# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 3 Azimuth Bias
### Propagation Error  
The index of refraction of the atmosphere changes with the altitude and this affects to the local speed of light. As the conversion between time and range is made with a constant speed of light, the determination of the position of the aircraft from the radar generally has biases (due to change of speed of light with altitude and weather conditions). As the bias is produced by accumulated error in the speed of light along the propagation path, the range bias will be a function of the target range. In order to model the relation between the ideal range and the bias we will use a spherically stratified atmosphere. The exponential model of the refractivity [23] is used because it models the typical atmosphere variations in the rank of altitude used in ATC applications:  
\[N(h)=N_{s}exp\left(-\frac{h}{H}\right) \tag{37}\]  
where \(N(h)\) is the refractivity at a given altitude (\(h\)) over mean sea level, \(H\) is a reference altitude (equal to 6950 m), and \(N(0)=N_{s}\) (with typical values between 300 and 350). The relation between the refractivity and the index of refraction (\(n(h)\)) is:  
\[n(h)=1+N(h)\times 10^{-6} \tag{38}\]  
The model in Equation (37) can be used also redefining altitude (h) to be referred to radar altitude, which just results in a change of the \(N_{s}\) constant, to be now the refractivity at the radar altitude over mean sea level. This will be the altitude reference and refractivity model to be used for the rest of this section. The height gradient of the index of refraction bends the ray, resulting in a slightly increased distance from that of the straight line used to model propagation on free space. Specifically, as it is deduced in [24] using a ray tracing procedure with the Snell's law for spherically stratified media [25], the geometrical distance of the bent ray path may be calculated as:  
\[s(h_{t},\varphi_{0t})=\int_{0}^{h_{t}}\frac{dh}{\sqrt{1-\left\{n_{0}\cos( \varphi_{0t})\left[\frac{n(h)}{1+\frac{h}{K_{e}}}\right]\right\}}} \tag{39}\]  
where \(\varphi_{0t}\) is the elevation ray angle observed by the radar, \(h_{t}\) is the target altitude relative to the radar, \(n_{0}\) is the index of refraction at radar altitude, and \(R_{e}\) is the Earth radius plus radar height. Additionally, the speed of light varies as it traverses different heights with different associated index of refraction. The effective distance (measured distance assuming constant speed of light, is affected by the propagation error), and can be calculated as proposed in [24]:  
\[r(h_{t},\varphi_{0t})=\int_{0}^{h_{t}}\frac{n(h)dh}{\sqrt{1-\left\{n_{0}\cos( \varphi_{0t})\left[\frac{n(h)}{1+\frac{h}{k_{e}}}\right]\right\}}} \tag{40}\]  
In the following we will work with the approximation that the geometrical distance between the target and radar is almost equal to a straight-line (free-space) path length [24] (\(\rho\approx s(h_{t},\varphi_{0t})\)). Then, the range bias due to propagation can be calculated as:  
\[\Delta\rho(h_{t},\varphi_{0t})=r(h_{t},\varphi_{0t})-s(h_{t},\varphi_{0t}) \tag{41}\]  
Both distances (geometrical distance and effective distances), and their difference, can be numerically estimated for different values of \(h_{t}\) and \(\varphi_{0t}\). With these pairs we can get a table of range biases (\(\Delta\rho\)) and geometrical distances (s). In these estimations we work with the approximation that the geometrical distance between the target and radar is equal to the ray path length [24]. Without this approximation the estimations are similar, but at distances lower than 20 Km the results of the biases are different. But in these places the total bias is very low (between 2 m and 6 m, depending on the altitude) and the approximation does not introduce significant error. The resultant range error is represented in the Figure 7, where each line represents the range error for a different altitude (\(h_{t}\)) as a function of slant range. As it is seen [26] two relevant characteristics can be observed.