Research Article Multistage Warning Indicators of Concrete ...downloads.hindawi.com/journals/mpe/2016/6581204.pdf · Research Article Multistage Warning Indicators of Concrete Dam

Research ArticleMultistage Warning Indicators of Concrete Dam underInfluences of Random Factors

Guang Yang12 and Meng Yang13

1State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering Hohai University Nanjing 210098 China2National Engineering Research Center of Water Resources Efficient Utilization and Engineering Safety Hohai UniversityNanjing 210098 China3College of Water-Conservancy and Hydropower Hohai University Nanjing 210098 China

Correspondence should be addressed to Meng Yang ymym 059126com

Received 25 November 2015 Revised 25 February 2016 Accepted 21 March 2016

Academic Editor Paolo Crippa

Copyright copy 2016 G Yang and M YangThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Warning indicators are required for the real-timemonitoring of the service conditions of dams to ensure safe andnormal operationsWarnings are traditionally targeted at some ldquosingle point deformationrdquo by deformation measuring points of concrete dam andscientific warning theory on ldquooverall deformationrdquo measured is nonexistent Furthermore the influences of random factors arenot considered In this paper the overall deformation of the dam was seen as a deformation system of single interactionalobservation points with different contribution degrees The spatial deformation entropy which describes the overall deformationwas established and the fuzziness indicator that measures the influence of complex random factors onmonitoring values accordingto cloud theory was constructed On this basis multistage warning indicators of ldquospatial deformationrdquo that consider fuzziness andrandomness were determined Analysis showed that the change law of information entropy of the damrsquo overall deformation isidentical to the real change law of the dam thus it reflects the real deformation state of the dam Moreover the identified warningindicators improved the warning ability of concrete dams

1 Introduction

Deformation is one of the major monitored items in damsafety Concrete dams are exposed to influences of variousnondeterministic settings such as the load effect of waterlevel uplift pressure and wind waves caused by hydrologicand hydraulic uncertainties as well as geological andmaterialuncertainties such as shearing and compressive strengthThus a concrete dam is a complicated system of nonde-terministic settings that are affected by various complexrandom factors [1 2] Considering a damrsquos long-term serviceconducting timely and effective warning against emergenciesthrough real-time monitoring is key to its safe operation [3]

The monitoring of dam safety is an important researchsubject in advanced mechanics and mathematics theories In1950 Tonini categorized the factors influencing the displace-ment of dam into water pressure temperature and effective-ness for a given period [4] These factors were expressed inthe polynomials of specific functions before a statistic model

with regression analysis was establishedThen the determin-istic model andmixedmodel were consulted for deformationof concrete dam and introduced finite element to monitorand evaluate the safety of dams [5] Furthermore manyscholars brought new achievements in diagnosing dam safetyfrom many aspects In 2009 Gu and Wang established thecatastrophe model of time-dependent component on thebasis of catastrophe theory and proposed the method todetermine the threshold value of the structure displacementof the dam [6] On the basis of the POT model in extremevalue theory in 2012 Su et al estimated the warningindicators by setting the threshold value and combiningthe probability of dam deformation with transfinite datasequence as the subject of modeling analysis [7] Warningindicators are sure to have some fuzziness and randomnessbecause of the influences of various complex random factorsOn the basis of the fuzzy finite element Chen (2006) realizedthe nondeterministic optimal control on roller compactedconcrete dam [8] Although the above-mentioned theories

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6581204 12 pageshttpdxdoiorg10115520166581204

2 Mathematical Problems in Engineering

and methods complement and improve traditional methodsin solving difficult problems in dam safety such approachesonly address the warning of ldquopointrdquo deformation and not thescientific ldquospatial and overallrdquo deformation Further studieson the deformation of concrete dams must consider theinfluences of complex random factors Thus the expressionof a damrsquos overall deformation should be constructed andscientific and accurate warning indicators that considerrandomness and fuzziness should be determined

This paper began with an analysis of an indicator offuzziness that affects the value of monitoring the dam bystudying the influences of fuzziness and randomness fromlong-term service of the dam based on cloud theory [9ndash11] Thereafter the relationship between a damrsquos overalldeformation and deformation of single observation pointswas analyzed through information entropy and synergeticsIn the analysis overall deformation refers to a systemwhereinsingle observation points with different contribution degrees(weights) influence one another A criterion indicator mea-suring the overall deformation conditions was constructedOn this basis multistage warning indicators for the overalldeformation of concrete dam considering fuzziness andrandomness were determined Therefore nondeterministicoptimal control was achieved [12] This paper concludes witha project case that verified the feasibility of the proposedtheory

2 Nondeterministic Optimal Control

Hydraulic engineering considers that some fuzziness andrandomness in dams are inevitable because of the influencesof various random factors such as the nondeterminacy ofmechanical parameters imposed load and boundary condi-tions Statistically the smaller probability of dam displace-ment indicates that the dam is in a more dangerous state120583 + 3120590 and 120583 minus 3120590 can be used as a warning indicator in oneconfidence coefficient of the dam if the displacement obeysthe normal distribution of mean value and variance and is inthe range of 120583plusmn3120590 in deterministic optimal control In realitygiven that dams are affected by random factors warningindicators will have a range of variations For example inFigure 1 the warning value of upstream displacement has amaximum control value and minimum control value

As the foundation of cloud theory the cloud model isprecisely both controlled and uncontrolled in microscopescale119880 is the time series ofmonitoring the damdeformationand 119862 is the qualitative judgment on dam safety If thequantitative value 119909 isin 119880 119906(119909) fall in [0 1] and follow theprobability distribution law

120583 119880 997888rarr [0 1] forall119909 isin 119880 119909 997888rarr 120583 (119909) (1)

The distribution of 119909 in 119880 is called cloud and (119909 119906) is thecloud droplet

In Figure 2 for observation point 119894 if the range whereinthe cloud droplet falls is given the upper bound and lower

bound of the cloud droplet in the cloud model 119910119906119894and 119910119897

119894can

be expressed as follows

119910119906

119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)

22(119864119899119894+3119867119890119894)

2

119910119897

119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)

22(119864119899119894minus3119867119890119894)

2

(2)

119909119894119895is the value 119895 of observation point 119894 and the fuzziness

Δ119894119895can be calculated by

Δ119894119895= 119910119906

119894119895minus 119910119897

119894119895 (3)

In this equation 119910119906119894119895is the upper limit value of 119909

119894119895in the

range and 119910119897119894119895is the lower limit value of 119909

119894119895in the range

Given the influence of random factors when 119909119894119895ge 0

119909119894119895changes in the range of [119909

119894119895minus 119909119894119895Δ119894119895 119909119894119895+ 119909119894119895Δ119894119895] when

119909119894119895lt 0 it is in the range of [119909

119894119895+ 119909119894119895Δ119894119895 119909119894119895minus 119909119894119895Δ119894119895]

When drawing up the warning indicator for downstreamdeformation the maximum and minimum control values ofthe indicator can be determined when the significance level is120572 thus indicating that the nondeterministic optimal controlhas been achieved

3 Methods of Characterizing Contributions ofSingle Observation Point

The overall deformation condition of the concrete dam isusually exposed to the influences of water pressure and tem-perature and is related to many factors including the physicaland mechanical properties of dam materials body structuregeology and hydrology and it could be referred to in Figure 3The principles of synergetics posit that a concrete dam isa synthesis of feature points with different contributions(weights) that influence one another The contribution of asingle observation point needs to be studied to construct areasonable expression of overall deformation

31 Construction of the Indicator Set of a Single ObservationPoint Weight Entropy [13ndash15] a basic concept in thermody-namics refers to a state function in a system The conceptof information entropy is a measurement of the systemrsquosdisorder and nondeterminacy [15] The measured value 119895 onobservation point 119894 is 119909

119894119895and its corresponding entropy is 119878

119894119895

According to entropy theory when an observation point is ina more dangerous state the system is in greater disorder andits entropy value is smaller Thus the following is obtained

119878119894119895= minus [120583

119894119895ln 120583119894119895+ (1 minus 120583

119894119895) ln (1 minus 120583

119894119895)] (4)

120583119894119895=

int

119909119894119895

minusinfin

119891 (120589) 119889120589 119909119894119895ge 0

int

+infin

119909119894119895

119891 (120589) 119889120589 119909119894119895lt 0

(5)

Formula (4) defines the information entropy of themeasurement value No matter what distribution 119909

119894119895 obeys

if the probability density function of the measured value is

Mathematical Problems in Engineering 3

War

ning

val

ue o

f dow

nstre

am

disp

lace

men

t

War

ning

val

ue o

f ups

tream

di

spla

cem

ent

Displacement

Probability density

Displacement

The m

axim

um co

ntro

l val

ue o

f war

ning

va

lue o

f dow

nstre

am d

ispla

cem

ent

The m

inim

um co

ntro

l val

ue o

f war

ning

va

lue o

f dow

nstre

am d

ispla

cem

ent

The m

axim

um co

ntro

l val

ue o

f war

ning

va

lue o

f ups

tream

disp

lace

men

t

The m

inim

um co

ntro

l val

ue o

f war

ning

va

lue o

f ups

tream

disp

lace

men

t

Probability density

Figure 1 Diagram of deterministic optimal control and nondeterministic optimal control

Degree of certainty

Upper limit of cloud dropletsLower limit of cloud droplets

Cloud dropletsEx

Figure 2 Diagram of the range where the cloud droplet falls

known the corresponding information entropy sequence canbe calculated Deformation of dam can be divided into threeparts water pressure component temperature componentand aging component Aging component comprehensivelyreflects the creep and plastic deformation of dam concreteand rock foundation and compression deformation of geo-logical structure of rock foundation At the same time italso includes the irreversible displacement caused by the damcrack and the autogenous volume deformation It changesdramatically in the early stage and gradually tends to be stablein the later stage The project selected in this paper is a damwhich has worked for many years and its aging componentstend to be stable and the deformation value is stable in annualperiod rule which obeys normal distribution

The indicator measuring how much information is con-tained in 119909

119894119895is the inverse entropy 119863

119894119895 119863119894119895= 1 minus 119878

119894119895

When one measured value reflects more information theentropy value 119878

119894119895will be smaller and its inverse entropy 119863

119894119895

will be greater Thus 119863119894119895can be used to measure how much

information is reflected by a single measured value

Suppose the weight distribution of all observation pointsis 120596119894| 119894 = 1 2 119899 where 119899 is the number of points

observed and 120596119894will meet the following requirement 120596

119894ge 0

and sum120596119894= 1 The entropy 119878

119894119895of the object matrix of the

deformation measured value 119909119894119895| 119894 = 1 sim 119899 119895 = 1 sim 119898

can be computed through (4) and (5) The inverse entropymatrix is expressed as follows

119863119894119895=

[[[[[[

[

1198631111986312sdot sdot sdot 119863

1119898

1198632111986322sdot sdot sdot 119863

2119898

11986311989911198631198992sdot sdot sdot 119863

119899119898

]]]]]]

]

(6)

In this matrix 119863119894119895is the inverse entropy of 119909

119894119895 The weight

of the feature points was traced by the projection pursuitmethod

32 Process of Calculating the Weight of a Single Obser-vation Point By using the projection pursuit method [1617] the high-dimensional data can be projected to lowdimension space and projection that reflects the structureor features of high-dimensional data is pursued to analyzehigh-dimensional dataThismethod is advantageous becauseit is highly objective robust resistant to interference andaccurate The steps are as follows

Step 1 The extreme value of the inverse entropy matrix wasnormalized through the following equation

119863lowast

119894119895=

119863119894119895minus [119863119895]min

[119863119895]max minus [119863119895]min

(7)

where [119863119895]max and [119863119895]min are the maximum and minimum

values of line 119895 in the matrix respectively


Weight of surveypoints 1

Overall deformation of concrete dam

Deformation ofsurvey points 1

Deformation ofsurvey points n



Weight of surveypoints n

middot middot middot


Figure 3 Diagram of overall deformation system of concrete dam

Step 2 The normalized value 119863lowast119894119895

was projected to unitdirection119875 119875 = (119901

1 1199012 119901

119895) and1199012

1+1199012

2+sdot sdot sdot+119901

2

119895= 1The

indicator function of the projection 119866(119894) was constructed

119866 (119894) =

119898

sum

119895=1

119901119895119863lowast

119894119895 (119894 = 1 2 119899) (8)

Step 3 The objective function of the projection was con-structed The best direction for the projection was estimatedby solving the maximization problem of the objective func-tion in the constraint condition

Objective function Max 119867 (119901) = 119878119866sdot 119876119866

Constraint condition119898

sum

119895=1

1199012

119895= 1

(9)

In this equation 119878119866is the divergent degree of the projection

119876119866is the local density of 1D data points along 119875 and is

expressed as follows

119878119866= [sum119899

119894=1(119866 (119894) minus 119892 (119894))

2

119899 minus 1]

05

119876119866=

119899

sum

119894=1

119899

sum

119895=1

(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903

119894119895)

(10)

where119892(119894) is themean value of this sequence119877 is the windowradius of the local density and 119877 = 01119878

119866in this paper 119903

119894119895is

the distance between two projection values and 119891(119905) is theunit step function 119891(119905) is equal to 1 as 119905 is greater than 0Otherwise 119891(119905) is equal to 0

Step 4 The projection value of one sample point was com-puted by substituting the best direction 119875lowast into (8) 120596

119894can be

calculated by normalizing the projection value

120596119894=

119866lowast

(119894)

sum119899

119895=1119866lowast (119895)

119894 = 1 2 119899 (11)

4 Study on Equivalent Model ofDamrsquos Overall Deformation

The overall deformation of a dam can be considered a defor-mation system of feature points with different contributionsthat influence one another as well as observation points ofthe deformation as feature pointsThe deformation conditionwas analyzed systemically and the overall deformation wasexpressed by the evolution of equation of all feature pointsThe deformation condition was described qualitatively bythe tectonic type of information entropy The absolute valueof the information entropy measures the danger level ofthe deformation value Smaller absolute value means higherdanger level The positive and negative values indicate thedirection of the deformation A positive value correspondsto downstream deformation whereas a negative value corre-sponds to upstream deformation The influences of randomfactors were considered and the fuzzy information entropywas constructed

41 Constructing Fuzzy Information Entropy of SingleMeasured Value The downstream deformation is positivewhereas the upstream deformation is negative

When the observation point moves downstream make120583119894119895= int119909119894119895

minusinfin

119891(120589)119889120589 and according to the definition of infor-mation entropy the information entropy of 119909

119894119895is expressed

as (4)Considering the influence of Δ

119894119895 120583119894119895will float in [1205830

119894119895 1205831

119894119895]

the following is then obtained

1205830

119894119895= int

119909119894119895minus119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

1205831

119894119895= int

119909119894119895+119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

(12)

When the observation point moves upstream make 120583119894119895=

int+infin

119909119894119895

119891(120589)119889120589 the information entropy of 119909119894119895is defined as

follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583

119894119895) ln (1 minus 120583

119894119895) (13)


0

Dow

nstre

am d

ispla

cem

ent

Ups

tream

di

spla

cem

ent

105

Sij

1205830

1 minus 1205830 120583

Figure 4 Diagram of 119878119894119895changes with the change of 120583

119894119895if 119864119909ge 0

Considering the influence ofΔ119894119895 120583119894119895floats in [1205830

119894119895 1205831

119894119895]The

following is then obtained

1205830

119894119895= int

+infin

119909119894119895minus119909119894119895Δ 119894119895

119891 (120589) 119889120589

1205831

119894119895= int

+infin

119909119894119895+119909119894119895Δ 119894119895

119891 (120589) 119889120589

(14)

Considering the influences of random factors the infor-mation entropy of 119909

119894119895minus119878119894119895floats in [1198780

119894119895 1198781

119894119895] whichwas defined

as the fuzzy information entropy of 119909119894119895

Take the downstream as an example Influenced by deter-mined and random factors the fuzzy entropy of 119878

119894119895changes

into the range of 119878119894119895in the range of [1205830

119894119895 1205831

119894119895]

42 Methods for Determining the Range of the InformationEntropy of Single Measured Value For Figure 4 the expecta-tion of one deformation monitoring sequence sample at oneobservation point (119864

119909ge 0 119878

119894119895) changes with the change of

120583119894119895 as shown in Figure 4 where 120583

0= int0

minusinfin

119891(120589)119889120589When the dam deforms downstream the change law of

119878119894119895is as follows 119878

119894119895will increase with increasing 120583

119894119895when 120583

119894119895

is in the range of (1205830 05) 119878

119894119895will decrease with decreasing

120583119894119895when 120583

119894119895is in the range of (05 1) when 120583

119894119895= 05 119878

119894119895will

reach themaximumvalue If 05 is in the range of [1205830119894119895 1205831

119894119895] the

maximum of 119878119894119895is 1198781119894119895when 120583

119894119895= 05 and its minimum value

is 1198780119894119895at the endpoint If 05 is not in the range of [1205830

119894119895 1205831

119894119895]

119878119894119895will have its maximum value 1198781

119894119895and minimum value 1198780

119894119895

at endpoints When the dam deforms upstream 119878119894119895will rise

with the rise of 120583119894119895and it will have its maximum value 1198781

119894119895and

minimum value 1198780119894119895at endpoints

The expectation of one deformationmonitoring sequencesample at one observation point (119864

119909lt 0 119878

119894119895) changes with

01

05

Dow

nstre

amU

pstre

am d

ispla

cem

ent

Sij

1205830

1 minus 1205830 120583

disp

lace

men

t


119894119895if 119864119909lt 0

the change of 120583119894119895as shown in Figure 5 where 120583

0= int0

minusinfin

119891(120589)119889120589

and Figure 5 is presented for 119878119894119895changes with the change of

120583119894119895When the damdeforms downstream the change law of 119878

119894119895

is as follows 119878119894119895will increase with decreasing 120583

119894119895and will have

its maximum value 1198781119894119895and minimum value 1198780

119894119895at endpoints

When the damdeforms upstream the figure shows that when120583119894119895is in the range of (1minus120583

0 05) 119878

119894119895will increase with the drop

of 120583119894119895 when 120583

119894119895is in the range of (05 1) 119878

119894119895will decrease with

the increase of 120583119894119895 when 120583

119894119895= 05 119878

119894119895will reach theminimum

value If 05 is contained in the range of [1205830119894119895 1205831

119894119895] 119878119894119895will have

its minimum value 1198780119894119895at 120583119894119895= 05 and have its maximum

value 1198781119894119895at endpoint if 05 is not contained in the range 119878

119894119895

will have its maximum value 1198781119894119895and minimum value 1198780

119894119895at

endpoints

43 Construction of the Fuzzy Information Entropy of OverallDeformation On the basis of the above results the expres-sion of information entropy of overall deformation can bededucedThe contribution of the order degree of observationpoint 119894 is 120596

119894120583119894119895 make 1205831

119894119895= 120583119894119895and 1205832

119894119895= 1 minus 120583

119894119895 According to

the broad definition of information entropy when the dammove deforms downstream the expression of informationentropy of overall deformation is expressed as follows

119878119895= minus

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895(ln120596119894+ ln 120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln120596119894minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln 120583119896119894119895


Choice of sample

Input sample eigenvalues

Calling cloud model program

Generating range of cloud fall

Calculating the fuzzinessof all measured value

Constructing deformed entropyof one measured value

Calculating weight of singleobservation point

Constructing fuzzy deformedentropy of overall deformation

Constructing fuzzy deformedentropy of one measured value

Ex EnHe

Figure 6 Computational process of fuzzy information entropy of overall deformation

= minus

119899

sum

119894=1

120596119894ln120596119894

2

sum

119896=1

120583119896

119894119895minus

119899

sum

119894=1

120596119894

2

sum

119896=1

120583119896

119894119895ln 120583119896119894119895

= minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895

(15)

When the dam move deforms upstream the expressionof the information entropy of overall deformation is

119878119895=

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895) =

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 (16)

Therefore the expression of information entropy of over-all deformation is defined as follows

119878119895=

minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878119894119895ge 0

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878

119894119895lt 0

(17)

The absolute value of the information entropy of the over-all deformationmeasures the danger level of the deformationthat is a smaller absolute value means a higher danger levelpositive and negative values stand for the direction of thedeformation a positive value means downstream deforma-tion whereas a negative value means upstream deformation

Considering the influences of random factors the fuzzyinformation entropy of 119909

119894119895is [1198780119894119895 1198781

119894119895] the fuzzy information

entropy of overall deformation can be illustrated through(17) Computational process of fuzzy information entropy ofoverall deformation is shown in Figure 6

5 Proposed Multistage FuzzyInformation Entropy of OverallDeformation Warning Indicators

Horizontal displacement of dam crest changes in an annualcycle ldquoupstream and downstream switchrdquo Therefore thisdisplacement should be in a certain scale and be controlledunder somemonitoring indicators for the safe damoperation

In the case of downstream displacement the primaryfuzzy warning indicator 1205751015840

1is defined as 1205751015840

1= (1205750

1 1205751

1) 12057501is

the lower limit of this indicator and 12057511is the upper limit

the secondary indicator 12057510158402is 12057510158402= (1205750

2 1205751

2) where 1205750

2is


When 12057511gt 1205750

2 a cross phenomenon appears in the primary

indicator and secondary indicator when both of them shouldbe categorized according to the membership of displacementmeasured 120575lowast was introduced because the membership ofdisplacement at this point is the same Figure 7 showsdiagram of multistage fuzzy information entropy warningindicators

The primary fuzzy warning indicator is 12057510158401= (1205750

1 120575lowast

) andthe secondary is 1205751015840

2= (120575lowast

1205751

2)

If the deformation value is in (12057501 1205751

1) or (1205750

1 120575lowast

) the damis in the state of primary warning if the value is in (1205750

2 1205751

2) or

(1205750

2 120575lowast

) the dam is in the state of secondary warningThe time sequence of deformation at each observation

point was analyzed by using the above theoretical methodThe lower and upper limits of the fuzzy information entropyof overall deformation affected by Δ

119894119895will be 1198780

119895 and 1198781

119895

Considering the damrsquos long-term service when the dammoves downstream the lower limit 1198780

119898119895 and upper limit


1Degree of membership

Displacement


Displacement

12057512120575021205751112057501

1205751212057502 1205751112057501 120575lowast

Figure 7Diagramofmultistage fuzzy information entropywarningindicators

1198781

119898119895 were selected and when the dam moves upstream

the lower limit 1198770119898119895 and upper limit 1198771

119898119895 were selected

1198780

119898119895 1198781119898119895 1198770119898119895 and 1198771

119898119895 are random variables and four

subsample spaces with the sample size of 119873 can be obtainedby the following

1198780

= 1198780

1198981 1198780

1198982 119878

0

119898119898

1198781

= 1198781

1198981 1198781

1198982 119878

1

119898119898

1198770

= 1198770

1198981 1198770

1198982 119877

0

119898119898

1198771

= 1198771

1198981 1198771

1198982 119877

1

119898119898

(18)

Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865

119899(119909)) with

the assumed theoretical probability distribution (119865119909(119909)) to

construct statisticsAccording to the method of empirical distribution func-

tion segmented cumulative frequency is obtained by usingthe following formula

119865119899(119909) =

0 119909 lt 119909119894

119894

119899 119909119894le 119909 lt 119909

119894+1

1 119909 ge 119909

(19)

In the formula 1199091 1199092 119909

119899is sample data after arrange-

ment The sample size is 119899

In the full range of random variable 119883 the maximumdifference between 119865

119899(119909) and 119865

119909(119909) is

119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)

1003816100381610038161003816 lt 119863120573

119899 (20)

In the formula 119863119899is a random variable whose distribu-

tion depends on 119899119863120573119899is critical value for a significant level 120573

It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected

The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878

119898is the extreme of the

information entropy of the upstream overall deformation if119878 gt 119878

119898 the probability of the dam failure is 119875(119878 gt 119878

119898) =

120572 = intinfin

119878119898

119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified

6 Example Analysis

61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m

The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013

In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99


G21 G9 G4G24

25

30

2631

28

27

3233

Figure 8 The arrangement of observation point

Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132

Figure 9 The process line of upstream water level

11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture

Figure 10 The process line of temperature

62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21

Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop

2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1

TemperatureDeformed entropy

0070

140210280

Tem

pera

ture

minus080minus076minus072minus068minus064minus060

Def

orm

ed en

tropy

Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011

2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27

Horizontal displacementTemperature

05

101520253035

Tem

pera

ture

minus16minus12minus08minus04004

Hor

izon

tal

disp

lace

men

t

Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011

The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1

63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780

119898119895

1198781

119898119895 1198770119898119895 and 1198771

119898119895 with K-S method The maximum 119863

119899

of each sequencewas obtained and comparedwith the criticalvalue 119863005

119899of the significant level 005 to determine the type


Table 1 The weight table of observation point

Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292

Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation

Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739

Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation


Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation

Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020



Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28

Figure 13 Range of cloud fall of the displacement at G21 27 and 28

of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7

K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771

119898119895 satisfy

normal distributionThe probability density function of the sequence is

119891 (119909) =1

radic21205871205902exp(minus

(119909 minus 120583)2

120590) (21)

Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary

warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as

the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)

64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)

Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise


Table 6 K-S test results

Probability distribution 1198780

1198981198951198781

1198981198951198770

1198981198951198771

119898119895

Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005

119899034 034 029 029

The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution

Table 7 Parameter values of the probability density function

Data series Parameter values120583 120590

2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245

Figure 14 Finite element model of the dam

The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe

In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in

minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003

XY

Z

Figure 15The results of finite element calculation ofworking condi-tion 1

XY

Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004

Figure 16 The results of finite element calculation of workingcondition 2

the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning

7 Conclusion

This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex


Table 8 Multistage fuzzy warning values of concrete dam

The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator

Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)

Table 9 Material parameter of the dam

Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07

Table 10 The results of finite element calculation of displacementof observation point (mm)

Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114

Table 11 The primary warning indicators of concrete dam

The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773

random factors The results of the specific studies are asfollows

(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model

(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established

(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam

Competing Interests

No conflict of interests exits in the submission of this paper

Acknowledgments

This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)

References

[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015

[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014

[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014

[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956


[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014

[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009

[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012

[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006

[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013

[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007

[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015

[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002

[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012

[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011

[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012

[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998

[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011

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


and methods complement and improve traditional methodsin solving difficult problems in dam safety such approachesonly address the warning of ldquopointrdquo deformation and not thescientific ldquospatial and overallrdquo deformation Further studieson the deformation of concrete dams must consider theinfluences of complex random factors Thus the expressionof a damrsquos overall deformation should be constructed andscientific and accurate warning indicators that considerrandomness and fuzziness should be determined

This paper began with an analysis of an indicator offuzziness that affects the value of monitoring the dam bystudying the influences of fuzziness and randomness fromlong-term service of the dam based on cloud theory [9ndash11] Thereafter the relationship between a damrsquos overalldeformation and deformation of single observation pointswas analyzed through information entropy and synergeticsIn the analysis overall deformation refers to a systemwhereinsingle observation points with different contribution degrees(weights) influence one another A criterion indicator mea-suring the overall deformation conditions was constructedOn this basis multistage warning indicators for the overalldeformation of concrete dam considering fuzziness andrandomness were determined Therefore nondeterministicoptimal control was achieved [12] This paper concludes witha project case that verified the feasibility of the proposedtheory

2 Nondeterministic Optimal Control

Hydraulic engineering considers that some fuzziness andrandomness in dams are inevitable because of the influencesof various random factors such as the nondeterminacy ofmechanical parameters imposed load and boundary condi-tions Statistically the smaller probability of dam displace-ment indicates that the dam is in a more dangerous state120583 + 3120590 and 120583 minus 3120590 can be used as a warning indicator in oneconfidence coefficient of the dam if the displacement obeysthe normal distribution of mean value and variance and is inthe range of 120583plusmn3120590 in deterministic optimal control In realitygiven that dams are affected by random factors warningindicators will have a range of variations For example inFigure 1 the warning value of upstream displacement has amaximum control value and minimum control value

As the foundation of cloud theory the cloud model isprecisely both controlled and uncontrolled in microscopescale119880 is the time series ofmonitoring the damdeformationand 119862 is the qualitative judgment on dam safety If thequantitative value 119909 isin 119880 119906(119909) fall in [0 1] and follow theprobability distribution law

120583 119880 997888rarr [0 1] forall119909 isin 119880 119909 997888rarr 120583 (119909) (1)

The distribution of 119909 in 119880 is called cloud and (119909 119906) is thecloud droplet

In Figure 2 for observation point 119894 if the range whereinthe cloud droplet falls is given the upper bound and lower

bound of the cloud droplet in the cloud model 119910119906119894and 119910119897

119894can

be expressed as follows

119910119906

119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)

22(119864119899119894+3119867119890119894)

2

119910119897

119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)

22(119864119899119894minus3119867119890119894)

2

(2)

119909119894119895is the value 119895 of observation point 119894 and the fuzziness

Δ119894119895can be calculated by

Δ119894119895= 119910119906

119894119895minus 119910119897

119894119895 (3)

In this equation 119910119906119894119895is the upper limit value of 119909

119894119895in the

range and 119910119897119894119895is the lower limit value of 119909

119894119895in the range

Given the influence of random factors when 119909119894119895ge 0

119909119894119895changes in the range of [119909

119894119895minus 119909119894119895Δ119894119895 119909119894119895+ 119909119894119895Δ119894119895] when

119909119894119895lt 0 it is in the range of [119909

119894119895+ 119909119894119895Δ119894119895 119909119894119895minus 119909119894119895Δ119894119895]

When drawing up the warning indicator for downstreamdeformation the maximum and minimum control values ofthe indicator can be determined when the significance level is120572 thus indicating that the nondeterministic optimal controlhas been achieved

3 Methods of Characterizing Contributions ofSingle Observation Point

The overall deformation condition of the concrete dam isusually exposed to the influences of water pressure and tem-perature and is related to many factors including the physicaland mechanical properties of dam materials body structuregeology and hydrology and it could be referred to in Figure 3The principles of synergetics posit that a concrete dam isa synthesis of feature points with different contributions(weights) that influence one another The contribution of asingle observation point needs to be studied to construct areasonable expression of overall deformation

31 Construction of the Indicator Set of a Single ObservationPoint Weight Entropy [13ndash15] a basic concept in thermody-namics refers to a state function in a system The conceptof information entropy is a measurement of the systemrsquosdisorder and nondeterminacy [15] The measured value 119895 onobservation point 119894 is 119909

119894119895and its corresponding entropy is 119878

119894119895

According to entropy theory when an observation point is ina more dangerous state the system is in greater disorder andits entropy value is smaller Thus the following is obtained

119878119894119895= minus [120583

119894119895ln 120583119894119895+ (1 minus 120583

119894119895) ln (1 minus 120583

119894119895)] (4)

120583119894119895=

int

119909119894119895

minusinfin

119891 (120589) 119889120589 119909119894119895ge 0

int

+infin

119909119894119895

119891 (120589) 119889120589 119909119894119895lt 0

(5)

Formula (4) defines the information entropy of themeasurement value No matter what distribution 119909

119894119895 obeys

if the probability density function of the measured value is


War

ning

val

ue o

f dow

nstre

am

disp

lace

men

t

War

ning

val

ue o

f ups

tream

di

spla

cem

ent

Displacement

Probability density

Displacement

The m

axim

um co

ntro

l val

ue o

f war

ning

va

lue o

f dow

nstre

am d

ispla

cem

ent

The m

inim

um co

ntro

l val

ue o

f war

ning

va

lue o

f dow

nstre

am d

ispla

cem

ent

The m

axim

um co

ntro

l val

ue o

f war

ning

va

lue o

f ups

tream

disp

lace

men

t

The m

inim

um co

ntro

l val

ue o

f war

ning

va

lue o

f ups

tream

disp

lace

men

t

Probability density


Degree of certainty


Cloud dropletsEx





119894119895 119863119894119895= 1 minus 119878

119894119895



119894119895





119894ge 0





119863119894119895=

[[[[[[

[

1198631111986312sdot sdot sdot 119863

1119898

1198632111986322sdot sdot sdot 119863

2119898

11986311989911198631198992sdot sdot sdot 119863

119899119898

]]]]]]

]

(6)


119894119895 The weight




119863lowast

119894119895=

119863119894119895minus [119863119895]min

[119863119895]max minus [119863119895]min

(7)
















1 1199012 119901

119895) and1199012

1+1199012


2

119895= 1The


119866 (119894) =

119898

sum

119895=1

119901119895119863lowast

119894119895 (119894 = 1 2 119899) (8)




sum

119895=1

1199012

119895= 1

(9)




119878119866= [sum119899

119894=1(119866 (119894) minus 119892 (119894))

2

119899 minus 1]

05

119876119866=

119899

sum

119894=1

119899

sum

119895=1

(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903

119894119895)

(10)



119894119895is



119894can be


120596119894=

119866lowast

(119894)

sum119899

119895=1119866lowast (119895)

119894 = 1 2 119899 (11)





minusinfin




119894119895 120583119894119895will float in [1205830

119894119895 1205831

119894119895]


1205830

119894119895= int

119909119894119895minus119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

1205831

119894119895= int

119909119894119895+119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

(12)


int+infin

119909119894119895


follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583

119894119895) ln (1 minus 120583

119894119895) (13)


0

Dow

nstre

am d

ispla

cem

ent

Ups

tream

di

spla

cem

ent

105

Sij

1205830

1 minus 1205830 120583


119894119895if 119864119909ge 0


119894119895 1205831

119894119895]The


1205830

119894119895= int

+infin

119909119894119895minus119909119894119895Δ 119894119895

119891 (120589) 119889120589

1205831

119894119895= int

+infin

119909119894119895+119909119894119895Δ 119894119895

119891 (120589) 119889120589

(14)


119894119895minus119878119894119895floats in [1198780

119894119895 1198781




119894119895changes


119894119895 1205831

119894119895]


119909ge 0 119878



0= int0

minusinfin


119878119894119895is as follows 119878


119894119895when 120583

119894119895

is in the range of (1205830 05) 119878


120583119894119895when 120583


119894119895= 05 119878

119894119895will


119894119895] the




119894119895 1205831

119894119895]



119894119895



119894119895and



119909lt 0 119878


01

05

Dow

nstre

amU

pstre

am d

ispla

cem

ent

Sij

1205830

1 minus 1205830 120583

disp

lace

men

t


119894119895if 119864119909lt 0


0= int0

minusinfin

119891(120589)119889120589



119894119895






0 05) 119878


of 120583119894119895 when 120583

119894119895is in the range of (05 1) 119878



119894119895= 05 119878



119894119895] 119878119894119895will have



119894119895


119894119895at

endpoints


119894120583119894119895 make 1205831

119894119895= 120583119894119895and 1205832

119894119895= 1 minus 120583



119878119895= minus

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895(ln120596119894+ ln 120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln120596119894minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln 120583119896119894119895


Choice of sample









Ex EnHe


= minus

119899

sum

119894=1

120596119894ln120596119894

2

sum

119896=1

120583119896

119894119895minus

119899

sum

119894=1

120596119894

2

sum

119896=1

120583119896

119894119895ln 120583119896119894119895

= minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895

(15)


119878119895=

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895) =

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 (16)


119878119895=

minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878119894119895ge 0

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878

119894119895lt 0

(17)



119894119895is [1198780119894119895 1198781






1is defined as 1205751015840

1= (1205750

1 1205751

1) 12057501is



2 1205751

2) where 1205750

2is


When 12057511gt 1205750




1 120575lowast


2= (120575lowast

1205751

2)


1) or (1205750

1 120575lowast


2 1205751

2) or

(1205750

2 120575lowast



119894119895will be 1198780

119895 and 1198781

119895





Displacement


Displacement

12057512120575021205751112057501

1205751212057502 1205751112057501 120575lowast


1198781




1198780

119898119895 1198781119898119895 1198770119898119895 and 1198771



1198780

= 1198780

1198981 1198780

1198982 119878

0

119898119898

1198781

= 1198781

1198981 1198781

1198982 119878

1

119898119898

1198770

= 1198770

1198981 1198770

1198982 119877

0

119898119898

1198771

= 1198771

1198981 1198771

1198982 119877

1

119898119898

(18)


119899(119909)) with




119865119899(119909) =

0 119909 lt 119909119894

119894

119899 119909119894le 119909 lt 119909

119894+1

1 119909 ge 119909

(19)

In the formula 1199091 1199092 119909




119899(119909) and 119865

119909(119909) is

119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)

1003816100381610038161003816 lt 119863120573

119899 (20)








119898) =

120572 = intinfin

119878119898


6 Example Analysis





G21 G9 G4G24

25

30

2631

28

27

3233


Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132


11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture




2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1


0070

140210280

Tem

pera

ture


Def

orm

ed en

tropy


2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27


05

101520253035

Tem

pera

ture


Hor

izon

tal

disp

lace

men

t




119898119895

1198781

119898119895 1198770119898119895 and 1198771


119899














Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










War

ning

val

ue o

f dow

nstre

am

disp

lace

men

t

War

ning

val

ue o

f ups

tream

di

spla

cem

ent

Displacement

Probability density

Displacement

The m

axim

um co

ntro

l val

ue o

f war

ning

va

lue o

f dow

nstre

am d

ispla

cem

ent

The m

inim

um co

ntro

l val

ue o

f war

ning

va

lue o

f dow

nstre

am d

ispla

cem

ent

The m

axim

um co

ntro

l val

ue o

f war

ning

va

lue o

f ups

tream

disp

lace

men

t

The m

inim

um co

ntro

l val

ue o

f war

ning

va

lue o

f ups

tream

disp

lace

men

t

Probability density


Degree of certainty


Cloud dropletsEx





119894119895 119863119894119895= 1 minus 119878

119894119895



119894119895





119894ge 0





119863119894119895=

[[[[[[

[

1198631111986312sdot sdot sdot 119863

1119898

1198632111986322sdot sdot sdot 119863

2119898

11986311989911198631198992sdot sdot sdot 119863

119899119898

]]]]]]

]

(6)


119894119895 The weight




119863lowast

119894119895=

119863119894119895minus [119863119895]min

[119863119895]max minus [119863119895]min

(7)
















1 1199012 119901

119895) and1199012

1+1199012


2

119895= 1The


119866 (119894) =

119898

sum

119895=1

119901119895119863lowast

119894119895 (119894 = 1 2 119899) (8)




sum

119895=1

1199012

119895= 1

(9)




119878119866= [sum119899

119894=1(119866 (119894) minus 119892 (119894))

2

119899 minus 1]

05

119876119866=

119899

sum

119894=1

119899

sum

119895=1

(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903

119894119895)

(10)



119894119895is



119894can be


120596119894=

119866lowast

(119894)

sum119899

119895=1119866lowast (119895)

119894 = 1 2 119899 (11)





minusinfin




119894119895 120583119894119895will float in [1205830

119894119895 1205831

119894119895]


1205830

119894119895= int

119909119894119895minus119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

1205831

119894119895= int

119909119894119895+119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

(12)


int+infin

119909119894119895


follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583

119894119895) ln (1 minus 120583

119894119895) (13)


0

Dow

nstre

am d

ispla

cem

ent

Ups

tream

di

spla

cem

ent

105

Sij

1205830

1 minus 1205830 120583


119894119895if 119864119909ge 0


119894119895 1205831

119894119895]The


1205830

119894119895= int

+infin

119909119894119895minus119909119894119895Δ 119894119895

119891 (120589) 119889120589

1205831

119894119895= int

+infin

119909119894119895+119909119894119895Δ 119894119895

119891 (120589) 119889120589

(14)


119894119895minus119878119894119895floats in [1198780

119894119895 1198781




119894119895changes


119894119895 1205831

119894119895]


119909ge 0 119878



0= int0

minusinfin


119878119894119895is as follows 119878


119894119895when 120583

119894119895

is in the range of (1205830 05) 119878


120583119894119895when 120583


119894119895= 05 119878

119894119895will


119894119895] the




119894119895 1205831

119894119895]



119894119895



119894119895and



119909lt 0 119878


01

05

Dow

nstre

amU

pstre

am d

ispla

cem

ent

Sij

1205830

1 minus 1205830 120583

disp

lace

men

t


119894119895if 119864119909lt 0


0= int0

minusinfin

119891(120589)119889120589



119894119895






0 05) 119878


of 120583119894119895 when 120583

119894119895is in the range of (05 1) 119878



119894119895= 05 119878



119894119895] 119878119894119895will have



119894119895


119894119895at

endpoints


119894120583119894119895 make 1205831

119894119895= 120583119894119895and 1205832

119894119895= 1 minus 120583



119878119895= minus

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895(ln120596119894+ ln 120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln120596119894minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln 120583119896119894119895


Choice of sample









Ex EnHe


= minus

119899

sum

119894=1

120596119894ln120596119894

2

sum

119896=1

120583119896

119894119895minus

119899

sum

119894=1

120596119894

2

sum

119896=1

120583119896

119894119895ln 120583119896119894119895

= minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895

(15)


119878119895=

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895) =

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 (16)


119878119895=

minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878119894119895ge 0

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878

119894119895lt 0

(17)



119894119895is [1198780119894119895 1198781






1is defined as 1205751015840

1= (1205750

1 1205751

1) 12057501is



2 1205751

2) where 1205750

2is


When 12057511gt 1205750




1 120575lowast


2= (120575lowast

1205751

2)


1) or (1205750

1 120575lowast


2 1205751

2) or

(1205750

2 120575lowast



119894119895will be 1198780

119895 and 1198781

119895





Displacement


Displacement

12057512120575021205751112057501

1205751212057502 1205751112057501 120575lowast


1198781




1198780

119898119895 1198781119898119895 1198770119898119895 and 1198771



1198780

= 1198780

1198981 1198780

1198982 119878

0

119898119898

1198781

= 1198781

1198981 1198781

1198982 119878

1

119898119898

1198770

= 1198770

1198981 1198770

1198982 119877

0

119898119898

1198771

= 1198771

1198981 1198771

1198982 119877

1

119898119898

(18)


119899(119909)) with




119865119899(119909) =

0 119909 lt 119909119894

119894

119899 119909119894le 119909 lt 119909

119894+1

1 119909 ge 119909

(19)

In the formula 1199091 1199092 119909




119899(119909) and 119865

119909(119909) is

119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)

1003816100381610038161003816 lt 119863120573

119899 (20)








119898) =

120572 = intinfin

119878119898


6 Example Analysis





G21 G9 G4G24

25

30

2631

28

27

3233


Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132


11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture




2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1


0070

140210280

Tem

pera

ture


Def

orm

ed en

tropy


2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27


05

101520253035

Tem

pera

ture


Hor

izon

tal

disp

lace

men

t




119898119895

1198781

119898119895 1198770119898119895 and 1198771


119899














Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















1 1199012 119901

119895) and1199012

1+1199012


2

119895= 1The


119866 (119894) =

119898

sum

119895=1

119901119895119863lowast

119894119895 (119894 = 1 2 119899) (8)




sum

119895=1

1199012

119895= 1

(9)




119878119866= [sum119899

119894=1(119866 (119894) minus 119892 (119894))

2

119899 minus 1]

05

119876119866=

119899

sum

119894=1

119899

sum

119895=1

(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903

119894119895)

(10)



119894119895is



119894can be


120596119894=

119866lowast

(119894)

sum119899

119895=1119866lowast (119895)

119894 = 1 2 119899 (11)





minusinfin




119894119895 120583119894119895will float in [1205830

119894119895 1205831

119894119895]


1205830

119894119895= int

119909119894119895minus119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

1205831

119894119895= int

119909119894119895+119909119894119895Δ 119894119895

minusinfin

119891 (120589) 119889120589

(12)


int+infin

119909119894119895


follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583

119894119895) ln (1 minus 120583

119894119895) (13)


0

Dow

nstre

am d

ispla

cem

ent

Ups

tream

di

spla

cem

ent

105

Sij

1205830

1 minus 1205830 120583


119894119895if 119864119909ge 0


119894119895 1205831

119894119895]The


1205830

119894119895= int

+infin

119909119894119895minus119909119894119895Δ 119894119895

119891 (120589) 119889120589

1205831

119894119895= int

+infin

119909119894119895+119909119894119895Δ 119894119895

119891 (120589) 119889120589

(14)


119894119895minus119878119894119895floats in [1198780

119894119895 1198781




119894119895changes


119894119895 1205831

119894119895]


119909ge 0 119878



0= int0

minusinfin


119878119894119895is as follows 119878


119894119895when 120583

119894119895

is in the range of (1205830 05) 119878


120583119894119895when 120583


119894119895= 05 119878

119894119895will


119894119895] the




119894119895 1205831

119894119895]



119894119895



119894119895and



119909lt 0 119878


01

05

Dow

nstre

amU

pstre

am d

ispla

cem

ent

Sij

1205830

1 minus 1205830 120583

disp

lace

men

t


119894119895if 119864119909lt 0


0= int0

minusinfin

119891(120589)119889120589



119894119895






0 05) 119878


of 120583119894119895 when 120583

119894119895is in the range of (05 1) 119878



119894119895= 05 119878



119894119895] 119878119894119895will have



119894119895


119894119895at

endpoints


119894120583119894119895 make 1205831

119894119895= 120583119894119895and 1205832

119894119895= 1 minus 120583



119878119895= minus

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895(ln120596119894+ ln 120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln120596119894minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln 120583119896119894119895


Choice of sample









Ex EnHe


= minus

119899

sum

119894=1

120596119894ln120596119894

2

sum

119896=1

120583119896

119894119895minus

119899

sum

119894=1

120596119894

2

sum

119896=1

120583119896

119894119895ln 120583119896119894119895

= minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895

(15)


119878119895=

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895) =

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 (16)


119878119895=

minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878119894119895ge 0

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878

119894119895lt 0

(17)



119894119895is [1198780119894119895 1198781






1is defined as 1205751015840

1= (1205750

1 1205751

1) 12057501is



2 1205751

2) where 1205750

2is


When 12057511gt 1205750




1 120575lowast


2= (120575lowast

1205751

2)


1) or (1205750

1 120575lowast


2 1205751

2) or

(1205750

2 120575lowast



119894119895will be 1198780

119895 and 1198781

119895





Displacement


Displacement

12057512120575021205751112057501

1205751212057502 1205751112057501 120575lowast


1198781




1198780

119898119895 1198781119898119895 1198770119898119895 and 1198771



1198780

= 1198780

1198981 1198780

1198982 119878

0

119898119898

1198781

= 1198781

1198981 1198781

1198982 119878

1

119898119898

1198770

= 1198770

1198981 1198770

1198982 119877

0

119898119898

1198771

= 1198771

1198981 1198771

1198982 119877

1

119898119898

(18)


119899(119909)) with




119865119899(119909) =

0 119909 lt 119909119894

119894

119899 119909119894le 119909 lt 119909

119894+1

1 119909 ge 119909

(19)

In the formula 1199091 1199092 119909




119899(119909) and 119865

119909(119909) is

119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)

1003816100381610038161003816 lt 119863120573

119899 (20)








119898) =

120572 = intinfin

119878119898


6 Example Analysis





G21 G9 G4G24

25

30

2631

28

27

3233


Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132


11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture




2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1


0070

140210280

Tem

pera

ture


Def

orm

ed en

tropy


2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27


05

101520253035

Tem

pera

ture


Hor

izon

tal

disp

lace

men

t




119898119895

1198781

119898119895 1198770119898119895 and 1198771


119899














Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

Dow

nstre

am d

ispla

cem

ent

Ups

tream

di

spla

cem

ent

105

Sij

1205830

1 minus 1205830 120583


119894119895if 119864119909ge 0


119894119895 1205831

119894119895]The


1205830

119894119895= int

+infin

119909119894119895minus119909119894119895Δ 119894119895

119891 (120589) 119889120589

1205831

119894119895= int

+infin

119909119894119895+119909119894119895Δ 119894119895

119891 (120589) 119889120589

(14)


119894119895minus119878119894119895floats in [1198780

119894119895 1198781




119894119895changes


119894119895 1205831

119894119895]


119909ge 0 119878



0= int0

minusinfin


119878119894119895is as follows 119878


119894119895when 120583

119894119895

is in the range of (1205830 05) 119878


120583119894119895when 120583


119894119895= 05 119878

119894119895will


119894119895] the




119894119895 1205831

119894119895]



119894119895



119894119895and



119909lt 0 119878


01

05

Dow

nstre

amU

pstre

am d

ispla

cem

ent

Sij

1205830

1 minus 1205830 120583

disp

lace

men

t


119894119895if 119864119909lt 0


0= int0

minusinfin

119891(120589)119889120589



119894119895






0 05) 119878


of 120583119894119895 when 120583

119894119895is in the range of (05 1) 119878



119894119895= 05 119878



119894119895] 119878119894119895will have



119894119895


119894119895at

endpoints


119894120583119894119895 make 1205831

119894119895= 120583119894119895and 1205832

119894119895= 1 minus 120583



119878119895= minus

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895(ln120596119894+ ln 120583119896

119894119895)

= minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln120596119894minus

119899

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln 120583119896119894119895


Choice of sample









Ex EnHe


= minus

119899

sum

119894=1

120596119894ln120596119894

2

sum

119896=1

120583119896

119894119895minus

119899

sum

119894=1

120596119894

2

sum

119896=1

120583119896

119894119895ln 120583119896119894119895

= minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895

(15)


119878119895=

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895) =

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 (16)


119878119895=

minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878119894119895ge 0

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878

119894119895lt 0

(17)



119894119895is [1198780119894119895 1198781






1is defined as 1205751015840

1= (1205750

1 1205751

1) 12057501is



2 1205751

2) where 1205750

2is


When 12057511gt 1205750




1 120575lowast


2= (120575lowast

1205751

2)


1) or (1205750

1 120575lowast


2 1205751

2) or

(1205750

2 120575lowast



119894119895will be 1198780

119895 and 1198781

119895





Displacement


Displacement

12057512120575021205751112057501

1205751212057502 1205751112057501 120575lowast


1198781




1198780

119898119895 1198781119898119895 1198770119898119895 and 1198771



1198780

= 1198780

1198981 1198780

1198982 119878

0

119898119898

1198781

= 1198781

1198981 1198781

1198982 119878

1

119898119898

1198770

= 1198770

1198981 1198770

1198982 119877

0

119898119898

1198771

= 1198771

1198981 1198771

1198982 119877

1

119898119898

(18)


119899(119909)) with




119865119899(119909) =

0 119909 lt 119909119894

119894

119899 119909119894le 119909 lt 119909

119894+1

1 119909 ge 119909

(19)

In the formula 1199091 1199092 119909




119899(119909) and 119865

119909(119909) is

119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)

1003816100381610038161003816 lt 119863120573

119899 (20)








119898) =

120572 = intinfin

119878119898


6 Example Analysis





G21 G9 G4G24

25

30

2631

28

27

3233


Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132


11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture




2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1


0070

140210280

Tem

pera

ture


Def

orm

ed en

tropy


2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27


05

101520253035

Tem

pera

ture


Hor

izon

tal

disp

lace

men

t




119898119895

1198781

119898119895 1198770119898119895 and 1198771


119899














Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Choice of sample









Ex EnHe


= minus

119899

sum

119894=1

120596119894ln120596119894

2

sum

119896=1

120583119896

119894119895minus

119899

sum

119894=1

120596119894

2

sum

119896=1

120583119896

119894119895ln 120583119896119894119895

= minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895

(15)


119878119895=

119898

sum

119894=1

2

sum

119896=1

120596119894120583119896

119894119895ln (120596119894120583119896

119894119895) =

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 (16)


119878119895=

minus

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878119894119895ge 0

119899

sum

119894=1

120596119894ln120596119894+

119899

sum

119894=1

120596119894119878119894119895 119878

119894119895lt 0

(17)



119894119895is [1198780119894119895 1198781






1is defined as 1205751015840

1= (1205750

1 1205751

1) 12057501is



2 1205751

2) where 1205750

2is


When 12057511gt 1205750




1 120575lowast


2= (120575lowast

1205751

2)


1) or (1205750

1 120575lowast


2 1205751

2) or

(1205750

2 120575lowast



119894119895will be 1198780

119895 and 1198781

119895





Displacement


Displacement

12057512120575021205751112057501

1205751212057502 1205751112057501 120575lowast


1198781




1198780

119898119895 1198781119898119895 1198770119898119895 and 1198771



1198780

= 1198780

1198981 1198780

1198982 119878

0

119898119898

1198781

= 1198781

1198981 1198781

1198982 119878

1

119898119898

1198770

= 1198770

1198981 1198770

1198982 119877

0

119898119898

1198771

= 1198771

1198981 1198771

1198982 119877

1

119898119898

(18)


119899(119909)) with




119865119899(119909) =

0 119909 lt 119909119894

119894

119899 119909119894le 119909 lt 119909

119894+1

1 119909 ge 119909

(19)

In the formula 1199091 1199092 119909




119899(119909) and 119865

119909(119909) is

119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)

1003816100381610038161003816 lt 119863120573

119899 (20)








119898) =

120572 = intinfin

119878119898


6 Example Analysis





G21 G9 G4G24

25

30

2631

28

27

3233


Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132


11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture




2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1


0070

140210280

Tem

pera

ture


Def

orm

ed en

tropy


2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27


05

101520253035

Tem

pera

ture


Hor

izon

tal

disp

lace

men

t




119898119895

1198781

119898119895 1198770119898119895 and 1198771


119899














Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Displacement


Displacement

12057512120575021205751112057501

1205751212057502 1205751112057501 120575lowast


1198781




1198780

119898119895 1198781119898119895 1198770119898119895 and 1198771



1198780

= 1198780

1198981 1198780

1198982 119878

0

119898119898

1198781

= 1198781

1198981 1198781

1198982 119878

1

119898119898

1198770

= 1198770

1198981 1198770

1198982 119877

0

119898119898

1198771

= 1198771

1198981 1198771

1198982 119877

1

119898119898

(18)


119899(119909)) with




119865119899(119909) =

0 119909 lt 119909119894

119894

119899 119909119894le 119909 lt 119909

119894+1

1 119909 ge 119909

(19)

In the formula 1199091 1199092 119909




119899(119909) and 119865

119909(119909) is

119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)

1003816100381610038161003816 lt 119863120573

119899 (20)








119898) =

120572 = intinfin

119878119898


6 Example Analysis





G21 G9 G4G24

25

30

2631

28

27

3233


Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132


11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture




2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1


0070

140210280

Tem

pera

ture


Def

orm

ed en

tropy


2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27


05

101520253035

Tem

pera

ture


Hor

izon

tal

disp

lace

men

t




119898119895

1198781

119898119895 1198770119898119895 and 1198771


119899














Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










G21 G9 G4G24

25

30

2631

28

27

3233


Ups

tream

wat

er le

vel

623

198

4

121

419

89

66

1995

112

620

00

519

200

6

119

201

1

11

1979

Date

120124128132


11

2007

11

2008

11

2006

123

120

08

11

2005

123

120

09

123

120

10

123

120

11

123

020

12

123

020

13

Data

05

101520253035

Tem

pera

ture




2011

11

2011

21

2011

31

2011

41

2011

51

2011

61

2011

71

2011

81

2011

91

2011

10

1

2011

11

1

2011

12

1


0070

140210280

Tem

pera

ture


Def

orm

ed en

tropy


2011

11

2011

13

1

2011

32

2011

41

2011

51

2011

53

1

2011

63

0

2011

73

0

2011

82

9

2011

92

8

2011

10

28

2011

11

27

2011

12

27


05

101520253035

Tem

pera

ture


Hor

izon

tal

disp

lace

men

t




119898119895

1198781

119898119895 1198770119898119895 and 1198771


119899














Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Cloud droplets

minus15

minus14

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

010

02

03

04

05

06

07

08

09 1

00102030405060708091

Cer

tain

ty

G21 27 28




119898119895 satisfy


119891 (119909) =1


(119909 minus 120583)2

120590) (21)









1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198981198951198781

1198981198951198770

1198981198951198771

119898119895


119899034 034 029 029




2

1198780

119898119895 0488355 00129

1198781

119898119895 0490627 00151

1198770

119898119895 minus050210 00249

1198771

119898119895 minus049674 00245





XY

Z


XY




7 Conclusion
















Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Competing Interests


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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