measurement a review of turbulence measurements using ......a. sathe and j. mann: lidar turbulence...

Atmos. Meas. Tech., 6, 3147–3167, 2013www.atmos-meas-tech.net/6/3147/2013/doi:10.5194/amt-6-3147-2013© Author(s) 2013. CC Attribution 3.0 License.

Atmospheric Measurement

TechniquesO

pen Access

A review of turbulence measurements using ground-basedwind lidars

A. Sathe and J. Mann

DTU Wind Energy, Risø campus, Roskilde, Denmark

Correspondence to:A. Sathe ([email protected])

Received: 29 May 2013 – Published in Atmos. Meas. Tech. Discuss.: 26 July 2013Revised: 7 October 2013 – Accepted: 16 October 2013 – Published: 19 November 2013

Abstract. A review of turbulence measurements usingground-based wind lidars is carried out. Works performedin the last 30 yr, i.e., from 1972–2012 are analyzed. Morethan 80 % of the work has been carried out in the last 15 yr,i.e., from 1997–2012. New algorithms to process the raw li-dar data were pioneered in the first 15 yr, i.e., from 1972–1997, when standard techniques could not be used to measureturbulence. Obtaining unfiltered turbulence statistics fromthe large probe volume of the lidars has been and still remainsthe most challenging aspect. Until now, most of the process-ing algorithms that have been developed have shown that bycombining an isotropic turbulence model with raw lidar mea-surements, we can obtain unfiltered statistics. We believe thatan anisotropic turbulence model will provide a more realisticmeasure of turbulence statistics. Future development in algo-rithms will depend on whether the unfiltered statistics can beobtained without the aid of any turbulence model. With thetremendous growth of the wind energy sector, we expect thatlidars will be used for turbulence measurements much morethan ever before.

1 Introduction

This study is motivated by the recent increase in the use ofwind lidars for wind energy purposes. Understanding andmeasuring atmospheric turbulence is vital to efficient har-nessing of wind energy and to measuring the structural in-tegrity of a wind turbine. Traditionally, meteorological mast(met-mast) anemometry has been used; in this method, eithercup or sonic anemometers are mounted on slender booms atone or several heights to measure turbulence over a certainperiod of time. For wind energy purposes, much interest is

focused on the turbulence of the wind and temperature, al-though some attention is also paid to other atmospheric vari-ables such as pressure, humidity, density, etc. In this arti-cle we focus our review only on the measurement of atmo-spheric turbulence of wind by ground-based wind lidars. Toour knowledge, there is no review article dedicated to sucha topic.Engelbart et al.(2007) provide an overall review ofdifferent remote sensing techniques for turbulence measure-ments including lidars, whereasEmeis et al.(2007) providea review of the use of lidars for wind energy applicationswithout focusing in particular on turbulence measurements.

Turbulence affects the wind turbines mainly in two ways:first, the fluctuations that are caused in the extracted windpower (Kaiser et al., 2007; Gottschall and Peinke, 2008),and second, the fluctuations in the loads on different com-ponents of a wind turbine (Sathe et al., 2012). These fluc-tuations result in inefficient harnessing of wind energy andhave the potential to inflict fatigue damage. Wind turbinesare generally designed for a period of twenty years (Bur-ton et al., 2001; IEC, 2005a). The size of a wind turbine hasgrown significantly over the past few decades. The upper tipof a modern wind turbine blade can easily reach heights upto 200 m above the ground. Thus measuring and understand-ing the turbulent wind field at great heights is essential. It isvery expensive to install and operate a met-mast at such greatheights for a sustained period of time. Especially offshore,the costs increase significantly owing to the large founda-tion needed to support the met-mast. Moreover, a met-mastcannot be moved from one place to another, thus limitingthe physical range of the studies. Because of all these fac-tors, measuring in the wake of a wind turbine (or multiplewakes) becomes quite a challenge. Lidars have the potentialto counter these disadvantages of the met-mast anemometry.

Published by Copernicus Publications on behalf of the European Geosciences Union.

3148 A. Sathe and J. Mann: Lidar turbulence measurements review

Recently, lidars have been used extensively for the measure-ment of the mean wind speed and wind profiling (Smith et al.,2006; Kindler et al., 2007; Peña et al., 2009; Wagner et al.,2011). However, despite having been researched for years allover the world (particularly for meteorological studies), theyhave not yet been accepted for turbulence measurements. Li-dars’ lack of acceptance can be attributed to different rea-sons, such as large measurement volumes leading to spatialaveraging of turbulence along the line-of-sight of its mea-surement axis, cross-contamination by different componentsof the wind field, low sampling rates, etc.

This article attempts to answer two research questionspertaining to measurement of atmospheric turbulence byground-based wind lidars:

1. What is the state of the art?

2. Are further improvements needed, either in lidar tech-nology or in data-processing algorithms that can maketurbulence measurements more reliable?

It is to be noted that our main focus is on reviewing pro-cessing algorithms that use raw lidar data and different scan-ning configurations. Although it is known that different li-dar parameters can also influence turbulence measurements(Frehlich, 1994; Banakh and Werner, 2005), we do not carryout a review with respect to the technology itself, but that canbe found inHardesty and Darby(2005) to a certain extent.

In general, for any variable (or a combination of differentvariables) turbulence is characterized in several ways: in thetime domain, as auto- or cross-correlation functions, turbu-lent kinetic energy dissipation rates, and structure functions;or in the Fourier space, as one- or multi-dimensional auto- orcross-spectrum. In the remainder of this article, we will delveinto these aspects in some detail. We believe that writing sucha review article without including any mathematics will pro-vide only a superficial explanation. Hence we have includedsome mathematics using a uniform set of notations in order toprovide a clear perspective of the past studies. To this end wedefine some mathematical preliminaries that characterize at-mospheric turbulence in Sect.2. In Sect.3, we provide someexplanation of the standard scanning configurations that havebeen used in the past. Section4 attempts to answer the firstresearch question posed above. It is divided into two subsec-tions, in which we first describe the pioneering works alongwith the corresponding mathematics, and then we classifythe past studies based on the investigated turbulence param-eters. Readers who are interested only in knowing the stateof the art without going into too much mathematical detailscan directly jump to Sect.4.2. In Sect.5, some perspectivesare provided on the specific turbulence parameters that areuseful for wind energy purposes. A summary is provided inSect.6, in which we attempt to answer the second researchquestion posed above.

Fig. 1.Schematic of the lidar operating in a staring mode.

2 Mathematical preliminaries

In this article we will often switch between the boldfacedvector notation and the Einstein indicial notation. We definethe wind field asv = (u,v,w), where we define the coor-dinate system to be right-handed such thatu (longitudinalcomponent) is in thex1 direction,v (transversal component)is in thex2 direction, andw is in the verticalx3 direction (seeFig. 1). If we consider that the fluctuations of the wind fieldare homogeneous in space then the auto- or cross-covariancefunctions can be defined only in terms of the separation dis-tance as

Rij (r) = 〈v′

i(x)v′

j (x + r)〉, (1)

where Rij (r) is the auto- or cross-covariance function,i, j = (1,2,3) are the indices corresponding to the compo-nents of the wind field,x is the position vector in the three-dimensional Cartesian coordinate system,r = (r1, r2, r3) isthe separation vector,〈〉 denotes ensemble averaging, and′ denotes fluctuations around the ensemble average. Equa-tion (1) denotes a two-point turbulent statistic. Atr = 0 weget a single-point turbulent statistic, which we can denote asthe variances and covariances. In matrix form it can be writ-ten as

R =

〈u′2〉〈u′v′

〉〈u′w′〉

〈v′u′〉〈v′2

〉〈v′w′〉

〈w′u′〉〈w′v′

〉〈w′2〉

, (2)

where the diagonal terms are the variances of the respectivewind field components and the off-diagonal terms are the co-variances. Here, it is implied thatR = R(0), and we drop theargument and the bracket for simplicity. From the definition

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A. Sathe and J. Mann: Lidar turbulence measurements review 3149

of R(r) andR, we can define integral length scale as

`ij =1

Rij

∞∫0

Rij (r1) dr1. (3)

Similar toRij (r), another useful two-point statistic to char-acterize turbulence is the velocity structure function, whichis defined as

Dij (r) = 〈(v′

i(x + r) − v′

i(x))(v′

j (x + r) − v′

j (x))〉. (4)

On many occasions it convenient to study turbulence in theFourier domain instead of the time domain. To this extent,we can define the spectral velocity tensor (or the three-dimensional spectral density) as the Fourier transform ofRij (r),

8ij (k) =1

(2π)3

∫Rij (r)exp(i k · r) dk, (5)

where 8ij (k) is the three-dimensional spectral velocitytensor, k = (k1,k2,k3) is the wave vector, and

∫dk =∫

∞

−∞

∫∞

−∞

∫∞

−∞dk1 dk2 dk3. From Eq. (5), it is obvious that

Rij (r) is the inverse Fourier transform of8ij (k). Practi-cally, it is not possible to measure a spectral velocity tensor,since we would need measurements at all points in a three-dimensional space. A one-dimensional velocity spectrum isthen used, which is defined as

Fij (k1) =1

2π

∞∫−∞

Rij (r1)exp(−ik1r1) dr1 (6)

=

∞∫−∞

∞∫−∞

8ij (k) dk2dk3. (7)

Another important statistic in the Fourier domain is the co-herence function defined as

cohij (k1) =|χij (k1, r2, r3)|

2

Fii(k1)Fjj (k1), (8)

where χij (k1, r2, r3) denotes the cross spectra betweenthe componentsi and j , and Fii(k1) = χii(k1,0,0) andFjj (k1) = χjj (k1,0,0) (no summation over repeated in-dices) are the one-dimensional spectra of thei andj com-ponents, respectively.

Ideally, we would like to measure one or more of the quan-tities in Eqs. (1)–(8) using a lidar. However, owing to inherentdifficulties in the lidar systems, this is quite often impossi-ble. We then have to resort to combining lidar measurementswith simplified turbulence models that are functions of sev-eral variables. As an example, according toMann(1994), theturbulence structure in the neutral atmospheric surface layerdescribed by8ij (k) can be modeled as a function of only

three parameters,Cε2/3 (which is a product of the univer-sal Kolmogorov constantC ≈ 1.5 (Pope, 2000) and the tur-bulent kinetic energy dissipation rate to the two-third powerε2/3), a characteristic length scale, and an anisotropy param-eter. Many studies in the past have attempted to estimateε

from the lidar measurements. Thus, measurement of one ormore of the model parameters using lidars is also a signifi-cant contribution to the measurement of turbulence.

3 Lidar measurement configurations

A lidar is an acronym for light detection and ranging, andsome fundamentals of its working can be found inMeasures(1984). Most of the past studies have used one of the follow-ing three measurement configurations:

1. Staring mode – the lidar beam is fixed at a certain anglewith respect to the vertical axis.

2. Scanning mode in a cone – this is called the velocityazimuth display (VAD, also called the plan positionindicator, PPI) technique.

3. Scanning mode in a vertical plane – this is called therange height indicator (RHI) scanning technique.

3.1 Staring mode

Figure1 shows the schematic of a lidar operating in staringmode. At a given instant of time – if we assume that a lidarmeasures at a point, and that the lidar beam is inclined ata certain angleφ (in some literature the complement ofφ isused, which is called the elevation angleα = 90◦

− φ) fromthe vertical axis, and makes an azimuth angleθ with respectto thex1 axis in the horizontal plane – then the radial velocity(also called the line-of-sight velocity) can be mathematicallywritten as

vr(φ,θ,df) = n(φ,θ) · v(n(φ,θ)df), (9)

where vr is the radial velocity measured at a point,n =

(cosθ sinφ,sinθ sinφ,cosφ) is the unit directional vector fora givenφ andθ , anddf is the distance from the lidar at whichthe measurement is obtained. In Eq. (9), we have implicitlyassumed thatvr is positive for the wind going away fromthe lidar axis, the coordinate system is right-handed, andu

is aligned with thex1 axis in a horizontal plane. In reality,a lidar never receives backscatter from exactly one point, butrather from all over the physical space. Fortunately the trans-verse dimensions of a lidar beam is much smaller than thelongitudinal dimensional, and for all practical purposes wecan consider the backscatter to be received only along thelidar beam axis. We can then mathematically represent theradial velocity as the convolved signal,

vr(φ,θ,df) =

∞∫−∞

ϕ(s) n(φ,θ) · v(n(φ,θ)(df + s)) ds, (10)

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Fig. 2.Schematic of the lidar operating in a VAD scanning mode.

wherevr is the weighted average radial velocity,ϕ(s) is anyweighting function integrating to one (that depends on thewhether the type of lidar being used is a continuous-wave(C-W) or a pulsed lidar), ands is the distance along the beamfrom the measurement point of interest.

3.2 VAD technique

Figure 2 shows the schematic of the VAD scanning tech-nique. It is an extension of a staring mode, where the lidarbeam rotates around a vertical axis, thus forming a cone withthe base at the measurement distance of interest and the apexat the lidar source.vr is thus measured at differentθ , andφ

is kept constant throughout the scan. At a givendf , the radialvelocity can be written as

vr(θ) = ucosθ sinφ + v sinθ sinφ + wcosφ. (11)

From Eq. (11), we see that whenφ, θ anddf are known,vr isonly a function of three unknown wind field components, i.e.,u, v andw. In principle, we then need three measurements ofvr at three differentθ to deduce theu, v andw components.For a standard VAD scan, we normally have much more thanthree measurements along the azimuth circle. We thus havemore equations and only three unknowns, if we assume hor-izontal homogeneity. Least squares analysis can be used todeduce the three unknown wind field components.

3.3 RHI technique

Figure3 shows the schematic of the RHI scanning technique.It is also an extension of staring mode, where the lidar beamrotates in a vertical plane at differentφ, andθ is kept con-stant throughout the scan. We can use the same Eq. (11) byvarying φ and keepingθ constant to deduce the wind field

Fig. 3.Schematic of the lidar operating in a RHI scanning mode.

components. Actually, from a single RHI scan, only two ve-locity components can be deduced, namely the vertical andthe horizontal in the plane of the RHI scan. Ifθ = 0 is keptconstant, then thev component cannot be determined (seeEq. 11). Owing to the fact that the lidar is placed on solidground,φ can vary only between 0◦ and 180◦ in a verticalplane. Usually, the scanning plane is aligned such that it isin the mean wind direction. As in the VAD technique, if wethen have more measurements at differentφ, the three un-known wind field components are estimated using the leastsquares analysis.

4 State of the art in turbulence measurements usingground-based lidars

Generally, measurement of atmospheric turbulence is a verychallenging prospect. With traditional instruments such asthe cup/sonic anemometers, great care has to be taken withregards to sampling frequencies, averaging periods, correc-tion for flow distortions, and orientation of the instrument.Because these instruments essentially measure at a point, de-ducing turbulence information from the raw data can be car-ried out using the standard procedures if these instrumentsare properly set up (Kaimal and Finnigan, 1994). Due to thefact that lidars measure in a much larger volume, and at dif-ferent points in space, standard techniques do not suffice evenif the instrument is correctly set up. Deducing turbulence in-formation from the raw lidar data has been and remains themost challenging aspect. In the following sections, we firstdiscuss the pioneering works that have demonstrated someof the techniques to process the raw lidar data in order tomeasure turbulence; this is followed by a discussion of re-cent studies that have used some of these techniques.

4.1 Pioneering works

Although much of the lidar turbulence work has been carriedout using the scanning configurations described in Sect.3,the ideas were taken from the pioneering works on radar me-



teorology (Lhermitte, 1962; Browning and Wexler, 1968).Discussing a bit of mathematics from the radar studies is es-sential, since they can and have been used in lidar studiestoo.Lhermitte(1962) was one of the first to explain the VADscanning technique using Doppler radars, wherevr is math-ematically represented as linear combination of the sine andcosine functions ofθ . Browning and Wexler(1968) were thefirst to conduct an experiment with a pulsed Doppler radarand estimate theu andv components of the wind field alongwith the mean horizontal divergence, stretching and shear-ing deformation, in the height range of 1.5–6 km. The latterterms were obtained using a Taylor series expansion aroundthe center of a scanning circle. An error analysis was alsocarried out to limit the errors in the estimated quantities be-low a certain level, which led to limiting the values ofφ. Theradar-estimated quantities were however not compared withany reference instrument. Based on the VAD scanning,Lher-mitte (1969) then suggested a technique of estimating turbu-lence componentsRij that was based on the measurementsof the variance of the radial velocity〈v′

r2〉. Mathematically,

by substituting the definition ofn(φ,θ) into Eq. (11), and bysquaring and ensemble-averaging, we get

〈v′r2〉 = 〈u′2

〉sin2φ cos2θ + 〈v′2〉sin2φ sin2θ + 〈w′2

〉cos2φ

+ 2〈u′v′〉sin2φ sinθ cosθ + 2〈u′w′

〉sinφ cosφ cosθ

+ 2〈v′w′〉sinφ cosφ sinθ. (12)

For ease of reading, we do not include the functional de-pendence ofvr on φ, θ anddf , but it is implicitly assumed.Wilson (1970) was the first to conduct an experiment usinga pulsed Doppler radar and estimateRij from the〈v′

r2〉 data

in the convective boundary layer (0.1–1.3 km). Only turbu-lence scales larger than the pulse volume but smaller thanthe scanning circle could be measured since all the data froma single scan was used. Also, no comparison with measure-ments from a reference instrument was carried out, and hencethe reliability of the radar measurements could not be veri-fied. Wilson (1970) demonstrated a mathematically equiva-lent way of performing the Fourier analysis where integralswere defined in four quadrants as

In =

nπ/2∫(n−1)π/2

〈v′r2〉 dθ, (13)

wheren = 1, ..,4. By combining these integrals he then ob-tained the following expressions:

sin2φ

(〈u′2

〉 + 〈v′2〉 +

2〈w′2〉

tan2φ

)=

1

π

(I1 + I2 + I3 + I4

), (14)

〈u′w′〉sin2φ =

1

4

((I1 + I2) − (I3 + I4)

), (15)

〈v′w′〉sin2φ =

1

4

((I1 + I4) − (I2 + I3)

), (16)

〈u′v′〉sin2φ =

1

4

((I1 + I3) − (I2 + I4)

). (17)

Using this method, we can thus estimate the covariances ofR at a givenφ. TheWilson (1970) method was extended byKropfli (1986) to also include the turbulence scales largerthan the scanning circle by using the data from multiplescans. Although the method was developed for Doppler radarstudies, it could also be used for Doppler lidar studies.

One of the first lidar studies to measure theu spectrum us-ing a C-W Doppler CO2 lidar was carried out byLawrenceet al. (1972). The lidar was oriented in the mean wind di-rection and the measurements were performed at 10 m abovethe ground, where the probe volume length (also called thefull width half maximum (FWHM) = 2l) of the weightingfunctionϕ(s) was about 30 cm. They concluded that the li-dar measurements of theu spectra were considerably bet-ter than those obtained using a cup anemometer. In this case〈u′2

〉 can be computed directly from theu fluctuations, sincethe lidar beam is oriented in the mean wind direction, andthe probe volume is quite small. The aforementioned studieswere based on detecting the Doppler shift in the frequencyof the reflected radiation. Using a non-Doppler effect tech-nique,Kunkel et al.(1980) was one of the first to estimate〈u′2

〉 using cross-correlation analysis and an aerosol lidar.The lidar beams were scanned in a sequence of three az-imuth angles. Turbulence was assumed to be isotropic andthe velocity fluctuations were assumed to have a Gaussiandistribution. The lidar-derived variances compared well withthe variances measured by a reference instrument mountedon a tower at 70 m. A technique was also demonstrated toestimateε from the lidar data; this requires measurementsof the boundary layer heightzi , 〈u′2

〉 and the radial velocityspectrum.Kunkel et al.(1980) estimatedzi using the lidarspectrum observations. As mentioned inSeibert et al.(2000),zi measurements are subjected to significant uncertainties;hence, one should be careful in using this method to estimateε from the lidar data.Hardesty et al.(1982) was one of thefirst to measure theu spectrum in the rotating plane of a windturbine of about 20 m diameter. A C-W lidar was placed ona ground and a rotating mirror was mounted on a meteoro-logical tower such that the laser beam directed towards themirror would focus the beam in a vertical plane at a certainφ. Due to the rotating action, VAD scanning was performedin a vertical plane. Taylor series expansion around the cen-ter of a scanning circle is then used for theu component, sothat the gradients in the vertical and horizontal directions areremoved. Owing to the small half-opening angles, the contri-butions by the cross components ofRij were assumed negli-gible. One has to be careful in using this assumption, as hasbeen explained in detail bySathe et al.(2011b).

Extending the work of Wilson (1970),Eberhard et al.(1989) derived a new set of equations to



estimateRij from the lidar data using VAD scanning, andthey termed their method the partial Fourier decompositiontechnique (the same name can also be used for theWilson,1970 method). By using standard trigonometric identities,they rearranged Eq. (12) as

〈v′r2〉 =

sin2φ

2

(〈u′2

〉 + 〈v′2〉 +

2〈w′2〉

tan2φ

)+ 〈u′w′

〉sin2φ cosθ + 〈v′w′〉sin2φ sinθ

+sin2φ

2

(〈u′2

〉 − 〈v′2〉)cos2θ + 〈u′v′

〉sin2φ sin2θ. (18)

If we denote Eq. (18) as a Fourier series with the correspond-ing Fourier coefficients, we then have

sin2φ

2

(〈u′2

〉 + 〈v′2〉 +

2〈w′2〉

tan2φ

)=

a0

2=

1

2π

2π∫0

〈v′r2〉 dθ, (19)

〈u′w′〉sin2φ = a1 =

1

π

2π∫0

〈v′r2〉cosθ dθ, (20)

〈v′w′〉sin2φ = b1 =

1

π

2π∫0

〈v′r2〉sinθ dθ, (21)

sin2φ

2

(〈u′2

〉 − 〈v′2〉)= a2 =

1

π

2π∫0

〈v′r2〉cos2θ dθ, (22)

〈u′v′〉sin2φ = b2 =

1

π

2π∫0

〈v′r2〉sin2θ dθ, (23)

wherea0 is the average,a1 anda2 are the Fourier cosines,andb1 andb2 are the Fourier sine coefficients. As an exam-ple, say for a given 30 min time series, if we have severalmeasurements ofvr at eachθ , then〈v′

r2〉 can then be com-

puted for eachθ , and hence so can the corresponding Fouriercoefficients. At one half-opening angle, we can thus computethe off-diagonal terms of Eq. (2), i.e., covariances. By mea-suring at two half-opening angles, and combining Eqs. (19)and (22), we can also compute the variances. As with theWilson (1970) method, no reference instrument was avail-able to verify the reliability of the measurements. Neverthe-less, the study was valuable as the method can potentially beused with the current lidar systems at those sites where thereference measurements are available.

In all of the above studies with a Doppler lidar (or radar),horizontal homogeneity is a key assumption that makes itpossible to combine lidar beam measurements from differentpoints in space and obtain turbulence statistics. This limitsthe application of such studies to homogeneous flat terrains.Frisch(1991) performed pioneering work on extending theanalysis ofWilson (1970) andEberhard et al.(1989) to alsoinclude horizontal inhomogeneities in the turbulence mea-

surements. He mathematically demonstrated that, by measur-ing at three half-opening angles, we can computeRij usingthe VAD scanning without assuming horizontal homogene-ity. This could potentially have huge implications on tur-bulence measurements in complex (non-homogeneous) ter-rain, where wind turbines are subjected to large turbulentforces. Using the Taylor series expansion around the centerof a scanning circle,Frisch(1991) denoted〈v′

r2〉 as a Fourier

series up to the third harmonic. For eachφ we then obtaina set of Fourier coefficients; i.e., forφ = φ1 we obtaina01 asthe Fourier coefficient of the zeroth harmonic,a11 andb11 asthe Fourier coefficients of the first harmonic,a21,b21 as theFourier coefficient of the second harmonic, anda31 andb31as the Fourier coefficients of the third harmonic. Similarly,we obtain the Fourier coefficients atφ = φ2 andφ = φ3. Thesecond index in the subscript of the Fourier coefficients de-notes the measurement at the correspondingφ. To computeRij , we only need the Fourier coefficients from the zerothup to the second harmonic given by Eqs. (19)–(23). The cor-responding expressions for the components ofRij are thengiven as follows.

〈u′2〉 = t1 + t2, (24)

〈v′2〉 = t1 − t2, (25)

where

t1 =

(a01cos(φ3)

(sin(φ2)sin(2φ2)cos(φ3) − 2sin2 (φ3)cos2 (φ2)

)+ a02cos(φ1)


)+ a03cos(φ2)


))/

(2sin2 (φ1)sin2 (φ2)(cos(φ2) − cos(φ1))cos2 (φ3)

+ 2sin2 (φ3)(sin2 (φ1)cos2 (φ2)(cos(φ1) − cos(φ3))

+ sin2 (φ2)cos2 (φ1)(cos(φ3) − cos(φ2))))

,

t2 =a22cos(φ1)csc2 (φ2) − a21cos(φ2)csc2 (φ1)

cos(φ1) − cos(φ2)(26)

〈w′2〉 =

(a01sin2 (φ2)sin2 (φ3)(cos(φ2) − cos(φ3))

+ sin2 (φ1)(a02sin2 (φ3)(cos(φ3) − cos(φ1))

+ a03sin2 (φ2)(cos(φ1) − cos(φ2))))

/

(sin2 (φ1)sin2 (φ2)(cos(φ1) − cos(φ2))cos2 (φ3)

+ sin2 (φ3)(sin2 (φ1)cos2 (φ2)(cos(φ3) − cos(φ1))

+ sin2 (φ2)cos2 (φ1)(cos(φ2) − cos(φ3))))

, (27)



〈u′w′〉 =

a11m1 + a12m2 + a13m3

1, (28)

〈v′w′〉 =

b11m1 + b12m2 + b13m3

1, (29)

where

m1 = sin(φ2)sin(φ3)(cos(2φ3) − cos(2φ2)) ,

m2 =1

2(sin(3φ1)sin(φ3) − sin(φ1)sin(3φ3)) ,

m3 = sin(φ1)sin(φ2)(cos(2φ2) − cos(2φ1)) ,

1 = 2sin3 (φ3)(sin(φ1)sin(2φ2)cos2 (φ1)−

sin(2φ1)sin(φ2)cos2 (φ2))

+ 2(sin(2φ1)sin3 (φ2) − sin3 (φ1)sin(2φ2)

)sin(φ3)cos2 (φ3)

+ sin(φ1)sin(φ2)sin(2φ3)(cos(2φ2) − cos(2φ1)) . (30)

〈u′v′〉 =

b22cos(φ1)csc2 (φ2) − b21cos(φ2)csc2 (φ1)

cos(φ1) − cos(φ2). (31)

Unfortunately, not much information is given regarding anyexperimental study, and hence, the validity and reliability ofthis technique remains unknown. Nevertheless, the techniqueremains a potential solution to measuring turbulence in com-plex terrain. Using a different scanning strategy,Gal-Chenet al.(1992) was one of the first to employ RHI scanning toestimateRij . They used a pulsed CO2 Doppler lidar in themean wind direction and perpendicular to the mean wind di-rection. The equations for〈v′

r2〉 are given as

〈v′r2〉 = 〈u′2

〉sin2φ + 〈w′2〉cos2φ ± 〈u′w′

〉sin(2φ), (32)

for the lidar beam aligned in the mean wind direction. The±

sign for 〈u′w′〉 indicates whether the wind is blowing away

from or towards the lidar beam. Similarly, for the cross-winddirection we have

〈v′r2〉 = 〈v′2

〉sin2φ + 〈w′2〉cos2φ ± 〈v′w′

〉sin(2φ), (33)

where the± sign indicates positive or negative cross windbeam direction. Equations (32) and (33) are then solved us-ing the least squares analysis to obtain components ofRij

(except〈u′v′〉). A method to estimateε is also provided us-

ing the one-dimensional longitudinal spectrum. In the inertialsubrange, the following relation is known (Pope, 2000):

F11(k1) = C1ε2/3k

−5/31 , (34)

whereF11(k1) is the one-dimensional spectrum of the lon-gitudinal wind field component, andC1 ≈ 0.5 is the Kol-mogorov constant related toF11(k1). The spectrum is mea-sured using a lidar at low elevation angle, and the inertialrange can be established by fitting the−5/3 slope to the spec-trum measurements.ε can then be estimated using Eq. (34),provided that the averaging is taken care of or can be ignored.

An innovative method was also provided to compute the sur-face heat flux using the third moment of the vertical velocityby the following equation:

∂

∂z

(1

2〈w′3

〉

)=

1

ρ〈w′

∂p′

∂z〉 −

ε

3+

g

θT

〈w′θ ′

T 〉, (35)

where〈w′3〉 is the third moment of the vertical velocity,z

is the height above the ground,∂/∂z is the vertical gradi-ent,p′ is the pressure fluctuation,ρ is the air density at thesurface,θT is the surface potential temperature, and〈w′θ ′

T〉

is the sensible heat flux.Wyngaard and Coté(1971) showedthat the pressure covariance term at the surface is negligible,and thus can be neglected in the surface measurements.〈w′3

〉

can be measured using lidar measurements, and hence〈w′θ ′〉

can be measured indirectly using Eq. (35). It should be notedthat the averaging time required for the third moments aresignificantly larger than those required to compute the lowerorder moments, owing to its influence on the systematic andrandom errors (Lenschow et al., 1994). Again, owing to themeasurement heights of interest, no reference instrument wasavailable, and hence, the reliability of lidar measurements isunknown. It should also be noted that at small elevation an-gles, the assumption of horizontal homogeneity may not bevalid, and one has to take this into account when interpretingthe lidar measurements.

In all of the above studies with a Doppler lidar (or radar),the estimated turbulence statistics from the lidar measure-ments will be subjected to different levels of volume aver-aging errors, depending on the type of lidar (C-W or pulsed),height above the ground, and the turbulence structure in theatmosphere (Sathe et al., 2011b). None of the aforemen-tioned studies have attempted to correct the turbulence statis-tics for the errors due to finite probe volume of a lidar, possi-bly because many were interested to measure in the convec-tive boundary layer. In this layer, the turbulence scales arequite large (Wyngaard, 2010), and perhaps probe volume av-eraging does not matter. However, if the measurements aredesired closer to the ground, particularly in the first 200 mabove the ground where the wind turbines operate, then onemust account for the averaging effects in the probe volume.Frehlich(1994) andFrehlich et al.(1994) demonstrated thisaveraging effect in the measurement of the structure function,where for smaller separation distances the averaging effectwas more pronounced.Smalikho(1995) was the first to de-rive explicit formulae to account for the small-scale filteringeffect of the finite probe volume for a C-W lidar. The formu-lae for the estimation ofε were derived using three differentmethods for a staring lidar, i.e., using

– the width of the Doppler spectrum,

– the velocity structure function, and

– the one-dimensional velocity spectrum.



He derived the following expression for the width of theDoppler spectrum:

〈σ 2s 〉 = 1.22Cε2/3l2/3, (36)

where 〈σ 2s 〉 is the second central moment of the Doppler

spectrum (or its width), andl is the Rayleigh length (whichfor a C-W lidar is the same as the half-width half-maximumof the weighting function of the probe volume). It shouldbe noted that there is a slight difference in the value of theKolmogorov constant used inBanakh et al.(1999), althoughthe same Eq. (36) is also stated inSmalikho(1995), i.e., inEq. (25) ofSmalikho(1995) the value of Kolmogorov con-stant is≈ 1.83, whereas in Eq. (13) ofBanakh et al.(1999),the value of Kolmogorov constant is≈ 2. For a continuouswave lidarϕ(s) is well approximated by a Lorentzian func-tion (Sonnenschein and Horrigan, 1971), andl = λbd

2f /πr2

b ,whereλb is the wavelength of the emitted radiation, andrbis the beam radius.〈σ 2

s 〉 can be measured andl is known,so ε can be estimated. The limitation of this method is thatEq. (36) can only be used whenl � L, whereL is the outerscale of turbulence. Moreover, the effect of mean radial ve-locity gradient within the probe volume has not been takeninto account. Equation (36) states that if there is no turbu-lence, then the Doppler spectral width should be zero. How-ever, if there is a mean change ofvr with s (within the probevolume) then there is an additional term proportional tol2. Ifthe lidar is C-W and the shear is linear, then the coefficientof l2 is infinite (Mann et al., 2010) and we cannot use thismethod.

The expression for the structure function was derived us-ing the assumption of local isotropy in the inertial subrange.Kristensen et al.(2011) re-derived the expression in great de-tail, where the probe volume weighting function is assumedbe Lorentzian. The expression is given as

D(r1) = Cε2/3l2/3 0(1/3)

5√

π0(5/6)

2π∫0

(1−

8

11cos2ξ

)9(r1,2,ξ) dξ, (37)

where D(r1) is the filtered radial velocity structure func-tion measured by the lidar,r1 = 〈u〉t is the separation dis-tance along thex1 axis,0(n) =

∫∞

0 xn−1exp(−x) dx is thegamma function,2 is the angle between the lidar beam andthe mean wind〈u〉, and

9(r1,2,ξ) =3

20

(1

3

)((cos2ξ +

( r1

l

)2cos2(ξ + 2

))1/3

·cos(2

3tan−1

( r1

l

∣∣∣cos(ξ + 2)

cosξ

∣∣∣))−

∣∣cosξ∣∣2/3

). (38)

r1 is computed using the Taylor’s hypothesis (Taylor, 1938),where turbulence is assumed to be advected by the meanwind 〈u〉 in time t . For the measured and known parametersD(r1), r1, l andC, the unknownε can be estimated, wherethe one-dimensional integral in Eq. (37) can be solved nu-merically. Using a similar approach,Smalikho(1995) and

Kristensen et al.(2011) also derived the expressions forthe one-dimensional velocity spectrum. However, due to theequivalence of the structure function and spectrum approach,the equation is not explicitly stated here.

The limitation of the structure function approach usinga staring Doppler lidar is that if there is little or no meanwind then Taylor’s hypothesis is violated and the structurefunctions cannot be estimated. In order to counter these limi-tationsBanakh et al.(1996) proposed a novel technique toestimateε using the VAD scanning. Instead of measuringthe structure function based on a separation distancer1 andusing the Taylor’s hypothesis, it is measured based on anangular separation distance,dfδ, on the base of the scan-ning cone, whereδ = 2sin−1(sinφ sinθ) is the angle sub-tended by the two lidar beams in a VAD scanning. Thereis, however, an assumption that the scanning speed is muchlarger than the advection speed of the turbulence.Kristensenet al. (2012) re-derived the expressions using this approachbut disregarded the contribution due to random instrumentalnoise that was considered inBanakh et al.(1996). For mod-ern lidar systems, the instrumental noise can be neglected(Mann et al., 2009); however, it was found to be signifi-cant for older systems (Frehlich et al., 1998; Drobinski et al.,2000), and hence one must be careful before neglecting it.Two approaches were chosen in the derivation byKristensenet al.(2012): time-domain autocorrelation approach, and theFourier-domain wave-number approach. The Fourier-domainapproach is derived for a C-W lidar (assuming a Lorentzianfunction), whereas the time domain approach provides ex-pressions as a function ofϕ(s). By using appropriateϕ(s),the time-domain expressions can be applied for a C-W ora pulsed lidar. The equations using both approaches are asfollows. In the time domain,

D(δ) = 2(1− cosδ)R(0)

+9

550

(1

3

)C(εdf)

2/3

∞∫−∞

∞∫−∞

ϕ(s′

1)ϕ(s′

2)

·

(3

(((s′

2 − s′

1)2+ 4s′

1s′

2sin2(δ/2))1/3cosδ − |s′

2 − s′

1|2/3

)+

s′

1s′

2sin2δ((s′

2 − s′

1)2 + 4s′

1s′

2sin2(δ/2))2/3

)ds′

1ds′

2, (39)

where D(δ) is the filtered radial velocity structure func-tion for a separation distance,dfδ, on the base of the cone,R(0) = 〈u′2

〉 = 〈v′2〉 = 〈w′2

〉 for isotropic turbulence, ands′

1 = s1/df , s′

2 = s2/df are non-dimensional variables. In theFourier domain, for a C-W lidar,



D(δ) = 2(1− cosδ)R(0) + C(εdf)2/3 3

550

(1

3

)·

(33√

2(1+ 7cosδ)sin2/3(δ/2) − 18

(df

l

)−2/3

+1

π

(2df

l

)−2/3π/2∫0

0(1/2)0(1/3)

0(5/6)

(7cosδ − 4cos(2ξ)

)·

(2cos

(2

3tan−1

( 4df sin(δ/2)sinξ

l(|cos(ξ + δ/2)| + |cos(ξ − δ/2)|

)))

·

((|cos(ξ + δ/2)| + |cos(ξ − δ/2)|

)2+ 16

(df

l

)2sin2(δ/2)sin2ξ

)1/3

−(4df

lsin(δ/2)sinξ

)2/3)

dξ

). (40)

As in the Smalikho (1995) method, the key to using thismethod is to appropriately selectdfδ � L, so that turbulenceis measured in the inertial subrange, and is locally isotropic.D(δ) can be measured using a lidar; then, by knowingR(0),we can estimateε. Banakh et al.(1996) did not include theR(0) term in their equation, perhaps because atδ � π/2,and df � L, this term is negligible. The advantage of us-ing Eq. (40) is that we need to solve only a single integralnumerically, whereas in Eq. (39) we need to solve a dou-ble integral numerically, and that may increase the numeri-cal error. The estimation ofR(0) can be quite challenging,since it also contains information about the large-scale tur-bulence.Kristensen et al.(2012) used empirical models forconvective turbulence (Kristensen et al., 1989) and estimatedthat R(0) = 1.74 ε2/3(df cosφ)2/3. Alternatively, one mayuse thevon Kármán(1948) energy spectrum and derive ex-pressions forR(0). The experimental verification of theKris-tensen et al.(2011, 2012) expressions remains to be seen, butthe experimental verification of theSmalikho(1995) andBa-nakh et al.(1996) expressions will be discussed later in thearticle.

One of the biggest limitations of a C-W lidar is thatl ∝ d2f ,

and hence measuring at greater heights becomes a problemowing to the large probe volume. A pulsed lidar is then ide-ally suited for this purpose, since the length of its probe vol-ume remains constant at all heights. To this end,Frehlich(1997) was one of the first to derive expressions for the fil-tered velocity correlation and structure function measured bya pulsed lidar. Numerical simulations were performed to ver-ify the model in which close agreement was observed. Thecovariances and structure functions of the radial velocitiesare expressed as a function of lidar parameters and single-point statistics. If the range gate length of a pulsed lidar isdefined asLp = cτ/2, wherec is the speed of light, andτ isthe pulse duration, thenFrehlich(1997) derived the follow-ing equation for the filtered covariance function of the radialvelocity:

R(r) = 〈v′r2〉

∞∫−∞

f (x,µ)(1− 3(χ |y − x|)

)dx, (41)

wherer is the separation distance along the beam,y = r/Lp,µ =

√2ln(2)Lp/l, χ = Lp/L,

3(x) = (ax)2/3(1+ (ax)b)−2/3b

, (42)

is supposedly the normalized structure function of the radialvelocity component, and

f (x,µ) =1

2√

πµ

(exp

(−µ2(x + 1)2)

+ exp(−µ2(x − 1)2))

+x

2

(erf

(µ(x + 1)

)+ erf

(µ(x − 1)

)− 2erf(µx)

)−

1√

πµexp(−µ2x2) +

erf(µ(x + 1)

)2

−erf

(µ(x − 1)

)2

, (43)

wheref (x,µ) is the filter function for a Gaussian transmit-ted pulse and a rectangular time window.Frehlich (1997)mentions that Eq. (42) is the universal function given byKaimal et al.(1972) (but we could not verify that), where,for neutral conditions,a = 0.26278 andb = 1.1948, anderf(x) = 2/

√π

∫ x

0 exp(−t2) dt is the error function. In or-der to use Eq. (41), it is necessary thatr � L. The filteredradial velocity structure function is given as

D(r) = 2〈v′r2〉

∞∫−∞

f (x,µ)(3(χ |y − x|) − 3(χ |x|)) dx. (44)

In both Eqs. (41) and (44) an empirical3(x) function is usedto expressR(r) and D(r) in terms of 〈v′

r2〉 andL, but in

principle we could also use thevon Kármán(1948) model.By measuringR(r) or D(r) using a lidar,L and〈v′

r2〉 can be

obtained by the fitting the measurements to Eqs. (41) or (44).Having obtained〈v′

r2〉, any/all of the Eqs. (13)–(33) can be

used to estimateRij .In an independent study,Banakh and Smalikho(1997b)

also derived expressions for the estimation ofε using a star-ing pulsed lidar. They followed the same structure functionapproach as inSmalikho(1995). Using numerical simula-tion, they compared the performance of their model with thenumerical results, and concluded that the relative errors in theestimation ofε are between 15–20 % for a signal-to-noise ra-tio equal to or greater than unity. Comparison of the modelwith the measurements will be more challenging, and possi-bly provide more confidence in the method. As for the C-Wlidar, Kristensen et al.(2011) re-derived the expressions ingreat detail. LikeFrehlich(1997), Kristensen et al.(2011) as-sumed a Gaussian transmitting pulse. Ifwp is defined as thepulse width, thenKristensen et al.(2011) introduced a length

scalelp =

√L2

p/12+ w2p and used the same same expression



as Eq. (37) also for the pulsed lidar, except thatl in Eq. (37)is replaced bylp and the9(r1,2,ξ) function is now givenas

9(r1,2,ξ) =3

20

(2

3

)|cosξ |

2/3(

1F1

(−

1

3;

1

2;−

r21 cos2(ξ + 2)

4l2p cos2ξ

)− 1

), (45)

where1F1(a;b;x) is the Kummer confluent hypergeometricfunction (Abramowitz and Stegun, 1965). It is to be notedthat using Eqs. (37) and (45), for some combinations ofα andr1/l (or r1/lp), D(r1) becomes negative, butKristensen et al.(2011) also provide the range within which Eqs. (37) and(45) are valid. An advantage of using a pulsed lidar is alsothat we do not need to apply Taylor’s hypothesis in order tocompute the separation distance. Thus, instead of usingr1 inEq. (45) we can use the separation distancer (provided thatr � L) along the lidar beam, since a pulsed lidar measuresat different range gates simultaneously, and hence measureD(r) along the lidar beam axis (Frehlich, 1997).

For a C-W lidar, it is reasonable to assume that the Dopplerspectrum obtains its width mainly due to velocity variations(and perhaps also the mean shear) inside the probe length ofthe lidar. For a pulsed lidar, this assumption is not reasonable,since some lidar parameters like the finite pulse width alsocontributes to the width of the Doppler spectra. Extractingturbulence information from the width of a Doppler spectrafor a pulsed lidar is then much more challenging. Neverthe-less,Smalikho et al.(2005) demonstrated that by depictingthe Doppler spectral width as a linear summation of contri-butions from atmospheric turbulence and lidar parameters,we can successfully measureε. Using numerical simulations,they concluded that the bias inε is very sensitive to the selec-tion of the optimal noise threshold level. Interestingly, theycompared the estimations ofε obtained by the spectral widthapproach and those obtained by the structure function ap-proach (Frehlich, 1997; Banakh and Smalikho, 1997b), andconcluded that at low turbulence levels, the structure func-tion approach results in lower random errors ofε, whereasat higher turbulence levels, the random errors inε obtainedby the spectral width approach are two times smaller thanthose obtained by the structure function approach. The ex-periment was carried out using the RHI scanning, but sinceno reference instruments were available, the reliability of thistechnique was unknown.

From the above, it can be seen that the major works on pro-cessing raw lidar data and obtaining unfiltered turbulence pa-rameters are based on the filtered radial velocity covariancesand structure functions. In all these works, the filter functionis obtained by assuming either the isotropy of turbulence inthe inertial sub-range or the entire range of turbulence scales.It is well-known, however, that turbulence is not isotropicon all scales of interest (Kaimal et al., 1972; Mann, 1994).Hence,〈v′

r2〉 andL obtained by fitting the modeled structure

function to the measurements (Frehlich et al., 1998) is notentirely reliable. A better solution would then be to use theanisotropic turbulence model (Mann, 1994) in modeling the

structure function measured by the lidar, and then fitting itto the measurements (Frehlich et al., 2006; Frehlich and Kel-ley, 2008). Even using an anisotropic turbulence model maynot provide reliable estimates of turbulence statistics underall conditions, e.g., theMann (1994) model is strictly validonly for homogeneous neutral surface layer. Alternatively, itwould be best if we do not need to combine turbulence mod-els with measurements, so that we get more reliable statisticsfrom lidar measurements.Mann et al.(2010) provided onesuch technique to obtain unfiltered radial velocity variancefor a C-W lidar without using any turbulence models. Theysuggested using the mean Doppler spectra given as

〈S(vr)〉 =f (η) + f (η∗)√

8π〈v′r2〉

, (46)

whereη = (Gl + ivr)/

√2〈v′

r2〉, G is the mean radial veloc-

ity gradient, ∗ denotes complex conjugation, andf (η) =

exp(η2)(1− erfη). Mann et al.(2010) assumed that the prob-ability density function ofvr at a given positions inside theprobe volume is Gaussian distributed, and that〈v′

r2〉 is con-

stant inside the probe volume. A systematic study of the in-fluence of these assumptions on the turbulence statistics hashowever not been carried out. By fitting Eq. (46) to measure-ments of〈S(vr)〉, 〈v′

r2〉 can be estimated.

4.2 Classification of the previous works according to theestimated turbulence quantity

From the previous section, it can be clearly understood thatprocessing raw lidar signals to extract turbulence informa-tion is an extremely challenging task. Until the mid- andlate 1990s, the focus was more on developing new data-processing methods to extract turbulence information. Newalgorithms for efficiently processing the raw lidar data arestill being developed, as seen in the recent work byMannet al.(2010). Nevertheless, many studies have benefited fromthe continuous developments in the past, where simulationstudies and measurement campaigns have been carried out.Because lidar is not yet an established technology to mea-sure atmospheric turbulence, it is important to compare lidarmeasurements with a reference instrument, as emphasized inthe review article byWilczak et al.(1996). In their review,lidar technology was termed to be a “young adult” in com-parison to sodars and radars. With the recent spurt in the mea-surement campaigns using lidars, we think that it has grownbeyond its status of “young adult”.

Table1 groups the studies that have focused on estimationof turbulence quantities using either simulation or lidar mea-surements. For each turbulence quantity, the total number ofstudies is also given. It is evident that significant effort hasbeen focused on estimation ofε, followed byRij , `ij , 〈v′

r2〉,

D(r), F (k1), andFij (k1).



Table 1.Grouping of the past studies according to the estimated turbulence quantity using a lidar.

No. Quantities Estimated List of references Total

1 Turbulent kinetic energy dissi-pation rate,ε

Kunkel et al. (1980); Gal-Chen et al.(1992); Frehlich et al.(1994); Banakh et al.(1995b, 1996); Frehlich(1997); Banakhet al. (1997); Banakh and Smalikho(1997a, b); Frehlich et al.(1998); Banakh et al.(1999); Drobinski et al.(2000); Frehlichand Cornman(2002); Davies et al.(2004, 2005); Collier et al.(2005); Banakh and Werner(2005); Smalikho et al.(2005);Frehlich et al.(2006); Frehlich and Kelley(2008); Davis et al.(2008); Frehlich et al.(2008); Lothon et al.(2009); Banakh et al.(2010); O’Connor et al.(2010); Chan(2011); Dors et al.(2011);Kristensen et al.(2011, 2012)

29

2 Components of the auto-covariance matrix,Rij

Kunkel et al.(1980); Eberhard et al.(1989); Gal-Chen et al.(1992); Frehlich et al.(1998); Cohn et al.(1998); Davies et al.(2003); Drobinski et al.(2004); Davies et al.(2005); Collieret al.(2005); Banta et al.(2006); Davis et al.(2008); Pichuginaet al.(2008); Wagner et al.(2009); Tucker et al.(2009); Mannet al.(2010); Sathe et al.(2011b); Lang and McKeogh(2011)

17

3 Integral turbulent length scale`ij , outer scale of turbulenceL

Frehlich(1997); Frehlich et al.(1998); Cohn et al.(1998); Ba-nakh et al.(1999); Drobinski et al.(2000); Frehlich and Corn-man(2002); Davies et al.(2004, 2005); Collier et al. (2005);Banakh and Werner(2005); Smalikho et al.(2005); Lothonet al.(2006); Frehlich et al.(2006); Frehlich and Kelley(2008);Frehlich et al.(2008); Lothon et al.(2009)

16

4 Radial velocity variance,〈v′r2〉 Eberhard et al.(1989); Gal-Chen et al.(1992); Frehlich(1997);

Mayor et al. (1997); Frehlich et al.(1998); Drobinski et al.(2000); Davies et al. (2004); Banakh and Werner(2005);Frehlich et al.(2006); Frehlich and Kelley(2008); Frehlich et al.(2008); Branlard et al.(2013)

12

5 Filtered radial velocity spec-trum, F (k1)

Banakh et al.(1997); Mayor et al.(1997); Frehlich et al.(1998);Drobinski et al.(1998); Banakh et al.(1999); Drobinski et al.(2000); Davies et al.(2004); Mann et al.(2009); Sjöholm et al.(2009); Kristensen et al.(2011); Dors et al.(2011); Angelouet al.(2012)

12

6 Filtered radial velocity struc-ture function,D(r) (or D(r1))

Frehlich et al.(1994); Frehlich(1997); Banakh and Smalikho(1997b); Frehlich et al.(1998); Banakh et al.(1999); Frehlichand Cornman(2002); Davies et al.(2004); Frehlich et al.(2008); Banakh et al.(2010); Chan(2011); Kristensen et al.(2011, 2012)

12

7 One-dimensional spectrum ofthe components of the windfield, Fij (k1)

Lawrence et al.(1972); Hardesty et al.(1982); Drobinski et al.(2004); Davies et al.(2005); Lothon et al.(2009); O’Connoret al.(2010); Canadillas et al.(2010); Sathe and Mann(2012)

8

8 Third order moments〈w′3〉 Gal-Chen et al.(1992); Cohn et al.(1998); Lenschow et al.

(2000)3

9 Kinematic heat flux,〈w′θ ′〉 Gal-Chen et al.(1992); Davis et al.(2008) 2

10 Coherence of the componentsof the wind field, cohij (k1)

Lothon et al.(2006); Kristensen et al.(2010) 2



4.2.1 ε, F (k1), D(r)

The greatest advantage of estimation ofε is that we can ex-ploit the universal behavior of isotropy in the inertial sub-range, either in the Fourier domain (using velocity spectrum)or the temporal domain (using structure function) (Pope,2000). Thus, estimation ofε involves estimation of eitherF (k1) or D(r) (we could also use the separation distancer1 instead ofr) in the inertial subrange from the lidar beamthat is oriented in any direction. The associated challengesare then threefold: proving the existence of the inertial sub-range, identifying the inertial subrange from the lidar data,and averaging within the probe volume. FromPope(2000),we understand that in order to have a well-defined iner-tial subrange, we need large Reynolds number flows. Fortu-nately, atmospheric flows are usually characterized by largeReynolds numbers (Wyngaard, 2010), especially during con-vective daytime conditions. Stable atmospheric conditionsthat normally occur during the late night and early morn-ing, can however present challenges since they are associatedwith low Reynolds number turbulence (Wyngaard, 2010). Wecan then assume that inertial subrange is well defined formost of the day, except during late night and early morningconditions.

The challenge associated with identifying the inertial sub-range from lidar measurements is mainly due to the probelength of a lidar. In principle, we need only one measure-ment – of eitherF (k1) or D(r) – in the inertial subrange.However, in order to avoid statistical uncertainty, it is recom-mended that one take multiple measurements, and use themall to fit a model. FromMann et al.(2009), Sjöholm et al.(2009) and Sathe(2012), it is clear that due to the probelength of a lidar, most of the turbulence scales in the inertialrange are filtered out. Modeling the lidar filter function thenbecomes inevitable, which has fortunately been carried outby Smalikho(1995), Banakh et al.(1996), Frehlich(1997),Smalikho et al.(2005), Sjöholm et al.(2009) andMann et al.(2009). In Sjöholm et al.(2009) and Mann et al.(2009),the goal was only to compare the lidar volume-averagedmeasurements of the radial velocity spectrum with referencepoint measurements; an estimation ofε was not carried out.These studies could be extended further to estimateε by us-ing the isotropic or anisotropic form of spectral tensor witha given energy spectrum. Apart from the filtering effect, wealso need to identify the cut-off low wavenumber range whenusingF (k1), and the maximum separation distance when us-ing D(r) in order to identify the inertial subrange.

In summary, there are four ways of estimatingε: widthof the Doppler spectra (Smalikho, 1995; Banakh et al.,1995a, 2010;Smalikho et al., 2005), radial velocity spectrum(Gal-Chen et al., 1992; Banakh et al., 1995b, 1997; Banakhand Smalikho, 1997a; Drobinski et al., 2000; Davies et al.,2005; Davis et al., 2008; Collier et al., 2005; Lothon et al.,2009; O’Connor et al., 2010; Dors et al., 2011; Kristensenet al., 2011), line-of-sight radial velocity structure function

(Frehlich et al., 1994; Frehlich, 1997; Banakh and Smalikho,1997a, b; Frehlich et al., 1998; Banakh et al., 1999; Frehlichand Cornman, 2002; Davies et al., 2004; Banakh and Werner,2005; Smalikho et al., 2005), and radial velocity azimuthalstructure function (Banakh et al., 1996, 1999; Banakh andSmalikho, 1997a; Frehlich et al., 2006, 2008; Frehlich andKelley, 2008; Chan, 2011; Kristensen et al., 2012). Very fewstudies have exploited the Doppler spectral width to estimateε; the reasons could be that for a C-W lidar the applicabil-ity of the Doppler spectral width is limited tol � L, andfor a pulsed lidar it is quite complicated to process the data(Smalikho et al., 2005). Nevertheless, as shown byBanakhet al. (2010), for a pulsed lidar it could be advantageous touse the Doppler spectral width approach, since the randomerrors inε can be reduced at higher turbulence levels thanthey can in the structure function approach, or equally, usingthe radial-velocity-spectrum approach.

4.2.2 〈v′r2〉, ìj , L

Apart fromε, another important parameter that characterizesturbulence is the length scale. The two most commonly useddefinitions of the length scale areìj andL, which have phys-ically different interpretations.L (also called the outer lengthscale of turbulence) is the length scale corresponding to themaximum spectral energy, whereasìj can be interpreted asthe length scale up to which turbulence is correlated. The twoscale lengths can, however, be shown to be related to eachother, as was done byFrehlich and Cornman(2002), Sma-likho et al.(2005) andLothon et al.(2006). Thus,ìj can beestimated from its relationship withL (Frehlich and Corn-man, 2002; Davies et al., 2004; Collier et al., 2005; Sma-likho et al., 2005; Lothon et al., 2006, 2009; Frehlich et al.,2006), or by using the definition given in Eq. (3) (Cohn et al.,1998). Practically, ij is estimated from the values of the au-tocorrelation function at the first zero crossing, butDavieset al.(2005) estimated the same using some properties of theautocorrelation function.L can be estimated using the struc-ture function approach (Frehlich, 1997; Frehlich et al., 1998,2008; Frehlich and Kelley, 2008). Drobinski et al.(2000) fol-lowed a slightly different approach, in which the radial ve-locity spectrum is split into two regions; one is the energy-containing range, and the other contains the inertial subrangeup to the dissipation range. Measurements of the radial ve-locity spectrum can thus be fitted to this model andL, ε es-timated simultaneously. Interestingly,Banakh et al.(1999)andBanakh and Werner(2005) also use the term outer lengthscale for ij , but we believe that it is important to distinguishbetween the two length scales.

Fewer studies have been carried out to estimate〈v′r2〉 than

to estimate toε (see Table1). This is perhaps because infor-mation of all turbulence scales is required to estimate〈v′

r2〉,

and a universal isotropic relation does not suffice. AlthoughEberhard et al.(1989) andGal-Chen et al.(1992) have es-timated〈v′

r2〉 from lidar measurements, no consideration to



probe volume averaging was given, and thus any other tur-bulence statistic derived using these measurements wouldnot contain information on small-scale turbulence. All subse-quent studies have followed the pioneering work ofFrehlich(1997), in which information about small-scale turbulencewas recovered by modeling the filter function. The maincontributions of theFrehlich(1997) method are first, that itpresents a technique to derive expressions of the radial ve-locity structure function (or, equivalently, the radial veloc-ity spectrum) for a lidar pulse with any given shape; sec-ond, it presents a turbulence model, with which we can es-timate〈v′

r2〉 and`ij . One can thus use a non-Gaussian shape

for the pulse and derive a different functional form of thespatial filter (Davies et al., 2004), or use a different turbu-lence model, e.g.von Kármán(1948) isotropic spectral ten-sor model (Frehlich and Cornman, 2002), or a more realisticanisotropicMann (1994) spectral tensor instead of the em-pirical Kaimal et al.(1972) models (Frehlich, 1997; Frehlichet al., 1998). UsingD(r) to estimate〈v′

r2〉 from a pulsed lidar

has the limitation of coarse vertical resolution. An azimuthalstructure function approach can then be used to improve thevertical resolution (Banakh et al., 1996; Frehlich et al., 2006;Frehlich and Kelley, 2008; Kristensen et al., 2012). Withoutusing any turbulence model,Mann et al.(2010) suggesteda technique (only for C-W lidars) to estimate〈v′

r2〉 using the

mean Doppler spectrum. The validity of this technique is suc-cessfully demonstrated inBranlard et al.(2013).

4.2.3 Rij , Fij (k1)

Rij is one of the most important turbulence statistics used inthe wind energy industry, due to the use of〈u′2

〉 in the defini-tion of turbulence intensity (IEC, 2005b). Unfortunately, it isalso one of the most challenging statistics to obtain from thelidar data, partly due to challenges in data processing, andpartly due to economic reasons. If economics is not a ma-jor constraint, then three lidars with beams intersecting atone point will provide spatially filtered turbulence statistics(Mann et al., 2009). With two lidars, we are restricted to es-timating the turbulence statistics of only two components,i.e., horizontal and vertical (Davies et al., 2005; Collier et al.,2005).

Normally, the economics of a project are important and weare then restricted to using only one lidar. In this case, a lidarbeam can be oriented in the direction of the turbulence statis-tic that we are interested in estimating. For example, if weare interested in estimating〈u′2

〉, then ideally the lidar beamshould be pointed horizontally in the mean wind direction atthe height of interest, and for the period within which〈u′2

〉

is obtained (Lawrence et al., 1972). For a ground-based lidarsystem this would be impossible since the beam would onlymeasure wind that is very close to the ground. Alternatively,we could point the lidar beam at a very small elevation angleand assume that the contributions from the vertical velocity

are negligible (Drobinski et al., 2004; Collier et al., 2005;Banta et al., 2006; Pichugina et al., 2008). An open questionthen is, how small the elevation should be so that the verti-cal velocity contributions can be neglected?Drobinski et al.(2004), Banta et al.(2006), andPichugina et al.(2008) ne-glected the vertical velocity contributions up to an elevationangle of 20◦, but provided no justification for the assump-tion of negligible vertical velocity contributions. This methodalso requires that the horizontal homogeneity assumption isvalid over a larger area, particularly if we are interested inmeasuring turbulence statistics at greater heights and/or sev-eral heights. Measurements of〈w′2

〉 can be relatively easierto take, since we only need to point the beam in the verticaldirection (Cohn et al., 1998; Tucker et al., 2009). In principle,following Frehlich(1997) andBanakh and Smalikho(1997b)approach, we can then obtain unfiltered〈w′2

〉 from 〈v′r2〉.

Rij can also be obtained using scanning lidar data, eitherusing RHI scanning (Gal-Chen et al., 1992; Davies et al.,2003; Davis et al., 2008) or VAD scanning (Eberhard et al.,1989; Mann et al., 2010). If, say for a VAD scanning, weuse high-frequencyvr measurements, deduce theu, v, andw components at every measurement time step, and obtain,say,〈u′2

〉 or F11(k1), then apart from the probe volume aver-aging effect, large systematic errors will also be introducedin the measurement of〈u′2

〉 due to the contamination by thediagonal and cross components ofR (Sathe et al., 2011b;Sathe and Mann, 2012). In such cases, one should be verycareful in using theRij measurements obtained from a scan-ning lidar, since removing only the probe volume filteringeffect (Wagner et al., 2009) without giving considerationto cross-contamination, or neglecting the effects of system-atic errors completely (Lang and McKeogh, 2011) will pro-vide erroneous values. Using〈v′

r2〉 instead of high frequency

vr measurements to obtainRij is then essential in order toavoid contamination by the components ofR (Eberhard et al.,1989; Gal-Chen et al., 1992; Mann et al., 2010; Sathe, 2012).The unfiltered〈v′

r2〉 can be obtained using methods suggested

by Frehlich(1997) andMann et al.(2010), and hence unfil-teredRij can also be obtained.

EstimatingFij (k1) from lidar data is even more challeng-ing than estimatingRij , since we need high frequency mea-surements ofvr. For a scanning lidar (e.g., VAD), combininghigh frequency measurements from the lidar beams orientedin different directions results in erroneous measurements ofFij (k1) (Canadillas et al., 2010; Sathe and Mann, 2012).Most studies in the past have thus used either a staring li-dar configuration (Lawrence et al., 1972; Davies et al., 2005;Lothon et al., 2009; O’Connor et al., 2010), or neglected con-tributions from thew component at small elevation angles(Hardesty et al., 1982; Drobinski et al., 2004).



4.2.4 〈w′3〉, 〈w′θ ′〉, cohij (k1)

Very little effort has been focused on the estimation of〈w′3〉,

〈w′θ ′〉 and cohij (k1). One of the reasons could be the com-

plexity of data processing and the associated errors thatpresent great challenges to their estimations. Particularly, anestimation of〈w′θ ′

〉, requires not only an estimation ofε,but also requires estimation of either〈w′3

〉 (Gal-Chen et al.,1992), or 〈w′2

〉 (Davis et al., 2008). Estimating higher ordermoments, particularly third and fourth order, introduce largeerrors in the measurements (Lenschow et al., 1994, 2000).Fortunately, we can reduce the errors in higher moments us-ing the autocorrelation technique (Lenschow et al., 2000) orthe spectral technique (Frehlich et al., 1998), which increasethe potential of estimating the heat flux using Eq. (35).

5 Turbulence quantities of interest for futureapplications in wind energy

Wind turbines have been and will be installed in differ-ent parts of the world where atmospheric conditions differsignificantly from each other. The reason why turbulenceis important for wind energy purposes is already given inSect.1. Here we will specifically discuss those parameterslisted in Table1 that are useful for wind energy. Accord-ing to IEC (2005a) standards, a wind turbine should be de-signed for different classes of turbulence intensities. The tur-bulence intensityI is defined as the ratio of standard devia-tion of theu component to the mean horizontal wind speed

(I =

√〈u′2〉/〈u〉). It is thus crucial to perform measurements

of 〈u′2〉. Apart fromI , it also important to measure the mean

wind speed profile, which is dependent on the velocity co-variances〈u′w′

〉 and〈v′w′〉 (Wyngaard, 2010). The diagonal

components ofR, i.e., 〈u′2〉, 〈v′2

〉 and 〈w′2〉 influence the

wind turbine loads significantly. Thus for wind energy pur-poses, it is very important to measureRij ; however, to dothis using lidars, we see from Eq. (12) that we need mea-surements of〈v′

r2〉. In order to get unfiltered〈v′

r2〉 measure-

ments (as seen in Sect.4) with the state-of-the-art methods,we need to fit lidar measurements ofD(r) and/orF (k1) tosome isotropic or anisotropic turbulence models.

A current practice in the wind energy industry to performload simulations is that a turbulent wind field is generatedusing either theMann(1994) model or an empiricalKaimalet al. (1972) spectrum is combined with some coherencemodel (IEC, 2005a). As discussed in Sect.2, the need tomeasureε andL is then clearly evident. These parametersare normally obtained by fitting theMann (1994) model tothe measurements ofFij (k1), which could be obtained us-ing lidars.L and cohij (k1) are important for estimating theloads and wake meandering (Larsen et al., 2008). The influ-ence of atmospheric stability on wind speed profile and onwind turbine loads is becoming increasingly evident (Sathe

1975 1980 1985 1990 1995 2000 2005 20100

1

2

3

4

5

Year

No.

of S

tudi

es

Fig. 4. Number of studies per year on lidar turbulence measure-ments.

et al., 2011a, 2012). For this reason, measurement of〈w′θ ′〉

is quite important for wind energy. According toLenschowet al. (1994), `ij is useful in estimating the averaging timerequired to keep the random errors below a certain thresholdfor a particular turbulence statistic, and hence is a desirablemeasurement quantity for wind energy purposes.

Recently, lidars are being contemplated to be used forwind turbine control. The concept is such that the lidar iseither placed on a nacelle of a wind turbine (Schlipf et al.,2012), or mounted inside a spinner (Mikkelsen et al., 2012;Simley et al., 2013) in order to detect the incoming wind fieldand carry out a feed-forward control to reduce the structuralloads on a wind turbine. The degree to which such a con-cept can be successfully applied depends on how well thelidars are able to detect the incoming turbulent structures.From Sathe et al.(2012) we understand that different com-ponents of a wind turbine are affected by different scales ofturbulent structures. It is thus important to be able to detectthe range of turbulence scales, up to the order of or less thanthe probe volume length.

6 Summary and discussion

Figure4 summarizes the number of studies that have signif-icantly contributed to the research on turbulence measure-ments using wind lidars from 1972–2012. Research on li-dar turbulence measurements dates back to 1972, but it wasnot until 1997 that the publication rate picked up pace. If weconsider that the lidar turbulence measurement research en-compasses the period 1972–2012, then more than 80 % ofthe research was carried out in the latter half of the 30 yrperiod, i.e., from 1997–2012. In the first 15 yr of develop-ment, barring the works ofSmalikho (1995) and Banakh



et al.(1996), focus was more on extracting turbulence infor-mation without taking into account probe volume averaging(see Sect.4). Since then substantial effort has been put intomodeling the averaging effect inside the lidar probe volume,mainly by Professor V. A. Banakh and Dr. I. N. Smalikhofrom the V. E. Zuev Institute of Atmospheric Optics of Rus-sian Academy of Sciences, Siberian Branch, Russia, and thelate Dr. R. Frehlich from the University of Colorado, USA.Interestingly, these scientists pioneered new processing algo-rithms independently of each other during roughly the sameperiod, i.e. from the mid 1990s until the mid 2000s, whereinthey demonstrated how to extract unfiltered turbulence pa-rameters (Smalikho, 1995; Banakh et al., 1996; Frehlich,1997; Banakh and Smalikho, 1997b; Smalikho et al., 2005;Frehlich et al., 2006). We believe that this development hassignificantly contributed to the number of research studiescarried out in the last 15 yr. Further development in process-ing algorithms will also greatly benefit from their works. Weexpect that the number of such studies will continue to in-crease due to increase in wind-energy development all overthe world.

That brings us to an obvious question:is there anythingnew to be discovered with regards to processing raw lidardata, scanning configurations, or the technology itself thatcan provide more reliable turbulence measurements using li-dars?We attempt to answer this question as follows:

1. Raw lidar data processing – up until now, the process-ing algorithms that have been developed have shownthat by combining an isotropic turbulence model withlidar measurements, we are able to estimateε, 〈v′

r2〉

and L (see Table1). However, turbulence is notisotropic in all range of scales. Anisotropy is partic-ularly observed on longer length scales, and thus itis more desirable to estimate〈v′

r2〉 and L by com-

bining an anisotropic turbulence model (Kristensenet al., 1989; Mann, 1994) with the lidar measurements.This recommendation was also made byFrehlich et al.(2006) andFrehlich and Kelley(2008). There is, how-ever, a need for developing algorithms that make as lit-tle use of models as possible in combination with themeasurements. Even an anisotropic turbulence modelsuch as that developed byMann (1994) is based ona set of assumptions, e.g. neutral atmospheric con-ditions, applicability in the surface layer, validity ofTaylor’s hypothesis, and it does not apply to com-plex terrain. If we then combine such a model withlidar measurements, and estimate turbulence parame-ters, then additional uncertainties may be introduced.In order to avoid such situations, further developmentof algorithms should also focus on making use of onlyraw lidar data to extract turbulence parameters, e.g., asshown inMann et al.(2010).

Furthermore, from the study bySathe et al.(2011b)and Sathe and Mann(2012), it is now clear that in

Fig. 5.Three lidars intersecting at a point.

a VAD scanning configuration, obtainingu, v andw

components fromvr data at each time step and thendeducing turbulence statistics will introduce large sys-tematic errors in the turbulence measurements. Onemight argue that under some conditions the lidar-to-sonic correlation may be close to 1, but, as shownin Sathe et al.(2011b), the correlation would dependon the type of lidar (C-W or pulsed), and the turbu-lence structure. The correlations with any referenceinstrument may not be repeatable if the experimentis conducted during different times of the day. It isthus recommended that such a data-processing methodshould not be followed. Alternatively, as shown byWilson (1970), Kropfli (1986), Eberhard et al.(1989)andMann et al.(2010), using〈v′

r2〉 instead ofvr to de-

duceRij is a fundamentally correct method to extractturbulence information from the raw lidar data.

2. Scanning configurations – three measurement config-urations have been used until now: staring, VAD, andRHI scanning (see Sect.3 for details of the scan-ning configurations). Ideally, using three staring lidarswith their beams crossing at a point (similar to sonicanemometer beams) would provide more reliable mea-surements as compared to using a single lidar in a VADor RHI scanning mode. Figure5 illustrates this con-cept. The necessity of making assumptions of horizon-tal homogeneity is reduced significantly, and poten-tially it can provide measurements in complex terrainwith increased reliability. The only challenge then isto tackle the probe volume averaging effect in the in-dividual beam direction. A step in this direction is theuse of two lidars (Davies et al., 2004, 2005; Collieret al., 2005). In VAD or RHI scanning, the assump-tion of horizontal homogeneity is inevitable, and thusrestricts the applicability of VAD and RHI scanningfor wind energy purposes, particularly in complex ter-rain. Bingöl et al.(2009) provided a technique to cor-rect for terrain-induced inhomogeneities in the meanflow. For turbulence measurements we can use Taylorseries expansion in the horizontal plane as shown by



Frisch(1991). The reliability of this algorithm has notbeen investigated thoroughly, but could form the basisof a potential study for the future.

On the other hand, using three lidars can increase thecost of a project significantly, a fact which hindersthe use of this method in wind energy development.If we use a single staring lidar at one height, then itbecomes almost unavoidable to combine a turbulencemodel with lidar measurements in order to extract tur-bulence information (Smalikho, 1995). Another aspectto be considered when deciding on the scanning con-figuration is the sampling frequency. In this regard,the staring configuration will again provide faster mea-surements than the scanning configurations. In a VADscanning, there is scope for reducing the number ofmeasurement points on a scanning circle. As shown byEq. (12) we need only six beams at different azimuthand half-opening angles to estimateRij . However, us-ing random six points will introduce random errors inthe turbulence measurements.Sathe(2012) providedsome initial calculations of the optimum configurationto minimize random errors. The sampling frequencywould thus be increased, but further investigations arerequired before this configuration can be put into use.

3. Improvement in lidar technology – new, cheaper solid-state lasers for coherent detection lidars with inte-grated optical amplification are being developed andtested (Hansen and Pedersen, 2008; Rodrigo and Ped-ersen, 2008). These may greatly expand the use of li-dars for wind measurements, but they are not specif-ically tailored for turbulence measurements. Prelimi-nary tests of these lidars have been carried out byRo-drigo and Pedersen(2012), and show good compar-ison with a sonic anemometer. The solid-state laserswith integrated amplification may in the near futurecompete with the more expensive lasers used in C-W Doppler lidars. Direct detection is still on an ex-perimental level (McKay, 1998) and has only beenused in the atmosphere sporadically (Xia et al., 2007;Dors et al., 2011). The simple design of these instru-ments may eventually lead to cheaper lidar systems.Non-coherent detection may also provide possible newways to estimate atmospheric turbulence (Mayor et al.,2012; Sela and Tsadka, 2011), but to our knowledge itdoes not, so far, challenge the capabilities of the coher-ent Doppler lidars.

Completely new principles could also drastically im-prove the turbulence-measuring capabilities of lidars.One suggestion is to exploit the translation of thespeckle pattern in the image plane of the lidar tele-scope. In this way, not only the line-of-sight velocitycould be estimated, but also the two transverse veloc-ity components. All components would be measured

in the same volume, reducing the problem of cross-contamination. In the laboratory, this method has beensuccessfully tested on translating hard targets (Iversenet al., 2011; Jakobsen et al., 2011), but it is muchharder to get the method to work with backscatterfrom atmospheric aerosols. Firstly, the return from theaerosols is much weaker and, secondly, the turbulencemay reduce the correlation time of the speckle pattern,which could adversely affect the transverse velocitydetermination.

In order to meet the objectives stated above, a simulationstudy can significantly help in better planning and design-ing the experiments (Frehlich and Cornman, 2002; Banakhand Werner, 2005). A possibly important aspect of turbu-lence measurements using lidars that we have not consideredin this review is the instrumental error, which is generally as-sumed to be uncorrelated (Frehlich et al., 1998; Lenschowet al., 2000). Fortunately for modern commercial lidar sys-tems, the magnitude of the instrumental error is not signifi-cant, and can be safely neglected (Mann et al., 2009). How-ever, for those lidar instruments that have significant instru-mental error and therefore could potentially bias turbulencemeasurements, the techniques suggested byFrehlich et al.(1998), Drobinski et al.(2000) andLenschow et al.(2000)can be used to perform corrections.

Table A1. Nomenclature.

C ≈ 1.5 universal Kolmogorov constantC1 ≈ 0.5 Kolmogorov constant related toF11(k1)

Dij (r) velocity structure functionFij (k1) one-dimensional velocity spectrumI longitudinal turbulence intensityIn Wilson (1970) integrals (n = 1, .,4) to esti-

mate the covariancesLp range gate length (cτ/2)Rij (r) cross covariance functionR = R(0) covariance matrixk wave vector in the Fourier domainn unit directional vectorr separation vector in three dimensionsx position vector in three dimensionscohij (k1) coherence function〈S(vr)〉 mean Doppler spectra〈u〉 mean wind speed〈w′θ ′

〉 sensible heat flux

〈u′2〉 variance of theu component

〈u′v′〉 covariance between theu andv components

〈u′w′〉 covariance between theu andw components

〈v′r2〉 radial velocity variance

〈v′2〉 variance of thev component

〈v′w′〉 covariance between thev andw components

〈w′2〉 variance of thew component

〈w′3〉 third moment of the vertical velocity



Table A1. Nomenclature.

v wind field vectorD(δ) filtered radial velocity structure function for

a separation distancedf δ

D(r) filtered radial velocity structure function fora separation distancer

D(r1) filtered radial velocity structure function fora separation distancer1

F (k1) filtered radial velocity spectrumR(r) filtered covariance function of the radial ve-

locity for a separation distanceran,bn,amn,bmn Fourier coefficientsc speed of lightdf focus distance for a C-W lidar and center of

the range gate for a pulsed lidari, j indices that take values 1, .,3 and denote the

component of the wind fieldk1,k2,k3 components of the wave vector along the

x1,x2,x3 axes respectivelyl Rayleigh lengthr separation distance along the lidar beamr1, r2, r3 separation distances along thex1,x2,x3 axes

respectivelyrb lidar beam radiusu longitudinal component of the wind field in

thex1 directionv transversal component of the wind field in

thex2 directionvr radial velocityw vertical component of the wind field in the

x3 directionwp pulse widthx1,x2,x3 axes defining the right handed cartesian co-

ordinate systemz height above the groundL outer length scale of turbulence8ij (k) three-dimensional spectral velocity tensor2 angle between the lidar beam and the mean

windα elevation angleχij (k1, r2, r3) cross spectra at separation distancesr2 and

r3δ angle subtended by two lidar beams in a

VAD scanning mode`ij integral length scale〈σ2

s 〉 second central moment of the Doppler spec-trum (Doppler spectrum width)

∂/∂z vertical gradientφ half-opening angleρ surface air densityτ pulse durationθ azimuth angleθT surface potential temperatureε energy dissipation rateFWHM full width half maximumPPI plan position indicatorRHI range height indicatorVAD velocity azimuth displayC-W continuous-wave

Acknowledgements.This work is carried out as a part of a researchproject funded by the Danish Ministry of Science, Innovation andHigher Education – Technology and Production, grant no. 11-117018. The resources provided by the Center for ComputationalWind Turbine Aerodynamics and Atmospheric Turbulence fundedby the Danish Council for Strategic Research grant no. 09-067216are also acknowledged. We also thank Erik Juul from the DTUWind Energy department for helping with the sketches of thescanning configurations.

Edited by: G. Ehret

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