Campbell Scientific CR5000 Measurement and Control Module - Manual (Page 126)

Section 6. Data Table Declarations and Output Processing Instructions
6-32
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 i
U S s S s S 2 2 2 2 2
= − − +
Summing ( ' )
Θ
i
2
over N samples and dividing by N yields the variance of
Θ
u.
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
= = − −
= =
∑ ∑
The term, (( ' ) ' ) /
Θ
i i
s NS
2
∑
, 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
( ) ( ( / )) ( / ) / / Θ
u U S U S
= − = −
2 1 81 1 1 2 1 2