We can now rewrite eq 3 as an upper-bound
percentage error in velocity, expressing each of
each selected time throughout the record. If the
the differentials as finite errors ∆:
actual vertical angle β representing any point in
the frequency band were known, the velocity at
∆λ ∆fd
∆v
= tan β ∆β + tan α ∆α +
+
(6)
the corresponding time could be obtained from eq
.
λ
v
fd
2. However, these angles are not generally known.
The apparent far and near edge returns are the
Our vertical angle measurement by inclinometer
most readily identifiable locations on the frequen-
of 25 was accurate to within 2, and the hori-
cy band. The width of this band depends on the
zontal angle alignment to the maximum velocity
antenna beam width and the threshold setting on
at 0 was accurate to about 5. The upper-bound
the power spectrum of the backscattered signal.
percentage error contributions of these terms are
Our initial step in data reduction is to replace the
then 0.018 and 0.0076, respectively. A source fre-
frequency scale of the data band by a velocity
quency of 5 GHz and an upper bound frequency
scale using eq 2 with the boresight angle as β. The
error of 100 MHz yield a percentage error in λ of
center of this band is the approximate ice velocity,
0.02. The percentage error in fd is
and it can be obtained immediately. For more pre-
cise work during postprocessing, the apparent far
∆fd
1
and near edge velocities v+ and v, corresponding
=
(7)
fd
N
to the upper and lower edges of the band, are re-
lated to each other and the desired velocity v as
where N is the number of samples acquired and
cos β
cos β
processed per second for each Doppler frequency
v+ =
v=
(9)
v ,
θa -
θa
determination. In our system, the sampling fre-
cosβ +
cosβ -
quency N was 1024 Hz, corresponding to a negli-
2
2
gible digital sampling error of less than 0.001.
where θa is an apparent beam-width angle. The
The sum of the absolute values of the upper-
bound error estimates in eq 6 yields ∆v/v equal
velocity v corresponding to the antenna boresight
to 0.047.
is bounded above by v+ and below by v . We ob-
The beam width of the radar antenna intro-
tain the difference between the edge velocities as
duces another potential error, in addition to those
a percentage of the velocity v from eq 9 as
given by eq 6. The antenna beam width corre-
sponds to an area on the surface of the ice that is
θ
θ
cosβ - a - cosβ + a
illuminated by the radar and returns backscatter
2
2
v+ - v-
=
(10)
.
cos β
v
band of Doppler frequencies represents the back-
scatter from across the illuminated area. The near
The near and far edge velocity ratios v/v and v+/
edge of the beam corresponds to the largest verti-
v are plotted in Figure 5 as a function of θa for a
cal angle and the upper-bound error. Following
range of boresight angles β. The potential beam-
the approach taken above, we obtain an upper-
bound estimate of this error by replacing β by β +
θ/2 in eq 2 and taking the partial derivative of
1.2
velocity with respect to θ/2:
θ
θ θ
v
= v tan β +
.
(8)
0.8
(θ / 2)
2
2 2
β = 35
β = 25
Far Edge
When is θ greater than 10, the beam-width error
β = 15
0.4
β = 15
given by eq 8 is potentially larger than the sum of
β = 25
Near Edge
β = 35
the upper-bound errors in eq 6.
0.0
0
10
20
30
40
50
VELOCITY FROM A DOPPLER FREQUENCY
θa ()
BAND AND A VIDEO RECORD
Figure 5. Relationship between the velocity ratio and
θa for both the near and far edges of the beam width
Systems with large beam width antennas re-
and a range of β values.
quire additional data analysis to obtain an accu-
5
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