## 2 Methodology
### Radio Interferometer Phase Detection  
The ground-based interferometer measurement system (Figure 2) retrieves information from an emitting source. It combines the signal received by different antennas at different locations. Each pair of receiving antennas forms a baseline and delivers an interferometric phase observation (\(\alpha_{ij}\)). More baselines can be added in order to improve the performance of the system. For orbit determination purposes, three channels and two baselines are required in order to retrieve the azimuth and elevation angle of arrival data. The interferometric signal is generated by means of a zero-lag correlation method, which processes the base-band of the downconverted radiofrequency signals. It compensates the coarse delay differences caused by the antenna locations and cabling. Uncorrelated receiver noise is strongly rejected in the correlation process, resulting in a high processing gain, defined as [28]:  
\[G_{\text{corr}}=\text{BW}\cdot\tau \tag{1}\]  
where BW corresponds to the bandwidth of the received channel and \(\tau\) is the integration time. Since Astra DVB-S TV transmissions of opportunity work with 30 MHz-bandwidth channels, using integration times in the order of one second results in a theoretical processing gain of 74.77 dB. Different GEOSAR concepts will allow working with similar values of processing gain owing to smaller bandwidths being able to be compensated by increasing the integration time. The correlators provide complex interferometric observations where the phase provides a high-resolution observation of satellite signals' angle of arrival. The magnitude of the correlation provides complementary information on the received signals' SNR and alignment of different channels delays before correlation. Since the displacements on the position of the satellite produce major changes in the interferometric phases of the available baselines, the interferometric phases will be used as observables to determine the satellite orbit. Mathematically, the interferometric phase acquisition of a single baseline is described as follows. The emitted complex signal by the target satellite along with its carrier is:  
\[x(t)=s(t)e^{i(2\pi f_{c}t+\phi_{0})} \tag{2}\]  
where \(s(t)\) is the low-pass complex modulation signal, \(f_{c}\) is the carrier frequency, and \(\phi_{0}\) is a constant phase. The arriving signal at the antenna, \(v(t)\), is assumed to be a time-delayed transmitted signal with added noise and amplitude attenuation caused by the path loss, \(A\). \[v(t)=A\cdot x(t-t_{d})+\omega(t) \tag{3}\]  
\[v(t)=A\cdot s(t-t_{d})\cdot e^{i(2\pi f_{c}(t-t_{d})+\phi_{0})}+\omega(t) \tag{4}\]  
\[v(t)=A\cdot s(t-t_{d})\cdot e^{i(2\pi f_{c}t-\alpha_{d}+\phi_{0})}+\omega(t) \tag{5}\]  
where \(t_{d}\) corresponds to the time delay, \(\alpha_{d}\) is the phase shift due to the time delay, and \(\omega(t)\) is the Additive White Gaussian Noise (AWGN) added by the receiver channel. Figure 2: Interferometric orbit retrieval block diagram. The system correlates the satellite signal received by different antenna sites at different locations. The resulting interferometric phase \(\alpha_{ij}\) is used as the input observable in the orbit determination module, which is based on the combination of a GEO dynamical model, which simulates the trajectory of the satellite, and an interferometric measurement model, which simulates the acquired phases. The estimated trajectory corresponds to the best fit between the experimental measurements and the modeled observations.