Radioactive Target Detection Using Wireless Sensor Network Tonglin Zhang

Abstract The detection of radioactive target is becoming more important recently in public safety and national security. By using the physical law for nuclear radiation isotopes, this chapter proposes a statistical method for wireless sensor network data to detect and locate a hidden nuclear target in a large study area. The method assumes multiple radiation detectors have been used as sensor nodes in a wireless sensor network. Radiation counts have been observed by sensors and each is composed of a radiation signal plus a radiation background. By considering the physical properties of radiation signal and background, the proposed method can simultaneously detect and locate the radioactive target in the area. Our simulation results have shown that the proposed method is effective and efficient in detection and location of the nuclear radioactive target. This research will have wide applications in the nuclear safety and security problems.

Keywords Decision and value fusion Likelihood ratio test Maximum likelihood estimates Signal plus background model Radiation and radioactive isotopes Wireless sensor network

31.1 Introduction Recent advances in wireless communications and electronics have enabled the development of low cost, lower-power, multifunctional sensor nodes that are small in size and efficiently communicate in short distance [1]. These tiny sensor nodes T. Zhang (&) Department of Statistics, Purdue University, 150 N. Unversity Street, West Lafayette 47906, USA e-mail: [email protected]

X. He et al. (eds.), Computer, Informatics, Cybernetics and Applications, Lecture Notes in Electrical Engineering 107, DOI: 10.1007/978-94-007-1839-5_31, Springer Science+Business Media B.V. 2012

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which consist of sensing, data processing, and communicating components, enhanced the ideas of wireless sensor networks (WSN). The goal of any WSN is to provide measures of a given physical process, which may have many physical parameters, to detect specific events [2]. Therefore, there are substantial amount of work on WNS for target detection [3, 4, 5]. A general question for a WSN is how to efficiently integrate the available information from individual sensors to reach a global decision about the presence of a target over a monitoring area. The past work on signal detection with WSN can be categorized into two groups of methods: decision fusion and value fusion methods, respectively [6]. In decision fusion, each sensor makes its own binary decision and then the network will make a consensus by fusing all decisions [5, 7, 8]. In value fusion, sensors collect measurements and the network will make a decision by fusing the collected values. In this chapter, we focus on the value fusion WSN methods because the original radiation data are counts, which are easily transmitted to the fusing center with low cost [9]. We assume a value fusion WSN has been used to monitor a large study area. In the monitored area, neither the location nor the strength of the radioactive target is known. A radiation detector, also known as a particle detector, is a device used to detect, track, and/or identify high-energy particles emitted from radioactive materials, including neutrons, alpha particles, beta particles, and gamma rays. In general, the observed radiation counts received by detectors can be modeled as a mixture of the signal from the radioactive source and the natural background radiation [10]. Let the area and efficiency of the radiation detector be A and e respectively. Then the total number of radiation count received by the detector (denoted by y) satisfies ms y PoissonðTAeðmb þ ÞÞ; ð31:1Þ 4pr 2 where T is the time duration, r is the distance between the detector and nuclear radioactive target, ms is the surface radiation intensity rate, and mb is the background radiation intensity rate. The research will have wide applications in the nuclear safety and security problem. We give the following two examples to highlight the importance. Example 1 Since the world trade center tragedies on September 11, 2001, the United States government has acknowledged the high possibility of terrorist attacks using weapons of mass destruction (WMD) [11]. A good summary of physical models of the nuclear weapons can be found in Fetter et al. [12]. Because a hidden nuclear weapon contains radioactive materials, which emit radiation from its surface, the method developed by this chapter will likely be used to the defense of nuclear terrorism country-widely. Example 2 The recent Fukushima nuclear power station disasters caused by the earthquake in Japan on March 11, 2001 have made serious nuclear pollution in the World. Currently, it is still too early to judge the final outcomes of the nuclear crisis that continues to Japan. This tragedy uncovers a huge safety and security problems that exist in all of the nuclear power plants in the world. In nuclear power

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plants, nuclear fuel and waste contain uranium or other radioactive materials. In order to improve the safety and security level of a nuclear power plant, it is important to develop a quickly and accurately detection system for the leakage of the nuclear units. The method developed by this chapter will provide a theoretical base for the detection system. This statistical model of the above examples is called the signal plus background model in the particle physics [13]. They can be written as special cases of Model (31.1). Based on the general framework of WSN, we may deploy many radiation detectors as sensor nodes at different locations. The statistical approach then can be proposed. The rest of the paper is organized as follows. In Sect. 31.2, we introduce our statistical methods. In Sect. 31.3, we introduce our numerical algorithms. In Sect. 31.4, we display our simulation results. In Sect. 31.5, we present our conclusion.

31.2 Statistical Method Assume a nuclear radiation target is hidden at an unknown location denoted by x ¼ ðx1 ; x2 ; x3 Þ: Suppose a WSN is used to detect and locate the target. Assume the WSN has m radiation detectors, which are deployed at ai ¼ ðai1 ; ai2 ; ai3 Þ for i ¼ 1; . . .; m respectively. Let qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ri ðxÞ ¼ kai xk ¼ ðai1 x1 Þ2 þ ðai2 x2 Þ2 þ ðai3 x3 Þ2 be the Euclidean distance between the target and the i-detector. Let Ai be the area and ei be the efficiency of the i-th detector respectively. Let T be the duration time of the detection (given by seconds). Let yi be the total number of radiation count observed by the i-th detector during the detection period. Then, we have yi PoissonðTni ðmb þ

ms ÞÞ; i ¼ 1; . . .; m; 4pri2 ðxÞ

ð31:2Þ

where ni ¼ Ai ei is the capability of detection for i-th detector. The unknown parameters in model (31.2) are mb ; ms and x: Based on model (31.2), the detection problem is interpreted as the hypothesis test of H0 : ms ¼ 0 vs H1 : ms [ 0;

ð31:3Þ

and the location problem is interpreted as the estimation and confidence interval for x: We propose a statistical method to test the significance of H0 : ms ¼ 0: The method is based on the famous likelihood ratio test, which is accessed by the likelihood ratio statistic. The likelihood ratio statistic is defined by the ratio of likelihood functions derived under H0 [ H1 : ms 0 and under H0 : ms ¼ 0

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respectively. When the test is significant, the location problem is accessed by the derivation of the estimate and confidence interval for x; which can be derived by the maximum likelihood method. In order to derive the likelihood ratio test statistic. We need to compute the maximum of the likelihood functions under H0 [ H1 : ms 0 and H0 : ms ¼ 0 respectively. In order to derive the estimate and confidence interval for x; we need to derive the limiting distribution of the maximum likelihood of x: The statistical approach is displayed in the remaining part of this section. The corresponding numerical algorithms will be displayed in the next section. Straightforwardly, the log likelihood function under model (31.2) is ‘ðms ; mb ; xÞ ¼

m X

logðYi !Þ þ

i¼1

þ

ms : 4pri2 ðxÞ

m X

Yi log½Tni ðmb þ

i¼1

m X ms ni ½mb Þ T 4pri2 ðxÞ i¼1

ð31:4Þ Under H0 : ms ¼ 0; the loglikelihood function ‘ðms ; mb ; xÞ in Equation (31.4) becomes ‘0 ðmb Þ ¼

m X i¼1

logðYi !Þ þ

m X

Yi logðTni mb Þ Tmb

i¼1

m X

ni :

ð31:5Þ

i¼1

Note that ‘0 ðmb Þ only contains parameter mb : It is not necessary to compute the estimate of xP under the null Pmhypothesis. Let ^mb;0 ¼ m i¼1 Yi =ðT i¼1 ni Þ be maximum likelihood estimate (MLE) of mb under H0 : ms ¼ 0: Then, ^mb can be analytically solved by ^mb;0 ¼ Pm Pm ^ be the MLE of ms ; mb and x under H0 [ H1 : ms ; ^mb and x i¼1 Yi =ðT i¼1 ni Þ: Let ^ ^ cannot be analytically solved, we have to ms 0: Because the MLE ^ms ; ^mb and x ^ This method will be introduced in develop a numerical algorithm of ^ms ; ^mb and x: ^ are derived. Then, ð^ms ; ^mb ; xÞ ^ as T ! 1 is the next Section. Suppose ^ms ; ^mb and x asymptotically normal [16]. This property will be used to locate nuclear radioactive target when the test is significant. We propose a loglikelihood ratio test to access the null hypothesis. By comparing Eqs. (31.4) and (31.5), we find that the location x is not present in (31.5). Therefore, this problem is nonstandard because the classical loglikelihood ratio test does not possess it usually asymptotic null distribution [1, 8]. To define the loglikelihood ratio test, we propose a conditional test statistic and use it to formulate the test by maximizing the conditional test statistic. Suppose x is pre-selected. Then, Eq. (31.4) only contains parameters ms and mb : In this case, the testing problem becomes standard. Therefore, the loglikelihood ratio test can be formulated by the conditional test statistic given by KðxÞ ¼ 2½‘ð^ms;x ; ^mb;x ; xÞ ‘0 ð^mb;0 Þ

ð31:6Þ

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where ^ms;x and ^mb;x are the conditional MLE of ms and mb under H0 [ H1 : ms 0 respectively. It is easily to see that K approximately follows v21 distribution if T is large [16]. Because x is unknown, we assume it belongs a set D R3 : Then, the loglikelihood ratio test statistic is defined by K ¼ sup KðxÞ:

ð31:7Þ

x2D

The null hypothesis H0 : ms ¼ 0 is rejected if K is large. Because neither the approximate nor the exact distribution of K is known, we propose a bootstrap method to access its p-value. If the p-value of K is less than a pre-selected significance level a; the test is significant; otherwise, the test is not significant. When the test is significant, we need to develop a method to locate the radioactive target in the study area. Since the problem becomes standard under ms [ 0; the location x of the radioactive target then can be estimated by the maximum likelihood method and its asymptotical distribution can be easily derived. Based on this property, we can compute the 100ð1 aÞ% elliptical confidence region for x by Ca ðxÞ ¼ fx :

1 ^ 0 Ið^ms ; ^mb ; xÞðx ^ ^ v2a;3 g; ðx xÞ xÞ T

^ is the 3 3 where v2a;3 is the upper a quantile of the v23 distribution, and Ið^ms ; ^mb ; xÞ Fisher Information matrix given by Ij1 ;j2 ¼

m X ^ j1 Þðaij2 x ^ j2 Þ 4ni^m2s ðaij1 x ; j1 ; j2 ¼ 1; 2; 3: 8 ^ 6 ^ ^ ^ r ð xÞ þ m r ð xÞ m b s i i i¼1

ð31:8Þ

31.3 Numerical Algorithms In this section, we introduce our numerical methods. The methods includes the ^ and the algoalgorithm for ^ms;x ; ^mb;x for a given x; the algorithm for ð^ms ; ^mb ; xÞ; ^ is developed rithm for the bootstrap p-value of K: The algorithm for ð^ms ; ^mb ; xÞ under the algorithm for ^ms;x ; ^mb;x : It is also used to develop the algorithm for the bootstrap p-value of K: When x is pre-selected, ^ms;x ; ^mb;x can be derived by solving the likelihood equation o‘=oms 0 rx ðms ; mb Þ ¼ ¼ : o‘=omb 0 Let Hx ðms ; mb Þ be the Hessian matrix of ‘ðms ; mb ; xÞ conditional on x: Then, Hx ðms ; mb Þ is always negative definite because its eigenvalues are always negative

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[17]. Therefore, ‘ðms ; mb ; xÞ as a function of ms and mb is concave, which induces the following Newton–Raphson algorithm for ^ms;x and ^mb;x always converges. Algorithm for ^ms;x and ^ms;x : ð0Þ

ð0Þ

• Derive an initial guess of ms;x and mb;x : ðuÞ

ðuÞ

• Let ms;x and mb;x be the solution in step u. Then, the solution of the next step is updated by ! ! ðuþ1Þ ðuÞ ms;x ms;x ðuÞ ðuÞ ðuÞ ¼ Hx1 ðmðuÞ ðuþ1Þ ðuÞ s;x ; mb;x Þrðms;x ; mb;x Þ: mb;x mb;x • Iterate the algorithm until convergence. The final solution from the above algorithm is the global maximizer of ‘ðms ; mb ; xÞ for a given x: It is the conditional MLE of ms and mb given x: This is useful in the computation of the unconditional MLE ð^ms ; ^mb ; xÞ of ðms ; mb ; xÞ: Because the ‘ðms ; mb ; xÞ is not concave as functions of ms ; mb and x; the big concern is convergence of the Newton–Raphson method. Therefore, we discard ^ below. In this this method and propose a bisection algorithm for ^ms ; ^mb and x algorithm, we assume x 2 D with D covered by a three dimension rectangle W, as D W ¼ fx : Cj hj xj Cj þ hj ; j ¼ 1; 2; 3g: The following algorithm is able to solve the global maximum of ms ; mb and x for any x 2 D: ^ Algorithm for ^ms ; ^mb and x: ð0Þ

– Let hj

ð0Þ

ð0Þ

ð0Þ

¼ hj for j ¼ 1; 2; 3 and xð0Þ ¼ ðx1 ; x2 ; x3 Þ ¼ ðC1 ; C2 ; Cj Þ: ðuÞ

ðuÞ

ðuÞ

ðuÞ

– Define xk1 k2 k2 ¼ ðx1;k1 k2 k3 ; x2;k1 k2 k3 ; x3;k1 k2 k3 Þ with ðuÞ

ðuÞ

ðuÞ

xj;k1 k2 ;k2 ¼ xj þ

ðkj 3Þhj ; k1 ; k2 ; k3 ¼ 1; 2; 3; 4; 5: 2

ðuÞ

Then, fxk1 k2 k3 : k1 ; k2 ; k3 ¼ 1; 2; 3; 4; 5g is composed of a 5 5 5 lattice ðuÞ

ðuÞ

centered at x3;3;3 with side unit increment hj =2 for j ¼ 1; 2; 3: ðuÞ

– Compute ^ms;x and ^mb;x by letting x ¼ xk1 k2 k3 for all k1 ; k2 ; k3 ¼ 1; 2; 3; 4; 5; and calculate all ‘ðms;xðuÞ

k1 l2 ;k3

; mb;xðuÞ

k1 l2 ;k3

ðuÞ

ðuÞ

; xk1 l2 ;k3 Þ values. Let xkm

k k 1 m2 m3

be the global

ðuÞ

maximizer for all of those x ¼ xk1 k2 k3 : ðuÞ

ðuþ1Þ

– Let xðuþ1Þ ¼ xkm

k k 1 m2 m3

ðuþ1Þ

wise let hj

ðuÞ

: If 1\km1 ; km2 ; km3 \5; then let hj

¼ hj : Iterate this algorithm until convergence.

ðuÞ

¼ hj =2; other-

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The above algorithm does not need the concavity of ‘ðms ; mb ; xÞ: It starts from an initial rectangle and then reduces the volume of the rectangle to 1=8 at each step. Therefore, it definitely converges. If the final result claims ms \0; then ^ms ¼ 0 which induces ^mb ¼ ^mb;0 : In this case K ¼ 0: Otherwise, K is positive. This algorithm is usedP in the computation the the bootstrap p-value of K: Pof m Let yþ ¼ m y and k ¼ þ i¼1 i i¼1 ki : Then under the null hypothesis, we have n ðy1 ; ; ym Þjyþ Multinomialðyþ ; ðPm 1

i¼1

ni

n ; . . .; Pmm

i¼1

ni

ÞÞ:

ð31:9Þ

Therefore, the algorithm for the bootstrap p-value is given as follows. Bootstrap method for the p- value of K Compute the observed value of K based on the observed counts y1 ; . . .; ym : Let it be K0 : (ii) Generate K independent random samples from Model (31.9). Denote the kth sample as ðy1;k ; ; ym;k Þ with k ¼ 1; . . .; K: (iii) Compute the value of K based on each generated ðy1;k ; ; ym;k Þ: Let it be Kk : (iv) The bootstrap p-value of K is derived by #fKk K0 : k ¼ 0; . . .; Kg=ðK þ 1Þ; where #ðSÞ is the number of elements contained in set S. (i)

We choose K ¼ 999 so that the bootstrap p-value is given by 0:001 increment. This p-value of K can be used to test the significance of the radioactive target. The null hypothesis will be rejected if the p-value is less than the significance level (e.g. 0.05).

31.4 Simulation Results We evaluated the detection method by the behavior of its power function. We also evaluated the location method by the behavior of the mean square error. The evaluation was based on Monte Carlo simulations. In order to save the computational time, we assumed the third dimension of the deployed sensors were 0 and the hidden nuclear radioactive target was also installed on the plane of the deployed sensors. That is, we assumed x3 ¼ ai3 ¼ 0 for all i. In our simulation, we assumed the WSN had 100 radiation sensors. They were identical and deployed at the 10 10 lattice. Assume the radioactive target was installed at (5.5,5.5). Consequently, the distance between the radioactive target and the i detector is ri ¼ ½ðai1 0:5Þ2 þ ðai2 0:5Þ2 1=2 with ðai1 ; ai2 Þ be the ith lattice points of the 10 10 lattice. The radiation count yi received by the ith detector then has the distribution yi PoissonðTni ðmb þ

ms ÞÞ; i ¼ 1; ; 100: 4pri2

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^ Table 31.1 Power function of K and MSE of x ms Power function of K ^ MSE of x

0

0.1

0.2

0.4

0.5

0.6

0.7

0.8

0.9

0.049 1.475

0.073 0.682

0.170 0.395

0.443 0.124

0.746 0.044

0.992 0.015

0.986 0.012

1.000 0.009

1.000 0.007

We assumed the sensors were identical so that ni were all the same. To make our simulation simple, we assumed ni ¼ 1 for all i. We fixed mb ¼ 1 and chose T ¼ 1000: We let ms vary from 0 to 0.8 with step ^ increment 0.1. We computed the power function of K and derived the MSE of x based on 1000 simulation repetitions. The power function was evaluated by the ^ percentage of the significance based on significance level a ¼ 0:05: The MSE of x 2 ^ was derived by the average of kx xk : The simulation result is given in Table 31.1 . Table 31.1 showed that the power function of K increased to 1 as ms became stronger. The type I error probability was displayed by the case when ms ¼ 0: As we expected, the power function was close to 0.05 when ms ¼ 0: In addition, the ^ approached to 0 as ms became stronger, which table also showed that the MSE of x indicated the radiation signal can be successfully located as T became large. In our simulation, we have found that the numerical results were almost identically if pﬃﬃﬃﬃﬃﬃﬃ Tms =mb kept constant. Therefore, Table 31.1 represented a group of numerical simulation scenarios.

31.5 Conclusion In this chapter, we have proposed a statistical method as well as the algorithm for nuclear radiation target detection based on value fusion data of WSN. Comparing with methods using single detectors, our method can simultaneously detect and locate nuclear radioactive targets. Numerical results based on Monte Carlo simulations showed that our algorithm could successfully fulfill our tasks. Even though our simulation only considered a particular set out of WSN, the idea presented by this chapter will be extensively used to real world nuclear safety and security problems. Acknowledgments This research was supported by the United States National Science Foundation Grant SES-07-52657.

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