## 3 The Proposed Algorithm Based on Novel Detection Probability
### An Novel Detection Probability Model  
The design of the underwater target tracking should be linked with a sonar equation [26]. Here, the detection probability model is modeled through the active sonar equation. The active sonar equation is defined as below. \[\mathrm{SL}-2\mathrm{TL}+\mathrm{TS}-(\mathrm{NL}-\mathrm{DI})=\mathrm{DT} \tag{21}\]  
describing source level (SL), one-way transmission loss (TL), target strength (TS), environmental noise level (NL), direction index (DI) and detection threshold (DT). TL can be expressed as a function of distance, indirectly, and the signal-to-noise ratio (SNR) and the detection threshold can also be expressed as a function of distance. According to the theory of sonar detection, the relationship between SNR and detection probability and false alarm probability can be expressed by the receiver operating characteristic (ROC) curve. Therefore, the relationship between detection probability and distance can be modeled by the sonar equation and ROC curve. According to the binary signal detection theory [27], the derivation process is described as follows and uses the two hypotheses as shown below. \[ \begin{split} H_{1}:x(t)&=s(t)+n(t)\\ H_{0}:x(t)&=n(t)\end{split}  \tag{22}\]  
where \(s(t)\) is known signal and \(n(t)\) is the Gaussian noise with mean zero and covariance \(\sigma_{n}^{2}\). The discrete time series \(X\) is obtained by \(\Delta t\) for the sampling interval. The likelihood ratio is denoted \(\lambda(X)\). \[\lambda(X)=\frac{p(X|H_{1})}{p(X|H_{0})}=\exp\ \left(\frac{1}{2\sigma_{n}^{2}} \sum\limits_{k=1}^{N}\left(2x_{k}s_{k}-s_{k}^{2}\right)\right)\ \ \ \ \ \begin{array}{c} \stackrel{{ H_{1}}}{{>}}\\ <\\ H_{0}\end{array} \ \ \ \lambda_{0} \tag{23}\]Simplifying the above Equation (23), we get the new Equation (24). \[\sum_{k=1}^{N}x_{k}s_{k} \begin{array}{c}\stackrel{{ H_{1}}}{{>}}\\ \stackrel{{ H_{0}}}{{<}}\\ \stackrel{{ H_{0}}}{{<}}\end{array} \sigma_{n}^{2}\ln\lambda_{0}+ \frac{1}{2}\sum_{k=1}^{N}s_{k}^{2} \tag{24}\]  
The left definition of the inequality is the detection statistic, and the right side is the test threshold \(\lambda_{0}\), which can be determined by different discriminate criteria. At the same time, the above results are generalized to the general situation, \(\Delta t\to 0,N\rightarrow\infty\), then the summation operation becomes integral operation and the statistical test quantity is \(G=\int_{0}^{T}x(t)s(t)dt\). An equivalent test is shown as blow. \[\int_{0}^{T}x(t)s(t)dt \begin{array}{c}\stackrel{{ H_{1}}}{{>}}\\ \stackrel{{ H_{0}}}{{<}}\\ \stackrel{{ H_{0}}}{{<}}\end{array} \frac{N_{0}}{2}\ln\lambda_{0}+ \frac{1}{2}\int_{0}^{T}s^{2}(t)dt=\frac{N_{0}}{2}\ln\lambda_{0}+\frac{E}{2} \tag{25}\]  
When \(H_{0}\) is true, \(E_{0}(G)=E_{0}\Big{(}\int_{0}^{T}x(t)s(t)dt\Big{)}=0\), \(D_{0}(G)=E_{0}\big{(}G^{2}\big{)}-E_{0}^{2}(G)=\frac{N_{0}E}{2}\). When \(H_{1}\) is true, \(E_{1}(G)=E_{1}\Big{(}\int_{0}^{T}x(t)s(t)dt\Big{)}=0\), \(D_{1}(G)=E_{1}\big{(}G^{2}\big{)}-E_{1}^{2}(G)=\frac{N_{0}E}{2}\). The probability density function of the test statistic \(G\) is obtained under the condition of binary hypothesis, which are shown in Equation (26). \[p(G|H_{0})=\frac{1}{\sqrt{\pi N_{0}E}}e^{-\frac{G^{2}}{N_{0}E}}\qquad p(G|H_{1} )=\frac{1}{\sqrt{\pi N_{0}E}}e^{-\frac{(G-E)^{2}}{N_{0}E}} \tag{26}\]  
Therefore, the false detection probability and detection probability are as below, respectively. \[ \begin{array}{c}P_{f}=\int_{V_{T}}^{\infty}p(G|H_{0})\,dG=\int_{V_{T}}^{ \infty}\frac{1}{\sqrt{\pi N_{0}E}}e^{-\frac{G^{2}}{N_{0}E}}dG\\ P_{d}=\int_{V_{T}}^{\infty}p(G|H_{1})\,dG=\int_{V_{T}}^{\infty}\frac{1}{\sqrt{ \pi N_{0}E}}e^{-\frac{(G-E)^{2}}{N_{0}E}}dG\end{array}  \tag{27}\]  
From the equation above, we can get the detection probability, as defined blow. \[P_{d}=1-\text{N}\Big{(}\text{N}^{-1}\Big{(}1-P_{f}\Big{)}-\sqrt{2\text{SNR}} \Big{)} \tag{28}\]  
where \(\text{N}(\cdot)\) express the standard normal distribution, \(\text{SNR}=E/N_{0}\).