## 4 Our Proposed GCI-IGGM-PMBM Scheme
### Recursion Based on Augmented States  
In the proposed GCI-IGGM-PMBM filter, the newborn intensity and detection probability are a priori unknown, which is unlike the standard PMBM filter. Gamma distribution is used to present the unknown detection probability, and target states comply with Gaussian distribution. The prediction and updating of IGGM-PMBM can be derived as described below. Proposition 1 (Predicted process). Given the posterior intensity of Poisson \(D_{k-1}(\hat{\mathbf{X}}^{u},\xi)\), the newborn intensity with \(B_{k}^{b}(\hat{\mathbf{X}}_{b},\xi)\) and MBM RFS with parameters \(\left\{w_{k-1}^{j},\left\{r_{k-1}^{j,i},p_{k-1}^{j,i}(\hat{\mathbf{x}})\right\} _{i\in\mathbb{I}}\right\}_{j\in\mathbb{I}}\) at time \(k-1\), representing the undetected targets and potentially detected targets, respectively, the predicted intensity of IGGM-PMBM can be obtained in two steps, as described below. 1. PPP process \[D_{k|k-1}^{u}(X_{k},\xi_{k})=B_{k}^{b}(X_{b},\xi_{k})+D_{k-1}^{u}(X_{k},\xi_{ k}),\] (40)  
\[B_{k}^{b}(X_{b},\xi_{k})=\int\int_{0}^{1}B_{k-1}^{b}(X_{k-1}^{b},\xi_{k-1}^{b })p_{S}(X_{k-1}^{b})f_{k|k-1}(X_{k}|X_{k-1}^{b})\hat{f}_{k|k-1}(\xi_{k}|\xi_{ k-1})\,dX_{k-1}^{b}d\xi_{k-1}^{b}, \tag{41}\]  
\[D_{k-1}^{u}(X_{k},\xi_{k})=\int\int_{0}^{1}D_{k-1}^{u}(X_{k-1},\xi_{k-1})p_{S} (\xi_{k-1})f_{k|k-1}(X_{k}|X_{k-1})\,\hat{f}_{k|k-1}(\xi_{k}|\xi_{k-1})\,dX_{k -1}d\xi_{k-1}, \tag{42}\]  
where \(f_{k|k-1}(X_{k}\Big{|}X_{k-1}^{b})\) denotes the transition function of target states, and \(f_{k|k-1}(\xi_{k}|\xi_{k-1})\) denotes the transition function of the augmented variate, presented as an unknown detection probability. 2. MBM process \[r_{k|k-1}^{j,i}=r_{k-1}^{j,i}\!\int_{0}^{1}p_{k-1}^{j,i}(X_{k-1},\xi_{k-1})p _{s}(\xi_{k-1})dX_{k-1}d\xi_{k-1},\] (43)\[p_{k|k-1}^{j,i}(X,\xi)=\int\int_{0}^{1}p_{k-1}^{j,i}(X_{k-1},\xi_{k-1})p_{s}(\xi_{ k-1})f_{k|k-1}(X_{k}|X_{k-1})\hat{f}_{k|k-1}(\xi_{k}|\xi_{k-1})dX_{k-1}d\xi_{k-1}, \tag{44}\]  
\[w_{k|k-1}^{j}(X,\xi)=w_{k-1}^{j}(X). \tag{45}\]  
Then, the predicted density is also a PMBM density, which is represented by the parameters \(D_{k|k-1}^{u}(X_{k},\xi_{k})\) and \(\left\{w_{k|k-1}^{j}\left\{r_{k|k-1}^{j,i},p_{k|k-1}^{j,i}(\hat{\mathbf{x}}) \right\}_{i\in\mathbb{U}}\right\}_{j\in\mathbb{J}}\). Proposition 2 (Updating process). Given the predicted PMBM density with parameters \(D_{k|k-1}^{u}(X_{k},\xi_{k})\) and \(\left\{w_{k|k-1}^{j}\left\{r_{k|k-1}^{j,i},p_{k|k-1}^{j,i}(\hat{\mathbf{x}}) \right\}_{i\in\mathbb{U}}\right\}_{j\in\mathbb{J}}\), as well as the measurements \(M_{k}\) at time \(k\), the updated density of IGGM-PMBM can be obtained in four stages as described below. 1. Update for undetected targets \[p_{k|k}(X,\xi)=(1-\xi)p_{k|k-1}(X,\xi).\] (46)
2. Update for potential targets detected for the first time \[r_{k,1}^{u}=\frac{\int_{0}^{1}\int D_{k|k-1}^{u}(X_{k-1},\xi_{k-1})l_{k}(M_{k} |X_{k-1})dX_{k-1}d\xi_{k-1}}{\kappa_{C}(M_{k},\xi_{k-1})+\int_{0}^{1}\int D_{k |k-1}^{u}(X_{k-1},\xi_{k-1})l_{k}(M_{k}|X_{k-1})dX_{k-1}d\xi_{k-1}},\] (47)  
\[p_{k,1}^{u}(X,\xi|M)=\frac{l_{k}(M|X)D_{k|k-1}^{u}(X,\xi)}{\int_{0}^{1}\int l_ {k}(M_{k}|X_{k-1})D_{k|k-1}^{u}(X_{k-1},\xi_{k-1})dX_{k-1}d\xi_{k-1}}, \tag{48}\]  
\[w_{k,1}^{j}=\frac{w_{k|k-1}^{j}\prod_{C\in A_{1}}\left(\kappa_{C}(M_{k},\xi_{ k-1})+\int_{0}^{1}\int D_{k|k-1}^{u}(X_{k-1},\xi_{k-1})l_{k}(M_{k}|X_{k-1})dX_{k-1 }d\xi_{k-1}\right)}{\sum_{j\in\mathbb{J}}\sum_{A_{1}\in\mathcal{A}^{j}}w_{k|k -1}^{j}\prod_{C\in A_{1}}\left(\kappa_{C}(M_{k},\xi_{k-1})+\int_{0}^{1}\int D_ {k|k-1}^{u}(X_{k-1},\xi_{k-1})l_{k}(M_{k}|X_{k-1})dX_{k-1}d\xi_{k-1}\right)},\] (49) where \(\kappa_{C}(M_{k},\xi_{k-1})\) denotes the clutter density originating from measurements based on the augmented variate unknown detection probability, and the subscript 1 denotes the update type. 3.