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 . Accordingly, uncertainties in lidar-derived mean wind velocity estimates have been well characterized and methods and procedures have been developed for error reduction and uncertainty control . However, use of lidar for turbulence measurements, while possible , is less established . Two methods are commonly used to derive the second-order moments (i.e., velocity variances and momentum fluxes) of turbulent flow from lidar data . The first method initially estimates the three orthogonal wind components from radial velocities acquired through a scan geometry (e.g., the conical scan) and then uses the eddy covariance method to derive the turbulence statistics. The second method involves three steps: (1) obtaining radial velocities to form a time series at different locations through a scan geometry, (2) estimating radial velocity variance from the time series obtained in (1), and (3) deriving the second-order moments based on their relationships with the radial velocity variance. Estimates from both methods have errors relative to the true or expected values defined as the ensemble means over all possible realizations, and these errors are commonly quantified in terms of the systematic and random components. Systematic errors are consistent deviations from the true values in all estimates, and they are also referred to as biases. Random errors are varying deviations from the true values arising for unknown reasons and are commonly modeled as random variables following Gaussian distributions. The first method can suffer from cross-contamination errors (biases) due to the correlation between the radial velocities used for wind velocity estimates . Therefore, the second method is considered to be more suitable for lidar turbulence measurements

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 ;

A brief description of lidar measurements is given below. For more details see and . Symbols introduced hereafter are explained in the nomenclature in Appendix B.

Based on the streamline coordinate system in Fig. , a wind velocity vector is defined as u=(u1,u2,u3), where u1, u2, and u3 are streamwise, transverse, and vertical wind components respectively at position x=(x1,x2,x3). Without loss of generality, we can assume that the mean streamwise velocity U1 has been removed, and u only consists of the fluctuating components of turbulence with zero means. A lidar measures the radial velocity (vr) from the Doppler frequency shift induced by the motion of scatterers along the line of sight (LOS), where the orientation of LOS is defined by the unit directional vector: n=(cos⁡ϕsin⁡θ,cos⁡ϕcos⁡θ,sin⁡ϕ), where θ is the azimuth angle that increases clockwise from being zero in a positive x2 direction and ϕ is the elevation angle relative to the x1–x2 plane (Fig. ). The radial velocity is the projection of wind velocity on the LOS and is defined as vr=n⋅u. The Galion lidar used here is a pulsed lidar that measures radial velocity with the step-stare technique in this application. Each radial velocity is measured over a dwell time of approximately 1.0 s, during which spectra of a large number of returned signals are averaged to improve the measurement accuracy. When operated with a scan geometry, the lidar steers its transceiver to probe at different sets of θ and ϕ. Hence, the sampling interval of two consecutive measurements at one location depends on the dwell time, the scan geometry, and the mechanical design of the lidar. The shortest sampling interval can be achieved with the staring scan for which the lidar measures with fixed θ and ϕ; that is, the sampling interval is close to the dwell time. The radial velocity (vR) is estimated from an averaged spectrum acquired over a range gate; hence, it is a weighted average of radial velocities along the LOS: vR(s)=∫-∞+∞Q(s′)vr(s-s′)ds′, where s denotes the range gate location and s′ is the range distance on the LOS. Here we use the subscript R to denote an average quantity and r to denote a point quantity. Note that vR defined in Eq. () is the expected value of a radial velocity measurement. The actual measured value differs from vR due to measurement errors as discussed in and Sect. . The weighting function Q in Eq. () can be approximated by a Gaussian function : Q(s′)=12πσQexp⁡-(s′-s)22σQ2, where the standard deviation σQ is a measure of the volumetric averaging size, which is 15.4 m for the Galion lidar used herein .

The covariance of vr (Rr) and vR (RR) along the x1 direction is defined as follows : Rr(r1)=ninjRij(r1e1)RR(r1)=ninj∫∫-∞+∞Q(s′)Q(s′′)Rij(s′′-s′)n+r1e1ds′ds′′, where s′ and s′′ denote the range distance, Rij is the velocity tensor for the ith and jth velocity components, ni and nj are the components of n, and e1 and r1 are the unit vector and the separation distance on the x1 axis, respectively. Note that i,j=1,2,3 and summation is assumed over repeated indices. We can map r1 to the temporal lag τ through the frozen turbulence hypothesis ; r1=U1τ. The spatial autocorrelation function of vr (ρr) and vR (ρR) separated by (r1,0,0) is then defined respectively as ρr(r1)=Rr(r1)/μ2,rρR(r1)=RR(r1)/μ2,R, where μ2,r=Rr(0) is the ensemble variance of vr, and μ2,R=RR(0) is the ensemble variance of vR. Due to the averaging given in Eq. (), RR(0)<Rr(0) and the ratio RR(0)/Rr(0) decreases when the size of volumetric averaging increases (i.e., σQ/L1 increases, where L1 is the integral length scale of u1) . However, RR(r1)=Rr(r1) when r1 is sufficiently large (e.g., r1≫L1) because values of Rr(r1) with large r1 are determined by eddies of large sizes that are not affected by the averaging. As illustrated in Fig. , with the assumption of isotropic turbulence, RR(r1) starts at a value lower than Rr(r1) at r1=0 and gradually converges to Rr(r1) as r1 increases. Because Rr(0) and RR(0) are the denominators in Eqs. () and (), respectively, ρR≥ρr and (ρR-ρr) increases with increasing σQ/L1.

As a result, if we denote Lr and LR respectively as the integral length scales for vr and vR, LR>Lr and LR/Lr increases with increasing σQ/L1 (Fig. ). Although quantitative evaluation of RR/Rr or (ρR-ρr) as a function of r1 requires the knowledge of velocity tensors Rij, the qualitative statement above is also true for non-isotropic turbulence and has implications for the errors of radial velocity variance estimates as demonstrated in the next section.

In the following, the analysis and notation used are from , and we use the word “error” to refer to the sampling error of radial velocity variance estimated from time series unless otherwise stated.

Radial velocity variance is estimated from time series of radial velocity that is related to u(t) which is a stationary and ergodic time series generated from a Gaussian process characterized by the mean wind speed U1 and covariance tensors. Assuming that the mean has been removed from u(t), it can be shown that both vr and vR are also from a stationary and ergodic time series of a Gaussian process that has the following properties:

zero ensemble mean for both vr and vR;

The measured radial velocity (v^R) from a lidar will deviate from the expected value (vR) as a result of the measurement error . The bias introduced by the measurement error is negligible for a lidar measurement with a high pulse accumulation number (e.g., 20 000 for the Galion lidar) . The random measurement error in v^R is modeled by an independent Gaussian random variable with zero mean and variance σR2, which is inversely proportional to the SNR . Turbulence can cause σR2 to increase by an amount that is proportional to the range gate size and the turbulent kinetic energy (TKE) dissipation rate . show that the amount of increase due to turbulence is about 20 % for a 48 m range gate size. For the Galion lidar with 30 m range gate size, the contribution from turbulence to σR2 is no more than 10 % with a relative weak SNR and a TKE dissipation rate on the order of 0.01 m2s-3 (based on the method in ). Therefore, it is a good approximation to estimate σR2 from the SNR data using Eq. (5) in for the Galion lidar .

Assuming zero bias in the measurement error, it can be shown that the radial velocity variance (μ^2,R(T)) estimated from lidar measurements over a sampling period of T can be represented as μ^2,R(T)=μ2,R(T)+σR2, where μ2,R(T) is the estimate of μ2,R from a time series over a period T without measurement errors. According to , μ2,R(T)=μ2,R+Es,R(T)+ϵr,R(T), where

Es,R(T) is systematic sampling error that is defined as 〈μ2,R(T)〉-μ2,R;

On the basis of empirical evidence presented in Sect. , in the following we use the theoretical framework presented in Sect.  to describe how the error in estimating radial velocity variance from lidar measurement that results from (i) autocorrelation function, (ii) sampling duration, and (iii) sampling interval can be minimized. The first factor (autocorrelation function) is determined by the underlying wind field, the lidar LOS orientation, and the size of volumetric averaging (i.e., σQ in Eq. ), and needs to be specified to estimate uncertainty in the variance estimates. It typically approximates an exponential but the precise functional form and the rate of decay varies depending on the flow field. Therefore, in practice, error reduction can only be achieved by adjusting the other two factors: sampling duration (T) and sampling interval (δt), with assumption or knowledge of the autocorrelation function ρ(τ).

Atmospheric turbulence is rarely isotropic, and for all the hours presented in Sect.  the three wind components had non-equal variance. However, here we use the isotropic turbulence model to model the autocorrelation function, noting that it is always possible to find an integral length scale to reproduce the observed autocorrelation function of radial velocity from lidar measurements with the isotropic turbulence model and the von Kármán spectra (e.g., Fig. ). Therefore, we argue that with a proper length scale, the isotropic turbulence model can generate an autocorrelation function that can be used to approximate non-isotropic turbulence conditions, justifying the use of the isotropic turbulence model here.

Based on the isotropic turbulence model and the von Kármán spectra, the relative systematic error (es) is negligible in comparison to the random error. Hence, only the analysis of random error variance er2 or standard deviation (er) is given below. The magnitude of er2 is not sensitive to the sampling interval (δt). Because the integral timescale (τ0) has the order of magnitude of 10 s and δt= 1–10 s, it is likely that δt/τ0<1, implying that in practice the sampling interval is a minor factor on error reduction (Fig. ). Increasing the size of volumetric averaging in terms of σQ can cause er2 to increase with a rate that decreases to nearly zero when the sampling duration increases and the LOS moves from being parallel to perpendicular to the wind direction (i.e., β from 0 to 90∘) (Fig. ). Therefore, it is also a minor factor on error reduction when the sampling duration is long. The LOS orientation relative to the wind direction (β) naturally affects the properties of random errors because the timescale of radial velocity varies with β. For example, for the isotropic turbulence model used here, streamwise velocity has the largest timescale and as a result, errors are large when the LOS is aligned with the wind direction (i.e., β=0∘) (Figs. a and a). In general er2 is not sensitive to β, but it must be acknowledged that the effect of β on er2 will change if a different turbulence model is applied. The factor that has the most significant effect on error reduction is the sampling duration (Figs.  and ). Thus optimization of lidar operation for retrieval of radial velocity variance can be considered through the lens of “how long is long enough?” . Provided that the optimum six-beam configuration proposed in (i.e., ϕ=45∘) is applied to the Galion lidar for which σQ=15.4 m under neutral atmospheric conditions over flat terrain, both the systematic and random errors are almost independent of LOS orientation and surface roughness length, which is used to predict the integral length scale and turbulence intensity (Fig. ). The systematic error is lower than 1.0 % when T>30 min (Fig. ). The standard deviation (er) of random errors can be reduced from 12.0 to 9.0 % by increasing the sampling duration from 30 to 60 min (Fig. ), which is consistent with the observed standard deviation 12.7–9.5 % in Sect. . The standard deviation remains higher than 6.0 % when T increases to 120 min (Fig. ). Note that the volumetric averaging increases er by less than 1.0 % because of its relatively small size (σQ/L1<0.1). The implication of the analysis above is that, given that 0.5–1.0 h is usually the length over which the stationarity assumption is valid in the ABL , the random error in radial velocity variance estimates will likely be around 10.0 % and it will be difficult to estimate radial velocity variance with random errors lower than 5.0 %.

The relatively high sampling error discussed above, in combination with measurement and attenuation errors described in Sect. , will propagate through and result in a bias and random error in the estimated turbulence parameters. The bias is mainly due to the attenuation error from volumetric averaging. The contribution from the measurement error to the bias can be ignored given that the SNR and the number of spectra used for radial velocity estimation are high (i.e., the dwell time is sufficiently long). The random error in the estimated turbulence parameters is mainly from the random sampling error. Thus, we can consider only attenuation and sampling errors in error quantification and scan geometry optimization for lidar turbulence measurements. The error quantification method is given in Appendix  for lidar-derived second-order moments (i.e., velocity variances and momentum fluxes), and is applied to evaluate the uncertainty in the second-order moments estimated with the optimal scanning geometry of the six-beam method from using the turbulence conditions, attenuation errors, and sampling errors observed during the PEIWEE experiment. The derived uncertainties of both the streamwise velocity variance and the vertical momentum flux, in terms of the relative standard deviation, decrease with increasing sampling duration (Fig. ). The uncertainty values are similar for the streamwise velocity variance derived from the six-beam method and the sonic data (Fig. ); they are around 10 % for T=30 min and 8 % for T=55 min. However, the vertical momentum flux derived from the six-beam method is much higher than that from the sonic data (Fig. ). For a 30 (55) min sampling duration, the median uncertainty is 38 (28) % for the vertical momentum flux derived from the six-beam method and 11 (8) % from the sonic data. Such large uncertainties in the derived turbulence statistics can introduce uncertainty when verifying lidar turbulence measurements with sonic data. Using the median uncertainty values in Fig.  and the linear model for verification, statistical simulations show that the coefficients of determination (i.e., R2) for the vertical momentum flux are 0.552±0.073 for a 30 min sampling duration and 0.694±0.065 for a 55 min sampling duration. The uncertainty is much lower for the streamwise velocity variance. The R2 values are 0.905±0.020 for a 30 min sampling duration and 0.946±0.012 for a 55 min sampling duration.

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 https://blogs.cornell.edu/barthelmie/files/2015/05/amt2016-83-1sqkw4c.zip.