# A Novel Ship-Tracking Method for GF-4 Satellite Sequential Images
## 2 Proposed Method Framework
### Ship Tracking  
#### 2.3.1 Ship Modeling  
To track ships using GF-4 satellite, an appropriate ship modeling is required before ship tracking. As the resolution of GF-4 satellite is far lower than that of LEO imaging satellites, ships are small and weak, and it is difficult to extract rich features. In this paper, only the target position and target amplitude are used for tracking in each frame. The measurement at time \(k\) can be expressed as  
\[z_{k}=(lon_{k},lat_{k},amp_{k})^{\mathsf{T}} \tag{9}\]  
where \(lon_{k}\) and \(lat_{k}\) are the longitude and latitude of the measurement in geographic coordinates, \(amp_{k}\) is the maximum amplitude of the target in the image. The state of a ship is defined at time \(k\) as  
\[\mathbf{x}_{k}=(\lambda_{k},\dot{\lambda}_{k},\varphi_{k},\dot{\varphi}_{k},a_{k}) ^{\mathsf{T}} \tag{10}\]  
where \(\lambda_{k}\), \(\varphi_{k}\) and \(\dot{\lambda}_{k},\dot{\varphi}_{k}\) represent the position and velocity components along longitude and latitude directions, respectively. The basic unit of longitude, latitude and course is the degree (\({}^{\circ}\)), and the velocity is determined in knots (kn). \(a_{k}\) is the amplitude. The measurement equation information is expressed as  
\[\mathbf{z}_{k}=\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{w}_{k},\mathbf{w}_{k}\sim\text{N}(\mathbf{0}, \mathbf{R}_{k}) \tag{11}\]  
where  
\[\mathbf{H}_{k}=\left[ \begin{array}{ccccc}1&0&0&0&0\\ 0&0&1&0&0\\ 0&0&0&0&1\end{array} \right] \tag{12}\]  
\[\mathbf{R}_{k}=\text{diag}(\sigma_{p}^{2},\sigma_{p}^{2},\sigma_{a}^{2}) \tag{13}\]  
where \(\mathbf{H}_{k}\) is the observation matrix, \(\mathbf{R}_{k}\) is the noise matrix, and \(\sigma_{p}\) is the standard deviation of positioning error after correction of position offset both in longitude and latitude. \(\sigma_{a}\) is the standard deviation of the amplitude. The speed \(s_{k}\) and the course \(\theta_{k}\) over ground (relative to true north) can be obtained that  
\[s_{k}=\sqrt{\dot{\lambda}_{k}{}^{2}+\dot{\varphi}_{k}{}^{2}} \tag{14}\]\[\theta_{k}=\left\{ \begin{array}{l}\arctan(\dot{\lambda}_{k}/\dot{\varphi}_{k}), \hskip 14.226378 pt\dot{\lambda}_{k}\geq 0,\dot{\varphi}_{k}>0\\ 360^{\circ}+\arctan(\dot{\lambda}_{k}/\dot{\varphi}_{k}),\dot{\lambda}_{k}<0, \dot{\varphi}_{k}>0\\ 180^{\circ}+\arctan(\dot{\lambda}_{k}/\dot{\varphi}_{k}),\hskip 14.226378 pt\dot{ \varphi}_{k}<0\\ 90^{\circ},\hskip 28.452756 pt\dot{\lambda}_{k}>0,\dot{\varphi}_{k}=0\\ 270^{\circ},\hskip 28.452756 pt\dot{\lambda}_{k}<0,\dot{\varphi}_{k}=0\end{array} \right. \tag{15}\]  
The state transition model in the dead reckoning of a rhumb line track from one point \((\lambda_{1},\varphi_{1})\) to the other point \((\lambda_{2},\varphi_{2})\) can be expressed as  
\[\left\{ \begin{array}{l}\varphi_{2}=\varphi_{1}+\dot{\varphi}T/60\\ \lambda_{2}=\lambda_{1}+\lambda T\sec(\varphi_{1})/60\end{array} \right. \tag{16}\]  
where \(T\) is the time interval, \(1/60\) is the factor used to convert the length (nm) to degree (\({}^{\circ}\)) since \(1^{\circ}\) of latitude equals approximately \(60\) nm. The motion model can be described as  
\[x_{k}=\mathbf{f}_{k-1}(x_{k-1})+\mathbf{G}_{k-1}\mathbf{\upsilon}_{k-1} \tag{17}\]  
where \(\mathbf{f}_{k-1}\) is state transition function, \(\mathbf{G}_{k-1}\) and \(\mathbf{\upsilon}_{k-1}\) is the process noise matrix and process noise, respectively. We define \(\mathbf{\upsilon}_{k-1}=\left(\ddot{\lambda}_{k-1},\ddot{\varphi}_{k-1},0 \right)^{T}\) and \(\mathbf{\upsilon}_{k-1}\sim\mathrm{N}(0,\mathbf{B}_{k})\), which represents a multivariate Gaussian function with zero mean and covariance \(\mathbf{B}_{k}\),  
\[\mathbf{B}_{k}=\mathrm{diag}(\mathbf{\upsilon}_{\mathbf{\upsilon}}^{2}, \mathbf{\upsilon}_{\mathbf{\upsilon}}^{2},0) \tag{18}\]  
where \(\mathbf{\upsilon}_{\mathbf{\upsilon}}\) is the process noise standard deviation. Derived from Equation (17), the state transition equation is  
\[x_{k}=\left[