# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 2 Measurement Model  
Radars used in ATC typically use a two dimensional rotating antenna and have only the capability to measure aircraft range and azimuth. The azimuth of the target is the angle of the antenna boresight while the target altitude is measured by the aircraft navigation system and can be communicated to radar through a data link. Primary Surveillance Radars (PSR) only can determine the range and the azimuth of the target due to the lack of data link capability. Nowadays in ATC systems primary radars are used as backup system to secondary radar network. SSR is based in the same concept that PSR but in this case, the SSR has data link capability. SSR sends an interrogation message and the aircraft responds with its Mode A/Mode S (identifying code) or mode C (barometric altitude). The azimuth and the range of the target are determined respectively with antenna boresight (usually corrected using monopulse [3]) and the time lapse between the radar interrogation and its response. Range and azimuth are typically referenced to a local Cartesian coordinated system: \(y\)-axis pointing to north, \(x\)-axis pointing to East and \(z\)-axis pointing to up (ENU system). Azimuth reference has its origin in the \(y\)-axis and it grows in clockwise. Barotmetric altitude is measured by an airborne barometric altimeter with reference at mean sea level. The relation between radar coordinates \((\rho_{m},\ \theta_{m},\ h_{m})\) and local Cartesian coordinates \((x_{m},y_{m})\) is:  
\[ \begin{split} x_{m}&=\rho_{m}\sin(\theta_{m})\cos( \varphi_{m})\\ y_{m}&=\rho_{m}\cos(\theta_{m})\cos(\varphi_{m}) \end{split}  \tag{1}\]  
where \(\rho_{m},\ \theta_{m},\ h_{m}\) are the range, azimuth and altitude of the measurement and \(\varphi_{m}\) is the target elevation with respect to \(i\)-th sensor horizontal plane, which can be calculated as:  
\[\varphi_{m}=arcsin\left(\frac{2R(h_{m}-h_{i})+h_{m}^{2}-h_{i}^{2}-\rho_{m}^{ 2}}{2\rho_{m}(R+h_{i})}\right) \tag{2}\]being \(h_{i}\) the geometric altitude of the \(i\)-th sensor with reference at mean sea level. \(R\) is the local Earth radius at the radar position, calculated as described in [18]. Local Cartesian coordinates will be used as middle step in the conversion between radar measures and common tracking coordinates for all sensors. The radar measurement model can be summarized as:  
\[ \begin{split}\rho_{m}&=\rho+n_{\rho}+\Delta\rho\\ \theta_{m}&=\theta+n_{\theta}+\Delta\theta\\ h_{m}&=h+n_{h}+\Delta h\end{split}  \tag{3}\]  
where \(n_{\rho}\), \(n_{\theta}\) and \(n_{h}\) are the noise measurement errors for range, azimuth and altitude respectively, \(\Delta\rho,\ \Delta\theta\) and \(\Delta h\) are systematic error terms due to bad system calibration, and \(\rho,\ \theta\) and \(h\) are the ideal range, azimuth and altitude of the target. Noise terms are modelled typically as uncorrelated additive white noises (usually considered Gaussian distributed), while systematic error terms can be considered constants or slow time variant (bias terms). Bias terms of range, azimuth and altitude depend on target position and will be deduced in Sections 3-5 for SSR sensors. They are composed by several environmental and equipment perturbations that are common to all the measurements of each sensor. The parameters of the mathematical model may change with time, due to variable weather conditions and hardware aging. In surveillance data fusion systems the measures obtained from different kind of sensor are fused in order to improve the information update ratio and the accuracy of the aircraft trajectory estimation. At the moment of the data fusion all measurement must be expressed in a common coordinated reference system. There are several coordinated systems that can be used in ATC. For small scenarios, flat Earth model can be assumed, therefore Cartesian coordinates are precise enough to track the trajectories with several sensors. When the scenario is larger and sensors are widely separated, the Earth curvature must be considered and a different coordinate system must be used.