## 3 Mathematical Model of Geolocation and Tracking Scenarios
### Geolocation and Target-Tracking Techniques
#### 3.3.1 Cross Ambiguity Function (CAF) Approach
Second step: Time update In the second step, it is necessary to evaluate and estimate state \[\mathbf{X}_{k-1|k-1}^{j}=\sqrt{\mathbf{P}_{k-1|k-1}}\xi^{j}+\hat{\mathbf{x}}_{k -1|k-1}\,\] (34) \[\mathbf{X}_{k|k-1}^{*(j)}=\mathbf{f}(\mathbf{X}_{k-1|k-1}^{j},\mathbf{u}_{k-1},k -1)\,\] (35) \[\hat{\mathbf{x}}_{k|k-1}=\sum_{j=1}^{m}\mathbf{w}^{j}\mathbf{X}_{k|k-1}^{*(j )}\,\] (36) \[\mathbf{P}_{k|k-1}=\sum_{j=1}^{m}\mathbf{w}^{j}\Big{(}\mathbf{X}_{k|k-1}^{*(j )}-\hat{\mathbf{x}}_{k|k-1}\Big{)}\times\Big{(}\mathbf{X}_{k|k-1}^{j}-\hat{ \mathbf{x}}_{k|k-1}\Big{)}^{\prime}+\mathbf{Q}_{k-1}\,\] (37) where \(\{\mathbf{X}_{k-1|k-1}^{j}\}_{j=1}^{m}\) is the quadrature points; \(\{\mathbf{X}_{k|k-1}^{*(j)}\}_{j=1}^{m}\) is the propagated quadrature points; \(\hat{\mathbf{x}}_{k|k-1}\) is the predicted state mean; and \(\mathbf{P}_{k|k-1}\) is the predicted error covariance. 3. Third step: Measurement update In the third step, it is necessary to evaluate and estimate measurement \[\mathbf{X}_{k|k-1}^{j}=\sqrt{\mathbf{P}_{k|k-1}}\xi^{j}+\hat{\mathbf{x}}_{k|k -1}\,\] (38) \[\mathbf{Y}_{k|k-1}^{j}=\mathbf{h}(\mathbf{X}_{k|k-1}^{j},\mathbf{u}_{k},k)\,\] (39) \[\hat{\mathbf{y}}_{k|k-1}=\sum_{j=1}^{m}\mathbf{w}^{j}\mathbf{Y}_{k|k-1}^{j}\,\] (40) \[\mathbf{P}_{\mathbf{yy}k|k-1}=\sum_{j=1}^{m}\mathbf{w}^{j}\Big{(}\mathbf{Y}_{ k|k-1}^{j}-\hat{\mathbf{y}}_{k|k-1}\Big{)}\times\Big{(}\mathbf{Y}_{k|k-1}^{j}- \hat{\mathbf{y}}_{k|k-1}\Big{)}^{\prime}+\mathbf{R}_{k}\,\] (41) \[\mathbf{P}_{\mathbf{xy}k|k-1}=\sum_{j=1}^{m}\mathbf{w}^{j}\Big{(}\mathbf{X}_{ k|k-1}^{j}-\hat{\mathbf{x}}_{k|k-1}\Big{)}\times\Big{(}\mathbf{Y}_{k|k-1}^{j}- \hat{\mathbf{y}}_{k|k-1}\Big{)}^{\prime}\,\] (42) \[\mathbf{X}_{k-1|k-1}^{j}=\sqrt{\mathbf{P}_{k-1|k-1}}\xi^{j}+\hat{\mathbf{x}}_{k -1|k-1}\,\]0 (43) \[\mathbf{X}_{k-1|k-1}^{j}=\sqrt{\mathbf{P}_{k-1|k-1}}\xi^{j}+\hat{\mathbf{x}}_{k -1|k-1}\,\]1 (44) \[\mathbf{X}_{k-1|k-1}^{j}=\sqrt{\mathbf{P}_{k-1|k-1}}\xi^{j}+\hat{\mathbf{x}}_{k -1|k-1}\,\]2 (45) where \(\{\mathbf{X}_{k|k-1}^{j}\}_{j=1}^{m}\) is the new quadrature points; \(\{\mathbf{Y}_{k|k-1}^{j}\}_{j=1}^{m}\) is the new propagated quadrature points; \(\hat{\mathbf{y}}_{k|k-1}\) is the predicted measurement; \(\mathbf{P}_{\mathbf{yy}k|k-1}\) is the innovation covariance matrix; \(\mathbf{P}_{\mathbf{xy}k|k-1}\) is the cross-covariance matrix; \(\mathbf{W}_{k}\) is the Kalman gain; \(\hat{\mathbf{x}}_{k|k}\) is the updated state; and \(\mathbf{P}_{k|k}\) is the corresponding error covariance. **Eventually**, this can obtain the posterior density; therefore, a nonlinear Kalman filter estimates the hidden state using a Gaussian distribution of probability density functions. \[\mathbf{X}_{k-1|k-1}^{j}=\sqrt{\mathbf{P}_{k-1|k-1}}\xi^{j}+\hat{\mathbf{x}}_{k -1|k-1}\,\]3  
where \(\mathbf{y}_{1:k}\) is the set of measurements \(\mathbf{y}_{1:k}=[\mathbf{y}_{1},\mathbf{y}_{2},\cdots\mathbf{y}_{K}]\). In this work, the visibility and availability of multiple LEO satellite constellations are assumed to be sequential, ensuring a minimum of one LEO satellite signal availability. A distributed sequential high-degree nonlinear filtering algorithm was developed and compared in the simulation with a distributed sequential filter such CKF 3 rd degree, CIF 3 rd degree, CIF-V1 5 th degree, CIF-V2 5 th degree, CIF-V3 5 th degree, and CIF-MY SOVSKKH 7 th degree.