# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 6 Bias Estimation with Opportunity Traffic  
As the bias error varies slowly with time, the error model parameters must be estimated dynamically. In a radar network the main real time information source are the measures taken from the opportunity traffic and they are used for bias estimation. With the measures of a single radar the biases cannot be determined because there is not a reference in order to get the projection of the biases over the horizontal measurement plane. The projection of the biases depends on the position of the aircraft and the radar.  
In this section, we will describe a method for the estimation of the bias parameters of a pair of radars with overlapped coverage. The purpose of the described method is not to describe an optimal/scalable process implementable in a real sensor network with more than two radars, but to analyze in the results section the adequacy of the proposed bias models and their effectiveness to remove systematic errors. More advanced methods may be found in the literature [6; 7; 8; 9; 13; 14; 15; 16; 17], and the application/extension of some of them will be part of our future research.  
From the previous sections, bias model parameters of the \(i\)-th radar can be arranged in a 12D vector:  
\[b_{i}=\left[\Delta\theta_{0}\;s_{ant}\;t_{axis}\;s_{axis}\;\Delta R_{x}\; \Delta R_{y}\;s_{enc,s}\;s_{enc,c}\;\Delta\rho_{0}\;\alpha_{1}\;\alpha_{2}\; \alpha_{3}\right]^{\prime} \tag{59}\]  
On the other side, altitude bias terms due to the atmosphere conditions are common to every radar and they are included in other bias vector:  
\[b_{atm}=\left[\Delta H_{p}\;\Delta T\right]^{\prime} \tag{60}\]  
In our simple example method biases will be estimated with the difference of measurements (pseudomeasurements) of the same aircraft taken by different sensors at the same time. This pseudomeasurements will be obtained in the common plane subtracting the horizontal projection of the same aircraft measures. As in real scenarios the measurements of the radars are not synchronized the measurements of one of the sensors need to be interpolated to the time of the other radar, following a process as proposed in [7]. This interpolation is made between two measures from the same radar to the measure time from the other radar (between the times from the other two measures).  
Figure 11: Representation of altitude bias for EUROCONTROL model, linearized model and expected estimation for \(\Delta H_{p}\)= 300 m and \(\Delta T\) = 15\({}^{\circ}\).  
Assuming that biases are small the projection of the bias terms may be approximated with a linearized function. With this linearization the measurement model for each of the plots projected in stereographic coordinates can be expressed as:  
\[X_{m,i}=\left(\begin{array}{c}x_{m,i}\\ y_{m,i}\end{array}\right)=\left(\begin{array}{c}x\\ y\end{array}\right)+H_{bi}b_{i}+H_{b,atm}b_{atm} \tag{61}\]  
where (\(x\),\(y\)) is the ideal horizontal position of the aircraft and (\(x_{m,i},y_{m,i}\)) is the measured positon of the aircraft by the \(i\)-th radar while \(H_{bi}\) and \(H_{b,atm}\) are the bias projection matrices.  
The projection matrices (\(H_{bi}\) and \(H_{b,atm}\)) are a composition of the linearized change of coordinates from Equation (1), the linearization of the projection to the stereographic plane, and the linearization of the bias models for each of the bias components, so \(H_{bi}\) and \(H_{bi,atm}\) may be expressed as:  
\[H_{bi}=\left[\begin{array}{cc}\frac{dx}{d\Delta b_{0}}&\frac{dy}{d\Delta b_{ 0}}\\ \frac{dx}{d\Delta y_{m}}&\frac{dy}{d\Delta y_{m}}\\ \frac{dx}{d\Delta x_{m}}&\frac{dy}{d\Delta x_{x}}\\ \frac{dx}{d\Delta x_{y}}&\frac{dy}{d\Delta x_{y}}\\ \frac{dx}{d\Delta x_{y}}&\frac{dy}{d\Delta x_{y}}\\ \frac{dx}{d\Delta x_{x}}&\frac{dy}{d\Delta x_{y}}\\ \frac{dx}{d\Delta x_{y}}&\frac{dy}{d\Delta x_{y}}\\ \frac{dx}{d\Delta x_{y}}&\frac{dy}{d\Delta x_{y}}\\ \end{array}\right]H_{bi,atm}=\left[\begin{array}{cc}\frac{dx}{d\Delta H_{p} }&\frac{dy}{d\Delta H_{p}}\\ \frac{dx}{d\Delta T}&\frac{dy}{d\Delta T}\end{array}\right] \tag{62}\]