Evaluation of a Procedure to Correct Spatial Averaging in TurbulenceStatistics from a Doppler Lidar by Comparing Time Series

with an Ultrasonic Anemometer

PETER BRUGGER,a KATJA TRÄUMNER, AND CHRISTINA JUNG

Karlsruhe Institute of Technology, Karlsruhe, Germany

(Manuscript received 3 July 2015, in final form 15 July 2016)

ABSTRACT

Doppler lidars are frequently used for wind measurements in the atmospheric boundary layer, but their

data are subject to spatial averaging due to the pulse length of the laser and sampling frequency of the return

signal. This spatial averaging also affects estimates of turbulence statistics like the velocity variance and outer

scale of turbulence fromDoppler lidar data. In this study a procedure from Frehlich and Cornman based on a

von Kármán turbulence model was systematically applied to correct these effects of spatial averaging on

turbulence statistics. The model was able to reduce the occurring bias of the velocity variance and outer scale

of turbulence in a comparison of time series from a Doppler lidar and an ultrasonic anemometer. The

measurements show that the bias of the velocity variance was reduced by 29% and that of the outer scale of

turbulence by 43%. But both turbulence parameters had a remaining systematic error that could not be

explained by the von Kármán model of the structure function.

1. Introduction

The atmospheric boundary layer is a vital element

within the interaction of the free atmosphere and the

earth’s surface (Stull 1988). Turbulence is a major

characteristic of the atmospheric boundary layer, and it

dominates the transport processes between the surface

and the free atmosphere (Kaimal and Finnigan 1994).

Descriptions of the turbulent flow on the basis of tur-

bulence parameters, such as velocity variance, dissipa-

tion rate of the turbulent kinetic energy, and length

scales, are widely applied in meteorology (Stull 1988).

Accurate measurements of turbulence parameters are

therefore desirable (O’Connor et al. 2010).

Pulsed Doppler lidars with heterodyne detection are

used for wind measurements in the atmospheric

boundary layer (Werner 2005). They emit a laser pulse

that is scattered and Doppler shifted by aerosols in the

atmosphere. The backscattered radiation is mixedwith a

local oscillator and the beat note signal is detected

(Grund et al. 2001). The frequency shift information of

each pulse is extracted and accumulated over multiple

pulses, and the mean Doppler shift is estimated to im-

prove precision (Rye and Hardesty 1993). The Doppler

shift yields the line-of-sight (radial) component of the

wind velocity. For a fixed-beam orientation, the spatial

resolution and measurement frequency of Doppler li-

dars allow for turbulence measurements of the ra-

dial velocity within the inertial subrange (Mann et al.

2008) and it is possible to derive turbulence parameters

from these measurements (Frehlich and Cornman 2002;

Lothon et al. 2006). Their limitation to measure only the

radial velocity makes the use of scanning techniques for

three-dimensional turbulence information for a single-

Doppler lidar necessary. Turbulence measurements

with these scanning techniques have limited precision

(Sathe et al. 2011, 2015). However, even with a fixed-

beam orientation the measurements of a Doppler lidar

entail significant spatial averaging, which affects the

estimation of turbulence parameters (Frehlich 1997).

The spatial averaging is caused by two processes.

First, the emitted laser pulse has a certain length and at a

given time the detector receives backscattered radiation

a Current affiliation: Institute of Meteorology and Climate

Research—Atmospheric Environmental Research (IMK-IFU),

KIT, Garmisch-Partenkirchen, Germany.

Corresponding author address: Peter Brugger, Institute of Me-

teorology and Climate Research—Troposphere Research (IMK-

TRO), KIT, Postfach 3640, 76021 Karlsruhe, Germany.

E-mail: peter.brugger@kit.edu

OCTOBER 2016 BRUGGER ET AL . 2135

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from all over the pulse volume (Frehlich et al. 1998).

Second, the Doppler shift of each pulse is extracted

from a backscatter time series that originates from an

elongated region due to the propagation of the laser

pulse through the atmosphere during the sampling of the

time series (Frehlich et al. 1998). The length of this re-

gion is the range gate length, given by

Dp5SPGc

2fS

, (1)

where c is the speed of light, SPG is the number of

samples per gate, and fS is the sampling frequency of the

backscattered radiation (Frehlich et al. 1998). Therefore,

measurements of the radial velocity from a Doppler lidar

are not point measurements, but a spatial average over

the pulse length and range gate length.

Spatial averaging has a turbulence-reducing effect on

velocity statistics from Doppler lidar measurements

(Frehlich 1997). This effect can be illustrated with an

example of white circles on black background (Fig. 1).

The high-resolution picture provides sharp differences

between black and white, and the size of the circles is

accurately visible. In the spatially averaged picture, the

small circles have become indistinguishable from the

background and large circles are blurred. This has re-

duced the black-and-white variance of the averaged

picture. Also, the average circle size is increased, be-

cause the smoothing of large circles spreads white into

areas that were black before and the small circles have

vanished. The spatial averaging of radial velocity mea-

surements from Doppler lidars shows a similar effect on

turbulence statistics. The solid line in Fig. 1 can be re-

garded as consecutive overlapping range gates along a

Doppler lidar’s laser beam or a time series from one

range gate assuming Taylor’s hypothesis of frozen tur-

bulence (Taylor 1938). The radial velocity variance gets

reduced as areas with high and low velocities are aver-

aged and the length scales of eddies (velocity patches

differing from the mean wind) are increased.

For this reason, a correction is needed for turbulence

parameters from Doppler lidar measurements. A cor-

rection is also necessary to make turbulence parameters

from Doppler lidars comparable to in situ point mea-

surements, as recent meteorological field campaigns

feature combinations of towers, eddy covariance sta-

tions, and Doppler lidars (Behrendt et al. 2011; Eder

et al. 2015; Klein et al. 2015). Frehlich et al. (1998) and

Frehlich and Cornman (2002) developed a procedure to

correct the effect of spatial averaging on estimates of

turbulence parameters from Doppler lidar measure-

ments using a von Kármán turbulence model.

The method from Frehlich and Cornman (2002) was

applied by Davies et al. (2004), Frehlich et al. (2006),

and Lothon et al. (2006) to calculate turbulent statistics

frommeteorological field experiments. A comparison to

spatially separated in situmeasurements from a tethered

lifting system was done by Frehlich et al. (2006) for

mean profiles of the boundary layer. Here, an evaluation

of the correction procedure from Frehlich and Cornman

(2002) is presented with an ultrasonic anemometer

measuring at the center of a Doppler lidar’s range gate,

which was not done in other studies. Turbulence pa-

rameters obtained by fitting a von Kármán model to

measurements of the structure function were compared

for both instruments. In the case of the Doppler lidar,

the von Kármán model has the modifications from

Frehlich and Cornman (2002) to account for spatial

FIG. 1. Effect of spatial averaging on white circles on black background. (left) The true field

and (right) the spatially averaged field. The averaging area was 4% of the domain size. In this

example the black–white variance along the solid line was reduced by 20% and the integral

length scale was increased by 18%.

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averaging. The paper is organized as follows: Section 2

describes the measurements on which the comparison is

based. Section 3 details the estimation of turbulence pa-

rameters. Section 4 presents the results of the evaluation.

2. Instruments and measurements

Doppler lidar measurements were compared with

ultrasonic anemometer measurements to evaluate the

procedure from Frehlich and Cornman (2002). The

measurements took place at the Karlsruhe Institute of

Technology (KIT) Campus North (CN) research facility

in Karlsruhe, Germany, from 1130 UTC 20 February

2013 until 0745 UTC 27 February 2013. The ultrasonic

anemometer wasmounted at a height of 100m above the

ground on a 200-m tower (49.09148N, 8.42488E). Thesurrounding area is flat with surface height variations of

less than 20m and is forested to the north, west, and

south; the height of the forest is approximately 35m.

East and northeast of the tower is the industrial area of

the research facility. The Doppler lidar was deployed

1.45 km northeast of the tower at 49.10388N, 8.43048E,and the laser was pointed toward the ultrasonic ane-

mometer with an elevation of el5 4:168 and an azimuth

of az5 196:68. The lidar beamwas pointed slightly to the

right of the ultrasonic anemometer using hard target

detection.

The Doppler lidar used was a WindTracer system

from Lockheed Martin Coherent Technology (year of

construction: 2004). It has a pulsed Tm:LuAG laser

with a wavelength of 2023nm, a spatial full width at half

maximumpulse length ofDr5 55:5m and a pulse energy

of 2.0mJ. The return signal was sampled at a frequency

of fS 5 250MHz, and the number of data samples used

for an estimate of one radial velocity was changed dur-

ing the measurements. The corresponding range gate

lengths ranged from Dp5 10:2m to Dp5 153:6m (see

Table 1). The pulse repetition frequency was 500Hz and

50 pulses were accumulated for estimates of the radial

velocity, resulting in an effective measurement fre-

quency of 10Hz.

The ultrasonic anemometer was an R2 from Gill In-

struments (serial number 0204R2). It measured the

three-dimensional wind vector with a frequency of 20Hz

and has an accuracy of 3% (Gill Instruments Ltd. 1995).

The weather conditions during the measurements were

within the recommended operating range. The ultra-

sonic anemometer is part of long-term turbulence

measurements at the tower and was mounted in 1995

(details described in Kalthoff and Vogel 1992).

3. Estimating velocity variances and outer scales

The velocity variance (s2y) is a measure of turbulence

intensity or turbulence kinetic energy (Stull 1988). The

outer scale of turbulence (L0) is the length scale from

which the turbulence cascade within the inertial subrange

draws its energy from the mean current (Kristensen and

Lenschow 1987). One way to estimate s2y and L0 is based

on the velocity structure function

Dy(s

0, s)5 h[y0(s

0)2 y0(s

01 s)]2i , (2)

where angle brackets indicate an ensemble average, y0 isthe turbulent velocity fluctuation of an arbitrary velocity

component y, s0 is a point in space, and s is a spatial

separation (Frehlich et al. 1998).

The longitudinal and transverse velocity structure

function can be described with a model introduced by

von Kármán (1948) for homogeneous and isotropic

turbulence, which includes only s2y andL0 as parameters

(Frehlich et al. 1998). The structure function is called

longitudinal if the direction of the s and y are parallel,

and transverse if they are perpendicular. The following

equations describe either the longitudinal or transverse

components of the structure function and a scalar sep-

aration s is used for them instead of s. Also, arguments

of a function are separated from model parameters by a

semicolon. The von Kármán model is then given by

D(s;s2y ,L0

)5 2s2yL

�s

L0

�(3)

TABLE 1. Doppler lidar settings during the measurements and mean wind conditions: Number of samples per gate, length of the range

gate [calculated from Eq. (1)], period of measurement, mean SNR of the Doppler lidar with standard deviation, mean horizontal wind

speed with standard deviation, and mean wind direction from the ultrasonic anemometer for each period.

SPG Dp (m) Start time End time SNR (dB) yH (ms21) Dir (8)

17 10,2 1252 UTC 21 Feb 2013 1428 UTC 22 Feb 2013 5.93 6 2.55 6.21 6 1.79 302.26

32 19,2 1545 UTC 26 Feb 2013 0745 UTC 27 Feb 2013 11.23 6 4.40 4.36 6 1.58 322.66

50 30,0 1428 UTC 22 Feb 2013 1457 UTC 23 Feb 2013 13.26 6 2.88 4.91 6 1.77 330.13

100 60,0 1127 UTC 20 Feb 2013 1250 UTC 21 Feb 2013 7.18 6 2.58 5.63 6 1.71 303.68

150 90,0 1458 UTC 23 Feb 2013 1259 UTC 24 Feb 2013 15.80 6 2.28 2.69 6 1.52 10.30

200 120,0 1259 UTC 24 Feb 2013 2011 UTC 25 Feb 2013 16.49 6 3.48 2.88 6 1.34 130.60

256 153,6 2011 UTC 25 Feb 2013 1544 UTC 26 Feb 2013 16.67 6 1.99 3.55 6 1.24 304.18

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with

L(x)5 1222/3

G(1/3)(x)1/3K

1/3(x) (4)

for the longitudinal structure function and

L(x)5 1222/3

G(1/3)

�s

L0

�1/3

��K

1/3

�s

L0

�2

s

2L0

K2/3

�s

L0

��

(5)

for the transverse structure function, where the gamma

function G is given by

G(x)5

ð‘0

t x21e2t dt for Re(x). 0 (6)

and the Bessel function Kp of the order p is given by

Kp(x)5

p[I2p(x)2 I

p(x)]

2 sin(pp)for x 2 (0,‘) (7)

with

Ip(x)5 �

‘

k50

1

k! � G(p1 k1 1)

�x2

�2k1p

(8)

(Bronstein et al. 2007). The von Kármán model is ac-

cording to Hinze (1975) an empirical interpolation for-

mula covering the range from the largest eddies to the

inertial subrange. It is independent of stratification or

the driving mechanism of turbulence. It shows good

agreement with observations from grid turbulence in

wind tunnels (von Kármán 1948) and has been used for

atmospheric turbulence in the boundary layer under a

wide range of conditions (Kristensen and Lenschow

1987; Shiau 2000; Frehlich et al. 2006). Values for s2y and

L0 can be estimated by fitting the von Kármán model to

the structure function of the measurements.

However, Doppler lidar measurements are not point

measurements, but an average over a slim cylinder ori-

ented along the beam direction (Frehlich et al. 1998).

The effect of spatial averaging can be seen as a lower

spectral kinetic energy density within the inertial sub-

range in the spectra of the turbulent kinetic energy

density (Fig. 2). It was shown by Stawiarski et al. (2013)

that spatial averaging due to pulse length and range gate

length can lead to errors on the order of 0.3–0.4m s–1 in

the standard deviation of the radial velocity for dual-

Doppler lidars. Furthermore, this will lead to an un-

derestimation of s2y and an overestimation of L0

(Frehlich et al. 1998). Therefore, Frehlich et al. (1998)

included the effects of spatial averaging in the von

Kármán model to correct estimations of s2y and L0.

a. Spatially averaged model of the velocity structurefunction

The effect of averaging due to the laser pulse and the

range gate can be predicted, because its effect on the

structure function is mathematically well understood.

Frehlich et al. (1998) averaged the von Kármán model of

the structure function [Eq. (3)] to include the spatial aver-

aging of the Doppler lidar, which is described by a convo-

lution of the true wind field with a weighting function

defined by the pulse length and the range gate length. As-

suming a linear average over the range gate length Dp and

aGaussianweighted average for the laser pulse with the full

width at halfmaximumDr, the averagedvonKármánmodel

of the structure function is given by (Frehlich et al. 1998)

Dwgt

(s;sy,L

0,Dp,Dr)5 2s2

yG

�s

Dp,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln(2)

p Dp

Dr,Dp

L0

�

(9)

with

G(y,m, x)5

ð‘2‘

F(x,m)[L(xjy2 xj)2L(xjxj)] dx, (10)

where L is given by Eq. (4) for a longitudinal structure

function or by Eq. (5) for a transverse structure function,

and

FIG. 2. Spectra of the turbulent kinetic energy density Sry of the

radial velocity against the frequency y from a Doppler lidar and an

ultrasonic anemometer. The spectra were smoothed with an av-

erage over 30 logarithmically spaced frequency intervals. The

lower kinetic energy of the Doppler lidar around 0.1Hz shows

a loss of turbulence compared to the ultrasonic anemometer due to

spatial averaging. The leveling of the Doppler lidar spectra at

higher frequencies is caused by noise. The spectra are taken from

the interval between 1527 and 1557 UTCwith 100 samples per gate

(see section 2 for details). The dashed line represents the25/3 law

from Kolmogorov (1941).

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F(x,m)51

2ffiffiffiffip

pmfexp[2m2(x1 1)2]1 exp[2m2(x2 1)2]g1 x

2ferf[m(x1 1)]1 erf[m(x2 1)]2 2erf(mx)g

21ffiffiffiffip

pmexp(2m2x2)1

erf[m(x1 1)]

22

erf[m(x2 1)]

2. (11)

The parametersDr andDp are dependent on theDoppler

lidar specifications. Fitting the averaged von Kármánmodel to Doppler lidar measurements of the structure

function produces estimates fors2y andL0 with correction

of spatial averaging.

b. Calculating structure functions from Doppler lidarmeasurements for comparison with in situmeasurements

The models from Eqs. (3) and (9) are based on a

longitudinal or transverse structure function. For the

ultrasonic anemometer, the longitudinal structure func-

tion can be calculated for the velocity component in the

mean wind direction using Taylor’s frozen turbulence

hypothesis (Taylor 1938). For a Doppler lidar with a

fixed beam, a longitudinal structure function can be

applied for the radial velocity along the lidar beam us-

ing range gates at different distances from the lidar for

sets of separations (Frehlich and Cornman 2002). The

structure function of the Doppler lidar spans a large

vertical distance due to the elevation angle of the laser

beam. A comparison of these two structure functions is

therefore not feasible, because the boundary layer tur-

bulence cannot be assumed to be homogeneous in the

vertical direction.

To achieve comparability, only the lidar range gate

closest to the ultrasonic anemometer was considered

and the structure function of the radial velocity was

computed for both instruments from the respective time

series using Taylor’s hypothesis. For the ultrasonic an-

emometer, the radial velocity was obtained by projec-

ting the wind vector onto the radial coordinate using

yua5 u

1cos(el) sin(az)1 u

2cos(el) cos(az)2 u

3sin(el) .

(12)

The center of the lidar range gate closest to the ultra-

sonic anemometer was less than 10m away and 5.9m

above the ultrasonic anemometer. The same velocity

component at the same height was compared, and dif-

ferences should therefore occur only from spatial aver-

aging and measurement errors with this approach. The

drawback is that the structure function is longitudinal

only if the mean wind is parallel to the laser beam. Er-

rors of s2y and L0 due to the nonlongitudinal structure

function are assumed to affect only absolute values, but

not the relative error between the ultrasonic anemometer

and the Doppler lidar. Experimental reasons for this as-

sumption are given in section 4.

The time series of theDoppler lidar was filtered with a

signal-to-noise ratio (SNR) threshold of 0 dB; that is,

data with an SNRbelow this threshold were discarded to

remove low-quality data points of the radial velocity.

The time series of the radial velocity from the ultrasonic

anemometer and the Doppler lidar were split into

30-min intervals. If more than 25% of the Doppler lidar

data within an interval was filtered due to too low SNR,

then the interval was discarded. For the remaining

intervals, a linear trend was removed from each interval

to obtain y0.The spatial separations of the structure function were

calculated for each interval in the mean wind direction

using Taylor’s hypothesis with Ds5 yDt, where y is the

horizontal mean wind speed from the ultrasonic ane-

mometer and Dt is the time between two samples of the

radial velocity. The structure function is calculated using

(Frehlich 1997)

D̂(kDs)5 2[B̂(0)2 B̂(kyDt)] , (13)

where the autocovariance function B̂ is given by

B̂(kyDt)51

NT2 k

�NT2k

l51

y0(lDt)y0[(l1 k)Dt] (14)

with NT as the number of data points in the interval. In

the case of the Doppler lidar, the autocovariance has a

bias s2e due to variance from uncorrelated noise

(Frehlich and Cornman 2002). This error was estimated

with the method described by Frehlich and Cornman

(2002) from the autocovariance with

s2e 5 B̂(0)2 [2B̂(Ds)2 B̂(2Ds)] (15)

and subtracted from B̂(0). In summary, the measured

structure function D̂ of the radial velocity with separations

in the mean wind direction for both instruments is fitted

with the longitudinal model given by Eq. (3) for the ul-

trasonic anemometer and by Eq. (9) for the Doppler lidar.

c. Model fit

The model used for the structure function of the

Doppler lidar is Eq. (9), and the model used for the ul-

trasonic anemometer is Eq. (3). They were fitted to the

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measured structure function of the radial velocity for

each 30-min interval [Eq. (13)] from the measurements

described in section 2. The range gate settings were

considered accordingly to Table 1. A nonlinear least

squares fit with a Levenberg–Marquardt algorithm was

used to fit themodel (Levenberg 1944;Marquardt 1963).

Initial values were set to s2y 5 1m2 s22 and L0 5 100m.

Separations between smin 5 10m and smax 5 1500m

were used to fit the model. This separation domain is

large enough to allow good estimates for s2y andL0 while

providing enough data points for the largest separations

of the structure function.

It is possible that the fit of the model does not con-

verge. A practical solution for this problem is limiting

the parameter space of the outer scale to L0 5 [0, 2000]

and s2y 5 [0, 10] with a sufficiently high number of iter-

ations for the fit and dismissing values close to the pa-

rameter space borders.

4. Comparison of s2y andL0 fromDoppler lidar and

ultrasonic anemometer

First, the 100-sample-per-gate subset of the data was

examined, because it is the recommended setting for the

Doppler lidar. The weather during this observation pe-

riod was influenced by a low pressure trough over

western Europe. The northeasterly flow on synoptic

scales was locally diverted to a northwesterly flow by

topographical effects of the Rhine Valley. The mean

wind direction ranged from 41.68 to 91.48 relative to the

lidar beam. During the measurements no precipitation

was detected and the sky was overcast. Tempera-

tures were between 3.88 and 21.28C. The stratification

was statically stable during nighttime and unstable

around noon.

The dataset of the radial velocity consists of 51 in-

tervals of 30min each for the ultrasonic anemometer

and the Doppler lidar. Three intervals were discarded

due to too low SNR. For the remaining intervals, the

structure function was computed and the values of s2y

and L0 were derived as described in section 3. A com-

parison between ultrasonic anemometer and corrected

Doppler lidar estimates for s2y is shown in Fig. 3 and for

L0 in Fig. 4.

Both parameters show outliers with considerably

larger 95% confidence bounds, shown as error bars in

the figures. Those data points are encircled in the figures

and were excluded from the linear regression and the

correlation coefficient. These data points show no sys-

tematic relation to the environmental conditions and

occur for different time intervals for s2y and L0.

Uncorrected values for s2y and L0 were obtained by

fitting the nonaveraged von Kármán model [Eq. (3)] to

the structure function of the Doppler lidar. The cor-

rection did reduce the systematic underestimation of the

variance in the slope of a linear regression from 0.52 to

0.64 by 25%. In the case of the outer scale, the over-

estimation was reduced from 1.93 to 1.28 by 71%. The

correlation between theDoppler lidar and the ultrasonic

anemometer is higher for s2y than forL0. The bias due to

spatial averaging between the Doppler lidar and the

FIG. 3. Radial velocity variances s2y from the Doppler lidar de-

rived with Eq. (9) vs s2y from the ultrasonic anemometer derived

with Eq. (3). The solid line is the identity, and the dashed line is

a linear regression of the plotted data excluding encircled data

points. Values of r and a of the linear regression with 95% confi-

dence bounds are shown in the text box. The dashed–dotted line is

the linear regression of the uncorrected Doppler lidar velocity

variances using model Eq. (3) for the Doppler lidar as well.

FIG. 4. Outer scale of turbulence L0 from the Doppler lidar

derived with Eq. (9) vsL0 from the ultrasonic anemometer derived

with Eq. (3). The plotted lines and numbers in the box are the same

quantities as in Fig. 3.

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ultrasonic anemometer depends on the intensity of the

turbulence and increases for stronger turbulence. This is

explained by the increase of small-scale patterns in the

wind field for more turbulent conditions. In contrast, the

percentage of correction is independent of the un-

derlying conditions. This is consistent with the expec-

tation, as the theory of the correction procedure makes

no distinction between shear and convective-driven

turbulence, and environmental parameters like mean

wind and stratification are not parameters of the model.

For s2y and L0, the correction procedure did not

eliminate the systematic error completely. The reasons

might be (i) different measurement frequencies,

(ii) beamwidth, (iii) influences of the tower, (iv) non-

longitudinal structure function, and (v) noncorrect Dpand Dr. Those effects will be regarded in detail. The

effect of different measurement frequencies of lidar and

ultrasonic was checked by averaging the ultrasonic an-

emometer to the sampling frequency of the Doppler li-

dar and was found to be negligible. Averaging due to the

beamwidth is considered to be negligible according to

Frehlich et al. (1998). Effects due to the influence of the

tower itself on the wind field were checked by compar-

ing the results for different wind directions, where the

ultrasonic anemometer and Doppler lidar range gate

were upwind or downwind of the tower, but a systematic

error remained for both cases. To investigate a con-

nection of the remaining systematic error to the non-

longitudinal structure function, the same procedures as

for Figs. 3 and 4 were carried out using only intervals

where the mean wind direction was in the direction of

the lidar beam direction or perpendicular to it. The

longitudinal model of the structure function was fitted

for intervals, where the wind direction was in the beam

direction and the transverse model was fitted for in-

tervals with a perpendicular wind direction. The whole

dataset from section 2 was used to get a sufficiently large

number of intervals for a correlation and linear re-

gression. In addition a 68 deviation of the mean wind

direction was allowed from the longitudinal or trans-

verse direction. The results are shown in Figs. 5 and 6

and have similar values for the correlation coefficients

and linear regressions as the ones for all mean wind di-

rections with a longitudinal model (Figs. 3 and 4). There-

fore, it was concluded that the remaining systematic error

was not caused by the nonlongitudinal structure function.

The source of the remaining systematic error is unknown

to us.

It was shown that the correction procedure only re-

duces the bias between the Doppler lidar and the ul-

trasonic anemometer, and the remaining bias could not

be explained with the experimental setup or further

averaging processes. For this reason it was investigated

next whether the remaining systematic error can be re-

duced by increasing Dr or Dp, both of which are pa-

rameters of themodel. The idea is that the parameters in

Eq. (9) had to be replaced by an effective pulse length

and range gate length. Therefore, the steps to calculate

the values of Figs. 3 and 4 were repeated for values

ranging from 10% to 200%of the pulse length and range

gate length. The same 48 intervals as for the comparison

from Fig. 3 were used for s2y for every combination of Dr

FIG. 5. Radial velocity variances s2y from the Doppler lidar de-

rived with Eq. (9) vs s2y from the ultrasonic anemometer derived

with Eq. (3) using only intervals where themeanwind directionwas

in beam direction with the longitudinal model or perpendicular to

it with the transverse model of the structure function. The plotted

lines and numbers in the box are the same quantities as in Fig. 3.

FIG. 6. Outer scale of turbulence L0 from the Doppler lidar

derived with Eq. (9) vsL0 from the ultrasonic anemometer derived

with Eq. (3) using only intervals where themeanwind directionwas

in beam direction with the longitudinal model or perpendicular to

it with the transverse model of the structure function. The plotted

lines and numbers in the box are the same quantities as in Fig. 3.

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and Dp, and for L0 the intervals from Fig. 4. The cor-

relation coefficient r and the slope a of a linear re-

gression were computed for each member of this

ensemble and are shown in Figs. 7–10 as a function of

pulse length and range gate length. The results show that

increasing the pulse length and range gate length does

improve the slope of the linear regression toward one.

However, this is accompanied by an increased scatter of

the velocity variance between the ultrasonic anemom-

eter and the Doppler lidar as indicated by the low cor-

relation coefficients for large range gate and pulse

lengths. The high values for the correlation coefficient of

the outer scale at large pulse lengths and range gate

lengths are misleading due to the lower border of the

parameter space for the outer scale at L0 5 0m, which

masks an increasing number of intervals that do not

converge for L0. It was concluded that increasing the

pulse length and range gate length is not sufficient to

correct the remaining systematic error; therefore, the

introduction of an effective Dp and Dr is not reasonable.For future investigations of this remaining systematic

error, a control experiment by deploying multiple ul-

trasonic anemometers along a Doppler lidar range gate

seems desirable. Averaging the ultrasonic anemometers

should lead to the Doppler lidar results, if spatial aver-

aging is the main problem behind the bias. An experi-

mental setup that allows for reanalyzing of the Doppler

lidar signal with different range gate lengths to in-

vestigate the dependency of the bias to spatial averaging

is also an interesting option for a future experiment.

Future research in general on a reduction of the bias in

turbulence parameters between those received from a

Doppler lidar and those from an ultrasonic anemometer

has to focus on the correction procedure itself, either by

using better suited turbulencemodels or by improving the

description of the measurement properties of a Doppler

lidar. Technical improvements of the instrument to re-

duce spatial averaging are also possible: The range gate

length can be reduced by a higher sampling frequency of

the return signal or by shorter pulse lengths. But at the

same time, a shorter pulse length of the laser will result

FIG. 8. Correlation coefficient of s2v from the ultrasonic ane-

mometer and the Doppler lidar as a function of the pulse length

and range gate length. Each square represents a correlation co-

efficient as in the text box of Fig. 3, but for values of Dp and Draccording to the figure axes. The cross represents the settings of our

Doppler lidar.

FIG. 7. Slope of a linear regression of s2y from the ultrasonic

anemometer and theDoppler lidar as a function of the pulse length

and range gate length. Each square represents a linear regression

like the dashed line in Fig. 3, but for values of Dp and Dr accordingto the figure axes. The cross represents the settings of our Doppler

lidar.

FIG. 9. Slope of a linear regression of L0 from the ultrasonic

anemometer and theDoppler lidar as a function of the pulse length

and range gate length. Each square represents a linear regression

like the dashed line in Fig. 4, but for values of Dp and Dr accordingto the figure axes. The cross represents the settings of our Doppler

lidar.

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in a broader frequency bandwidth of the laser, which will

lead to errors in measurements of the radial velocity.

Before closing this discussion, it should be noted that

the separation domain used to fit the model, especially

the maximum separation, has an influence on the results

previously shown. In theory, the separations of the

structure function from Eq. (13) range from zero to in-

finity. In practice, the time series is limited and a finite

domain of separation is used to fit the model. Davies

et al. (2004) used separations between smin 5 300m and

smax 5 1500m, and Frehlich and Cornman (2002) and

Frehlich et al. (2006) used smin 5 10m and smax 5 500m

as a separation domain for the fit of the von Kármánmodel. A smaller maximum separation leads to a sys-

tematically smaller s2y , because it is mostly determined

at large separations as L in Eq. (9) converges to one for

large separations. A variation of the smallest separation

between 0 and 300m has nearly no influence on the

variance. Increasing the smallest separation increases

large values of L0, while small values of L0 (L0 , 300m)

are not strongly affected. Different maximum separa-

tions showed no systematic influence on L0.

5. Summary

An experimental verification of the procedure from

Frehlich andCornman (2002) to correct the effects of spatial

averaging in estimations of the radial velocity variance s2y

and outer scale of turbulence L0 from Doppler lidar mea-

surementswas presented.Acomparisonof time series of the

radial velocity from an ultrasonic anemometer and a nearby

range gate of a Doppler lidar was used for the validation.

Derivations of s2y and L0 from Doppler lidar data

using a von Kármán model with the corrections given by

Frehlich et al. (1998) lead to better agreement with the

ultrasonic anemometer than without correction of the

spatial averaging. The systematic error in the slope of a

linear regression was reduced by 71% for L0 and 25%

for s2y . However, the results have a remaining systematic

error of 28% for L0 and 36% for s2y , which still leads to

an underestimation of the radial velocity variance and

an overestimation of the outer scale of turbulence

(Figs. 3 and 4). Different sources for the remaining error

were checked, but they could not provide an explana-

tion. Further research is therefore needed to determine

the cause of the remaining error and to develop a tur-

bulence model that enables correct estimation of tur-

bulence parameters from Doppler lidar measurements.

Acknowledgments. This scientific work was done

within the Young Investigator Group ‘‘Exploring co-

herent structures using Dual Doppler lidar systems’’ of

the Karlsruhe Institute of Technology (KIT), Karslruhe,

Germany.We are grateful to the Institute ofMeteorology

and Climate Research (IMK) of the Karlsruhe Institute

of Technology (KIT) for the data from the measurement

tower at theCampusNorth (CN) research facility.Weare

grateful to Dr. Andreas Wieser for his help with setting

up the Doppler lidar and executing the measurements.

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