Differences between Lidar and Sonic Measurements

Comparison of sonic data from meteorological masts and processed data from ground based lidar systems are usually done by linear regression of 10-minute-mean values [60]. Current lidar systems are able to achieve coefficients of determination of R² > 0.99 for the processed horizontal wind speed in flat terrain. However, the signals provided by commercial systems are not direct measurements, as explained later in Section 4.1.2. They are calculated based on the assumption of homogeneous flow. This is also the main reason for the lower performance of such systems in complex terrain.

Contrary to comparing processed lidar and sonic data, the following examination focuses on the line-of-sight wind speed in order to demonstrate the two main differences between both measurement principles:

1. On the one hand a sonic anemometer is able to measure a three-dimensional wind vector, while a lidar system is limited to line-of-sight wind speeds.

2. On the other hand a sonic anemometer is measuring within a small volume, which can be considered as a point measurement, while a lidar system is measuring over a probe volume defined by the laser pulse.

For this investigation, the SWE scanning lidar system (see Appendix B) has been installed on the nacelle of a stopped turbine (see Figure 4.1), pointing towards three sonic anemometers installed on different heights on a nearby meteorological mast. Part of this investigation has been presented in [79], where more details of the experimental setup can be found.

The first part of this examination focuses on the difference between line-of-sight and three-dimensional wind speed measurements. For this purpose, the 10-minute-mean values from the three sonic anemometers are projected on the corresponding normalized lidar vector. The

4.1 Model-Based Approach to Wind Field Reconstruction 53

Table 4.1:Linear regression between projected sonic and lidar measurement.

height [m] 16.5 34.5 52.5 slope [-] 1.001 1.020 1.030 offset [m/s] 0.058 -0.016 -0.270 R² [-] 0.998 0.999 0.999

v_los,L [m/s]

at 52.5 m

v_los,L [m/s]

at 34.5 m

vlos,S[m/s]

vlos,L [m/s]

at 16.5 m

0 10

0 10

0 10

0 10

Figure 4.2: Linear regression between projected sonic and lidar measurement.

reduced line-of-sight wind speed for each of the three sonic anemometers is calculated using (3.31):

v_los,S =x_n,i,I u_S,I +y_n,i,I v_S,I+z_n,i,I w_S,I, (4.1) with the normalized laser vector [xn,i,I y_n,i,Iz_n,i,I]^Tdefined in (3.32) for each height and the wind vector [uS,I v_S,I w_S,I]^T measured by the sonic anemometers in the corresponding positions. In the case of the central anemometer, the line-of-sight wind speed is equal to theu_S,I component.

The reduced wind speeds v_los,S from the sonic anemometers are then compared to the line-of-sight wind speeds v_los,L from the lidar system. Table 4.1 and Figure 4.2 show that both signals correspond very well. All data of the campaign is used without further filtering. The agreement demonstrates the effect of the first difference between lidar and sonic measurements: After the reduction to line-of-sight wind speed, 10-minute-mean values of the sonic anemometer become directly comparable to the lidar data.

The second part of this examination focuses on the difference between point and volume mea-surements. For this purpose, 6 h of high resolution data from 11 a.m. to 5 p.m. on December 27, 2011 is analyzed. This period is chosen, because the wind is blowing almost constantly from the meteorological mast towards the lidar system during the 6 h, and the means of the components vS,I and wS,I are close to zero. Figure 4.3 shows an excerpt of 10 minutes. The sonic anemometer data is collected at 35 Hz and is reduced again to line-of-sight wind speeds,

while the lidar system measures with 0.5 Hz. Both signals have a similar time progression.

However, the sonic data show more variation at higher frequencies. This becomes more obvious comparing the auto-spectra in Figure 4.4.

Due to the special setup of the experiment, it can be assumed that for this wind direction and for the central point, the lidar volume includes all aerosols passing the sonic anemometer.

Thus, the spacial filtering effect of the lidar pulse volume measurement can be imitated by a time filter using Taylor’s Frozen Turbulence and the Gaussian weighting function (3.37) by

v_los,S,f(t) = Z∞

−∞

v_los,S(τ)fRW(¯u(t−τ)) dτ =v_los,S(t)∗f_RW(¯ut), (4.2) where ∗ denotes convolution and ¯u is the mean wind speed. The filtered sonic signal v_los,S,f is still not exactly the same as the lidar signal (see central plot in Figure 4.3), but more similar than the unfiltered one.

The spectrum of the filtered sonic signal can be calculated from the time signal. Here, the spectrum of the filtered signal is calculated from the spectrum of the unfiltered one in order to explain the relationship of both spectra directly. Based on the convolution theorem the convolution of two functions in the time domain is the product of the Fourier transforms:

F{vlos,S(t)∗fRW(¯ut)}=F{vlos,S(t)}F{fRW(¯ut)}. (4.3) With the auto-spectrumS_SSofv_los,S and with the Fourier transform of the normalized Gaussian weighting function with standard deviation σL

F{f_RW(¯ut)}= exp − 2πf

¯ u

2

σ²_L 2

!

(4.4)

the corresponding auto-spectrum can than be calculated by

S_SS,f=S_SSF²{f_RW(¯ut)}. (4.5) It shows a similar drop-off at high frequencies compared to the auto-spectrum of the lidar signal in Figure 4.4. This agreement demonstrates the effect of the second difference between lidar and sonic measurements: After applying the range weighting function also to the sonic data projected on the laser beam direction, the high resolution data of sonic anemometers and lidar systems become comparable.

Eventually, with this investigation, the lidar model described in Section 3.3 is validated by reproducing the main difference between lidar and sonic measurements.

4.1 Model-Based Approach to Wind Field Reconstruction 55

time [s]

vlos[m/s] at52.5mvlos[m/s] at34.5mvlos[m/s] at16.5m

0 100 200 300 400 500 600

4 6 8 10 12 4 6 8 10 12 4 6 8 10 12

Figure 4.3: Time line of line-of-sight wind speeds. Sonic (black), lidar (dark gray) and filtered sonic (light gray).

frequency [Hz]

S[m2 /s2 /Hz]at34.5m

10⁻³ 10⁻² 10⁻¹ 10⁰

10⁻³ 10⁻¹ 10¹ 10³

Figure 4.4: Auto-spectra of line-of-sight wind speeds. Sonic (black), lidar (dark gray) and filtered sonic (light gray).

x_I

y_I z_I

α_L



 u_I v_I w_I





v_los,W

v_los,S v_los,E

v_los,N

Figure 4.5: Sketch of the DBS scan.