# Geolocation and Tracking by TDOA Measurements Based on Space-Air-Ground Integrated Network
## 2 Theoretical Performance  
### Problem Statement and Mathematical Formulation  
Suppose the target is on the surface of the Earth with known constant height \(h\). The original position and velocity of the target are \(\mathbf{u}_{o}\) and \(\dot{\mathbf{u}}_{o}\), which are defined in the ECEF coordinate system. The velocity is assumed to be horizontal and fixed during the whole observation time. The constrained conditions of \(\mathbf{u}_{o}\) and \(\dot{\mathbf{u}}_{o}\) are shown as follows. \[\left\{ \begin{array}{c}\mathbf{u}_{o}^{T}\mathbf{u}_{o}=\left(N+h\right)^{2 }\\ \mathbf{u}_{o}^{T}\dot{\mathbf{u}}_{o}=0\end{array} \right. \tag{1}\]  
where \(N\) is the radius of curvature in prime vertical, \(N=\frac{a}{\sqrt{1-e^{2}\sin^{2}B}}\). \(a\) is the semi-major axis of the earth and \(e\) is the first eccentricity. \(B\) is the latitude of the target; \(\dot{\mathbf{u}}_{o}\) and \(h\) will be zeros when the emitter is stationary. Assume there are \(M\) moving observers in the SAGI network. Every observer intercepts \(K\) chips of signal. The true position of the \(j\)th observer at the time of intercepting the \(k\)th chip of signal is \(\mathbf{s}_{j,k}^{o}\). The noisy value \(\mathbf{s}_{j,k}\) is available and the random error is expressed as \(\Delta\mathbf{s}_{j,k}\). Then, the TDOA measurement between the \(j\)th observer and the 0 th observer for the \(k\)th chip of signal is  
\[t_{j,0,k}=\frac{\left\|\mathbf{u}_{k}-\mathbf{s}_{j,k}^{o}\right\|}{c}-\frac{ \left\|\mathbf{u}_{k}-\mathbf{s}_{0,k}^{o}\right\|}{c}+\Delta t_{j,0,k},j=0, \ldots,M-1;k=1,\ldots,K. \tag{2}\]  
where \(\Delta t_{j,0,k}\) is the TDOA measured noise, and \(j=0,\ldots,M-1\), \(k=1,\ldots,K\). Due to the assumption of uniform motion, the position and velocity of the emitter at other times are expressed as \(\mathbf{u}_{k}=\mathbf{u}_{0}+(k-1)T\hat{\mathbf{u}}_{0}\) and \(\hat{\mathbf{u}}_{k}=\hat{\mathbf{u}}_{0}\). \(T\) is the transmitted interval of signal chips. Collecting the TDOA measurements and observer locations yields  
\[ \begin{split}&\mathbf{z}=\mathbf{z}^{o}+\mathbf{n}\\ &\mathbf{z}=\left[\mathbf{z}_{t}^{T},\mathbf{z}_{nav}^{T}\right] ^{T}\\ &\mathbf{z}_{t}=\left[\mathbf{z}_{t,1}^{T},\mathbf{z}_{t,2}^{T},\ldots,\mathbf{z}_{t,K}^{T}\right]^{T}\\ &\mathbf{z}_{t,k}=\left[t_{1,0,k},t_{2,0,k},\ldots,t_{M-1,0,k} \right]^{T}\\ &\mathbf{z}_{nav}=\left[\mathbf{z}_{nav,1}^{T},\mathbf{z}_{nav,2 }^{T},\ldots,\mathbf{z}_{nav,K}^{T}\right]^{T}\\ &\mathbf{z}_{nav,k}=\left[\mathbf{s}_{0,k}^{T},\mathbf{s}_{1,K}^ {T},\ldots,\mathbf{s}_{M-1,k}^{T}\right]^{T}\end{split}  \tag{3}\]  
where \(\mathbf{z}^{o}\) represents the true values of TDOA measurements and observer locations and \(\mathbf{n}\) is the measurement noise vector of \(\mathbf{z}^{o}\). \[ \begin{split}&\mathbf{z}^{o}=\left[\mathbf{z}_{t}^{oT},\mathbf{z}_{ nav}^{oT}\right]^{T}\\ &\mathbf{z}_{t}^{o}=\left[\mathbf{z}_{t,1}^{oT},\mathbf{z}_{t,2 }^{oT},\ldots,\mathbf{z}_{t,K}^{oT}\right]^{T}\\ &\mathbf{z}_{t,k}^{o}=\left[t_{1,0,k}^{o},t_{2,0,k}^{oT},\ldots,t _{M-1,0,k}^{o}\right]^{T}\\ &\mathbf{z}_{nav}^{o}=\left[\mathbf{z}_{nav,1}^{oT},\mathbf{z}_{ nav,2}^{oT},\ldots,\mathbf{z}_{nav,K}^{oT}\right]^{T}\\ &\mathbf{z}_{nav,k}^{o}=\left[\mathbf{s}_{0,k}^{oT},\mathbf{s}_{1,K}^{oT},\ldots,\mathbf{s}_{M-1,k}^{oT}\right]^{T}\\ &\mathbf{n}=\left[\mathbf{n}_{t}^{T},\mathbf{n}_{nav}^{T}\right] ^{T}\\ &\mathbf{n}_{t}=\left[\mathbf{n}_{t,1}^{T},\mathbf{n}_{t,2}^{T}, \ldots,\mathbf{n}_{t,K}^{T}\right]^{T}\\ &\mathbf{n}_{t,k}=\left[\Delta t_{1,0,k},\Delta t_{2,0,k},\ldots, \Delta t_{M-1,0,k}\right]^{T}\\ &\mathbf{n}_{nav}=\left[\mathbf{n}_{nav,1}^{T},\mathbf{n}_{nav,2 }^{T},\ldots,\mathbf{n}_{nav,K}^{T}\right]^{T}\\ &\mathbf{n}_{nav,k}=\left[\Delta\mathbf{s}_{0,k}^{T},\Delta \mathbf{s}_{1,K}^{T},\ldots,\Delta\mathbf{s}_{M-1,k}^{T}\right]^{T}\end{split}