# Underwater Acoustic Target Tracking: A Review
## 4 Theory of Target Tracking
### State Filtering and Estimation  
Target tracking is a typical uncertainty problem. Due to the development of monitoring and anti-surveillance technologies as well as the improvement of target mobility, the problem of uncertainty in target tracking becomes more serious. The source of uncertainty mainly comes from the uncertainty of target motion represented by process noise, the uncertainty of the source of measurements (measurement noise) and the false noise incurred by multi-target and the dense cluttered environment. Therefore, the essence of target tracking is estimating and predicting the state of the target by filtering algorithms to eliminate the uncertainty associated with the target. According to the dynamic motion model and observation model established, how to choose the appropriate filtering algorithm or improve the existing ones to meet the specific requirements of the underwater target tracking is the key to designing target tracking algorithms. To this end, we will introduce the basic theory of filtering and the commonly used filtering algorithms in underwater target tracking:  
_Bayesian filter_: The core design idea of the Bayesian filter is obtaining the posterior probability by using observation data obtained at the current moment to correct the prior probability density function (PDF) [44]. Bayesian theory considers that the posterior PDF obtained by prior PDF and the current system information can better reflect the characteristics of the system. Therefore, the study of the system should be based on the posterior PDF. The Bayesian filter consists of two processes: prediction and update. The goal of prediction is to obtain the prior PDF by the target motion model. The process of update introduces the measurement value obtained from the observation model into the output of the prediction step to correct the probability so that the posterior probability of the target state is acquired. Let \(z_{1:k}=\{z_{1},z_{2},\cdots,z_{k}\}\) denote the measurements obtained from time instant 1 to \(k\), \(p(x_{0:k}|z_{1:k})\) denote the posterior PDF before time \(k\), and the probability of the initial state of the target is assumed to be known. Since the state vector \(x_{k}\) obeys the first-order Markov process generally, the posterior PDF can be recursively obtained using the measurements \(z_{1:k}=\{z_{1},z_{2},\cdots,z_{k}\}\). Assuming that \(p(x_{k-1}|z_{1:k-1})\) has been obtained, the probability of a one-step prediction is described as:  
\[p(x_{k}|z_{1:k-1})=\int p(x_{k}|x_{k-1})\,p(x_{k-1}|z_{1:k-1})\mathrm{d}x_{k-1} \tag{7}\]  
\(p(x_{k}|x_{k-1})\) in Equation (7) represents the state transition probability. The update process utilizes the observation value obtained at time instant k to update the prior probability. The posterior PDF can be derived from Equations (8) and (9):  
\[p(x_{k}|y_{1:k})=\frac{p(y_{k}|x_{k})\,p(x_{k}|y_{1:k-1})}{p(y_{k}|y_{1:k-1})} \tag{8}\]  
\[p(y_{k}|y_{1:k-1})=\int p(y_{k}|x_{k})\,p(x_{k}|y_{1:k-1})\mathrm{d}x_{k} \tag{9}\]  
where \(p(y_{k}|y_{1:k-1})\) denotes a normalized constant, while \(p(y_{k}|x_{k})\) represents the likelihood probability. In linear conditions, the Bayesian filter can acquire the approximately optimal solution, namely the Kalman filter (KF). While in nonlinear conditions, the approximate optimal solution is hard to acquire owing to the infinite dimensional integral calculation. Then some sub-optimal algorithms are proposed such as the extended Kalman filter (EKF), the unscented Kalman filter (UKF) and the particle filter (PF). _EKF_: This filtering algorithm is developed based on the original KF formulation for nonlinear problems [45]. EKF adopts a Taylor series to linearize the nonlinear part to obtain a sub-optimal estimation. When the maneuvering of the target is not high, EKF can achieve satisfactory results:  
\[ \begin{array}{l}x(k+1)\approx f(k,\hat{x}(k|k))+f^{\prime}_{x}(k)[x(k)-\hat {x}(k|k)]+w(k)\\ \qquad\qquad+\frac{1}{2}\sum\limits_{i=1}^{n}e_{i}[x(k)-\hat{x}(k|k)]^{T}f^{ \prime}_{xx}(k)[x(k)-\hat{x}(k|k)]\end{array}