# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 3 Azimuth Bias
### Azimuth Bias Composition  
The global azimuth bias model is a composition of the three bias elements previously discussed. Neglecting noise effects, in a radar, the measured azimuth of the antenna is the azimuth where the rotation axis is pointing plus the antenna bias. The azimuth of the rotation axis is the azimuth indicated by the encoder plus the axis bias. Finally, the azimuth indicated by the encoder is the ideal azimuth plus the encoder bias. With this:  
\[\begin{array}{c}\theta_{enc}=\theta+\Delta\theta_{enc}\\ \theta_{axis}=\theta_{enc}+\Delta\theta_{axis}\\ \theta_{m}=\theta_{ant}=\theta_{axis}+\Delta\theta_{ant}\end{array} \tag{31}\]  
and then:  
\[\begin{array}{c}\theta_{m}=\theta+\Delta\theta=\theta+\Delta\theta_{ant}+ \Delta\theta_{axis}+\Delta\theta_{enc}\end{array} \tag{32}\]  
With this, the complete model for azimuth bias is:  
\[\begin{array}{c}\Delta\theta=\Delta\theta_{ant}+\Delta\theta_{axis}+\Delta \theta_{enc}\\ =\theta_{0}-s_{ant}\tan(\varphi)+\beta_{axis}\sin(\theta-\alpha_{axis})\tan (\varphi)-\frac{1}{4}\sin(2\theta-2\alpha_{enc})s_{enc}^{2}-\frac{\Delta R}{ R}\sin(\theta-\alpha_{exc})\end{array} \tag{33}\]  
This model should not be used in general in the bias estimation algorithms due to instability problems due to lack of linearity of some of part of the model, although its physical meaning is clear. The alternative equivalent model (preferred for bias estimation due to its linearity with respect to bias parameters) should be obtained composing (11), (16), (29) and (32), resulting:  
\[\begin{array}{c}\Delta\theta=\theta_{0}-s_{ant}\tan(\varphi)+[t_{axis}\sin (\theta)-s_{axis}\cos(\theta)]\tan(\varphi)+s_{enc,s}\sin(2\theta)\\ +s_{enc,c}\cos(2\theta)+\Delta R_{y}\sin(\theta)-\Delta R_{x}\cos(\theta) \end{array} \tag{34}\]  
**4. Range Bias**  
Range biases are again due to several factors. The main factors that compose the biases are:  
* Transponder delay: every aircraft has a different transponder delay (typically between \(-75\) m and \(75\) m, usually assumed to be uniformly distributed). This aircraft dependent bias must be estimated for each aircraft in the tracking phase. The range model developed in this paper is for biases due to sensor and environment. As the estimation will be made with many targets, the mean of the delays of all the aircraft tends to be near zero. For more details on this bias term and methods to estimate it, see [8].
* Temporal reference error: radars measure the difference of time between the interrogation and the arrival of the response. Usually there is a small constant error in the reference time and it produces a constant error (\(\rho_{0}\)) in the range estimation.  
* Clock error: A small error in the clock calibration may produce a linear term in the range estimation [22]. In modern radar this term can be considered negligible due to the precision of the clocks.
* Propagation error: The conversion between time and distance is made using the speed of light in the ISA (International Standard Atmosphere) at mean sea level. However, the speed of light changes along the propagation path due to the change in the refraction index with altitude and weather conditions. Any change in the real speed of light will produce a range bias term (\(\Delta\rho_{prop}\)). Considering the range biases terms previously explained, the range bias (\(\Delta\rho\), as defined (3)) results: \[\Delta\rho=\rho_{0}+\Delta\rho_{prop}\] (35)  
In cases when the clock has a bad calibration, the bias has additionally a linear term dependent on the clock error, and parameterized through a range gain factor (\(\alpha_{clk}\)), being the complete model:  
\[\Delta\rho=\rho_{0}+\alpha_{clk}\,\rho+\Delta\rho_{prop} \tag{36}\]  
Next, we will focus on the propagation term.