Campbell CR800 Specifications Page 152

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Section 6. Data Table Declarations and Output Processing Instructions
6-34
The algorithm for σ(θu) is developed by noting (Figure 6.4-4) that
Cos U / s ; where
ii
(') '
Θ
Θ
Θ
Θ
iii
u
=
=
FIGURE 6.4-5. Standard Deviation of Direction
The Taylor Series for the Cosine function, truncated after 2 terms is:
Cos ( ') ( ' ) /ΘΘ
ii
≅−12
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,
ss'S
ii
=
+
Equating the two expressions for Cos (θ) and using the previous equation for
s
i
;
12
2
−=+(')/ /(' )Θ
iii
UsS
Solving for
(')Θ
i
2
, one obtains;
(') / (')'/ '/ΘΘ
iiiii
U S sS sS
22
22 2=− +
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
ΘΘ ΘuNUSsNS
iii
i
N
i
N
22 2
11
21==
==
s
i
Θ
'
i
Θ
u
U
i
U
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