Behavior and mechanisms of Doppler wind lidar error in varying stability regimes

. Wind lidars are widespread and important tools in atmospheric observations. An intrinsic part of lidar measurement error is due to atmospheric variability in the remote-sensing scan volume. This study describes and quantiﬁes the distribution of measurement error due to turbulence in varying atmospheric stability. While the lidar error model is general, we demonstrate the approach using large ensembles of virtual WindCube V2 lidar performing a proﬁling Doppler-beam-swinging scan in quasi-stationary large-eddy simulations (LESs) of convective and stable boundary layers. Error trends vary with the stability regime, time averaging of results, and observation height. A systematic analysis of the observation error explains dominant mechanisms and sup-ports the ﬁndings of the empirical results. Treating the error under a random variable framework allows for informed predictions about the effect of different conﬁgurations or conditions on lidar performance. Convective conditions are most prone to large errors (up to 1.5 m s − 1 in 1 Hz wind speed in strong convection), driven by the large vertical velocity variances in convective conditions and the high elevation angle of the scanning beams (62 ◦ ). Range-gate weighting induces a negative bias into the horizontal wind speeds near the surface


Introduction
Effectively and efficiently collecting observations of atmospheric winds poses an ongoing, multifaceted challenge for the atmo-20 spheric science community. Wind-profiling lidar offers a cheaper, more easily deployable, and higher-profiling alternative to traditional meteorological towers while scanning lidar allows for collection of data over broad regions of the atmosphere. Over the last few decades, lidar technology has grown into maturity, with several commercial wind lidar systems becoming available since the late 2000s. Lidar systems are widely employed in scientific studies of atmospheric boundary layer meteorology have unencumbered access to full knowledge of the flow field, allow for control over the the case parameters (terrain, forcing, 60 boundaries), and can 'deploy' instruments in ways that are not physically or financially possible in reality (e.g. re-sampling the same flow field or testing many locations in a domain). The comprehensive flow field data also opens up the discussion about the appropriate reference 'truth' for lidar observations. Measurements may be better thought of as representing volume averages, which we cannot directly measure in the field but can compute a reference for in LES.
Earlier virtual lidar studies have generally considered complex lidar behavior and have been built on a range of different LES 65 models. The coordinated use of multiple lidar to simultaneously probe spanning vectors of the wind in a volume was studied by Stawiarski et al. (2013) using the parallelized large-eddy simulation model (PALM) (Maronga et al., 2015). The dependence of profiling lidar on horizontal homogeneity complicates its use in complex terrain; Klaas et al. (2015) investigated observation deviations due to terrain and choice of instrument location. Gasch et al. (2019) implemented an airborne virtual lidar with PALM and studied errors due to flow inhomogeneities. Wind energy applications have been a notable driver of virtual lidar 70 studies. Simley et al. (2011) modeled scanning continuous wave lidar to optimize their upwind measurements for use in wind turbine control. The measurement of turbine wakes with profiling lidar was explored in Lundquist et al. (2015) (using SOWFA) and Mirocha et al. (2015) (using WRF-LES). Turbine wakes are also considered in Forsting et al. (2017) which focuses on the volume averaging along lidar beams in these high-gradient regions. Only recently has virtual lidar been employed for baseline studies of profiling wind lidar in favorable (flat, uniform, quasi-stationary) conditions. Rahlves et al. (2021) used virtual lidar 75 in PALM LES to compare the bulk performance of various scan types (DBS and VAD at varying cone angles) across a suite of convective conditions.
We have developed a virtual lidar model tool in Python to run on output from the Weather Research and Forecasting largeeddy simulation (WRF-LES). WRF-LES boasts a user base of over 48,000 and is attractive for its accessibility as an opensource, documented model. It can be configured for ideal simulations or coupled with mesoscale nesting to simulate case 80 studies of real sites (Mazzaro et al., 2017;Haupt et al., 2019), and offers a range of subfilterscale turbulence models for use in LES (Mirocha et al., 2010;Chow et al., 2005). In validations, WRF-LES has also compared well to observations of boundary layers in varying stabilities (Peña et al., 2021). Though the virtual lidar tool is targeted at WRF-LES, with minor adjustments to accommodate for the different output formats, it could be easily adapted for use with other LES models.
In this first demonstration of the virtual lidar tool, we consider a specific case of the Leosphere WindcubeV2 profiling lidar 85 ( Figure 1) measuring mean winds in ideal simulations of stable and convective conditions over flat terrain. As in Rahlves et al. (2021), the configuration allows for a baseline assessment of the lidar performance by omitting external sources of inhomogeneities, like complex terrain or wind turbines, and isolates the system error arising from complex but statistically stationary turbulent boundary layer flow. Depending on the quantity of interest, Rahlves et al. (2021) found that the configuration choices (scan type, cone angle, averaging length) have distinct effects on the lidar retrieval error. Additionally, profiling in strongly 90 convective conditions, absent other sources of inhomogeneity, the lidar exhibited markedly larger errors than in more moderate convection. Our work extends that study for a single DBS profiling scan to include a range gate weighting in the lidar model, a further stable stability regime, and the disaggregation of the vertical profile heights. The idea of using ensembles to gauge the uncertainty of the error (Rahlves et al., 2021) is expanded to using larger ensembles to characterize an error distribution particular to the flow conditions and scan geometry. Further, we present an analytic treatment of the observation system error 95 to explain the dominant error mechanisms and trends, supporting the findings of the empirical results. The framework further enables informed, a priori predictions of how different configurations or conditions might be expected to impact lidar performance, without relying on the full virtual lidar model. Section 2 presents the lidar model and its configuration to represent the WindcubeV2 lidar. It then describes the WRF-LES cases against which ensembles of the virtual instruments were tested. The resulting observation error distributions are summa-100 rized in section 3, focusing on empirical trends across stability conditions, height, and time-averaging. Section 4 analyzes the mechanisms behind the errors and the conditions which drive the trends found by the virtual lidar model. Here we address the influence of the range gate weighting function and analytically represent the impact of violations of the horizontal homogeneity assumption. Further treatment investigates the wind speed and direction measurements derived from the wind components and time-averaged measurements.

Generalized virtual lidar model
The virtual lidar was designed to create a configurable, general model that can be modified to replicate most lidar instruments.
The observing system is decomposed into modular components common across lidar systems: the retrieval of radial wind velocities along an individual beam via a range gate weighting function, the scanning pattern the beam moves through, and 110 the internal post-processing of these measurements. The handling of each of the components can be easily modified and new definitions substituted to allow for customization. This initial study focuses on a common commercial system: the vertical profiling Leosphere WindcubeV2 performing a Doppler-Beam-Swinging (DBS) scan ( Figure 1). Its parameters and geometry, summarized in table 2, form the basis of the description of the model stages. Thorough discussions of the Windcube, other wind lidar systems, and the underlying technology 115 can be found in Courtney et al. (2008); Lindelöw (2008); Cariou and Boquet (2010), and Thobois et al. (2015). 5 https://doi.org/10.5194/amt-2022-73 Preprint. Discussion started: 9 March 2022 c Author(s) 2022. CC BY 4.0 License.

Sampling along a single lidar beam
The basis of wind lidar is the retrieval of radial (line-of-sight) wind velocities along an emitted laser beam using the backscatter off aerosols entrained in the flow. Doppler lidar diagnose the shift in the frequency of the backscattered light to measure radial wind velocity (equation 1).
120 v r =b · u = u sin γ cos φ + v cos γ cos φ + w sin φ The radial velocity, v r , is the dot product of the wind velocity vector, u = (u, v, w) with u along the zonal direction, and the beam unit direction vectorb. The beam direction points along azimuthal angle, γ ∈ [0 • , 360 • ) clockwise from north, and elevation angle from the horizon, φ ∈ [0 • , 90 • ]. With this convention, positive radial velocities move away from the instrument.
In the context of our model, we assume 'perfect' conditions in the sense of ignoring factors like aerosol type, size, and 125 density distribution and conditions like humidity, fog, or precipitation that can affect the quality of the return signal in the optical measurement of the radial velocity (Aitken et al., 2012;Boquet et al., 2016;Rösner et al., 2020). We instead focus on the representation of the averaging introduced by the sampling process.
Although the scattering cannot be explicitly resolved on an LES scale, previous studies have found that the full sampling procedure (collection and internal processing of backscattered light) is well-approximated by the application of a range-gate 130 weighting function (RWF) (Frehlich, 1997;Gryning et al., 2017;Simley et al., 2018). Prevalent lidar technologies employ either continuous wave (e.g. ZephIR) or pulsed (e.g. Windcube) lasers to target distances along the beam at which to retrieve velocities. Continuous wave systems set a focal distance to center returns, while pulsed lidar release a rapid sequence of pulses and separate the returns into a series of spatio-temporal 'range gates' along the beam. In both cases, the process acts like a weighted, volume average of radial velocities along the beam about the target distance. The cross-sectional area of the beam is 135 negligibly small compared to the along-beam length scale, so that the averaging may be described by a one-dimensional line integral. The system-observed radial velocity at a distance r, v r (r), is given by the convolution of projected wind velocities with the weighting function (equation 2).
is the beam direction unit vector, and u the velocity field.

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For a pulsed lidar, the weighting function arises from the convolution of the range-gate profile with the beam pulse profile (Frehlich, 1997); equation 3 gives the integral in space.
A top hat, normalized indicator function, χ, represents the time span of the range gate and the pulse shape, g, is assumed to be Gaussian (Banakh et al., 1997). In the lidar operation, the parameters for the pulse and range gate are temporal quantities.

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To transform into their representation for the spatial integral, assume that propagation is at the speed of light and note that the originating signal must travel to a point in space and back to the instrument receiver to be collected (Lindelöw, 2008;Cariou and Boquet, 2010). The indicator function for the spatial range gate with temporal interval τ m is given in equation 4.
We can also express the spatial Gaussian pulse in terms of the temporal full-width half-maximum (FWHM) parameter, τ 150 (equation 5).
Here, c (0.29979 m/ns) is the speed of light, τ m the temporal range gate, and τ the full width half maximum (FWHM) pulse duration. Other representations of the pulse and range gate (i.e. not top hat and Gaussian) are not necessarily valid under this approximation. Lindelöw (2008), for example, adapted the form to account for a focused beam which scales the RWF by the focusing efficiency. The basic, unadapted form presented here and implemented in our model for this study is also used in several other virtual lidar models (Stawiarski et al., 2013;Lundquist et al., 2015;Gasch et al., 2019;Forsting et al., 2017).

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Using parameters from the WindcubeV2 (table 2), the shape and extent of the RWF (equation 6) can be made concrete (figure 2a). The weights drop to half their peak value about 20 m to either side of the target distance and are non-negligible up to around 40 m. The choices of pulse and range gate parameters in a coherent lidar system must balance the desire for spatial locality (reducing the 'width' of the RWF) and the need for adequate accumulation time (τ m ) to accurately resolve frequencies used for the radial velocity measurement. The application of the RWF to the radial velocity profile may be thought of as a 165 smoothing/low-pass filter (figure 2(b,c,d)).
The form of the pulsed lidar RWF is distinct from that of continuous wave lidar (Gryning et al., 2017;Forsting et al., 2017).
A major difference is the consistency of the pulsed RWF over target distances; the RWFs of continuous wave lidar vary with distance, giving better locality close to the instrument than far away. Closer distances will be sampled at higher resolution by a CW instrument and further distances by a pulsed lidar.

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To compute the RWF-weighted retrieval from an LES flow field, the wind components are first interpolated to points along the beam and the projection onto the beam direction found. The virtual lidar uses a linear barycentric interpolation from a triangulation of the LES grid (i.e. linear interpolation on tetrahedrons using Virtanen et al. (2020), Spe). The numerical approximation of the convolution integral from the interpolated radial velocities treats the continuous weighted average as a discrete weighted average (equation 7). The form is a slightly modified formulation of that used in Lundquist et al. (2015) and  Forsting et al. (2017) Parameterizing along the beam, the {s i } nodes are the points where the winds have been interpolated (s i = 0 is at r 0 ). If the nodes are taken as midpoints of intervals with corresponding lengths {h i } partitioning the integral range [−T, T ], then the quadrature formulation is a normalized midpoint rule. The normalization ensures the result is a weighted average (avoiding 180 over-or under-estimation due to the numeric weights not summing to unity). The placement of the nodes {s i } is free to be chosen for convenience or, as recommended by Forsting et al. (2017), to optimize utility of the interpolated points in the convolution so that fewer points need be interpolated. Our current implementation uses equi-spaced points one meter apart along the beam (see appendix E for discussion).
Interpolation dominates the computational work in most virtual lidar models, making it a prime target for performance opti-185 mization. Some preliminary efforts have been made in our implementation, e.g. computing and saving interpolation weights to reuse for the different velocity components and for repeating beams on a static grid. Further improvements to the implementation and choice of nodes are possible and will be targeted in subsequent developments.

Time-resolved scanning patterns
Based on the type of scan they perform, lidars are categorized as profiling or scanning systems. Profiling lidars are designed 190 to provide a vertical profile of the three-dimensional wind velocity, much as would be reported by a meteorological tower.
To reconstruct a three-dimensional wind vector, the instrument needs spanning radial velocity samples from at least three different directions. The scanning head rotates through the necessary positions quickly, which limits the intervening evolution of the wind field. Scanning lidars perceive the atmosphere in a fundamentally different way; they move the beam more slowly through a slice of the atmosphere, resolving the radial velocity of features in the spatial selection. Common configurations 195 include vertical slices (range height indicator or RHI scan) which fix the azimuthal angle while varying elevation; conical slices (vertical azimuthal display or VAD scan) which fix the elevation angle but slowly complete a full rotation, or a partial conical section (plan position indicator or PPI scan), as visualized in Clifton et al. (2015). Any scanning geometry, as described above or a more complex configuration, arises from 'pointing' the beam and is most naturally and compactly represented as a time series of elevation, φ, and azimuthal, γ, angles in spherical coordinates with the beam source at the origin.

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For the purposes of this study, we consider the Doppler-beam-swinging (DBS) profiling scan used by the WindcubeV2, which moves through the four cardinal directions, angled 28 • from the vertical, before pointing the beam straight vertically ( Figure 1). The total scan takes approximately 5 seconds, spending about a second at each of the scan positions (Bodini et al., 2019). The range gates correspond to equi-spaced heights above the ground. As the beam rotates through the scan, radial velocities are measured at the center and four points around the circular perimeter of the scanning cone for each given height.

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At each second in the post-processing stage, the most recent set of radial velocities is used to reconstruct an estimate of the vertical profile of three-dimensional velocities.
The beam accumulation time for the WindcubeV2 is about a second, while the LES model time steps are on the order of a tenth of a second. The additional averaging due to the longer accumulation time is ignored in the current version of the virtual lidar; it handles the scan by performing the beam sampling on snapshots of the flow field output at one second intervals. It 210 is assumed that in the WindcubeV2 the temporal average is less significant than the spatial averaging; future versions of the model will account for accumulation times by performing this averaging step explicitly.

Internal processing: 3-D velocity reconstruction
In the WindcubeV2, the post-processing stage reconstructs the three-dimensional velocity from the radial velocities collected across the scan cycle. Under the assumptions of horizontal homogeneity and invariance of the flow field over the scan duration, 215 the radial velocities collected by each of the beams at a given height are all projections of the same three-dimensional velocity vector. Omitting the vertical beam, we solve for the vector components at a given range gate height (equation 8) as in, e.g. Cariou and Boquet (2010).
At a fixed height, u E,W,N,S are the most recently measured radial velocities (v r (r)) from beams pointed in each of the 220 cardinal directions. The elevation angle of the beams from the horizon is φ = 62 • .
Later versions of Leosphere's Windcube instruments use a modified reconstruction (Krishnamurthy, 2020) for the vertical velocity (equation 9), which weights the beams in the reconstruction using the estimated wind direction, Θ, measured clockwise from due north, as presented in Newman et al. (2016) and Sathe et al. (2011). Re-weighting emphasizes beams along the mean wind direction, exploiting the fact that decorrelation distances along the mean wind direction are typically longer than along 225 the cross-stream direction.
When the mean wind direction is at a 45 • angle to the lidar axes (delineated by the south-north and east-west beam pairs), the weights reduce to the uniform 1 4 in the original reconstruction (equation 8). When the mean wind aligns directly with one of the lidar axes, only the two respective beams on that axis are used. For our tests, we use reconstruction with wind-direction 230 weighting to represent currently utilized versions of the instrument. The grey region demarcates the WindcubeV2 vertical range including the region of influence for the range gate weighting.

Idealized atmospheric boundary layer simulations of varying stability
Realistic atmospheric flow fields were generated using large-eddy simulation (LES) configurations of the Advanced Research Weather Research and Forecasting (WRF-ARW) model v4.1 (Skamarock et al., 2019). WRF-ARW is a finite difference numerical model which solves the flux form of the fully compressible, nonhydrostatic Euler equations for high Reynolds number 235 flows. The model runs on a staggered, Arakawa-C grid with stretched, terrain-following hydrostatic pressure coordinates in the vertical. The simulations in this study employed a third-order Runge-Kutta time integrator and fifth-and third-order horizontal and vertical advection. All simulations used a nonlinear backscatter and anisotropy (NBA2) sub-filter scale stress model (Mirocha et al., 2010).
To establish a baseline reference for lidar operation in ideal conditions, all simulations in this study used uniform flat, 240 grassy terrain (roughness length z 0 = 0.1m), periodic boundary conditions, with temporally and spatially invariant forcing. The idealized scenarios isolate fundamental characteristics of the atmospheric flows, removing potential influence from additional complexities and inhomogeneities from e.g. mesoscale forcing, varied terrain and land cover, and the influence of the diurnal cycle or nearby obstructions like wind turbines. None of the simulations in this study incorporated models for moisture, clouds, radiation, microphysics, or land surface. The simulations were distinguished by varying stability: two convective cases and one 245 stable stratification case, detailed in table 1 and figure 3. For each of the simulations, we used ten minutes of simulated time after spin-up was achieved, output at one second intervals.
For the convective boundary layers, we use data from precursor simulations in Rybchuk et al. (2021). Following the labeling therein, we designate the cases as the 'strong' and 'weak' convective boundary layers. Although both are considered strongly  (Salesky et al., 2017), the cases differ meaningfully in the relative strength of the surface heating and geostrophic winds. The strong CBL case features stronger heating and slower winds than the weak CBL.
The stable boundary layer simulation closely follows the configuration of Sanchez Gomez et al. (2021). The surface condition for the stable case is driven with a cooling rate, rather than a negative heat flux (Basu et al., 2007). Spinning up a stable case to relatively steady turbulence statistics can also be computationally expensive; a set of two one-way nested domains was   The mean background states of the LES cases are spatially and temporally consistent across the domain, including the direction of the prevailing winds. To account for potential differences due to the relative orientation of the lidar axes in the flow, the ensemble of virtual lidar were re-oriented at three additional offset angles {15 • , 30 • , 45 • }, and allowed to sample the LES fields again. The small sensitivity of the error to the relative orientation is discussed in section 4.4.1.

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Determining the error in the lidar observation depends on defining a reference 'truth'. Profiling lidar are often thought of as replacing meteorological towers, returning a vertical profile of three-dimensional velocities similar to a tower fitted with instruments, but what value the lidar should actually be thought of as 'measuring' is not so straightforward. The samples used to estimate the wind components lack the precise locality of tower instruments; the beams collecting line-of-sight data span an increasingly large area with height, each incorporating additional vertical extent via the RWF. These factors suggest that 285 a volume average might be a more appropriate reference truth (as suggested in e.g. Courtney et al. (2008)). The lidar reflects pieces of both representations: it has the spatial spread of the volume average, but depends on only a handful of points on the edge of the volume which impart higher variability similar to a pointwise profile.
Along with a 'pointwise', tower-style truth profile of interpolated velocities above the instrument, we determined a volumeaveraged profile for each lidar. The volume-average was computed as the mean of all LES points which fall inside cylinders 290 tracing the lidar scan radius (figure 1). Centered at each range gate, the cylinders were defined to have radius equal to that of the scan cone and height corresponding the the vertical projection of the RWF range resolution. For the WindcubeV2 the range resolution is the FWHM of the RWF (≈ 40m) so that the cylinders are 40 sin(φ) ≈ 35.3m tall. At the lowest levels with the smallest volumes, a minimum of around 80 LES points are used which increased to several hundred points in the top cylinders.

Observation error trends 295
The error incurred in any individual measurement does not necessarily represent general behaviors; deducing useful, generalizable trends entails focusing instead on distributions of the observation error. Each virtual instrument in the ensemble provides one instance of the ways the WindcubeV2 might interact with turbulent features in each flow regime, thereby sampling the error distribution. We characterize the resulting error distribution, and trends in its behavior, from the raw 1Hz lidar output as well as for time-averaged quantities.

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We define the lidar error as the difference between the lidar-observed value and the reference truth, u err = u lidar − u ref .
Along with errors in each of the reconstructed wind components (u, v, w) we consider errors in the horizontal wind speed and direction derived from the components (equation 10) which are also reported by the lidar and commonly used.
The wind direction, placed in the appropriate quadrant, is compactly represented by the two-argument inverse tangent (the

Raw 1-second velocities
A lidar reports a vertical profile of velocities each second for the ten minute duration of the simulation. Each distribution 310 consists of ten minutes of data combined over the 45 ensemble members and four orientation angles, giving a total of 108,000  regarded as having more 'outliers' with respect to the width than would be found in a normal distribution.
The rest of discussion focuses on trends in the the mean and standard deviation of the error. For the sake of space, they are only shown for the wind speed and direction (figure 5) and for the vertical velocity (figure 6); see appendix A for the first four moments for all variables. Patterns in the error behavior are considered over stability cases, with respect to height, and as they appear in particular measured quantities.

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In all cases, the distribution of the error with respect to the pointwise truth displays larger standard deviation than the error using the volume-averaged truth. With the beam measurements at the perimeter of the scan volume, the lidar reconstruction has no way to predict small-scale variations at the center of the volume where the pointwise truth resides. It can only reconstruct an average representation, and comparison with the point value incorporates additional uncertainty into the error through the pointwise truth reference. The stability case and corresponding structure of the boundary layer powerfully influence the error behavior. The strong convective case consistently suffers from more significant error, with greater bias and standard deviation in the wind speed, wind direction and vertical velocity error. It is followed by the weak CBL, with the stable case being the best behaved. The primary mechanisms identified in section 4 justify the relative ordering. The heterogeneity, especially in the vertical velocity, of the convective plumes strongly drives the error while high wind speeds in the stable case help to reduce it.

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Height trends depend on the change in volume circumscribed by the scan, but also on the vertical structure of the boundary layer ( figure 3) and corresponding scale and character of the turbulent structures as noted by Wainwright et al. (2014) for sodar measurements. In the test convective conditions, the standard deviation of the error grows with height up to the top of the Windcube range. The error growth tracks the growth in the vertical velocities from the lower layers the middle of the boundary layer where the convective plumes are strongest. In the stable case, the standard deviation reaches a maximum just below the 350 boundary layer height at 170m. The peak in the error spread seems to occur close to the infection point in the v mean velocity and the maximum vertical velocity magnitude. In these quasi-stationary conditions, the height trends depend strongly on the vertical structure of the flow, not just the size of the scan volume.
Some error traits are particular to the quantity being measured, outside of the general trends identified above. The wind speed measurement exhibits a bias toward over-estimation (positive error mean), particularly in convective conditions. In the stable 355 case, the distribution is more centered except close to the surface where the mean becomes negative (underestimate). Under strong convection, the wind direction observations also suffer from a bias; the direction lists more and more counter-clockwise Though the error magnitudes are not generally large, the behavior is far from uniform and exhibits strong dependence on the flow itself, even with respect to height within the same stability regime. The mechanisms driving the observed trends in error are broken down in depth in section 4.

Ten-minute time averaged velocities 380
Time averaging is often used to filter out random error 'noise' in the raw measurements output with each beam update. Common averaging intervals may be over two, ten, or even thirty minutes, with experimental evaluations of the system accuracy often reported in terms of the ten-minute average in the wind energy context. As with the error in the high(er) frequency wind measurements, we characterize the error distribution of the ten-minute averaged measurements (figure 8).
Velocity component observations are averaged over the ten-minute simulation span and compared to the time-averaged truth 385 references. The wind speed and direction are not averaged directly but re-computed from the time-averaged wind components (equation 10) so that they correspond to a vector average of the wind. Wainwright et al. (2014) and Gasch et al. (2019) distinguish the 'scalar' average from this kind of 'vector' average while more detailed analysis comparing the averages have been undertaken by Clive (2008) and Rosenbusch et al. (2021). We omit time-averaged vertical velocity error here. Each distribution in the ten-minute average consists of an ensemble of a total of 180 values (ensemble and offset angles).

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Under quasi-stationary conditions, the notions of pointwise and volume-averaged truth start to converge to a general spatiotemporal average, which is reflected in the merging of the two error distribution metrics. The correspondence suggests that field studies comparing against time-averaged 'point' tower measurements can effectively reflect the error with respect to the volume-average as well. The overall error magnitudes found by the virtual lidar are consistent with those in field deployments of lidar compared against tower measurements. In select, flat conditions typical mean discrepancies in the range ±0.2 m/s 395 with standard deviations of 0.20 m/s  have been reported, which encompass all but the more extreme errors in the strong convection case. The virtual ten-minute averaged wind speeds have mean errors within ±0.2m/s with standard deviation < 0.3m/s and wind direction have mean error bias within 2 • and standard deviations in 2.5 • except for in the strongly convective case where it can reach up to 12 • .
Many of the qualitative trends identified in the raw, 1 Hz measurement error persist in this averaged assessment; the ordering 400 of relative error between stability conditions and their respective height trends are similar to the previous section. While respecting these general trends, the form of the moment curves shift a little compared to the original distribution, e.g. the curvature of the standard deviation with height.
The major benefit of the time average lies in the marked reduction (by a factor of around 5) of the error standard deviation across cases and heights. The erroneous rotation in the strong CBL wind direction mean remains although the time average 405 lessens the over-estimation bias in the wind speed. The distribution shapes in the ten-minute average also grow closer to a normal distribution shape.
The following analysis systematically addresses how turbulent variations induce error in the wind reconstruction and how that error propagates into derived quantities. Much of the analysis presented in this section is quite general and applies to any DBS Values computed using the estimated wind components can take on their own, non-trivial error behavior. We characterize the error in wind speed and direction in terms of the u and v error distributions and examine its dependence on the background wind speed and direction. A time average is also a function acting on the raw velocity data and we trace and quantify its 420 expected and observed effect on the error distributions.
Finally, an empirical comparison is made between the error in the different Windcube vertical velocity reconstruction techniques and the direct measurement made by the vertical beam.

Random variable model of turbulent variation in wind component error
To break down the driving sources of error in the reconstruction of the velocity components, we start by deriving the form 425 of the error. The formulation allows the contribution to the error due to the turbulent variations to be explicitly delineated and tracked. For a fixed height, let U = (U, V, W ) T be the mean velocity across the scan volume, i.e. the volume-averaged truth, and assume that it is constant through the five second scan duration. (Different notions of the volume average could be used here, but the disk seems the most useful average representation to measure). Each angled beam samples a perturbed velocity, u E,W,N,S = U + u E,W,N,S where the subscript denotes the cardinal direction of the beam azimuthal direction. The 430 measurement of the projection of the perturbed, point velocity is subject to an additional perturbation, r E,W,N,S , due to the range gate weighting function. Then the radial velocity measured by the beam pointed east, for example, is given by: Carrying the forms through the reconstruction (equation 8), the error in the wind components is given by the difference with the volume-averaged value U . (This analysis is similar to that of Newman et al. (2016), which extends the derivation to turbulent 435 variances). For the horizontal u and v components, the resulting representation of the error is given in equations 11 and 12.
The vertical velocity error form depends on whether the wind-direction weighting is used; we limit our analysis to the equally  weighted version (equation 13).
In a perfectly horizontally homogeneous wind field, the velocity perturbation values all individually vanish (i.e. not due to cancellations). However, even in that perfectly horizontally homogenous case, a non-   The variance (σ 2 ) can also be decomposed into a linear combination of constituent terms, but introduces covariance terms (equations 16-17). (The variance is preferred here to the standard deviation (σ) reported in section 3 so that the contributions are additive , i.e. no square root). We do not explicitly expand the covariance terms which quantify correlations between the perturbations.
As before, the mean of the error distribution describes offsets, or biases, in the error; it represents how quantities are con-

Effects of range gate weighting
The size of discrepancies in the radial velocity measurement due to weighting by the RWF may be analytically bound. The bound serves to illuminate the conditions under which the perturbations from the point value can become large.
Assume any RWF, ρ(s), is a non-negative, symmetric function which monotonically decays as s → ±∞ and satisfies 485 ∞ −∞ ρ(s) = 1. Let v r (r) be the radial velocity profile along the beam and R > 0 a threshold distance from the target r 0 .
Then under the RWF model, the size of the discrepancy between the range-gate-weighted measurement, v r (r 0 ), and the actual, pointwise radial velocity, v r (r 0 ), is constrained in equation 18 (derivation in appendix D).
The bounding terms can be forced to be small by selecting R to manipulate the coefficients. The magnitude of the radial 490 velocities in the atmosphere can practically be expected to be finitely bound. Taking R to be large drives the integral of ρ(s) to one and thus the first term to zero. The second term does the opposite: the coefficient grows rapidly with R and is small for small R. The tension between the requirements picks out the conditions which allow for potentially large deviations in the weighted measurement from the true point value.
Radial velocity profiles with constant gradient do not incur error in the RWF application; symmetry leads the linear contri-

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The impact of the RWF on the total error in the test cases is generally limited compared to that of the horizontal inhomogeneities, except where the RWF interacts with the surface layer. Its contribution (r ) to bias and variability of the wind component error is separated in each stability case according to the model in the previous section. In the convective regimes, the proportion of error variance from RWF perturbations is negligible across heights and wind components (figures 9,10). The relative impact is larger in the stable case, especially for vertical velocity, though the absolute magnitude is still small (figure 505 11). The most prominent influence of the RWF is near the surface layer (due to strong shear) and manifests as a negative shift in the mean error in the horizontal velocities. The weak CBL and stable BL, with strong winds and surface shear, exhibit the effect most strongly. Assuming a logarithmic profile, a large second derivative is to be expected close to the surface, which our analysis suggests is conducive to larger error in the radial velocity measurement. The logarithmic curve in u has a consistent shape which induces an repeated underestimates by the RWF, r E < 0. Accounting for the negative projection of u on the west 510 beam, r W > 0 so that the difference r E −r W is systematically negative, leading to a negative bias in u measurements. It follows that if the the mean winds were easterly, the sign of the bias would also flip (r E − r W > 0) so that the magnitude of the wind component would still be underestimated. Our findings linking the effect of shear on lidar error due to the RWF echo those found by Courtney et al. (2014), and would be applicable in other strong-shear regions of the atmosphere such as at the top of a wind turbine wake. scan radius occur due to heterogeneity on a larger spatial scale than those along a single beam, potentially permitting larger 520 turbulent structures with larger variations.
To describe the error due to the perturbation terms, we consider what they represent and how they relate to the turbulence in the flow. In the lidar model, the velocity perturbations (u beam ) are taken with respect to the average over the scan volume; they are a kind of turbulent fluctuation under the high-pass filter based on the scale of the scan volume (i.e. filtering out turbulence at the larger scales). The variance of the perturbations (σ 2 (u beam )), which determines their magnitude, is a filtered fraction of the 525 full turbulent Reynolds velocity variance (e.g. u u ). As the size of the scan volume increases, the volume average approaches an ensemble mean so that the perturbations are Reynolds fluctuations. The variance expected in the lidar perturbations is determined by the proportion of turbulent variances in each direction above the filter scale (figure 13), with the full variance constituting a 'cap' on the total possible variance.
It is tempting, with usual conventions about turbulent perturbations, to assume that the lidar velocity perturbations will We can now see how these behaviors play out in the virtual lidar data. Using the derived forms (equations 14-17), the contributions from the horizontal (u , v ) and vertical (w ) perturbations to the wind component errors are delineated for each test case (figures 9, 10, and 11). For the most part, the error mean biases can be attributed to RWF effects and the velocity 545 perturbation terms do have close to zero mean, but important deviations from that assumption do arise. The largest deviations from zero mean error (except for the RWF) occur in the strong CBL case and stable BL case, though they do arise elsewhere.
We attribute these deviations to the presence of large coherent structures on the order of the scan volume as discussed above, which agrees with the presence of the large convective plumes as well as the larger-scale structures above the boundary layer in the stable BL.

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In convective conditions, the weighted perturbations in vertical velocity dominate the other sources of variance in the error, indicating that the vertical velocity perturbations become larger than the other terms and dominate the occurrences of large magnitude error. The stronger the convection, the more dominant the role of the vertical velocity terms. By contrast, the error in the stable BL arises from a more even interplay in the horizontal and vertical velocity perturbations. The covariance between the beam perturbations also generally serves to temper the overall error variances.

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The relative contributions of the vertical and horizontal perturbation terms is due to the variance of the perturbations themselves, which we identified as the filtered portion of the full turbulent velocity variance, and the weighting of the terms in the error (equations 16 and 17). Physically, we might expect convective plumes to violate the desired horizontal homogeneity in the flow 12), but it is an even more outsized effect that creates the error. The convective plumes induce large but localized vertical velocity variances so that the lidar scan volume does not filter out the bulk of vertical velocity variance while more of 560 the horizontal turbulence is filtered out (figure 13). The cone angle then over-weights the vertical velocity variance terms. The compounding effects conspire to make the vertical velocity dominant in creating large errors (figures 9 and 10). Even in the stable boundary layer, which features much smaller vertical velocity variances, the vertical velocity perturbations contribute significantly to the error. The smaller scales of the vertical velocity variations in stable conditions (figure 12) coincide with smaller turbulent variance, but also allow a larger portion of the variance to be filtered into the lidar perturbations ( figure 13).

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The result is again over-weighted according to the cone angle. The contributions from the horizontal perturbations, meanwhile, are lessened by more of the variations being filtered out by the scan volume and lack of relative weighting.  As non-linear functions of u and v, the error in the wind speed and direction do not drop out as cleanly as they did for the individual wind components. We can, however, still estimate the breakdown of contributions so we can analyze the behavior.
Consider the error of the squared wind speed, To estimate the wind speed error itself, suppose that the lidar-derived estimate is close to the actual volume-averaged wind 595 speed, |U h |. Then we can factor and approximate (first order in |U h | err ) as follows.
Combining with the squared speed error and solving for the |U h | err , we find the error in the computed wind speed with respect to the horizontal velocity component errors (equation 19).
The wind speed error consists of a projection of the horizontal wind errors and a strictly positive term. The more closely the mean wind aligns with an axis, the more heavily the corresponding component error is weighted. We observed slightly wider error distributions in cross-stream velocity estimates compared to streamwise (figure A2). In that case, the potentially larger error is weighted less heavily.

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The squared terms account for the positive bias observed in the wind speed error distributions (figure 5) since it consistently shifts errors to be more positive. We can explicitly find a theoretical mean of the wind speed error (appendix C) and simplify by assuming the u and v error random variables have a mean of zero (although this assumption can break down, e.g. in the presence of coherent structures or due to the RWF near the surface).
As with the wind speed, the mean value does not directly cancel. Applying the difference identity for arctan and simplifying, Turn this into looser bound that is simpler to interpret. The derivative of arctan is continuous and bound above by one so that we may bound, | arctan(x)| ≤ |x|. For the error, The bound is tighter the smaller the error and the sign of the bounding expression should match that of the error creating an 'envelope' for the error. As opposed to the wind speed, in the wind direction error form, the mean velocity components Under the error approximation, we can similarly estimate the mean error (equation 23). If we again assume the u and v reconstruction errors to have zero mean, then the wind direction mean error should also be zero.

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The idea that lidar might manifest smaller error at higher winds seems intuitive. On top of potential implicit effects on the homogeneity over the scan volume, the derived error forms draw out explicit dependencies on wind speed and direction. The trends predicted by the analytic form convincingly describe those observed in the virtual lidar data. We find that wind speed powerfully influences error in the lidar observations, while the orientation of the lidar with respect to the mean wind has negligible impact.  It should be noted that although no strong trends were found with respect to the relative offset between the mean wind and the lidar axes, the signs of some of the biases can change with the signs of u and v. The derived error forms for the wind speed and direction and discussion of the RWF (section D) explain these dependencies.  The lidar error varies along with the 'random' turbulence in the flow which we have reflected by describing the error as a random variable drawn from a distribution dependent on the character of the turbulence and the lidar scan geometry. In preceding analysis we considered how the physical and mathematical mechanisms might influence the distribution of the error in the raw lidar measurement. Now we consider how time averaging acts on the full error distribution.

Reducing error through time averaging
Time averages were performed over the wind components (which is mathematically equivalent to averaging over the beam That is, the error in the time-averaged measurement can be expressed as the sum of (scaled) random variables drawn from the distribution of the original sample errors. Using linearity, the mean of the time-averaged error distribution, u T err , is equal to the mean for the original sample errors; i.e. time averaging does not change the mean ensemble error of the wind components.
The result holds for any linear reconstruction. The mean error in individual measurements may benefit from time averaging, however, as the time average approaches the potentially less biased 'ensemble' error from the selective sub-sample (e.g. figure   7).
The time average does, however, reduce the error spread, as observed in the virtual lidar data. The variance of a sum of 705 independent, identically distributed variables is well known; the variance of the original random variable (the un-averaged error) is scaled by one over the number of samples in the average. In a time series, the samples cannot simply be treated as independent since subsequent samples are highly correlated. In the lidar, the pattern of turbulent structures that gave rise to a particular error continue to influence the error in the following samples as well so that the errors are quite similar. The reduction factor on the error standard deviation may be estimated by a range. The lower bound is given by treating all of the T s samples The standard deviation reduction factor (assuming τ s , τ c fixed) scales proportionally to T −1/2 . The marginal utility of longer time average in terms of reducing the standard deviation shrinks rapidly; just four independent samples are needed to halve the standard deviation but 100 are needed to bring the standard deviation to a tenth of its original value.
The error in the derived quantities of wind speed and direction were derived in terms of the wind component errors. For the 720 time-averaged quantities, we may simply use the modified component distributions, which we determined above had (1) no change in mean and (2) scaled variance.
The wind speed error experiences an improvement not only in the spread of the error but also in the mitigation of the positive bias. In the mean, the first terms in the wind speed error (19) remain unaltered, but the positive bias term is proportional to the sum of the u and v error variances and is accordingly scaled by the factor (T /τ c ) −1 . Though the mean estimates of the wind 725 components themselves do not improve under the time average, the reduction in the spread of their errors leads to a marked improvement in the mean error of the wind speed. The variance for the wind speed was not explicitly computed but estimated to be roughly equal to a weighted combination of the u and v error variances. The reduction factor for the wind speed variance (and thus standard deviation) is about that of the wind components. Using the rough approximation for the decorrelation time, the projections for the change in the time averaged behaviors hold up against the virtual lidar data (figure 16).

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The mean wind direction error should experience negligible change under the time average since it arises from the component error means. The estimate of the error magnitudes project it to be proportional to the component standard deviation over the mean horizontal wind speed. Since we assume the wind speed to be consistent through the time window, the wind direction

Note on vertical velocity
The vertical velocity measurement demands separate treatment from the horizontal winds. The vertical velocity itself behaves distinctly from the horizontal winds: it varies much more rapidly and the the features of interest occur at smaller spatial and temporal scales, the background signal being close to zero. The WindcubeV2 offers two possibilities to reconstruct the vertical velocity from the measured radial velocities by either equally weighting the beams or using the wind direction to selectively 740 weight them (equations 8,9). The measurement from the vertical beam is a third option that samples just the vertical velocity and does not require reconstruction.
Since the variation in the vertical velocity generally occurs at a smaller scale than the scan volume, the most interesting information will tend to be lost in the reconstruction process. Even if the volume average were perfectly recovered, the average itself loses information. The error induced in the different sensing cases were separately examined from the virtual instrument 745 data. Figure 17 compares the two reconstruction techniques (equations 8 and 9). The equally-weighted reconstruction was examined in depth with the analytic break down of the error; here we empirically consider the effect of instead using the wind direction weighting. At least with respect to the disk-averaged truth, the lessened dependence on the full four beams using wind direction weighting seems to outweigh the beneficial effects. When the wind is directed between the lidar axes (45 degree angle), the two 750 reconstructions are identical; the difference arises when the wind direction lies more closely with one of the axes so that two beams are weighted more heavily, shrinking the contribution from the other two beams. In our test cases, the reconstruction relies primarily on the east and west beams. With respect to the volume-averaged vertical velocity, the bias and standard deviation of the error is on the whole larger under the reconstruction using wind direction weights than with equally weighted beams ( figure 17). In the context of the random variable error model (equation 13), weakening or removing dependence on 755 some of the beams removes the chance for cancellation of the perturbations; the mean over two points will tend to be a poorer representation of the volume mean than that over four points. The concentrated dependence can also magnify the influence of variations experienced the two beam locations. In light of the empirical error behavior, the benefits of incorporating more points seems to supersede the benefits of streamwise correlations.
The vertical beam sidesteps the implicit volume average as well as the need for any reconstruction. In this case, the error 760 incurred in measuring a pointwise vertical velocity arises purely from the effects of the range-gate-weighting in the measurement process. The RWF produces errors with magnitude and character that are distinct from the reconstruction errors ( figure   18). Across stability cases, the bias shrinks to be negligible. The difference in standard deviation between the convective cases vanishes and the overall magnitude is significantly reduced compared to the error in the reconstruction. As opposed to the error in the reconstruction estimate, the standard deviation with only the RWF decreases with height. In the stable case, the RWF was 765 a larger portion of the error in the vertical velocity reconstruction ( figure 11). Under stable conditions, the difference between the reconstruction and the vertical beam is less dramatic; the standard deviation magnitudes are about halved and the decrease in the standard deviation with height exists with both but is more pronounced with the vertical beam. Even using the vertical beam, the standard deviation can still be relatively quite large compared to typical values.

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Atmospheric variability inextricably influences error in wind lidar measurements. By using virtual instruments acting on LES flows, error mechanisms can be isolated and explicitly tracked and analyzed to better understand the error behavior as a whole.
In this study, we considered profiling lidar measurements in quasi-stationary conditions. Even in the absence of explicit sources The quantification of the error aligns with values found in field studies (e.g. Courtney et al. (2008)) and in similar estimates of virtual lidar performance in baseline convective conditions (Rahlves et al., 2021). To define the error, we compared virtual 780 measurements against both a pointwise truth reference and the average over the scan volume. Under ten-minute averages in the quasi-stationary conditions of the test cases, the distinction between the kinds of spatio-temporal averages fades and the two error distributions seem to converge. The magnitudes of the overall errors in the virtual measurements fall generally within experimentally determined ranges in favorable conditions: ten-minute averaged wind speeds have mean errors within ±0.2m/s with standard deviation < 0.3m/s and wind direction have mean error bias within 2 • and standard deviations in 2.5 • except 785 for in the strongly convective case where it can reach up to 12 • .
The character of the error in the reconstructed wind vector components is driven by the form of the turbulence, so that the lidar accuracy is dependent on the flow regime and vertical structure of the boundary layer. This derivation explains findings from other sensitivity studies (Klaas and Emeis, 2021;Rahlves et al., 2021) in that unstable conditions are prone to larger error than stable conditions. The error variances were connected to weighted, spatially filtered turbulent variances in  a cap determined by the full, unfiltered turbulence. We propose that the competing effects of the cone angle lead to different optimal angles depending on the flow conditions and that reasonable estimates can be performed to minimize error if rough numbers for any gradients and the magnitudes of the turbulent variances / spectra are known.
Some profiling scans use a different number of off-vertical beams to diagnose the mean winds. The beams are usually preferred to be symmetrically spaced to remove potential bias (Sathe et al., 2015;Teschke and Lehmann, 2017). Common 835 reconstructions of the three-dimensional velocity fit sinusoids to the radial velocity measurements using a least squares process (Newsom et al., 2015)). The result is a linear operation on the beam radial velocities of which the DBS scan presented in this study is a special case. The error in these alternative profiling scans should take on a similar error form to that derived here, with perturbations averaged over a greater number of beams. Under a simple stochastic model, Teschke and Lehmann (2017) showed the standard deviation of the error should be proportional to N −1/2 with N the number of beams. The form our error The wind direction has no explicit bias except that arising from the u and v reconstructions. The standard deviation is 855 roughly that of the u and v errors over the average wind speed (i.e. the error magnitudes are reduced at higher winds). The observed error magnitudes strongly depend on mean wind speeds (especially the wind direction) but are only weakly related to the relative orientation of the lidar. As in Rahlves et al. (2021), our results suggest no predominant direction of the random Appendix C: Derivation of wind speed and direction error means The random variable equation for wind speed error was derived in equation 19. Compute the expected value. We can further simplify if the component errors are assumed to have zero mean.
The approximated wind direction function 22 is the ratio of two random variables, each of which is a linear function of the component errors. Compute the expected value of the approximated form.
If the component errors have zero mean, then the wind direction error is zero. If they are nonzero, large wind speeds can temper the bias and high wind speeds compound it. Appendix D: Derivation of error bound on RWF weighted radial velocity measurement Let v r (r 0 ) be the actual radial velocity at radius r 0 and v r (r 0 ) the observed, range-gate-weighted radial velocity centered at r 0 . Let T > 0 be an arbitrary threshold to split the integral.
We'll assume the v r (s) profile has at least two continuous derivatives. The range gate weighting function, ρ(s), should generally be non-negative, symmetric, and satisfy ∞ −∞ ρ(s)ds = 1 by definition so we assume these properties as well.

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Using the triangle inequality, integral mean value theorem, and Taylor series expansion, we have the following derivation. The numeric computation of the range-gate-weighted radial velocity involves approximating a convolution integral for a continuous weighted average. The estimate should ideally maintain the weighted average nature of the operation to prevent, e.g., underestimating the result by virtue of only incorporating a sub-unity set of weights. For this reason, previous implementations (Forsting et al., 2017;Lundquist et al., 2015) have implemented the RWF convolution as a discrete weighted average using re-normalized RWF values at the corresponding points.
The quadrature form amounts to a re-scaled midpoint rule when the nodes ({s k }) are equispaced; rewrite to show explicitly.
For nodes that are not equispaced, it suggests that the variable interval width h k should be incorporated into the approximation such that it is still a rescaled midpoint rule. First, we'll truncate the domain over which we try to estimate the integral to a finite interval. The omitted contribution from the ends of the infinite integration interval can be bound small using that fact that the the tails of the RWF must vanish in order for the infinite integral ∞ −∞ ρ(s)ds = 1 to converge. ρ(s k )vr(r + s k ) + (ρ(s k )vr(r + s k )) (s − s k ) + 1 2 (ρ(ξ k )vr(r + ξ k (s))) (s − s k ) 2 ds = 1 i hiρ(si) h k ρ(s k )vr(r + s k ) − s k +h k /2 s k −h k /2 ρ(s k )vr(r + s k ) + 1 2 (ρ(ξ k )vr(r + ξ k (s))) (s − s k ) 2 ds = 1 i hiρ(si) h k ρ(s k )vr(r + s k ) − ρ(s k )vr(r + s k )h k − 1 2 s k +h k /2 s k −h k /2 (ρ(ξ k )vr(r + ξ k (s))) (s − s k ) 2 ds ≤ 1 i hiρ(si) − 1 h k ρ(s k )vr(r + s k ) + 1 2 s k +h k /2 s k −h k /2 (ρ(ξ k )vr(r + ξ k (s))) s 2 ds All together, the error of the numeric approximation of the integral may be bounded by: In addition to picking the interval [−T, T ] large enough so that the contribution from the tails is small and the usual second order error term from the midpoint rule, the points should be selected so that the midpoint approximation of integral of the RWF (i.e the first term) is close to one.
Equispaced points are easy to determine and implement but do not maximize the utility of each point included. For a virtual 960 lidar, which has to interpolate the winds for every node, inefficiently using points can be computationally expensive before the quadrature is even computed. The idea behind the other node distributions is to sample more heavily near the RWF center, where each point has greater impact on the result so that we reduce the number of 'low utility' nodes.
Exponentially spaced nodes (e.g. {−16, −8, −4, −2, −1, 0, 1, 2, 4, 16}) are easy to compute independent of the particular RWF function like the equispaced nodes, but crudely achieves the goal of clustering points more heavily close to the RWF 965 center. Forsting et al. (2017) recommend using nodes spaced so that the integral of the RWF between nodes is constant. The result is nodes clustered more heavily toward the center and more equal weighting on each node (not 'oversampling' the tails).
Based on the midpoint form and error bound derived above, we propose an alternative in the same vein. We can explicitly set the h k ρ(s k ) to be constant so that every point has equal weight. Starting with s 0 = 0 and h 0 = 1 N where N is the total (odd) number of points, we can iteratively solve out for the symmetric nodes. In practice, the last nodes usually do not seem to be 970 able to attain the same h k ρ(s k ) = 1/N weighting. Either the sum k h k ρ(s k ) must be reduced to less than one or the constant weighting relaxed for the end nodes. The latter is preferable to reduce the overall error.
Selecting nodes for computational expediency in the case of multiple range gates along a beam introduces further considerations than those for a single range gate. For the WindcubeV2 the intervals of dependency for the 20m-spaced range gates overlap (E1; nodes should be reused where possible rather than interpolating a separate set of points for each range gate. Meth-975 ods using spacing by RWF area or imposing constant weighting are likely to produce unusual numbers that do not necessarily coincide between multiple range gates. Exponential spacing makes it easier to impose overlap in position. The cluster of points at the center of one range gate appear at the tail of another range gate and would have to be subselected to the desired points. This research has been supported by the US National Science Foundation (grand nos. AGS-1554055 and AGS-1565498).