3.2 Wind Field Simulation
3.3.3 Low-pass Filtering of Dropwindsonde Profiles
As introduced above, low-pass filters are commonly employed to smooth dropwind- sonde profiles. Although their characteristics obviously impact the statistics, especially the turbulence intensity, of the measured wind field derived by compositing dropwind-
0 100 200 300 400 500 600 2 3 4 5 6 7 Height (m) Turbulent Velocity (m/s) BD CD Wind
(a) Differentiation Comparison
0 100 200 300 400 500 600 2 3 4 5 6 7 Height (m) Turbulent Velocity (m/s) MA-2s BF-2s MA-3s BF-3s Wind (b) Filter Comparison
Figure 3.11: Turbulent velocity profile comparison showing the impact of various dif- ferentiation schemes and filter designs, where ”BD” stands for the 1st order backward
differentiation, ”CD” stands for the 2nd order central differentiation,”MA” stands for the
moving average, ”BF” stands for the 1st order Butterworth filter (the cut-off time scale
is specified after these symbols), ”Wind” equals the ”true” wind velocity interpolated from the pseudo-stochastic wind field.
0 100 200 300 400 500 600 0 2 4 6 8 10 12 Height (m) Squared Difference (m2/s2) BD FD CD
(a) Differentiation Error Comparison
0 100 200 300 400 500 600 0 2 4 6 8 10 12 Height (m) Squared Difference (m2/s2) MA-2s MA-3s BF1-2s BF1-3s BF2-2s BF2-3s
(b) Filter Error Comparison
Figure 3.12: Error profile comparisons showing the impact of various differentiation schemes and filter designs, where ”BD” stands for the 1st order backward differentia-
tion, ”CD” stands for the 2nd order central differentiation, ”FD” stands for the 1st order
forward differentiation, ”MA” stands for the moving average, ”BF” stands for the But- terworth filter (the filter order and the cut-off time scale are specified after the filter symbol.
sonde measurements, a 5− s filter, without any further specifications, is used in previous studies with no theoretical or experimental justification (Franklin et al., 2003; Vickery et al., 2009), and its influence on the composition result has not been thoroughly dis- cussed. To explore the influence of low-pass filters on dropwindsonde profiles, individual pseudo dropwindsonde profiles filtered by two types of filters, a moving average filter and a Butterworth filter, with 2 seconds and 3 seconds taken as cut-off time scales, were used to derive the mean and turbulent wind velocity profiles of the measured pseudo-stochastic wind field. Due to the limitation of the pseudo-stochastic wind field height, 600m, the widely adopted cut-off time scale, 5 seconds, is considered too large since it produces a 60msmoothing scale given that the falling rate of the dropwindsonde is 12m/s. However, the influence of using a larger cut-off time scale, such as 5 second, is predictable based on the trends found here. Since the frequency response of the filters clearly impacts the fluctuations remaining in the filtered measurements, this aspect should be investigated. Because the frequency response of the Butterworth filter performs better, in a comparison with the moving average filter, in filtering out high frequency fluctuations, it is expected to see that it produces improvements in reconstructing the turbulent wind velocity profile of the measured wind field.
Similar to the comparison for evaluating finite difference schemes, Fig. 3.10(b) shows that the mean wind profiles composited from pseudo measurements processed using dif- ferent filter designs are indistinguishable. This indicates that the influence of the low-pass filter characteristics, including the cut-off time scale and the type of the filter, on the mean wind structure found by the dropwindsonde is negligible, and therefore the design of the filter is not important when only the mean wind profile is of interest. In contrast, the influence of filter characteristics on the turbulent velocity profile composition is obvi- ous. Figure 3.11(b) shows the difference of the turbulent wind velocity profiles calculated using different filter designs can go up to 0.6m/s, or 12% in the relative sense.
The comparison of the error, defined exactly as in the discussion of the influence of the differentiation scheme, is shown in Fig. 3.12(b). The comparison indicates the first-order
Butterworth with the cut-off scale of 2− s gives the best estimate, in general, of the instantaneous wind speed of the measured wind field. This implies that the currently used 5 − s cut-off time scale is too large. In addition, the moving average generally gives larger errors, as expected, comparing to the results smoothed by the Butterworth filter, although the improvement of using Butterworth filter is not significant. It should be noted that this finding is made based on the assumption that the dropwindsonde measurement error can be described by a series of independent, log-normally distributed random variables. This error assumption is rather crude and unvalidated, and therefore the conclusion presented should be used with caution. A further investigation, taking various causes of the measurement error into account, is needed to yield a more sound suggestion on the filter design in processing dropwindsonde measurements.
Since the filter is commonly used in post-processing dropwindsonde measurements, the influences of filter characteristics and finite difference scheme are combined when the mea- surement needs to be dynamically corrected by the wind finding equations. Since there is no evidence so far indicating that their influences can be superimposed, their combi- nation should be examined in addition to individual investigations. In Figs. 3.10 and 3.11, the profiles showing the influence of the finite difference schemes investigated are calculated using the Butterworth filter with 2 second cut-off scale while those showing the influence of low-pass filter characteristics are calculated using the first-order backwards scheme. Since the influences of both finite difference schemes and filter characteristics on the mean wind profile composition are negligible, discussions in following paragraphs are concentrated on their combined influence on the turbulent wind velocity profile com- position and the turbulent wind velocity definition can be found in the discussion of the wind finding equations’ validity.
Seen in Fig. 3.11(a), the second-order central difference yields an overall better es- timate when pseudo measured profiles are smoothed by the best available filter, which follows the trend found in the analysis of the error introduced solely by finite difference schemes, as shown in Fig. 3.12(a). The improvement, however, is not as obvious as in Fig.
3.12(a). This implies that higher order difference schemes, which are better in capturing the trend of high order variations within a finite step, does not outperform the lower order differentiation scheme. The reason for this is that the variation of a pseudo drop- windsonde measured profile, including the trend resolved by a high order finite difference, is filtered out by a low-pass filter. Therefore, the selection of the differentiation scheme is not as important as indicated by Fig. 3.11(a). Consistent with the findings made in analyzing filter characteristics individually, the Butterworth filter with cut-off scale of 2 second produces the turbulent wind velocity profile close to the one calculated from the pseudo-stochastic wind field, as shown in Fig. 3.11(b). However, the Butterworth filter with cut-off scale of 3s also produces a profile approximating the ”true” profile. This indicates the importance of time scales used in the filter design may not be as significant as indicated by the individual examination articulated above. This is because extra small scale variations introduced by the dropwindsonde acceleration compensate the smoothing given by the filter with a larger cut-off scale.