# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 5 Altitude Bias  
Civil surveillance radars used in ATC do not measure elevation of the targets. The reply of the aircraft transponder to the SSR interrogation codifies the altitude of the aircraft in the message (Mode C), completing the 3D position determination by the radar sensor. Aircraft altitudes may be of two different types:  
* Geometric altitude: it is the Euclidean distance between the mean sea level (MSL) and the position of the aircraft. * Baporometric altitude: this altitude is calculated from the air pressure of the atmosphere at aircraft location. The conversion between pressure and altitude is made assuming International Standard Atmosphere (ISA) model (temperature 288.15 K and pressure 101,325 Pa, at MSL) [28]. Baporometric altitude is the same that geometric altitude only when the atmospheric conditions are the ISA conditions. Baporometric altitude is used in ATC systems, because all aircraft must always measure the pressure to fly and its flight performance depends critically on this pressure, barometers are easy to be calibrated and vertical separation can be done easily using this magnitude. On board certified sensors for geometric altitude usable in all phases of flight (such as GPS) have only recently become widely available, and this information is not always available in the ground. All the surveillance systems send barometric altitude but only some of them can send the geometric altitude. Using the barometric altitude usually implies systematic errors in altitude determination that are almost equal for every aircraft and those depends on the pressure and temperature. This altitude bias term, when the polar measure is transformed to the horizontal plane, introduces a systematic error in the 2D position. The projection of this bias will be negligible when the elevation of the aircraft is close to zero, but it will be an important factor when the aircraft is near to the sensor and the elevation is big. This situation is typical in some air traffic control systems, as that of a radar close to an airport with several others far away of that airport. Then the horizontal projection of the measures from the near radar are heavily distorted due to the systematic altitude errors. The projection in the horizontal plane is implemented using the target elevation, as calculated in Equation (2). This equation is equivalent to search the cross point between the circle defined with radius \(\rho\) and the line defining the constant measured altitude. This is shown the Figure 10, where \(\rho\) is the measured range, \(h\) and \(h_{m}\) are the ideal and measured altitude respectively. \(\rho_{p}\) is the projected range calculated with the ideal altitude and \(\rho_{mp}\) is the projected range calculated with the measured altitude. In [28] a model of the atmosphere for its use in Air Traffic Management (ATM) trajectory prediction, based on ISA model, is defined. The ranges of altitudes of civil aircraft cover the troposphere and the lower stratosphere. The tropopause is the boundary between both layers. Hence there are two different models for different altitudes. In this model, the tropopause is allocated where the barometric altitude is 11,000 m. Below the tropopause the atmospheric model of barometric altitude is a non-linear function that depends on the temperature and the pressure at MSL. Above the tropopause, the model is a linear function because the gradient of temperature is zero. In the model, the barometric altitude (\(h_{p}\)) is the measured altitude and the geometric altitude (\(h_{g}\)) will be the ideal altitude. The difference between both altitudes will be considered the altitude bias.