Research Article Seepage Monitoring Models Study of Earth ...downloads.hindawi.com/journals/mpe/2016/1656738.pdf · the earth-rock dams in coastal areas su er from rainstorms, the

Research ArticleSeepage Monitoring Models Study of Earth-Rock DamsInfluenced by Rainstorms

Jianchun Qiu123 Dongjian Zheng123 and Kai Zhu23

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 Jianchun Qiu 121267184qqcom

Received 20 November 2015 Revised 2 March 2016 Accepted 10 March 2016

Academic Editor Sajid Hussain

Copyright copy 2016 Jianchun Qiu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For earth-rock dams influenced by rainstorms seepage status monitoring is very important and provides the basis for the safe andeffective operation of earth-rock dams The most influential factors concerning the seepage of earth-rock dams are the reservoirwater level precipitation temperature and timeliness and the influence of the reservoir water level and precipitation on the seepageof an earth-rock dam exhibits hysteretic effects The reservoir water level of an earth-rock dam abruptly increases and may exceedthe historically highest water level therein causing new deformations of the earth-rock dam or even plastic deformation Thusthe permeability coefficient for parts of an earth-rock dam changes and we present the exceeded water level factor Consideringthe complexity of the seepage monitoring of earth-rock dams based on the hysteretic reservoir water level and precipitationtemperature timeliness and the exceeded water level factor a statistical model based on an explicit function and an artificialwavelet neural network model based on an implicit function are established Based on these two models an integrated monitoringmodel based onmaximum entropy theory is established At the end of this paper three monitoring models are used for the seepagemonitoring of a measuring point of an earth-rock dam influenced by rainstorms and the results show that the three monitoringmodels obtain satisfactory predication accuracy

1 Introduction

Due to the characteristics of low cost fine environmentaladaptability and lower construction difficulty earth-rockdams have been widely used and rapidly developed in theworld whichmake up over 80 percent of all dams [1] Duringthe construction and running period of earth-rock damssafety problems such as seepage [2] cracks [3] and landslide[4]may happen Seepage has considerable influence on earth-rock dams and often increases from small range to largerange which may cause dam settlement collapse and con-centrated leakage passage for earth-rock damsThe structuraldamage may be a single form or multiple forms of damagein one part or different parts of the earth-rock dams Forthe earth-rock dams in coastal areas suffer from rainstormsthe rapidly increased reservoir water lever and large amountof rain within a short time may cause threatened structural

problems Therefore it is of great significance to study theseepage status of earth-rock dams influenced by rainstormsBecause earth-rock dams and the surrounding environmentare rather complex and fickle the potential seepage diseasesare difficult to find out Seepage monitoring analysis [5] ofearth-rock dams could help to judge the existence of seepagedamage and grasp the running status of earth-rock damswhich provides basis for the safety running of earth-rockdams

To monitor and analyze the seepage status of earth-rockdams influenced by rainstorms accurately and timely seepagemonitoringmodels should be built up to help to find out seep-age diseases conveniently and ensure the stable operation ofthe dams [5 6] The influence factors concerning earth-rockdam seepage are the reservoir water level precipitation tem-perature timeliness and so forth In fact the influence of thereservoir water level and precipitation on earth-rock dam

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1656738 11 pageshttpdxdoiorg10115520161656738

2 Mathematical Problems in Engineering

seepage exhibits hysteretic effect Moreover rainstorms maylead the reservoir water level to exceed the historical highestwater level which causes new deformation or even plasticdeformation of earth-rock dams Then the permeabilityproperty of earth-rock material would change which influ-ences the seepage state of earth-rock dams The exceededreservoir water level factor is put forward to consider theabrupt increased reservoir water level

Therefore the seepage statistical model considering thehysteretic effect of reservoir water level and precipitation andthe exceeded water level factor is established Meanwhilethe influence factors on seepage monitoring indexes arerather complex which make it difficult to simulate withexplicit function Artificial wavelet neural network [7ndash9] isthe implicit function which has the advantage to explaincomplex relationship Thus earth-rock dam seepage mon-itoring model based on artificial wavelet neural networkis established Based on the two monitoring models andmaximum entropy theory [10ndash12] an integrated seepagemonitoring model is set up to optimize the earth-rockdam seepage monitoring further Finally the three seepagemonitoring models are applied to analyze the seepage statusof an earth-rock dam influenced by rainstorms The resultsshow the threemodels with fine precision successfully used inearth-rock dam seepagemonitoring which provide technicalsupport for seepage monitoring of other earth-rock dams

2 Earth-Rock Dam Seepage StatisticalModel considering the Hysteretic Effectof Reservoir Water Level and Precipitationand the Exceeded Water Level Factor

Themost influential factors concerning earth-rock dam seep-age include reservoir water level precipitation temperatureand timeliness Given that the effect of reservoir waterlevel and precipitation on seepage has the hysteretic effectin the traditional seepage statistical model the previousreservoir water level and precipitation are categorized basedon averages over a number of previous days [5] for examplethe previous two days the previous five days and the previousten days Practice has proven that the influences of reservoirwater level and precipitation on seepage rise in the first stageand then decrease which are presented as normal distribu-tion [5]The normal distribution curve is used to simulate thehysteretic effect and the hysteretic days and influence daysof reservoir water level and precipitation influence on earth-rock dam seepage are hard to determine Considering theefficiency of general calculation methods is low quantumgenetic algorithm [13 14] is used to calculate the hystereticdays and influence days to obtain the optimal earth-rock damseepage statistical model Moreover due to the influence ofrainstorms the reservoir water levelmay exceed the historicalhighest water level As a result the abrupt increased reservoirwater level is difficult to simulate Hence the exceeded waterlevel factor is added

21 ReservoirWater Level Component For the seepage indexwe take the piezometric tube level as an example The pie-zometric tube level is hysteretic influenced by the reservoir

water level and the seepage index at the time 119905 is given asfollows

ℎ (119905) = 119865 [119905119867 (119905) 119867 (119905 minus 1205911(119905)) 119867 (119905 minus 120591

2(119905))

119867 (119905 minus 120591119899(119905))]

(1)

where 120591 is the retardation time 120591119896(119905) ge 0 119896 = 1 2 119899

and119867(119905) and119867(119905 minus 120591119899(119905)) are the reservoir water level at the

corresponding timeEquation (1) reflects the hysteretic relationship between

the reservoir water level and the piezometric tube leveland the piezometric tube level at the time 119905 is continuouslyinfluenced by the previous reservoir water level Because thehysteretic time is difficult to determine the previous reservoirwater level is often categorized based on averages over anumber of previous days for example the previous two daysthe previous five days and the previous ten days Howeverthese factors are fuzzy and are unable to accurately reflectthe hysteretic effect of reservoir water levels Moreover theinfluence of the reservoir water level on the piezometric tubelevel may be from the previous two days the previous tendays or longer

Suppose that the seepage index is influenced by the reser-voir water level from the previous 119899 days119867

119894(119894 = 1 2 119899)

and the equivalent water level119867119889is expressed as follows

119867119889= 120593 (119867

1 1198672 119867

119896 1205771 1205772 120577

119896) (2)

where 120577119894 119894 = 1 2 119896 (119896 le 119899) is the weight of the 119894th water

levelrsquos influence on the equivalent water level and 120593 is thefunction reflecting the hysteretic influence of the reservoirwater level on the seepage index

Considering the characteristic of the weight vector 120577 =

[1205771 1205772 120577

119896] the following equation is then obtained

119896

sum

119894=1

120577119894= 1 119896 le 119899 (3)

Therefore the equivalent water level 119867119889is obtained as

follows

119867119889=

119896

sum

119894=1

120577119894119867119894 (4)

Numerous studies have shown that 120577(119905) presents a normaldistribution [5] Combinedwith the characteristics of normaldistributions the hysteretic days and influenced days are setas 1199091and 119909

2 The process that describes the influence of the

reservoirwater level on the seepage index is shown in Figure 1and the hysteretic influence function is given as follows

120577 (119905) =1

radic21205871199092

119890minus(119905minus119909

1)221199092

2 (5)

If a water storage starts at the time 1199050 then at the fixed

time 1199051 the following equation is obtained

int

1199051

1199050

120577 (119905) 119889119905 = 1 (6)

Mathematical Problems in Engineering 3

Todayt

120577

x1x2 = 120590

Figure 1 Normal distribution curve of the hysteretic water level

For the piezometric tube level at a specific measuringpoint during a certain period of time the hysteretic days 119909

1

and influenced days 1199092are constant

120577(119905) in (5) can be treated as a function reflecting theinfluence of the previous water level on the equivalent waterlevel 119867

119889 Suppose that the equivalent reservoir water level

at the time 1199051is 119867119889and that the influenced weight function

120577(119905) ge 0 Then the following equation is obtained

119867119889= int

1199051

1199050

120577 (119905)119867 (119905) 119889119905

= int

1199051

1199050

1

radic21205871199092

119890minus(119905minus119909

1)221199092

2119867(119905) 119889119905

(7)

where 1199091is the undetermined hysteretic reservoir water level

days 1199092is the undetermined influenced normal distribution

standard deviation and119867(119905) is the reservoir water level at thetime 119905

Quantum genetic algorithm is used to obtain the hys-teretic days and influenced distributed parameter Generallythe continuous integration is replaced by a discrete integra-tion and an integration range with the value of 3-4 times 119909

2

meets the accuracy requirements

22 Precipitation Component To reasonably consider thehysteretic effects of precipitation on earth-rock dam seepagea lognormal distribution function is used to depict the pre-cipitation effect on the earth-rock dam seepage Specificallythe hysteretic influencing function of precipitation is given asfollows

119908 (119905) =1

radic21205871199094119905

119890minus(ln 119905minus119909

3)221199092

4 (8)

where 1199093is the hysteretic days of precipitation and 119909

4is the

influencing distribution parameterSuppose that the observation day is 119905

1and that the starting

day is 1199050 The influence range can be 1-2 months and then

int

1199051

1199050

119908 (119905) 119889119905 = 1 (9)

Similarly an exponential transformation is adopted forthe influence of precipitation infiltration Suppose that theequivalent precipitation at the time 119905

1is 119875119889 The hysteretic

influenced function of precipitation meets the followingcondition

119875119889= int

1199051

1199050

119908 (119905) [119875 (119905)]120573119889119905

= int

1199051

1199050

1

radic21205871199094119905

119890minus(ln 119905minus119909

3)221199092

4 [119875 (119905)]120573119889119905

(10)

where 119875(119905) is the precipitation at the time of 119905 120573 is theinfiltration transformation index 0 lt 120573 lt 1 and theremaining symbols have the same meaning as in (8)

Therefore the precipitation component is expressed asfollows

ℎ119875= 119887119875119889= 119887int

1199051

1199050

1

radic21205871199094119905

119890minus(ln 119905minus119909

3)221199092

4 [119875 (119905)]120573119889119905 (11)

where ℎ119875is the precipitation component 119887 is the regression

coefficient 1199093is the undetermined hysteretic days of pre-

cipitation 1199094is the undetermined distributed parameter of

precipitation 119875(119905) is the precipitation at the time of 119905 and 119875119889

is the equivalent precipitationSimilarly quantum genetic algorithm is used to optimally

obtain 1199093and 119909

4

23 Temperature Component Although seepage has nodirect relation with the water temperature air temperatureand batholith temperature the temperature variation changesthe viscosity of water and the batholith crack opening asa result the seepage status may change The temperaturecomponent of seepage is mainly caused by the temperaturevariations in the dam foundation and the batholith on bothsides of the dam and the temperature variation exhibitsannual periodic variations or semiannual periodic variationsHowever sufficientmeasured temperature data have not beenprovided Therefore a temperature component consisting ofsine and cosine functions with periods of a year and half ayear is given as follows

ℎ119879=

119899

sum

119894=1

(1198881119894sin 2120587119894119905

365+ 1198882119894cos 2120587119894119905

365) 119894 = 1 2 (12)

where 119894 = 1 2 corresponds to the period of a year and half ayear 119899 = 2 and 119888

1119894and 1198882119894are the regression coefficient

24 Timeliness Component During operation of the earth-rock dam the structure of soil grainsmay changeMeanwhilethe gradually produced deposit in front of the dam forms anatural blanket The influence of these factors on the seepageexhibits the timeliness process and can be modeled using thefollowing equation

ℎ120579= 1198891120579 + 1198892ln 120579 (13)

where 1198891and 119889

2are the regression coefficients of the timeli-

ness component and 120579 is the cumulative number of days fromthe starting day to the measured day dividing 100

4 Mathematical Problems in Engineering

25 Exceeded Eater Level Factor For earth-rock dams influ-enced by rainstorms the reservoir water level may exceed theprevious historical highest water level The reservoir waterlevel is unable to accurately reflect abrupt increases in thereservoir water level If the reservoir water level for a certainday exceeds the previous historical highest water level theexceeded water level factor is used The factor relates to theexcess of the reservoir water level the reservoir water level onthe day of operation and the rate of change of the water levelThe greater the excess reservoir water level is the greater theinfluence on the earth-rock dam seepage will be Thereforethe exceeded water level factor is given as follows

ℎ119890=

2

sum

119894=1

[119890119894(DH119894)] +

3

sum

119895=1

119891119894

Δ119867

Δ119905

1003816100381610038161003816100381610038161003816Δ119905=119895

(14)

where DH1is the excess reservoir water level DH

2is the

product of DH1and the water level119867 on the measuring day

and Δ119867Δ119905 is the rate of change over the previous 119894 days

26 Earth-Rock Dam Seepage Statistical Model By studyingthe earth-rock dam seepage statistical model considering thehysteretic effect of precipitation and reservoir water level andthe exceeded water level factor combined with the temper-ature component and timeliness component the earth-rockdam seepage statistical model is obtained as follows

119867 = ℎ119867+ ℎ119875+ ℎ119879+ ℎ120579+ ℎ119890

= 1198600+ 119886119867119889+ 119887119875119889

+

119899

sum

119894=1

(1198881119894sin 2120587119894119905

365+ 1198882119894cos 2120587119894119905

365) + 1198891120579

+ 1198892ln 120579 +

2

sum

119894=1

[119890119894(DH119894)] +

3

sum

119895=1

119891119894

Δ119867

Δ119905

1003816100381610038161003816100381610038161003816Δ119905=119895

(15)

where 1198600is a constant term and the other parameters have

the same meaning as mentioned aboveBy selecting multiple correlation coefficients or the resid-

ual standard deviation as the objective function the quantumgenetic algorithm is used to obtain the optimal hystereticparameters and the coefficient in (15) The calculation pro-cedure is shown in Figure 2

3 Earth-Rock Dam Seepage Monitoring ModelBased on Artificial Wavelet Neural Network

Therelationship between environment factors and earth-rockdam seepage is rather complex and it is difficult for an explicitfunction to simulate this complex relationship Artificialneural network algorithm is a type of implicit function withstrong nonlinear fitting ability and adaptability The wavelettransformation has the ability to better analyze the localdetails of the measured data and reflects the characteristics ofthemeasured dataTherefore combinedwithwavelet analysisand the artificial neural network algorithm an earth-rockdam seepagemonitoringmodel based on the artificial waveletneural network is established

Start

Input monitoring data

Calculate the corresponding equivalentreservoir water level and precipitation

Initialization hysteretic parameters of precipitation

Quantum genetic algorithm is usedto optimize the hysteretic parameters

Calculate the coefficient of the statisticalmodel the multiple correlation coefficient

and residual standard deviation

Attain the termination requirement

No

Obtain the coefficient and hystereticparameters of the optimal statistical model

Output

End

Yes

and reservoir water level x1 x2 x3 and x4

Figure 2 Calculation process of the optimal earth-rock damseepage statistical model

31 Wavelet Theory Wavelet analysis [15 16] is a type ofmultiresolution data analysis method with the ability toanalyze any details of an object Through wavelet analysis asignal is resolved into different frequency bands Supposingthat the wavelet function120593(119905) transformswith amagnitude of120591 the inner product between the original signal 119909(119905) and thewavelet function using a scaling of 119886 is obtained as follows

119891119909(119886 120591) =

1

radic119886int

+infin

minusinfin

119909 (119905) 120593 (119905 minus 120591

119886) 119889119905 119886 gt 0 (16)

Through the transformation of the wavelet basis functionand the analysis of the local features of the signal the localcharacteristics of the seepage data are obtained which helpto reflect the multiscale change law of the measured data

32 Earth-Rock Dam Seepage Monitoring Model Based onArtificial Wavelet Neural Network Because it fuses the arti-ficial neural network algorithm and wavelet theory artificialwavelet neural networks have the advantage of providingmultiscale analysis and implicit function Therefore theseepage monitoring model based on the artificial waveletneural network more objectively reflects the seepage statusThe transfer function of the nodes in the hidden layer ofthe neural network is the wavelet basis function The weightvalue and threshold value are adjusted through error back

Mathematical Problems in Engineering 5

X1

X2

X5

wij

j

Y

Figure 3 Topological structure of artificial wavelet neural network

propagation Figure 3 shows the topological structure of theartificial wavelet neural network

In Figure 3 119883119894(119894 = 1 2 5) are the input factors

which correspond to the reservoir water level componentprecipitation component temperature component timeli-ness component and the exceeded water level factor 119884 is theoutput value that is the seepage index and119908

119894119895and V119895are the

weight valuesThe data series of input factors are119883

119894(119894 = 1 2 5) and

the output in the hidden layer is

ℎ (119895) = ℎ119895[

sum4

119894=1119908119894119895119883119894minus 119887119895

119886119895

] 119895 = 1 2 119897 (17)

where ℎ(119895) is the output value of the 119895th node in the hiddenlayer119908

119894119895is the weight between the input layer and the hidden

layer ℎ119895is the wavelet basis function 119887

119895is the shift factor of

the wavelet basis function and 119886119895is the scaling factor of the

wavelet basis functionAs the excitation function in the network the selection

of wavelet function is very important to the fitting andpredicated results SinceMorlet wavelet has the characteristicof calculation stability small error and fine robustness onerror interference a Morlet wavelet [17] with the functionimage shown in Figure 4 is used in the artificial wavelet neuralnetwork and its specific formula is given as follows

ℎ (119909) = cos (175119909) 119890minus11990922 (18)

Then the output of the seepage index is

119884 =

119897

sum

119895=1

V119895ℎ (119895) (19)

where119884 is the exporting seepage index and the other parame-ters are similar to those in (17) and (18) and Figure 4

The gradient descent method is used to modify theweights in the artificial wavelet neural network throughwhich the output values gradually approach the expectation

h(x)

10

08

06

04

02

0

minus02

minus04

X

minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5

Figure 4 Morlet wavelet function

until the termination condition is satisfiedThe concrete stepsare given as follows

(1) Parameter initialization the scaling factor 119886119895 the shift

factor 119887119895 and the weights 119908

119894119895and 119908

119895119896are initialized

randomly and the learning rate 120578 and themomentumcoefficient 120572 are selected appropriately

(2) Classification of original samples the original sam-ples are classified as either the training set or the testset The training set is used to train the network andthe test set is used to test the network

(3) Comparison of output values the test set is importedinto the network and the mean squared error 119890 isobtained by subtracting the predicted output from theexpected output as follows

119890 =1

119873

119873

sum

119894=1

(119884 minus )2

(20)

where is the expected output and 119873 is the samplesize

(4) Modification of weight based on the error 119890 thegradient descentmethod is used tomodify theweightscaling factor and shift factor to make the predictedoutput approximate the expected output

(5) Convergence judgment when the computation isconverged the computation is halted Otherwisethe computation should return to Step (3) and thenetwork parameters should be adjusted as follows

Δ119908119894119895

119899+1= minus120578

120597119890

120597119908119894119895

119905+ 120572Δ119908

119894119895

119899

Δ119886119895

119899+1= minus120578

120597119890

120597119886119895

119905+ 120572Δ119886

119895

119899

Δ119887119895

119899+1= minus120578

120597119890

120597119887119895

119905+ 120572Δ119886

119895

119899

(21)

6 Mathematical Problems in Engineering

where Δ119908119894119895

119899 Δ119886119895

119899 and Δ119886119895

119899 are the changes in thenetwork parameters in the 119899th iterationIn addition

119908119894119895

119899+1= 119908119894119895

119899+ Δ119908119894119895

119899+1

119886119895

119899+1= 119886119895

119899+ Δ119886119895

119899+1

119887119895

119899+1= 119887119895

119899+ Δ119887119895

119899+1

(22)

where 119908119894119895

119905 119886119895

119905 and 119887119895

119905 are the network parameters inthe 119899th iteration

The weight and valve values are modified until the ter-minal condition is satisfied The finally obtained parametersare substituted into the artificial wavelet neural network andcombined with the import of reservoir water level com-ponent precipitation component temperature componenttimeliness component and the exceeded water level factorthe seepage index series are then exported Finally the moni-toringmodel based on the artificial wavelet neural network isobtained and can be used tomonitor earth-rock dam seepage

4 The Integrated Earth-Rock DamSeepage Monitoring Model Based onMaximum Entropy Theory

The two above-mentioned earth-rock dam seepage monitor-ing models mentioned above have their own strengths andweaknessesThrough the consideration of the hysteretic effectof seepage and the exceeded water level factor the statisticalmodel reflects the seepage status of the earth-rock dam to acertain extent However a reasonable explicit function thatreflects the complex influencing factors of seepage is difficultto select Artificial wavelet neural network helps to addressthis difficulty and is quite capable of determining the localdetails ofmeasured data Simultaneously the artificial waveletneural network suffers from artificial factors Thereforebased on the application of maximum entropy theory toobtain the advantages of the two above-mentioned monitor-ing models mentioned the integrated earth-rock dam seep-age model is established This model is of great significanceto the monitoring of the seepage status of earth-rock dams

Maximum entropy theory originates from classical ther-modynamics which is an important theory in the frontierof modern physics Entropy is applied to information theorywhich has been successfully applied to different subject areasSpecifically the expression of entropy is given as follows

119878 (119909) = minus

119899

sum

119894=1

119901119894ln119901119894 (23)

where 119901119894is the probability that the signal 119909

119894appears in a

signal source and 119878(119909) is the magnitude of the entropy whichmeasures the uncertainty of the system status

Under the given condition to solve an ill-posed problemone probability distribution among all possible probabilitydistributions is found to have the maximum value of entropy

For the seepage index of earth-rock dams the optimiza-tion problem inmaximum entropy theory is given as follows

max 119878 (119883) = minus

119899

sum

119894=1

119901119894ln119901119894 (24)

In addition the constraint condition is119899

sum

119894=1

119891119896(119909119894) 119901119894= 119865119896

119896 = 1 2

119899

sum

119894=1

119901119894= 1 119901

119894ge 0

(25)

When the entropy is maximized the following equationholds

119901119894= exp[120582

0+

2

sum

119896=1

120582119896119891119896(119909119894)] 119894 = 1 2 119899 (26)

where 119878(119883) is the entropy of the seepage index 119901119894is the

probability when the value of the seepage index119883 is 119909119894119891119896(119909119894)

is a function such as the first-order central moment andsecond-order central moment of 119883 119865

119896is the mean value

of 119891119896(119909119894) 120582119896are Lagrange multiplier 119896 = 1 corresponds

to the earth-rock dam seepage statistical model and 119896 =

2 corresponds to the earth-rock dam seepage monitoringmodel based on the artificial wavelet neural network

The seepage index of earth-rock dams can be regarded asa discrete series continuously changing time which is usedin the two monitoring models to obtain predicted values andthe difference between the predicted value and the measuredvalue With the obtained results regarding the constrainedinformation of the predictor maximum entropy theoryis applied to solve the constraint information problem toimprove the precision of the model predicationThe concretecalculation steps are given as follows

(1) Calculation of Model Eigenvalues To obtain informationabout the earth-rock dam seepage statistical model and theearth-rock dam seepage monitoring model based on the arti-ficial wavelet neural network eigenvalues of the computationresults of the models are obtained Suppose that119873measuredvalues are provided and the 119894th measured data set is 119909

119894(119894 =

1 2 119873) and the calculated values of the seepage indexbased on the two monitoring models are

119894119896(119896 = 1 2) The

variance of the calculated values based on the two models is

119890119896=

1

119873

119873

sum

119894=1

(119909119894minus 119894119896

119894119896

)

2

119896 = 1 2 (27)

(2) Solution of the Probability Density Function Based onMaximum Entropy The earth-rock dam seepage index 119883 isa discrete random variable and the following equation basedon maximum entropy theory is given

max 119878 (119883) = minus

119873

sum

119894=1

119901119894ln119901119894 (28)

Mathematical Problems in Engineering 7

The constraint condition is119873

sum

119894=1

(119909119894minus 119894119896

119894119896

)

2

119901119894= 119890119896

119873

sum

119894=1

119901119894= 1 119901

119894ge 0

(29)

Substituting (26) into (29)

119873

sum

119894=1

(119909119894minus 119894119896

119894119896

)

2

exp[1205820+

2

sum

119896=1

120582119896(119909119894minus 119894119896

119894119896

)

2

] = 119890119896 (30)

119873

sum

119894=1

exp[1205820+

2

sum

119896=1

120582119896(119909119894minus 119894119896

119894119896

)

2

] = 1 (31)

Combining (30) and (31) the Lagrangemultipliers 1205820and

120582119896(119896 = 1 2) are obtained and the probability function of the

119894th measured value is

119901 (119883 = 119909119894) = exp[120582

0+

2

sum

119896=1

120582119896(119909119894minus 119894119896

119894119896

)

2

] (32)

(3) Prediction of Seepage Index For the seepage index inthe upcoming 119879 days the predicated results of the 119905th(119905 = 1 2 119879) seepage index based on the two monitoringmodels are

119905119896(119896 = 1 2) and the probability of the 119905th

predicated seepage index is

119901 (119883 = 119909119905) = exp[120582

0+

2

sum

119896=1

120582119896(119909119894minus 119894119896

119894119896

)

2

] (33)

By integrating the probability the desired value whichis the predicated value on the 119905th day is obtained Thenrepeating Step (3) the predicated series of the 119879 seepagemonitoring values of the earth-rock dam is obtained

5 Case Study

51 Project Profile To verify the three models applied to themonitoring of the earth-rock dam seepage status we take areservoir located in Zhejiang China for analysis The reser-voir is an integrated large reservoir which has the functionsof flood protection water supply irrigation and electricitygeneration It controls drainage area of 258 square kilometersand has a total storage of 114million cubicmetersThenormalwater level and the maximum flood level are 115127m and123527m The project consisted of diversion dam normalspillway emergency spillway flood diversion sluice watersupply tunnel and power station The reservoir dam is ClayCore Wall Sand Dam with the top elevation of dam topelevation of wave wall maximum dam height width of damcrest and length of dam crest being 124527m 125727m369m 6m and 560m respectively The upstream slope anddownstream slope of the dam are 1 20ndash280 and 1 185ndash20

Every year there is abundant rain around the reservoirSometimes the reservoir suffered from rainstorm In theperiod between 1 January 2012 and 31 December 2013

11

2012

14

2012

17

2012

110

201

2

11

2013

14

2013

17

2013

110

201

3

11

2014

0

40

80

120

160

200

Prec

ipita

tion

(mm

)

Date (year-month-day)Precipitation

Figure 5 Monitoring data of precipitation (mm)

11

2012

11

2014

14

2012

17

2012

110

201

2

11

2013

14

2013

17

2013

110

201

3

108110112114116118120122124

Rese

rvoi

r wat

er le

vel (

m)

Reservoir water levelDate (year-month-day)

Figure 6 Monitoring data of reservoir water level (m)

the reservoir suffered from rainstorm due to the influenceof Typhoon Haikui which happened in August 2012 andTyphoon Fitow which happened in October 2013 and themaximumprecipitation in one day is 1372mm and 1830mmThe corresponding increased reservoir water levels are 843mand 789m After the rainstorm the reservoir water levelreduces gradually

Therefore we take the measuring point I-1 in the crosssection of 0ndash600 of the earth-rock dam for analysis andthe measured interval of measuring point I-1 is 7 days Bysetting the period between 1 July 2012 and 31 October 2013 asthe modeling period the three monitoring models are usedto fit the piezometric tube level of the measuring point andpredicate the seepage status of the measuring point between 1November 2013 and 31 December 2013 Figures 5 and 6 showthemonitoring data for precipitation and reservoirwater levelbetween 1 January 2012 and 31 December 2013 and theirmeasured interval is 1 day

52 The Application of Three Models in Monitoringthe Seepage Status

521 Statistical Model considering the Hysteretic Effect ofReservoir Eater Level and Precipitation and the Factor ofthe Exceeded Water Level Based on (15) and on the fitnessfunction with the multiple correlation coefficients 119877 a quan-tum genetic algorithm with a population of 40 is used to

8 Mathematical Problems in Engineering

Table 1 Statistical metrics of the statistical model

Coefficient ofdetermination

Root meansquare error

Mean biasederror

0934 0422 1043

0 10 20 30 40 50080082084086088090092094096098

Fitn

ess v

alue

Iterations

Figure 7 Iterative curve of quantum genetic algorithm

optimize the hysteretic parameters of the reservoir water leveland precipitation Figure 7 illustrates the iterative curve ofthe quantum genetic algorithm and the optimal multiplecorrelation coefficient is 0966 Table 1 shows the statisticalmetrics of the statistical model

Through the solution obtained using the optimizationalgorithm the hysteretic parameters 119909

1 1199092 1199093 and 119909

4are

19 days 16 days 40 days and 40 days respectively Theequivalent reservoir water level and precipitation based on (7)and (11) are shown in Figures 8 and 9 Figure 10(a) illustratesthe comparison between the fitting values and the measuredvalues and Figure 10(b) shows the scatter plot of measuredversus fitted with regression line As shown in Figures 10(a)and 10(b) we can clearly see howmodel behaves in lower andhigher regions Table 2 shows the coefficients of the model

For the prediction series between 1 November 2013 and 31December 2013 the predicted values and measured values ofpiezometric level are shown in Table 3

522 Seepage Monitoring Model Based on Artificial WaveletNeuralNetwork The input of themonitoringmodel based onthe artificial wavelet neural network includes the equivalentreservoir water level equivalent precipitation temperaturetimeliness and the exceededwater level factor and the outputis the measuring point I-1 series According to (15) 13 factorsare input factors By setting the number of network nodesof the hidden layer as 10 the topological structure of thenetwork is 13-10-1

Since the measured interval of measuring point I-1 is 7days we select 7 days 14 days and 21 days as the samplingtime to have the comparative analysis Table 4 shows themeansquared error for the three cases and the artificial waveletneural network model with the sampling time of 7 days hasthe greater precision Therefore we select all the measuredvalues of measuring point I-1 to have analysis

17

2012

19

2012

111

201

2

11

2013

13

2013

15

2013

17

2013

19

2013

111

201

3

109110111112113114115116117

Equ

ival

ent r

eser

voir

wat

erle

vel (

m)

Equivalent reservoir water levelDate (year-month-day)

Figure 8 Equivalent reservoir water level (m)

17

2012

19

2012

111

201

2

111

201

3

11

2013

13

2013

15

2013

17

2013

19

2013

0607080910111213

Equi

vale

nt p

reci

pita

tion

(mm

)

Equivalent precipitationDate (year-month-day)

Figure 9 Equivalent precipitation (mm)

Figure 11 shows the mean squared error variability forthe 30 epochs in the network training process using artificialwavelet neural network and the minimum mean squarederror is 0426 Figure 12(a) illustrates the comparison betweenthe fitted values of the monitoring model based on theartificial wavelet neural network and measured values andFigure 12(b) shows the scatter plot of measured versus fittedwith regression line As shown in Figures 12(a) and 12(b)we can clearly see how model behaves in lower and higherregions Table 5 shows the predicated values and measuredvalues of piezometric tube level

In addition Table 6 shows the coefficients of artificialwavelet neural network model Through the comparisonbetween the coefficients of the statistical model and theartificial wavelet neural network we can see the statisticalmodel has a higher precision as a whole

523 The Integrated Seepage Monitoring Model Based onMaximum Entropy Theory Based on maximum entropytheory the Lagrangianmultipliers 120582

0 1205821 and 120582

2are obtained

through the solution of (28)ndash(32) 1205821corresponds to the

statisticalmodel and1205822corresponds to themonitoringmodel

based on the artificial wavelet neural networkAfter applying the probability function the predicated

values in the predication period are obtained Table 7 shows

Mathematical Problems in Engineering 9

17

2012

19

2012

111

201

2

11

2013

13

2013

15

2013

17

2013

19

2013

111

201

3100101102103104105106107108109110

Measured valuesFitted values

Piez

omet

ric le

vel (

m)

Date (year-month-day)

(a)

Measured values

DataFitY = T

101 102 103 104 105 106 107 108 109

Fitte

d va

lues

109

108

107

106

105

104

103

102

101

R = 096654

(b)

Figure 10 (a) Comparison between fitted values of the statistical model and measured values (mm) and (b) the scatter plot of measuredversus fitted with regression line

Table 2 Coefficients of the statistical model

Coefficients 1198600

119886 119887 11988811

11988821

11988812

11988822

Value 130968 minus0308 6465 minus34432 75351 52705 minus35038Coefficients 119889

11198892

1198901

1198902

1198911

1198912

1198913

Value 387371 0399 0708 4304119890 minus 5 0324 minus0959 1261

Table 3 The predicated values of the statistical model

Date Measured value Fitted value2013116 101580 10122520131127 101370 10133620131218 101590 10112320131113 101500 1014542013124 101320 10116020131225 101360 10085420131120 101430 10131320131211 101220 100910

Table 4 Mean squared error in the cases of different sampling time

Sampling time 7 days 14 days 21 daysMean squared error 0426 0613 0768

the predicated values of the integrated model based onmaximum entropy theory

0 1 2 3 4 5 6 7 8 9

10

1Column number

Mea

n sq

uare

d er

ror

Figure 11 Mean squared error variability for 30 epochs in networktraining process by artificial wavelet neural network

To compare the prediction precision of the three modelswe show the statistical metrics of the predicated values ofthe three models in Table 8 and we find that the integratedmonitoring model based on maximum entropy theory fuses

10 Mathematical Problems in Engineering

17

2012

19

2012

111

201

2

111

201

3

11

2013

13

2013

15

2013

17

2013

19

2013

100

102

104

106

108

110

Measured valuesFitted values

Piez

omet

ric le

vel (

m)

Date (year-month-day)

(a)

R = 09189

Fitte

d va

lues

109

108

107

106

105

104

103

102

101

Measured values

DataFitY = T

101 102 103 104 105 106 107 108 109

(b)

Figure 12 (a) Comparison between fitted values of the monitoring model based on artificial wavelet neural network and measured valuesand (b) the scatter plot of measured versus fitted with regression line

Table 5 Comparison between predicated values of the monitoringmodel based on maximum entropy theory and measured values

Date Measured value Fitted value2013116 101580 10189720131127 101370 10157420131218 101590 10107020131113 101500 1021392013124 101320 10138220131225 101360 10106020131120 101430 10131320131211 101220 101137

Table 6 Statistical metrics of the artificial wavelet neural networkmodel

Coefficient ofdetermination

Root mean squareerror

Mean biasederror

0843 0652 0964

the advantages of the two models and thus obtains greaterprediction accuracy

6 Conclusions

Thiswork studied the threemonitoringmodels applied to theseepage status of earth-rock dams influenced by rainstormsThe main content of this paper was as follows

Table 7 Comparison between predicated values of the integratedmodel and measured values

Date Measured value Fitted value2013116 101580 10156720131127 101370 10136520131218 101590 10137120131113 101500 1014662013124 101320 10135320131225 101360 10118920131120 101430 10137620131211 101220 101468

Table 8 Statistical metrics of the predicated values of the threemodels

Model Root meansquare error

Mean biasederror

Statistical model 0304 0249Artificial wavelet neuralnetwork model 0342 0280

Integrated model 0134 0097

(1) Based on the research on the hysteresis effect ofreservoir water levels and precipitation the reservoirwater level and precipitation were equivalently pro-cessed to obtain equivalent reservoir water levels and

Mathematical Problems in Engineering 11

precipitation To easily simulate the rapidly increasedreservoir water lever the exceeded water level factorwas introduced Combined with the components oftemperature and timeliness a statistical model wasestablished

(2) By analyzing the characteristics of wavelet theory andneural networks a monitoring model that fuses bothwavelet theory and neural network was establishedThemonitoring model benefits from a strong nonlin-earitymapping ability and thus can be used to analyzethe detailed characteristics of seepage monitoringdata

(3) Considering the deficiency of a single monitoringmodel an integrated monitoring model based onmaximum entropy theory was established and foundto improve the predication accuracy of seepage statusdetermination for earth-rock dams

(4) Through the application of the three monitoringmodels to a measuring point of an earth-rock daminfluenced by rainstorms we found that the threemodels have acceptable precision in fitting and pred-icating the seepage status thereby providing techno-logical support for the seepage monitoring of similarearth-rock dams

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant nos 51579085 4132300151139001 and 51279052) Project Funded by China State KeyLaboratory of Hydrology-Water Resources and HydraulicEngineering (Grant no 20145028312) Jiangsu Province ldquo333High-Level Personnel Training Projectrdquo (Grant no 2016-B1307101) Open Fund of Key Laboratory of Earth-RockDamFailure Mechanism and Safety Control Techniques MinistryofWater Resources (Grant no YK914022) andHuairsquoanWaterConservancy Academician Workstation

References

[1] M Li and FWangDesign and Construction of Earth Rock DamChina Waterpower Press Beijing China 2011

[2] T V Panthulu C Krishnaiah and J M Shirke ldquoDetection ofseepage paths in earth dams using self-potential and electricalresistivity methodsrdquo Engineering Geology vol 59 no 3-4 pp281ndash295 2001

[3] M Lamea andHMirzabozorg ldquoSimulating nonlinear behaviorof AAR-affected arch dams including detection of crack pro-filesrdquoArabian Journal for Science and Engineering vol 40 no 2pp 329ndash341 2014

[4] C-H Wu S-C Chen and Z-Y Feng ldquoFormation failureand consequences of the Xiaolin landslide dam triggered byextreme rainfall from Typhoon Morakot Taiwanrdquo Landslidesvol 11 no 3 pp 357ndash367 2014

[5] ZWu SafetyMonitoringTheory and Its Application of HydraulicStructures Higher Education Press Beijing China 2003

[6] H Huang and B Chen ldquoDam seepage monitoring modelbased on dynamic effect weight of reservoir water levelrdquo EnergyProcedia vol 16 pp 159ndash165 2012

[7] B J Li and C T Cheng ldquoMonthly discharge forecastingusing wavelet neural networks with extreme learning machinerdquoScience China Technological Sciences vol 57 no 12 pp 2441ndash2452 2014

[8] H Loussifi K Nouri and N B Braiek ldquoA new efficienthybrid intelligent method for nonlinear dynamical systemsidentification the Wavelet Kernel Fuzzy Neural NetworkrdquoCommunications in Nonlinear Science amp Numerical Simulationvol 32 pp 10ndash30 2016

[9] B Doucoure K Agbossou and A Cardenas ldquoTime seriesprediction using artificial wavelet neural network and multi-resolution analysis application to wind speed datardquo RenewableEnergy vol 92 pp 202ndash211 2016

[10] H Gzyl E ter Horst andGMolina ldquoApplication of themethodof maximum entropy in the mean to classification problemsrdquoPhysica A vol 437 Article ID 16220 pp 101ndash108 2015

[11] H Cui and V P Singh ldquoMaximum entropy spectral analysis forstreamflow forecastingrdquo Physica A Statistical Mechanics and ItsApplications vol 442 pp 91ndash99 2016

[12] F A N Palmieri and D Ciuonzo ldquoObjective priors frommaximum entropy in data classificationrdquo Information Fusionvol 14 no 2 pp 186ndash198 2013

[13] A SaiToh R Rahimi and M Nakahara ldquoA quantum geneticalgorithm with quantum crossover and mutation operationsrdquoQuantum Information Processing vol 13 no 3 pp 737ndash7552014

[14] H-L Liu ldquoAcoustic partial discharge localization methodologyin power transformers employing the quantum genetic algo-rithmrdquo Applied Acoustics vol 102 pp 71ndash78 2016

[15] E Pomponi A Vinogradov and A Danyuk ldquoWavelet basedapproach to signal activity detection and phase picking applica-tion to acoustic emissionrdquo Signal Processing vol 115 pp 110ndash1192015

[16] A Alhasan D J White and K De Brabanterb ldquoContinuouswavelet analysis of pavement profilesrdquoAutomation in Construc-tion vol 63 pp 134ndash143 2016

[17] M A Goulart L Sanches M T Vilani and O B P JuniorldquoAnalysis of evapotranspiration by Morlet wavelet in area ofVochysia divergens Pohl in Pantanalrdquo Revista Brasileira deEngenharia Agricola e Ambiental vol 19 no 2 pp 93ndash98 2015

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