# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 3 Azimuth Bias
### Propagation Error
The first one is that the range error depends on the altitude of the aircraft. The second one is that a second-grade polynomial seems to be good enough to approximate the range error dependency with the slant range for every altitude [27]. Considering only this polynomial, the range error can be expressed as:  
\[\Delta\rho(\rho,h)=\beta_{1}(h)\rho+\beta_{2}(h)\rho^{2} \tag{42}\]  
where \(\beta_{1}(h)\) and \(\beta_{2}(h)\) are polynomial coefficients, different for each altitude. This model does not allow to make a good estimation of the parameters with opportunity traffic, because the number of aircraft in each altitude layer is reduced. We must find a model which allows the simultaneous use of aircraft measures at all altitudes in the estimation process. Figure 7: Range bias for aircraft at different altitudes. A key idea in the derivation of such model is to search for decoupled range and altitude dependencies in \(\Delta\rho(\rho,h)\):  
\[\Delta\rho(\rho,h)=f(\rho)\cdot g(h) \tag{43}\]  
To do so, the quotient between a parabola in an altitude and the parabola at an arbitrary reference altitude (\(h_{r}=\) 14,000 m in the following) for the same slant range is made:  
\[g(\rho,h)=\frac{\Delta\rho(\rho,h)}{\Delta\rho(\rho,h_{r})} \tag{44}\]  
This quotient is shown as a function of the slant range in Figure 8 and can be observed that for each altitude it is almost constant. The right end of the horizontal lines in this figure corresponds to the radio-wave propagation horizon. This result demonstrates that the range bias at a specific altitude is the range bias at the reference altitude multiplied by a constant, supporting Equation (43), as the dependency of \(g(\rho,h)\) with the slant-range is negligible (i.e., it could be substituted by \(g(h)\)), and \(f(\rho)\cdot\) would be directly \(\Delta\rho(\rho,h_{r})\), a function not depending of \(h\). From those ideas, we have that:  
\[f(\rho)=\beta_{1}(h_{r})\rho+\beta_{2}(h_{r})\rho^{2}=\alpha_{1}\rho+\alpha_{2 }\rho^{2} \tag{45}\]  
In order to derive \(g(h)\) we have represented in the Figure 9 the quotient between the range bias at an altitude and the bias at a reference altitude as a function of altitude. In this figure, it can be seen  
\[n(h)=1+N(h)\times 10^{-6} \tag{38}\]0 Quotient between range bias at an altitude and bias at 14,000 m for the same slant range. \[n(h)=1+N(h)\times 10^{-6} \tag{38}\]1 Ratio between range bias at an altitude and bias at reference altitude (14,000 m) for the same slant range as a function of altitude. that a linear model is a good approximation. A second order polynomial would be an almost exact approximation but it implies the addition of another parameter that would make the convergence of the bias estimation processes slower with reduced accuracy gains. Then \(g(h)\) can be modelled with a linear function normalized to reference altitude (\(h_{r}\)):  
\[n(h)=1+N(h)\times 10^{-6} \tag{38}\]2  
Finally, the contribution of the tropospheric propagation to the range bias can be modelled, combining Equations (43), (45) and (46):  
\[n(h)=1+N(h)\times 10^{-6} \tag{38}\]3