# 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 signal-to-noise 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^(1-1/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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##### Conference Paper: Adaptive Calibration for Fusion-based Wireless Sensor Networks.

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**ABSTRACT:**Wireless sensor networks (WSNs) are typically com- posed of low-cost sensors that are deeply integrated with phys- ical environments. As a result, the sensing performance of a WSN is inevitably undermined by various physical uncertainties, which include stochastic sensor noises, unpredictable environment changes and dynamics of the monitored phenomenon. Traditional solutions (e.g., sensor calibration and collaborative signal pro- cessing) work in an open-loop fashion and hence fail to adapt to these uncertainties after system deployment. In this paper, we propose an adaptive system-level calibration approach for a class of sensor networks that employ data fusion to improve system sensing performance. Our approach features a feedback control loop that exploits sensor heterogeneity to deal with the aforementioned uncertainties in calibrating system performance. In contrast to existing heuristic based solutions, our control- theoretical calibration algorithm can ensure provable system stability and convergence. We also systematically analyze the impacts of communication reliability and delay, and propose an optimal routing algorithm that minimizes the impact of packet loss on system stability. Our approach is evaluated by both experiments on a testbed of Tmotes as well as extensive simulations based on data traces gathered from a real vehicle detection experiment. The results demonstrate that our calibration algorithm enables a network to maintain the optimal detection performance in the presence of various system and environmental dynamics.INFOCOM 2010. 29th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 15-19 March 2010, San Diego, CA, USA; 01/2010 - SourceAvailable from: Huan Li[Show abstract] [Hide abstract]

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Page 1

Data Fusion Improves the Coverage of Wireless Sensor

Networks

Guoliang Xing1; Rui Tan2; Benyuan Liu3; Jianping Wang2; Xiaohua Jia2; Chih-Wei 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 signal-to-noise 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 [Computer-Communication 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

personal or classroom use is granted without fee provided that copies are

not made or distributed for profit or commercial advantage and that copies

bear this notice and the full citation on the first page. To copy otherwise, to

republish, to post on servers or to redistribute to lists, requires prior specific

permission and/or a fee.

MobiCom’09, September 20–25, 2009, Beijing, China.

Copyright 2009 ACM 978-1-60558-702-8/09/09 ...$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 large-scale 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 network-level 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 cut-off 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 large-scale 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 signal-to-noise ratio (SNR). To the best of our knowl-

edge, this work is the first to study the coverage performance

of large-scale 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 signal-to-noise 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 low-SNR 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 large-scale fusion-

based WSNs has received little attention.

There is a vast of literature on stochastic signal detec-

tion based on multi-sensor 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 small-scale 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 large-scale 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 k-coverage

(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 signal-to-noise 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 closed-form 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., yi|H1 ∼ 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 signal-to-noise 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 two-dimensional

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 two-dimensional

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 ρ||A||, i.e., N(A) ∼ Poi(ρ||A||), 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 large-scale 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 lower-bounds 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 upper-bounded 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 multi-hop 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 large-scale 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,

mission-critical 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 signal-to-noise 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 large-scale

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 real-world

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 ≥ t|H0) = Q

„t − µ

„t − µ − si

σ

«

,(5)

PD = P(yi ≥ t|H1) = 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 non-decreasing 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