The Spanish SEPRONA [1] agency fights against poaching with considerable success. However, the problem remains relevant. Since 2007, SEPRONA has detained over 150 people for hunting felonies [2, 3]. Consequently, the authors have proposed the DiANa project, Detección de caza furtiva con Armas de fuego en parques NAcionales (Detection of illegal hunting with gunfires in national parks), to automatically detect and locate gunshots, which is endorsed by Cabañeros National Park [4].

The DiANa system consists of a network of acoustic sensors that locate gunshots. There exist some commercial solutions for gunshot detection, although they do not locate sound sources [5]. There also exist sound location tools for the military, but they are too costly and cannot be deployed in large numbers in civilian applications [6]. This paper presents an original method for gunshot location and a planning algorithm for its deployment in large terrain extensions (e.g., a national park). Two key design issues are implementation cost minimization and performance maximization in real-time scenarios. Therefore, sensor protocols and tasks (such as time synchronization and gunshot position estimators), as well as network planning for node placement, have been taken into account.

Assuming a group of sensors has correctly detected an acoustic signal (e.g., using Gaussian Mixture Models, GMM [7, 8]), the location of this signal results from the combination of sensor data. A sensor only knows its own position and local time, and therefore these measures must suffice to locate the gunshot.

In the literature there are several location estimators, such as triangulation [9] and trilateration [10] methods. In the triangulation schema, every sensor determines the direction from which the acoustic event is detected, and then the location of that event is calculated as the intersection of the detection directions. We discarded this alternative, because determining the direction of the acoustic event would make the hardware design too complex and costly, and the solution would be highly sensitive to terrain shape. A trilateration schema determines the location of the event from the distance between the source of the acoustic event and a fixed sensor. These distances can be obtained if the generation time of the event is known. However, in our scenario, only the detection time is available to the nodes.

A directly observable acoustic signal between a couple of microphones is the time difference of arrival (TDoA) [11–13]. The TDoA technique exploits the relationship between distance and transmission time when the propagation speed is known. Once the time delays are calculated, they are processed in order to estimate the location of the source [14–16]. Due to the distances between the deployed sensors (in the order of hundreds of meters), and the smooth landscape in Cabañeros National Park (Figure 1), we have assumed a two-dimensional scenario.

The hyperbolic location method [17] minimizes an error measure that is a nonlinear function of the potential source location. This approach is scalable, since location accuracy increases with the number of nodes that detect the gunshot (see Section 5.). Its domain can be a plane or a three-dimensional space.

TDoA-based ranging techniques require accurate clock synchronization. Every sensor knows its own position exactly, and it records the arrival time of the sound event. So, the sensor clocks must be as tightly synchronized as possible, using dedicated time-synchronization algorithms. Since a gunshot location network must scale to large sizes, it is necessary to minimize the number of exchanged messages to attain convergence, keeping energy consumption at reasonable levels if the electric grid is not available.

Selecting a synchronization schema for a real application like ours is not easy. The best known synchronization schemas implement network mechanisms to adjust all local clocks to the same value. This is achieved by exchanging time stamps between node pairs. The more frequent the exchanges, the higher the time accuracy. Two representative examples are Reference Broadcast Synchronization (RBS) [18] and the Timing-Sync Protocol for sensor Networks (TSPN) [19]. We discarded GPS receivers due to their high cost. In Section 2.3. we propose a new ad-hoc flood method to set the clock times in every network node to the same value. In this method, the nodes do not exchange synchronization messages, and thus they save power. Once a node detects a gunshot, the time of the event is transmitted to the sink node through a previously generated path. The method performs a cooperative backward time adjustment, so that every node along the path is able to estimate the event time.

Regarding network planning, since the areas under surveillance are wide, and electric power is seldom available, it is necessary to both maximize detection coverage and minimize system cost. Therefore, we model sensor network planning as an unconstrained problem with two objective functions. We provide a set of candidate solutions of interest by combining a derivative-free descent method we have recently proposed and a Pareto front approach.

Due to the inherent difficulties exhibited by the thus far formulated models in sensor network planning, several heuristic optimization strategies have been proposed in the literature: variants of simulated annealing [20], genetic algorithms [21], gradient descent (when applicable) [22] and others [23, 24].

Some of these approaches (simulated annealing, genetic algorithms and the like) do not guarantee theoretical convergence. Regarding gradient descent methods, the gradient is often unavailable or too costly to compute. Therefore, we have adapted a non-monotone derivative-free optimization technique with guaranteed convergence [25] to formulate and solve a computationally efficient optimization model with a dual objective: maximization of acoustic network coverage and minimization of power infrastructure cost. The results are presented as a Pareto front, revealing solutions that are clearly superior to random seeding.

Our notation is as follows: Lower case Greek letters are scalars, lower case Latin letters are vectors in IR², x k j is the j-th component of the vector x_k, and ‖x‖ is the Euclidean norm. A capital Latin letter, say S, stands for a collection of vectors in IR²; if S = {s₁, …, s_p} we also say that S ∈ IR^2p; τS = {τs₁, …, τs_p}, and the sum Z = S + D means that S and D have the same number of elements, say p, and z_k ∈ Z if and only if ∃k ∈ {1, …, p} | z_k = s_k + d_k. In general the subindex i is the value of an entity (scalar, vector, set, and so forth) at the i-th iteration of an algorithm; for instance S_i = {s_i1, …, s_ip} is a set of p vectors in IR², at the i-th iteration.

The rest of the paper is organized as follows:

Section 2. describes the sound location and node synchronization procedures. In Section 3. we present the optimization model to plan the acoustic sensor network. In Section 4. we verify that the problem fulfills the conditions in [26] to apply the advanced derivative-free algorithms in [25]. In Section 5. we perform numerical tests to evaluate the approach. Finally, Section 6. concludes the article.

In this section we describe the location procedure for acoustic events, which we have implemented for MicaZ motes. The goal of our system is gunshot location by means of a sensor network. Location is based on hyperbolic positioning [17]. Hyperbolic positioning requires the sensor clocks to be synchronized, in order to apply the TDoA technique. For this reason we have implemented a synchronization protocol in the MICAz motes. Next, we describe the system architecture, the location method and, finally, the synchronization schema.

The sound spectrum of a gunshot is dominated by the 130 Hz to 3 kHz frequency range [29]. According to [30], at these frequencies there is an extra attenuation of 3 dB each ∼25 m in woods, yielding 12 dB vs. 9 dB in open space. Given the sensitivity threshold of the sensing node, its maximum ranges are 1 Km in open space and 750 m in a wood.

We consider a large outdoor flat scenario with open space areas and wood patches. The scenario is crossed by a few power lines. In it, we wish to deploy a given number of fixed sensing nodes, so that the detection coverage is maximum. A point gets covered if it is reachable by one sensing node at least, although this capability increases if more sensing nodes see the point (to achieve source signal location we require a coverage of at least three nodes, as explained in Section 2.2.).

At the same time, we wish to minimize the distance between the sensing nodes and the power lines so that the cost of the power infrastructure is minimal in case the nodes are not autonomous. The nodes communicate through the electric grid itself, so transmission coverage is not an issue of interest.

This scenario clearly prevents an optimal educated guess, specially when only a small number of sensing nodes is available.

We define δ(s, x) = ω₁(s, x) + ω₂(s, x), where:

ω₁(s, x) corresponds to propagation distance through wood space.

Given a sensor location s ∈ X and a point x ∈ X we say that x is visible from sensor s if: (5) 1 Km 750 m ω 1 ( s , x ) + ω 2 ( s , x ) < 1 Km

Let S = {s₁ …, s_p}, s_k ∈ X, k = 1, …, p, be the positions of p acoustic sensors on X. We denote by V(x) the set of all sensors s ∈ S that are visible from the location x. We also define an arbitrary grid G ⊆ X. Objective function f₁, which measures acoustic sensor coverage, is defined as follows: (6) f 1 ( S ) = ∑ x ∈ G { 0.5 ⋅ card ( V ( x ) ) if card ( V ( x ) ) < 3 3 + 0.01 ⋅ ( card ( V ( x ) ) − 3 ) if card ( V ( x ) ) ≥ 3

This function penalizes grid points that see fewer than three sensors (a gunshot in those points cannot be located with highest precision), and gives a small bonus to grid points that see more than three sensors (the minimum number of sensors for highest precision location).

A second objective function f₂ measures the cost of the sensor deployment. As in the case of sensor coverage, there are many ways to model this. In this paper we assume that the cost of a sensor unit is negligible compared to the cost of a permanent power line. As we previously said, the scenario is crossed by m power lines. We define: (7) f 2 ( S ) = ∑ s ∈ S min i ∈ { 1 , … , m } δ i ( s )where δ_i(s) is the Euclidean distance between s and the i-th power line, i.e., between s and the point in that line that is closest to s in Euclidean distance.

Our ultimate task is to place p sensors on X in such a way that the coverage f₁ on X is maximized and the cost f₂ on X is minimized. Clearly, these objectives are contradictory. Minimizing −f₁ (i.e., maximizing f₁) tends to spread the sensors, whereas minimizing f₂ tends to concentrate them around the power lines. For that reason, it is desirable to produce the Pareto front [31] of these two functions, which represents a pool of candidate solutions. A point x* ∈ X belongs to the Pareto front of a set of functions in X if a further decrease in one of them is not possible without causing an increase in some of them. The methodology in [31] obtains joint descent directions for all the objective functions in a set, but it requires all of them to be differentiable, and that is not the case with f₁.

In our case, since there are only two objective functions, we define the following unconstrained optimization problem: (8) minimize s 1 , … , s p f ( s 1 , … , s p ) = ( θ − 1 ) f 1 ( s 1 , … , s p ) + θ f 2 ( s 1 , … , s p )

By solving problem (8) repeatedly, assigning random values to θ in [0, 1], we obtain a collection of points of the Pareto front of f₁ and f₂.

Many optimization algorithms are iterative. Starting with a solution estimate S₁ = {s₁₁, …, s_1p}, a subsequence {S_i}_i∈I = {s_i1, …, s_ip}_i∈I is generated that hopefully converges to the solution of the problem. As it is common in all implementations, there are several parameters (magic numbers) the user must set. Some of them will notably influence the performance of the algorithm, and often depend upon the structure of the objective function.

We model the 10 km × 10 km flat wood scenario in Figure 5. The light areas represent open space, and the dark ones represent wood patches. The scenario is crossed by three “vertical” power lines, respectively at 1.66, 5 and 8.33 Km from the left side. This terrain representation partly encloses the difficulties in modelling Cabaneros national park.

Grid G is given by a uniform discretization of the scenario in 50-m steps. In it we want to install 200 acoustic sensors (p = 200) with maximum detection coverage, at a minimum power cost. In order to obtain the Pareto front, we obtain the solution of problem (8) several times, starting from different random points S₀. We normalize the objective function by dividing it by its value at the starting points.

Since an algorithm can lead to a local optimum of problem (4) or to a solution that differs from the global optimum, we have chosen a brute force approach that computes the function at every grid point. Real-time system response is not affected, as the calculation time in a 10 km × 10 km scenario is in the order of a tenth of millisecond.

Following the results in [33], instead of applying the non-monotone derivative-free search as described in Section 4., we applied a zone search variant. If we simply “move” one sensor at a time, instead of “moving” them all, the evaluation of the objective function is significantly less time consuming, since most computations in f(S) do not change.

To formalize this approach, we split the scenario into q non-overlapping zones X_j, j = 1, …, q, such that X = ∪_j=1..qX_j and X_k ∩ X_j = ∅, k ≠ j. Let V(s_i) = {x ∈ G | s_i ∈ V(x)}. When we move the k-th sensor, the remaining sensor positions do not change, i.e., the group moves from S = {s₁, …, s_k, …, s_p} to S ′ = { s 1 , … , s k ′ , … , s p }. When this happens, V ( S ′ ) = V ( S ) ∪ V ( s k ′ ) − V ( s k ). To quickly obtain a relaxed estimate T_v of V ( s k ) ∪ V ( s k ′ ), we discard all X_j | ∀x ∈ X_j min ( δ ( s k , x ) , δ ( s k ′ , x ) ) > 1 Km. Let T_nv be the set of discarded zones and let T_v = V(S′) − T_nv. Then, we compute Σ_x∈Tv f(S′), and keep Σ_x∈Tnv f(S′) = Σ_x∈Tnv f(S) from the previous iterate. In the numerical tests that follow, we divided our scenario in 100 [1 Km × 1 Km] zones (q = 100). We compute one objective function component per zone at the beginning of the algorithm execution. When only one sensor is moved, there is no need to recompute the objective function in zones that are not affected by the sensor movement.

This way, the time to compute an objective function value drops from 20 seconds on average to just 1.5 seconds on a Pentium IV.

Let S = {s₁, …, s_k, …, s_p}. In order to move only one AP at a time, say s_k, we generate a set of unit search directions d_j ∈ IR², j = 1, …, n such that the set D = {d₁, …, d_n} positively spans IR². We recall that n ≥ 3. We declare a success when f ( s 1 , … , s k − 1 , s k + τ i d j , s k + 1 , … , s p ) ≤ f ( S ) + α i ( φ i − f ( S ) ) − ν ( τ i )for some d_j ∈ D and some s ∈ S. The point S is blocked if the algorithm is unable to move a single s ∈ S. We observe that this schema may be carried out simultaneously in a multi processor environment and it is straightforward to show that after we try all s ∈ S we have searched on a set of directions that positively span IR^2p, although we are using at least 3p directions of search.

We tuned the method in preliminary trials and determined the following parameter values:

Δ = 0.1 Km, i.e., the method stops when the maximum sensor displacement is under 100 m.

Figures 6, 7 and 8 show the results. Instead of representing f₂ versus f₁, we represent f₂ versus the coverage areas (in percentage) that correspond to V(x) ≥ 1, V(x) ≥ 2 and V(x) ≥ 3, respectively. Algorithm execution is repeated 20 times, from 20 different starting points. Each starting point consists of a random deployment of 200 acoustic sensors in the scenario in Figure 5, with a uniform distribution.

The results reveal that the Pareto front approach is useful. The solutions show a compromise between cost and coverage. In the V(x) ≥ 1 case there is no real advantage over a random seeding in terms of coverage, although the cost drops considerably for 100% coverage. However, in the V(x) ≥ 3 case, the optimization result is clearly superior to random seeding across the whole Pareto front. At maximum cost, there is a 20% increase in coverage, and at 85% coverage (the best coverage of random seeding) there is a 50% decrease in cost.

Figures 9, 10 and 11 show three sensor deployments in our synthetic scenario in Figure 5, for three different choices of θ = {0.1, 0.5, 0.9} (red points represent sensor positions, blue areas zones with less coverage, dark blue areas indicate no coverage at all). The results are consistent with the fact that coverage improves for low θ values. As the values of θ become higher, the sensors tend to concentrate around the power lines. Therefore, power cost decreases, but so does the coverage (Figure 11 for θ = 0.9).

Figure 12 represents the percentage of grid points covered by different numbers of sensors in the three previous deployments. In the θ = 0.9 case, where the cost function has more weight, over 35% grid points are out of coverage, and, due to the concentration of the sensors around the power lines, there are grid points that are covered by 13 sensors or more. As θ decreases, grid points covered by less than 3 sensors are rare, and the majority of the grid points are covered by 3-5 sensors.

Table 2 shows location precision and expected coverage results. The location algorithm described in Section 2.2. has been tested in the three previous deployments (θ = {0.1, 0.5, 0.9}). Table 2 shows the average location error in meters (distance between estimated and real positions) for 10, 000 random gunshot positions in the simulation area, for each θ value and three levels of variation in sensor clocks. These variations are generated by adding a normal Gaussian random toss to the arrival time at each node, for three different values of σ between 0 and 10 ms. The results show that location accuracy increases with the number of detecting nodes, as well as with a better clock synchronization. In addition, this table provides information on the expected coverage at each scenario. For instance, in the θ = 0.1 case, the probability that one or more nodes detect the event is 0.99973, for two or more nodes it is 0.99777, and so on. Clearly, there must be a compromise between detection accuracy, coverage and deployment cost.

In this paper, we have proposed a gunshot location procedure based on sensor networks and a Pareto front approach to optimize large-scale deployments. The location procedure is based on hyperbolic multi-lateration using data from a synchronized sensor network. We also propose a practical distributed synchronization algorithm for that purpose, with low energy consumption. Sensor network planning follows a Pareto front approach, using a monotone descent method without derivatives that is compatible with realistic optimization functions. Our results are clearly superior to random placement, achieving a 50% cost reduction for 85% coverage. In the two-dimensional scenario, event detection by at least four nodes is required to achieve satisfactory gunshot location accuracy.