# Maritime Over the Horizon Sensor Integration:
## 4 Track Modeling Process
When the first pair of the data set (\(Lat_{o}(i)\), \(Long_{o}(i)\)) is available for a given fusion track estimator, it tries to determine the median \(m\) of the slopes using Equation 4:  
\[m=\frac{Long_{o}(i)-Long_{o}(i-1)}{Lat_{o}(i)-Lat_{o}(i-1)} \tag{6}\]  
With \(m\) determined, the \(b\)_coefficient_ may be determined as the median of the following values:  
\[b=Long_{o}(i)-mLat_{o}(i) \tag{7}\]  
The projected vessel's course is calculated as:  
\[course=arctan(m) \tag{8}\]  
The estimator will then enlarge the data set with each available pair of \(Lat_{o}(i)\) and \(Long_{o}(i)\) and repeat the procedure. It is important to note that the data set will not be enlarged indefinitely, but only until a certain number of data pairs (T) is reached and the so-called data window is formed. The reception of any new data pairs after the threshold is reached will cause the erasure of the oldest data pair, thus sliding the window forward. In this way, if the vessel starts any maneuver, the old data points will have less influence in the tracking process. Next, the true vessel's velocity (\(Vel\)) needs to be calculated. This is done by dividing the distance the vessel covered by the elapsed time, in the following manner:  
\[V_{el}=\frac{\sqrt{\left(K_{1}(Lat_{o}(i)-Lat_{o}(1)\right)^{2}+\left(K_{2}(Long _{o}(i)-Long_{o}(1)\right)^{2}}}{t} \tag{9}\]  
where _Lat\({}_{o}\)(i) and Long\({}_{o}\) (i)_ represent the vessel's parameters at the current time, _Lat\({}_{o}\)(1) and Long\({}_{o}\) (1)_ represent the initial vessel's parameters in the current window (not initial in the entire track) and t represents the elapsed time. The coefficients K\({}_{1}\) and K\({}_{2}\) are calculated according to [29]. In order to predict the new vessel's position, the current location on the slope is moved for a distance which the vessel is able to cover in one detection cycle. In this case, the detection cycle is 33 s. At the end, the gating radius around the predicted vessel's position is a circle of a radius equal to half of the covered distance. At the end of this section, the computational complexity of this algorithm needs to be addressed. There are two factors that influence the computational complexity of the algorithm: N (number of active tracks per radar) and M (number of radars with overlapping zones). The functions that form the algorithm have different complexities ranging from O(N), O(N \(\times\) M), O(N\({}^{2}\)), O(N\({}^{2}\)\(\times\) M) all the way to the most complex with the complexity: O(N\({}^{2}\)\(\times\) M\({}^{2}\)). Since the number of radars that have overlapping zones shouldn't be more than 3, we can consider that M << N, and that M and M\({}^{2}\) are constant factors in the complexity analysis. Therefore, the complexity of the algorithm can be considered to be: O(N\({}^{2}\)). One additional thing to note here is that this complexity is valid only during the initial phase of the algorithms' operation. One N in N\({}^{2}\) factor is for old/stable tracks, while the other represents new tracks which arrived during the last period of integration. The number of newly appeared tracks in the stable mode of operation is usually at least an order of magnitude lower than the number of old tracks. Therefore, we may assume that the overall algorithm complexity in the stable mode of operation is close to O(N \(\times\) log(N)).