At GM Falkenberg a StreamLine DL from the manufacturer HALO Photonics with the specifications given in Table  was used and operated using a conical-scan-mode configuration to apply the turbulence retrieval approach by . This configuration is defined by three key parameters, namely the elevation angle (ϕ=35.3°), the azimuthal resolution (Δθ∼1°), and the time duration for one single scan (Tscan=72 s). In order to realize this scanning strategy the DL was configured to be in continuous scan motion (CSM) while sampling data. A custom scan file (see Appendix ) has been defined for the scanner configuration including information about the angular rotation rate ωs, the start and end positions of the scanner, and the elevation angle ϕ. In analogy to the work of we set ωs=5° s-1 to nearly satisfy Δθ∼1°. Note that the latter implies measurements on an irregular grid which for analysis purposes later on requires the transfer of the data to an equidistant grid with Δθ=1°. The specific value for ϕ goes back to an earlier theoretical work of and , who focused on Doppler-radar-based turbulence measurements. In addition, have shown that using DL this value is also an optimum beam elevation angle for a mean wind retrieval with a minimum in the retrieval error. With the specifications for Δθ and ωs and due to the pulse repetition frequency fp=10 kHz (see Table ) we had to adjust the configuration setting for the number of pulses per ray to Na=2000 using the relation Na=Δθfp/ωs . This is a minor difference compared to the value suggested in , i.e., Na=3000, which is due to a higher pulse repetition frequency, i.e., fp=15 kHz, characterizing their DL system. Note that for StreamLine DL systems the system-specific parameter fp cannot be changed by the user. The low value for Na is non-favorable if a high measurement quality is needed. For best possible measurement quality in the lower ABL it is therefore important to use the focus setting option to improve the signal intensities within a selected height range. For the DL used in our studies (DL78 hereafter) the focus was set to 500 m. Working with StreamLine DL systems, the range resolution ΔR along the line of sight (LOS) can also be adjusted. For reasons of compatibility with the pulse length of τp=180 ns the range resolution was set to ΔR=cτp/2≈30 m, where c denotes the speed of light.

Note that with StreamLine XR DL systems HALO Photonics by Lumibird offers a further development of the StreamLine series. XR systems operate with larger pulse length in order to increase the range, depending on the presence of scattering particles in the atmosphere. The larger pulse length, however, reduces the spatial resolution of the measurements along the line of sight (LOS), which is not an option for measurements in the ABL if the focus is on the detection and investigation of small-scale structures.

For the measurements carried out in this work the relevant DL output variables are the radial velocity estimates Vr along each single LOS of the conical scan and the associated SNR values. The estimation of Vr is based on the determination of the Doppler shift fd of the backscattered signal by an onboard signal processor. A number of methods are available to determine fd , but the DL manufacturers usually do not disclose to the customer the details of the implemented algorithm. The performance of the estimation algorithms and thus the quality of Vr may vary. The assessment of the performance of the estimation algorithms is generally based on the probability density function (PDF) of the velocity estimates. According to , the PDF of velocity estimators performing well is characterized by a localized distribution of “good” estimates centered around the true mean velocity and a fraction of uniformly distributed bad estimates. This leads to the following frequent distinction in DL radial wind measurements: 1Vr=Vr+Vein the case of good estimateVbin the case of bad estimate, with Ve denoting a random instrumental error . In the literature, bad estimates are mostly described as random outliers or noise uniformly distributed over the resolved velocity space . It can be shown that the occurrence of noise in a series of radial velocity measurements based on a conical scan can be determined by means of the autocorrelation function (ACF) evaluated at lag 1 (Appendix ). In particular, for a conical scan with high azimuthal resolution of Δθ∼1° as used in this work, ACF = 1 indicates noise-free measurements, while ACF < 1 gives an indication of the occurrence of noise.

Typical examples for DL78 measurements that differ in terms of their noise characteristics are shown in Fig. . Each measurement example reflects a different 30 min time period and range gate height. Per column, different analysis diagrams are provided for each measurement example. Noise-free measurements indicated by ACF = 1 (Fig. c) are shown in the column on the left. Apart from the superimposed small-scale fluctuations which reflect natural turbulent fluctuations, the time series plot of radial velocities (Fig. a) and the corresponding velocity–azimuth display (VAD) plot (Fig. c) show a clean sinusoidal course without any random outlier. The sinusoidal course is typical for measurements with conically scanning DL systems and manifests itself in a U-shaped bimodal distribution of the radial velocities provided the wind field was stationary (Fig. d). In the middle and right columns the measurements are contaminated with noise, which is indicated by ACF = 0.8 and ACF = 0.3, respectively (Fig. g, k). Here, the periodic signals are temporarily interrupted by bad estimates randomly representing “any” value in the velocity range ±19 m s-1 (Fig. e, i). Furthermore, differences in the distribution of bad estimates are noticeable. In contrast to the measurements in the middle column where the bad estimates appear quite uniformly distributed (type A noise hereafter), an additional higher aggregation of bad estimates around zero (type B noise hereafter) is noticeable in the right column. This becomes particularly clear by comparing the corresponding panels with the VAD diagrams (Fig. g, k) and those with the histograms of the radial velocities (Fig. h, l). Note that contrary to Fig. d the characteristic U-shaped distribution in Fig. l can no longer be recognized because it mixes with a Gaussian-like distribution of bad estimates. Finally, for each measurement example a clear difference in the level of the signal intensities is noticeable (Fig. b, f, j). With SNR values around -10 dB the signals are strong in the noise-free case, and with values smaller than -15 dB the signals are weak in the noisy cases. It is important to point out that for the noisy cases the signal levels are mostly the same and thus do not provide any indication about the type of noise distribution.

All three measurement examples have been taken with the same DL system under identical configuration (e.g., Na = 2000 pulse accumulations). Despite the low pulse accumulations there are measurement cases with and without noise. This can be explained by the natural variability in the atmospheric aerosol content over the course of a day and with altitude. Aerosols act as backscattering targets and their atmospheric loading influences the quality of the DL signals and therewith the amount of noise in the measurements. A sufficiently large amount of aerosol can contribute to noise-free DL measurements even for low pulse accumulations. Little aerosol combined with low pulse accumulation, however, represents an unfavorable constellation for achieving good data quality.

Concerning the differences in the bad estimate distributions, to the authors' knowledge up to now there have been no user reports about nonuniform bad estimate distributions in DL measurements available. A uniform distribution of bad estimates indicates that the noise component of the spectrum of the lidar signal is white noise . It is believed that additional non-white DL noise sources such as shot noise, detector noise, relative intensity noise (RIN), and speckles cause these nonuniform type B noise characteristics. At this point more in-depth investigations would be necessary but cannot be carried out within the scope of this work. It is worth pointing out, however, that the occurrence of type B noise is not a system-specific DL78 problem. During the (Field Experiment on Sub-mesoscale Spatio-Temporal Variability in Lindenberg) campaign , there was the opportunity to compare the measured data from three StreamLine and four StreamLine XR Doppler lidars (see Sect. ) positioned side by side and configured identically using the scan mode outlined in . The comparison revealed type-B-like noise contamination within the measured data for several systems albeit to varying degrees (see Figs.  and in Appendix ). This suggests that this type B noise issue is at least typical for StreamLine DL systems.

The signal-to-noise ratio (SNR) is determined from the Doppler spectra and is defined as the ratio between the signal power and the noise power. The first bears the meaningful information in a measurement and the latter is considered to be an unwanted signal contribution that is blurring this information. The higher the level of signal power and the smaller the level of noise power, the better the SNR and thus the quality of the radial velocity estimate.

In practice, DL users are often faced with deciding on a suitable SNR threshold value (SNRthresh hereafter) to separate good from bad estimates. Depending on how the measurement data are used later on, the expected uncertainty of the radial wind velocity also plays a role in this decision. provide a guideline on that issue based on an experimental approach. The results showed good agreement with theoretical results based on an approximate equation introduced by , reading 2σ=2π0.5/α0.5(1+0.16α)Δν/Np0.5, with α=SNR/((2π)0.5(Δν/B)) and Np=MNa(SNR). Here, σ denotes the error estimate of the radial velocity in the weak signal, multipulse-averaged regime, Na the number of accumulated pulses, B the bandwidth, Δν the signal spectral width, and M the gate length in points. For more details see Appendix . Note that Eq. () can be used in two ways. On the one hand, it provides an estimate for the uncertainty of the radial velocity estimate depending on the SNR. On the other hand, it provides guidance to calculate SNRthresh for a prescribed acceptable uncertainty in the Doppler lidar estimate. Examples for an evaluation of Eq. () for different numbers of Na are given in Fig. . The curves basically show how the uncertainty of the measurements decreases with increasing SNR. Additionally, the effect of pulse accumulation becomes visible. For the same requirement on the uncertainty of the Doppler estimate, e.g., σ<0.5 m s-1 or σ< 0.1 m s-1, the corresponding SNR threshold value for reliable data would be lower for Doppler estimates based on higher pulse accumulations (e.g., SNRthresh = -24 dB or SNRthresh = -17 dB for Na = 30 000) than for Doppler estimates based on a lower number of pulse accumulations (e.g., SNRthresh = -18.5 dB or SNRthresh = -11 dB for Na=2000). Another approximate equation to determine SNRthresh is suggested in . Taking into account the number of accumulated pulses Na only, they propose the following equation for an SNR threshold determination: 3SNRthresh=1Na+2Na. Example results for SNRthresh derived from Eq. () for different Na are also given in Fig. .

The turbulence retrieval proposed by requires measurements with a probability of bad estimates close to zero. The example shown in Fig.  clearly illustrates that with SNRthresh = -12.7 dB calculated by means of Eq. (), a universally valid first-guess SNR threshold satisfying this requirement across all measuring height ranges can be obtained. Here, estimates for Vr from range gate number 4 to 99 are displayed against their associated SNR values. While bad estimates randomly filling the entire search band ±19 m s-1 can be observed to the left of the threshold line, none can be found to the right. If one further relaxed this threshold, the probability of bad estimates would increase. From Eq. () it can be additionally inferred that with SNRthresh = -12.7 dB measurement uncertainties less than 0.1 m s-1 can be expected. Though it is important to note that by applying SNRthresh=-12.7 dB to DL radial velocity measurements, as shown with the examples given in Fig.  as well as Fig. a, c, and e, a huge fraction of obviously good estimates would be discarded. A reduction of data availability would be the consequence, making a representative derivation of wind and turbulence products often difficult or even impossible. This is a limiting factor of this kind of approach to distinguish between good and bad estimates .

Methodically different from the SNR thresholding technique is the consensus averaging (CNS) method introduced by . The method was originally developed to exclude outliers from radar wind profiler data. A schematic that explains the CNS approach is shown in Fig. . Here noise-contaminated measurements of Vr from several single conical scans executed one after the other are displayed using the VAD perspective. Separating the range of measurement directions (0 to 360°) into equidistant intervals Ii=[(i-1)Δθ,iΔθ] (i=1,2,3,…,n with n∈N), the basic idea is to seek within each Ii along the Vr axis for this subset of data satisfying both (i) the occurrence within a prescribed interval ΔVr, which is assumed to be a typical value for the atmospheric wind variance, and (ii) the provision of data availability Ximax⁡ which, however, must not fall below a prescribed value Xthresh. Similar to the SNR thresholding technique, the difficulty exists in a meaningful choice of ΔVr and Xthresh as will be shown in more detail below.

If the focus is on the derivation of turbulence variables, including the determination of variances caused by eddies in turbulent flow, the problem of this CNS approach is that it requires an a priori estimate of the variance which is actually being attempted to be derived. If ΔVr does not correspond to the true atmospheric situation, e.g., the assumed value for ΔVr is too small or too large, it may happen that either measurements bearing relevant wind information are rejected or that bad estimates remain in the dataset. Examples of this are given in Fig. b, d, and f assuming ΔVr=3 m s-1 and Xthresh=60 %. The early morning 30 min measurement example from 25 May 2021 with the timestamp 04:30 UTC (Fig. b) shows noise-contaminated DL measurements during weak wind and turbulence conditions. At this time the actual wind variance was obviously lower than assumed by ΔVr so that the prescribed interval ΔVr gave room for the inclusion of bad estimates which had to be accepted as reliable due to the CNS concept. The afternoon example from 25 May 2021 with the timestamp 16:00 UTC (Fig. d) shows DL measurements during stronger wind and turbulence conditions than in the morning. Additionally, at some point during the 30 min interval a change in the wind direction contributes to a phase shift in the sine signal represented by some of the scan circles. Mainly because of this nonstationarity the variability of the Doppler velocity measurements is obviously larger than assumed by ΔVr=3 m s-1 in some azimuth sectors so that relevant information characterizing this nonstationarity remains outside of the interval ΔVr and is discarded by the CNS. Note that at this point the focus is on the performance of the CNS and not on reconstructed wind and turbulence variables. Hence, this non-applicability of the CNS method during nonstationary measuring intervals needs to be considered apart from the question of whether a derivation of wind and turbulence variables is meaningful if nonstationarity occurs. In Sect.  it will be shown that a wrong inclusion of bad estimates or a false exclusion of good estimates because of a non-compatible ΔVr compared to the actual atmospheric situation has the consequence that turbulence products (e.g., TKE) calculated based on improperly pre-filtered measurement data may be either overestimated or underestimated.

Another limitation of the CNS filtering technique is that it expects a uniform distribution of bad estimates for a successful application. This becomes evident by considering the CNS filtering results for the measurement example shown in Fig. f. This example represents the type B noise case shown in Fig. . Here, the subsets of Doppler velocities found by the CNS for some of the azimuthal sectors often do not represent the desired good estimates because the high density of bad estimates around zero erroneously shifts the range ΔVr bearing a maximum of data availability towards zero. In this case the nonuniform distribution of the bad estimates makes a successful application of the CNS impossible. Note that the disadvantages worked out here can be generalized to all statistical methods used for outlier detection which require additional assumptions about the distribution and the variance of the quantity of interest.

The VV90D perspective represents a diagram in a rectangular coordinate system where each radial velocity value Vr obtained from a conically scanning DL is plotted versus its counterpart measured at an azimuthal shift of 90°. In particular, for a time series of radial velocity measurements Vr(R,θ;t) this means that V=Vr(R,θ;t) is plotted along the y axis and V90=Vr(R,θ-90°;t∗) is plotted along the x axis. Here, R denotes the range gate, θ the azimuth angle along the scan circle, and t the timestamp of the measurement. Note that t∗ denotes the timestamp of the shifted counterpart value. The motivation underlying this graphic representation of DL measurements will be explained in more detail using the two noise-free measurement examples shown in Fig. . In particular, measurements from a conically scanning DL visualized in both a Vr-t diagram and a VV90D plot are shown in Fig. a and d and Fig. b and e, respectively. The plots in the upper line reflect a homogeneous and stationary measurement example (case 1) which can be clearly seen by the smooth sinusoidal course of the radial velocity, i.e., V∼sin⁡(θ), with a nearly constant amplitude (Fig. a). The same measurement example visualized in a VV90D diagram shows clear circular patterns (Fig. b). The latter can be explained by taking the phase shift identity sin⁡(θ-90°)=-cos⁡θ into account, yielding V90∼-cos⁡θ. Therewith paired data points (x=V90,y=V) plotted in a rectangular coordinate system describe a circle. Note that this way of looking at DL measurements shows analogies to the harmonic oscillator where the time evolution of both displacement and motion are frequently visualized in a phase-diagram plot to show the 90° phase relationship between velocity and position much more clearly . The plots in the lower line of Fig.  represent a nonstationary measurement example (case 2). Due to the sinusoidal course superimposed with smaller fluctuations and a varying amplitude (Fig. d) the corresponding VV90D diagram (Fig. e) of the same measurements shows a slightly wider and more blurred ring structure.

A quantitative description of the VV90D ring structures is provided with the diagrams shown in Fig. c and f. These diagrams are referred to as ri-count{Vi} diagrams hereafter. Here, ri denotes the radius of a pre-defined circular ring ri-Δr≤ri≤ri+Δr with origin at (x=0,y=0) and width Δr in the VV90 plane. The quantity count{Vi} denotes the number of measurement data that can be found in this ring. Note that these data represent a circular-ring-related subset Vi of the whole measurement series, i.e., Vi⊂{V}. Taking the equation of a circle into account, i.e., r=x2+y2, in practice both the subset Vi and count{Vi} can be determined by identifying the radial velocities V=Vr(R,θ;t) of the measurement time series that satisfy the relation 4r=V902+V2, with the range of radii r defined through the boundaries of the circular ring ri-Δr≤ri≤ri+Δr. In order to generate the ri-count{Vi} diagrams shown in Fig. c and f the full area of the VV90 plane has been subdivided into closely spaced circular rings of increasing radius (i=0,…,n with n∈N) with discrete fixed steps Δr=0.5 m s-1. Note that this value turned out to be a viable choice if using the ri-count{Vi} diagram as a tool in a filtering procedure (see Sect. ). For case 1, the data availability of measured radial velocities is constrained to only a few circular rings with ri ranging between 5 and 6.5 m s-1, with the largest fraction of measurements in the circular ring with ri = 5.5 m s-1 (see Fig. c). Additionally, due to the stationary wind field conditions the obtained availability distribution is strictly unimodal and symmetric. For case 2 the measurements are distributed over a broader range of circular rings with ri taking values between 0.5 and 12.5 m s-1. The largest fraction of the measurements occurs within a circular ring with ri = 4.5 m s-1 (see Fig. f). The distribution of data availability is nearly unimodal but asymmetric. The examples shown in Fig.  represent just two specific situations, and a great variety of VV90D and associated ri-count{Vi} diagrams may result for different atmospheric and lidar signal conditions, including multi-modal distributions (not shown here). In the following we refer to the VV90D diagram and the associated ri-count{Vi} diagram as the framework of the VV90D perspective.

Compared to the commonly used VAD visualization technique, the framework of the VV90D perspective represents an alternative way of displaying radial velocity measurements from a conically scanning DL and opens up new possibilities for data analysis at the same time. In the next section it will be shown how this framework can be used to develop suitable filtering techniques of bad estimates in noisy DL data.

Two different coarse filtering techniques are presented next. The underlying ideas are motivated by characteristic features of good and bad estimates in the VV90D diagram, which will be briefly explained. For this purpose the noise-contaminated measurement examples of type A and type B from Sect.  are used. Compared to noise-free DL measurements, noticeable features of noise-contaminated DL measurements in the VV90D are the greater spread of paired DL data (x=V90,y=V) and a lack of clear circular patterns (Fig. a, d). In the associated ri-count{Vi} diagrams (Fig. b, e) this also goes along with a broader distribution of available data points over a larger range of circular rings. Furthermore it is obvious that for the type B noise measurement example the more densely distributed bad estimates around zero make a cross-shaped region visible in the VV90D (Fig. d) and cause a pronounced secondary peak in the ri-count{Vi} diagram (Fig. e). Additionally, by examining in more detail the properties of the data occurring in the three color-coded circular rings, it becomes apparent that circular rings with a high data number mostly contain reliable DL radial velocities, i.e., good estimates. This can be seen from the fact that the data occurring in these rings mostly follow the expected sinusoidal course of the DL measurements (see the black dots in Fig. c, f). It is also striking that this happens in a dense sequence of data points. For circular rings with an increasingly lower data number the associated data subsets Vi contain increasingly more measurements that deviate from the sine, i.e., radial velocities which reflect bad estimates taking any value within the velocity space ±19 m s-1 (see the orange dots in Fig. c, f). It is noticeable here that bad estimates in such subsets Vi mostly occur as singular points having no further data points in the immediate environment. Note again that the type B measurement example represents an exception here. While the circular ring ri with the global peak in the ri-count{Vi} diagram contains mostly good data, the circular ring with the secondary peak contains mostly bad estimates (see the red dots in Fig. f).

The coarse filter results presented in Sect.  and are not yet satisfactory for the following reasons. Firstly, the frequently unjustified rejection of good estimates after applying coarse filter I results in an unnecessary reduction of reliable measurement data. Secondly, the number of remaining bad estimates after applying coarse filter II is still too high. Hence, additional efforts are required to further optimize the filter results. Therefore, the results of coarse filters I and II will be treated as intermediate results only at this point. Possible further optimization steps are considered in more detail next. The entirety of these steps represents the filter for post-processing. Note that all analyses are from now on carried out using the VAD perspective.

In the previous subsections the limits of the usability of approach I and approach II depending on the type of noise have been discussed. The type of noise, however, is not the only factor affecting the applicability of the two different filtering techniques. Their success is also linked to the strength and temporal evolution of the wind during the measurement period. This becomes obvious by comparing the filter results of approach I and approach II for both type A and type B noise (Figs.  and ) while considering the following categories: (I) weak and stationary wind, (II) strong and stationary wind, (III) weak and nonstationary wind, and (IV) strong and nonstationary wind.

Comparing the filter results for measurements with type A noise the outcomes of approach I and approach II are equally good for category I and II (Fig. b, e and c, f), but for category III and IV the results based on approach I (Fig. h, k) turn out better than based on approach II (Fig. i, l). The differences in the results for category III and IV are not because bad estimates have not been correctly detected by approach II but rather due to the wrong rejection of a substantial number of obviously good estimates. This error can be traced back to a bimodal distribution in the measurement-related ri-count{Vi} diagram (not shown) caused by the nonstationarity of the wind field during the 30 min measurement interval. Here, the secondary peak was incorrectly interpreted as noise around zero (see Sect. ). Note that coarse filter II of approach II is not designed to make this distinction and thus fails when applied to situations belonging to category III and IV. In contrast, when comparing the filter results for type B noise measurements (Fig. ) approach II is better than approach I for category I and II. This can be seen not only visually but also numerically by means of the ACF. While the ACF values were between 0.23 and 0.37 in the unfiltered time series (Fig. a, d), after applying the filtering technique the ACF takes values between 0.98 and 0.99 for approach II (Fig. c, f) but only values around 0.92 for approach I (Fig. b, e). Finally, two more findings are considered worth mentioning here: firstly, measurements during wind conditions belonging to category III are obviously difficult to manage for both approaches, indicated by the comparably poor ACF values of the fully filtered time series in a range between 0.87 and 0.89 (Fig. h, i). Secondly, it becomes obvious that, particularly during weak wind conditions (i.e., category I and III), the bands of good estimates sometimes seem too narrow from a purely visual point of view. This observation holds for both of the filter approaches (Fig. b, c and h, i). The reason for this lies in the method to determine the envelope in connection with the final re-activation step of previously discarded good estimates (see Sect. ). As a result, some information on the actual variability is lost in the filtered dataset. This in turn may result in an underestimation of variances as will be shown in Sect. .

Knowledge and insights gained from the overview given in this section are important to develop a strategy to implement the filtering techniques for operational use. More detailed information on a strategy that can be used for an implementation of approach I and approach II is given in Appendix .

In order to be able to assess the quality of TKE variables based on differently pre-filtered DL measurements, as an independent reference, sonic data from a 99 m tall meteorological mast are used. The measurements were performed with a USA-1 sonic anemometer (Metek GmbH) at a sampling rate of 20 Hz, and the raw data were processed with EddyPro (LiCor Inc.) software. The mast is operated at GM Falkenberg at a distance of about 80 m towards SSW from the DL system.

Results of an intercomparison of TKE retrievals based on differently pre-filtered DL data versus sonic TKE measurements are summarized in Fig. . Three different cases are analyzed: TKE retrievals calculated by means of (i) SNR-threshold-based filtered input data using SNRthresh = -12.7 dB (Fig. a–d), (ii) CNS-based filtered data (Fig. e–h), and (iii) CF-based filtered data (Fig. i–l). Here CF (combined filter) stands for a combined application of approach I and approach II because of the occurrence of type A and type B noise in the DL78 measurements (see Sect. – and Appendix ). For each case an additional distinction is made between TKE products which are subject to different quality control (QC) steps. Here, the minimum requirement for a TKE value representing a 30 min mean is that its retrieval is based on more than 60 % of reliable measurements of Vr during the measurement interval (lev_c hereafter). Since the TKE reconstruction method relies on a variety of theoretical assumptions (e.g., see Eq. 22 in ) a further QC step proves the fulfillment of these requirements (lev_b hereafter). To obtain meaningful results the evaluations are based on a 2-month dataset of DL measurements at GM Falkenberg which were collected during from 18 May to 17 July 2021.

Considering the Doppler lidar TKE retrieval based on a pre-filtering of the measurements using the SNR threshold approach, the problems that arise in connection with routine turbulence measurements (see Sect. ) can be supported here numerically. Although good data quality is achieved compared to sonic measurements (R2 = 0.99 and RMSD = 0.21; Fig. a), data availability is very low (36.77 %; Fig. d). TKE values based on CF pre-filtered DL data have almost comparable data quality (R2 = 0.98 and RMSD = 0.23; Fig. i) but score additionally with a significantly higher data availability (82.57 %; Fig. l). TKE results based on CNS pre-filtered DL data are also characterized by a higher data availability (78.85 %; Fig. h) but have poorer data quality (R2 = 0.97 and RMSD = 0.27; Fig. e). To explain the latter, the time series plot comparisons between DL data and sonic data displayed in Fig. b and f for the example day 25 May 2021 are helpful. This day is a typical example day with noise-contaminated DL measurements over the whole day but with an obvious higher density of bad estimates in the early morning between 04:00 and 07:00 UTC and in the afternoon from 16:00 UTC onwards (see Fig. , 2 – Appendix ). Note that during these time intervals no reliable TKE values are available if an SNR threshold pre-filtering is used. In contrast, using CNS pre-filtered data, retrieved TKE values are available for these time periods, which, however, are partly subject to errors if compared with sonic data. For instance, DL-based TKE values obtained at 04:30 and 16:00 UTC show either a pronounced overestimation or underestimation if compared with sonic-based TKE measurements. With a closer look into the DL radial velocity measurements from which these 30 min TKE values have been retrieved, this overestimation and underestimation can be easily explained (see Fig. b, d). For the measurement period between 04:00 and 04:30 UTC the prescribed search interval ΔVr=3 m s-1 was too large so that bad estimates remained in the dataset and introduced an additional variance contribution, yielding an overestimation of the TKE. For the measurement period between 15:30 and 16:00 UTC the value for ΔVr was too small so that relevant features of the wind field were not captured. As a consequence, the dataset and thus the derived variances are not representative for the 30 min measurement interval, yielding an underestimation of the TKE value. These examples clearly show the weaknesses of the CNS filtering technique if the ΔVr search interval is inadequately selected. With a fixed ΔVr during 24/7 routine measurements such situations occur quite frequently, which explains why the data quality of TKE values retrieved from CNS pre-filtered DL data is worse compared to TKE retrieved from pre-filtered DL data using the SNR thresholding technique. Note that with the newly introduced CF filtering technique the above-described issues can be avoided and therewith more reliable TKE values can be derived (Fig. j, l).

One additional interim note should be given on the high sonic TKE value of ∼ 7 m2 s-2 at 16:00 UTC (Fig. f). This value is likely caused by an instationary wind field during the 30 min measuring interval rather than due to eddies in the turbulent flow. For that reason this value should be flagged as non-reliable. The problem, however, is that with the CNS filtering technique relevant wind information characterizing this nonstationarity would be rejected so that the identification of such nonstationarity would not be possible.

Another way to compare TKE variables retrieved from differently pre-filtered DL measurements is based on the use of a so-called Bland and Altman plot (Fig.  c, g and k). This kind of plot was introduced by to statistically assess the comparability between two measurement methods. The advantage of this statistical evaluation method is that it not only provides insight into systematic deviations (bias) and limits of agreement but also reveals how the differences between two measurement methods depend on the magnitude of the measurement value. For the Bland–Altman plots shown in Fig.  c, g, and k, the y axis shows the percentage error (PE) between two paired measurements, i.e., 100×(TKEDL-TKESonic)/TKESonic, and the x axis represents the average of these values, i.e., (TKEDL+TKESonic)/2. Additionally, the horizontal lines indicate the mean percentage error (PE‾) and twice the standard deviation (±2σ) from the mean. The latter denotes the statistical limits of agreement. For the Bland–Altman plot shown in Fig. c the mean percentage error is close to zero (PE‾ = -0.84 %). This indicates that there are hardly any systematic differences between DL-based TKE retrievals using the SNR threshold method for noise filtering and sonic-based TKE measurements. Additionally it can be seen that there is a nearly symmetric distribution of the paired points around the horizontal line denoting PE‾, with a slight tendency towards a more frequent underestimation by the DL for small TKE values and a more frequent overestimation for higher TKE values. As a result, in 95 % of the cases the DL delivers a TKE value that is up to 39 % smaller or up to 37 % larger than the one measured by the sonic anemometer. With the exception of an offset of about 2.7 % in bias and the limits of agreement, qualitatively similar results can be derived when comparing DL-based TKE retrievals using the CF filter with sonic-based TKE (Fig. k). The Bland–Altman comparisons for DL-based TKE values using the CNS filter reveal less good correspondence (Fig. g). Although the mean percentage error (PE‾ = -0.63 %) does not significantly differ from the results shown in Fig.  c, the range between the limits of agreement is larger (PE‾±2σ={+40%,-41%}). Furthermore, a slightly different point cloud distribution can be seen. Especially for larger TKE values there seems to be a greater underestimation of DL-based TKE retrievals compared to sonic TKE.

With increasing altitudes the probability of bad estimates in DL radial velocity measurements increases. This is due to the typically decreasing aerosol density, which is also noticeable in weaker SNR values. Therefore, the actual robustness of the filtering methods can be tested much better at larger measuring heights, where the methods have to cope with an increased occurrence of noise. Unfortunately, the tower-based measurements at GM Falkenberg are only provided up to 99 m height so that for higher altitudes no independent references for a comparison are available. For that reason an alternative way based on the following strategy is used. In the subsequent analysis, DL TKE values obtained using the SNR threshold method for noise filtering are used as an alternative reference intended to replace the missing sonic data at higher altitudes. This can be motivated by two arguments. On the one hand, it could be shown that the comparison of TKE values based on SNR pre-filtering with independent sonic data provided the best results (see Fig. ). On the other hand the SNR thresholding approach is a standard method to exclude bad estimates from DL measurements. For that reason confidence in the validity of TKE products based on SNR pre-filtering is very high and TKE products based on differently pre-filtered input data should be comparable by similar performance of the filtering method.

Comparisons of TKE products based on SNR pre-filtered DL data with those based on CNS filtering and CF filtering for measurement heights between 45 and 500 m are shown in Fig. a–b and c–d, respectively. There are two obvious issues with the CNS filtering that become apparent in Fig. a and b: on the one hand, the clear underestimation of larger DL-based TKE values for values above 2 m2 s-2 and, on the other hand, the occurrence of a significant number of strongly overestimated TKE values in a range of 0.06 m2 s-2 < TKE < 0.5 m2 s-2. These errors are caused by the deficiencies of the CNS filter method already described in Sect. , i.e., an inappropriate choice of the search interval ΔVr. In contrast, the comparisons shown in Fig. c and d reveal better agreement between TKE retrievals based on SNR pre-filtered DL data with those based on CF filtering. This is evident from both a visual and statistical perspective. For instance, the higher sensitivity of the CF filter with respect to the rejection of bad estimates and the acceptance of good estimates contributes with PE‾±2σ={+10%,-17%} to much better limits of agreement than observed for TKE retrievals based on CNS pre-filtered DL data (PE‾±2σ={+38%,-39%}).