Specifications

Section 6. Data Table Declarations and Output Processing Instructions
The algorithm for σ(θu) is developed by noting (Figure 6.4-4) that
Cos U / s ; where
ii
(') '
Θ
Θ
Θ
Θ
ii
u
i
=
=
−
s
i
Θ
'
i
Θ
u
U
i
U
FIGURE 6.4-5. Standard Deviation of Direction
The Taylor Series for the Cosine function, truncated after 2 terms is:
Cos ( ') ( ') /ΘΘ
ii
≅−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,
ss'S
ii
=
+
Equating the two expressions for Cos (θ‘) and using the previous equation for
;
s
i
12
2
−=(')/ /(' )Θ
iii
UsS+
i
Solving for
, one obtains;
(')Θ
i
2
(') / (')'/ '/ΘΘ
iiii
U S sS sS
22
22 2=− − +
Summing
over N samples and dividing by N yields the variance of Θu.
Note that the sum of the last term equals 0.
(')Θ
i
2
(( )) ( ')/ ( /) (( ') ')/ σ ΘΘ ΘuNUSs
ii
i
N
i
N
22 2
11
21==−−
==
∑∑
NS
i
6-34