# Detection and Tracking of Weak Exoatmospheric Target with Elliptical Hough Transform
## 5 Simulations
### Field Data Analysis  
To evaluate the effectiveness of the presented EHT method, we also use filed data to investigate the overall performance. The radar is a phased-array radar, which is manufactured by the China Electronics Technology Group Corporation (CETC) and is operated at the X-band. The data we analyzed involves a constellation comprising three identical scientific satellites of the European Space Agency (ESA), which are designed to provide the best survey of the geomagnetic field and its temporal evolution of the Earth system. To meet the need for the detection accuracy of the magnetic field of the Earth system, the satellite constellation consists of three circular orbits, with satellites A and C approximately 450 km above the ground, and satellite B 530 km above the ground. Because these data span different periods, we use only the real tracking data of satellite A to investigate the performance. The shape of satellite A is shown on the left of Figure 15.  
Figure 14: Accumulation effect of EHT with two targets (Scheme II, Plane-constrained integration, with \(q=0.001\), (150–250) s), showing in \(a-b\) space. (**a**) Accumulation effect in Hough parameter. (**b**) Detection result in THIS-CS. Figure 13: Comparison of detection and tracking performance with conventional Kalman filters. (**a**) Presented EHT method. (**b**) Conventional Kalman filter with NN association strategy. The orbital elements of satellite A are listed in Table 2. It can be seen from Table 2 that, the trajectory is approximately a part of a circular orbit, as the eccentricity \(e=0.003368\) is close to zero. According to the basic knowledge of astronomy, we can calculate the true elliptical parameters (\(a\),\(b\),\(\theta\)) = (6811.4487 km, 6811.4483 km, 0\({}^{\circ}\)). Since the target is a satellite and it has a circular orbit, the range angle \(\theta\) indeed can be of any value. Figure 16 shows the raw measurements of the satellite when tracked in the radar centered spherical CS. The data is collected on 8 May 2017. The total observation time is approximately 272 s of its exoatmospheric flight, with the range in the interval [1296 km, 1950 km], azimuth in the interval [225.7384\({}^{\circ}\), 300.6767\({}^{\circ}\)], and elevation in the interval [4.4123\({}^{\circ}\),14.6599\({}^{\circ}\)]. Since the radar is a phased-array radar and has the ability of adaptive scheduling, the maximum revisiting period is 3.94 s, the minimum period is 0.067 s, and the average is 0.5 s. The measurement accuracies of the range, azimuth, and elevation of the radar are approximate \(\sigma_{R}=200\) m, \(\sigma_{A}=0.00228\) rad, and \(\sigma_{E}=0.0011769\) rad, respectively. \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Quantity** & **True Value** \\ \hline a & semi-major axis & 6811.44871 km \\ \hline e & eccentricity & 0.0003368 \\ \hline i & inclination & 87.3681\({}^{\circ}\) \\ \hline \(\Omega\) & right ascension of ascending node & 173.0810\({}^{\circ}\) \\ \hline \(\omega\) & argument of periapsis & 96.9899\({}^{\circ}\) \\ \hline f & true anomaly & 263.1742\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of Satellite A.  
Figure 16: Real satellite measurement data in the radar-centered spherical CS. Figure 15: Shapes of satellites A and B. With a series of coordination transformations, the satellite trajectory in the presented THIS-CS can be obtained. As can be seen in Figure 17, the trajectory of the orbit is indeed a part of a circular orbit, with an average height of approximately 449 m. Due to the radar line-of-sight limitation, the radar can only observe a small part of the entire circular orbit. Since the raw data set is the independent tracking data containing the raw measurements of range, azimuth, and elevation, and for some reason lacking the SNR information, obviously, it cannot be used directly to evaluate the performance of the elliptical Hough transform, as the Hough transform usually needs target data, as well as clutter data.