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 ∆*f*d

∆*v*

= tan β ∆β + tan α ∆α +

+

(6)

the corresponding time could be obtained from eq

.

λ

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 *f*d is

and it can be obtained immediately. For more pre-

cise work during postprocessing, the apparent far

∆*f*d

1

and near edge velocities *v*+ and *v*, corresponding

=

(7)

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

(9)

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

=

(10)

.

cos β

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-

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 *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

θa ()

θa for both the near and far edges of the beam width

Systems with large beam width antennas re-

quire additional data analysis to obtain an accu-

5