11 60119906 = 70 10 7 8 11 10 8 54











(2)

6 Complexity

0

01

02

03

04

05

06

07

08

09

1

10 12 14 16 18

Poss

ibili

ty

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

10 1812 14 16Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 1810Hs




(a)

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty12 14 16 1810

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

12 14 16 1810Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 18 2010Hs

(b)


Complexity 7



x y

z




+ 0118119865119910119905(3)



119878 = 1198881 (119867 + 1198883V)1198882 (4)


X

YZ

110m

70 m

54 m 27 m

142m

56 m


Conductors


015

30

45

60

75

90

105

120

135

150165

180195

210

225

240

255

270

285

300

315

330345



119878 = 004143 (119867 + 86745)20786 (5)


8 Complexity


Element 455

Element 452


08

06

04

02

0






(6)


119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)



119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)




envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)


Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5



119906 = 24 31 18 31 28 31 31 170119906 = 27 30 17 28 28 27 28 158119906 = 30 26 17 27 28 23 30 151119906 = 33 26 16 25 30 22 30 149119906 = 36 24 14 23 25 21 27 134119906 = 39 21 14 23 27 20 26 131119906 = 42 20 13 23 26 19 21 122119906 = 45 20 16 22 27 19 19 123119906 = 48 18 15 18 27 17 17 112119906 = 51 17 12 16 23 15 16 99119906 = 54 16 12 14 21 14 18 95119906 = 57 16 10 13 19 13 16 87119906 = 60 15 10 12 16 16 13 82119906 = 63 12 8 11 13 13 16 73119906 = 66 10 8 9 11 11 11 60119906 = 70 10 7 8 11 10 8 54











(2)

6 Complexity

0

01

02

03

04

05

06

07

08

09

1

10 12 14 16 18

Poss

ibili

ty

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

10 1812 14 16Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 1810Hs




(a)

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty12 14 16 1810

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

12 14 16 1810Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 18 2010Hs

(b)


Complexity 7



x y

z




+ 0118119865119910119905(3)



119878 = 1198881 (119867 + 1198883V)1198882 (4)


X

YZ

110m

70 m

54 m 27 m

142m

56 m


Conductors


015

30

45

60

75

90

105

120

135

150165

180195

210

225

240

255

270

285

300

315

330345



119878 = 004143 (119867 + 86745)20786 (5)


8 Complexity


Element 455

Element 452


08

06

04

02

0






(6)


119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)



119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)




envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)


Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity

0

01

02

03

04

05

06

07

08

09

1

10 12 14 16 18

Poss

ibili

ty

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

10 1812 14 16Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 1810Hs




(a)

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty12 14 16 1810

Hs

0

01

02

03

04

05

06

07

08

09

1

Poss

ibili

ty

12 14 16 1810Hs

0010203040506070809

1

Poss

ibili

ty

12 14 16 18 2010Hs

(b)


Complexity 7



x y

z




+ 0118119865119910119905(3)



119878 = 1198881 (119867 + 1198883V)1198882 (4)


X

YZ

110m

70 m

54 m 27 m

142m

56 m


Conductors


015

30

45

60

75

90

105

120

135

150165

180195

210

225

240

255

270

285

300

315

330345



119878 = 004143 (119867 + 86745)20786 (5)


8 Complexity


Element 455

Element 452


08

06

04

02

0






(6)


119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)



119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)




envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)


Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7



x y

z




+ 0118119865119910119905(3)



119878 = 1198881 (119867 + 1198883V)1198882 (4)


X

YZ

110m

70 m

54 m 27 m

142m

56 m


Conductors


015

30

45

60

75

90

105

120

135

150165

180195

210

225

240

255

270

285

300

315

330345



119878 = 004143 (119867 + 86745)20786 (5)


8 Complexity


Element 455

Element 452


08

06

04

02

0






(6)


119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)



119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)




envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)


Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity


Element 455

Element 452


08

06

04

02

0






(6)


119883119868 997888rarr 119875119891119868 = 119875119891 | 119875119891 isin [119875119891119897 119875119891119906] (7)



119865119883 (119909) = 1 minus 119875 (119883 gt 119909) 119865119883 (119909) = 119875 (119883 le 119909)

(8)




envelope (1198751119891 119875119899119891)

= [min(1198751119891 119875119899119891) max (1198751119891 119875119899

119891)]

(9)


Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 9



011

10

50

90

99

999

Perc

enta

ge



0

5

10

15

20

Freq

uenc

y

minus0010

minus0005

0000

0005

0010

Resid

ual

minus0010

minus0005

0000

0005

0010

Resid

ual

000 001minus001

Residual

0000 0004 0008minus0004minus0008

Residual



200E + 01

400E + 01

600E + 01

800E + 01

100E + 02

120E + 02

140E + 02

5030 402010

S(M

N)

Hs





6 Conclusion


10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




10 Complexity




Residual

001

10

50

90

99

9999

Perc

enta

ge

minus08

minus04

00

04

08

Resid

ual

20 4minus4 minus2

Fitted value

minus08

minus04

00

04

08

Resid

ual

100 200 300 400 500 600 700 800 9001 1000Observed value

0

25

50

75

100


Residual

Freq

uenc

y


1

0

F F

X





Maximum

Minimum

00102030405060708091



CDF

F(x

)




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 11






0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(a)

0

01

02

03

04

05

06

07

08

09

1



CDF

F(x

)

(b)

00102030405060708091



CDF

F(x

)

(c)





References



12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




12 Complexity





































Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 13






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Managing Information Uncertainty in Wave Height Modeling ...

Documents

Transcript of Managing Information Uncertainty in Wave Height Modeling ...