## 2 Materials and Methods
Applying the Cauchy-Schwarz (CS) inequality to Equation (26), we have  
\[\eta_{1}\geq\eta_{\mathrm{e}}\eta_{\mathrm{P}} \tag{28}\]  
where  
\[\eta_{\mathrm{e}}=\frac{1}{N^{4}}E\Bigg{[}\sum_{l=1}^{N}\sum_{k=1}^{N}\sum_{l^ {\prime}=1}^{N}\sum_{k^{\prime}=1}^{N}\chi\big{(}\Delta t_{k,l}-\Delta t_{k^{ \prime},l^{\prime}},0\big{)}\Bigg{]} \tag{29}\]  
and  
\[\eta_{\mathrm{P}}=\frac{1}{N^{4}}E\Bigg{[}\sum_{l=1}^{N}\sum_{k=1}^{N}\sum_{l^ {\prime}=1}^{N}\sum_{k^{\prime}=1}^{N}\mathrm{e}^{\mathrm{i}(\Delta\phi_{k}- \Delta\phi_{k^{\prime}})}\mathrm{e}^{\mathrm{i}(\Delta\phi_{l}-\Delta\phi_{k^{ \prime}})}\mathrm{e}^{-\mathrm{i}2\pi f_{0}(\delta\tau_{k}-\delta\tau_{k^{ \prime}}+\delta\tau_{l}-\delta\tau_{l^{\prime}})}\Bigg{]} \tag{30}\]  
are the envelope coherent integration efficiency and phase coherent integration efficiency, respectively. Notice that the position error is negligible relative to the detection range. By utilizing the first-order Taylor approximation, the position error is obtained by  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]0where  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]1  
represents the observation direction vector of the kth satellite with the target position \(\mathbf{p}=\left(p_{x},p_{y},p_{z}\right)^{\mathrm{T}}\). We assume that the synchronization errors are uncorrelated from each other. The relative position errors are given by \(\Delta x_{k}\sim N(0,\sigma_{x}^{2})\), \(\Delta y_{k}\sim N(0,\sigma_{y}^{2})\), \(\Delta z_{k}\sim N(0,\sigma_{z}^{2})\), and \(k=1,2,\cdots,N\), yielding  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]2  
where \(\sigma_{\tau}^{2}\triangleq\left(\sigma_{x}^{2}\alpha^{2}+\sigma_{y}^{2} \beta^{2}+\sigma_{z}^{2}\gamma^{2}\right)/c^{2}\) is calculated with \(a_{k}\approx a\), \(b_{k}\approx b\), and \(c_{k}\approx c\), \(k=1,2,\cdots,N\) implying almost the same observation direction among radars. When the relative time, frequency, and phase errors obey \(\Delta f_{k}\sim N(0,\sigma_{f}^{2})\), \(\Delta t_{k}\sim N(0,\sigma_{t}^{2})\), and \(\Delta\overline{\phi}_{k}\sim N(0,\sigma_{\overline{\phi}}^{2})\), respectively, we obtain  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]3  
where \(\sigma_{\phi}^{2}=\sigma_{\overline{\phi}}^{2}+4\pi^{2}f_{0}^{2}\sigma_{t}^{2} +4\pi^{2}\sigma_{f}^{2}\sigma_{t}^{2}/\Big{(}\sigma_{f}^{2}+\sigma_{t}^{2} \Big{)}\). Notice that if \(x\sim N(0,\sigma^{2})\), then \(E[\mathrm{e}^{\mathrm{i}\pi}]=\mathrm{e}^{-\sigma^{2}/2}\). By substituting Equations (33) and (34) into (30), \(\eta_{\mathrm{P}}\) simplifies to  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]4  
Once the product of the envelope coherent integration efficiency \(\eta_{\mathrm{e}}\) and phase coherent integration efficiency \(\eta_{\mathrm{P}}\) satisfies the requirement of the indicator of arranged coherent integration efficiency, we can declare that the resulting coherent integration efficiency satisfies the arranged indicator also using Equation (28). (b) The impact of other non-ideal factors on coherent integration efficiency. 1 Beam pointing error.