# Geolocation and Tracking by TDOA Measurements Based on Space-Air-Ground Integrated Network
## 2 Theoretical Performance
### Crlb
#### 2.2.2 CRLB in Geodetic Coordinate System  
The position and velocity of the target in the ECEF coordinate system can be transformed to the geodetic coordinate system. The dimension of the initial location vector of the emitter can be reduced from 6 to 4 by the use of the constrained conditions in (1). The initial position and velocity of the emitter in the geodetic coordinate system are \([L_{o},B_{o},h]^{T}\) and \([V_{L},V_{B},0]^{T}\), where \(h\) is the known constant. Hence, the unknown position and velocity vectors are \(\boldsymbol{\alpha}_{o}=[L_{o},B_{o}]^{T}\) and \(\dot{\boldsymbol{\alpha}}_{o}=[V_{L},V_{B}]^{T}\), respectively. Due to the assumption of uniform motion, the position and velocity of the target at other times are expressed as \(\boldsymbol{\alpha}_{k}=\boldsymbol{\alpha}_{0}+(k-1)T\dot{\boldsymbol{ \alpha}}_{0}\) and \(\dot{\boldsymbol{\alpha}}_{k}=\dot{\boldsymbol{\alpha}}_{0}\). \(T\) is the transmitted interval of signal chips. The transformation relation from geodetic to ECEF can be expressed as  
\[\left\{ \begin{array}{l}x_{o}=(N+h)\cos B_{o}\cos L_{o}\\ y_{o}=(N+h)\cos B_{o}\sin L_{o}\\ z_{o}=\left(N\big{(}1-e^{2}\big{)}+h\right)\sin B_{o}\end{array} \right. \tag{26}\]  
Define the unknown parameter \(\theta_{2}= \begin{bmatrix}\theta_{a}^{T},\theta_{nav}^{T}\end{bmatrix} ^{T}\), where the vector of emitter location is \(\theta_{a}= \begin{bmatrix}\boldsymbol{\alpha}_{o}^{T},\dot{\boldsymbol{\alpha }}_{0}^{T}\end{bmatrix} ^{T}\), and the vector of observer location is \(\theta_{nav}=\boldsymbol{\alpha}_{nav}^{o}\). Hence, the FIM can be defined as  
\[\textbf{I}_{2}(\theta_{2})=\textbf{J}_{2}^{T}\textbf{C}^{-1}\textbf{J}_{2} \tag{27}\]  
where  
\[\textbf{J}_{2}=\frac{\partial\textbf{z}}{\partial\textbf{\theta}_{2}}= \begin{bmatrix} \textbf{J}_{a}&\textbf{J}_{t,nav}\\ 0&\textbf{E}_{nav}\end{bmatrix}  \tag{28}\]  
\[\textbf{J}_{a}=\frac{\partial\textbf{z}_{t}}{\partial\theta_{a}}\textbf{J}_{t,nav }=\frac{\partial\textbf{z}_{t}}{\partial\theta_{nav}},\textbf{E}_{nav}=\frac{ \partial\textbf{z}_{nav}}{\partial\theta_{nav}} \tag{29}\]Then, the CRLB of \(\theta_{2}\) will be  
\[\text{CRLB}(\theta_{2})=\left\{ \begin{bmatrix}\mathbf{J}_{\alpha}&\mathbf{J}_{t, nav}\\ 0&\mathbf{E}_{nav}\end{bmatrix} ^{\text{T}} \begin{bmatrix}\mathbf{C}_{t}^{-1}&0\\ 0&\mathbf{C}_{nav}^{-1}\end{bmatrix}\begin{bmatrix}\mathbf{J}_{\alpha}&\mathbf{ J}_{t,nav}\\ 0&\mathbf{E}_{nav}\end{bmatrix} \right\}^{-1}= \begin{bmatrix}\mathbf{A}_{2}& \mathbf{U}_{2}\\ \mathbf{V}_{2}&\mathbf{D}\end{bmatrix} ^{-1} \tag{30}\]  
Hence, the CRLB of the emitter location \(\theta_{\alpha}\) can be expressed as  
\[\text{CRLB}(\theta_{\alpha})=A_{2}^{-1}+A_{2}^{-1}\mathbf{U}_{2}\Big{(} \boldsymbol{D}-V_{2}A_{2}^{-1}\mathbf{U}_{2}\Big{)}^{-1}V_{2}A_{2}^{-1} \tag{31}\]  
\(A_{2}^{-1}\) is the CRLB of \(\theta_{\alpha}\) if the observer locations are known accurately. \(A_{2}^{-1}\mathbf{U}_{2}\Big{(}\boldsymbol{D}-V_{2}A_{2}^{-1}\mathbf{U}_{2} \Big{)}^{-1}\)\(V_{2}A_{2}^{-1}\) represents the performance degradation due to random errors of observer locations. \(\mathbf{D}\) is defined in (15), and \(\mathbf{A}_{2}\), \(\mathbf{U}_{2}\) and \(\mathbf{V}_{2}\) are defined as follows.