August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011 Journal of Earthquake and Tsunami Vol. 7, No. 2 (2013) 1350011 (11 pages) c World Scientific Publishing Company DOI: 10.1142/S1793431113500115 OPTIMIZATION OF SEISMIC SENSOR LOCATIONS ALONG HIGHWAY LINKS TADANOBU SATO International Institute for Urban System Engineering Southeast University 2 Sipailou Street Nanjing, Jiansu 210096, China [email protected]IKUMASA YOSHIDA Department of Urban and Civil Engineering Tokyo City University 1-28-1 Tamatsutsumi Setagaya Tokyo 158-8557, Japan YUKIO ADACHI Hanshin Express Public Corporation University Osaka 541-0056, Japan Received 15 January 2013 Accepted 25 March 2013 Published 30 June 2013 Optimum positioning of newly installing seismographs along highway lines is discussed. First, we introduce a method to estimate the spatial distribution of earthquake intensity under the condition that several observation data of earthquake intensity are provided at given sensor locations. We also define a spatial index which indicates the estimation error level of the spatial distribution of earthquake intensity of which value becomes zero at the sensor locations. The optimum sensor location problems can be formulated as a minimization problem of this index value which is a function of the sensor locations and the prior information of the special distribution of earthquake intensity in the target area. The design parameters are locations of the sensors. Genetic Algorithm (GA) is used for the optimization. After a brief introduction of real coded GA, optimum sensor locations along highway lines are determined as to minimize the spatial index value by using this GA code. Keywords : Optimum positioning; seismographs; highway lines; spatial distribution; ground motion intensity; Kalman filter; genetic algorithm. 1. Introduction Nowadays civil infrastructures are composed of many structures and facilities are monitored by sets of sensor systems for maintenance and emergency management purposes. The locations of sensors are usually determined by engineer’s experience or subjective judgement. The sensor locations should be determined as to minimize uncertainties of the judgement and estimated special distribution of earthquake intensity based on the observation data. There are two aspects in this problem, 1350011-1
Transcript
2nd Reading
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
Department of Urban and Civil EngineeringTokyo City University 1-28-1 Tamatsutsumi Setagaya
Tokyo 158-8557, Japan
YUKIO ADACHI
Hanshin Express Public Corporation UniversityOsaka 541-0056, Japan
Received 15 January 2013Accepted 25 March 2013Published 30 June 2013
Optimum positioning of newly installing seismographs along highway lines is discussed.First, we introduce a method to estimate the spatial distribution of earthquake intensityunder the condition that several observation data of earthquake intensity are providedat given sensor locations. We also define a spatial index which indicates the estimationerror level of the spatial distribution of earthquake intensity of which value becomes zeroat the sensor locations. The optimum sensor location problems can be formulated as aminimization problem of this index value which is a function of the sensor locations andthe prior information of the special distribution of earthquake intensity in the targetarea. The design parameters are locations of the sensors. Genetic Algorithm (GA) isused for the optimization. After a brief introduction of real coded GA, optimum sensorlocations along highway lines are determined as to minimize the spatial index value byusing this GA code.
Nowadays civil infrastructures are composed of many structures and facilities aremonitored by sets of sensor systems for maintenance and emergency managementpurposes. The locations of sensors are usually determined by engineer’s experienceor subjective judgement. The sensor locations should be determined as to minimizeuncertainties of the judgement and estimated special distribution of earthquakeintensity based on the observation data. There are two aspects in this problem,
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
T. Sato, I. Yoshida & Y. Adachi
(1) how to define a proper index evaluating appropriate sensor locations in the con-cerned infrastructure system, by which the priority of sensor locations distributingin the system can be indicated, (2) how to minimize this index with respect to thesensor locations by selecting from the prior suggested candidate locations.
In this study, the optimum positioning of seismographs along highway lines isdiscussed. The purpose of monitoring the highway lines is to take proper actions forquick recovery from the damage to the structure system caused by events and toprevent secondary disaster after the events. It is important to estimate seismic inten-sity properly immediately after an earthquake occurrence along the highway linesfrom the observed data. Because highway lines cover wide area it takes long timeto finish complete inspection of structural damage in the highway lines. Thereforeestimation of damage to structure systems before starting the action to repair thedamage to structures is essential using the estimated special distribution of seismicintensity conditioned by limited observation data. Inter/extrapolation algorithm isneeded to estimate the seismic intensity at the site without a sensor. Kriging isone of methods that can estimate spatial distribution of seismic intensity in theconcerned area from the limited number of observed points based on the proba-bilistic concept [Journel, 1977; Cressie, 1991]. In this method the uncertainty of theestimation is also obtained. The index that expresses the advantage of the sensorlocation can be set up based on the estimation uncertainty.
Once the index is defined, optimum sensor locations can be obtained by mini-mizing the index with respect to the position of sensors. This minimization problemis expected to have many local minimums so that ordinary minimization methodsuch as Gauss–Newton method, DFP or BFGS cannot be used. Recently a lot ofglobal minimization methods have been studied. One of the most prospective meth-ods is Genetic Algorithm (GA) which is assimilated from the way of evolution oflife, especially mechanism of genes. Some researchers proposed GA with real codedcrossover operators and reported advantage of their methods [Tsutsui et al., 1997;Kitai et al., 1999].
In this paper, the sensor location index is proposed based on the uncertaintiesof the estimated spatial distribution, and the optimum seismograph positions alongnewly constructed highway lines are determined by minimizing the location indexwith respect to the sensor locations.
2. Objective Function to Solve Optimal Sensor Location Problem
2.1. Formulation to estimate special distribution of state variable
The problem of stochastic field can be expressed by a pair of the state transfer andobservation equation as follows:
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
Optimization of Seismic Sensor Locations Along Highway Links
where x is the discretized stochastic field variable vector (defined as the statevector), x̄ is its deterministic component and w is random component of x. zis the vector composed of the discrete observed variable. The observation Eq. (2) isassumed to be expressed as the sum of a linear function of x and a contaminatingnoise vector v. Both of the random vectors v and w are assumed to be Gaussianand independent each other as follows:
E
[(w
v
)(xT vT )
]=
[M 0
0 R
]. (3)
Under the above assumptions and information, we can easily define a simple objec-tive function J based on the analogy of least square error method as given inEq. (4) which is expressed by the sum of two quadratic terms of system noise andobservation noise vectors.
J =12(x − x̄)TM−1(x − x̄) +
12(z − H(x))T R−1(z − H(x)). (4)
We can derived the best estimator of vector x by minimizing the objectivefunction J . From the condition of ∂J/∂x = 0, the best estimator x̂ and the errorcovariance matrix P of the state vector x are given by
x̂ = x̄ + PHT R−1(z − Hx̄), (5)
P = (HTR−1H + M−1)−1. (6)
Equations (5) and (6) are transformed into following expression using Kalmangain K [Kalman, 1960].
x̂ = x̄ + K(z − x̄), (7)
K = MHT (HMHT + R)−1, (8)
P = M − KHM. (9)
Equations (7)–(9) are well known as the observation updating process in Kalmanfilter algorithm.
We here consider a special case in which a part of state vector x is directlyobserved. The vector x is divided into two part, observed part x1 and nonobservedpart x2 as follows:
xT = {xT1 ,xT
2 }. (10)
In this special case, the observation equation is rewritten as
z = Hx = [I 0]
{x1
x2
}+ v. (11)
1350011-3
2nd Reading
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
T. Sato, I. Yoshida & Y. Adachi
According to this division of state vector x into x1 and x2, a priori covariancematrix M is also sub-divided into four parts.
M =
[M11 M12
MT12 M22
]≡ E
[(x1 − x̄1)(x1 − x̄1)T (x1 − x̄1)(x2 − x̄2)T
(x2 − x̄2)(x1 − x̄1)T (x2 − x̄2)(x2 − x̄2)T
]. (12)
With these notations, Kalman gain K defined in Eq. (8) is re-expressed as follows:
K =
[M11 M12
MT12 M22
][I
0
][[I 0]
[M11 M12
MT12 M22
][I
0
]+ R
]−1
=
[M11
MT12
][M11 + R]−1. (13)
Substituting this formula of Kalman gain into Eq. (7), we have the best estimatorof state vector as follows:{
x̂1
x̂2
}=
{x̄1
x̄2
}+
[M11
MT12
][M11 + R]−1{z− x̄1}. (14)
From Eq. (9), the covariance matrix of state vector is also given by the followingsubdivide formula
P =
[P11 P12
PT12 P22
]
≡ E
[M11 − M11(M11 + R)−1M11 M12 − M11(M11 + R)−1M12
MT12 − MT
12(M11 + R)−1M11 M22 − MT12(M11 + R)−1M12
]. (15)
If the observation noise is negligible, R = 0, Eqs. (14) and (15) become simple andcorrespond to the simple Kriging estimation [Hoshiya and Yoshida, 1996].
The matrix M is defined by spatial autocorrelation function R(d1d2). In thispaper, we use the following Gauss type formula
R(d1, d2) = ρ2exp
[−((
d1
a1
)2
+(
d2
a2
)2)]
,
where σ is the standard deviation in the field, d1 and d2 are distance of two pointsalong each coordinate, a1 and a2 are autocorrelation distance along each coordinate.
2.2. Index to determine the appropriate sensor location
Covariance matrix P22 in Eq. (15) expresses the uncertainty of unobserved partof the state vector which means the estimated part of state vector along highwaylines, if predicting points x2 are set along the lines. The index for sensor locationcan be defined based on this covariance matrix P22. Several indices such as thedeterminant (which is often referred as the information entropy) and trace of the
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
Optimization of Seismic Sensor Locations Along Highway Links
matrix are proposed as sensor location index [Hoshiya and Yoshida, 1996]. In thispaper, an index J is used to determine appropriate position of sensor group. Thisis defined by the ratio between the trace of posteriori and that of priori covariancematrix as follows:
J =traceP22
traceM22=
trace(M22 − MT12(M11 + R)−1M12)
trace(M22). (16)
The second term in the trace operator of numerator in Eq. (16) is composed of thecorrelation matrix between the observed and unobserved state vectors M12 and thecovariance matrix of observed part of the state vector M11. If there is no correlationbetween the observed and unobserved state vectors M12 becomes zero matrix. Andthe determinant of M12 is smaller than that of M11. We can easily understand thatthe value of this index J moves within the range between 0 and 1. When this valueis close to 1, that means the correlation between observed and unobserved partsof the state vector is very low, information of observation data is not useful forestimating x2. If this index value becomes smaller, the observation data becomesmore useful, and consequently the better sensor location is determined. We hereminimize this index to determine the most appropriate sensor locations.
3. Optimization of Sensor Location by Gentic Algorithm
3.1. Two objective optimization
In civil engineering problems, there are many multi-objective optimization prob-lems. A typical example is trade-off relationship between safety level and cost ofa structure. Reducing uncertainty to estimate seismic intensity and increasing thenumber of seismographs allocating inside of an infrastructure is a typical exampleof the trade-off problem because we need another extra cost to increase the numberof seismographs. In multi-objective optimization problem, the solutions are givenas a set of optimal points, namely pareto solutions. It is said that GAs are suitablefor this kind of problem [Hiroyasu et al., 1997]. In this section, we illustrate thealgorithm to solve multi-objective optimization problems by GA.
The scalar value which represents the fitness of an individual must be estimatedbased on the plural objective functions. In this paper, we apply the method ofpareto-ranking proposed by Fonseca [Fonseca and Fleming, 1993] to solve the prob-lem. The concept of the pareto-ranking is shown in Fig. 1. Considering the rectanglecomposed of the location of an individual and the origin in objective function space,the fitness of the individual is defined as the number of other individuals withinthe rectangle. Consequently, the fitness of pareto-optimum solutions is 0, while thefitness of the other solutions is integer number more than 0, and less than thepopulation size.
The individuals who have good fitness are preserved to next generation withoutcrossover or mutation. The preserved individuals are called as elite. The numberof elite individuals is one of the GA parameters to be specified. When the number
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
T. Sato, I. Yoshida & Y. Adachi
Fig. 1. (Color online) Concept of pareto ranking.
of pareto-optimum solutions in a generation is less than the specified number, onlythe pareto-optimum solutions are preserved to the next generation in this study.On the other hand, when the number of the pareto-optimum solutions is largerthan the specified number, the specified numbers of individuals with large distancefrom other pareto-optimum solutions in design variable space are selected from thepareto-optimum solutions to maintain the diversity of the solutions.
Performance index Jp is proposed to represent the performance of a set ofpareto-optimum solutions [Yoshida, 2000]. This index is needed for convergencejudgement in numerical calculations as stated below. As shown in Fig. 2, theperformance index Jp is defined based on the area S1, S2 in objective functionspace.
Jp =S2
S1 + S2× 100. (17)
As the value of Jp is the smaller, the performance of a set of pareto-optimumsolutions is the better.
For the effective crossover, two individuals with close objective functionsshould be selected, because solutions of multi-objective optimization problem, eventhe pareto-optimum solutions have diversity. For example, design variables of apareto-optimum solution with large objective function z1 and small objective func-tion z2 are very different from those of the pareto-optimum solution with small z1
and large z2. As shown in Fig. 3, considering the isosceles triangle with the bothends of the pareto-optimum solutions, the angle θ between the solutions can be
Fig. 2. (Color online) Outline of crossover in real coded GA.
1350011-6
2nd Reading
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
Optimization of Seismic Sensor Locations Along Highway Links
Fig. 3. (Color online) Outline of crossover in real coded GA.
defined by the vector from the vertex to the solution. With using the angle θ, par-ents for crossover are selected as follows, (1) A couple of individuals are randomly(roulette strategy) selected as temporary parents, (2) The angle θ between the tem-porary parents is calculated, (3) The temporarily selected parents are accepted withthe probability Pa.
Pa = exp(−(θ/θa)2). (18)
Here, θa is one of the GA parameters to be specified. When small value is speci-fied, crossover between close individuals is performed. When a very large value isspecified, temporarily selected parents are always accepted.
3.2. Two objective optimization
Basically, the outline of real coded GA is same as that of binary coded GA, exceptcrossover operator and mutation. There are several models proposed for real codedcrossover operator with Gaussian noise. The concepts of the crossover operatorwhich are used in this paper are illustrated in Fig. 4 [Honseca and Fleming, 1993].In the case of Type-1, the children are derived by following equation.
c̄ =p̄1 + p̄2
2+ α
p̄1 − p̄2
2, here, α ≈ N(0, 1), (19)
c̄ is a child made from parents p̄1 and p̄2. The children are generated in one-dimensional subspace of the parameter space in the case of Type-1. In Type-2,
Fig. 4. Outline of crossover in real coded GA.
1350011-7
2nd Reading
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
T. Sato, I. Yoshida & Y. Adachi
n dimensional perturbation is given to Type-1 children, consequently Type-2 chil-dren are generated in n dimensional space, when there are n design variables. Themagnitude of the disturbance is defined based on the distance from Parent-3 to theline with Parent-1 and 2. In Type-3, the children are generated in two-dimensionalsubspace made by 3 parents as shown in Fig. 4. In all types, the generated childrenare distributed in the neighborhood of the gravity center of their parents.
Disturbance ν is added to each design variable as mutation. In this study,Cauchy distribution is used.
4. Optimum Location for Newly Installed Sensor
4.1. Two objective optimization
Newly constructed highway lines in Kansai area of Japan are illustrated in Fig. 5.The locations of existent five sensors are also shown. Now we want to determine thebest location to install a new single sensor. Candidates of the installation points arenumbered from 1 to 73. For each point, the location index defined by Eq. (17) is cal-culated and all index values are shown in Fig. 6. Four lines in the figure are the casesfor assuming different auto-correlation distances. The point with minimum locationindex is appeared around the candidate point 44 regardless of auto-correlation dis-tances. This point is a cross point of upper two lines. Point 14 is another cross pointof bottom two lines, but four index values with different auto-correlation distancesat this point are larger than those of point 44. Because the configuration of existingfour observation sites is much closer to the point 14 therefore if we allocate the
Fig. 5. Newly constructed highway and location number for new sensor.
1350011-8
2nd Reading
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
Optimization of Seismic Sensor Locations Along Highway Links
Fig. 6. Newly constructed highway and location number for new sensor.
new observation sensor at the point 44 the uncertainty of seismic intensity alongthe highway lines is effectively reduced. This is the main reason the location indexvalue at the point 44 is much smaller than that of the point 14.
4.2. Optimum location for several sensors
In the case of single sensor, it is simple to determine its best position as far as thelocation index is defined. However, it becomes difficult and complicates to determinethe locations of several sensors simultaneously. The more sensors are installed, wehave more information but need more budget. The number of sensor and uncertaintyof information, namely location index, are in the trade-off relationship. In thispaper, two-objective-optimization problem, the location index and the number ofsensor is solved by using real coded GA introduced in the former section.
The convergence process of the optimization is shown in Fig. 7. The vertical axisrepresents the performance of pareto solutions defined by Eq. (17). The performanceis improved up to around 70th generation. After that, effective improvement is notexpected. Figure 8 shows solutions of generation 1st, 10th, 40th, and 140th. The
Fig. 7. Minimum sensor index of each generation.
1350011-9
2nd Reading
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
T. Sato, I. Yoshida & Y. Adachi
Fig. 8. Pareto solutions of 1st, 10th, 40th, and 140th (final) generation.
(1) Three new sensors (2) Seven new sensors
Fig. 9. Optimum sensor locations obtained by minimizing the sensor location index.
vertical axis represents location index and horizontal axis represents the numberof sensors including currently active five sensors. The optimum sensor locationobtained from the final generation is shown in Fig. 9, for each case of the number ofnew sensors being three (total eight) and seven (total twelve), as examples. Thus,the optimum sensor locations depending on the number of sensor are determined,taking into account locations of existent sensors.
1350011-10
2nd Reading
August 1, 2013 9:40 WSPC/S1793-4311/238-JET 1350011
Optimization of Seismic Sensor Locations Along Highway Links
5. Concluding Remarks
In this paper, the index for evaluating suitability of sensor locations was intro-duced and optimal sensor locations along the highway lines were determined. Thedemonstrated method can determine the optimal points of sensor locations from thestandpoint of probabilistic interpolation. Structural or geographical local conditionsuch as a bridge, edge of mountain or river is also very important. We need to takeinto account this kind of local effect for positioning newly installed sensors. But hereonly the special correlation of earthquake intensity defined by a simple correlationfunction was used for positioning sensors. The precise estimation of seismic motionintensity along the highway lines is essential not only for positioning sensors butalso for damage estimation or total disaster prevention/mitigation plan. These areour future topics.
References
Cressie, N. [1991] Statistics for Spatial Data (John Wiley & Sons, USA).Fonseca, C. M. and Fleming, P. J. [1993] “Genetic algorithms for multiobjective opti-
mization: Formulation, discussion and generalization,” Proc. 5th Int. Conf. GeneticAlgorithms, ed. Forrest, S. (Urbana-Champaign, IL), pp. 416–423.
Hiroyasu, T. and Miki, M. and Watanabe, S. [1999] “Distributed genetic algorithms witha new sharing approach in multiobjective optimization problems,” Proc. Congress onEvolutionary Computation, 1, 69–76.
Hoshiya, M. and Yoshida, I. [1996] “Identification of conditional stochastic Gaussian field,”J. EM. ASCE, 122(2), 101–108.
Journel, A. G. [1977] “Kriging in terms of project,” Math. Geol. 9(6), 563–586.Kitai, H., Ono, I. and Kobayashi, S. [1999] “Theoretical analysis of the unimodal normal
distribution crossover for real-coded genetic algorithms (in Japanese),” Trans. Soc.Instru. Contr. Eng., 35(11), 1333–1339.
Kalman, R. E. [1960] “A new approach to linear filtering and prediction problem,” ASCSJ. Basic Eng. 82D(1), 35–45.
Tsutsui, S., Ghosh, A., Corne, D. and Fujimoto, Y. [1997] “A real coded genetic algo-rithm with an explore and an exploiter populations,” Proc. 7th Int. Conf. GeneticAlgorithms, pp. 238–345.
Yoshida, I. [2000] “Comparison of real coded GA(Genetic Algorithm) and binary codedGA,” Int. Conf. Monte Carlo Simulation, pp. 81–87.
Yoshida, I., Toyoda, K. and Hoshiya, M. [1997] “Appraisal of observation allocation basedon information entropy,” 7th Int. Conf. Structural Safety and Reliability, pp. 955–958.
