## 1 Introduction
### Motivation and Contributions of This Paper  
The motivation of this paper is to develop an analytical non-stationary RS-GBSM based channel model for S2A communication over an open area, with considerations of both large scale attenuation and small scale fading. The contributions are as follows. 1. The time variant geometrical structure of the S2A system including a ellipsoidal ground scattering region is presented with regional UD of the ground scatterers. 2. An analytical non-stationary channel model is developed, including three paths, i.e., the LOS path, the ground specular reflection path, and the ground scattering path. Non-stationary and stationary analytical expressions of channel impulse responses (CIRs), transfer functions (TFs), auto-correlation functions (ACFs), and power spectrum density (PSD) are derived. 3. The non-stationary characteristics of the channel is observed through the simulation of CIRs, TFs, ACFs, PSD, and etc. 4. Based on the UD assumption of the scatterers, distributions of the path attenuation, the time delay, and the Doppler shift of the scattering path are derived, simulated, and fitted. The rest of this paper is organized as follows. In Section 2, we present the geometrical structure and the assumptions. In Section 3, we develop the non-stationary S2A channel model, formulate the analytical expressions of the CIRs, the TFs, and the ACFs. In Section 4, we deduce the cumulative distribution functions (CDFs) and probability density function (PDFs) of the path delay, the path attenuation, and the Doppler shift based on UD of the scatterers. In Section 5, we present the simulation results, analyze them, and present the fitting functions of the CDFs of the path attenuation and Doppler shift. Finally, in Section 6, we provide the conclusions. Notation: boldface uppercase letters stand for matrices while boldface lowercase letters represent vectors. The superscript \((\cdot)^{\text{T}}\) denotes the transpose operator. The superscript \((\cdot)^{\text{*}}\) denotes the conjugate operator. \(|\mathbf{x}|\) is the norm of a vector \(\mathbf{x}\). \(E\{\cdot\}\) denotes expectation operation. \(\exp{(\cdot)}\) denotes the natural  
In Figure 1, \(\mathbf{T}\) is the satellite transmitter, \(\mathbf{R}\) is the aircraft receiver. The altitudes of \(\mathbf{R}\) and \(\mathbf{T}\) are \(h_{\mathbf{R}^{\prime}}\) and \(h_{\mathbf{T}^{\prime}}\) respectively. \(\mathbf{O}^{\prime}\) is the center of the earth, and \(\mathbf{O}\) is the ground specular reflection point. In this paper, \(\mathbf{O}\) is determined with C.Wagner algorithm [22], and is set as the origin of the Cartesian coordinates. \(\theta=\frac{1}{2}\angle ROT\) is the reflection angle. The YOZ plane is determined with \(\mathbf{R}\), \(\mathbf{O}\), and \(\mathbf{T}\), and the Z axis is set alone \(\mathbf{O}^{\prime}\mathbf{O}\). Y axis is set to be perpendicular to Z axis and directing to \(\mathbf{T}\). \(h_{\mathbf{T}}\) and \(h_{\mathbf{R}}\) are the heights and also the Z axis coordinates of \(\mathbf{T}\) and \(\mathbf{R}\). \(\Phi=\gamma_{1}+\gamma_{2}=\angle RO^{\prime}T\). According to the concerned literatures, such as [2; 23], the earth surface should be modeled as the spherical surface. Since the atmospheric refraction should be considered, effective earth radius \(r_{e}\) is adopted, as shown in Figure 1. In this paper, the longitudes, latitudes, and altitudes of the satellite and the aircraft are assumed to be known. And the satellite is assumed to be a Geosynchronous Earth Orbit (GSO) satellite. Similar to [2], the effects of tropospheric are not considered in this paper since ground scattering is what we focus on and the effects of tropospheric is quite stable in the scenario we discuss according to [24]. In Figure 1, since the transmitter and the receiver might moving during communication, the distances and the AOAs are time variant. Thus, in most of the situations the channel is non-stationary. According to