# Underwater Acoustic Target Tracking: A Review
## 4 Theory of Target Tracking
### Target Model Construction  
Most of the target tracking problems are investigated based on a model, in other words, based on two descriptions. One is the target behavior represented by the dynamic motion model; the other is the observation of the target called the observation model. The establishment of underwater target dynamic motion model and observation model is the precondition of target tracking [42]. The dynamic motion model and observation model can be collectively referred to as the state space model. The commonly used one is as Equation (1) shows:  
\[\left\{\begin{array}{c}x_{k+1}=f\left(x_{k},w_{k}\right)\\ z_{k}=h\left(x_{k},v_{k}\right)\end{array}\right. \tag{1}\]  
where \(x_{k}\), \(z_{k}\), \(w_{k}\) and \(v_{k}\) are the state, measurement, process noise and measurement noise vector at time instant \(k\) respectively. The transfer function \(f(\cdot)\) is determined by the target motion, and the measurement function \(h(\cdot)\) is constructed by the corresponding relationship between measurements obtained from sensors and the state of the target. For example, if the measurement information is the bearings obtained from two passive sensors as Figure 4 shows.  
The relationship between the measured bearings and the position of the target can be described as Equation (2) shows:  
\[\begin{array}{c}B1=\tan\left(\frac{y(t)-Y_{1}}{x(t)-X_{1}}\right)\\ B2=\tan\left(\frac{y(t)-Y_{2}}{x(t)-X_{2}}\right)\end{array} \tag{2}\]  
Figure 4: The scheme of obtaining bearing information.  
The measurement function \(h(\cdot)\) can be represented by:  
\[h(x_{k},v_{k})=\left[\begin{array}{c}\tan^{-1}(\frac{y(t)-Y_{1}}{x(t)-X_{1}})\\ \tan^{-1}(\frac{y(t)-Y_{2}}{x(t)-X_{2}})\end{array}\right]+\left[\begin{array} []{c}Bn1\\ Bn2\end{array}\right] \tag{3}\]  
The second term to the right of the Equation (3) represents the measurement noise \(v_{k}\). The accuracy of target motion model directly affects the tracking precision. If the model is far different from the actual situation, it may lead to the divergence of the subsequent filtering process [43]. However, it makes no sense to put numerous efforts into modeling as accurately as possible, because the more sophisticated the model is, the more complex it is and the more difficult it is to implement the filtering process. For underwater targets such as submarines, the changing course and speed are the most common maneuver types. Thus, in the marine environment, there are three frequently used target motion models: the constant velocity (CV) model, the constant acceleration (CA) model and the turning model. To provide readers with a preliminary understanding of target motion models, the CV model will be introduced in detail.  
In three-dimensional space, the state of a target at time \(k\) can be represented as \(x_{k}=(x,v_{x},y,v_{y},z,v_{z})\), where \((x,y,z)\) is the position vector and \((v_{x},v_{y},v_{z})\) is the velocity vector. The state equation is described in discrete time with sampling interval \(T\) as Equation (4) shows:  
\[x_{k+1}=F_{k}x_{k}+Q_{k}w_{k}=\left[\begin{array}{cccccc}1&T&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&T&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&T\\ 0&0&0&0&0&1\end{array}\right]x_{k}+\left[\begin{array}{cccc}\frac{1}{2}T^{2} &0&0\\ T&0&0\\ 0&\frac{1}{2}T^{2}&0\\ 0&T&0\\ 0&0&\frac{1}{2}T^{2}\\ 0&0&T\end{array}\right]w_{k} \tag{4}\]  
where the process noise \(w_{k}\in R^{3}\) can be seen as the random acceleration in three dimensions obeying zero-mean Gaussian distribution. If we take the \(y\) component as an example, the range change in \(y\) component and the velocity change in \(y\) component as Equations (5) and (6) show:  
\[y_{k+1}=y_{k}+v_{k,y}\times T+\frac{1}{2}T^{2}\times w_{k,y} \tag{5}\]  
\[v_{k+1,y}=v_{k,y}+T\times w_{k,y} \tag{6}\]