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where x is the center of the scanning circle and f(x) is any function of x. The corresponding
spectral transfer function is given as,
ˆTf(k1) = sinc 2
k1Lf
, (299)
2
where sinc(x) = sin(x)/x. The variances of uqq, vqq and wqq are given as,
〈u ′2
qq〉sin 2
φ = Φij(k)βi(k)β ∗ j(k) ˆ Tf(k1) dk, (300)
〈v ′2
qq〉sin 2 φ =
〈w ′2
qq〉cos 2 φ =
Φij(k)γi(k)γ ∗ j(k) ˆ Tf(k1) dk, (301)
Φij(k)αi(k)α ∗ j(k) ˆ Tf(k1) dk. (302)
13.2.2 Systematic turbulence errors for the WindCube lidar
The assumption made in section 13.2.1 that the mean wind direction comes from the North
cannot be made for the WindCube, since it measures at four azimuth angles only (refer Fig.
164), e.g. North, East, South and West. In this case the coordinate system is such that u is
aligned in the mean wind direction. Thus,
uwc = uNScosΘ+uEW sinΘ, (303)
vwc = uNSsinΘ−uEW cosΘ, (304)
whereuNS anduEW denotewindspeedsintheNorth-SouthandEast-Westdirectionsrespectively,
Θ denotes the wind direction, and the subscript wc denotes the velocity components
measured by the WindCube. From simple geometrical considerations (refer Fig. 164),
uNS = ˜vrN − ˜vrS
, (305)
2sinφ
uEW = ˜vrE − ˜vrW
2sinφ
, (306)
where ˜vrN, ˜vrS, ˜vrE, ˜vrW are the weighted average radial velocities in the North, South,
East and West directions respectively. For the w component,
wwc = P(˜vrN + ˜vrS)+Q(˜vrE + ˜vrW)
, (307)
2cosφ
where P and Q are the weights associated with the wind direction such that P + Q = 1.
Leosphere uses P = cos 2 Θ and Q = sin 2 Θ, and hence, we use the same in our calculations.
We proceed by deriving expressions for the uwc variance. The expressions for the (co-)
variances of the remaining components of wind velocity can be derived in a similar manner.
Substituting Eqs. (305), (306) into Eq. (303) we get,
uwc = 1
2sinφ [(˜vrN − ˜vrS)cosΘ+(˜vrE − ˜vrW)sinΘ]. (308)
We define unit vectors in the four directions as,
nN = n(−Θ),
nS = n(π −Θ),
nE = n( π
2 −Θ),
(309)
nW = n( 3π
2 −Θ),
where nN, nS, nE and nW are the unit directional vectors in the North, South, East and
West directions respectively. From Eq. (281), for the North direction,
∞
˜vrN =
ϕ(s)nN ·v(snN +dfnN) ds. (310)
−∞
DTU Wind Energy-E-Report-0029(EN) 247