Section 7. Installation
193
Standard Deviation of Direction
Figure 59: Standard Deviation of Direction
The Taylor Series for the Cosine function, truncated after 2 terms is:
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,
Equating the two expressions for Cos (θ') and using the previous equation for s
i
;
Solving for (Θ
i
')
2
, one obtains;
Summing (Θ
i
')
2
over N samples and dividing by N yields the variance of Θu.
Note The sum of the last term equals 0.
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 Campbell Scientific;
the Air Resources Laboratory, NOAA, Idaho Falls, ID; and MERDI, Butte, MT.
In these tests, the maximum differences in
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