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reducing the covariance of the state vector not only at low altitudes but also at higher altitudes, due to the dynamic
processes involved. This filtering-smoothing process works both ways, i.e. for a landing case as well as a take-off
case. Only when applying the smoother this increase in accuracy can be obtained for the higher altitudes; with a
filter only this would not have happened, or only to a much lesser degree. A typical example is given of the estimate
of the inertial vertical velocity z , as well as its standard deviation  z taken from the covariance matrix Pi /N , as
function of time as it developed for a particular landing approach in Figure 1.
5
0 50 10 0 150 200 250 300 350 400
TIME [S]
-4
-2
0
2
4
6
8
10
VERTICAL VELOCITY[M/S]
(>0 DESCENT)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
STD. DEV. [M/S]
VERT. VEL.
ST D. DEV.
TouchDown
0 50 100 150 200 250 300 350 400
TIME [S]
1400
1200
1000
800
600
400
200
0
ALTITUDE [M]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
STD. DEV. [M/S]
ALTITUDE
STD.DEV.
200 FT
TouchDown
a) vertical velocity and std. dev. b) altitude and ‘zdot’ std. dev.
Figure 1 Envelope of vertical velocity ‘zdot’, its std.dev. and altitude
As one can see the vertical velocity varies from +4 m/s (descent at 790 fpm) to near zero at t=250s, and finally
back to zero again at the end (landing). The standard deviation in the estimate of the vertical velocity starts off at
about 0.7 m/s, then drops quickly to 0.5 m/s, where it more or less stays constant at this value, and at the end it drops
further to 0.2 m/s just before landing (i.e. from the moment the radio altimeter signal is being used in the calculated
altitude, which is at 200 ft AGL or lower). This shows that the overall accuracy in the estimated vertical velocity,
and hence wind component, is in the order of 0.5 m/s (1 Kt) or better, but it also shows that it is a dynamic process.
Due to the fairly long times involved the filtered-smoothed covariance reaches steady-state values for most of the
time.
2.4Angle of attack calibration
One of the primary sources of information for the QAR data analysis is the calibrated angle-of-attack, obtained
from the angle-of-attack vane (AOA-vane). There is a relationship between the AOA-vane and the “ true”
aerodynamic angle of attack α, which is used in the calibration. This calibration, usually for a number of aircraft
configurations (i.e. different flap settings), is normally not available, and has to be derived from the QAR data.
The calibration equation generally is:
ci ao a AOAi a Fi a AOAi   Fi    1.   2.  3.  . (4)
It is assumed there is a time lag ‘ ’ between the measured vane angl e AOA and the actual calibrat ed angle of
attack ci  due to QAR recording time delays, pneumatic line time lags and all other sorts of factors that could
introduce delays. In general the delay found varied between 0.25s and 1s. The time lag was found from the peak
values in the cross-correlation between the measurement of AOA at time t and the computed inertial angle of attack
at time t+. The calibration coefficients a0 – a3 are determined once per aircraft type through a multi-linear
regression analysis.
2.5Sideslip angle estimation
2.5.1 Original estimate
A second aerodynamic parameter for calculating meteorological variabl es is the aerodynamic sideslip angle,
usually denoted by β. In all the aircraft QAR data considered so far, there is no measurement of the sideslip angle, so
an estimation process had been developed (Ref. 1). Principally it is derived from the original approximation that the
lateral force on the aircraft comes from the tail fin due to the sideslip angle. The original sideslip angle was derived
from the lateral acceleration Ay minus the bias in the lateral acceleration as:
m
Va S c y
m
Y fin
Ay y fin
2
2 1

    (5)
6
Here the side force coefficient cy is linearized with respect to β and the gradient approximated using slender airfoil
theory applied to the tailfin as
  , where   5.73

 y c y c y c (6)
and it is based on the tailfin surface area Sfin. The resulting sideslip angle was then computed, correcting the lateral
acceleration with the estimated bias y, during a flight according to
　
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