## 4 Discussion
### Multi-Sensor Detection Information Fusion Method  
In this paper, we adopt a detection information fusion method based on adaptive weight allocation for multi-sonar detection probability fusion processing. We consider that high detection probability corresponds to high confidence and should be given higher weights. Therefore, the proposed method allocates weights based on real-time detection probability values from multiple sonars. The detection probability is calculated through the signal excess, and there is a one-to-one correspondence between the two. The relationship between the signal excess and detection probability is illustrated in Figure 16.  
According to Figure 16, when the signal excess is \(-10\) dB, the detection probability is almost \(0\); when the signal excess is \(15\) dB, the detection probability is almost \(1\). The detection probability is directly proportional to the signal excess. As the signal excess increases, the detection probability also increases, and the corresponding confidence level also increases. In this work, we assume a linear relationship between the signal excess and confidence level, that is, when the signal excess is less than or equal to \(-10\) dB, the confidence level is \(0\); when the signal excess is greater than or equal to \(15\) dB, the confidence level is \(1\). The relationship between the signal excess and confidence level is illustrated in Figure 17.  
Figure 16: Relationship curve between the signal excess and detection probability.  
Figure 15: Cumulative detection probability mutual information diagram between each sonar and the whole under two working modes.  
When the signal excess and detection confidence corresponding to each sonar are known, we normalize them according to the calculation method of Formula (8) and comprehensively consider the influence of signal excess and detection confidence on the weights. We use the weighted sum method to obtain the formation detection probability value.  
To quantify the advantages of adaptive weight allocation, we compare it with the average weighting fusion method. We take a discrete time point on a certain target trajectory as an example and explain it via variance as the evaluation index.  
At a discrete time point on the trajectory of a certain target, the relative situation between the sonar and the target is known. At this point, the signal excesses of the three sonars for the target are \(-5\) dB, \(0\) dB, and \(5\) dB, respectively. The detection probabilities for the target are \(0.1518\), \(0.5\), and \(0.7937\), respectively. The detection probability variances for the three sonars are \(\delta_{1}^{2}=0.05\), \(\delta_{2}^{2}=0.03\), and \(\delta_{3}^{2}=0.01\), respectively.  
(1) If the average weighting method is used for multi-sonar detection information fusion processing, the weight assigned to each sensor is: \(weight_{A1}=weight_{A2}=weight_{A3}=1/3\). The variance of the fusion result is: \(\delta_{A}^{2}=\sum_{i=1}^{3}weight_{Ai}^{2}\times\delta_{i}^{2}\), and the calculated value is \(0.01\).  
(2) If the method proposed in this paper is used for multi-sonar detection information fusion processing, according to the information fusion process in Section 2.3, the weight assigned to each sensor is: \(weight_{B1}=0.0667;weight_{B2}=0.1373;weight_{B3}=0.7960\). The variance of the fusion result is: \(\delta_{B}^{2}=\sum_{i=1}^{3}weight_{Bi}^{2}\times\delta_{i}^{2}\), and the calculated value is \(0.0071\).  
Based on the simulation results, we can conclude that the detection information fusion method based on adaptive weight allocation has lower variance and better stability. However, in this method, we assume a simple linear relationship between signal excess and confidence level, which has certain limitations and requires further research on the mathematical model between them.