Campbell Scientific CR10X Measurement and Control System - Manual (Page 177)

SECTION 11. OUTPUT PROCESSING INSTRUCTIONS
11-3
and Ux and Uy are as defined above.
Resultant mean horizontal wind speed, U :
U =(Ue
2
+Un
2
)
1/2
U
Un
Ue
FIGURE 11-2. Mean Wind Vector
where for polar sensors:
Ue=( ΣS
i
Sin Θ
i
)/N
Un=( ΣS
i
Cos
Θ
i
)/N
or, in the case of orthogonal sensors:
Ue=(
ΣUe
i
)/N
Un=( ΣUn
i
)/N
Resultant mean wind direction, Θ Θ Θ Θu:
Θu=Arctan (Ue/Un)
Standard deviation of wind direction,
σσσσ(Θ Θ Θ Θu),
using Campbell Scientific algorithm:
σ(Θu)=81(1-
U /S)
1/2
The algorithm for σ(θu) is developed by noting
(Figure 11-3) that
Cos U / s ; where
i i
( ') '
Θ Θ Θ Θ
i i i
u
= = −
s
i
Θ'
i
Θu
U
i
U
FIGURE 11-3. Standard Deviation
of Direction
The Taylor Series for the Cosine function,
truncated after 2 terms is:
Cos ( ') ( ') /
Θ Θ
i i
≅ −
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,
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
Θu. 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 than a few degrees.
The final form is arrived at by converting from
radians to degrees (57.296 degrees/radian).
σ( ) ( ( / )) ( / ) / / Θu U S U S = − = − 2 1 81 1 1 2 1 2
*** 70 SAMPLE ***
FUNCTION
This instruction stores the value from each
specified input location. The value(s) stored are
those in the input location(s) when Instruction
70 is executed with the Output Flag set high.
PARAM. DATA
NUMBER TYPE DESCRIPTION
01: 4 Repetitions
02: 4 Starting input location no.
Outputs Generated: 1 for each repetition