Aerospace and Electronic Systems Magazine February 2018 - 27
Holland
be calculated given the known frequency of the downlink and the
known satellite state at any given time.
Equation (1) can be then be simplified to
hkBFO ( x k , s k ) = ΔFkup ( x k , s k ) + δ f kcomp ( x k ) + δ f kdet ( s k ) ,
(2)
where δ f kdet ( s k ) is effectively a known deterministic value for any
time step k. The other terms in (2) couple the aircraft state xk by
way of the aircraft position and velocity to the BFO as per the following equations adapted from [2]4:
ΔFkup ( x k , s k ) =
δ f kcomp ( x k ) =
F up ( v s − v x ) ( p x − p s )
,
c
p x − ps
T
(3)
F up ( vˆ x ) ( pˆ x − pˆ s )
,
c
pˆ x − pˆ s
T
(4)
where the dependence on the time step k on the right-hand side of
the equations has been removed for simplicity of notation. In (3)
and (4),
C
· is the three-dimensional Cartesian distance;
C
Fup is the uplink carrier frequency;
C
c is the speed of light;
C
vs is the velocity vector of the satellite;
C
vx is the velocity vector of the plane;
C
ps is the position vector of the satellite;
C
px is the position vector of the plane;
C
C
C
Figure 3.
BTO rings during the MH370 flight [3]. The latitude of the arrow shown
for each ring is arbitrary.
ΔFkup ( x k , s k ) =
vˆ x is the SDU's estimate of the plane's velocity vector, which
is obtained using the plane's track angle and ground speed,
while assuming the vertical speed is zero;
pˆ s is the SDU's estimate of the position vector of the satellite,
which assumes the satellite is at its nominal oribital slot of
0° N and 64.5° E; and
pˆ x is the SDU's estimate of the position vector of the plane,
which is obtained using the plane's latitude and longitude
while assuming the plane is at sea level.
EFFECT OF UNCOMPENSATED VERTICAL VELOCITY
The vertical speed of the plane is not used in the SDU Doppler
compensation. As such, there is a direct contribution of Doppler because of the proportion of the vertical velocity vector
projected onto the radial direction from the aircraft to the satellite. It is straightforward to understand that if the plane was
directly below the satellite, the vertical velocity vector would
be fully toward or from the satellite if the plane was climbing or
descending. The direct contribution of the Doppler to the BFO
in that case would be governed by the following standard Doppler equation:
4
The sign convention used in [2] is opposite that used in (3).
FEBRUARY 2018
vz ·F up
,
c
(5)
where Fup and c are as previously defined and vz is the vertical
speed of the plane. Substituting an uplink frequency of 1,646.6525
MHz (the uplink frequency stated in [1]) and a vertical velocity
of 100 feet per minute (fpm), equivalent to 0.508 meters per second, (5) implies that the predicted BFO would increase by 2.8 Hz
per 100 fpm of climb rate or decrease by 2.8 Hz per 100 fpm of
descent rate if the plane were directly below the satellite. This is
the maximum possible contribution of the plane's climb or descent
rate to the BFO. In the more general case, (5) is moderated by the
sine of the elevation angle θ from the aircraft to the satellite. This
is expressed as
ΔFkup ( x k , s k ) =
vz ·F up sin(θ )
.
c
(6)
As such, at 00:19Z (the time at which the plane crosses the
seventh arc, provided by Figure 3),5 where the elevation to the
satellite is 38.8°, the contribution to the BFO of climb or descent
rate is reduced to approximately +1.7 or −1.7 Hz per 100 fpm,
respectively.
EFFECT OF TRACK ANGLE ON BFO TOWARD END OF FLIGHT
Another factor that needs to be considered in interpreting
the BFOs at 00:19Z is the track angle of the aircraft. In [2],
Figure 5.6, a set of curves was shown, illustrating the relationship of the BFO error and the aircraft track angle at 18:39Z. That
5
The arcs referred to in this article and other MH370 literature are
segments of the rings in Figure 3.
IEEE A&E SYSTEMS MAGAZINE
27
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