A high-fidelity lidar turbulence measurement technique relies on accurate estimates of radial velocity variance that are subject to both systematic and random errors determined by the autocorrelation function of radial velocity, the sampling rate, and the sampling duration. Using both statistically simulated and observed data, this paper quantifies the effect of the volumetric averaging in lidar radial velocity measurements on the autocorrelation function and the dependence of the systematic and random errors on the sampling duration. For current-generation scanning lidars and sampling durations of about 30 min and longer, during which the stationarity assumption is valid for atmospheric flows, the systematic error is negligible but the random error exceeds about 10 %.

Coherent Doppler lidars (hereafter called lidars) are increasingly being deployed to
measure flow in the atmospheric boundary layer (ABL) particularly for applications
to wind engineering

Using the second method mentioned above, errors in the estimated variances and
momentum fluxes are accumulations of the following three types of errors in the
estimated radial velocity variance:

measurement errors caused by radial velocity estimator uncertainty and
atmospheric turbulence

attenuation errors due to the volumetric averaging effect of lidar measurement

sampling errors as a result of estimating the ensemble mean by the time
average

The approach taken and the format of this paper are as follows. The
theoretical framework used herein to quantify errors in radial velocity
variance from lidar measurements leverages the theory developed to
characterize uncertainties in statistical moments estimated from a time series
of sonic anemometer measurements in

A brief description of lidar measurements is given below. For more details see

The schematic of lidar line of sight (LOS) orientation in the streamline coordinate system

Based on the streamline coordinate system in Fig.

The covariance of

Examples of turbulence statistics of point radial velocity
(

As a result, if we denote

In the following, the analysis and notation used are from

Radial velocity variance is estimated from time series of radial velocity that is related to

zero ensemble mean for both

ensemble variance

ensemble autocorrelation

The measured radial velocity (

Assuming zero bias in the measurement error, it can be shown that the radial
velocity variance (

Radial velocities from a pulsed lidar naturally form a discrete time series
(see Sect.

It is clear from Eqs. (

Covariance (

The relative systematic error (

Thus both the statistically simulated wind data and physical reasoning provide evidence that volumetric averaging increases the autocorrelation of radial velocity and inflates the errors in radial velocity variance estimates. Further confirmation will be provided in the next section, where data from a field experiment are used to show the effects of volumetric averaging and sampling duration on the errors.

Measurements presented herein were obtained during the Prince Edward Island
Wind Energy Experiment (PEIWEE) conducted at the Wind Energy Institute of
Canada (WEICan) site on the North Cape of PEI

The lidar conducted automatic cleaning at the beginning of each hour,
resulting in a 60

Time series of

Stationarity is the fundamental assumption required to obtain theoretical
estimates of the errors (as in Sect.

The systematic error

A new time series of size of

A subset of size of

Sequences of

Per the definition in Eq. (

To calculate the random error variance, the expected value

An example of the autocorrelation and errors in radial velocity
variance estimate using data from the hour starting at 19 May 2015 04:00 at
40

Two methods are used to estimate the errors associated with different
sampling durations after the means are removed from hourly time series of the
point radial velocity from sonic anemometers (

Comparison of the values of radial velocity autocorrelation function
from sonic data (

Comparison of systematic errors estimated from sonic data
(

The relative variance of random error

Comparison of lidar radial velocity systematic errors

Consistent with expectations, error estimates from both

Relationships between

Contours of the relative variance

Contours of the relative variance

On the basis of empirical evidence presented in
Sect.

Atmospheric turbulence is rarely isotropic, and for all the hours presented in
Sect.

Based on the isotropic turbulence model and the von Kármán spectra,
the relative systematic error (

Variation of the systematic error (

Box plots of the relative standard deviation of the streamwise
velocity variance

The relatively high sampling error discussed above, in combination with
measurement and attenuation errors described in Sect.

Use of lidar for estimation of turbulence fields if realized could revolutionize atmospheric boundary layer characterization studies and has applications to many fields. Accurate radial velocity variance estimates are necessary (but not sufficient) to obtain robust turbulence statistics from lidar. The accuracy of radial velocity variance estimates and their relationship to pseudo-point measurements from sonic anemometers are determined by (i) the applicability of the stationarity assumption, (ii) the effect of volumetric averaging on the radial velocity autocorrelation function, (iii) the sampling interval, and (iv) the sampling duration. Of these factors, (i), the stationarity assumption, is determined only by atmospheric conditions, but it is most likely to be achieved within the period of 1 h in environments where the surface conditions are homogeneous. The second factor, (ii), the volumetric averaging, is dictated by the probe length that is determined by the lidar properties; it causes the radial velocity autocorrelation function to increase, and thus increases errors in radial velocity variance estimates. Large probe length can result in high errors. The third factor, (iii), the sampling interval, is determined partly by the scan geometry which is needed to sample radial velocities with different LOS orientations to reconstruct the wind field, and partly by the lidar configurations of e.g., the dwell time of each measurement and the scanning speed. Errors are not sensitive to the sampling interval because the sampling interval for lidar turbulence measurement is commonly smaller than the turbulence integral timescale. The last factor, (iv), the sampling duration, which together with the sampling interval determines the number of samples available for radial velocity estimates, can only be extended to the limit implied by the stationarity assumption, but in principle, as sampling duration increases, the errors associated with the radial velocity variance decrease. Given these constraints on radial velocity variance estimates, this paper uses theories and empirical observations to show that for sample periods for which stationarity can reasonably be asserted (approximately 1 h), the systematic error can be reduced to a level lower than 1 %, and the standard deviation of random errors will be around 10 %. These errors will propagate through to estimation of turbulence statistics from lidar measurements, and thus provide a fundamental limit on the likely accuracy of those estimates.

The raw lidar and sonic data used in this paper can be download from

According to the linear relationship between the wind velocity and radial
velocity in Eq. (

This work was funded by the US National Science Foundation (award no. 1464383 and 1540393) and the US Department of Energy (award no. DE-EE0005379). Edited by: A. Stoffelen Reviewed by: two anonymous referees