Campbell Scientific CR9000X Measurement and Control System User Manual
Page 223
Section 6. Data Table Declarations and Output Processing Instructions
For deviations less than 40 degrees, the error in this approximation is less than
1%. At deviations of 60 degrees, the error is 10%.
The speed sample may be expressed as the deviation about the mean speed,
s
s' S
i
i
= +
Equating the two expressions for Cos (
θ‘) and using the previous equation for
;
s
i
1
2
2
−
=
+
( ') /
/ ( '
)
Θ
i
i
i
U
s S
( ')
Θ
i
2
( ')
/
( ')
'/
'/
Θ
Θ
i
i
i
i
U S
s S
s S
2
2
2 2
2
= −
−
+
( ')
Θ
i
2
Solving for
, one obtains;
i
Summing
over N samples and dividing by N yields the variance of
Θu.
Note that the sum of the last term equals 0.
( (
))
( ') /
(
/ )
(( ')
') /
σ
Θ
Θ
Θ
u
N
U S
s
NS
i
i
i
i
N
i
N
2
2
2
1
1
2 1
=
=
−
−
=
=
∑
∑
) /
Θ
s
NS
∑
(( ')
'
i
i
2
The term,
, is 0 if the deviations in speed are not
correlated with the deviation in direction. This assumption has been verified in
tests on wind data by CSI; the Air Resources Laboratory, NOAA, Idaho Falls,
ID; and MERDI, Butte, MT. In these tests, the maximum differences in
σ
σ
(
) (
( ') / )
(
) ( (
/ ))
/
/
Θ
Θ
Θ
u
N
u
U S
i
=
=
−
∑
2
1 2
1 2
2 1
and
have never been greater tan a few degrees.
The final form is arrived at by converting from radians to degrees (57.296
degrees/radian).
σ
(
) ( (
/ ))
(
/ )
/
/
Θu
U S
U S
=
−
=
−
2 1
81 1
1 2
1 2
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