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 supports 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

This work was authored by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under contract no. DE-AC36-08GO28308. Funding was provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the US Government. The US Government retains and the publisher, by accepting the article for publication, acknowledges that the US Government retains a nonexclusive paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for US Government purposes.

Effectively and efficiently collecting observations of atmospheric winds poses an ongoing, multi-faceted challenge for the atmospheric science community. Wind-profiling light detection and ranging (lidar) instruments offer a cheaper, more easily deployable and higher-ranging alternative to traditional meteorological towers while scanning lidar systems allow for collection of data over broad regions of the atmosphere. Over the last few decades, lidar technology has matured, 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

All wind lidar instruments function on the fundamental basis of sampling the flow along an emitted beam. With a single lidar's beam, only a one-dimensional line-of-sight projection of the velocity can be measured. In light of this sampling limitation, dual- and triple-lidar methods have been explored to allow concurrent measurement of the necessary spanning wind vectors

The error of remote-sensing instruments like lidar, sodar, and radar depends not just on the system itself but is a statistical distribution arising from the interplay of the system with the turbulent atmospheric flow. Sources of error in profiling lidar measurement are delineated by

The study of instrument error using numerical large-eddy simulations (LESs) was introduced by

LES enables the generation of realistic turbulent atmospheric flows with which to study likely interactions and resulting error behavior of remote-sensing instruments. The spatial resolution of LES is typically on the order of 1 m to tens of meters and is designed to explicitly capture the most critical length scales in the atmospheric boundary layer while parameterizing the effects of the smallest turbulent scales. The resolution is not sufficient to explicitly compute the underlying optical measurement of scattering in profiling lidar; however, the salient effects of volume averaging and reconstruction over the scanning volume occur at a scale that can be supported by the LES data. Compared to field studies of instrument accuracy, studies with virtual instruments in LES have unencumbered access to full knowledge of the flow field. This knowledge enables control over the case parameters (terrain, forcing, boundaries), and so users can “deploy” instruments in ways that may not be physically or financially possible in reality (e.g. re-sampling the same flow field or testing many locations in a domain)

Earlier virtual lidar studies have generally considered complex lidar behavior and have been built on a range of different LES models. The coordinated use of multiple lidar devices to simultaneously probe spanning vectors of the wind in a volume was studied by

We have developed a virtual lidar tool in Python to run on output from WRF-LES. WRF-LES boasts a user base of over 48 000 and is attractive for its accessibility as an open-source documented model. It can be configured for ideal simulations or coupled with mesoscale nesting to simulate case studies of real sites

In this first demonstration of the virtual lidar tool, we consider a specific case of the Leosphere WindCube V2 profiling lidar (Fig.

Geometry of a DBS scan performed by a Leosphere WindCube V2 to estimate a vertical profile of the 3D wind velocity. At a frequency of 1

Section

The virtual lidar is 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 an RWF, the scanning pattern the beam moves through, and 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 WindCube V2 performing a DBS scan (Fig.

The basis of wind lidar technology 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 devices diagnose the shift in the frequency of the backscattered light to measure radial wind velocity (Eq.

In the context of our model, we assume “perfect” conditions in the sense of ignoring factors like aerosol type, size, and 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

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 an RWF

For a pulsed lidar, the weighting function arises from the convolution of the range-gate profile with the pulse profile

Parameters used in the model to configure a representative WindCube V2 lidar performing a DBS scan.

The RWF for the modeled WindCube V2 (Fig.

To compute the RWF-weighted retrieval from an LES flow field, the wind components are first interpolated to points along the beam and then projected onto the beam direction. The virtual lidar uses a linear barycentric interpolation from a triangulation of the LES grid (i.e. linear interpolation on tetrahedrons using

Based on the type of scan they perform, lidars are categorized as profiling or scanning systems. Profiling lidars are designed to provide a vertical profile of the 3D wind velocity, much as would be reported by a meteorological tower. To reconstruct a 3D wind vector, the instrument needs spanning radial velocity samples from at least three different directions. The scan is completed quickly to limit the intervening evolution of the wind field. Any scanning geometry, such as that described here or other common or complex options as in

For the purposes of this study, we consider the DBS profiling scan used by the WindCube V2, which moves through the four cardinal directions, angled 62

The beam accumulation time for the WindCube V2 is about a second, whereas the LES model time steps are on the order of a 10th 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 1

In the WindCube V2, the post-processing stage reconstructs the 3D velocity from the radial velocities collected across the scan cycle. Under the assumptions of horizontal homogeneity (i.e. constant winds) over the scan volume and invariance over the scan duration, the radial velocities collected by each of the beams at a given height are all projections of the same 3D velocity vector. Omitting the vertical beam, we solve for the vector components at a given range-gate height (Eq.

At a fixed height,

Later versions of Leosphere's WindCube instruments use a modified reconstruction (Raghavendra Krishnamurthy, personal communication, 2020) for the vertical velocity (Eq.

When the mean wind direction is at a 45

Realistic atmospheric flow fields are generated using LES configurations of the Advanced Research Weather Research and Forecasting (WRF-ARW) model v4.1

Parameters for WRF-LES runs used to represent different stability regimes.

For each of the idealized LES cases, mean profiles of

To establish a baseline reference for lidar operation in ideal conditions, all simulations in this study use uniform flat, grassy terrain (roughness length

For the convective boundary layers (CBLs), we use data from precursory simulations in

The stable-boundary-layer simulation closely follows the configuration of

Mean profiles, computed with data from the valid regions of the LES cases, are characteristic of the respective stability regimes (Fig.

A virtual WindCube V2 is created in the lidar model as described in Sect.

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 instruments is re-oriented at three additional offset angles (15, 30, 45

Determining the error in the lidar observation depends on defining a reference truth. Profiling lidars are often thought of as replacing meteorological towers, returning a vertical profile of 3D 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 a vertical extent via the RWF. These factors suggest that a volume average might be a more appropriate reference truth (as suggested in

Along with a “pointwise” tower-style truth profile of interpolated velocities above the instrument, we determined a volume-averaged profile for each lidar. The volume average is computed as the mean of all LES points that fall inside cylinders tracing the lidar scan radius (Fig.

Alongside the virtual model of the lidar, we develop an analytic model of the measurement error which serves to help explain the mechanisms at work and interpret the results of the virtual measurements. 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 reconstruction of the form of Eq. (

Two elements directly introduce error into the observation model: the application of the RWF in the radial velocity measurement and the assumption of horizontal uniformity in the reconstruction. Using a random variable model, we identify the contributions of the RWF and horizontal velocity variations to the error in the wind component reconstructions. Duration of the scan cycle and time staggering of the beams are not explicitly addressed in the error model, though they are included in the implementation of the virtual instrument.

Quantities derived from the estimated wind components can take on their own non-trivial error behavior. Natural derived quantities that are often computed from lidar data include horizontal wind speed and direction and time-averaged winds. We characterize the error in wind speed and direction in terms of the

We start by deriving the form of the error in the reconstructed velocity components. The formulation allows the contribution to the error due to the turbulent variations to be explicitly delineated and tracked. For a fixed height, let

In a perfectly horizontally uniform wind field, the velocity perturbation values all individually vanish (i.e. not due to cancelations). However, even in that perfectly horizontally uniform case, a non-linear vertical profile can still induce non-zero error through the RWF.

Instantaneous slice of vertical velocity across the west–east plane in

In the presence of turbulence in the flow, the velocity over the scan volume is no longer uniform, and the beams sample perturbed variations, violating the assumption underlying the exact reconstruction. The perturbation values may be regarded as random variables with distributions resulting from the character of the atmospheric variations and the lidar scan geometry (Fig.

As functions of the random perturbations, the wind component errors are themselves random variables. The mean,

The mean operator is linear and directly decomposes the overall error mean into constituent parts for the horizontal (Eq.

The relative weighting of the perturbations is controlled by the scan cone elevation angle from the horizon,

Along with the relative weights, the elevation angle controls the spatial separation of the beams, thus implicitly influencing the distributions of the perturbations themselves. The beam separation can be of particular importance in the presence of background spatial variation in the flow in which larger separation can induce a greater mean error, as explored in terms of linear variations in vertical velocity in

More off-vertical beams may be used with a linear least-squares reconstruction process

The virtual lidar model uses the LES to indirectly predict the perturbation distributions and the complex ways in which the perturbations can be inter-related with each other and with respect to the volume averages. Random variable theory can then be used to describe the propagation of uncertainty into the error from the attributes of the perturbation distributions.

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,

Radial velocity profiles with constant gradient do not incur error in the RWF application; symmetry leads the linear contributions to cancel. Indeed, visual inspection confirms this behavior in regions of constant gradient about

Variations in the velocity across the scan volume are directly represented in the error model by the velocity perturbations in the wind component errors (Eq.

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 (

It is tempting, with usual conventions about turbulent perturbations, to assume that the lidar velocity perturbations will have zero mean and identical distributions at each of the beam locations. Under these assumptions, the mean error due to the horizontal homogeneity violations would be zero. However, the volume average over the disk is neither the direct mean of the beam velocities nor the turbulent ensemble mean. The velocity perturbations can produce a non-zero mean because of consistently occurring spatial patterns in how the velocities vary at the edges of the scan volume with respect to the average velocity over that volume. The spatial structures at play in the LES cases with respect to the lidar scan volume can be seen in the cross sections in Fig.

The horizontal wind vector is commonly represented not by its components but by the wind speed,

Conventions used for the horizontal wind vector direction and signs of the wind direction error.

The derived values do not inherit the error from the wind component errors directly; rather the quantities should be thought of as functions of the

We may expand the lidar-sensed horizontal wind speed (Eq.

We can explicitly find a theoretical mean of the wind speed error and simplify by assuming the

Without explicitly computing the variance, we can estimate the magnitude of the wind speed error. Based on the leading order terms, the error should generally be on the order of the individual component errors (i.e. their standard deviation), though the bias term has the potential to become more prominent in adverse conditions (large

Now consider the wind direction error. To simplify the analysis, we set aside the quadrant correction and consider just the traditional inverse tangent function to find the angle in

Time averaging is a tool used to reduce the variation in the error in the raw high-frequency measurements made by the lidar, leaving a more reliable mean measurement. Under conditions in which the background flow continues to evolve in time, the utility of time averaging must be weighed against the length of the interval during which quasi-stationary conditions exist and the sacrificed resolution of shorter timescale dynamics. Making an informed assessment of an appropriate time window length rests on quantifying the expectation of the improvement of the measurement accuracy.

The lidar error varies along with the “random” turbulence in the flow, which we have reflected by describing the error as a random variable that is drawn from a distribution dependent on the character of the turbulence and the lidar scan geometry. Here, we consider how time averaging acts on the error distribution of the raw 1

First, we consider a time average (arithmetic mean) performed over the wind components (which is mathematically equivalent to averaging over the beam radial velocities when the reconstruction is linear as in Eq.

The primary effect of the time average on the velocity components is to reduce the width of the error distribution, i.e. the typical magnitude of the errors. The variance of the arithmetic mean of independent, identically distributed random variables is well-known

In the horizontal wind speed and direction, the time average can be computed either from the time-averaged vector components (a vector average) or directly over the scalar speed and direction computed each second (a scalar average) (Eqs.

For the vector-averaged quantities, we determine the effect on the wind direction and speed error by carrying the changes in the time-averaged wind component error distributions through into the error forms (Eqs.

The discrepancy between the scalar- and vector-averaged lidar quantities arises from the persistence of the positive bias term in the scalar average and its corresponding decay in the vector average. By mathematical analog of a Reynolds decomposition to the error fluctuation on the volume-average winds, the wind speed error derivation (Eq.

Assume that the vector time average acts like an ensemble Reynolds average (as it does over a long enough time window), so that the vector time averages of the pointwise and lidar measurements both reduce to

Let the volume-average reference be the Reynolds-averaged wind,

The error incurred in any individual measurement depends on the specific realization of turbulence during the measurement and is not necessarily representative of the full variability of possible error behavior. To deduce useful information about bias and typical error magnitudes that can be generalized to other measurements in the same conditions, we focus instead on the distribution of the observation error. Each virtual instrument in the ensemble provides instances of the way the WindCube V2 might interact with turbulent features in each flow regime, thereby sampling the error distribution. The raw 1

A lidar reports a vertical profile of velocities each second over the duration of the simulation. Each distribution consists of 10 min of data combined over the 45 ensemble members and four orientation angles, giving a total of 108 000 error samples. Disaggregating by height and stability, kernel density estimates (KDEs) of the error histogram visualize the resulting distribution. Collating the KDEs into a ridgeline plot (e.g. wind speed in Fig.

Kernel density estimates of the 1

Statistical moments serve to summarize and quantify the properties of the distributions, facilitating intercomparison and the identification of trends in the error behavior. We consider the first four moments: unbiased estimators of the mean, centered variance/standard deviation, and the adjusted Fisher–Pearson standardized moment coefficients for skewness and excess kurtosis

We start by examining the reconstructed horizontal velocity components. The skewness and kurtosis metrics suggest generally normal behavior except for the excess kurtosis (

Mean and standard deviation of error in the

In all cases, the distribution of the error with respect to the pointwise truth displays a larger standard deviation than the error using the volume-averaged truth (Fig.

Contributions to the mean

In general, the mean biases are close to zero (

For the most part, in homogeneous turbulence, non-zero mean biases in

The mean vertical profiles of

The most prominent influence of the RWF is near the surface layer due to strong shear, manifesting as an under-estimate of the magnitude of the horizontal velocities. As shown for a general RWF (Eq.

The variability of the measurement errors, shown in the variance and standard deviation, is a consequence of the velocity perturbations, with negligible contribution from the RWF. In convective conditions, the weighted vertical velocity perturbations dominate the other sources of variance in the error, indicating the

The proportion of the full turbulent velocity variances (

The velocity perturbation terms arise from a weighted, filtered portion of the full turbulent velocity variance. Physically, we might expect convective plumes to violate horizontal uniformity in the flow (Fig.

The coupling of the error with the turbulent structure, and the vertical velocity in particular, helps explain the cause of the correlation of the error height trends with the boundary layer structure. The error variance in the stable boundary layer peaks near the center of the boundary layer where the turbulent vertical velocity variances are large and more of the horizontal variances pass through the filter. The diminishing error variance at higher altitudes relies on the decay of both the horizontal and vertical perturbations and the increase in turbulent length scales, which combats the increase in scan volume size. The lidar range does not extend to the top of the boundary layer in the convective test cases, but we might expect the error variance to also peak in the middle of the boundary layer and decrease with height as the vertical velocity variance tapers back toward zero. The dependence of the error height trends not only on the volume circumscribed by the scan but also on the vertical structure of the boundary layer and corresponding scale and character of the turbulent structures was also noted by

The choice of cone angle determines the degree of projection of the horizontal and vertical perturbations (manifest in the weighting in the error form) as well as the spatial separation of the sampling beams. In the strong and weak CBL test cases in particular, the error demonstrates the adverse impacts of heavy weighting on the vertical perturbations.

The horizontal wind speed and direction are computed from the lidar-measured wind components and compared against those of the volume-averaged winds (Eq.

The distributions are again roughly normal. There is a slight positive skewness (long tail on the positive side of the distribution) in the wind speed error (0.25 in strong convection) (Fig.

Mean and standard deviation of error in the 1

The height and stability trends in the mean and variance of the errors (Fig.

We derived a systematic positive bias term in the wind speed measurement (Eq.

We anticipated that the wind direction bias should be close to zero assuming the

The idea that lidar might manifest a smaller error at higher winds seems intuitive. In addition to potential implicit effects on correlations across the scan volume, the derived error forms (Eqs.

Trends in

The wind speed trends across all the virtual lidar wind speed and direction data are shown in Fig.

According to the wind direction error form (Eq.

Potential differences in error as a function of the lidar orientation are due to projection of the error vector onto the mean wind parallel or transverse directions (Eqs.

Errors in the 100

As in

Common time-averaging intervals used with lidar data may be over 2, 10, or even 30 min, with experimental evaluations of the system accuracy often reported in terms of the 10 min average in wind energy contexts. As with the error in the high(er)-frequency wind measurements, we characterize the error distribution of the 10 min averaged measurements (Fig.

Mean and standard deviation of error in (vector) 10 min averaged wind speed and wind direction. For wind speed, the primary axis gives absolute error, and colored secondary axes designate relative error with respect to 100

We first characterize the error distributions of the vector-averaged lidar measurement compared to the vector-averaged point and volume references for the wind speed and direction (Fig.

Under stationary and homogeneous flow 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 profiles. 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 (ignoring spatial displacement of the tower from the lidar). 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

As anticipated, the time-averaged errors reflect a decrease in the wind speed bias in the convective cases, little change in the wind direction bias, and a reduction of the standard deviations (by a factor of around 5). The degree of reduction in the biases and standard deviation is not uniform, but varies somewhat with stability and height, likely depending on the decorrelation scales in the error time series. This leads to some shift in the shape of the moment profiles compared to the original distribution, e.g. the curvature of the standard deviation with height.

Wind speed error at 100

Based on the error model, not only a reduction in error magnitude and bias (in the wind speed) were predicted, but also the rate of reduction. The wind speed error distribution at 100

Comparison of time-averaged lidar wind speed estimates at 100

We also compared 10 min vector-, scalar-, and hybrid-averaged lidar wind speeds (Eqs.

The theory behind the hybrid average leverages the expected inflation in a scalar-averaged wind speed (compared to the vector average) in a lidar and a pointwise measurement (Eqs.

Decomposition of the lidar and pointwise scalar-average inflation factor (

The lidar wind speed inflation is decomposed into horizontal velocity fluctuation terms and terms due to vertical or mixed vertical–horizontal velocity fluctuations. Note that the lidar inflation factor is due to perceived variances in the horizontal velocity components, and its decomposition echoes the decomposition of the

The derivation of the hybrid scheme and the weightings shown above are predicated on the assumption that the bias term vanishes completely in the 10 min vector-averaged pointwise and lidar measurements. The vector averages were assumed to be equal approximations of the speed of a Reynolds-averaged wind. The behavior of the bias in the vector-averaged winds suggests that, at least for the upper range gates in the strong CBL, a non-negligible positive bias (0.1–0.2

The vertical velocity measurement demands separate treatment from the horizontal winds. The vertical velocity itself behaves distinctly from the horizontal winds because it varies more rapidly, and the features of interest occur at smaller spatial and temporal scales, with the background (spatiotemporal average) signal tending close to zero. The WindCube V2 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 weight them (Eqs.

Figure

Comparison of the error in vertical velocity measurement using the evenly weighted vertical velocity reconstruction (Eq.

Comparing the reconstruction techniques, 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

In the context of the random variable error model (Eq.

The vertical beam sidesteps the implicit volume average along with the need for any reconstruction. In this case, the error 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 (Fig.

Vertical velocity measurement error moments for the vertically pointed beam. Error is with respect to the pointwise truth.

The quantification of the error found here aligns with values found in field studies

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. Our derivation explains findings from other sensitivity studies

The range-gate weighting in the radial velocity measurement has minimal relative effect on the total lidar error except in high-shear regions near the surface layer. Deviations incurred in the radial velocity measurement by the weighted volume average along the beam should vanish under constant gradients but can grow in the presence of large second derivatives in the radial velocity projection along the beam. For the most part, the impact of the larger variations over the scan volume dominate any RWF effects. In the bottom few range gates near the surface, however, the virtual lidar data reflect a prominent interaction of the RWF with shear near the surface layer, leading to measurement bias. The persistent curvature in the profile results in significant (around 0.2

Within the class of DBS and VAD profiling scans, any control over the reconstruction error comes from adjusting the cone angle,

The cone angle determines the degree of projection of the horizontal and vertical perturbations (manifest in the weighting in the error form) as well as the spatial separation of the sampling beams. In the strong and weak CBL test cases in particular, the error demonstrates strong adverse impacts of resulting heavy weighting on the vertical perturbations. The dominance of the vertical perturbation terms can be tempered by reducing the elevation angle.

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

The error in wind component reconstructions propagates into the error in the corresponding computation of horizontal wind speed and direction. The error was formulated in terms of the

The wind direction has no explicit bias except that arising from the

Individual measurements can suffer from larger errors, which can be reduced through time averaging. While time averaging cannot correct for biases in the wind component measurements, the standard deviations of the error are reduced by a factor proportional to

Vertical velocity, with features of interest existing on smaller spatial and temporal scales, is a greater challenge to lidar measurements. A vertically pointing beam omits the need for reconstruction or the implicit large-scale spatial average over the scan volume. Instead, only the smaller-scale averaging from the range gate is applied. The errors associated with the vertical beam with respect to the pointwise values are significantly smaller and represent a more useful value that captures more of the small-scale variability in

Atmospheric variability influences error in wind lidar measurements. By using virtual instruments acting on LES flow fields, 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, quasi-homogeneous conditions. Even in the absence of explicit sources of inhomogeneity, observation error emerges, tightly coupled to the character of turbulence in the flow. The error distributions of a virtual WindCube V2 lidar performing a DBS scan were estimated from ensembles of virtual instruments run in uniform, ideal WRF-LES scenarios in convective and stable boundary layer regimes. An analytic error model leverages random variable representations to describe how the turbulent variability propagates into the lidar error, decomposing the contributions from velocity perturbations at each beam from the volume average and from deviations in the point measurement due to range-gate weighting.

The resulting errors depend on the stability and height of the measurement. Strongly convective conditions exhibit the largest errors, reaching a standard deviation of 1.5

The errors in the

Time averages of the 1

Fully leveraging the access to the flow field afforded by an LES model, virtual lidar tools allow for not only predicting instrument error but also for separating and analyzing potentially competing mechanisms that give rise to the error. Performance optimization of the model implementation would reduce the computational cost and allow longer scan times and larger ensembles to be studied. The results would benefit by comparison to field data and investigating ways to identify the mechanisms and possible behavior of error in the data. For specifically targeted quantities and heights, optimizations of the scan using knowledge of likely mechanisms should be tested to confirm expected behaviors. Working from this baseline study, two streams of modifications are envisioned. First, additional complications to the flow field could be introduced, e.g. complex terrain and heterogeneous flows like turbine wakes or canopy flows. Second, we can consider more complicated virtual measurements, such as modified scans from scanning lidars employing different scan topologies, or deployment of lidars on moving platforms such as ships, buoys, vans, or aircraft.

Comparison of

Comparison of error distribution moments over disaggregated lidar orientation angles. No offset, same axes as LES domain (solid), rotated 15

The mean wind speed error value is computed by taking the expected value of the random variable equation for wind speed error (Eq.

The argument of the inverse tangent function in the wind direction error (Eq.

Similarly, we can obtain the variance with the same assumptions about the means of the

Even in the presence of small biases in

The mean wind direction error is given by

Let

We will assume the

Using the triangle inequality, integral mean value theorem, and Taylor series expansion, we have the following derivation.

Where we have introduced

Growth and decay of the coefficients in the bounding terms of the radial velocity measurement error with increasing threshold distance,

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 under-estimating the result by virtue of only incorporating a sub-unity set of weights. For this reason, previous implementations

Sum the error accumulated over all the intervals. If the sub-intervals partition the full interval

All together, the error of the numeric approximation of the integral may be bounded by

Choices for nodes include (1) equispaced, (2) exponentially spaced, (3) equal RWF area, and (4) equal mid-point area

Exponentially spaced nodes (e.g.

Using 15 points per range gate, the node locations

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 WindCube V2, the intervals of dependency for the 20

Virtual lidar code may be found at

JKL was responsible for the conceptualization, with RR responsible for the software and analysis and investigation. Both authors contributed to the methodology and the writing and editing.

The contact author has declared that none of the authors has any competing interests.

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Much thanks to Alex Rybchuk for use of his idealized convective boundary layer LES data, to Miguel Sanchez Gomez for his work to produce robust and stable LES case runs, and to Raghavendra Krishnamurthy for his guidance with the WindCube V2 velocity reconstruction. The authors would also like to express appreciation to Andrew Black and the two anonymous reviewers for their helpful comments and reviews of the manuscript.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under award number DE-SC0021110.

This research has been supported by the US National Science Foundation (grant no. AGS-1565498).

We would like to acknowledge high-performance-computing support from Cheyenne (

This work utilized resources from the University of Colorado Boulder Research Computing Group, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University.

This research has been supported by the Advanced Scientific Computing Research (grant no. DE-SC0021110), the National Science Foundation (grant nos. AGS-1565498, ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University.

This paper was edited by Ulla Wandinger and reviewed by two anonymous referees.