In order to be as close as possible to a realistic measurement environment, this study utilizes time-varying LES generated wind fields. The LES fields are simulated using the parallelized LES model (PALM) and provided by the University of Hanover .

When using LES, a trade-off between the resolution of the simulation and the domain extent has to be realized, as computational power is limited. On the one hand, the resolution has to be sufficient to ensure a realistic simulation of turbulence and associated Doppler lidar wind measurements. Doppler lidar measurements are assumed to be represented realistically when the lidar range gate length Δp is much larger than the grid spacing Δx of the LES simulation . On the other hand, the domain has to be large enough to ensure a sufficient number of independently sampled wind profiles for statistical analysis, given the investigated turbulence characteristics, system setup, retrieval strategy and sampling procedure.

The LES used in this study employs a grid spacing of Δx=10m (corresponding to a resolution finer than O(100m)) and a domain size of 5km×5km×1.8km. The grid spacing fulfills the condition Δp≫Δx at the assumed range gate length Δp=72m (Sect. ). Further, the domain size is sufficiently large to sample a sufficient number of independent wind profiles for statistical analysis, given the turbulence characteristics present, an adequate system setup, retrieval strategy and sampling procedure, as is shown in Sect. . The ADLS can be easily adopted for use with other LES wind fields (e.g., a larger or longer simulation domain), should this be necessary for future studies.

The present LES is driven with a geostrophic background wind of uG=0, 5, 10 and 15ms-1 and a constant kinematic surface heat flux w′Θ′‾=0.03Kms-1 (w′Θ′‾=0.23Kms-1 for the 0ms-1 background wind case). Data output started with fully developed turbulence at a temporal resolution of 1s after a spin-up time of 1h. The LES wind fields are the same as used by , and so a detailed description is available in Sect. 5.1 and . The convective situation is classified as unstable stratification with the stability criteria pointing to the development of organized convective structures for uG>0ms-1. The boundary layer height is approximately 600m (1200m for the 0ms-1 background wind case), with the entrainment zone extending from 600 to 1000m. Gravity waves are present in the entrainment zone and above. The stream-wise integral length scales for u and v are in the range of 200–500m, whereas the span-wise integral length scales are much smaller, in the range of 50–200m . Due to the in-depth description in , the used LES data are not detailed further here.

In the second section, all main system components consisting of aircraft, scanner and lidar are simulated geometrically. An illustrative overview of the results obtained after system simulation is shown in Fig. .

For the measurement, the aircraft and atmospheric motion vectors need to be combined to give the measured radial Doppler velocities. Additionally, in order to obtain the atmospheric motion vector from the LES, the locations of the lidar range gates need to be calculated. Both operations are conducted based on the state of the system components, meaning the current aircraft orientation and motion, as well as scanner position and lidar setting. Conveniently, this calculation is achieved by defining two separate coordinate systems (Fig. ), following the theory outlined in and . The LES wind fields are based in earthbound coordinate system (superscript E) oriented east-north-up (ENU). The aircraft coordinate system (superscript A) is oriented along-right wing-down (ARD). The scanner is emulated and the measurement is performed in the aircraft coordinate system. To transfer between the two systems, coordinate transformations are required; the details are given in Appendix .

The measured Doppler velocity vD can be calculated through projecting the velocity vectors onto the lidar beam vector. The radial Doppler velocity measured by the lidar has a contribution from the lidar motion vector vLA and the atmospheric motion vector vpA. Both are projected onto the beam direction vector b, which is defined as a unit vector: 1vD=bA⋅vpA+vLA. The lidar motion can be split into two contributions, the aircraft motion vaA and an aircraft movement moment arm due to aircraft rotation ω with the moment arm r (specifying the distance between the position of the inertial navigation unit in the aircraft and the position of the final mirror turning the lidar beam): 2vD=bA⋅vpA-vaA-ωA×rA. The above assumes that motion towards the lidar is negative, whereas motion away from the lidar is positive. Depending on the lidar beam direction, an airborne Doppler lidar system measures foremost the aircraft speed as it presents a vector with very large magnitude compared to the atmospheric motion vector and the aircraft movement moment arm. The atmospheric motion vector needs to be transformed into the aircraft coordinate system where the measurement is performed using the rotation matrix T′AE (see Appendix ). The same is true for the aircraft motion vector which is originally calculated with respect to the LES coordinate system. Consequently, the measurement is achieved according to the following formula: 3vD=bA⋅T′AEvpE-T′AEvaE-ωA×rA. Appendix details how the weighted and averaged particle velocity vpE is obtained from the LES considering the range gate position and motion during the measurement process.

Before the application of the retrieval, the contribution of the aircraft motion and aircraft movement moment arm to the measured radial velocity is then removed again: 4vCOR=vD+bA⋅T′AEvaE+ωA×rA. This calculation isolates the motion-corrected velocity vCOR, which is the atmospheric wind contribution to the measured radial velocity (the projection of the atmospheric wind vector on the beam vector). The ADLS makes it possible to add inaccuracies in any of the components relevant in the measurement or motion correction process. However, please note that throughout this study no system inaccuracies are introduced in the measurement or motion correction process in order to focus on problems in wind profiling due to atmospheric inhomogeneity. In addition, it is assumed that for the real system the optical alignment and beam direction can be reliably quality controlled using a beam calibration procedure based on ground returns. The theory and procedure are outlined in . Consequently, for the ideal ADLS system, the motion-corrected Doppler velocity due to the particle velocity is equal to the particle velocity projection itself. Therefore, compared to a ground-based system, only the measurement geometry is altered due to the moving system. The above-discussed rotations and transformations do not influence the accuracy of the motion correction. As a result, all wind profiling errors discussed in the next sections stem purely from the inhomogeneous atmospheric flow conditions due to boundary layer turbulence. In a homogeneous wind field, the retrieved wind profile is exact.

An illustrative example of the ADLS capability is given by simulated nadir retrievals. In Fig. , the simulated vertical wind measurement is compared to the LES input along two transects for an ideal system with a lidar measurement frequency of 10Hz. The aircraft is flying at an altitude of 1700m with an IAS of 65ms-1 and a superimposed 8s, 5∘ roll oscillation, combined with a 15s, 3∘ pitch oscillation.

The first transect, a crosswind flight (Fig. a, b), reveals the spanwise boundary turbulence structure for the 10ms-1 background wind case. Along-stream organization of turbulence into rolls occurs, and gravity waves are present above the boundary layer. The ADLS results show that the simulated lidar is able to capture turbulent structures in the boundary layer, although smoothing occurs for the finest scales. Vertically, the resolution of the measurement is defined by the range gate length, whereas horizontally it is defined through the combination of aircraft speed and measurement frequency. Another noticeable effect in the measurement is the movement correction capability of the scanner, which is disabled for the first half of the transect and enabled for the second half. During the first half, the lidar beam is not pointing exactly nadir due to the superimposed, artificial aircraft roll and pitch oscillation. This deviation causes a portion of the horizontal wind to be projected into the measurement. For the crosswind case, this effect is caused by the (rather strong) roll oscillations of the aircraft. Consequently, the measured vertical wind shows some additional superimposed structures compared with the LES for the first half of the transect. This contribution of the mean horizontal wind can be removed, as a second step, if the horizontal wind profile is known . However, variability in the mean horizontal wind profile will still manifest in the vertical wind measurement. The roll and pitch oscillations also result in a distorted curtain location, which is not directly below the aircraft anymore (illustrated in Fig. ). The winding curtain results in a non-equidistant sampling of turbulence, complicating analysis further. For the second half, the scanner movement compensation is enabled. Consequently, the beam is pointing nadir at all times and no horizontal wind contribution is visible in the measurements.

The second transect, an upwind flight (Fig. c, d), reveals the streamwise boundary turbulence structure for the 10ms-1 background wind case. The flight is conducted in an updraft region of the along-stream organized boundary layer convection; therefore, positive vertical wind speeds dominate. In this case, the superimposed pitch movement contaminates the retrieved vertical wind measurement with a contribution from the horizontal wind if the scanner movement correction is disabled. Thereby, nadir-pointing measurements provide a good opportunity to check the accuracy of the scanner movement compensation and beam direction accuracy using the motion-corrected wind measurement and ground return velocity. Should the movement compensation or pointing accuracy not be satisfactory, it is still possible to calibrate them using the ground return velocity . Using this information, the amount of horizontal wind mapped into the vertical wind can be removed in a further post-processing step if the vertical profile of the horizontal wind is known. In order for this method to yield reliable results, an accurate horizontal wind speed estimation is necessary. For this reason, the reliability of wind profiling measurements is the focus of this study.

The theory and problems associated with airborne Doppler wind profiling are very similar between lidar and radar systems. Therefore, no distinction is made between the studies using either of the two instruments in the following, unless necessary due to explicit differences. Similar to ground-based wind profiling, airborne wind profiling is usually conducted by scanning in conical scans along the flight path. Retrieval of the mean wind vector can be achieved through inversion of the beam matrix, yielding a least-squares solution to the problem . and apply the original velocity azimuth display (VAD) analysis directly, while neglecting the altered beam geometry due to aircraft movement. Other methods at higher computational cost exist as well. discuss a variational approach which can improve the traditional solution by adding anelastic mass continuity constraints on the estimated solution. Accumulation of the Doppler spectra can be conducted for Doppler lidar and has been shown to extend the retrieval limits in clear air conditions with little return signal .

In this study, a standard inversion-based approach outlined by for airborne wind profile retrieval is followed. In this method, multiple radial velocity measurements under different beam-pointing directions are sampled from an atmospheric wind field, e.g., by scanning (see Fig.  for an illustration of the scan trajectory). The inverse problem is then solved by calculating the inverse of the beam-pointing matrix (for the associated formulas and details, see Appendix ). Thereby, a least-squares (LSQ) estimate of the wind vector, which was responsible for the observations, is obtained (see Fig.  for an illustration of the LSQ-fit procedure). In this study, a singular value decomposition (SVD) is performed to calculate the inverse. The SVD solution yields benefits compared to the direct solution, as a number of reliability control parameters become available .

The retrieval error is defined as the difference between the true wind speed VmT, calculated from the input LES wind field as discussed below, and the ADLS retrieved wind speed VmR, obtained from wind profiling (see Fig.  for an example on the obtained wind profiles and associated errors): 5ΔVm=VmR-VmT. In this study, the wind speed Vm is calculated from all three components, Vm=u2+v2+w2. For the quantitative analysis, four metrics are employed, consisting of the root mean squared error (RMSE), the relative root mean squared error (REL), the number of available wind profile points (N) and the retrieval bias, defined in Appendix .

To compare the simulated retrieved wind speed, an LES-based model truth has to be defined. This study follows the approach described in to define the model truth. In this, the model truth is the average over the points in the sampling volume touched by the lidar beam after weighting by the lidar weighting function (Sect. ). In this study, the vector average is used as the wind speed averaging method. Vector averaging is more representative of the lidar-measured wind speed than scalar averaging, as the lidar averages the wind field over a large area rather than summing up individual contributions without respect for directional change. The method used makes a difference especially for lower wind speed cases (illustrated in Fig. , discussed in depth in Sect. ). The advantage of the direct, equal-volume-based LES–lidar comparison, as noted by , is that differences which occur between the simulated measurement and model truth are exclusively traceable to the wind field inhomogeneities. Therefore, optimization of the measurement system setup with respect to the impact of wind field inhomogeneities is possible. When an ideal system is assumed, with no measurement system inaccuracy and no co-location problem, the results are also useful as they present a lower bound to the observable Doppler lidar error. A lower error cannot be observed from real in situ comparisons (assuming equal atmospheric conditions, measurement system setup and retrieval strategy), as the error given in this study specifies the inherent error due to the Doppler lidar viewing geometry and retrieval settings.

The simulation results presented in the following are not directly transferable to real-world wind profiling comparisons. Today, no measurement system is able to trace the exact volume touched by the lidar beam and thereby measure the input truth, as is done in the ADLS. Therefore, a real-world comparison should result in larger error levels than what is presented here, due to additional co-location problems and due to the different wind vector measurement principles . It seems worthwhile to extend the analysis in this direction at a later point, as the underlying statistical foundation exists already . The topic is not addressed here, as it is beyond the scope of this work.

The retrieved wind profiles are sampled from the LES domain in temporal and spatial proximity. Independence of the retrieved wind profiles is important to ensure a sufficiently large and independent dataset for robust analysis with reliable values. Therefore, correlation is minimized through an adequate measurement system setup, retrieval strategy and sampling procedure explained in the following. However, it should be noted that any potential correlation in wind profiling error does not affect the reported errors levels systematically (RMSE, REL, bias). Correlation reduces the number of effectively available wind profiles and thereby influences the uncertainty with which the error levels are estimated, but it does not impact the error levels themselves. Nevertheless, in order to achieve a sufficiently large statistical sample size, correlation is minimized through an adequate measurement system setup, retrieval strategy and sampling procedure. To this end, all profiles are sampled from non-overlapping measurement domains in time and space (termed “checkerboard approach” in the following), illustrated in Fig.  and explained in the following.

The measurement system setup is simulated based on an intended upcoming airborne Doppler lidar system for boundary layer research. The simulator settings are adjusted accordingly, an overview of parameter settings is given in Fig. . The aircraft characteristics are based on an unpressurized, medium-range turboprop aircraft, flying above the boundary layer at measurement speed under visual flight rules. Therefore, the aircraft is flying at an IAS of 65ms-1 and the aircraft altitude is set to 1100m. Combined with the chosen scan elevation angle, this setting gives equal along- and across-track retrieval volume averaging distances.

The scan pattern used in this study is based on the scan geometry of existing systems . It consists of a 20 s full circle scan time (scan speed 18∘s-1) at 60∘ elevation (30∘ off-nadir). The scan trajectory starting point is chosen randomly at each time step in order to ensure sampling of different locations by the lidar beam. Additionally, the scan rotation direction is reversed for subsequent transects through the LES.

The lidar measurement frequency is set to 1Hz, with pulse width and range gate settings according to Sect. . Changing the measurement frequency to 10Hz does not alter the results significantly, as the additional measurements are strongly autocorrelated (see Fig. ). Consequently, the greater number of measurements does not provide new information to the retrieval, nor does it result in a better fulfilled homogeneity assumption.

The wind profile retrieval consists of u, v, w component retrieval using a volume-based profile separation. Retrievals are obtained from along-track averaging over Xa=1300m and the same across-track averaging distance of Xc=1300m. The along-track averaging distance is based on the distance covered by the aircraft during one full scan rotation. The across-track averaging distance corresponds to the maximum across-track distance covered by the lidar beam at 60∘ elevation and a flight altitude of 1100m above ground. The vertical wind profile resolution is chosen as 62m, yielding one range gate within every vertical layer. Only wind profiles within the boundary layer and entrainment zone are compared as this study focuses on retrieval error under inhomogeneous, turbulent conditions. Therefore, the maximum wind profile altitude is set to 800m, preventing the impact of larger scale features such as gravity waves with longer de-correlation scales.

Three flight trajectories traverse the LES domain in parallel from south to north at a horizontal distance of 1750m to each other, in a crosswind flight direction. Each of the three trajectories is repeated 28 times at 1 min temporal spacing, giving a total number of 84 actual transects. To avoid correlation among the analyzed profiles from different transects, all profiles are sampled from non-overlapping measurement domains in time and space, illustrated in Fig. .

To this end, only one wind profile is retrieved for each transect. The retrieval volumes for the parallel transects at each time step are arranged in a checkerboard pattern to ensure maximum spatial independence. The distance between spatially neighboring transects and measurements is larger than the integral scale of turbulence in the LES, generating statistical independence.

The checkerboard pattern is shifted by Xa along flight track for subsequent transects, thereby also ensuring temporal independence of the retrieved profiles. After three time steps, the retrieval volume shift procedure is repeated. Thereby, the same location is sampled only every 3 min. The decorrelation of the retrieved profiles is aided by crosswind advection and the pencil-beam nature of the lidar beam. While large-scale structures in the flow, e.g., due to coherent turbulence, may persist over some time and distance, small-scale variation is more rapid. The small-scale variation is expressed by the short integral length scales of the LES flow at 50–500m, denoting the spatial distance over which turbulence is statistically independent from the previous sample. A random offset within one retrieval volume length is added to all retrieval volume start points within one retrieval volume shift procedure. The random start point prevents a systematic influence of the transect beginning location on measurement quality for the different flight directions.

An analysis of the spatiotemporal correlation of the obtained wind profile retrieval errors ΔVm shows that the obtained errors are independent to a high degree for all background wind speed cases, time steps and retrieval altitudes (Figs. , , and ). The autocorrelation functions allow for evaluation of both, the spatial as well as the temporal correlation of profiling error, at the same time for the chosen checkerboard approach. On the one hand, the autocorrelation values at lags 1, 2, 4, 5, 7, 8 and so on reveal the spatial correlation of wind profiling error between neighboring retrieval volumes. On the other hand, the autocorrelation values at lags 3, 6, 9 and so on reveal the temporal correlation of retrieval error for the first retrieval volume (as the retrieval volume shift procedure is repeated after a cycle of three volumes, the same volume is sampled again at these lags). Taking into account the small number of points used to calculate the autocorrelation functions, they do not display concerning systematic structure (e.g., repetitive with cycle durations 2 or 4, or similar between trajectory or altitudes). The marginal structure present beyond the noise resulting from the small sample size can be explained by large-scale organization of turbulence on scales similar to the spacing of the retrieval volumes. Examples of weak structures are the alternating correlation coefficients for the first lags of trajectory 1 in the 0ms-1 background wind case at low altitudes (Fig. ), as well as the slightly negative correlation for lag 1 in the 15ms-1 background wind case at low altitudes (Fig. ). As the associated correlation coefficients remain small (<0.5 in almost all cases), the number of effectively available wind profiles is not affected strongly by these effects.

Please note that despite investigating an overall independent error (between spatially and temporal different profiles), the error of vertically adjacent wind profile points is not random for an individual profile (see Fig. ). The error between neighboring vertical profile layers is not random because turbulent structures are correlated between vertically adjacent layers. This makes an individual wind profile appear smooth, despite overall random error being present. The magnitude of the random error only becomes noticeable when looking at profiles sampled at different locations or times (this disguise is likely one of the reasons why the error due wind field inhomogeneities has received little attention in literature so far). The vertical correlation effect is present in both the ADLS and real measurements with equal magnitude.

The previous section showed that a statistical analysis of error in wind profiling is possible using the given system setup, retrieval strategy and sampling procedure.

The ADLS simulation allows for retrieval of 336 individual wind profiles giving 4032 wind profile points for all altitudes (12 wind profile points per wind profile at the different retrieval altitudes). These consist of 84 wind profiles (1008 wind profile points) for each background wind case, yielding a sufficient statistical basis for evaluation. The 336 wind profiles (4032 wind profile points) are more than what is typically available for comparison in real-world measurements, as co-located validation measurements are very difficult and costly to conduct. For example, 33 wind profiles (740 wind profile points) are compared to dropsonde data in , approximately 10 wind profiles to a ground-based wind profiler in , a single wind profile to dropsonde data in and approximately 49 wind profiles (2056 wind profile points) to dropsonde data in , with each of them mentioning the importance of the spatially differing sampling volumes and associated problems.

The wind profiling quality of the standard system setup and retrieval settings in the investigated inhomogeneous flow conditions can be evaluated from Fig. . As discussed above, the wind profile retrieval is made assuming an ideal measurement system; thereby all deviations are directly traceable to AVAD assumption violation due to wind field inhomogeneities. For quality filtering purposes, a minimum coefficient of determination of Rmin2=0.90 and a maximum condition number CNmax<9 are applied. Similar values have been recommended and used in the past by . The effect of quality filtering on retrieval quality is investigated in detail in Sect. .

A total number of 3006 (of the 4032 retrieved) wind profile points pass quality filtering and the results provide a number of interesting observations. Firstly, the wind speed retrieval is unbiased provided that an appropriate retrieval strategy is used and appropriate quality filtering criteria are applied (discussed in Sect. ). Overall retrieval quality is high, with R2=0.99 and RMSE=0.36ms-1. Yet, even with strict quality filtering applied, deviations above 1ms-1 can occur. The LES cases with higher wind speeds show higher wind variability and increased absolute wind speed retrieval error. With decreasing wind speed (separately for each of the cases), associated with measurements in the boundary layer under turbulent conditions, deviations increase but remain unbiased. Interestingly, the 1008 retrieved wind profile points for the 0ms-1 background wind case are completely excluded by the quality filtering criteria (as well as 18 wind profile points from the 5ms-1 background wind case). They also show much higher wind speed retrieval scatter and can introduce a positive bias if inappropriate quality filtering criteria are applied, although the individual retrieved components themselves are unbiased (Fig. ). The reasons behind this are discussed in Sect. , where the effect of quality filtering is thoroughly investigated.

The vertical distribution of wind profile error mirrors that of the responsible boundary layer turbulence (Fig. ). Errors are largest in the middle of the boundary layer where up- and downdrafts have maximum intensity. Towards the ground, a slight reduction in wind profiling error is observable for all wind speed cases. There, the size of the turbulent elements becomes smaller and consequently the lidar scan averages over more eddies, thereby better fulfilling the homogeneity assumption. This is a theoretical result; in real measurements, other effects such as ground chirp will degrade near-ground retrieval of wind profiles . Towards the top of the boundary layer, wind profile error decreases as the homogeneity assumption is better fulfilled. Nevertheless, entrainment and detrainment processes can still cause noticeable retrieval error, especially for higher wind speed cases. In the free, homogeneous atmosphere, wind profile error vanishes (not shown).

The wind direction retrieval (Fig. ) is of similarly good quality compared to the wind speed retrieval. The wind profile points which pass quality filtering cluster in a small area around 270∘, representing the westerly wind direction. The wind direction retrievals for the 0ms-1 background wind case scatter throughout the full range as the wind direction is not meaningful without a background wind speed. As before, these values are completely eliminated by quality filtering. The retrieval exhibits slightly degraded quality criteria (lower R2, small y-axis intercept) compared to the wind speed retrieval. However, this behavior is due to the data not being distributed over a wide range, as is the case for wind speed.

Assessing the reliability of the parameters retrieved through the inversion process is a common problem in inverse theory, and two metrics are investigated as a part of this study.

The coefficient of determination R2 (Appendix , Eq. ) is often used to detect a violation of the homogeneity assumption . When using this approach, it is assumed that deviations from the homogeneous state show up as deviations of the measurements from the LSQ fit. Wind field inhomogeneities smaller than the scan volume size decrease the R2. Therefore, it is commonly assumed that the R2 captures the degree of violation of the homogeneity assumption. Often-used filtering criteria are R2>0.8 or R2>0.95 for ground-based wind profiling . However, as shown already by , a linear change in the vertical wind biases the retrieved horizontal components and a linear change in the horizontal components biases the retrieved vertical component. Both deviations are not detectable as deviations from the LSQ fit.

If the matrix inversion is performed based on a singular value decomposition, additional quality criteria such as the condition number (CN) (Appendix , Eq. ), describing the degree of collinearity among the model geometry, become available and are frequently utilized . The CN provides a measure of the spread of the model space, thereby diagnosing collinearity . A high CN indicates high collinearity in the model geometry. Consequently, the real system state is not well explored in at least one direction, and as a result the inferred system state is prone to a large error amplification . The CN is often used as a quality filtering criteria to exclude measurements where the spread of the beam-pointing directions is not sufficient to explore the wind field adequately . In this section, the performance of the most commonly used quality criteria (R2 and CN) is evaluated, specifically their relation to wind profile quality and their adequacy in detecting violations of the homogeneity assumption and collinearity in the model geometry.

The analysis of the simulator results for the ideal measurement system setup and standard retrieval setting reveals a number of interesting findings (Figs.  and  for the full range of non-quality-filtered values). Deviations from the geostrophic background wind towards lower wind speeds, associated with stronger turbulence in the boundary layer, generally decrease the value of the R2 but do not influence the CN, as it is expected (Fig. c, f). Even at high R2>0.90, significant wind profile errors up to 1ms-1 do exist. The CN values cluster in a narrow range with values between 3 and 4, as the sampling volume is generally well explored by the lidar for all retrievals due to the checkerboard approach.

A noteworthy feature occurs for the 0ms-1 background wind case. Here, a clear dependence of wind speed error on the R2 value is observable (Fig. a, b, c). The estimated wind speeds are only slightly biased for coefficients of determination in the range close to 0. However, with increasing R2, the bias of the retrieved wind speed increases linearly to values in the range 1–2ms-1. This problem exists despite the fact that the individual estimated components themselves are unbiased (Fig. ). Coefficients of determination above 0.9 are non-existent for the 0ms-1 background wind speed; therefore, the biased values can be filtered completely. The observed behavior is caused by the lidar measurement method: at low wind speeds, the measured radial velocity and thereby also the retrieval is strongly influenced by the vertical wind and less by the horizontal wind, especially at elevation angles closer to nadir (see also Fig. ). For a horizontally sheared vertical wind field, exactly in phase with the measurement geometry at scan elevation angle φ and with magnitude w (a worst-case scenario, leading to no detectable variation in the LSQ fit), the erroneous mapping follows ΔVm=w⋅tan⁡(φ). For high coefficients of determination, the vertical wind variations are more in phase (with an expected horizontal wind contribution from the simplified horizontal wind field model) and thereby mapped more into horizontal wind, causing a stronger positive bias. The values of the horizontal wind speed in the range of 0–3ms-1 correspond with a mean vertical wind of 0–1.5ms-1 amplitude being mapped into the horizontal wind at 60∘ elevation. At lower coefficients of determination, the noisy (sub-scan-volume) vertical wind variations overwhelm the smaller horizontal wind speed signal. However, they do not lead to a mean wind speed being estimated as they cancel out due to averaging, thereby causing no bias. If the scalar averaged LES wind speed is used for lidar comparison instead of the vector-averaged LES wind speed discussed here, the described linear trend is equally present but with an intercept offset of -1ms-1 along the y axis. However, for elevation angles closer to the horizontal, giving more accurate retrievals, the estimated lidar wind speed tends towards the vector average, making it the correct choice for comparison.

In summary, for low wind speeds, the application of a higher R2 mainly filters for smoother in-phase variations of the vertical wind. Consequently, selecting an inadequate threshold for R2 can cause a bias, an important finding that is not expected and, to our knowledge, undocumented so far. For low wind speed cases, the R2 is not an appropriate quality filtering criteria, unless an appropriately high threshold is chosen which then successfully filters the low wind speed cases.

For the retrievals investigated here, a clear separation between biased retrievals from the 0ms-1 background wind case and the non-biased retrievals from the 5ms-1 background wind case is possible, based on R2>0.9 threshold filtering. However, in reality, this separation is not necessarily possible, as intermediate background wind speeds in the range of 0–5ms-1 might be present. These cases may also result in biased retrievals, as long as the radial velocity contribution of the vertical wind is of comparable magnitude to that of the horizontal wind, giving the possibility for non-negligible erroneous mapping. A possible bias for unknown reasons, varying by day, has been reported by already. They stress the need for further investigation of this phenomena using simulated lidar data. The bias at low wind speeds is also noticeable in studies comparing lidar-measured wind speeds to in situ measurements; however, it is without discussion or explanation so far. Some recent examples of the observed overestimation at low wind speeds are, besides others, visible in (ground based; Figs. 4, 5), (airborne; Figs. 2, 3b under turbulent conditions inside the boundary layer) and (ground based; Fig. 5e).

Therefore, based on the above discussion, we recommend the following procedure in wind profile analysis. In doubtful situations, where the approximate magnitude of the measured wind speed is unknown, one should always analyze a long spatial average over multiple scan rotations first (approximately 10 times the expected maximum eddy size). For longer averaging, the mapping of vertical wind does not influence the retrieval significantly, as the structure of the vertical wind is not in phase with the measurement geometry over long distances (except for extended regions of horizontally sheared vertical wind, e.g., in complex terrain). Thereby, the approximate magnitude of the wind speed can be analyzed reliably, although at coarse spatial resolution. If the outcome of this analysis is a wind speed magnitude below 5ms-1, and a non-negligible vertical wind with similar magnitude is present, a further analysis at higher spatial resolution is not advisable due to the possibility of erroneous mapping. If, on the other hand, the wind speed magnitude is above this threshold, or the vertical wind magnitude is negligible, the spatial resolution of the retrieval can be refined further. In the case of a sheared wind speed profile, e.g., low wind speeds close to the ground and higher wind speeds above, the required averaging distance varies vertically. As along- and across-track averaging distances used in the retrieval are freely selectable parameters, they can also be adjusted vertically in order to allow for an optimal retrieval with the highest resolution. In the future, if advanced and flexible measurement system are available, it also appears possible to adjust the measurement system setup based on the encountered wind field conditions. For this procedure, an analysis of preferable measurement system setups and retrieval strategies which are less influenced by erroneous mapping appears worthwhile to conduct in future studies.

Leaving the low wind speed problem aside, even when using strict quality filtering criteria, considerable wind speed retrieval error can still exist due to inhomogeneities. Or, stated in another way, even relatively small deviations from the homogeneity assumption that do not lead to significant degradation of the quality filtering criteria can cause noticeable retrieval error. For these cases, the possibility of a data-driven uncertainty estimation (e.g., based on the measured radial velocity variance) appears promising to investigate.

Overall, it can be concluded that wind profiling error due to violation of the homogeneity assumption can be of non-negligible magnitude. The errors reported in this study, solely due to flow inhomogeneity, are of comparable magnitude to, or even above, what has been reported as overall wind profiling error in other studies . Wind profiling error due to flow inhomogeneity should therefore not be neglected as an important source of error when evaluating the measurement accuracy of airborne Doppler lidar systems.