Chapter 31Radioactive Target Detection UsingWireless Sensor Network

Tonglin Zhang

Abstract The detection of radioactive target is becoming more important recentlyin public safety and national security. By using the physical law for nuclearradiation isotopes, this chapter proposes a statistical method for wireless sensornetwork data to detect and locate a hidden nuclear target in a large study area. Themethod assumes multiple radiation detectors have been used as sensor nodes in awireless sensor network. Radiation counts have been observed by sensors and eachis composed of a radiation signal plus a radiation background. By considering thephysical properties of radiation signal and background, the proposed method cansimultaneously detect and locate the radioactive target in the area. Our simulationresults have shown that the proposed method is effective and efficient in detectionand location of the nuclear radioactive target. This research will have wideapplications in the nuclear safety and security problems.

Keywords Decision and value fusion � Likelihood ratio test � Maximum likeli-hood estimates � Signal plus background model � Radiation and radioactiveisotopes � Wireless sensor network

31.1 Introduction

Recent advances in wireless communications and electronics have enabled thedevelopment of low cost, lower-power, multifunctional sensor nodes that are smallin 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, USAe-mail: tlzhang@purdue.edu

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

293

which consist of sensing, data processing, and communicating components,enhanced the ideas of wireless sensor networks (WSN). The goal of any WSN is toprovide measures of a given physical process, which may have many physicalparameters, to detect specific events [2]. Therefore, there are substantial amount ofwork on WNS for target detection [3, 4, 5]. A general question for a WSN is howto efficiently integrate the available information from individual sensors to reach aglobal decision about the presence of a target over a monitoring area. The pastwork on signal detection with WSN can be categorized into two groups ofmethods: decision fusion and value fusion methods, respectively [6]. In decisionfusion, each sensor makes its own binary decision and then the network will makea consensus by fusing all decisions [5, 7, 8]. In value fusion, sensors collectmeasurements 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 originalradiation data are counts, which are easily transmitted to the fusing center with lowcost [9]. We assume a value fusion WSN has been used to monitor a large studyarea. In the monitored area, neither the location nor the strength of the radioactivetarget is known. A radiation detector, also known as a particle detector, is a deviceused to detect, track, and/or identify high-energy particles emitted from radioac-tive materials, including neutrons, alpha particles, beta particles, and gamma rays.In general, the observed radiation counts received by detectors can be modeled as amixture of the signal from the radioactive source and the natural backgroundradiation [10]. Let the area and efficiency of the radiation detector be A and erespectively. Then the total number of radiation count received by the detector(denoted by y) satisfies

y�PoissonðTAeðmb þms

4pr2ÞÞ; ð31:1Þ

where T is the time duration, r is the distance between the detector and nuclearradioactive target, ms is the surface radiation intensity rate, and mb is the back-ground radiation intensity rate. The research will have wide applications in thenuclear safety and security problem. We give the following two examples tohighlight the importance.

Example 1 Since the world trade center tragedies on September 11, 2001, theUnited States government has acknowledged the high possibility of terroristattacks using weapons of mass destruction (WMD) [11]. A good summary ofphysical models of the nuclear weapons can be found in Fetter et al. [12]. Becausea hidden nuclear weapon contains radioactive materials, which emit radiation fromits surface, the method developed by this chapter will likely be used to the defenseof nuclear terrorism country-widely.

Example 2 The recent Fukushima nuclear power station disasters caused by theearthquake in Japan on March 11, 2001 have made serious nuclear pollution in theWorld. Currently, it is still too early to judge the final outcomes of the nuclearcrisis that continues to Japan. This tragedy uncovers a huge safety and securityproblems that exist in all of the nuclear power plants in the world. In nuclear power

294 T. Zhang

plants, nuclear fuel and waste contain uranium or other radioactive materials. Inorder to improve the safety and security level of a nuclear power plant, it isimportant to develop a quickly and accurately detection system for the leakage ofthe nuclear units. The method developed by this chapter will provide a theoreticalbase for the detection system.

This statistical model of the above examples is called the signal plus back-ground model in the particle physics [13]. They can be written as special cases ofModel (31.1). Based on the general framework of WSN, we may deploy manyradiation detectors as sensor nodes at different locations. The statistical approachthen can be proposed.

The rest of the paper is organized as follows. In Sect. 31.2, we introduce ourstatistical 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 byx ¼ ðx1;x2;x3Þ: Suppose a WSN is used to detect and locate the target. Assumethe WSN has m radiation detectors, which are deployed at ai ¼ ðai1; ai2; ai3Þ fori ¼ 1; . . .;m respectively. Let

riðxÞ ¼ kai � xk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðai1 � x1Þ2 þ ðai2 � x2Þ2 þ ðai3 � x3Þ2q

be the Euclidean distance between the target and the i-detector. Let Ai be the areaand ei be the efficiency of the i-th detector respectively. Let T be the duration timeof the detection (given by seconds). Let yi be the total number of radiation countobserved by the i-th detector during the detection period. Then, we have

yi�PoissonðTniðmb þms

4pr2i ðxÞÞÞ; i ¼ 1; . . .;m; ð31:2Þ

where ni ¼ Aiei is the capability of detection for i-th detector. The unknownparameters in model (31.2) are mb; ms and x: Based on model (31.2), the detectionproblem 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 intervalfor x:

We propose a statistical method to test the significance of H0 : ms ¼ 0: Themethod is based on the famous likelihood ratio test, which is accessed by thelikelihood ratio statistic. The likelihood ratio statistic is defined by the ratio oflikelihood functions derived under H0 [ H1 : ms� 0 and under H0 : ms ¼ 0

31 Radioactive Target Detection 295

respectively. When the test is significant, the location problem is accessed by thederivation of the estimate and confidence interval for x; which can be derived bythe maximum likelihood method.

In order to derive the likelihood ratio test statistic. We need to compute themaximum of the likelihood functions under H0 [ H1 : ms� 0 and H0 : ms ¼ 0respectively. In order to derive the estimate and confidence interval for x; we needto derive the limiting distribution of the maximum likelihood of x: The statisticalapproach is displayed in the remaining part of this section. The correspondingnumerical algorithms will be displayed in the next section.

Straightforwardly, the log likelihood function under model (31.2) is

‘ðms; mb;xÞ ¼ �X

m

i¼1

logðYi!Þ þX

m

i¼1

Yi log½Tniðmb þms

4pr2i ðxÞÞ� � T

X

m

i¼1

ni½mb

þ ms

4pr2i ðxÞ�:

ð31:4Þ

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

‘0ðmbÞ ¼ �X

m

i¼1

logðYi!Þ þX

m

i¼1

Yi logðTnimbÞ � Tmb

X

m

i¼1

ni: ð31:5Þ

Note that ‘0ðmbÞ only contains parameter mb: It is not necessary to compute theestimate of x under the null hypothesis.

Let m̂b;0 ¼Pm

i¼1 Yi=ðTPm

i¼1 niÞ be maximum likelihood estimate (MLE) of mb

under H0 : ms ¼ 0: Then, m̂b can be analytically solved by m̂b;0 ¼Pm

i¼1 Yi=ðTPm

i¼1 niÞ: Let m̂s; m̂b and x̂ be the MLE of ms; mb and x under H0 [ H1 :ms� 0: Because the MLE m̂s; m̂b and x̂ cannot be analytically solved, we have todevelop a numerical algorithm of m̂s; m̂b and x̂: This method will be introduced inthe next Section. Suppose m̂s; m̂b and x̂ are derived. Then, ðm̂s; m̂b; x̂Þ as T !1 isasymptotically normal [16]. This property will be used to locate nuclear radio-active target when the test is significant.

We propose a loglikelihood ratio test to access the null hypothesis. By com-paring 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 ratiotest does not possess it usually asymptotic null distribution [1, 8]. To define theloglikelihood ratio test, we propose a conditional test statistic and use it to for-mulate 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 loglikelihoodratio test can be formulated by the conditional test statistic given by

KðxÞ ¼ 2½‘ðm̂s;x; m̂b;x;xÞ � ‘0ðm̂b;0Þ� ð31:6Þ

296 T. Zhang

where m̂s;x and m̂b;x are the conditional MLE of ms and mb under H0 [ H1 : ms� 0respectively. It is easily to see that K approximately follows v2

1 distribution if T islarge [16]. Because x is unknown, we assume it belongs a set D � R3: Then, theloglikelihood ratio test statistic is defined by

K ¼ supx2D

KðxÞ: ð31:7Þ

The null hypothesis H0 : ms ¼ 0 is rejected if K is large. Because neither theapproximate nor the exact distribution of K is known, we propose a bootstrapmethod to access its p-value. If the p-value of K is less than a pre-selected sig-nificance 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 theradioactive target in the study area. Since the problem becomes standard underms [ 0; the location x of the radioactive target then can be estimated by themaximum likelihood method and its asymptotical distribution can be easilyderived. Based on this property, we can compute the 100ð1� aÞ% ellipticalconfidence region for x by

CaðxÞ ¼ fx :1Tðx� x̂Þ0Iðm̂s; m̂b; x̂Þðx� x̂Þ� v2

a;3g;

where v2a;3 is the upper a quantile of the v2

3 distribution, and Iðm̂s; m̂b; x̂Þ is the 3 3Fisher Information matrix given by

Ij1;j2 ¼X

m

i¼1

4nim̂2s ðaij1 � x̂j1Þðaij2 � x̂j2Þm̂br8

i ðx̂Þ þ m̂sr6i ðx̂Þ

; j1; j2 ¼ 1; 2; 3: ð31:8Þ

31.3 Numerical Algorithms

In this section, we introduce our numerical methods. The methods includes thealgorithm for m̂s;x; m̂b;x for a given x; the algorithm for ðm̂s; m̂b; x̂Þ; and the algo-rithm for the bootstrap p-value of K: The algorithm for ðm̂s; m̂b; x̂Þ is developedunder the algorithm for m̂s;x; m̂b;x: It is also used to develop the algorithm for thebootstrap p-value of K:

When x is pre-selected, m̂s;x; m̂b;x can be derived by solving the likelihoodequation

rxðms; mbÞ ¼o‘=oms

o‘=omb

� �

¼ 00

� �

:

Let Hxðms; mbÞ be the Hessian matrix of ‘ðms; mb;xÞ conditional on x: Then,Hxðms; mbÞ is always negative definite because its eigenvalues are always negative

31 Radioactive Target Detection 297

[17]. Therefore, ‘ðms; mb;xÞ as a function of ms and mb is concave, which induces thefollowing Newton–Raphson algorithm for m̂s;x and m̂b;x always converges.

Algorithm for m̂s;x and m̂s;x:

• Derive an initial guess of mð0Þs;x and mð0Þb;x:

• Let mðuÞs;x and mðuÞb;x be the solution in step u. Then, the solution of the next step isupdated by

mðuþ1Þs;x

mðuþ1Þb;x

!

¼ mðuÞs;x

mðuÞb;x

!

� H�1x ðmðuÞs;x; m

ðuÞb;xÞrðmðuÞs;x; m

ðuÞb;xÞ:

• 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 isuseful in the computation of the unconditional MLE ðm̂s; m̂b;xÞ of ðms; mb;xÞ:

Because the ‘ðms; mb;xÞ is not concave as functions of ms; mb and x; the bigconcern is convergence of the Newton–Raphson method. Therefore, we discardthis method and propose a bisection algorithm for m̂s; m̂b and x̂ below. In thisalgorithm, 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 forany x 2 D:

Algorithm for m̂s; m̂b and x̂:

– Let hð0Þj ¼ hj for j ¼ 1; 2; 3 and xð0Þ ¼ ðxð0Þ1 ;xð0Þ2 ;xð0Þ3 Þ ¼ ðC1;C2;CjÞ:– Define xðuÞk1k2k2

¼ ðxðuÞ1;k1k2k3;xðuÞ2;k1k2k3

;xðuÞ3;k1k2k3Þ with

xðuÞj;k1k2;k2¼ xðuÞj þ

ðkj � 3ÞhðuÞj

2; k1; k2; k3 ¼ 1; 2; 3; 4; 5:

Then, fxðuÞk1k2k3: k1; k2; k3 ¼ 1; 2; 3; 4; 5g is composed of a 5 5 5 lattice

centered at xðuÞ3;3;3 with side unit increment hðuÞj =2 for j ¼ 1; 2; 3:

– Compute m̂s;x and m̂b;x by letting x ¼ xðuÞk1k2k3for all k1; k2; k3 ¼ 1; 2; 3; 4; 5; and

calculate all ‘ðms;xðuÞk1 l2 ;k3

; mb;xðuÞk1 l2 ;k3

;xðuÞk1l2;k3Þ values. Let xðuÞkm1 km2 km3

be the global

maximizer for all of those x ¼ xðuÞk1k2k3:

– Let xðuþ1Þ ¼ xðuÞkm1 km2 km3: If 1\km1 ; km2 ; km3\5; then let hðuþ1Þ

j ¼ hðuÞj =2; other-

wise let hðuþ1Þj ¼ hðuÞj : Iterate this algorithm until convergence.

298 T. Zhang

The above algorithm does not need the concavity of ‘ðms; mb;xÞ: It starts froman initial rectangle and then reduces the volume of the rectangle to 1=8 at eachstep. Therefore, it definitely converges. If the final result claims ms\0; then m̂s ¼ 0which induces m̂b ¼ m̂b;0: In this case K ¼ 0: Otherwise, K is positive. This algo-rithm is used in the computation of the the bootstrap p-value of K:

Let yþ ¼Pm

i¼1 yi and kþ ¼Pm

i¼1 ki: Then under the null hypothesis, we have

ðy1; � � � ; ymÞjyþ �Multinomialðyþ; ðn1

Pmi¼1 ni

; . . .;nm

Pmi¼1 ni

ÞÞ: ð31:9Þ

Therefore, the algorithm for the bootstrap p-value is given as follows.Bootstrap method for the p- value of K

(i) 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 kthsample as ðy1;k; � � � ; ym;kÞ with k ¼ 1; . . .;K:

(iii) Compute the value of K based on each generated ðy1;k; � � � ; ym;kÞ: Let it beKk:

(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.

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. Thenull 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 alsoevaluated the location method by the behavior of the mean square error. Theevaluation was based on Monte Carlo simulations. In order to save the compu-tational time, we assumed the third dimension of the deployed sensors were 0 andthe hidden nuclear radioactive target was also installed on the plane of thedeployed 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 wereidentical and deployed at the 10 10 lattice. Assume the radioactive target wasinstalled 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; ai2Þ be the ith latticepoints of the 10 10 lattice. The radiation count yi received by the ith detectorthen has the distribution

yi�PoissonðTniðmb þms

4pr2i

ÞÞ; i ¼ 1; � � � ; 100:

31 Radioactive Target Detection 299

We assumed the sensors were identical so that ni were all the same. To make oursimulation 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 stepincrement 0.1. We computed the power function of K and derived the MSE of x̂based on 1000 simulation repetitions. The power function was evaluated by thepercentage of the significance based on significance level a ¼ 0:05: The MSE of x̂

was derived by the average of kx̂� xk2: The simulation result is given inTable 31.1 .

Table 31.1 showed that the power function of K increased to 1 as ms becamestronger. The type I error probability was displayed by the case when ms ¼ 0: Aswe expected, the power function was close to 0.05 when ms ¼ 0: In addition, thetable also showed that the MSE of x̂ approached to 0 as ms became stronger, whichindicated the radiation signal can be successfully located as T became large. In oursimulation, we have found that the numerical results were almost identically ifffiffiffiffiffiffiffi

Tmsp

=mb kept constant. Therefore, Table 31.1 represented a group of numericalsimulation scenarios.

31.5 Conclusion

In this chapter, we have proposed a statistical method as well as the algorithm fornuclear radiation target detection based on value fusion data of WSN. Comparingwith methods using single detectors, our method can simultaneously detect andlocate nuclear radioactive targets. Numerical results based on Monte Carlo sim-ulations showed that our algorithm could successfully fulfill our tasks. Eventhough our simulation only considered a particular set out of WSN, the ideapresented by this chapter will be extensively used to real world nuclear safety andsecurity problems.

Acknowledgments This research was supported by the United States National Science Foun-dation Grant SES-07-52657.

Table 31.1 Power function of K and MSE of x̂

ms

0 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9

Power function of K 0.049 0.073 0.170 0.443 0.746 0.992 0.986 1.000 1.000MSE of x̂ 1.475 0.682 0.395 0.124 0.044 0.015 0.012 0.009 0.007

300 T. Zhang

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