Specifications

Section 11. Programming Resource Library
11-31
The algorithm for σ(θu) is developed by noting (FIGURE 11.5-2) that
u' where;/sU)'( Cos
iiiii
Θ−Θ=
Θ
=
Θ
FIGURE 11.5-3. Standard Deviation of Direction
The Taylor Series for the Cosine function, truncated after 2 terms is:
2/)'(1)'( Cos
2
ii
Θ−≅Θ
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 can be expressed as the deviation about the mean speed,
S'ss
ii
+
=
Equating the two expressions for Cos (θ‘) and using the previous equation for
i
s ;
)S's/(U2/)'(1
ii
2
i
+=Θ−
Solving for
2
i
)'(Θ , one obtains;
S/'s2S/'s)'(S/U22)'(
ii
2
ii
2
i
+Θ−−=Θ
Summing
2
i
)'(Θ over N samples and dividing by N yields the variance of Θu.
Note that the sum of the last term equals 0.
∑∑
==
Θ−−=Θ=Θσ
N
1i
N
1i
i
2
i
2
i
2
NS/)'s)'(()S/U1(2N/)'())u((
The term,
∑
Θ NS/)'s)'((
i
2
i
, 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
∑
−=ΘσΘ=Θσ
2/12/12
i
))S/U1(2()u( and )N/)'(()u(
s
i
Θ
'
i
Θ
u
U
i
U