## 2 Research on the Collaborative Search Efficiency Evaluation Model of Multi-Sonar System
### Sonar Detection Model  
We assume that the sonar has detection blind spots within a certain opening angle at both the head and tail during the working process, as shown in Figure 2[18; 19].  
Figure 1: Schematic diagram of the target motion model.  
The sonar detection probability is closely related to various factors such as target radiation noise, self-noise, marine environmental noise, relative situation, blind zone location, etc. Based on the sonar equation, the signal excess is calculated to solve the detection probability. The relationship between the signal excess and detection probability can be expressed as [20]:  
\[\begin{cases}SNR=\frac{\lg P_{f}}{\lg P_{d}}-1\\ \frac{\lg P_{f}}{\lg P_{d}}=10^{\frac{1}{(SNR_{0.5}+0.1+0.1+0.1)}}\end{cases} \tag{2}\]  
where \(P_{f}\) represents the false alarm probability, usually taken as \(10^{-4}\), and \(SNR_{0.5}\) is the signal-to-noise ratio obtained when \(P_{d}\) equals 0.5. \(SE\) represents the signal excess, which can be calculated using sonar equation.  
In the passive working mode of sonar, when noise is the main interference type, the signal excess is expressed as [21]: \(SE=SL-TL-NL-DT+DI\). Where \(SL\) represents the target radiation sound source level, \(TL\) represents the transmission loss between the target and the receiver, \(NL\) represents the marine environmental noise level, \(DT\) represents the sonar detection threshold, and \(DI\) represents the receiving directionality index. Among them, \(TL\) is calculated according to the empirical formula: \(20 lgr+\alpha\cdot r\), where \(r\) represents the distance between the target and the receiver, \(\alpha\) represents the attenuation coefficient, and its relationship with frequency is expressed as follows [22]:  
\[\alpha=\frac{0.1f^{2}}{1+f^{2}}+\frac{40f^{2}}{4100+f^{2}}+2.75\times 10^{-4}f^{ 2}+0.003 \tag{3}\]  
In the active working mode of sonar [23], when noise is the main type of interference, the signal excess is expressed as: \(SE=SL_{1}-2 TL+TS-NL-DT+DI\). Among them, \(SL_{1}\) represents the emission sound source level, \(TS\) represents the target strength, which is strongly correlated with the real-time situation and \(TS\) is constantly changing. The meanings of the other parameters are the same as those in the previous text.  
In the sonar dual-base working mode [24; 25; 26; 27; 28], when noise is the main type of interference, the signal excess is represented as: \(SE=SL_{1}-TL_{1}-TL_{2}+TS-NL-DT+DI\). Among them, \(TL_{1}\) represents the transmission loss between the transmitting node and the target, \(TL_{2}\) and represents the transmission loss between the target and the receiving node. The meanings of the other parameters are the same as those in the previous text.  
In the sonar dual-base working mode, in addition to the detection blind zone at the head and tail positions, there is also a pulse width blind zone, as shown in the gray area in Figure 3[29]. Assuming that the pulse width of the actively emitted sound wave is \(\tau\), the time required for the sound wave to transmit from the emitting node to the target is \(\tau_{TS}\), the time required for the sound wave to transmit from the target to the receiving node is  
Figure 2: Schematic diagram of the sonar detection range.  
\(\tau_{SR}\), and the time required for the sound wave to directly reach the receiving node from the emitting node is \(\tau_{TR}\), the condition of the pulse width blind zone is: \(\tau_{TS}+\tau_{SR}\leq\tau+\tau_{TR}\).