Campbell Scientific Wireless Sensor Network (CWB100, CWS220, and CWS900) User Manual
Page 62
Appendix B. Measurement Names and Meanings
Standard deviation of wind direction,
σ(Θu), using Campbell Scientific
algorithm:
σ(Θu)=81(1-
U
/S)
1/2
The algorithm for
σ(θu) is developed by noting (Figure B.3-3) that
Cos U / s ;
where
i
i
(
')
'
Θ
Θ
Θ
Θ
i
i
u
i
=
=
−
FIGURE B.3-3. Standard Deviation of Direction
The Taylor Series for the Cosine function, truncated after 2 terms is:
Cos (
')
(
') /
Θ
Θ
i
i
≅ −
1
2
2
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
+
Solving for
( ')
Θ
i
2
, one obtains;
(
')
/
( ' )
'/
'/
Θ
Θ
i
i
i
i
U
S
s S
s S
2
2
2 2
2
= −
−
+
i
Summing
( ')
Θ
i
2
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
i
i
i
N
i
N
2
2
2
1
1
2 1
=
=
−
−
=
=
∑
∑
NS
i
B-6