k [rad/m]
SLL[(m/s)2 /Hz]
10−3 10−2 10−1 100
10−4 10−2 100 102
Figure 5.5: Effect of temporal averaging: Auto-spectrum of the longitudinal wind component (light gray) and of the temporal averaged lidar measurement calculated directly with the analytical model
(dark gray) and estimated from the signal of the simulation (black).
5.3.1 Temporal Averaging
Lidar systems need to average spectra over a certain time to determine the Doppler frequency shift, see Section 2.3. This can be modeled by a running average. For example, the point mea-surement of a staring lidar system aligned with the mean wind direction and with meamea-surement acquisition timeTACQ is represented by:
v0L(t) = Z∞
−∞
uW(τ)rect(t−τ)−TACQ/2 TACQ
dτ
=uW(t)∗rect
t−TACQ/2 TACQ
, (5.18)
where rect() is the rectangular function defined at time t as rect(t) =
( 1, |t| ≤ 12
0, |t|> 12 (5.19)
and∗denotes convolution, which is translated by the Fourier transformation to a multiplication of the individual Fourier transforms. With the normalized cardinal sine function defined as
sinc(f) = sin(πf)
πf (5.20)
at the frequency f, the resulting auto-spectrum is
SLL =F{v0L}F∗{v0L}= sinc2(f TACQ)Sii,u. (5.21)
5.3 Correlation of a Real Scanning Lidar System 91
k [rad/m]
SLL[(m/s)2 /Hz]
10−3 10−2 10−1 100
10−2 10−1 100 101 102 103
Figure 5.6: Effect of spatial averaging: Auto-spectrum of the longitudinal wind component (light gray) and of the pulsed lidar measurement with a discretized range weighting function calculated directly with the analytical model (dark gray) and estimated from the signal of the simulation (black).
Figure 5.4 shows the effect for a lidar measurement with an acquisition time TACQ = 2 s.
The averaged signal (dark gray) is delayed compared to the original one (black with dots) by TACQ/2 = 1 s. The estimated auto-spectrum in Figure 5.5 using Welch’s averaged periodogram method agrees well with the calculated auto-spectrum based on (5.21). This validates the overall approach: Instead of the time consuming process of generating wind fields, simulating lidar measurements, and only then estimating the spectra, the proposed method is able to calculate directly the spectra with less computational effort.
5.3.2 Spatial Averaging
Due to the pulse length (for pulsed systems) or the optical focusing (for continuous wave sys-tems) and the data processing, real lidar systems average the wind speeds along the laser beam according to a weighting function fRW, see Section 3.3. For pulsed systems it can be assumed that fRW depends only on the distance a from the measurement point. For continuous wave systems there is an additional dependency on the distance from the lidar. With the weighting function fRW(a) it is possible to calculate the line-of-sight wind speed of each measurement pointi by a spatial integral, see Equation (3.34). With Taylor’s Frozen Turbulence Hypothesis this could be translated into an integral over time similar to (4.2) and then the spectrum could be calculated analytically by a convolution with the Fourier transform F{fRW} of the weight-ing function similar to (5.21), dependweight-ing on the wavenumber k. However, here a discretized approach is used that is consistent with the overall semi-analytic approach. For this purpose, the weighting function is evaluated at discrete distances and the spatial separations are treated similar to the time delays of a temporal filter.
If, for example, a discrete weighting function with the valuesfRW1, fRW2, and fRW3 is used for
a staring lidar aligned with the mean wind direction, the measurement can be modeled by v0L = fRW1u1,W+fRW2u2,W+fRW3u3,W
fRW1+fRW2+fRW3 , (5.22)
whereu1,W,u2,W, andu3,W are the longitudinal wind at the distancesa1,a2 = 0, anda3 =−a1 from the measurement point. By treating the spatial shift as a delay in the wavenumber domain similar to a temporal shift, the resulting spectrum is represented with the imaginary unit i:
SLL =F{v0L}F∗{v0L}= Sii,u
(fRW1+fRW2+fRW3)2
fRW12 +fRW1fRW2exp(ika1) +fRW1fRW3exp(2ika1) +fRW2fRW1exp(−ika1) +fRW22 +fRW2fRW3exp(ika1) +fRW3fRW1exp(−2ika1) +fRW3fRW2exp(−ika1) +fRW32
. (5.23)
Figure 5.4 shows the effect for a lidar measurement with a1 = 20 m and the Gaussian range weighting function (3.37). Due to the mean wind speed of ¯u = 20 m/s, the resulting signal (light gray) is a weighted sum of the original signal (black with dots) at the current time, and at 1 s before and after the current time. Due to the symmetry of the range weighting function, there is no time delay for the spatial averaged signal relative to the original signal, in contrast to the time averaged signal. Again, the estimated spectrum from the signal follows the spectrum from the analytic model in Figure 5.6. If more discretization points for the range weighting function are used, the spectrum will monotonously drop down similar to Figure 4.4, where the spectrum (4.5) is based on the Fourier transform of the range weighting function.
5.3.3 Discrete Scanning
Real lidar systems with only one laser source measuring in different points have to scan sequen-tially. Thus the lidar measurement in a certain measurement point is only available in certain time points. If the lidar estimate of the rotor effective wind speed is calculated from an average over the last full scan as described in Section 4.3.1, the values of previous time points need to be available at the current time. This can be achieved by buffering the values. Following the considerations in [8] and personal communications with Eric Simley, a staring lidar point measurement evaluating the longitudinal wind component only after each full scan (Tscan) and then holding the value, is described by
v0L = uW(t) X∞
h=−∞
δ(t−hTscan)
| {z }
∆(t)
!
∗rect
t−Tscan/2 Tscan
, (5.24)
5.3 Correlation of a Real Scanning Lidar System 93
k [rad/m]
SLL[(m/s)2 /Hz]
10−3 10−2 10−1 100
10−2 10−1 100 101 102 103
Figure 5.7: Effect of discrete scanning: Auto-spectrum of the longitudinal wind component (light gray) and of the lidar measurement only evaluated at discrete time points calculated directly with the
analytical model (dark gray) and estimated from the signal of the simulation (black).
where δ() is the Dirac delta function with
δ(t) =
( ∞, t= 0
0, t6= 0 with Z∞
−∞
δ(t)dt = 1. (5.25)
For each part of the sum, the multiplication of the signal uW(t) with the Dirac delta function δ(t−hTscan) yields a signal, which at the time t = hTscan has the value of uW(t) at this time multiplied with∞and which is zero for all other times. Due to this selective effect, the infinite sum of the Dirac delta functions is denoted as Dirac comb ∆(t). The convolution with the rectangular function then produces a signal, which holds the value uW(hTscan) from the time t = hTscan to t = hTscan+Tscan due to the unity integral of the Dirac delta function. Finally, the sum generates the signal of a lidar measurement at discrete times while holding the value until the next measurement is performed.
For the analytic model, the following points need to be considered:
• A multiplication is translated by the Fourier transformation into a convolution and a convolution into a multiplication.
• The Fourier transform of the Dirac comb is again a Dirac comb:
F{∆(t)}= ∆(f) = X∞
h=−∞
δ
f− h Tscan
. (5.26)
• Since convolution of a function g(f) with a delta function δ
f −Tscanh
corresponds to shifting the function by Tscanh , the convolution with the Dirac comb is equivalent to an
infinite sum of shifted functions:
(∆∗g)(f) = X∞
h=−∞
g
f − h Tscan
. (5.27)
• Different frequency components of the signals uW are uncorrelated [8].
With these considerations, the auto-spectrum for discrete lidar measurements is SLL =F{v0L}F∗{v0L}
= (F{uW} ∗ F{∆})F{rect}(F∗{uW} ∗ F∗{∆})F∗{rect}
= X∞
h=−∞
F{uW}
f− h Tscan
X∞ h=−∞
F∗{uW}
f − h Tscan
!
(F{rect}F∗{rect})
= X∞
h=−∞
Sii,u
f− h Tscan
sinc2(f Tscan). (5.28)
Figure 5.4 shows the effect for a lidar measurement, where a measurement is taken every Tscan = 2 s and held. The resulting signal (black with circles) is thus equivalent to the original signal (black with dots) at multiples of Tscan.
Again, the analytic model in Figure 5.7 can reproduce the estimated spectrum. Due to the wind field’s discretization time of 0.25 s, the signal is band limited to the Nyquist frequency of fmax = 2 Hz. Thus in this case, the infinite sum in (5.28) can be limited to those values of h, where |(f− Tscanh )| ≤fmax holds.
5.3.4 Wind Field Reconstruction
If the lidar beam is not perfectly aligned with the mean wind direction, the line-of-sight will be not only determined by the longitudinal wind speed component, but also by the lateral and vertical wind speed components, as described by the lidar point measurement model (3.31).
As discussed in Chapter 4, assumptions are necessary to reconstruct the rotor effective wind speed. The reconstruction method together with measurements not aligned with the mean wind direction yield a distortion of the rotor effective wind speed estimate by the lateral and vertical wind components.
Considering for example a lidar system measuring simultaneously two horizontal points (zn,i,W = 0), the line-of-sight wind speeds are
vlos,1 =xn,1,W u1,W+yn,1,W v1,W
vlos,2 =xn,2,W u2,W+yn,2,W v2,W. (5.29)
5.3 Correlation of a Real Scanning Lidar System 95
k [rad/m]
SLL[(m/s)2 /Hz]
10−3 10−2 10−1 100
10−2 10−1 100 101 102 103
Figure 5.8: Effect of wind reconstruction: Auto-spectrum of the longitudinal wind component (light gray) and a lidar estimate of the rotor effective wind speed from simultaneous measurement of two
points based on the analytical model (dark gray) and from a simulation (black).
Assuming that the lidar system is aligned with the mean wind direction, the rotor effective wind speed can be estimated by a mean over both line-of-sight wind speeds corrected byxn,i,W:
v0L = 1 2
vlos,1 xn,1,W
+ vlos,2 xn,2,W
= 1 2
u1,W + yn,1,W xn,1,W
v1,W+u2,W+ yn,2,W xn,2,W
v2,W
. (5.30) With the coherence γij,ur from (5.9), the resulting spectrum is
SLL = 1
4 Sii,u(2 + 2γij,ur) +Sii,v
y2n,1,W
x2n,1,W + y2n,2,W x2n,2,W
!!
. (5.31)
In Figure 5.8, the spectrum from the analytical model agrees well with the simulation for the two points measured at x1,W =x2,W = 80 m andy1,W =−y2,W = 20 m.
5.3.5 Wind Evolution
Wind evolution is considered here by the simple exponential model of coherence γij,ux (2.22).
If, for example, a second point at ∆xij,W downwind is added to the perfect staring lidar system (5.4), the rotor effective wind speed estimate can be calculated by shifting the measurement downwind in time considering Taylor’s Frozen Turbulence Hypothesis:
v0L = 1
2(u1,W(t) +u2,W(t−∆xij,W/¯u)). (5.32) Due to wind evolution, the resulting spectrum is
SLL= 1
4Sii,u(2 + 2γij,ux). (5.33)
time [s]
v0[m/s]
0 60 120 180 240 300
5 10 15 20
Figure 5.9: Wind speed from the nacelle anemometer (dark gray) and rotor effective wind speed estimates from the CART2 (light gray) and the scanning lidar system (black).