Data fusion improves the coverage of wireless sensor networks.
ABSTRACT Wireless sensor networks (WSNs) have been increasingly available for critical applications such as security surveillance and environmental monitoring. An important performance measure of such applications is sensing coverage that characterizes how well a sensing field is monitored by a network. Although advanced collaborative signal processing algorithms have been adopted by many existing WSNs, most previous analytical studies on sensing coverage are conducted based on overly simplistic sensing models (e.g., the disc model) that do not capture the stochastic nature of sensing. In this paper, we attempt to bridge this gap by exploring the fundamental limits of coverage based on stochastic data fusion models that fuse noisy measurements of multiple sensors. We derive the scaling laws between coverage, network density, and signaltonoise ratio (SNR). We show that data fusion can significantly improve sensing coverage by exploiting the collaboration among sensors. In particular, for signal path loss exponent of k (typically between 2.0 and 5.0), rho_f=O(rho_d^(11/k)), where rho_f and rho_d are the densities of uniformly deployed sensors that achieve full coverage under the fusion and disc models, respectively. Our results help understand the limitations of the previous analytical results based on the disc model and provide key insights into the design of WSNs that adopt data fusion algorithms. Our analyses are verified through extensive simulations based on both synthetic data sets and data traces collected in a real deployment for vehicle detection.

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ABSTRACT: This paper proposes an information quality IQ aware tracking algorithm that reduces energy consumption for moving target tracking in uncertain sensor networks with guarantee of given quality constraints. We first model the uncertainty in sensor networks. Then, we map information quality to the combination of sensed data error, residual energy and energy cost of sensor nodes. Moreover, the problem of IQaware nodes selection is induced and an energyefficient tracking algorithm based on nodes selection is presented. Finally, a comprehensive set of simulations is presented. We conclude that the proposed tracking algorithm yields excellent tracking performance in sensor networks under uncertainty.International Journal of Sensor Networks. 09/2013; 14(1):3341.  [Show abstract] [Hide abstract]
ABSTRACT: Volcano monitoring is of great interest to public safety and scientific explorations. However, traditional volcanic instrumentation such as broadband seismometers are expensive, power hungry, bulky, and difficult to install. Wireless sensor networks (WSNs) offer the potential to monitor volcanoes on unprecedented spatial and temporal scales. However, current volcanic WSN systems often yield poor monitoring quality due to the limited sensing capability of lowcost sensors and unpredictable dynamics of volcanic activities. In this article, we propose a novel qualitydriven approach to achieving realtime, distributed, and longlived volcanic earthquake detection and timing. By employing novel innetwork collaborative signal processing algorithms, our approach can meet stringent requirements on sensing quality (i.e., low false alarm/missing rate, short detection delay, and precise earthquake onset time) at low power consumption. We have implemented our algorithms in TinyOS and conducted extensive evaluation on a testbed of 24 TelosB motes as well as simulations based on real data traces collected during 5.5 months on an active volcano. We show that our approach yields nearzero false alarm/missing rate, less than one second of detection delay, and millisecond precision earthquake onset time while achieving up to sixfold energy reduction over the current data collection approach.ACM Transactions on Sensor Networks (TOSN). 03/2013; 9(2).
Page 1
Data Fusion Improves the Coverage of Wireless Sensor
Networks
Guoliang Xing1; Rui Tan2; Benyuan Liu3; Jianping Wang2; Xiaohua Jia2; ChihWei Yi4
1Department of Computer Science & Engineering, Michigan State University, USA
2Department of Computer Science, City University of Hong Kong, HKSAR
3Department of Computer Science, University of Massachusetts Lowell, USA
4Department of Computer Science, National Chiao Tung University, Taiwan
glxing@msu.edu; {tanrui2@student., jianwang@, csjia@}cityu.edu.hk;
bliu@cs.uml.edu; yi@cs.nctu.edu.tw
ABSTRACT
Wireless sensor networks (WSNs) have been increasingly
available for critical applications such as security surveil
lance and environmental monitoring.
formance measure of such applications is sensing coverage
that characterizes how well a sensing field is monitored by
a network. Although advanced collaborative signal process
ing algorithms have been adopted by many existing WSNs,
most previous analytical studies on sensing coverage are con
ducted based on overly simplistic sensing models (e.g., the
disc model) that do not capture the stochastic nature of sens
ing. In this paper, we attempt to bridge this gap by explor
ing the fundamental limits of coverage based on stochastic
data fusion models that fuse noisy measurements of multi
ple sensors. We derive the scaling laws between coverage,
network density, and signaltonoise ratio (SNR). We show
that data fusion can significantly improve sensing coverage
by exploiting the collaboration among sensors. In particu
lar, for signal path loss exponent of k (typically between 2.0
and 5.0), ρf = O(ρ1−1/k
ties of uniformly deployed sensors that achieve full coverage
under the fusion and disc models, respectively. Our results
help understand the limitations of the previous analytical re
sults based on the disc model and provide key insights into
the design of WSNs that adopt data fusion algorithms. Our
analyses are verified through extensive simulations based on
both synthetic data sets and data traces collected in a real
deployment for vehicle detection.
An important per
d
), where ρf and ρd are the densi
Categories and Subject Descriptors
C.2.1 [ComputerCommunication Networks]: Network
Architecture and Design—Network topology; G.3 [Probability
and Statistics]: Stochastic processes
Permission to make digital or hard copies of all or part of this work for
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permission and/or a fee.
MobiCom’09, September 20–25, 2009, Beijing, China.
Copyright 2009 ACM 9781605587028/09/09 ...$10.00.
General Terms
Performance, Theory
Keywords
Data fusion, target detection, coverage, performance limits,
wireless sensor network
1.INTRODUCTION
Recent years have witnessed the deployments of wireless
sensor networks (WSNs) for many critical applications such
as security surveillance [16], environmental monitoring [25],
and target detection/tracking [21]. Many of these applica
tions involve a large number of sensors distributed in a vast
geographical area. As a result, the cost of deploying these
networks into the physical environment is high. A key chal
lenge is thus to predict and understand the expected sensing
performance of these WSNs. A fundamental performance
measure of WSNs is sensing coverage that characterizes how
well a sensing field is monitored by a network. Many recent
studies are focused on analyzing the coverage performance
of largescale WSNs [4,19,24,33,38,41,43].
Despite the significant progress, a key challenge faced by
the research on sensing coverage is the obvious discrepancy
between the advanced information processing schemes adopted
by existing sensor networks and the overly simplistic sens
ing models widely assumed in the previous analytical stud
ies. On the one hand, many WSN applications are designed
based on collaborative signal processing algorithms that im
prove the sensing performance of a network by jointly pro
cessing the noisy measurements of multiple sensors. In prac
tice, various stochastic data fusion schemes have been em
ployed by sensor network systems for event monitoring, de
tection, localization, and classification [8, 10, 11, 16, 20, 21,
29, 34]. On the other hand, collaborative signal process
ing algorithms such as data fusion often have complex com
plications to the networklevel sensing performance such as
coverage. As a result, most analytical studies1on sensing
coverage are conducted based on overly simplistic sensing
models [3,4,14,18,19,23,24,33,38,39,43]. In particular, the
1Among the total six papers on the coverage problem of
WSNs that have been published at MobiCom since 2001,
five of them adopted the disc sensing model. Similarly, the
disc model is also assumed by seven out of nine relevant
papers published at MobiHoc since 2001.
Page 2
1
0.8
0.6
0.4
0.2
200150100 500
Detection probability PD
Distance from the vehicle (meters)
t = 0.01
t = 0.05
Figure
probability vs.
tance from the vehicle.
1: Detection
the dis
0.10
0.08
False alarm rate PF
0.06
0.04
0.02
0
0.150.10 0.05 0.01
Detection threshold t
Figure 2:
rate vs. detection thresh
old.
False alarm
sensing region of a sensor is often modeled as a disc with
radius r centered at the position of the sensor, where r is
referred to as the sensing range. A sensor deterministically
detects the targets (events) within its sensing range. Al
though such a model allows a geometric treatment to the
coverage problem, it fails to capture the stochastic nature of
sensing.
To illustrate the inaccuracy of the disc sensing model, we
plot the sensing performance of an acoustic sensor in Fig. 1
and 2 using the data traces collected from a real vehicle de
tection experiment [1]. In the experiment, the sensor detects
moving vehicles by comparing its signal energy measurement
against a threshold (denoted by t). Fig. 1 plots the probabil
ity that the sensor detects a vehicle (denoted by PD) versus
the distance from the vehicle. No clear cutoff boundary be
tween successful and unsuccessful sensing of the target can
be seen in Fig. 1. Similar result is observed for the rela
tionship between the sensor’s false alarm rate (denoted by
PF) and the detection threshold shown in Fig. 2. Note that
PF is the probability of making a positive decision when no
vehicle is present.
In this work, we develop an analytical framework to ex
plore the fundamental limits of coverage of largescale WSNs
based on stochastic data fusion models. To characterize the
inherent stochastic nature of sensing, we propose a new cov
erage measure called (α,β)coverage where α and β are the
upper and lower bounds on the system false alarm rate and
detection probability, respectively. Compared with the clas
sical definition of coverage, (α,β)coverage explicitly cap
tures the performance requirements imposed by sensing ap
plications. For instance, the full (0.05,0.9)coverage of a
region ensures that the probability of detecting any event
occurring in the region is no lower than 90% and no more
than 5% of the network reports are false alarms.
The main focus of this paper is to investigate the fun
damental scaling laws between coverage, network density,
and signaltonoise ratio (SNR). To the best of our knowl
edge, this work is the first to study the coverage performance
of largescale WSNs based on collaborative sensing models.
Our results not only help understand the limitations of the
existing analytical results based on the disc model but also
provide key insights into designing and analyzing the large
scale WSNs that adopt stochastic fusion algorithms. The
main contributions of this paper are as follows.
• We derive the (α,β)coverage of random networks un
der both data fusion and probabilistic disc models.
Based on these results, we can compute the minimum
network density that is required to achieve a desired
level of sensing coverage. Moreover, the existing ana
lytical results based on the disc model can be naturally
extended to the context of stochastic event detection.
• We study the fundamental scaling laws of (α,β)coverage.
Let ρd and ρf denote the minimum network densities
for achieving full coverage under the disc and fusion
models, respectively. We prove that ρf = O(2r2
where r is the radius of sensing disc and R is the fu
sion range within which the measurements of all sen
sors are fused2. As fusion range can be much greater
than sensing range, ρf is much smaller than ρd. Fur
thermore, when the optimal fusion range is adopted,
ρf = O(ρ1−1/k
ponent that typically ranges from 2.0 to 5.0. In par
ticular, when k = 2 (which typically holds for acoustic
signals), ρf = O(√ρd). This result shows that data
fusion can effectively reduce the network density com
pared with the disc model. Furthermore, the existing
analytical results based on the disc model significantly
overestimate the network density required for achiev
ing coverage.
R2 · ρd)
d
) where k is the signal’s path loss ex
• We study the impact of signaltonoise ratio (SNR) on
the network density when full coverage is required. We
prove that ρf/ρd = O(SNR2/k). This result suggests
that data fusion is more effective in reducing the den
sity of lowSNR network deployments, while the disc
model is suitable only when the SNR is sufficiently
high.
• To verify our analyses, we conduct extensive simula
tions based on both synthetic data sets and real data
traces collected from 20 sensors. Our simulations show
that our analytical results can accurately predict the
stochastic coverage of WSNs under a variety of realis
tic settings.
The rest of this paper is organized as follows. Section 2
reviews related work. Section 3 introduces the background
and problem definition. We study the (α,β)coverage under
the disc and fusion models in Section 4 and 5, respectively.
In Section 6, we investigate the impact of data fusion on
asymptotic sensing coverage. Section 7 presents simulation
results and Section 8 concludes this paper.
2. RELATED WORK
Many sensor network systems have incorporated various
data fusion schemes to improve the system performance. In
the surveillance system based on MICA2 motes [16], the
system false alarm rate is reduced by fusing the detection
decisions made by multiple sensors. In the DARPA Sen
sIT project [1], advanced data fusion techniques have been
employed in a number of algorithms and protocols designed
for target detection [8,21], localization [20,34], and classi
fication [10,11]. Despite the wide adoption of data fusion
in practice, the performance analysis of largescale fusion
based WSNs has received little attention.
There is a vast of literature on stochastic signal detec
tion based on multisensor data fusion. Early works [5,37]
2We adopt the following asymptotic notation: 1) f(x) =
O(g(x)) means that g(x) is the asymptotic upper bound of
f(x); 2) f(x) = Θ(g(x)) means that g(x) is the asymptotic
tight bound of f(x).
Page 3
focus on smallscale powerful sensor networks (e.g., several
radars). Recent studies on data fusion have considered the
specific properties of WSNs such as sensors’ spatial distri
bution [10, 11, 29] and limited sensing/communication ca
pability [8]. However, these studies focus on analyzing the
optimal fusion strategies that maximize the system perfor
mance of a given network. In contrast, this paper explores
the fundamental limits of sensing coverage of WSNs that
are designed based on existing data fusion strategies. Re
cently, irregular sampling theory has been applied for re
constructing physical fields in WSNs [30,31]. Different from
these works that focus on developing sampling schemes to
improve the quality of signal reconstruction, we aim to ana
lyze sensors’ spatial density for achieving the required level
of coverage.
As one of the most fundamental issues in WSNs, the cov
erage problem has attracted significant research attention.
Previous works fall into two categories, namely, coverage
maintenance algorithms/protocols and theoretical analysis
of coverage performance. These two categories are reviewed
briefly as follows, respectively.
Early work [22,26,27] quantifies sensing coverage by the
length of target’s path where the accumulative observations
of sensors are maximum or minimum [22,26,27]. However,
these works focus on devising algorithms for finding the tar
get’s paths with certain level of coverage. Several algorithms
and protocols [41,42] are designed to maintain sensing cov
erage using the minimum number of sensors. However, the
effectiveness of these schemes largely relies on the assump
tion that sensors have circular sensing regions and determin
istic sensing capability. Several recent studies [2,17,32,40]
on the coverage problem have adopted probabilistic sensing
models. The numerical results in [40] show that the coverage
of a network can be expanded by the cooperation of sensors
through data fusion. However, these studies do not quantify
the improvement of coverage due to data fusion techniques.
Different from our focus on analyzing the fundamental lim
its of coverage in WSNs, all of these studies aim to devise
algorithms and protocols for coverage maintenance.
Theoretical studies of the coverage of largescale WSNs
have been conducted in [4,14,18,19,23,24,33,38,43]. Most
works [18,19,23,33,38,43] focus on deriving the asymptotic
coverage of WSNs. The critical conditions for full kcoverage
(i.e., any physical point is within the sensing range of at
least k sensors) over a bounded square area [19,33,38,43] or
barrier area [18,23] are derived for various sensor deployment
strategies. The coverage of randomly deployed networks is
studied in [24]. The existing theoretical results on coverage
for both static and mobile sensors/targets are surveyed in
[4]. However, all the above theoretical studies are based on
the deterministic disc model. In this paper, we compare
our results obtained under a data fusion model against the
results from [4,24].
3.BACKGROUND AND PROBLEM DEFI
NITION
In this section, we first describe the preliminaries of our
work, which include sensor measurement, network, and data
fusion models. We then introduce the problem definition.
3.1Sensor Measurement and Network Model
We assume that sensors perform detection by measuring
Table 1: Summary of Notation∗
DefinitionSymbol
Soriginal signal energy emitted by the target
mean and variance of noise energy
peak signaltonoise ratio (PSNR), δ = S/σ
path loss exponent
signal decay function, w(x) = Θ(x−k)
distance from the target
attenuated signal energy, si= S · w(di)
noise energy, ni∼ N(µ, σ2)
signal energy measurement, yi= si+ ni
false alarm rate / detection probability
upper / lower bound of PF / PD
hypothesis that the target is absent / present
network density
the set of sensors within fusion range of point p
the number of sensors in F(p)
upper bound of target localization error
∗The symbols with subscript i refer to the notation of sensor i.
µ, σ2
δ
k
w(·)
di
si
ni
yi
PF / PD
α / β
H0 / H1
ρ
F(p)
N(p)
ǫ
the energy of signals emitted by the target3. The energy
of most physical signals (e.g., acoustic and electromagnetic
signals) attenuates with the distance from the signal source.
Suppose sensor i is dimeters away from the target that emits
a signal of energy S. The attenuated signal energy si at the
position of sensor i is given by
si = S · w(di), (1)
where w(·) is a decreasing function satisfying w(0) = 1,
w(∞) = 0, and w(x) = Θ(x−k). The w(·) is referred to as
the signal decay function. Depending on the environment,
e.g., atmosphere conditions, the signal’s path loss exponent
k typically ranges from 2.0 to 5.0 [15, 20]. We note that
the theoretical results derived in this paper do not depend
on the closedform formula of w(·). We adopt the following
signal decay function in the simulations conducted in this
paper:
w(x) =
1
1 + xk. (2)
The sensor measurements are contaminated by additive
random noises from sensor hardware or environment. De
pending on the hypothesis that the target is absent (H0) or
present (H1), the measurement of sensor i, denoted by yi, is
given by
H0 :
H1 :
yi = ni,
yi = si+ ni,
(3)
(4)
where ni is the energy of noise experienced by sensor i. We
assume that the noise ni at each sensor i follows the nor
mal distribution, i.e., ni ∼ N(µ,σ2), where µ and σ2are
the mean and variance of ni, respectively. We assume that
the noises, {ni∀i}, are spatially independent across sensors.
Therefore, the noises at sensors are independent and iden
tically distributed (i.i.d.) Gaussian noises. In the presence
of target, the measurement of sensor i follows the normal
3Several types of sensors (e.g., acoustic sensor) only sample
signal intensity at a given sampling rate. The signal energy
can be obtained by preprocessing the time series of a given
interval, which has been commonly adopted to avoid the
transmission of raw data [8,10,11,20,34].
Page 4
distribution, i.e., yiH1 ∼ N(si+ µ,σ2). Due to the inde
pendence of noises, the sensors’ measurements, {yi∀i,H1},
are spatially independent but not identically distributed as
sensors receive different signal energies from the target. We
define the peak signaltonoise ratio (PSNR) as δ = S/σ
which quantifies the noise level. Table 1 summarizes the
notation used in this paper.
The above signal decay and additive i.i.d. Gaussian noise
models have been widely adopted in the literature of multi
sensor signal detection [2, 5, 8, 20, 24, 27, 29, 34, 37, 40] and
also have been empirically verified [15,20]. In practice, the
parameters of these models (i.e., S, w(·), µ, and σ2) can
be estimated using training data. The normal distribution
might be an approximation to the real noise distribution in
practice. As discussed in Section 5.1, the assumption of i.i.d.
Gaussian noises can be relaxed to any i.i.d. noises.
We consider a network deployed in a vast twodimensional
geographical region. The positions of sensors are uniformly
and independently distributed in the region. Such a deploy
ment scenario can be modeled as a stationary twodimensional
Poisson point process. Let ρ denote the density of the under
lying Poisson point process. The number of sensors located
in a region A, N(A), follows the Poisson distribution with
mean of ρA, i.e., N(A) ∼ Poi(ρA), where A repre
sents the area of the region A. We note that the uniform
sensor distribution has been widely adopted in the perfor
mance analysis of largescale WSNs [4,19,24,33,38]. There
fore, this assumption allows us to compare our results with
previous analytical results.
3.2Data Fusion Model
Data fusion can improve the performance of detection sys
tems by jointly considering the noisy measurements of mul
tiple sensors. There exist two basic data fusion schemes,
namely, decision fusion and value fusion. In decision fusion,
each sensor makes a local decision based on its measurements
and sends its decision to the cluster head, which makes a sys
tem decision according to the local decisions. The optimal
decision fusion rule has been obtained in [5]. In value fu
sion, each sensor sends its measurements to the cluster head,
which makes the detection decision based on the received
measurements. In this paper, we focus on value fusion, as
it usually has better detection performance than decision
fusion [37]. Under the assumptions made in Section 3.1,
the optimal value fusion rule is to compare the following
weighted sum of sensors’ measurements to a threshold (the
derivation can be found in Appendix A):
Yopt =
X
i
si
σ· yi.
However, as sensor measurements contain both noise and
signal energy (see (4)), the weight
ceived by sensor i, is unknown.
to adopt equal constant weights for all sensors’ measure
ments [8,29,40]. Since the measurements from different sen
sors are treated equally, the sensors far away from the target
should be excluded from data fusion as their measurements
suffer low SNRs. Therefore, we adopt a fusion scheme as
follows.
For any physical point p, the sensors within a distance of
R meters from p form a cluster and fuse their measurements
to detect whether a target is present at p. R is referred
to as the fusion range and F(p) denotes the set of sensors
si
σ, i.e., the SNR re
A practical solution is
within the fusion range of p. The number of sensors in F(p)
is represented by N(p). A cluster head is elected to make
the detection decision by comparing the sum of measure
ments reported by member sensors in F(p) against a detec
tion threshold T. Let Y denote the fusion statistics, i.e.,
Y =P
We assume that the cluster head makes a detection based
on snapshot measurements from member sensors without us
ing temporal samples to refine the detection decision. Note
that such a snapshot scheme is widely adopted in previous
works on target surveillance [8,20,29,34,40]. Fusion range R
is an important design parameter of our data fusion model.
As SNR received by sensor decays with distance from the
target, fusion range lowerbounds the quality of information
that is fused at the cluster head. In Section 5.2, we will dis
cuss how to choose the optimal fusion range. The above data
fusion model is consistent with the fusion schemes adopted
in [8,29,40]. If more efficient fusion models are employed,
the scaling laws proved in this paper still hold as discussed
in Section 6.5.
We assume that the target keeps stationary after appear
ance and the position of a possible target can be obtained
through a localization algorithm. For instance, the target
position can be estimated as the geometric center of a num
ber of sensors with the largest measurements. Such a simple
localization algorithm is employed in the simulations con
ducted in this paper. The localized position may not be
the exact target position and the distance between them is
referred to as localization error.
calization error is upperbounded by a constant ǫ. The lo
calization error is accounted for in the following analyses.
However, we show that it has no impact on the asymptotic
results derived in this paper.
The above data fusion model can be used for target detec
tion as follows. The detection can be executed periodically
or triggered by user queries. In a detection process, each sen
sor makes a snapshot measurement and a cluster is formed
by the sensors within the fusion range from the possible tar
get to make a detection decision. The cluster formation may
be initiated by the sensor that has the largest measurement.
Such a scheme can be implemented by several dynamic clus
tering algorithms [6]. The fusion range R can be used as an
input parameter of the clustering algorithm. The communi
cation topology of the cluster can be a multihop tree rooted
at the cluster head. As the fusion statistics Y is an aggrega
tion of sensors’ measurements, it can be computed efficiently
along the routing path to the cluster head. In this work, we
are interested in the fundamental performance limits of cov
erage under the fusion model and the design of clustering
and data aggregation algorithms is beyond the scope of this
paper.
i∈F(p)yi. If Y ≥ T, the cluster head decides H1;
otherwise, it decides H0.
We assume that the lo
3.3 Problem Definition
The detection of a target is inherently stochastic due to
the noise in sensor measurements.
mance is usually characterized by two metrics, namely, the
false alarm rate (denoted by PF) and detection probability
(denoted by PD). PF is the probability of making a positive
decision when no target is present, and PD is the probabil
ity that a present target is correctly detected. In stochastic
detection, positive detection decisions may be false alarms
caused by the noise in sensor measurements. In particular,
The detection perfor
Page 5
Figure 3: Coverage un
derthe disc
Sensing range r = 17m,
which is computed by
(7).
model.
Figure
under the fusion mo
del.Grayscale repre
sents the value of PD.
4: Coverage
although the detection probability can be improved by set
ting lower detection thresholds, the fidelity of detection re
sults may be unacceptable because of high false alarm rates.
Therefore, PF together with PD characterize the sensing
quality provided by the network. For a physical point p,
we denote the probability of successfully detecting a tar
get located at p as PD(p). Note that PF is the probability
of making positive decision when no target is present, and
hence is location independent.
Our focus is to study the coverage of largescale WSNs.
We introduce a concept called (α,β)coverage that quantifies
the fraction of the surveillance region where PF and PD are
bounded by α and β, respectively.
Definition 1
α ∈ (0,0.5) and β ∈ (0.5,1), a physical point p is (α,β)
covered if the false alarm rate PF and detection probability
PD(p) satisfy
((α,β)coverage). Given two constants
PF ≤ α,PD(p) ≥ β.
The (α,β)coverage of a region is defined as the fraction of
points in the region that are (α,β)covered.
The full coverage of a region refers to the case where
the (α,β)coverage of the region approaches one, i.e., the
false alarm rate is below α and the probability of detect
ing a target present at any location is above β. In practice,
missioncritical surveillance applications [11–13,16] require
a low false alarm rate (α < 5%) and a high detection prob
ability (β ≫ 50%).
We now illustrate the (α,β)coverage by an example, where
δ = 1000 (i.e., 30dB), α = 5%, β = 95%, and R = 50m.
Fig. 3 and 4 illustrate the coverage under the disc and fusion
models, respectively. In Fig. 4, when a target (represented
by the triangle) is present, the sensors within the fusion
range from it fuse their measurements to make a detection.
The gray area is (α,β)covered, where grayscale represents
the value of PD at each point. As shown in Fig. 3, the cov
ered region under the disc model is simply the union of all
sensing discs. As a result, when a high level of coverage is
required, a large number of extra sensors must be deployed
to eliminate small uncovered areas surrounded by sensing
discs. In contrast, data fusion can effectively expand the
covered region by exploiting the collaboration among neigh
boring sensors.
In the rest of this paper, we consider the following prob
lems:
1. Although a number of analytical results on coverage
[4,19,24,33,38,41–43] have been obtained under the
classical disc model, are they still applicable under the
definition of (α,β)coverage which explicitly captures
the stochastic nature of sensing? To answer this ques
tion, we propose a probabilistic disc model such that
the existing results can be naturally extended to the
context of stochastic detection (Section 4).
2. How to quantify the (α,β)coverage when sensors can
collaborate through data fusion? Answering this ques
tion enables us to evaluate the coverage performance of
a network and to deploy the fewest sensors for achiev
ing a given level of coverage (Section 5).
3. What are the scaling laws between coverage, network
density, and signaltonoise ratio (SNR) under both
the disc and fusion models? The results will provide
important insights into understanding the limitation
of analytical results based on the disc model and the
impact of data fusion on the coverage of largescale
WSNs (Section 6).
4.COVERAGE UNDER PROBABILISTIC
DISC MODEL
As the classical disc model deterministically treats the de
tection performance of sensors, existing results based on this
model [4,19,24,33,38,41–43] cannot be readily applied to
analyze the performance or guide the design of realworld
WSNs.In this section, we extend the classical disc mo
del based on the stochastic detection theory [37] to capture
several realistic sensing characteristics and study the (α,β)
coverage under the extended model.
In the probabilistic disc model, we choose the sensing range
r such that 1) the probability of detecting any target within
the sensing range is no lower than β, and 2) the false alarm
rate is no greater than α. As we ignore the detection prob
ability outside the sensing range of a sensor, the detection
capability of sensor under this model is lower than in reality.
However, this model preserves the boundary of sensing region
defined in the classical disc model. Hence, the existing re
sults based on the classical disc model [4,19,24,33,38,41–43]
can be naturally extended to the context of stochastic de
tection.
We now discuss how to choose the sensing range r under
the probabilistic disc model. The optimal Bayesian detec
tion rule for a single sensor i is to compare its measurement
yi to a detection threshold t [37]. If yi exceeds t, sensor i
decides H1; otherwise, it decides H0. Therefore, the PF and
PD of sensor i are given by
PF = P(yi ≥ tH0) = Q
„t − µ
„t − µ − si
σ
«
,(5)
PD = P(yi ≥ tH1) = Q
σ
«
, (6)
where P(·) is the probability notation and Q(·) is the comple
mentary cumulative distribution function (CDF) of the stan
dard normal distribution, i.e., Q(x) =
As PD is nondecreasing function of PF [37], it is maxi
mized when PF is set to be the upper bound α.
the optimal detection threshold can be solved from (5) as
topt = µ+ σQ−1(α), where Q−1(·) is the inverse function of
1
√2π
R∞
xe−t2/2dt.
Hence