- freely available
Sensors 2009, 9(12), 9493-9512; doi:10.3390/91209493
2. Gunshot Location
2.1. System architecture
- Sensor nodes: The sensor nodes in known positions are equipped with the necessary hardware for the detection of acoustic events. They can discriminate between normal and shot segment classes in audio streams. When a sensor node detects a sound event, it transmits a packet with information about the type of sound event and a sound timestamp to a special node, the sink.
- Sink nodes: Sink nodes collect the packets sent by sensor nodes and deliver them to the GIS server to calculate the position of the sound event. Sink nodes may be sensing nodes as well.
- GIS server: Using the information from the sink nodes, the GIS server estimates the position of the acoustic event by means of a hyperbolic method, described in Section 2.2.
2.2. Source location procedure
- The speed of sound varies depending on altitude, humidity and air temperature. As we have mentioned, multi-path propagation affects the accuracy of acoustic signal detection. Single spread-spectrum techniques such as those in  largely mitigate it.
- The microphone directionality or polar pattern affects the result.
- The clock drift may drastically vary in time due to environmental temperature and humidity changes. In Section 2.3. we propose an approach to reduce sensor clock deviations.
2.3. Synchronization schema
- Level discovery: This step is similar to the level discovery stage in TPSN . Before the synchronization process takes place, the network has to organize itself as a hierarchical tree, beginning at a root node (in our case we choose the sink). According to the minimum number of hops to the sink, a level is assigned to each node (level 0 to the root). To compute the tree, the process starts at the root, broadcasting a level discovery packet to the nodes at level 0. The nodes that receive this packet are marked as children of the root node, and they set their level to 1. The nodes ignore further level discovery packets with greater or equal level numbers. Then, level 1 nodes broadcast their level discovery packets, and so on. Note that this process also permits discover of optimal communication paths (in number of hops) to the root, and, thus, it is valid for network routing.
- Synchronization: Once the hierarchical network structure is completed, the synchronization process may start. In general, level k nodes synchronize their children (of level k + 1).Besides its own local clock, a sensor node will maintain an estimation of its synchronizer node clock in the upper hierarchical level. The approximation consists of calculating the regression line of those two clocks. Previously, the level k node receives several synchronized time-stamps of level k – 1 (see Figure 4), which are broadcast following the tree structure that was created at the level discovery step. Figure 4 shows the regression line used to calculate the parent node clock in a level k node. Value αk represents the clock offset at reference time t = 0, and the slope βk is the rate of change (clock drift) of the local clock.Once a node detects a gunshot, it sends the event to its parent node in the upper level, according to the parent time clock. After one or more hops, level 0 (sink node) will receive estimations of the detection time that are synchronized with the sink clock, from one or more level 1 nodes. This way, local clock exchanges do not spend power. Since clock drift varies slowly, the regression line must only be calculated every 6 or 8 hours, according to our tests with MicaZ motes. Only large temperature variations affect the regression line slope, requiring node re-synchronization.
3. Optimization Model
- ω1(s, x) corresponds to propagation distance through wood space.
- ω2(s, x) corresponds to propagation distance through open space.
4. Solving the Optimization Model
4.1. Alternative approaches
4.2. Derivative-free unconstrained minimization
- A1. f(S) : IR2p → IR is bounded below, and remains in a compact set,
- A2. f(S) has directional derivatives f′(S, D) everywhere defined:
- A3. The unit directions D1, …, Dn positively span R2p.
- A4. The function ν(·) : IR+ → IR+ is little-o of τ, that is: lim ν(τ)/τ = 0
- A5. The reference values are upper bounds of f(·), i.e., φi ≥ f(Si) for all i, and decrease sufficiently after a given finite number of successful iterations.
- A6. f(S) is strictly differentiable or locally convex at all limit points of the sequence generated by the algorithm.
5. Numerical Tests
- Δ = 0.1 Km, i.e., the method stops when the maximum sensor displacement is under 100 m.
- ετ = 1
- τ = 5
- max_success= 40 (20%p), but we do not allow τ > 8.
- We initially set φ to the value of the objective function at the starting point.
- α = 0, i.e., we perform a monotone derivative-free search.
- Finally, we set ν(τΔ) = 0.001(τΔ)2.
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|Parameters: ετ, μ, τ, φ|
|Get S, let fS = f(S)|
|DO success= 0|
|Choose Dk, k = 1, …, n that positively span IR2p|
|FOR j = 1 TO n|
|[Z fZ] = INTERPOLATE (S, Dk, τΔ)||Remark 2|
|α = min(τ, α), φ = fS + α(φ − fS)||Remark 3|
|IF (fZ ≤ ϕ − ν(τΔ))|
|S = Z, fS = fZ|
|φ = fS||Remark 4|
|IF (success>max_success)||Remark 5|
|τ = τ + 1|
|τ = τ − 1|
|WHILE (τ > ετ)|
|>1 [0.99973%]||>2 [0.99777%]||>3 [0.99448%]||>4 [0.75258%]||>5 [0.45608%]|
|θ = 0.1||sinc. < 1ms||64.40||64.40||64.40||5.30||1.60|
|>1 [0.99988%]||>2 [0.99730%]||>3 [0.98856%]||>4 [0.70991%]||>5 [0.43301%]|
|θ = 0.5||sinc. < 1ms||139.20||138.90||123.50||8.30||0.00|
|>1 [0.64476%]||>2 [0.59659%]||>3 [0.57273%]||>4 [0.50632%]||>5 [0.47687%]|
|θ = 0.9||sinc. < 1ms||114.00||66.80||20.20||11.00||1.50|
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