## 2 Problem Formulation
Using the above notations, the distance from the \(m\)th transmitter to the target of interest is equal to  
\[R_{t,m}^{\mathrm{o}}=\left\|\mathbf{u}^{\mathrm{o}}-\mathbf{s}_{t,m}^{\mathrm{ o}}\right\| \tag{4}\]  
the distance from the target of interest to the \(n\)th receiver is equal to  
\[R_{r,n}^{\mathrm{o}}=\left\|\mathbf{u}^{\mathrm{o}}-\mathbf{s}_{r,n}^{\mathrm{ o}}\right\| \tag{5}\]  
and the baseline distance with respect to the mth transmitter and nth receiver is  
\[R_{t,m,r,n}^{\mathrm{o}}=\left\|\mathbf{s}_{t,m}^{\mathrm{o}}-\mathbf{s}_{r,n }^{\mathrm{o}}\right\| \tag{6}\]  
According to this, the BR measurement with respect to the \(m\)th transmitter and \(n\)th receiver, i.e., the sum of the distances from the \(m\)th transmitter to the target and the target to the \(n\)th receiver, can be formulized as  
\[ \begin{array}{l}r_{m,n}=r_{m,n}^{\mathrm{o}}+\Delta r_{m,n}\\ =R_{t,m}^{\mathrm{o}}+R_{r,n}^{\mathrm{o}}-R_{t,m,r,n}^{\mathrm{o}}+\Delta r_ {m,n}\\ =\left\|\mathbf{u}^{\mathrm{o}}-\mathbf{s}_{t,m}^{\mathrm{o}}\right\|+\left\| \mathbf{u}^{\mathrm{o}}-\mathbf{s}_{r,n}^{\mathrm{o}}\right\|-\left\|\mathbf{ s}_{t,m}^{\mathrm{o}}-\mathbf{s}_{r,n}^{\mathrm{o}}\right\|+\Delta r_{m,n} \end{array}  \tag{7}\]  
where \(r_{m,n}^{\mathrm{o}}=R_{t,m}^{\mathrm{o}}+R_{r,n}^{\mathrm{o}}-R_{t,m,r,n}^{ \mathrm{o}}\) represents the true BR with respect to the \(m\)th transmitter and \(n\)th receiver, \(\Delta r_{m,n}\) is the BR measurement noise. Herein, it is important to emphasize that the time delay measurement comes from the cross correlation operation between the target signal and the direct path reference signal [7], and its error characteristics are not affected by the transmitter/receiver position error since the transmitter/receiver position error is not involved in the estimation of time delay. Thus, the BR measurement noise is only related to the time delay measurement noise, and not related to the transmitter/receiver position error. Obviously, there will be \(MN\) BR measurements to be produced with respect to the \(M\) transmitters and \(N\) receivers, which can be recast into a \(MN\)-by-\(1\) vector as  
\[\mathbf{r}=\mathbf{r}^{\mathrm{o}}+\Delta\mathbf{r} \tag{8}\]  
Figure 1: Practical scenario geometry of multi-static passive radar in the presence of transmitter/receiver position error and calibration targets. where \(\mathbf{r}=[\mathbf{r}_{1}^{\mathrm{T}},\mathbf{r}_{2}^{\mathrm{T}},\ldots,\mathbf{r }_{M}^{\mathrm{T}}]^{\mathrm{T}}\) with \(\mathbf{r}_{m}=[r_{m,1},r_{m,2},\ldots,r_{m,N}]^{\mathrm{T}}\) is the BR measurement vector, \(\mathbf{r}^{\mathrm{o}}=[(\mathbf{r}_{1}^{\mathrm{o}})^{\mathrm{T}},(\mathbf{ r}_{2}^{\mathrm{o}})^{\mathrm{T}},\ldots,(\mathbf{r}_{M}^{\mathrm{o}})^{ \mathrm{T}}]^{\mathrm{T}}\) with \(\mathbf{r}_{m}^{\mathrm{o}}=[r_{m,1}^{\mathrm{o}},r_{m,2}^{\mathrm{o}},\ldots,r_{m,N}^{\mathrm{o}}]^{\mathrm{T}}\) is the true BR vector, and \(\mathbf{\Delta r}=[\mathbf{\Delta r}_{1}^{\mathrm{T}},\mathbf{\Delta r}_{2}^{ \mathrm{T}},\ldots,\mathbf{\Delta r}_{M}^{\mathrm{T}}]^{\mathrm{T}}\) with \(\mathbf{\Delta r}_{m}=[\mathbf{\Delta r}_{m,1},\mathbf{\Delta r}_{m,2},\ldots,\mathbf{\Delta r}_{m,N}]^{\mathrm{T}}\) is the BR measurement noise vector, which is usually assumed follow a Gaussian distribution with zero-mean and covariance \(\mathbf{Q}_{\mathbf{r}}\). As presented in Figure 1, to alleviate the transmitter/receiver position error and enhance localization accuracy, \(K\) calibration targets located at \(\mathbf{c}_{k}^{\mathrm{o}}=[x_{c,k^{\prime}}^{\mathrm{o}},y_{c,k^{\prime}}^{ \mathrm{o}},z_{c,k}^{\mathrm{o}}]^{\mathrm{T}}\) (\(k=1,2,\ldots,K\)) are employed, and the BRs among the calibration targets and the transmitter-receiver pairs are also measured.