## 2 System Model  
In this section, we introduce the proposed STAR-RIS-aided NOMA system model which is shown in Figure 1. Without loss of generality, a two-dimensional (2D) coordinate system is considered. In this scenario, direct communications between the two users and the BS are blocked. To overcome the blocking, a UAV equipped with STAR-RIS establishes a communication link between the users and the BS.  
The coordinates for user 1 (U1), user 2 (U2), and STAR-RIS are \((L,0)\), \((-L,0)\), and \((r,h_{uav})\). We use \(h_{uav}\) to denote the height of the UAV and \(r\) to represent the horizontal coordinate of the UAV. In the system, the BS is located far away, and its coordinate is \((h_{bs},r_{bs})\). The \(O\) point which is located in \((0,0)\) denotes the origin point of the 2D coordinate system. \(d_{BS}\) denotes the distance from the BS to the STAR-RIS, and \(d_{r}\) and \(d_{t}\) represent the distances from the STAR-RIS to each user, which are, respectively, shown as  
\[d_{r}=\sqrt{\left(L-r\right)^{2}+h_{uav}^{2}}, \tag{1a}\]  
Figure 1: Illustration of the system model.  
\[d_{t}=\sqrt{\left(L+r\right)^{2}+h_{uw}{}^{2}}, \tag{1b}\] \[d_{BS}=\sqrt{\left(r_{bs}-r\right)^{2}+\left(h_{bs}-h_{uav}\right) ^{2}},\] (1c) \[r\in(0,L]\text{.} \tag{1d}\]  
As shown in Figure 1 and Equations (1a)-(1d), the system model simplifies the position by ignoring the y-axis component. This simplification is reasonable because the horizontal position of the UAV is not involved in the optimization process. Since the focus of this work is on optimizing the UAV altitude and STAR-RIS configuration, modeling in two dimensions suffices, without loss of generality.  
The wireless signal incident on a particular STAR-RIS element is split into transmitted and reflected signals, as shown in Figure 1. The signal incident on the \(m\)th element of the STAR-RIS is represented as \(s_{m}\), where \(m\in\{1,2,\ldots,M\}\) with \(M\) representing the total number of elements. In addition, the signals transmitted and reflected by the \(m\)th element can be, respectively, denoted as \(t_{m}=\sqrt{\beta_{m}^{t}}e^{i\phi_{m}^{t}}s_{m}\) and \(r_{m}=\sqrt{\beta_{m}^{t}}e^{i\phi_{m}^{t}}s_{m}\), where \(\sqrt{\beta_{m}^{t}}\), \(\sqrt{\beta_{m}^{t}}\in(0,\,1)\) and \(\phi_{m}^{t},\phi_{m}^{t}\in[0,\,2\pi)\). \(\beta_{m}^{t}\) and \(\beta_{m}^{t}\) denote the transmission and reflection coefficients for the \(m\)th element, respectively: \(\phi_{m}^{t}\) and \(\phi_{m}^{t}\) denote the phase shift for the \(m\)th element, respectively. All STAR-RIS elements are assumed to work in the energy splitting protocol [6], which means the signal arriving on each element is divided into the transmitting signal and reflection signal, with an energy splitting ratio of \(\beta_{m}^{t}:\beta_{m}^{r}\). Since the total signal energy equals the transmitted and reflected energies, the transmission and reflection coefficients always satisfy \(\beta_{m}^{t}+\beta_{m}^{r}=1\) for each element. In this case, the phase shift matrix of the transmission link \(\mathbf{\Theta}_{t}=diag(\sqrt{\beta_{1}^{t}}e^{i\phi_{1}^{t}}\),..., \(\sqrt{\beta_{M}^{t}}e^{i\phi_{M}^{t}})\) and the phase shift matrix of the reflection link \(\mathbf{\Theta}_{r}=diag(\sqrt{\beta_{1}^{r}}e^{i\phi_{1}^{t}}\),..., \(\sqrt{\beta_{M}^{r}}e^{i\phi_{M}^{t}})\) and \(diag(\cdot)\) represents a diagonal matrix.