Detecting and Locating Radioactive Signals with
Wireless Sensor Networks
Tonglin Zhang
Department of Statistics, Purdue University
West Lafayette, Indiana 47907-2066, USA
Email: [email protected]
Fax: 1-765-494–0558
Abstract—Methods of detecting and locating nuclear radioactive targets via wireless sensor networks (WSN) are proposed
in this paper. In real applications, a nuclear radioactive target
can be interpreted either as a hidden nuclear weapon held by
nuclear terrorists or as a leak of nuclear radioactive materials at
decision by fusing the collected values.
There are several reasons to believe that value fusion is
more appropriate for nuclear radioactive target detection. First,
radioactive particles are counts. They are usually given by
a unit of nuclear power stations. Assume a WSN is composed of
the total number of particles observed during the detection
radiation detectors deployed at different locations over the area
period, which can be easily reported to the fusion center at
of interest. Detection and location methods have been proposed
the end of the detection period. Second, radioactive particles
based on physical law that radiation received by a sensor is
composed of a background plus a signal, in which the signal
rate decreases as the distance increases. Based on the simulation
are received sequentially at each sensor. Their arrival time can
be easily and cheaply reported to the fusion center as soon as
evaluations, the proposed methods have been shown efcient and
they are observed. Sensors are mostly idled since they only
effective. This research will have wide applications in the nuclear
need to report their status when particles are observed. Third,
safety and security problems.
background uncertainty usually varies spatially in the study
area. There is no requirement to know the background in the
Index Terms—Condence interval, likelihood ratio test, maximum likelihood estimates, signal plus background model, radiation and radioactive isotopes, wireless sensor network.
I. I NTRODUCTION
usage of a value fusion WSN, which gains a lot of advantages
than the usage of a decision fusion WSN. Therefore in this
paper, we focus on the methodological development for a value
fusion WSN.
Suppose a value fusion WSN has been used to detect and
Currently, using wireless sensor networks (WSN) is becom-
locate a nuclear radioactive target. Assume the sensor nodes
ing less expensive since recent advances in wireless commu-
are radiation detectors. In general, a radiation detector, also
nications and electronics have enabled the development of low
known as a particle detector, is a device used to detect, track,
cost, lower-power, multifunctional sensor nodes that are small
and identify high-energy particles emitted from radioactive
in size and efciently communicate in short distance [20].
materials, including neutrons, alpha particles, beta particles,
These tiny sensor nodes which consist of sensing, data pro-
and gamma rays. In general, the observed radiation count
cessing, and communicating components, enhanced the ideas
received by a detector is a mixture of the signal emitted from
of WSN. A WSN is composed of many sensor nodes that are
the radioactive source and the background emitted from natural
densely deployed in a study area. It has wide applications in
radionuclides [16]. In this model, neither the location nor
sensing building and structures monitoring [27], trafc control
the strength of the radioactive target is known. The classical
[10], oceanographic data collection and pollution monitoring
approach is to assume a specic statistical model for the signal
[2], and chemical source localization [24].
and frame it as a hypothesis test of the null hypothesis
Among those, one of the important applications of WSN is
target is present in the study area vs the alternative
H0 : no
H1 : there
to detect and locate targets over an area of interest. There are
is a target in this area [19]. Following this idea, we propose to
substantial amount of work on WSN for target detection [4],
test
[6], [7], [8], [9], [19]. A general question for a WSN is how to
is positive. We will develop an optimal statistical approach
efciently integrate the available information form individual
based on this idea.
H0 :
the signal strength is zero vs
H1 :
the signal strength
sensors to reach a global decision about the presence of
The rest of the paper is organized as follows. In Section II,
a target over a monitoring area. The past work on signal
we briey review the physical models for radioactive signals.
detection with sensor networks can be categorized into two
In Section III, we propose our statistical detection and location
groups of methods: decision fusion and value fusion methods,
methods. Because the detection method is nonstandard [11],
respectively. In decision fusion, each sensor makes its own
we propose a bootstrap method to access the method in Section
binary decision and then the network will make a consensus
IV. In Section V, we display our numerical evaluations for
by fusing all decisions [5], [10], [19]. In value fusion, the
the proposed methods based on Monte Carlo simulations. In
sensor collects measurements and the network will make a
Section VI, we present our conclusion.
II. P HYSICAL BACKGROUND
The classical approach to address this problem assumes only
Nuclear weapons or nuclear power plants contain radioac-
one detector is used. Here we assume multiple detectors are
tive materials, which can sustain chain reactions. There are
used. Each is deployed at a different location. In this case,
several basic ways to detect radioactive materials. Among
the background intensity rates are the same but the signal
those, passive detection is the preferred technique because
intensity rates are different as they depend on the distances.
of its simplicity and safety. In passive detection, radioactive
By comparing the spatial distribution of the observations from
signals can be detected by measuring neutrons or gamma
the detectors, our method can simultaneously detect and locate
rays emitted from the radioactive materials. The count of
a hidden nuclear radioactive target. The detail of method will
neutrons or gamma rays follows a Poisson distribution with a
be introduced in the next Section.
certain intensity rate (per second), which varies from isotope
III. D ETECTION AND L OCATION M ETHOD
to isotope depending on the halife and the types of radiation
emitted during radioactive decay. A summary of several wellknown radioactive materials can be found in [15].
Assume a nuclear radiation target is hidden at an unknown
location denoted by
As a detector moves always from the radioactive target,
ω = (ω1 , ω2 , ω2 ).
Suppose a value
fusion WSN is used to detect and locate the target. Assume
the expected rate of neutrons or gamma rays decreases in-
the WSN has
versely with the square of the distance [15]. Suppose a
at
the detector and radioactive target. Then, the radiation count
m radiation detectors, which are deployed
ai = (ai1 , ai2 , ai3 ) for i = 1, · · · , m respectively. Let
ri = ri (ω) = kω − ai k be the Euclidean distance between
the target and the i-detector. Let ξi = Ai ei , where Ai is the
area and ei is the efciency of the i-th detector respectively.
Let yi be the total number of radiation counts observed by
the i-th detector during the detection with duration period T
received by the detector follows a homogeneous Poisson
(seconds). Then, the statistical model is
nuclear radioactive target is hidden at unknown location
ω = (ω1 , ω2 , ω3 ) and a radiation detector is deployed at
ã = (ã1 , ã2 , ã3 ). Let r(ω) = kω − ãk = [(ω1 − ã1 )2 +
(ω2 − ã2 )2 +(ω3 − ã3 )2 ]1/2 be the Euclidean distance between
process with intensity rate equal to
second), where
νs
Aeνs /(4πr2 (ω))
(per
from the target, and
A
and
e
are the area and the efciency
of the detector respectively. Suppose the time of the detection
period is
T
yi ∼ P oisson(T ξi (νb +
is the surface radiation rate (per second)
(seconds). Then, the total number of signal count
received by the detector follows a Poisson distribution with
expected value equal to
T Aeνs /(4πr2 (ω)).
νs
)), i = 1, · · · , m.
4πri2
The unknown parameters contained in Model (3) are
ω.
νb , νs and
Based on Model (3), the detection problem is interpreted
by Equation (2). The location problem is interpreted as the
estimation and condence interval for
Besides the radiation from the target, the detector also
(3)
H0
accessed when
ω.
It will be only
is rejected.
receives radiation from the natural background. In general,
We propose a statistical method to test the signicant of the
the neutron background is due to cosmic rays and increases
null hypothesis. The method is based on the famous likelihood
with altitude and geomagnetic latitude, especially during the
ratio test, which is accessed by the likelihood ratio statistic
maximum phase of the solar activity cycle [15]. Most of the
[18]. The likelihood ratio statistic is dened by the ratio of
gamma-ray background is due to radionuclides in soils and
likelihood functions under
rock [16]. The background is independent of the signal and
H0 : νs = 0
also follows a homogeneous Poisson process. In the detection
estimate and the condence interval for
with during period
T
H0 ∪ H1 : νs ≥ 0
and under
respectively. When the test is signicant, the
ω
can be accessed
(seconds), the number of background
by the asymptotical normality of the maximum likelihood
radiation counts received by the detector follows a Poisson
estimator (MLE). In oder to derive the likelihood ratio test
distribution with expected value equal to
Let
Then,
Y
Y
T Aeνb .
statistic. We need to compute the maximum of the likelihood
be the total radiation count observed by the detector.
is the combination of the signal and background,
which satises
y ∼ P oisson(T Ae(νb +
νs
)).
4πr2 (ω)
functions under
parameter
ω∈R
3
νs ,
background intensity
νb ,
(1)
νs = 0
and location
`(νs , νb , ω) =
m
X
yi log[T ξi (νb +
i=1
νs > 0, which makes the
detection problem into a statistical hypotheses testing problem
H1 : νs > 0.
(2)
(4)
m
X
νs
−T
ξi [νb +
]
−
log(yi !).
4πri2
i=1
i=1
model, a signal is combined with a background so that the total
observation is their combination. The presence of the nuclear
νs
)]
4πri2
m
X
signal plus background model [17], [21], [25], [26]. In this
vs
ω , we need to
ω . This will be
The loglikelihood function under Model (3) is
.
target is interpreted by the case when
respectively. In order the
introduced in the remaining part of this Section.
This statistical model given by Equation (1) is called the
H0 : νs = 0
and
derive the limiting distribution of the MLE of
The unknown parameters contained in Model (1) includes
the signal intensity
νs ≥ 0
derive the estimate and condence interval for
Under
H0 : νs = 0, `(νs , νb , ω)
`0 (νb ) = −
m
X
i=1
log(yi !) +
m
X
i=1
in Equation (4) becomes
yi log(T ξi νb ) − T νb
m
X
i=1
ξi .
(5)
Because
estimate
`0 (νb ) only depends
ω under νs = 0.
νb ,
on
it is not necessary to
IV. B OOTSTRAP M ETHOD FOR T HE T EST S TATISTIC
Because neither the exact nor the approximate distribution
We propose a loglikelihood ratio test to access the null
of
hypothesis. By Comparing Equations (4) and (5), we nd that
its
the location
ω
Λ is known,
p-value. The
we propose a bootstrap method to access
bootstrap idea is a statistical realization of
is not present in (5). Therefore, this problem is
the simulation concept to optimize the utilization of limited
nonstandard because the classical loglikelihood ratio test does
data: one ts a statistical model to the data and treats the
not possess it usually asymptotic null distribution [3], [11],
tted model as true [14]. The bootstrap algorithm draws
[12]. In order to well-dene the loglikelihood ratio test, we
samples from the tted model. This is done many times, each
rst propose a conditional test statistic for a given
ω
and then
maximize the conditional test statistic for all possible
ω.
This
method has been previously used by [11].
ω
Suppose
νs
and
νb .
In this case, the testing problem
becomes standard. Therefore, the loglikelihood ratio test can
be formulated by the conditional test statistic given by
Raphson algorithm to compute their values [1]. This algorithm
`(νs , νb , ω) may have
more than one local maxima. Because the limit of this paper,
we omit the detail of the algorithm.
Because ω is unknown, we assume it belongs to a set A ⊆
R3 . Then, the loglikelihood ratio test statistic is dened by
Λ = sup Λ(ω).
statistic. The bootstrap procedure is most useful when we do
not know the exact or asymptotic distributions of a statistic.
Under the null hypothesis,
the
α,
p-value
of
Λ
H0 : νs = 0
(7)
T ξi νb for i = 1, · · · , m,
νb . To eliminate νb ,
we use the distribution of (y1 , · · · , ym ) conditional on its total
Pm
y+ = i=1 yi . This conditional distribution is well-known as
(y1 , · · · , ym )|y+
ξ1
∼Multinomial(y+ , ( Pm
i=1 ξi
Λ
is large. If
is less than a pre-selected signicance level
the test is signicant; otherwise, the test is not signicant.
Λ(ω) approximately follows χ21
under H0 : νs = 0 if T is large [22]. However,
approximate nor the exact distribution of Λ is
It is easily to see that
distribution
neither the
known [3]. Therefore, we propose a bootstrap method to
access its
p-value.
The detail of this method will be given
in the next Section.
standard under
νs > 0,
the location
ω
of the radioactive target
then can be easily derived by its asymptotical normality. Let
(ν̂s , ν̂b , ω̂)
(νs , νb , ω).
region for ω is
be the MLE of
elliptical condence
Then, the
100(1 − α)%
χ2α,3
I(ν̂s , ν̂b , ω̂)
is the upper
is the
3×3
Ij,k = 4ν̂s2
for
j, k = 1, 2, 3.
α
quantile of
(9)
Λ.
not change. Based on the generated data set in the bootstrap
method, we can mimic the null distribution of
p-value.
Λ
to access its
Therefore, we display our bootstrap method below.
p-value of Λ.
Λ based on the observed
it be Λ0 .
Bootstrap method for the
i) Compute the observed value of
y1 , · · · , ym . Let
K independent
ii) Generate
random samples from Model
Λ based on each generated
k = 1, · · · , K .
iv) The bootstrap p-value of Λ is derived by #{Λk ≥ Λ0 :
k = 0, · · · , K}/(K + 1), where #(S) is the number of
elements contained in set S .
We choose K = 999 so that the bootstrap p-value is given by
0.001 increment. The null hypothesis will be rejected if the
bootstrap p-value is less than the signicance level (e.g. 0.05).
(9). Compute the value of
data. Let them be
Λk
for
V. S IMULATION R ESULTS
We evaluated the test statistic
Λ based on the behavior of its
power function. We evaluated the location method based on 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
1
{ω : (ω − ω̂)0 I(ν̂s , ν̂b , ω̂)(ω − ω̂) ≤ χ2α,3 },
T
where
i=1 ξi
)).
We use Equation (9) to generate the null distribution of
When the test is signicant, we need to locate the radioactive target in the study area. Since the problem becomes
ξm
, · · · , Pm
In this case, the total count in each bootstrap repetition does
counts
is rejected if
are independently Poisson dis-
which contains an unknown parameter
ω∈A
The null hypothesis
yi
tribution with expected value equal to
Λ(ω) = 2[`(ν̂s,ω , ν̂b,ω , ω) − `0 (ν̂b,0 )]
(6)
Pm
Pm
where ν̂b,0 =
i=1 Yi /(T
i=1 ξi ) is the MLE of νb under
H0 : νs = 0, and ν̂s,ω and ν̂b,ω are the conditional MLE of
νs and νb under H0 ∪ H1 : νs ≥ 0 respectively. There is
no analytic solution for ν̂s,ω and ν̂b,ω . We propose a Newtonneeds to be developed carefully because
essential idea of the bootstrap is that it assumes that the sets of
ctional samples can approximate the distribution of the test
is pre-selected. Then, Equation (4) only con-
tains parameters
time producing a ctional sample from the tted model. The
χ23
were all
distribution, and
Fisher Information matrix given by
m
X
ξi (aij − ω̂j )(aik − ω̂k )
,
ν̂b ri8 (ω̂) + ν̂s ri6 (ω̂)
i=1
0
and the nuclear radioactive target was hidden on
the same plane of the deployed sensors. That is, we assumed
ω3 = ai3 = 0
for all
i.
sensors. They were identical and deployed at the
tice. Assume the radioactive target was
(8)
100 radiation
10 × 10 latinstalled at (5.5, 5.5).
In our simulation, we assumed the WSN had
Consequently, the distance between the radioactive target and
the
i
detector was
ri = ri (ω) = kai − ωk =
p
(ai1 − 5.5)2 + (ai2 − 5.5)2 ,
TABLE I
P OWER
FUNCTION OF
0
0.049
12.089
Power
MSE
Λ AND
MSE
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