# Wildlife Monitoring Using a Multi-UAV System with Optimal Transport Theory
## 3 Method
### Sample Point Generation and Propagation: Time-Varying Case  
In the previous section, the OT-based multi-UAV exploration strategy is proposed for the time-invariant case, which is not appropriate for the animal herds wandering around their habitat. Therefore, the reference distribution (or the sample points) needs to be time-varying as well to reflect the time-varying nature of the animal herd locations. This section will provide the method to generate and propagate the sample points.  
At any time \(T\), let \(Z=\{z_{T}^{q}\}_{q=1}^{G}\) be the tracking information containing the locations of \(G\) numbers of tracked animals obtained by the GPS trackers. If the distance between any two tracked animals is within a specific distance given as a threshold, they are considered as the same herd. Otherwise, they will be members of different herds.  
Since the animal herd locations are mostly unknown, clusters of sample points need to be assigned to the herds which are determined from the available tracking information. For the animal herd with tracked animals, its distribution is given as Gaussian distribution initially. The center of each distribution is assigned to the tracked animal locations in the herd. If more than one tracked animal is in the herd, the center of the corresponding Gaussian distribution is considered as the mean of the locations of the tracked animals. The covariance of the distribution is considered as a user-defined parameter.  
The next step is to propagate the sample points for the estimation of the animal herds wandering around. To this end, the Correlated Random Walk model in (3) is employed to propagate each sample point, since the CRW model is associated with the drift of the animal herds. The variables \(u_{T+1}\), \(r_{u,T+1}\), \(\theta_{u,T+1}\) and \(v_{u,T+1}\) in (3) can be replaced by variables \(y_{j,T+1}\), \(r_{j,T+1}\), \(\theta_{j,T+1}\) and \(v_{j,T+1}\), followed by the sample point propagation based on (3).  
The sample point propagation using the CRW model alone cannot improve the performance of the monitoring as it does not incorporate an estimation correction procedure if the agents detect any animals during the monitoring mission. Hence, the center of the sample points associated with the herd of the detected animal is relocated to \(z_{T+\tau}^{q}\) when an animal located at \(z_{T+\tau}^{q}\) is detected by the UAV at time \(T+\tau\), where \(\tau\) represents the time elapsed after the tracking information is received. After this relocation, the sample points in the distribution with the mean located at \(z_{T+\tau}^{q}\) continue propagating using (3). As the proposed method is for the centralized scheme, the sample point propagation, animal herd detection, and sample point correction are shared with all agents through communication and information sharing.  
Regarding the communication between UAVs, either air-to-air or air-to-ground wireless communications, several different communication technologies can be adopted such as direct link, satellite, _ad hoc_ network, and cellular network (see [19] for more details of each technology). For example, the satellite-based communication technology provides global coverage, which might be useful for wildlife monitoring applications as it enables the multi-UAV system to communicate with each other through satellites even in remote areas without terrestrial network (e.g., Wi-Fi or cellular).