## 3 Materials and Methods
### Velocity Vector Association (VVA) Discriminator  
According to the spatial correlation of the physical target, the velocity vector is obtained first. Considering that the velocity information obtained in each receiver is only the radial velocity, it will require the measurements of at least three receivers to calculate the velocity vector. From Equation (7), the relationship between the radial velocity and the target's velocity vector \(\mathbf{v}_{PT}=[v_{x},v_{y},v_{z}]^{T}\) is represented as  
\[\begin{bmatrix}v_{0}\\ v_{1}\\ v_{2}\end{bmatrix}=-\frac{\mathbf{A}}{2}\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\end{bmatrix} \tag{10}\]  
where \(\mathbf{A}\) is the transformation matrix.  
\[\mathbf{A}=\begin{bmatrix}2\cos\alpha_{0}\cos\beta_{0}&2\sin\alpha_{0}\cos \beta_{0}&2\sin\beta_{0}\\ \cos\alpha_{0}\cos\beta_{0}+\cos\alpha_{1}\cos\beta_{1}&\sin\alpha_{0}\cos \beta_{0}+\sin\alpha_{1}\cos\beta_{1}&\sin\beta_{0}+\sin\beta_{1}\\ \cos\alpha_{0}\cos\beta_{0}+\cos\alpha_{2}\cos\beta_{2}&\sin\alpha_{0}\cos \beta_{0}+\sin\alpha_{2}\cos\beta_{2}&\sin\beta_{0}+\sin\beta_{2}\end{bmatrix} \tag{11}\]  
Therefore, the target's velocity vector \(\mathbf{v}\) and its error covariance matrixes \(\mathbf{P_{v}}\) can be obtained as  
\[\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\end{bmatrix}=-2\mathbf{A}^{-1}\begin{bmatrix}v_{0}\\ v_{1}\\ v_{2}\end{bmatrix} \tag{12}\]  
and  
\[\mathbf{P_{v}}=E[d\mathbf{v}\;d\mathbf{v}^{T}] \tag{13}\]  
By applying Equations (12) and (13), two velocity vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) are obtained from the data measured by three receivers with T/R station, and the corresponding error covariance matrixes are \(\mathbf{P_{v1}}\) and \(\mathbf{P_{v2}}\), respectively. The velocity vector association method selects the Mahalanobis distance between velocity vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) as the discrimination factor  
\[M_{\mathbf{v}}=(\mathbf{v}_{1}-\mathbf{v}_{2})^{T}\mathbf{\Sigma_{v}^{-1}}( \mathbf{v}_{1}-\mathbf{v}_{2}) \tag{14}\]  
where  
\[\begin{split}\mathbf{\Sigma}_{V}&=E\Big{[}(\mathrm{d }\mathbf{v}_{1}-\mathrm{d}\mathbf{v}_{2})(\mathrm{d}\mathbf{v}_{1}-\mathrm{d} \mathbf{v}_{2})^{T}\Big{]}\\ &=\mathbf{P_{v1}}+\mathbf{P_{v2}}-E\Big{[}\mathrm{d}\mathbf{v}_{1} \mathrm{d}\mathbf{v}_{2}^{T}\Big{]}-E\Big{[}\mathrm{d}\mathbf{v}_{2}\mathrm{d }\mathbf{v}_{1}^{T}\Big{]}\end{split} \tag{15}\]  
When the velocity vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) correspond to the same physical target, namely hypothesis H\({}_{0}\), the Mahalanobis distance \(M_{\mathbf{v}}\) between \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) approximately follows the Chi-square distribution with 1 degree of freedom, namely \(M_{\mathbf{v}}\sim\chi_{1}^{2}\). The discriminator based on the velocity vector association is established by  
\[M_{\mathbf{v}}\lessapprox\limits_{H_{12}}^{H_{0}}\eta \tag{16}\]  
where \(\eta\) is the discrimination threshold.  
From the derivation process, not only two different velocity vectors of a target and the corresponding error covariance matrixes are required in the velocity vector association discriminator, but also the cross-variance matrixes \(E[\mathrm{d}\mathbf{v}_{1}\mathrm{d}\mathbf{v}_{2}^{T}]\) and \(E[\mathrm{d}\mathbf{v}_{2}\mathrm{d}\mathbf{v}_{1}^{T}]\) are needed to be calculated for the measurements of two receivers shared in the computation of the two velocity vectors.