# Research on Passive Ranging Technology of Moving Ship Based on Vertical Hydrophone Array
## 2 Basic Theory
### Data Fusion Technology Based on Weighted Least Squares  
Due to the complex ocean environment, the range localization result for a single hydrophone is not accurate. Therefore, we adopt data fusion technology based on the weighted least squares to improve the localization accuracy. Here, we assume that a vertical hydrophone array is composed of \(N\) elements. Each element collects sound pressure information on the ship noise separately. The signal received by each hydrophone is processed by the DEMON spectrum, and then the estimated range is obtained by the sound intensity of the pressure difference. We regard \(\hat{\theta}_{j}\) as the estimated range of the \(j\)-th hydrophone, and the estimation range vector of the vertical hydrophone line array can be written as:  
\[\overline{\mathbf{r}}=[\hat{r}_{1},\hat{r}_{2},\cdots,\hat{r}_{N}]^{T} \tag{19}\]  
The estimated ranges of different sensors are fused to obtain a more precise target location. According to the linear fusion model, the estimated range from the source to the sensors can be denoted as [21]:  
\[\overline{\mathbf{r}}=\mathbf{H}x+\mathbf{e} \tag{20}\]  
where \(x\) denotes the state parameters to be measured. \(\mathbf{e}=[e_{1},e_{2},\cdots e_{N}]^{T}\) represents the \(N\times 1\) dimensional error and satisfies the Gaussian distribution, namely \(E\big{(}\mathbf{e}_{j}=0\big{)}\), \(j=1,2\cdots N\). \(\mathbf{H}\) is the \(N\times 1\) dimensional mapping matrix between \(\overline{\mathbf{r}}\) and \(x\). Generally, \(\mathbf{H}=[1,1,\cdots 1]^{T}\) because of the same type of each hydrophone. However, the estimation range of each hydrophone has a different accuracy due to complex environments in the deep sea. Thus, we adopt the weighted least squares (WLS) method to eliminate the uncertainty. That is, for the high-accuracy element the weight is set to be larger, but for the low-accuracy element the weight is smaller. After using the WLS method, the estimation range becomes more accurate. We denote the weighted coefficient of \(N\) hydrophones as \(\mathbf{w}=[w_{1},w_{2},\cdots,w_{N}]\), which satisfies the relation \(\sum\limits_{j=1}^{N}w_{j}=1,\ 0\leq w_{j}\leq 1\). In addition, the weight needs to satisfy the relationship \(E\big{(}\mathbf{w}^{T}\overline{\mathbf{r}}-x\big{)}=0\). Figure 3: The trajectory of the moving source. Moreover, the estimation error is lower than the variance of estimated ranges of any single element. The sum of the square of weighted errors can be expressed as:  
\[J_{w}(\hat{x})=\left[\overline{r}-\mathbf{H}\hat{x}\right]^{T}\mathbf{W}[\overline{r}- \mathbf{H}\hat{x}] \tag{21}\]  
where \(\mathbf{W}\) is a weight vector and satisfies the expression \(\mathbf{W}=diag(w_{1},w_{2},\cdots,w_{N})\). According to the WLS rule, the optimal weight \(\mathbf{W}\) can be acquired when \(J_{w}(\hat{x})\) reaches the minimum. Take the partial derivative of Equation (21) and set it to zero, namely:  
\[\frac{\partial J_{w}(\hat{x})}{\partial\hat{x}}=-\mathbf{H}^{T}\Big{(}\mathbf{W}+\mathbf{W} ^{T}\Big{)}(\overline{r}-\mathbf{H}\hat{x})=0 \tag{22}\]  
The value of the estimated range after optimal WLS fusion can be written as:  
\[\hat{x}=\Big{(}\mathbf{H}^{T}\mathbf{W}\mathbf{H}\Big{)}^{-1}\mathbf{H}^{T}\mathbf{W}\overline{r}= \frac{\sum\limits_{j=1}^{N}w_{j}\hat{r}_{j}}{\sum\limits_{j=1}^{N}w_{j}}\quad j =1,2\cdots,N \tag{23}\]  
Further, the variance after WLS data fusion is expressed as:  
\[\sigma^{2}=E\Big{[}(x-\hat{x})^{2}\Big{]}=\sum\limits_{j=1}^{N}{w_{j}}^{2}{ \sigma_{j}}^{2} \tag{24}\]  
where \({\sigma_{j}}^{2}\) is the estimated variance of the \(j\)-th sensor. Based on the extreme value theory, the weight corresponding to the minimum total mean square error (MSE) is:  
\[w_{j}=\frac{1/{\sigma_{j}}^{2}}{\sum\limits_{j=1}^{N}\frac{1}{{\sigma_{j}}^{2} }}\quad j=1,2\cdots,N \tag{25}\]  
Substituting Equation (25) into Equation (23), the ultimate range result after fusion is derived.