## 4 Modeling of Acoustic Target Tracking
### Geometric Representation of Acoustic Target Tracking  
In 2-dimensional space and ideal environmental conditions we can assume that sound propagates in a circular form from a target's location with respect to time. If we consider the third dimension as time, the propagation of sound waves in 2-dimensional space with respect to time makes a vertical circular right cone. Figure 3 shows a vertical right circular double cone, half of its aperture angle, and the coordinates of its apex point. Each piece of double cones that are placed apex to apex is called a nappe. Equation (15) shows the general equation of a vertical circular right double cone whose apex point has \((x_{o},y_{o},z_{o})\) coordinate:  
\[\frac{(X-x_{o})^{2}+(Y-y_{o})^{2}}{a^{2}}=\frac{(Z-z_{o})^{2}}{c^{2}} \tag{15}\]  
\(X\), \(Y\) and \(Z\) are free variables in Equation (15), and half of the opening angle of the cone is computed as in Equation (16):  
\[\theta=tan^{-1}\left(\frac{a}{c}\right) \tag{16}\]  
We need to solve simultaneous equations in Equation (2) to compute the time when and the location where a target object had generated a sound. Solving these simultaneous equations can be done using numerical methods. Algebraic geometry can also be used preferably to solve this problem. Algebraic geometry can convert degree two equations to degree one equations and ease the solution of simultaneous equations. It also permits us to visualize and interpret the solution method and results. We can better represent the target tracking problem as a geometric problem if we rewrite Equation (2) in the form of Equation (17):  
Figure 3: A vertical right circular double cone. \[\left\{ \begin{aligned} \frac{(x_{1}-x)^{2}+(y_{1}-y)^{2}}{v^{2}}& =(t_{1}-t)^{2}\\ \frac{(x_{2}-x)^{2}+(y_{2}-y)^{2}}{v^{2}}&=(t_{2}-t)^ {2}\\ \frac{(x_{3}-x)^{2}+(y_{3}-y)^{2}}{v^{2}}&=(t_{3}-t)^ {2}\end{aligned} \right. \tag{17}\]  
Let us assume time to be the third dimension that grows in upward direction. Now we can visualize sound propagation in 2-dimensional space with respect to time as a 3-dimensional right circular cone with a specific aperture angle. We have a cone equation and three known points on the surface of its up nappe and we want to determine the coordinate of its apex point that is \((x,y,t)\) in Equation (17). Since sound is broadcasted in 2-dimensional space, motes can sense it after some time delay. Motes cannot detect the sound of a target object before the sound is made. For this reason, 3-dimensional sensing information of sensing nodes belongs to the up nappe. Figure 4 shows a double right cone that is generated with sound propagation in 2-dimensional space. Logically, the up nappe is of interest to target tracking, while degree two equations in simultaneous equations in Equation (2) contain the down nappe too. Aperture of this cone is the angle \(2\theta\) and is related to the sound propagation speed in space and is considered constant and equal to 344.0 m/s in this paper. The aperture of sound propagation cone is calculated as follows:  
\[tan(\theta)=v,\qquad v=344.0\ m/s \tag{18}\]  
Point \((x,y,t)\) is the spatio-temporal information of target object and represents the target at time \(t\) was at position \((x,y)\) and has generated a sound. Sensor nodes \(i\) where \(i=\) 1,2,3 are distributed on a 2-dimensional surface with known \((x_{i},y_{i})\) coordinates. If we draw vertical lines parallel with the time axis, the points at which these lines meet with the surfaces of the up nappe represent the times at which t  
Figure 4: Sound propagation with respect to time in 2-dimensional space. sensor nodes hear the sound of the target object. Sensors can hear the sound of target after some time delay that directly depends on their distances from the target. Information that is sent by each sensor node to its neighboring nodes for computing the spatio-temporal information of the target is a record that can be shown as a point \(P_{t}(x_{i},y_{i},t_{i})\) in Figure 4, where \(t_{i}\) represents the time at which sensor \(i\) hears the sound of the target.