# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 1 Introduction  
Nowadays radars are the basic element in Air Traffic Control (ATC) networks. Modern sensors as Wide Area Multilateration (WAM) [1] and Automatic Dependent Surveillance-Broadcast (ADS-B) [2] have higher performance than radars with lower cost, so in the future it's expected they will partially take the place of radars. In any case, radars will still be used as a backup network for many years. The type of radar most used in ATC sensor network is the Secondary Surveillance Radar (SSR), being Mode S radar an enhanced version of this system [3]. SSR is a rotating 2D radar sending an interrogation though an antenna with high directivity in azimuth and low directivity in elevation (2D antenna). Afterwards the aircraft transponder replies with an answer codifying barometric altitude (Mode C) and identification (Mode S and/or Mode A) data. SSR radars estimate the position of the aircraft in polar coordinates, using the distance, the azimuth (both measured by the radar) and the altitude (measured by an airborne barometric altimeter). To enable surveillance of wide areas radars with overlapped coverages are deployed. Therefore, aircraft measures (plots) from different sensors are sent to a control center where the data from all sensors is fused in order to get a unique trajectory estimation (track) for each aircraft. All the measures are transformed from radar polar local coordinates to a common coordinated system, usually Cartesian coordinates projected over the stereographic plane [4]. To make this data fusion process stable and accurate, the systematic errors affecting the measures from each radar must be corrected before the change of coordinates. If this correction isn't applied the measures of the same aircraft from different sensors are misaligned and the track could be either very unstable (apparent zig-zag maneuvers induced by measurements misalignment), or in extreme cases it could even be split into several tracks (i.e., one per sensor). Radar data fusion systems usually have an algorithm that estimates the misalignment error using the measures from the current aircraft in the airspace (opportunity traffic). These algorithms are usuallybased in the least squares (LS) method (or similar approaches, such as WLS, or MSE estimators [5]) and have the need of a parametrized mathematical model that links the systematic errors in azimuth, distance and altitude with the sensor physical non-idealities. In the abundant literature about biases estimation methods in ATC networks such as [6; 7; 8], the mathematical bias models used are simple (modeling just a range and azimuth offset and a range bias term proportional to range) because the objective of the papers was showing the estimation methods. Recently in [9] a novel registration algorithm has been presented where the track state and sensor biases are estimated simultaneously modeling the sensors with range and azimuth offset. Other related examples can be found in [10; 11]. In real applications, these models are too simple, resulting in reduced performance when used with real data. At the same time, the development of mathematical models for biases is a classic topic for radio electric signals. For instance, in [12] a model is developed for measurement errors in azimuth for antennas in Deep Space Networks. Meanwhile, [13; 14; 15; 16; 17] it's developed an azimuth bias model for 3D antennas. Due to the dispersion of the literature about the systematic error modelling, it is apparent quite often both in the literature and in real systems there is much effort on improving estimation processes but little effort is given to rigorous modelling of the error sources. The objective of this paper is to derive a complete systematic error model for ATC radars enabling an important improvement of the bias estimation processes. In Section 2 we will introduce a measurement model for azimuth, distance and altitude measurements. Then, in Sections 3-5, we will derive the relation between the measurement error and the measurement biases terms.