## 4 Information Update and Fusion
### Target Probability Map  
In the multi-UAV system, each UAV \(U_{i}\) keeps the target probability map \(\mathcal{P}_{i,t}\) of the whole mission region, which can be represented as :  
\[\mathcal{P}_{i,t}=\Big{[}\mathcal{P}_{i,t}^{x,y}\Big{]}_{L_{x}\times L_{y}}. \tag{9}\]  
\(\mathcal{P}_{i,t}^{x,y}\in[0,1]\) indicates target existence probability in cell \(G_{x,y}\) at timestep \(k\) in the understanding of UAV \(U_{i}\). Initially, \(\mathcal{P}_{i,0}^{x,y}=0.5\), which means that there is no prior information about the distribution of targets.  
Fundamentally, every UAV should possess the capability to autonomously modify its target probability map solely relying on its own observations, without the need for information exchange with neighbors. This would ensure that the multi-UAV system remains robust to any disruption in communication. Therefore, we will first introduce the target probability update by observations for each individual UAV without cooperation. Bayesian rule [39; 40] is most popular technique to update the probability map based on observations, which is represented as  
\[\begin{split}\mathcal{P}_{i,t}^{x,y}&=P\Big{(} \theta_{x,y}=1\ |\ Z_{i,t}^{x,y}\Big{)}\\ &=\frac{P\Big{(}Z_{i,t}^{x,y}\ |\ \theta_{x,y}=1\Big{)} \mathcal{P}_{i,t-1}^{x,y}}{P\Big{(}Z_{i,t}^{x,y}\ |\ \theta_{x,y}=1\Big{)} \mathcal{P}_{i,t-1}^{x,y}+P\Big{(}Z_{i,t}^{x,y}\ |\ \theta_{x,y}=0\Big{)} \Big{(}1-\mathcal{P}_{i,t-1}^{x,y}\Big{)}}\\ &=\left\{\begin{array}{ll}\frac{P_{0}\mathcal{P}_{i,t-1}^{x,y} }{P_{0}\mathcal{P}_{i,t-1}^{x,y}+P_{t}\big{(}1-\mathcal{P}_{i-1}^{x,y}\big{)}},&\text{if }Z_{i,t}^{x,y}=1\\ \frac{(1-P_{0})\mathcal{P}_{i,t-1}^{x,y}}{(1-P_{0})\mathcal{P}_{i,t-1}^{x,y}+( 1-P_{t})\big{(}1-\mathcal{P}_{i-1}^{x,y}\big{)}},&\text{if }Z_{i,t}^{x,y}=0\\ \mathcal{P}_{i,t-1}^{x,y},&\text{otherwise}.\end{array}\right.\end{split} \tag{10}\]It is evident that the probability map update does not follow a linear pattern. To reduce computational complexity, a linear update method is introduced, which is initially suggested in [41]. Firstly, Equation (10) is equivalent to  
\[\frac{1}{\mathcal{P}_{i,t}^{x,y}}-1=\left\{\begin{array}{ll}\frac{P_{F}}{P_{ D}}\bigg{(}\frac{1}{P_{i,t-1}^{x,y}}-1\bigg{)},&\text{if }Z_{i,t}^{x,y}=1\\ \frac{1-P_{F}}{1-P_{D}}\bigg{(}\frac{1}{P_{i,t-1}^{x,y}}-1\bigg{)},&\text{if }Z_{i,t}^{x,y}=0\\ \frac{1}{P_{i,t-1}^{x,y}}-1,&\text{otherwise}.\end{array}\right. \tag{11}\]  
A nonlinear conversion of \(\mathcal{P}_{i,t}^{x,y}\) is adopted:  
\[Q_{i,t}^{x,y}\triangleq\text{ln}\bigg{(}\frac{1}{P_{i,t}^{x,y}}-1\bigg{)}. \tag{12}\]  
Then, Equation (11) can be rewritten as:  
\[Q_{i,t}^{x,y}=Q_{i,t-1}^{x,y}+\Delta_{i,t}^{x,y}, \tag{13}\]  
where  
\[\Delta_{i,t}^{x,y}=\left\{\begin{array}{ll}\ln\frac{P_{F}}{P_{D}},&\text{ if }Z_{i,t}^{x,y}=1\\ \ln\frac{1-P_{F}}{1-P_{D}},&\text{ if }Z_{i,t}^{x,y}=0\\ 0,&\text{otherwise}.\end{array}\right. \tag{14}\]  
Compared to the Bayesian method in Equation (10), the update in Equation (13), which is a linear function of \(Q_{i,t}^{x,y}\), is more efficient in calculation.  
When multiple UAVs are deployed, each UAV can broadcast the cognitive information to its neighbors for information fusion and cooperation, which will result in better performance. The UAV starts by examining the cells in its sensing area and then passing the observations to its neighboring UAVs. Following the receipt of observations from other UAVs, \(Q_{i,t}^{x,y}\) undergoes the following update:  
\[R_{i,t}^{x,y}=\eta Q_{i,t-1}^{x,y}+\sum_{j\in\mathbb{N}_{i,t}}\Delta_{j,t}^{x,y}, \tag{15}\]  
where \(\eta\in(0,1)\) is the information decaying factor. Then, each UAV \(U_{i}\) transmits \(R_{i,t}^{x,y}\) of the whole area to the neighbors for information fusion, which is represented by:  
\[Q_{i,t}^{x,y}=\sum_{j\in\mathbb{N}_{i,t}}w_{i,j,t}R_{j,t}^{x,y}, \tag{16}\]  
where  
\[w_{i,j,t}=\left\{\begin{array}{ll}1-\big{(}(d_{i,t}-1)/N_{U}\big{)},&\text{ if }j=i\\ 1/N_{U},&\text{if }j\
eq i\text{ and }j\in\mathbb{N}_{i,t}\\ 0,&\text{otherwise}.\end{array}\right. \tag{17}\]