A LiDAR Statistical Barnes Objective Analysis (LiSBOA) for the optimal design of lidar scans and retrieval of the velocity statistical moments is proposed. LiSBOA represents an adaptation of the classical Barnes scheme for the statistical analysis of unstructured experimental data in

Reliable measurements of the wind velocity vector field are essential for understanding the complex nature of atmospheric turbulence and providing valuable data sets for the validation of theoretical and numerical models. However, field measurements of wind speed are typically characterized by large uncertainties due to the generally unknown and uncontrollable boundary conditions

Wind speed has been traditionally measured through local sensors, such as mechanical, sonic, and hot-wire anemometers

In the last few decades, remote sensing instruments have been increasingly utilized to probe the atmospheric boundary layer

Besides the mentioned capabilities, lidars present some important limitations, such as reduced range in adverse weather conditions

In the realm of wind energy, early lidar measurements were limited to the qualitative analysis of snapshots of the line-of-sight (LOS) velocity, i.e., the velocity component parallel to the laser beam

Besides the mean field, the calculation of higher-order statistics from lidar data to investigate atmospheric turbulence is still an open problem. In this regard,

For remote sensing instruments, data are typically collected based on a spherical coordinate system, and then interpolated over a Cartesian reference frame oriented with the

The scope of this work is to define a methodology to postprocess scattered data of a turbulent velocity field measured through a scanning Doppler wind lidar (or eventually other remote sensing instruments) to calculate mean, standard deviation and even higher-order statistical moments on a Cartesian grid. The proposed methodology, referred to as the LiDAR Statistical Barnes Objective Analysis (LiSBOA), represents an adaptation of the classic Barnes scheme to

It will be shown in the following that the revisited Barnes scheme offers several advantages compared to the above-cited techniques for lidar data analysis: (i) it allows one to explicitly select the cut-off wavenumber to filter out small-scale variability, while retaining relevant modes in the flow field; (ii) the distance-based weighting function provides smoother fields than linear interpolation or window average, while still being simpler and computationally inexpensive compared to more sophisticated techniques (e.g., optimal interpolation and variational methods); and (iii) it provides guidance for the optimal design of lidar scans to investigate specific wavelengths in the flow. On the other hand, the procedure requires estimates of input parameters for the flow under investigation and the lidar system used. In case these parameters cannot be obtained from existing literature or preliminary tests, a sensitivity study on the variability in the LiSBOA results to the input parameters can be carried out.

The remainder of the paper is organized as follows: in Sect.

The Barnes scheme was originally conceived as an iterative algorithm aiming to interpolate a set of sparse data over a Cartesian grid

In the literature, there is a lack of consensus regarding the selection of the total number of iterations

In the frequency domain, the Barnes objective analysis is tractable as a low-pass filter applied to a scalar field,

We consider a continuous scalar field,

By leveraging the convolution theorem, it is possible to calculate the response function of the mean of the

For real applications, the actual LiSBOA response function can depart from the abovementioned theoretical response (Eq.

the convolution integral in Eq. (

the distribution of the sampling points is usually irregular and nonuniform, leading to larger errors where a lower sample density is present

an error is introduced by the back-interpolation function,

Before proceeding with further analysis, it is necessary to address the applicability of LiSBOA to anisotropic and multi-chromatic scalar fields. Generally, the application of LiSBOA with an isotropic weighting function is not recommended in the case of severe anisotropy of the field and/or the data distribution. At the early stages of objective analysis techniques, the use of an anisotropic weighting function was proved to be beneficial for increasing accuracy while highlighting patterns elongated along a specific direction, based on empirical

The scaling factor,

Regarding the reconstruction of the flow statistics through LiSBOA, two categories of error can be identified. The first is the statistical error due to the finite number of samples of the scalar field,

The spectral response of LiSBOA is studied through the Monte Carlo method. The goal of the present section is twofold, namely validating the analytical response of mean and variance (Eq.

An experimental sampling process is mimicked by evaluating the scalar field

Visualization of LiSBOA applied to a Monte Carlo simulation of the synthetic field in Eq. (

Since the response is only a function of

For the error quantification, the 95th percentile of the absolute error calculated at each grid point

Pearson correlation coefficient between the

The small positive correlation

Median of the

To verify the analytical response of the mean and variance of the scalar field,

Validation of the 3D theoretical response of LiSBOA for the case

The trend of the response of the mean (Fig.

Finally, the link between error and the random data spacing,

An efficient application of LiSBOA to lidar data relies on the appropriate selection of the parameters of the algorithm, namely the fundamental half wavelengths,

First, the integral quantities of the flow under investigation required for the application of LiSBOA need to be estimated, such as extension of the spatial domain of interest, characteristic length scales, integral timescale,

Then, it is necessary to define the fundamental half wavelengths,

the smaller

an excessively large

the higher

the radius of influence

Response of the fundamental mode in the scaled coordinates as a function of the number of iterations and the smoothing parameter.

Selected combinations of

As a final remark about the selection of

The optimal lidar scanning strategy aimed to characterize atmospheric turbulent flows implies finding a trade-off between a sufficiently fine data spacing, which is quantified through

The design of a lidar scan aiming to reconstruct turbulent statistics of an ergodic flow through LiSBOA can be formalized as a two-objective (or Pareto front) optimization problem. The first cost function of the Pareto front, which is referred to as

The second cost function for the optimal design of lidar scans,

Standard deviation of the sample mean normalized by the standard deviation of velocity as a function of the number of realizations,

Schematic of the LiSBOA procedure for the optimal design of lidar scans and reconstruction of the statistics for a turbulent ergodic flow.

The whole procedure for the design of a lidar scan and retrieval of the statistics is reported in the flow chart of Fig.

By following the steps outlined in the present section, the mean, variance, or even higher-order statistical moments of the velocity field can be accurately reconstructed for the wavelengths of interest. It is worth mentioning that the LiSBOA of wind lidar data should always be combined with a robust quality control process of the raw measurements. Indeed, the space–time averaging operated by LiSBOA makes the data analysis sensitive to the presence of data outliers, which need to be identified and rejected beforehand to prevent contamination of the final statistics. The interested reader is referred to

The LiSBOA algorithm is applied to a synthetic data set generated through the virtual lidar technique to assess accuracy in the calculation of statistics for a wind turbine wake probed through a scanning lidar installed on the turbine nacelle. For this purpose, a simulator of a scanning Doppler pulsed wind lidar is implemented to extract the line-of-sight velocity from a numerical velocity field produced through high-fidelity large eddy simulations (LES). Due to their simplicity and low computational costs, lidar simulators have been widely used for the assessment of postprocessing algorithms of lidar data and scan design procedures

As a case study, we use the LES data set of the flow past of a single turbine with the same characteristics of the 5-MW NREL (National Renewable Energy Laboratory) reference wind turbine

For the estimation of the flow characteristics necessary for the scan design, the azimuthally averaged mean and standard deviation of streamwise velocity, as well as the integral timescale are considered (Fig.

Azimuthally averaged statistics of the LES streamwise velocity field.

The streamwise LES velocity field shows the presence of a higher-velocity jet surrounding the nacelle, while

To reconstruct the mentioned flow features, the fundamental half wavelengths in the spanwise and vertical directions selected for this application of LiSBOA are

Azimuthally averaged energy spectra of the LES velocity fields.

The availability of the LES data set allows the testing of the relevance of the selected

The analysis of the flow statistics reported in Fig.

A main limitation of lidars is represented by the spatiotemporal averaging of the velocity field, which is connected with the acquisition process. Three different types of smoothing mechanisms can occur during the lidar sampling. The first is the averaging along the laser beam direction within each range gate, which has commonly been modeled through the convolution of the actual velocity field with a weighting function within the measurement volume

A total of three versions of a lidar simulator are implemented for this work. The simplest one is referred to as ideal lidar, which samples the LES velocity field at the experimental points through a nearest-neighbor interpolation. This method minimizes the turbulence damping while retaining the geometry of the scan and the projection of the wind velocity vector onto the laser beam direction. The second version of the lidar simulator reproduces a step-stare lidar, i.e., the lidar scans for the entire duration of the accumulation time at a fixed direction of the lidar laser beam. A total of two filtering processes take place for this configuration, namely beam-wise convolution and time averaging. To model the beam-wise average, the retrieval process of the Doppler lidar is reproduced using a spatial convolution

The third version of the lidar simulator mimics a pulsed lidar operating in continuous mode and performing PPI scans, where, in addition to the beam-wise convolution and time averaging, azimuth-wise averaging occurs due to the variation in the lidar azimuth angle of the scanning head during the scan. The latter is taken into account by adding an azimuthal averaging to the time average, among all data points included within the following angular sector:

It is noteworthy that the accuracy estimated through the present analysis only includes error due to the sampling in time and space and data retrieval. Other error sources, such as the accuracy of the instrument

Figure

Snapshot at the hub height horizontal plane of the wake generated by the 5-MW NREL reference wind turbine.

Pareto front for the design of the optimal lidar scan for the LES data set for different

The application of LiSBOA requires the provision of technical specifications of the lidar, specifically accumulation time,

With the information provided about the flow under investigation and the lidar system, it is possible to draw the Pareto front for the optimization of the lidar scan as a function of different combinations of angular resolutions of the lidar scanning head,

Error analysis of LiSBOA applied to virtual radial velocity fields:

For the optimization of the lidar scan, the lidar angular resolution,

Virtual lidar simulations are performed for all the values of angular resolution utilized in the Pareto front reported in Fig.

Figure

From a more technical standpoint, the error on the mean velocity field,

This error analysis confirms that the optimal configurations selected through the Pareto front (i.e.,

Mean streamwise velocity for

As in Fig.

Azimuthally averaged profiles of mean streamwise velocity and turbulence intensity for three downstream locations.

The 3D fields of mean velocity and turbulence intensity calculated over

Mean streamwise velocity fields obtained through the ideal lidar simulator with

Same as Fig.

Finally, Figs.

LiSBOA can be applied to lidar data sets that are statistically homogeneous as a function time,

The results of LiSBOA for the optimal design of wind lidar scans are affected by the selection of the input parameters, such as the total sampling time,

The total sampling time,

For this sensitivity study, the characteristic integral timescale,

Regarding the characteristic velocity variance,

Pareto fronts for the design of the volumetric scan for different inputs.

The choice of the fundamental half wavelength,

For the sake of completeness, the influence of the different fundamental half wavelengths on the statistics is assessed by calculating the

We acknowledge that the technical specifications required by LiSBOA (namely

Statistics retrieved from a step-stare virtual lidar scan of the LES data set by means of different techniques.

For the sake of completeness, LiSBOA is compared with other widely used techniques for statistical postprocessing of wind lidar data, specifically the Delaunay triangulation

Overall, all the methods, except for the window average, have similar accuracy in the retrieval of the mean velocity (see the mean absolute percentage error, MAPE, in Table

Comparison between LiSBOA and other techniques for the retrieval of mean velocity and turbulence intensity from the LES data set through a virtual lidar scan.

On a final note, it is noteworthy that LiSBOA is currently formulated for a single scalar field, namely a velocity component (radial or equivalent horizontal). However, in principle, this procedure can be extended to vector fields, such as fully 3D velocity fields. Furthermore, other constraints can be added for the optimal scanning design, such as imposing a divergence-free velocity field for incompressible flows. Also, some modifications could extend the application of LiSBOA to other remote sensing instruments, such as sodars and scanning radars.

A revisited Barnes objective analysis for sparse, nonuniform distributed, and stationary lidar data has been formulated to calculate mean, variance, and higher-order statistics of the wind velocity field over a structured

LiSBOA has been validated against volumetric synthetic 3D data generated through Monte Carlo simulations. The results of this test have shown that the sampling error for a monochromatic scalar field is mainly driven by the data spacing being normalized by the half wavelength.

The LiSBOA framework provides guidelines for the optimal design of scans performed with a scanning Doppler pulsed wind lidar and the calculation of wind velocity statistics. The optimization problem consists of providing background information about the turbulent flow under investigation, such as total sampling time, expected velocity variance, and integral length scales, technical specifications of the lidar, such as range gate and accumulation time, and spatial wavelengths of interest for the velocity field. The formulated optimization problem has two cost functions, namely the percentage of grid nodes not satisfying the Petersen–Middleton constraint for the smallest half wavelength of interest (i.e., lacking adequate spatial resolution to avoid aliasing in the statistics), and the standard deviation of the sample mean. The outputs of the optimization problem are the lidar angular resolution and, for a given response of the mean field, the allowable smoothing parameters and number of iterations to use for LiSBOA.

LiSBOA has been validated using a data set obtained through the virtual lidar technique, namely by numerically sampling the turbulent velocity field behind the rotor of a 5

In the companion paper

The first iteration of LiSBOA produces a weighted average in space of the scalar field,

The LiSBOA algorithm has been implemented in a publicly available code which can be downloaded at

SL and GVI developed LiSBOA and prepared the paper. The lidar data were generated as part of a team effort, which included contributions from all three authors. SL implemented LiSBOA in a MATLAB code under the supervision of GVI.

The authors declare that they have no conflict of interest.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors.

This research has been funded by the National Science Foundation CBET Fluid Dynamics (grant no. 1705837). This material is based upon work supported by the National Science Foundation (grant no. IIP-1362022; Collaborative Research – I/UCRC for Wind Energy, Science, Technology, and Research) and from the WindSTAR I/UCRC Members of Aquanis, Inc., EDP Renewables, Bachmann Electronic Corp., GE Energy, Huntsman, Hexion, Leeward Asset Management, LLC, Pattern Energy, EPRI, LM Wind, Texas Wind Tower, and TPI Composites. The Texas Advanced Computing Center is acknowledged for providing computational resources. The authors thank Stefano Leonardi and Umberto Ciri for sharing the LES data set.

This research has been supported by the National Science Foundation, Directorate for Engineering (grant nos. 1705837 and IIP-1362022).

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