Active Detection With A Barrier Sensor Network Using A Scan Statistic

Transcription

1 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX Active Detection With A Barrier Sensor Network Using A Scan Statistic Xiufeng Song, Student Member, IEEE, Peter Willett, Fellow, IEEE, Joseph Glaz, and Shengli Zhou, Senior Member, IEEE Abstract The cooperation of an active acoustic source and a large number of distributed passive sensors offers an opportunity for active sonar detection. The system works as follows: first, each sensor compares its matched filter output with a given threshold to obtain a binary local decision 0 or ; then a fusion center collects them to make a system-level inference. How effectively to combine these local results distributed detection fusion is the concentration of this paper. Suppose that the sensor network is unaware of the target s reflection model. Then the local detection probabilities cannot be obtained; therefore, the optimal fusion rule is unavailable. The obvious detection strategy is a counting rule test CRT, which simply counts the total number of s and compares it to a threshold. This approach does not require the knowledge of sensor locations, and equally considers all network subareas. However, the reflected signal from some targets, such as a submarine, can be highly aspect dependent, and in many instances only sensors in a particular zone can receive its echoes. This paper focuses on the scan statistic, which slides a window across the sensor field, and selects the subarea with the largest number of s to make a decision. The scan statistic integrates the aspect-dependence characteristic of the target into detection fusion. With a proper window size, it may suppress the subarea interference, and improve the system-level performance. Index Terms Detection, multistatic, detection fusion, scan statistic, counting rule test, sensor network, sonar, submarine. I. INTRODUCTION It is well known that the acoustic reflection of many sonar targets such as submarines, unmanned underwater vehicles, and weapons is aspect dependent, and much of the reflected energy will be concentrated within a particular conical angle []. A reliable detection is possible only if some receivers are within that area, particularly at low signal-noise-ratio SNR [2]. If a system, such as a classical monostatic [3] or multistatic sonar [4] [5], merely has a few source-receiver pairs, the possibility that at least one receiver falls into the conical area is low. Therefore, a smart target can effectively hide itself by properly adjusting its location or orientation. An alternative Manuscript received July 26, 20; accepted October 26, 20. Date of publication XXX; date of current version XXX. This work was supported by the U.S. Office of Naval Research under Grant N Associate Editor: Gopu Potty. X. Song, P. Willett and S. Zhou are with the Department of Electrical and Computer Engineering, University of Connecticut, 37 Fairfield Way U-257, Storrs, CT 06269, USA xiufeng.song@gmail.com, willett@engr.uconn.edu, and shengli@engr.uconn.edu. J. Glaz is with the Department of Statistics, University of Connecticut, 25 Glenbrook Road, Storrs, CT 06269, USA joseph.glaz@uconn.edu. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier XXX sensor field source target detectable zone Fig.. A barrier sensor network is composed of an acoustic source and a narrow passive sensor band, and it could be used to detect a target out of the sensor band. Since the acoustic reflection of some sonar targets is angle dependent, only those located in a particular zone are able to reliably detect the echoes if the transmission energy is not large. configuration is a distributed sensor network [2] [6]. If it has sufficient spatial span and receiver density, the aggregated detection sensitivity to source-target-receiver geometry may be reduced, and the overall detection performance could be improved. We are interested in multistatic detection with a barrier sensor network, which is composed of an acoustic source and a band of passive sensors as shown in Fig.. Such a setup is very useful for region denial and coastline monitoring [2] [6]. Let the sensors have only limited computational capacity, such that they can make only a binary decision 0 or via a simple comparison of their matched filter outputs with a given threshold. A fusion center FC thereafter collects the local results to make a system-level inference. The focus of this paper is how to fuse these distributed decisions, and our main contribution is the comparison of a counting rule test CRT to a scan statistic. Distributed detection fusion is an active research area. In [7], the authors showed that the optimal fusion rule under the Neyman-Pearson criterion was the likelihood ratio test, which utilized the probabilities of local detection and false alarm rate to combine these binary results. Since sensor networks do not have knowledge of the location and reflection model of the target, local detection probabilities are generally unknown. Therefore, the optimal fusion rule is irrelevant. A suboptimal alternative is the CRT [8] [0], which counts the total number of s and compares it to a given threshold to infer whether a target exists. As the CRT requires neither local detection

2 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 2 probabilities nor sensor locations, it is very suitable in this application. The CRT equally weights all the local decisions without considering their spatial distribution. However, since most reflection energy from some complex targets is focused in a narrow conical area, only a subset of sensors can produce reliable local detections, while others are generally dominated by background noise [2]. If one could find that significant area and utilize only it to make a final decision, the system-level performance would be improved, especially at low SNR. An emerging local detection fusion approach is the scan statistic [] [2]. It slides a window with proper size around the sensor field, and picks up the most significant area to make a system-level detection. Here, the significance is defined by the number of s falling into the window. The key feature of a scan statistic is that its threshold can be set accurately to produce any desired system-level false alarm rate. In this paper, we will show that the scan statistic can outperform the CRT in barrier sensor networks based detection fusion. To recapitulate, the estimation of the location of an aspectdependent target with the configuration in Fig. was explored in [2]. Supposing that a target was present, the processing center could estimate its position with the binary decisions if the full local detection information were available. On the other hand, the idea of the application of a scan statistic in target detection was introduced in [3]. In [3], an isotropic target was within a large square sensor field. Its significant area would be a circle and it is invariant to the target location a square window was used. In our problem, the target is external to the sensor network. The significant area is angle and distance dependent, and a rectangular scanning window is adopted. The system-level performance with square and rectangular windows have different analyzing approaches [] [2]. In all, this paper follows these two works: it applies the scan statistic to the problem of [2]. The rest of this paper is as follows. Section II introduces the target reflection model and the local detection behaviors. Section III gives the global detection fusion approaches, while the scan statistic based detection fusion is proposed in Section IV. The system-level performance of a scan statistic is analyzed in Section V, while numerical examples are shown in Section VI. Conclusions are drawn in the end. II. PROBLEM STATEMENT We are interested in target detection with a barrier sensor network, where an acoustic source and N passive sensors cooperate to monitor a suspicious area as shown in Fig.. The system works as follows. First, the acoustic source emits a probing signal st, which can reach both the surveillance area and the distributed sensors. If a target invades that area, reflections will be concentrated in a certain direction with a narrow beamwidth [, p.309]. Secondly, all sensors apply local matched filers for detection. As an operational matter they may be awoken from a sleep mode by the direct signal. A. Local Sensor Node Detection Let a sensor have limited communication capacity, and it transmits a binary local decision instead of raw data to the FC. Suppose the sensor noises to be independent and identically distributed i.i.d. zero-mean complex Gaussian with variance σw, 2 and suppose that each sensor makes its test based on the magnitude square of its matched filter output r i. Under the null hypothesis H 0, r i has an exponential distribution with probability density function pdf [4] fr i H 0 = 2ξσw 2 exp r i 2ξσ 2 w, where ξ = st 2 dt denotes the waveform energy. Under the alternative hypothesis H, the received signal is composed of two parts: echoes plus noise. Let the echoes be independent of sensor noise. With a Rayleigh fading model [, pp. 89 and 382], we have [4] fr i H = 2ξσw 2 + 2ξ 2 ε 2 i exp r i 2ξσ 2 w + 2ξ 2 ε 2 i, 2 where ε 2 i stands for the contribution of target reflection to the ith sensor, and it is a function of source-target-sensor geometry. Here, we assume that σw 2 is estimable by other means to avoid concerns of the local decisions being constant false alarm rate CFAR. Let all the sensors employ the same threshold τ in decision making. With the Neyman-Pearson Lemma [5, p.47], the local false alarm rate and probability of detection for the ith sensor can be written as i = p di = τ τ fr i H 0 dr i = exp τ fr i H dr i = exp τ, + β i where τ τ/2ξσw, 2 and β i ξε 2 i /σ2 w relates to the reflection geometry and transmission power. Clearly, all i s are equal, and we will use to refer them for notational simplicity. The probabilities of local detection p di s are different, because the β i s are geometry dependent. The binary local detection results for sensor node i is I i = {0, }, where I i takes the value of if there is a detection; otherwise it takes the value of 0. B. Local Detection Modeling Underwater sound propagation is complex. An acoustic signal can be delayed, weakened, and distorted during propagation. Parameter β i is a complicated function of propagation distance, source-target-sensor geometry, as well as the underwater environmental parameters including temperature and salinity; a precise formula is usually unavailable in real time. Here we only consider propagation loss and angular dependence for simplicity. In general, the effects of propagation loss and angle dependent reflection might be separately modeled [2], and this yields 3 β i = c 0 f f 2, 4 where c 0 is a geometry independent constant, f denotes the power loss due to the propagation distance, and f 2 describes the angle dependent reflection.

3 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 3 y x source: x,y barrier sensor field s s target: x,y principal reflection direction a vi t t normal line sensor i: x,y i i Fig. 2. An illustration of angle dependent power loss of a submarine in two dimensions [2], where y t > y s. The reflection of an acoustic source focuses on a principal direction, say a in the coordinates, while the angle of sensor i is v i. If x i x t, we have v i [ π/2, 0] based on 7; otherwise, v i [0, π/2]. The power of the received signal at sensor i is a function of a v i. We assume that the source and the sensors are at the same depth as the target. Therefore, the propagation distance for the path source-target-the ith sensor is d i = x s x t 2 + y s y t 2 + x i x t 2 + y i y t 2, 5 where x s, y s, x t, y t, and x i, y i respectively denote the coordinates of source, target, and the ith sensor. In this paper, a cylindrical spreading loss model [, p.02] is employed wherein the power loss, f, is said to be proportional inverse first power of the propagation distance d i ; that is: f d i d i. 6 Note that such power loss may exist at moderate and long ranges whenever the sound is trapped by an underwater acoustic channel [, p.02]. The angle dependent reflection is more complicated than the propagation loss. Here we adopt the two-parameter aspectdependent target a submarine modeling approach of [2]. Denote the principal reflection angle as a and the angle from the ith sensor to submarine as v i, as in Fig. 2 xi x t v i = arctan. 7 y i y t As the surveillance area is on the single side of the barrier sensor network, we have y i < y t for i. As a consequence, there is no ambiguity in determining the v i s from 7, and they are always in [ π/2, π/2]. The aspect dependent reflection gain f 2 may be determined by the difference between a and v i, and can be well matched by a butterfly pattern [, p.3]. Its wing sides have significant reflection gains, and this will lead to a detectable zone in a sensor network. Unfortunately, this pattern does not have a closedform expression, and a Butterworth filter based approximation is suggested in [2] f 2 v i, a = + vi a, 8 U 2K where 2U denotes the 3dB bandwidth, and K determines the order of the Butterworth filter. Substituting 4 into 3, we have p di θ = exp τ, 9 + c 0 f d i f 2 v i, a where θ = {x t, y t, a} summarizes the set of unknown parameters under H. Due to the directionally selective reflection, only those sensors within the detectable zone are likely to declare detections under hypothesis H. III. GLOBAL DETECTION FUSION APPROACHES In underwater surveillance, distributed sensors periodically transmit their local decisions I i s to a FC; the latter properly combines I = {I, I 2,, I N } to make a system-level decision. Based on how to use I, detection fusion can be divided into two categories: global and local. In the global fusion, a fused decision includes all the local decisions I i s, while a local one may use only part of them. Here, we will review some typical global approaches and discuss their feasibility in angle-dependent sonar target detection. A. Chair-Varshney Rule For a binary detection fusion problem, the optimal Neyman- Pearson test statistic is given by the Chair-Varshney rule [7] Λ CVR = log pi Hs pi H0 s N = i= [ I i log p d i θ + I i log p θ di ], 0 where H s 0 denotes the system-level null hypothesis, and H s represents the system-level alternative hypothesis. The optimal fusion rule is essentially a weighted sum of local detections, and it requires the knowledge of local sensor s performance. Unfortunately, since the target s parameter set θ is unknown, p di θ s are not available. As a result, the optimal fusion rule may not be performed at FC for this problem. B. Counting Rule Test In [8] and [9], a suboptimal counting rule test CRT is suggested, where the FC counts the total number of local decision s and compares it with a given threshold T CRT to decide whether a target appears. Mathematically, its statistic is written as Λ CRT = N i= H s I i T CRT, H s 0 where all the local decisions are equally weighted. As the CRT does not require knowledge of probabilities of local detection, it is applicable here. The closed-form system-level false alarm rate of the CRT can be found in [8] and [9]. However, its system-level probability of detection for this problem could not be analytically derived, and it could only be analyzed numerically. Note that the utilization of a Butterworth pattern is for illustrating the idea of angle dependent reflection; a real radiation pattern can be readily used if an exact formulation is available.

4 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 4 sensor declaring detection scanning window H: s a nonempty zone Z has a locally significant detection cluster. In general, the zone Z specified in H s is unknown; some heuristic conjectures such as the size of Z are necessary. c0,0 Fig. 3. Two dimensional Bernoulli grid of a barrier sensor network, where M = 20 and M 2 = 4. If no target appears in the surveillance area, the sensors declaring detection follow a uniform distribution. Otherwise, certain local detection clusters may happen. A scan statistic test slides a rectangle window across the sensor field to check whether a cluster exists. c0, 0 denotes the cell of a certain position. C. Other Modifications and Discussions An interesting GLRT based decision fusion is proposed in [0], where the maximum likelihood ML estimate of θ is utilized to modify the Chair-Varshney rule. Since the FC does not have knowledge of target radiation model for our problem, an explicit joint conditional pdf pi θ is unavailable, and hence ML is infeasible. Furthermore, the target is assumed isotropic, and located within a dense sensor field [0], so that sensors declaring detection may surround the target, such a favorable geometrical complementarity could result in a good estimation. In our problem, the target is outside the sensor network; the geometrical complementarity of sensor nodes is poor. Even though the FC strives to estimate it with a mismatched model, the estimation accuracy would be low. In [6], the authors suggested to assign a prior distribution to θ and hence construct a Bayesian test. As this method desires the knowledge of pi θ, it is of limited applicability here. IV. SCAN STATISTIC BASED TARGET DETECTION A. From Target Detection To Cluster Detection A scan statistic is developed to search clusters in a spatial or and time spanned data set. It has been widely used in bioinformatics, ecology, and medicine [] [2]. Briefly, a test based on scan statistic slides a window across the observation domain, and utilizes the most significant data subset to infer the hypothesis. The shape of the window is usually problem dependent. It can be a square, rectangle, circle, or ellipse, and no uniformly best one exists. The key feature of a scan statistic is that despite the sliding behavior of the window powerful methods exist for false-alarm rate calculation. Note, however, that only for some window shapes especially rectangular are these formulae presently known. Suppose that the sensors are uniformly distributed in the barrier band. With the i.i.d. background noise assumption, the sensors declaring are uniformly dispersed under H s 0. If a target invades the surveillance area and if the mainlobe of its reflection pattern falls into the sensor field, a certain number of sensors within this zone will declare. Therefore, a local detection cluster may form, and the detection problem could be converted to a cluster detection one: H s 0: no significant cluster exists within the sensor network; B. Bernoulli Sensor Field In this case, the total number of sensors N is known, and the sensors are regularly deployed in a two-dimensional barrier band. The barrier band can be uniformly divided into N small sub-square cells, and each cell contains exactly one sensor as shown in Fig. 3. Let the barrier sensor field be composed of M M 2 = N small cells, where M and M 2 respectively denote the numbers of grid columns and rows. Let X lk denote the binary local decision of the sensor in cell cl, k, where l and k are respectively the row and column indexes; we have N M M 2 I i = X lk, 2 i= l= k= which means that a one-dimensional local decision stream I is reorganized into two dimensions. A shape match between the local detection cluster and scanning window would improve the test performance. Unfortunately, since the cluster varies with the target location and the skewness of its pattern mainlobe, a perfect footprint coincidence may not be guaranteed. Here, a suboptimal rectangle window is employed. As the width of the sensor band is thin, it will be included in the window for scanning convenience, see Fig. 3. Let the discrete length of the window be W, where W M, and then the total number of within the window is Y m = m+w l=m M 2 X lk = k= m+w l=m Z l, 3 where m M W +, and Z l M 2 k= X lk. The two dimensional summation for X lk s can be equivalently converted into a one-dimensional problem on Z l ; this is an important difference from [3]. The scan statistic SW, M is defined as the largest count within the event set Y = {Y, Y 2,, Y M W +} [, p.273] SW, M = max {Y m ; m M W + }, 4 and the FC makes a system-level decision based on { SW, M < T B, declare H s 0 SW, M T B, declare H s, 5 where T B corresponds to a specified significance level in discriminating a cluster. Intuitively, the scan statistic divides the sensor field into several partially overlapped subareas, and then selects the most significant one to make a system-level decision. If W = M, the scan statistic becomes a CRT. C. Poisson Sensor Field In the previous scenario, the number of sensors is known, and they are regularly deployed within the preassigned cells. Such an ideal deployment enables a comprehensive sensor field coverage; however, it is hard to maintain. For example,

5 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 5 a division slice scanning window Simulation Approximation sensor declaring detection Fig. 4. Two dimensional slicing of a barrier sensor network. If no target appears in the surveillance area, the sensors declaring detection are uniform and sparse. Otherwise, local detection clusters may happen. A scan statistic slides a rectangle window across the sensor field to check whether a cluster exists. system level false alarm rate =0.0, W=6 =0.0, W=7 =0.05, W=7 =0.05, W=6 some sensors may be out of battery power, or communication in data collection; hence, the number of activated sensors may differ from that deployed. Furthermore, the currents may displace the sensors, and disarray the deployment. Here, we are interested in a random barrier sensor network. Both the number of activated sensors and their locations are random: the total number of activated sensors, say N, has a Poisson distribution; and those activated sensors are uniformly distributed within the barrier band. These two assumptions have been adopted in many random sensor network treatments [2], [8] [0], [3], [6]. Suppose that the initial distribution of activated sensors is a homogeneous Poisson process with density λ; we have p N = λa N e λa, 6 N! where A denotes the area of the sensor band. Let the sensors be within x x x 2 and y x y 2, and then we get A = x 2 x y 2 y. The locations of sensors are i.i.d., and have a uniform pdf { fx, y = A x x x 2 and y x y otherwise. Each sensor makes a local decision based on its matched filter output. With an i.i.d background noise assumption, the number of sensors declaring under hypothesis H0 s also has a Poisson distribution with density λ f = λ, and they are uniformly located within the sensor field. Under H, s the distribution of sensors declaring is no longer uniform, and a cluster may form. We use a window to scan the sensor band to check whether a cluster exists. Continuous scanning would be the best choice for a given window size; however, we prefer a discrete scanning scenario again for analysis simplicity: uniformly dividing the sensor band into M contiguous narrow slices as shown in Fig. 4. Let Xi stand for the number of sensors declaring in the ith slice, and let the window cover exactly W slices; hence, the total number of within in the window is Y m = m+w l= m X l, where m M W +. Here the two-dimensional data summation can also be reduced to a scalar case. The scan statistic SW, M is defined as the largest count within the event set threshold T B Fig. 5. A comparison of simulated and approximated system-level false alarm rates of scan statistic in a Bernoulli sensor field. Each simulated curve is based on runs. Ȳ = { Y, Y 2,, Y M W +}: SW, M = max { Y m ; m M W + }, 8 and the FC makes the system-level decision based on { SW, M < T P, declare H s 0 SW, M T P, declare H s, where T P corresponds to a given significance level again. V. PERFORMANCE ANALYSIS FOR SCAN STATISTIC A. Bernoulli Sensor Field 9 The system-level performance including the false alarm rate and the probability of detection will be discussed for scanning a Bernoulli sensor network. With the i.i.d. sensor noise assumption, the X lk s are i.i.d. Bernoulli random variables with parameter in the absence of a target. Therefore, the Z l s are independent and described by a binomial distribution M2 Pr{Z l = i} = p i f i M 2 i 20 under the null hypothesis H s 0. The system-level false alarm rate is defined as P B f = Pr { SW, M T B H s 0}, 2 and it is a function of Pr{Z l = i} and window size W. The analysis of P f B is intricate, and a closed-form expression is not available. In [7], the authors suggested that P f B could be accurately approximated with [ ] M 2W P f B GTB,W 2W G TB,W 2W, 22 G TB,W 2W where G TB,W 2W Pr{SW, 2W < T B } G TB,W 2W Pr{SW, 2W < T B } 23 can be calculated by the procedure in Appendix I. Intuitively, G TB,W 2W and G TB,W 2W denotes the statistical

6 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 6 system level false alarm rate =0.0, W=6 =0.0, W=7 Simulation Approximation =0.05, W=7 =0.05, W= threshold T B Scan SNR=7dB Scan SNR=5dB Scan SNR=3dB CRT SNR=7dB CRT SNR=5dB CRT SNR=3dB system level false alarm rate Fig. 6. A comparison of simulated and approximated system-level false alarm rates of scan statistic in a Poisson sensor field. Each simulated curve is based on runs. Fig. 7. A comparison of ROCs for scan statistic and CRT at different SNRs in a Bernoulli sensor field, where = 0.05 and W = 7. results for a W -size window scanning two random sequences respectively with length 2W and 2W. To verify the approximation accuracy of 22, we plot several curves of the system-level false alarm rate P f B versus the scan statistic threshold T B in Fig. 5. We choose M = 0 and M 2 = 6. Each simulated curve is obtained by Monte Carlo runs. From this figure, we see that 22 exhibits good approximation performance for different W and values, and would be effective in false alarm analysis. The system-level probability of detection is defined as P B d = Pr { SW, M T B H s }. 24 However, since the conditional pdf of pi H s is unavailable, P d B could not be analytically derived or accurately approximated. In general, it can only be analyzed via simulations; since it is target dependent in any case, this seems acceptable. B. Poisson Sensor Field With an i.i.d. sensor noise assumption, the X l s are i.i.d. Poisson random variables Pr{ X l = k} = λ f A ke λ f A k! 25 under hypothesis H s 0, where A represents the area of each slice, and k 0 dentes an integer. The system-level false alarm rate is P P f = Pr { SW, M T P H s 0}. 26 A closed-form expression for P f P is not available, but again it can be approximated with 22. Actually, approximation 22 is a general result.. To approximate 26, we should employ 25 instead of 20 in the calculation of G TP,W 2W and G TP,W 2W. The approximation accuracy for the Poisson scenario with different parameter values is demonstrated by Fig. 6. In the simulation, we choose M = 0 and λ f A/pf = λ A = 25. Each simulated curve is obtained by runs window size W Fig. 8. The system-level probability of detection P d B as a function of window size W for scan statistic in a Bernoulli sensor field, where = 0.05, P f B = 0.0, and SNR = 5 db. A. Bernoulli Sensor Field VI. NUMERICAL RESULTS This subsection numerically compares the performance of CRT and the scan statistic in target detection with a Bernoulli sensor network. Let the sensor band cover area R = { x, y 2500m x 2500m, 0 y 250m } 27 in two dimensions. The rectangular sensor field is uniformly divided into 6 0 cells, with each cell containing exactly one sensor node. The total number of sensors is N = 606. The acoustic source is located at x s = 0, y s = 0. Under the H s hypothesis, a target is fixed at x t = 0, y t = 000m, while its heading is uniformly distributed within [ π, π]. The power control constant c 0 is chosen as c 0 = 2ρ x t x s 2 + y t y s 2, 28

7 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 7 where 2 x t x s 2 + y t y s 2 normalizes the maximum propagation loss due to f i d i as shown in 9, and ρ is defined as the source-to-receiver SNR. In the following simulations, the SNR refers to ρ. Finally, the beampattern parameters are chosen as U = 4 and K = 5 [2], which enable f 2 to have a good approximation to the Butterfly one in [, p.3]. The detection performance is compared with the help of the receiver operating characteristic ROC. Let the local false alarm rate be = Fig. 7 gives the ROCs for the CRT and scan statistic with different SNRs, while Fig. 8 depicts the relationship between the system-level probability of detection and window size W for scan statistic at a given SNR. From these figures, we see that: For a given SNR, the scan statistic with a properly selected window size has better detection performance than the CRT. The reason for this is that if a target can only affect a part of sensor network the detectable zone, picking up that subarea may be better than interrogating the entire field. The window size of the scan statistic can affect the ROC. Since the shape of local detection cluster is not available, the best window size is unavailable. From Fig. 8, we see that when the window size varies from 4 to 7, the detection probabilities do not exhibit a significant change. In other words, the scan statistic has a certain amount of robustness in window size selection. B. Poisson Sensor Field with Fixed Density This subsection compares the detection performance of the CRT and the scan statistic for a Poisson sensor network. The sensor band shares the same coverage as that in the previous example. The number of sensors is a Poisson variable with density parameter λ = /2500. The sensor field is sliced into M = 00 rectangles, and each with size m 2. The acoustic source and the target setup under H s hypothesis are the same as the previous subsection. Let the local false alarm rate be = Fig. 9 gives the ROCs of those approaches with different SNRs, while Fig. 0 depicts the ROCs for the scan statistic approach with different window sizes at a given SNR. The observations are similar to those of the Bernoulli case. C. Poisson Sensor Fields with Different Densities In the previous subsection, the sensor density λ is fixed. Here, we are interested in the system-level detection performance with varying λ. The sensor field s spatial span and the target location and orientation settings are the same as the previous subsection. The SNR is 5 db while the local false alarm rate is = Let the system-level false alarm rate be 0.0, the relationship between the sensor density λ and system-level probability of detection is shown in Fig.. We have the following observations: The system-level probabilities are increasing functions of λ for both CRT and the scan statistic. This is not surprising as a higher density means that more sensors would fall in the conical area and report reliable local decisions Scan SNR=7dB Scan SNR=5dB Scan SNR=3dB CRT SNR=7dB CRT SNR=5dB CRT SNR=3dB system level false alarm rate Fig. 9. A comparison of ROCs for scan statistic and CRT at different SNRs in a Poisson sensor field, where = 0.05 and W = window size W Fig. 0. The system-level probability of detection P d P as a function of window size W for scan statistic in a Poisson sensor field, where = 0.05, P f B = 0.0, and SNR = 5dB. The scan statistic may outperform the CRT for a given sensor density. This coincides with the observation of the previous subsection. Fig. may be useful for resource allocation, for example, determining the sensor density for a given performance level. VII. CONCLUSIONS This paper investigated distributed decision fusion in detecting a sonar target with barrier sensor networks. As the reflected signal from some targets such as submarines, unmanned underwater vehicles, and weapons, is aspect dependent, most of the reflected energy would be concentrated within a narrow conical area. If the barrier band is with sufficient length span, only part of its nodes will be significantly affected, particularly at low SNR. Suppose that each sensor performs a local detection with a simple comparison of its matched filter output with a given threshold. In the absence

8 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX scan CRT Poisson sensor density λ x 0 4 Fig.. The system-level probability of detection as a function of sensor density λ in a Poisson sensor field, where = 0.05, P f B = 0.0, SNR = 5 db, and W = 7. of probabilities of local detection, two approaches were used in fusing those local results: the counting rule test CRT [8] [0] and the scan statistic. The CRT does not require sensor location information, and equally weights all the local detections in the system-level calculation. The scan statistic slides a window to pick up the significantly affected area and uses it to make a system-level decision. In other words, the scan statistic utilizes the spatial distribution properties of local detections in decision making, while the CRT does not. Numerical examples showed that scan statistic based detection fusion may outperform the CRT at the expense of collecting more information sensor locations. In this treatment, environmental effects have been idealized: cylindrical spreading and no medium-caused distortion, plus a Butterworth beampattern. A fortunate feature of the scan statistic is that to relax these has no effect whatever on the false alarm rate. A dilution caused by propagation effects or target mis-modeling of the region of activated sensors within the barrier will somewhat affect the local probabilities of detection. However, as a scan statistic requires neither the propagation nor the target model in detection fusion, we would expect this not to be major. Some limitations of the proposed method and suggested future research might include the following: A scan statistic currently desires an identical local false alarm rate homogeneous clutter. A scan statistic with heterogeneous local false alarm rates is an interesting topic. This paper utilizes a scan statistic with a single window. Without knowledge of adversary strategy, it is hard to choose an optimal window size. Robust scanning with multiple windows [8] would be an interesting extension, and it may be also a good tool to tackle multiple targets. This paper suggests the use of a barrier configuration, which forms a narrow rectangular surveillance band in open water. For a given number of sensors, to design the density and the bandwidth to guarantee a large and reliable surveillance span is of practical interest. APPENDIX I CALCULATION OF G k,w 2W AND G k,w 2W This part briefly outlines an iterative procedure to calculate two basic quantities, G k,w 2W and G k,w 2W, in approximating the system-level false alarm rate of scan statistic [9]. Let fy be the event pdf. Specifically, for the Bernoulli sensor field, we have fy Pr{Z l = y}, while fy Pr{ X l = y} for the Poisson case. Define c as the pdf threshold c sup y {fy > 0}, 29 and then we can initialize the procedure with { k b k fy 2 y = j=0 fj, 0 y minc, k 0, otherwise. 30 For 2 i W, the instrumental variables b k 2i y s can be recursively calculated with y b k 2i y = η=0 k y+η v=0 b k v 2i y ηfvfη, 0 y k 0, otherwise. 3 Once b k 2i y s are obtained, G k,w 2W can be obtained via G k,w 2W = k b k 2W y. 32 y=0 With equation 32, we can also get G k,w 2W s. Specifically, G k,w 2W can be calculated via G k,w 2W = k fxg k x,w 2W. 33 x=0 Note that this procedure is general, and it does not depend on the distribution of fy. REFERENCES [] R. J. Urick, Principles of Underwater Sound, 3rd ed. New York: McGraw-Hill, 983. [2] S. Zhou and P. Willett, Submarine location estimation via a network of detection-only sensors, IEEE Trans. Signal Process., vol. 55, no. 6, pp , Jun [3] H. Van Trees, Detection, Estimation, and Modulation Theory, Part III: Radar-Sonar Signal Processing and Gaussian Signals in Noise, st ed. New York: John Wiley & Sons, Inc., 200. [4] D. Abraham, Distributed active sonar detection in dependent Kdistributed clutter, IEEE J. Ocean. Eng., vol. 34, no. 3, pp , Jul [5] S. Coraluppi, Multistatic sonar localization, IEEE J. Ocean. Eng., vol. 3, no. 4, pp , Oct [6] S. Barr, B. Liu, and J. Wang, Barrier coverage for underwater sensor networks, in Proc. of MILCOM Conf., San Diego, CA, Nov [7] Z. Chair and P. Varshney, Optimal data fusion in multiple sensor detection systems, IEEE Trans. Aerosp. Electron. Syst., vol. 22, no., pp. 98 0, Jan [8] R. Niu and P. K. Varshney, Distributed detection and fusion in a large wireless sensor network of random size, EURASIP J. Wireless Commun. Netw., no. 4, pp , [9], Performance analysis of distributed detection in a random sensor field, IEEE Trans. Signal Process., vol. 56, no., pp , Jan

9 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 9 [0], Joint detection and localization in sensor networks based on local decisions, in Proc. of Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, Oct [] J. Glaz, J. Naus, and S. Wallenstein, Scan Statistics. Springer, 200. [2] J. Glaz, V. Pozdnyakov, and S. Wallenstein, Scan Statistics Methods and Applications. Birkhauser, [3] M. Guerriero, P. Willett, and J. Glaz, Distributed target detection in sensor networks using scan statistics, IEEE Trans. Signal Process., vol. 57, no. 7, pp , Jul [4] C. Rago, P. Willett, and Y. Bar-Shalom, Detection-tracking performance with combined waveforms, IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp , Apr [5] N. Mukhopadhyay, Probability and Statistical Inference, st ed. New York: Marcel Dekker Inc., [6] M. Guerriero, L. Svensson, and P. Willett, Bayesian data fusion for distributed target detection in sensor networks, IEEE Trans. Signal Process., vol. 58, no. 6, pp , Jun [7] J. Glaz and J. Naus, Tight bounds and approximations for scan statistic probabilities for discrete data, Ann. Appl. Probab., vol., no. 2, pp , May 99. [8] J. Glaz and Z. Zhang, Multiple window discrete scan statistics, J. Applied Stat., vol. 3, no. 8, pp , Oct [9] V. V. Karwe and J. Naus, New recursive methods for scan statistic probabilities, Comput. Stat. Data Anal., vol. 23, no. 3, pp , 997. Joseph Glaz received the Ph.D. degree in statistics from Rutgers University, Piscataway, NJ, in 978. He is Professor, Director of Graduate Programs and Associate Head, Department of Statistics, University of Connecticut, Storrs. Areas of research interest include geometrical probability, parametric bootstrap, scan statistics, sequential analysis, and simultaneous inference. Dr. Glaz was elected Ordinary Member of the International Statistical Institute in 996, Fellow of American Statistical Association in 2000, and Fellow of the Institute of Mathematical Statistics in He is a recipient of Abraham Wald Prize in Sequential Analysis in 2006 and the AAUP Excellence in Research Award in He is Editor-in-Chief of a scientific journal Methodology and Computing in Applied Probability, published by Springer. Xiufeng Song S 08 received the B.S. degree from Xidian University, Xi an, China, in 2005 and the M.S. degree from Institute of Electronics, Chinese Academy of Sciences CAS, Beijing, China, in 2008, both in electrical engineering. He is currently working towards the Ph.D. degree with the Department of Electrical and Computer Engineering, University of Connecticut, Storrs. His research interests lie in signal processing, detection, and estimation theory. Peter Willett F 03 received his BASc Engineering Science from the University of Toronto in 982, and his PhD degree from Princeton University in 986. He has been a faculty member at the University of Connecticut ever since, and since 998 has been a Professor. His primary areas of research have been statistical signal processing, detection, machine learning, data fusion and tracking. He has interests in and has published in the areas of change/abnormality detection, optical pattern recognition, communications and industrial/security condition monitoring. Dr. Willett is editor-in-chief for IEEE Transactions on Aerospace and Electronic Systems, and until recently was associate editor for three active journals: IEEE Transactions on Aerospace and Electronic Systems for Data Fusion and Target Tracking and IEEE Transactions on Systems, Man, and Cybernetics, parts A and B. He is also associate editor for the IEEE AES Magazine, editor of the AES Magazines periodic Tutorial issues, associate editor for ISIF s electronic Journal of Advances in Information Fusion, and is a member of the editorial board of IEEE Signal Processing Magazine. He has been a member of the IEEE AESS Board of Governors since He was General Co-Chair with Stefano Coraluppi for the 2006 ISIF/IEEE Fusion Conference in Florence, Italy, Program Co-Chair with Eugene Santos for the 2003 IEEE Conference on Systems, Man, and Cybernetics in Washington DC, and Program Co-Chair with Pramod Varshney for the 999 Fusion Conference in Sunnyvale. Shengli Zhou SM received the B.S. degree in 995 and the M.Sc. degree in 998, from the University of Science and Technology of China USTC, Hefei, both in electrical engineering and information science. He received his Ph.D. degree in electrical engineering from the University of Minnesota UMN, Minneapolis, in He has been an assistant professor with the Department of Electrical and Computer Engineering at the University of Connecticut UCONN, Storrs, , and now is an associate professor. He holds a United Technologies Corporation UTC Professorship in Engineering Innovation, His general research interests lie in the areas of wireless communications and signal processing. His recent focus is on underwater acoustic communications and networking. Dr. Zhou served as an associate editor for IEEE Transactions on Wireless Communications, Feb Jan. 2007, and IEEE Transactions on Signal Processing, Oct Oct He is now an associate editor for IEEE Journal of Oceanic Engineering. He received the 2007 ONR Young Investigator award and the 2007 Presidential Early Career Award for Scientists and Engineers PECASE.