# Dynamic Data Filtering of Long-Range Doppler LiDAR Wind Speed Measurements
## 4 Results
### Evaluation of Filtering Based on Staring Measurements
#### 4.1.3 Error Analysis  
In order to gain a better understanding of the error behaviour and insight into the resulting error, we performed an error analysis. For this, the frequency distribution of the errors is calculated. Figure 14 illustrates histograms for the RMSE of the mean velocity and the velocity standard deviation of all 4325 10 min intervals with a non-constant bin width increasing exponentially. It can be seen that the errors are subject to a double log-normal distribution or Pareto distribution. Explaining the cause of this specific distribution is out of the scope of this paper. Nevertheless, we do a qualitative analysis supported by the cumulative distribution presented in Figure 15. While the distribution of absolute average velocity error of the unfiltered LiDAR data (red line) follows this behaviour very well, local deviations of all used filters can be found from a value on of approximately 3 m/s (Figure 14a). The error distribution of the Gaussian kernel dynamic data filter seems to be displaced towards higher errors. We fitted a double logarithm distribution to the histogram to determine the most probable error of the fitted distribution which is provided in Table 3.  
The error behaviour of the standard deviation shows double peaks at 0.1 m/s and 4.4 m/s for the unfiltered case and suggests that two functions overlap here. The frequencies of the velocity standard deviation error, for the filtered data, show as well a second peak shifted to ca. 1 m/s. These error behaviours are also confirmed by Figure 15a,b that shows the resulting errors for error values below a certain threshold (x-axis). It turns out that Figure 15 is equivalent to the cumulative distribution of error from Figure 14. While the resulting RMSEs increase up to 3 m/s error threshold for all filters, this is a turning point followed by a split in behaviour. As expected, the unfiltered LiDAR data results in the highest error up to a threshold of 17 m/s. This error is exceeded from the combined filter approach of [PERSON] et al. (2016) and the CNR-threshold filter respectively the combined filter of [PERSON] et al. (2017) at the error thresholds of 26 m/s and 29 m/s. While the average error of those three filters are below the unfiltered data, it turns out that the RMSE, as a measure of velocity dynamic accuracy, are the \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & **Dyn. Data** & **Dyn. Data** & **CNR** & **Std.** & **Iter.** & **Quartile** & **[PERSON]** & **[PERSON]** & **Unfiltered** \\  & **Hist.** & **Gauss.** & **Threshold** & **Dev.** & **Std** & **Quartile** & **[PERSON]** & **[PERSON]** & **Unfiltered** \\ \hline Velocity [m/s] & 0.09 & 0.20 & 0.09 & 0.09 & 0.09 & 0.10 & 0.11 & 0.09 & 0.11 \\ Vel. std. dev. [m/s] & 0.09 & 0.12 & 0.06 & 0.12 & 0.06 & 0.10 & 0.11 & 0.06 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Most probable velocity and standard deviation error of fitted double log-normal distribution to 10 min error histogram. Figure 15: Influence of maximum error threshold to the resulting error (**a**) RMS velocity error over velocity error threshold and (**b**) RMS velocity standard deviation error over velocity standard deviation error threshold. highest with in the test case shown in Table 1. A possible explanation may be that all three filters are based on the CNR-threshold filter. While these three filters produce the smallest error up to a threshold of 13 m/s, an enormously increase is followed till the maximum error is reached. The maximum error can be determined by following the error threshold to the maximum value. By comparing the error behaviour from Figure 15a,b with the theoretical accumulated function of a Pareto distribution (root function), the assumption of multiple overlapping distributions may be confirmed. We see the typical increase of a root function several times in Figure 15a,b.