## 2 Research on the Collaborative Search Efficiency Evaluation Model of Multi-Sonar System
### Multi-Sensor Detection Information Fusion Model  
Information fusion refers to the comprehensive processing of information from multiple sensors or data sources to obtain more comprehensive and accurate results. It is of great significance in improving the detection accuracy of sensor systems, enhancing system robustness, and improving decision reliability [30; 31; 32; 33; 34; 35]. The multi-sensor fusion center fuses the detection results reported by each sensor to obtain the overall detection probability for the target. The detection probability values of each sensor can be fused into the overall detection probability value via the following methods:  
(1) Product method: This method is suitable for scenarios where each sensor detection process is an independent event [36]. \[Pd=1-\prod_{i=1}^{N}(1-Pd_{i}) \tag{4}\]  
(2) Weighted average method: Assign a certain weight to each detection probability, and then calculate the weighted average probability. \[Pd=\omega_{1}Pd_{1}+\omega_{2}Pd_{2}+\cdots+\omega_{i}Pd_{i} \tag{5}\]  
(3) Maximum value method: Take the maximum detection probability of each sensor as the overall detection probability. \[Pd=\max\{Pd_{1},Pd_{2},\cdots,Pd_{i}\} \tag{6}\]  
The collaborative search process of multi-sonar system has the following characteristics: (1) The scene situation changes in real time, that is, at any time, the signal excess received by each sonar changes in real time; (2) At each moment, each sonar has a detection probability for the target, and each detection probability corresponds to a certain confidence level. Generally, the larger the signal excess, the higher the confidence level of the detection result. At each moment, there is a certain correlation between the detection results of each sonar on the target. On the basis of the above characteristics, we propose a multi-sensor detection probability fusion method based on dynamic adaptive weight calculation. The flowchart of this fusion method is shown in Figure 4.  
Figure 3: Schematic diagram of the pulse width blind zone. The specific calculation steps are as follows:  
(1) Dynamic calculation of the confidence level under uncertain situation conditions  
In real-time situations, the signal excess is first obtained based on the sonar equation. Then, we define a relationship between signal excess and confidence level, which can be empirical, linear, or nonlinear. We assume a simple linear relationship between the signal excess and confidence level: \(C_{i}=aSE_{i}+b\), where \(C_{i}\) represents the detection confidence level of the \(i\)-th sensor, \(SE_{i}\) represents the signal excess of the \(i\)-th sensor, \(a\) or \(b\) represents the coefficient. The detection confidence level should meet the requirement that \(0\leq C_{i}\leq 1\), if \(C_{i}\) is less than 0, it should be uniformly calculated as 0, if \(C_{i}\) is greater than 1, it should be uniformly calculated as 1. Therefore, to be precise, the confidence level is a 'piecewise function', expressed as follows:  
\[C_{i}= \begin{cases}0,&SE_{i}\leq-\dfrac{b}{a}\\ aSE_{i}+b,&-\dfrac{b}{a}\leq SE_{i}\leq\dfrac{1-b}{a}\\ 1,&SE_{i}\geq\dfrac{1-b}{a}\end{cases}  \tag{7}\]  
(2) Normalization of the signal excess and detection confidence level  
When normalizing a set of signal excesses, it is inevitable that some signal excesses may be less than 0. To solve this problem, the first step is to perform 'exponential power' processing. \[ \begin{cases}SE^{\prime}_{i}=\dfrac{e^{S_{E_{i}}}}{\sum_{i=1}^{n}e^{S_{E_{i}}} }\\ C^{\prime}_{i}=\dfrac{\bar{C}_{i}}{\sum_{i=1}^{n}\bar{C}_{i}}\end{cases}  \tag{8}\]  
Figure 4: Flow chart of the multi-sensor detection probability fusion method based on dynamic adaptive weight calculation. (3) Comprehensive weight calculation of each sensor  
The weighted sum method is used to obtain the weights of each sensor, namely \(W_{i}=\rho SE_{i}^{\prime}+(1-\rho)C_{i}^{\prime}\), where \(\rho\) represents a fixed weight coefficient, \(0<\rho<1\).