## 1 Introduction  
Satellite communication (Satcom) has become an outgrowth of the continuing demand for higher capacity, real-time communication and wider coverage, due to its unique ability to provide seamless connectivity and high data rate [1; 2; 3]. In addition, Satcom is a more economical solution to provide a reliable and high speed connectivity than deploying other terrestrial networks, especially in some remote and sparsely populated locations [4; 5]. However, Satcom systems are prone to the practical masking effect, where the line-of-sight (LOS) communication between the satellite and terrestrial user may be blocked due to obstacles and shadowing. For the reasons mentioned above, a hybrid satellite-terrestrial cooperative network (HSTCN) has been provided to overcome the disadvantages [6; 7; 8; 9; 10; 11; 12; 13; 14]. In [7], the authors analyzed the system performance of the satellite-terrestrial networks by using maximal ratio combining (MRC) over shadowed-Rician fading channel. In [8], the authors addressed the problem of amplify-and-forward (AF) relaying in HSTCN, where a masked destination node could receive both the direct transmitted signal and relayed signal from a terrestrial link and the symbol error rate (SER) of the considered system is derived. In [9], the authors proposed the beamforming (BF) and combining scheme for a two-way AF protocol based communication between two multi-antenna earth stations, where the asymptotic expression for the SER of the considered system is also obtained. In [10], the authors analyzed the SER for the HSTCN with AF protocol. Besides, the authors in [11] derived the analytical expression for the ergodic capacity of the HSTCN. In [12], the authors examined the problem of AF based relaying in a hybrid satellite terrestrial link and derived the novel expressions for the SER of the considered system. In [13], the authors investigated the performance of integrated wireless sensor and multi-beam satellite networks under terrestrial interference and derived the closed-form approximations of capacity per beam. In [14], the authors proposed the energy efficient optimal power allocation schemes in the cognitive satellite terrestrial networks for non-real-time and real-time applications and maximized the energy efficiency of the cognitive satellite user. Multiple terrestrial relays can bring significant system performance growth by increasing the diversity gain of the network. Relay selection scheme can get the balance between the complexity and efficiency. Among the selection schemes, full relay and partial relay selection schemes are two important parts [15; 16]. Opportunistic relay selection scheme is the special case of the partial relay selection scheme [17; 18]. Until now, several open literatures have incorporated the relay selection scheme into the HSTCN. For example, in [19], the authors derived the exact outage probability of the HSTCN, where a selection scheme of the best relay terminal was performed. In [20], the authors analyzed the multiple terrestrial relay and multiple user HSTCN with max-max selection scheme, where the analytical expression for the outage probability (OP) was derived. In [21], the authors studied the ergodic capacity of the HSTCN with multiple terrestrial relays using full relay selection scheme, where all terrestrial relays cooperate with each other in transmitting the source signal. Furthermore, the authors in [22] analyzed the system performance of the multiple secondary networks in a cognitive HSTCN environment with partial selection scheme. However, in practice, hardware is not always ideal. Many types of impairments may affect the hardware, for example, phase noise, I/Q imbalance, and high power amplifier nonlinearities [23; 24; 25; 26; 27; 28; 29; 30]. In [26; 27], the authors studied the outage performance of orthogonal frequency-division multiplexing dual-hop opportunistic AF relaying in the presence of I/Q imbalance (IQI) in all nodes.