# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 5 Altitude Bias
The model proposed in [28] is:  
Below tropopause:  
\[h_{g}=h_{p}-\Delta H_{p}+\frac{\Delta T}{\beta_{t}}\ln\biggl{(}\frac{T_{0}+ \beta_{t}h_{p}}{T_{0}+\beta_{t}\Delta H_{p}}\biggr{)} \tag{49}\]  
Above tropopause:  
\[h_{g}=h_{g,trop}+\frac{T_{0}+\Delta T+\beta_{t}h_{p,trop}}{T_{0}+\beta_{t}h_{ p,trop}}\left(h_{p}-h_{p,trop}\right) \tag{50}\]  
where \(\Delta H_{p}\) is the barometric altitude offset with respect to ISA (which depends on the difference between current pressure and standard pressure at mean sea level), \(\Delta T\) is the difference between current temperature and standard temperature at mean sea level, \(T_{0}\) is the temperature of the standard atmosphere at mean sea level, \(\beta_{t}\) is the temperature gradient and \(h_{g,trop}\) is the geometric altitude where the barometric altitude is \(h_{p,trop}=\) 11,000 m (which should be calculated with Equation (49) to guarantee geometric height continuity). To reduce the impact of altitude biases in measurement projection in the horizontal plane \(\Delta T\) and \(\Delta H_{p}\) must be estimated and corrected. Next we will derive a parametric model for these terms. Note the altitude bias effect is important when the elevation of the aircraft is big (high altitude and short  
Figure 10: Difference between the projection in the slant range for the same range measure and different altitude. range). We will next fit and show the consistency of a linear model between the geometric altitude and the barometric altitude based on the previously presented model, resulting in:  
Below tropopause: we may approximate linearly \(ln(1+x)\approx x\) when \(x\approx 0\). Using this approximation in \(\ln\Bigl{(}1+\frac{T_{0}+\beta_{t}h_{p}}{T_{0}+\beta_{t}\Delta H_{p}}-1\Bigr{)}\) from (49) we can derive:  
\[h_{g}=h_{p}-\Delta H_{P}+\frac{\Delta T}{\beta_{t}}\left(\frac{T_{0}+\beta_{t} h_{p}-T_{0}+\beta_{t}\Delta H_{p}}{T_{0}+\beta_{t}\Delta H_{p}}\right) \tag{51}\]  
After some algebra we have:  
\[h_{g}=\left(h_{p}-\Delta H_{P}\right)\left(1+\frac{\Delta T}{T_{0}+\beta_{t} \Delta H_{p}}\right) \tag{52}\]  
and then we can solve for \(h_{p}\), resulting:  
\[h_{p}=\frac{h_{g}}{1+\frac{\Delta T}{T_{0}+\beta_{t}\Delta H_{p}}}+\Delta H_{p} \tag{53}\]  
Above tropopause: Regrouping the terms in Equation (50), and approximating \(h_{g,trop}\) using (52), the following relation is obtained:  
\[h_{g}=h_{g,trop}+\left(1+\frac{\Delta T}{T_{0}+\beta_{t}h_{p,trop}}\right) \left(h_{p}-h_{p,trop}\right) \tag{54}\]  
where:  
\[h_{g,trop}=\left(h_{p,trop}-\Delta H_{p}\right)\left(1+\frac{\Delta T}{T_{0}+ \beta_{t}\Delta H_{p}}\right) \tag{55}\]  
Then, we can solve again for \(h_{p}\), resulting:  
\[h_{p}=\frac{h_{g}-h_{g,trop}}{1+\frac{\Delta T}{T_{0}+\beta_{t}\Delta H_{p,trop }}}+h_{p,trop} \tag{56}\]  
In this section \(h_{g}\) is always the ideal geometric altitude (called \(h\) in Equation (3)) and \(h_{p}\) is the barometric altitude (measured by the barometer, called \(h_{m}\) in Equation (3), assuming negligible altitude noise). Changing the notation to that of Equation (3) the resultant model is:  
Below tropopause:  
\[h_{m}=\frac{h}{1+\frac{\Delta T}{T_{0}+\beta_{t}\Delta H_{p}}}+\Delta H_{p} \tag{57}\]  
Above tropopause:  
\[h_{g}=h_{g,trop}+\frac{T_{0}+\Delta T+\beta_{t}h_{p,trop}}{T_{0}+\beta_{t}h_{ p,trop}}\left(h_{p}-h_{p,trop}\right) \tag{50}\]0  
In both cases, the model has an offset height and a height gain. In Figure 11 the bias associated to the more exact EUROCONTROL model [28] in Equations (49) and (50), and the linearized model are represented (in Equations (57) and (58)) vs. the geometric altitude (\(h\)). Those two models use the same values for the parameters \(\Delta H_{p}=300\) m and \(\Delta T=15^{\circ}\), and the linearization has very low error at low altitudes but at high altitudes the errors between both models can be around 100 m. Meanwhile, the black line (labelled expected estimation) represents a least squares regression of the EUROCONTROL model using the model in Equations (57) and (58) with parameters \(\Delta H_{p}\) and \(\Delta T\).