# Acoustic Localization for a Moving Source Based on Cross Array Azimuth
## 3 Localization Analysis of Double Arrays
### Precision Analysis for Localization  
Compared to the single array, the variable \(L\) has been added to the localization expression for double arrays. However, the time delay remains the key variable. As the structure and sensors of the two arrays are identical, the standard time delay errors of both are equal (\(\sigma_{\tau}=\sigma_{\tau 12}=\sigma_{\tau 13}=\sigma_{\tau 14}=\sigma_{\tau 15}= \sigma^{{}^{\prime}}_{\tau 12}=\sigma^{{}^{\prime}}_{\tau 13}=\sigma^{{}^{\prime}}_{ \tau 14}=\sigma^{{}^{\prime}}_{\tau 15}\)). As the direction was determined by Array 1, the azimuth error was analyzed according to Equation (24). Meanwhile, range error \(\sigma_{R}\) is influenced by azimuth error \(\sigma_{\varphi}\) and array distance \(L\). Range precision is obviously determined by the azimuth precision according to Equation (24). Range precision can be expressed with error transmission as  
\[\sigma_{R}=\frac{\partial R}{\partial\tau_{i}}=\frac{\partial R}{\partial \varphi}\frac{\partial\varphi}{\partial\tau_{i}}=(\frac{\partial R}{\partial \varphi_{1}}+\frac{\partial R}{\partial\varphi_{2}})\frac{\partial\varphi}{ \partial\tau_{i}} \tag{25}\]  
\[\sigma_{R}=\frac{\sqrt{2}cr(\sin(\varphi_{1}+\varphi_{2})+\sec\varphi_{1} \cos\varphi_{2})}{\sin(\varphi_{1}-\varphi_{2})}\sigma_{\tau} \tag{26}\]  
Applying the \(\sin\) theorem on \(\Delta\)TO\({}_{1}\)O\({}_{2}\) gives:  
\[\frac{\sin\angle T\Omega_{2}O_{1}}{R}=\frac{\sin\angle\Omega_{2}TO_{1}}{L} \tag{27}\]  
\[\frac{\sin\varphi_{2}}{R}=\frac{\sin(\varphi_{2}-\varphi_{1})}{L} \tag{28}\]  
\[R=\frac{\L\sin\varphi_{2}}{\sin(\varphi_{2}-\varphi_{1})} \tag{29}\]  
Taking partial derivative in Equation (29) yields:  
\[\sigma_{R}=\frac{\sqrt{2}cL\cos\varphi_{2}}{D\sin(\varphi_{2}-\varphi_{1})} \sigma_{\tau} \tag{30}\]Substituting in Equation (29) gives  
\[\sigma_{R}=\frac{\sqrt{2}cR}{D\tan\varphi_{2}}\sigma_{\tau} \tag{31}\]  
In Equation (31), range precision is determined by sound velocity \(c\), array size \(D\), azimuth \(\varphi_{2}\) of Array 2, error of time delay \(\sigma_{\tau}\), and range \(R\). The distribution of the range error is shown in Figure 5. In Figure 5a, \(R\) = 100 m, \(c\) = 343 m/s, \(\sigma_{\tau}\) = 100 \(\upmu\)s. In Figure 5b, \(D\) = 3 m, \(R\) = 100 m, \(c\) = 343 m/s. Figure 5 reveals that \(\varphi_{2}\) affected range precision substantially. In (1\({}^{\circ}\), 20\({}^{\circ}\)) and (160\({}^{\circ}\), 179\({}^{\circ}\)), the range error remains very high. In (20\({}^{\circ}\), 160\({}^{\circ}\)), the error was much lower and acceptable. In Figure 5a, array size \(D\) significantly affected precision when \(D\) < 3 m. Error was 10.78% when \(\varphi_{2}\) was 8.12\({}^{\circ}\). The error reduced as angle \(\varphi_{2}\) is increased. When \(\varphi_{2}\) was 15.24\({}^{\circ}\), range error was 5.65% and it reduced to 3.78% as \(\varphi_{2}\) was increased to 22.36\({}^{\circ}\). In Figure 5b, distribution was the same to Figure 5a, and the error stayed below 5% when \(\varphi_{2}\) is above 20\({}^{\circ}\). Since array distance \(L\) is independent to time \(t\), the error expression of \(L\) is:  
\[\sigma_{R}^{L}=\frac{\partial R}{\partial L} \tag{32}\]  
Take partial derivatives of \(L\) in Equation (24):  
\[\sigma_{R}^{L}=\frac{k_{2}\sqrt{1+{k_{1}}^{2}}}{2(k_{2}-k_{1})}=\frac{k_{2}k _{1}\sqrt{1+\frac{1}{k1^{2}}}}{2(k_{2}-k_{1})} \tag{33}\]  
Substituting \(\left\{ \begin{array}{l}y=R\sin\varphi_{1}\\ k_{1}=\tan\varphi_{1}\end{array} \right.\) into Equation (33) gives  
\[\sigma_{R}^{L}=\frac{R}{L} \tag{34}\]  
Equation (34) was the expression of the range error with factor \(L\). The error was affected by range \(R\) and array distance \(L\). The relative error is \(1/L\); therefore, the relative error theoretically stays constant when \(L\) was designated. The error is below 6.67% when \(L\geq 15\) m. The distribution is shown in Figure 6.  
Figure 5: Distributions of the range error of \(R\) and \(D\). In summary, the locating ability of the double array is good.