## 2 Problem Formulation  
Address a typical multi-static passive radar localization scenario as presented in Figure 1, where \(M\) non-cooperative transmitters located at \(\mathbf{s}_{\text{t},m}^{\text{o}}=[x_{\text{t},m}^{\text{o}},y_{\text{t},m}^{ \text{o}},z_{\text{t},m}^{\text{o}}]^{\text{T}}\) (\(m=1,2,\ldots,M\)) are employed to illuminate the surveillance area, and \(N\) receivers located at \(\mathbf{s}_{\text{r},n}^{\text{o}}=[x_{\text{r},n}^{\text{o}},y_{\text{r},n}^ {\text{o}},z_{\text{r},n}^{\text{o}}]^{\text{T}}\) (\(n=1,2,\ldots,N\)) are deployed to determine a single target's position denoted by \(\mathbf{u}^{\text{o}}=[x^{\text{o}},y^{\text{o}},z^{\text{o}}]^{\text{T}}\). In fact, the exact positions of the transmitters and receivers might not be known, and only the inaccurate measured versions, i.e., \(\mathbf{s}_{\text{t},m}=[x_{\text{t},m},y_{\text{t},m},z_{\text{t},m}]^{\text{T}}\) and \(\mathbf{s}_{\text{r},n}=[x_{\text{r},n},y_{\text{r},n},z_{\text{r},n}]^{\text {T}}\), are available for processing. Formulaically, we arrive at  
\[\mathbf{s}_{\text{t},m}=\mathbf{s}_{\text{t},m}^{\text{o}}+\mathbf{\Delta}\mathbf{ s}_{\text{t},m} \tag{1}\]  
\[\mathbf{s}_{\text{r},n}=\mathbf{s}_{\text{r},n}^{\text{o}}+\mathbf{\Delta}\mathbf{ s}_{\text{r},n} \tag{2}\]  
where \(\mathbf{\Delta}\mathbf{s}_{\text{t},m}\) and \(\mathbf{\Delta}\mathbf{s}_{\text{r},n}\) are the position error of the \(m\)th transmitter and the \(n\)th receiver respectively and also referred to as position uncertainty. Stacking (1) and (2) with respect to all the transmitters and receivers, yields a \(3(M+N)\)-by-\(1\) transmitter and receiver position vector as  
\[\mathbf{s}=\mathbf{s}^{\text{o}}+\mathbf{\Delta}\mathbf{s} \tag{3}\]where \(\mathbf{s}=\left[\mathbf{s}_{t}^{\mathrm{T}},\mathbf{s}_{r}^{\mathrm{T}}\right]^{ \mathrm{T}}\) with \(\mathbf{s}_{t}=\left[\mathbf{s}_{t,1}^{\mathrm{T}},\mathbf{s}_{t,2}^{\mathrm{ T}},\dots,\mathbf{s}_{t,M}^{\mathrm{T}}\right]^{\mathrm{T}}\) and \(\mathbf{s}_{r}=\left[\mathbf{s}_{r,1}^{\mathrm{T}},\mathbf{s}_{r,2}^{\mathrm{ T}},\dots,\mathbf{s}_{t,N}^{\mathrm{T}}\right]^{\mathrm{T}}\) is the noisy transmitter and receiver position vector, \(\mathbf{s}^{\mathrm{o}}=\left[\left(\mathbf{s}_{t}^{\mathrm{o}}\right)^{ \mathrm{T}},\left(\mathbf{s}_{r}^{\mathrm{o}}\right)^{\mathrm{T}}\right]^{ \mathrm{T}}\) with \(\mathbf{s}_{t}^{\mathrm{o}}=\left[\left(\mathbf{s}_{t,1}^{\mathrm{o}}\right)^{ \mathrm{T}},\left(\mathbf{s}_{t,2}^{\mathrm{o}}\right)^{\mathrm{T}},\dots, \left(\mathbf{s}_{t,M}^{\mathrm{o}}\right)^{\mathrm{T}}\right]^{\mathrm{T}}\) and \(\mathbf{s}_{r}^{\mathrm{o}}=\left[\left(\mathbf{s}_{r,1}^{\mathrm{o}}\right)^ {\mathrm{T}},\left(\mathbf{s}_{r,2}^{\mathrm{o}}\right)^{\mathrm{T}},\dots, \left(\mathbf{s}_{r,N}^{\mathrm{o}}\right)^{\mathrm{T}}\right]^{\mathrm{T}}\) is the true transmitter and receiver position vector, and \(\mathbf{\Delta s}=\left[\mathbf{\Delta s}_{t}^{\mathrm{T}},\mathbf{\Delta s}_ {r}^{\mathrm{T}}\right]^{\mathrm{T}}\) with \(\mathbf{\Delta s}_{t}=\left[\mathbf{\Delta s}_{t,1}^{\mathrm{T}},\mathbf{ \Delta s}_{t,2}^{\mathrm{T}},\dots,\mathbf{\Delta s}_{t,M}^{\mathrm{T}}\right]^ {\mathrm{T}}\) and \(\mathbf{\Delta s}_{r}=\left[\mathbf{\Delta s}_{t,1}^{\mathrm{T}},\mathbf{ \Delta s}_{t,2}^{\mathrm{T}},\dots,\mathbf{\Delta s}_{t,N}^{\mathrm{T}}\right]^ {\mathrm{T}}\) is the transmitter and receiver position error vector that can be assumed zero-mean Gaussian with covariance \(\mathbf{Q}_{\mathbf{s}}\) without loss of generality.