Introduction
Turbulence within the planetary boundary layer (PBL) transports
and disperses heat, moisture, momentum, and other quantities. Additionally,
atmospheric turbulence affects several disciplines and industries, such as
wind energy, aviation, and air quality. For example, wind turbines may
perform poorly and have a lower power output when turbulence intensity is
large , and turbulence can shorten the
lifespans of wind turbines . Pollutant dispersion from
factories and other sources is primarily driven by advection and turbulent
mixing within the PBL. Precise measurements are necessary to understand the
role of turbulence within these disciplines and to validate the turbulence
generated or parameterized in numerical weather prediction (NWP) models and
simulations. Scanning Doppler lidars are capable of addressing this need by
measuring vertical profiles of turbulent quantities throughout the entire
PBL.
Many different approaches have been used to measure turbulence quantities with
Doppler lidars . Prior to the
availability of commercial Doppler lidars, it was necessary to employ
techniques for single lidars, which are the techniques evaluated in the
present study. In the most simple case, data from fixed-beam vertical staring
have been used to directly calculate vertical velocity w statistics in the
unstable convective boundary layer
e.g.,.
More sophisticated techniques take advantage of scanning to provide other
turbulence components and quantities. adapted a
technique developed for scanning Doppler radar
that used 360∘ azimuth
conical (plan position indicator, PPI) scans to measure profiles of
turbulence kinetic energy (TKE), individual velocity variances
(u′2‾, v′2‾, w′2‾), and momentum
fluxes (u′w′‾, v′w′‾, u′v′‾).
used elevation (range–height indicator, RHI)
scans, which provide a vertical cross section of radial wind-velocity data
points, to calculate vertical profiles of the streamwise variance, which was
found to be approximately equal to TKE in the stable conditions studied.
evaluated these values against sonic anemometers
at several levels on a 120 m tower and found good correlations. These
scanning techniques will be further evaluated in the present study.
For instrument systems that lack full scanning capability, a simpler approach
is the so-called Doppler beam swinging (DBS) technique typically used by
radar wind profilers and Doppler sodar. In this method the transmitted beam
cycles among (typically) five discreet fixed look angles, one vertical and
four beams tilted at some elevation angle but aimed in four orthogonal
horizontal directions, such as the four cardinal directions. While
identified problems with the computational procedure using
these orthogonal beams to measure turbulence quantities, the technique has
been modified in various ways to correct for these limitations
e.g.,. This problem can
also be addressed using a six-beam pattern: five tilted beams instead of four
plus a vertical . This six-beam technique will also be
evaluated in this study.
These single-lidar, multiple-look-angle techniques are potentially very
powerful, given the importance of being able to measure vertical profiles of
turbulence , but a crucial assumption is that the mean flow
and turbulence need to be homogeneous over the horizontal sampling footprint
at each measurement height in space and time. When commercial Doppler lidars
became available, it became possible to deploy three or more lidars to
simultaneously and continuously sample a given volume in space,
thereby enabling all turbulence components to be measured directly without
assumption. , , and
used triple-Doppler-lidar arrays in this way to measure the six components of
the Reynolds stress tensor. Although we do not evaluate multiple Doppler
techniques in this study, it is important to be aware of these capabilities
in designing future multiple Doppler measurement programs, since they would
be available as a component of a turbulence verification effort.
Turbulence measurements are needed to address a range of problems, which
involve different breadths of the turbulence spectrum. All applications
require accurate measurements of fluctuations by the largest
energy-containing turbulent eddies and at least the lowest wavenumbers of the
inertial subrange. Within the inertial subrange the magnitude of the
fluctuations drops off quickly (exponentially) with increasing wavenumber, so
high wavenumbers make correspondingly smaller contributions to the total
variances . Detailed studies of turbulence dynamics,
which may include studies of inflows to wind turbines or turbulence generated
by them, may require accurate representation of fluctuations over the entire
turbulence spectrum from large-eddy to dissipation scales
e.g.,. For such studies, employing the best data
acquisition strategies and understanding the errors involved is important.
Other studies may not require this degree of precision. For example,
evaluating the ability of NWP models to
predict TKE involves values of 2–4 m2 s-2 in convective conditions
and 1–2 m2 s-2 in weakly stable conditions. Such accuracies are
achievable without measuring the entire spectrum. Many field programs are
employing scanning lidar remote sensing in arrays to investigate spatial and
temporal variations of the mean wind, as recommended in . In
such cases, the measurement of turbulence is not the primary goal, so the
data-acquisition and scanning approaches are not optimized for turbulence
measurement. It is still desirable to obtain quantitative turbulence
information (e.g., for NWP verification) from the scans that are performed.
It is essential to understand the error properties of these techniques to
know whether the calculated values are useful for the intended purpose.
To systematically evaluate these different turbulence measurement techniques,
a Doppler lidar cycled each hour continuously through the methods during the
last 2 weeks of the eXperimental Planetary boundary layer
Instrumentation Assessment (XPIA) field campaign. These
measurements are compared with measurements from sonic anemometers, a
commonly used reference instrument , at six heights on a
300 m meteorological tower located 540 m from the lidar. Through this
comparison, the following questions will be addressed in this study.
How accurate are the various single-Doppler-turbulence measurement strategies in determining turbulence characteristics? Does
the accuracy vary depending on the measurement height?
What main caveats need to be considered when applying each technique? How should random errors and instrument noise be characterized and treated?
What is the optimal operational scanning strategy to derive turbulence estimates? Should different strategies be used for different objectives?
To address these questions, the paper is organized as follows. The various
scanning strategies and methods to measure turbulence, including specific
details of implementation, are described in Sect. . An
overview of the experiment and the instrumentation used is detailed in
Sect. . Within Sect. , the techniques are
statistically compared through validation with sonic anemometry. Implications
for future studies and possible future research directions are discussed
within Sect. . A summary and the conclusions are provided in
Sect. .
Turbulence measurement strategies
The scanning procedures used most often by Doppler lidars are azimuthal
scanning, elevation scanning, and stares at a particular look angle. Each of
these approaches can be used to measure one or more of the velocity variances
and covariances. The theory for turbulence measurements is based on the
relationship between the observed radial velocity vr and the flow within
the resolution volume given by
vr=ucosθcosϕ+vsinθcosϕ+wsinϕ+ϵ,
wherein u is the streamwise horizontal velocity, v is the crosswise
horizontal velocity, w is the vertical velocity, ϕ is the elevation
angle above the horizon, θ is the angle between u and the azimuth of
the lidar, and ϵ is the uncorrelated random error in the measurement.
The value of ϵ typically increases with range from the lidar, as the
signal-to-noise ratio (SNR) decreases. By squaring Eq. () and
removing the mean from each quantity, the radial velocity variance is given
by
vr′2‾=u′2‾cos2θcos2ϕ+v′2‾sin2θcos2ϕ+w′2‾sin2ϕ+2u′v′‾sinθcosθcos2ϕ+2u′w′‾cosθcosϕsinϕ+2v′w′‾sinθcosϕsinϕ+ϵ2‾,
where the covariance terms involving ϵ are 0 since it is
uncorrelated. All of the turbulence measurements techniques are ultimately
based on Eq. (). Brief derivations and details of how these
measurements are made, in addition to modifications introduced within this
study, are described here. Complete derivations for each method can be found
in the works cited.
Velocity–azimuth display
While PPI scans have been used to take accurate measurements of the mean wind
through velocity–azimuth display (VAD) analysis , these
scans can also be used to quantify turbulence. details a
technique for measuring turbulence from PPI scans, based on pioneering work
by and wherein turbulence is measured
using Doppler radar observations. From PPI scans at two sufficiently
different elevation angles, all six components of the Reynolds stress tensor
can be retrieved using the residuals of the VAD fitting by utilizing a
partial Fourier decomposition of Eq. (). However, the covariances
or momentum fluxes u′v′‾, u′w′‾, and
v′w′‾ can be measured from any single PPI scan, and TKE can be
obtained from a single scan if ϕ=35.3∘ for mathematical
basis, see Eq. 4a in.
Sample PPI scan (a) during a turbulent time period, with the VAD fitting and its residuals to vr observations at
the range ring denoted by the red circle shown in (b). Turbulence structures can be visualized in the
residuals across the entire scan (c).
A sample PPI scan for a turbulent time period is shown in
Fig. a. For each range ring, the mean wind speed and direction
are determined using VAD analysis. The complete VAD analysis described by
includes terms for the vertical velocity of the
scatterers as well as horizontal divergence, stretching deformation, and
shearing deformation. However, a more simplified variation of the VAD
analysis is often used by neglecting divergence and the deformation terms.
For the results presented here, the simplified form is used since it yields
more accurate estimates of the measured turbulent quantities when compared
with sonic anemometer measurements. This may be due to variability from large
turbulent motions being incorrectly partitioned into the divergence or
deformation terms. However, in complex terrain or other locations these terms
may not be negligible. An example of the fitting of this equation and its
residuals vr′, which are deviations from the expected mean vr, is shown
in Fig. b. If the mean flow (i.e., u‾,
v‾, and w‾) is homogeneous over the scanning circle,
then the residuals of the fitting are results of turbulent motions and
ϵ. This is visualized within Fig. c, wherein coherent
areas of positive and negative vr′ represent turbulent eddies. Since
turbulent structures are correlated spatially, ϵ can be quantified
and removed by applying a structure-function fit to the autocovariance of
vr′ across radials for a given range gate, similar to the method outlined
using Eq. (32) in . To our knowledge, this is the
first time that the autocorrelation technique has been used to remove noise
variance from a scan, as it is typically used for a time series from prolonged
stares . This technique can lead to an overestimate of
ϵ when the inertial subrange is smaller than the distance between
adjacent azimuths, which is more likely at long ranges from the lidar as the
spatial separation between adjacent beams increases.
Previously, measurements using the technique described by
have not been evaluated against in situ
observations. used a variation of this technique by applying
it to a 30∘ sector PPI and assumed isotropic turbulence to relate
vr′2‾ to TKE. Estimates of TKE from the arc scan showed good
agreement (r2=0.89) with those from sonic anemometers on a linear scale.
Other studies e.g., have used a loose
variation of this VAD technique to quantify turbulence by using only a small
number of beams (4–6) spaced around the entire 360∘, which is
substantially different from using more than 100 beams around the sampling
ring.
Six-beam technique
propose a technique to measure all six components of
the Reynolds stress tensor by continuously cycling between measurements at
six different angles. One beam is vertical, and the other five are at a set
elevation angle (45∘ herein) and are equally spaced 72∘ apart
in azimuth. For each beam, the time series of vr are linearly detrended
over a fixed time window, which is 20 min here, and vr′ is computed as its
residual. While the 20 min detrending window may filter out large convective
eddies when the wind speed is small, a shorter window or a higher-order
detrending would exacerbate this filtering effect for smaller eddies. Values
of vr′2‾ are computed for each beam separately. Thus, there are
six known values of vr′2‾, one for each beam, and each is a
function of differently weighted velocity variances and covariances based on
the scan elevation and azimuth, as in Eq. (). This can be
represented by the matrix relationship
Mu′2‾v′2‾w′2‾u′v′‾u′w′‾v′w′‾=vr1′2‾vr2′2‾vr3′2‾vr4′2‾vr5′2‾vr6′2‾,
where M is a six by six matrix of coefficients based on different combinations
of θ and ϕ, as in Eq. (). Thus, it is possible to
solve for the six unknown components of the Reynolds stress tensor through an
inversion of Eq. ().
For each beam, the lidar stared at the given location for 1 s, collecting two
samples 0.5 s apart before advancing to the next position. To remove
uncorrelated noise ϵ2 from the observed vr′2‾ for
each beam, the autocovariance at the first lag for the samples that were
0.5 s apart was taken as vr′2‾, following the technique
presented by . This likely results in a slight
underestimate of vr′2‾, since contributions from small eddies
that are uncorrelated over short timescales are removed. In the future, it is
recommended that more samples be collected along each beam so that a
structure-function or linear fitting may be applied to the autocovariance for
a more robust measurement of vr′2‾ for each beam. On average,
the scanner took ≈3.6 s to slew between beam positions, so that the
scanner returned to the same beam every ≈27 s.
The measured vr′2 on the right-hand side of Eq. () include the
desired turbulent fluctuations but also variations due to spatial and
temporal sampling by beams aimed at very different directions. Under some
conditions, mostly when the turbulence was weak, the matrix-inversion
calculation in Eq. () can lead to negative values for one or more
of the calculated variances, which should be positive-definite quantities.
This result, which has also been reported by , is a
nonphysical computational artifact, thought to be primarily due to sampling
errors and nonturbulent variations within each beam variance being propagated
through the matrix inversion in Eq. (), resulting in spurious
values of the calculated variances. Nonphysical negative variance estimates
have been removed in the following analyses.
RHI scans and vertical stares
Shallow RHI scans have also been used as a means to measure horizontal
velocity variances
e.g.,.These scans are
conducted by scanning from the horizon up to ≈30∘, typically at
two angles orthogonal to each other. Since the scans are mostly at low
angles, it is assumed that the observed vr are due to the horizontal wind
and that the contribution of w is negligible. To ensure that measurements
at different elevation angles are comparable, values of vr are
normalized by ϕ using
vrH=vrcosϕ,
where vrH is the radial velocity projected in the horizontal. For each RHI, observations are binned by height (30 m bins used
herein),
which is used to make a mean profile of vrH. The profile of vrH‾ is used to calculate deviations from the mean flow
vrH′. This is done by simply taking the difference between vrH and vrH‾ for the given height, where
vrH‾ is linearly interpolated between the center of mass of each height grid. The variance of vrH′ is
calculated using the same height grid to produce a profile of vrH′2‾, which is the horizontal wind variance
within the RHI plane. An example of this process and each derived product is provided in Fig. .
Sample RHI scan (a) showing instantaneous values of vrH over the scan plane. (b) Vertical profile
of the mean vrH for the scan, which is used to calculate vrH′ (c). For each height bin between the
solid horizontal black lines on (a) and (c), the variance of vrH′ is calculated (d).
When one of these scans is oriented with the mean flow and the other
transversely, the two measured profiles of vrH′2‾ can be
treated as u′2‾ and v′2‾ respectively. If the scans
are not oriented in such a way or if large directional shear is present, it
is possible to rotate the variances to be aligned with the mean flow by
u′2‾=vrH1′2‾cos2Θ1+vrH2′2‾cos2Θ2-vrH1′vrH2′‾sin2Θ1andv′2‾=vrH1′2‾sin2Θ1+vrH2′2‾sin2Θ2+vrH1′vrH2′‾sin2Θ1,
wherein Θ is the angle between the RHI scan azimuth and the mean flow,
the subscripts denote the two different RHI scan planes, and the scans are
orthogonal. Although the covariance term vrH1′vrH2′‾ cannot
be measured with this method, it is typically small compared to the other
terms and can be neglected. Thus, values of u′2‾ and
v′2‾ can be computed directly through the rotation. The mean
wind profile, including the wind speed and direction necessary for the rotation,
is directly computed using the two profiles of vrH‾. Using
this technique, there is no straightforward way to remove contamination from
ϵ2‾ in the variances. Thus, data were removed if the
SNR <-27 dB to reduce contamination from highly noisy data. The SNR values
used for filtering were taken as the carrier-to-noise ratio
produced by
the lidar manufacturer's processing algorithms.
To calculate TKE, values of w′2‾ also need to be known. For
quantification of w′2‾, vertical stares were used in conjunction
with the shallow RHI scans. Vertical stares are the most straightforward
method to measure any vertical turbulent quantity with a Doppler lidar. Since
the w profile is continuously measured, it is simple to take the variance
of time series of w to obtain w′2‾. However, ϵ2
contaminates the measurement and needs to be removed to improve the accuracy
of the measurement. As described earlier, the autocovariance technique
described by is used to remove instrument noise.
Herein, values of σw2 are taken as the extrapolated -2/3 structure-function fit
to the autocovariance of the time series at lags 1–5. Using this
technique removes contamination by ϵ2 and mitigates volume
averaging effects, which otherwise reduce the observed w′2‾
.
Summary of measured variables for each type of scanning
strategy.
u′2‾
v′2‾
w′2‾
u′v′‾
u′w′‾
v′w′‾
TKE
VAD (single ϕ)
✓
✓
✓
✓(ϕ=35.3∘)
VAD (two ϕ's)
✓
✓
✓
✓
✓
✓
✓
Six beam
✓
✓
✓
✓
✓
✓
✓
RHI and vertical stare
✓
✓
✓
✓
Experimental overview
A Leosphere Windcube 200S® was operated at
the now-defunct Boulder Atmospheric Observatory (BAO) during 15–31 May 2015,
a total of 17 days. The sampling period for the present study immediately
followed the eXperimental Planetary boundary layer Instrumentation Assessment
field campaign during which a significant complement of remote-sensing
instruments, including six Doppler lidars, were operated in the vicinity of
the tower . The BAO featured a 300 m meteorological
tower instrumented at multiple levels. The Doppler lidar system was deployed
540 m to the south–southwest of the 300 m tower, as shown in
Fig. .
The BAO was located in Erie, CO, approximately 25 km east of the foothills of
the Rocky Mountains, and was designed primarily for PBL research as well as
testing and calibration of various atmospheric sensors
. Within the immediate vicinity, the terrain is
relatively flat with gently rolling terrain. The 300 m tower was located on
the property. For this experiment, 3-D Campbell CSAT3 sonic anemometers were
installed on northwest (NW, 334∘) and southeast (SE, 154∘)
booms at six levels (50, 100, 150, 200, 250, and 300 m). Data were recorded
at 20 Hz. A tilt-correcting algorithm that used a planar fit
was applied to the measurements after the experiment was
finished. Data were filtered to remove time periods when the turbulence may
be affected by the wake of the tower, following the results of
. Specifically, data from the NW and SE sonics are
removed when the wind direction is from between 100–170 and
300–20∘ respectively. Turbulence statistics from the sonic
anemometers were averaged over 20 min blocks for comparison, similar to the
averaging time for the various lidar scanning strategies discussed below. Any
20 min averages where the statistics between the two sonic anemometers at the
same height differed by a factor of 2 or more were removed to ensure the
statistics were comparable and not affected by the tower.
The lidar operated with 50 m range gate spacing and at a nominal pulse length
of 50 m, meaning that most of the transmitted pulse energy lies within a 50 m
window. The remnant of pulse energy outside this window is too weak to
significantly affect the velocity calculation for normal atmospheric
aerosol-backscatter conditions, although interceptions of the pulse by hard
targets (such as clouds or wind turbines) can contaminate the return signal
in adjacent range gates see . The accumulation time for
each beam was 0.5 s and the pulse repetition frequency (PRF) was 20 kHz. Due
to the high PRF, the maximum unambiguous range of the lidar was only 7500 m.
Hence, range-folded echoes from clouds or other strong targets were
occasionally apparent in the signal. Characteristics of these erroneous
echoes and a discontinuity-based algorithm used to remove them are described
by . Over the 17 days, 5.6 % of the data were conservatively
detected as possible range-folded echoes and removed. Each hour, a sequence
of scanning strategies were conducted. For 20 min, the lidar cycled between
PPI scans at a ϕ of 35.3 and 50.8∘ scanning at
3∘ s-1 in azimuth. Over this 20 min period, five scans were
completed at each elevation angle. Following the PPI scans, three shallow
RHIs were performed at perpendicular angles (θ of 330 and
60∘), followed by a 10 min vertical stare. For the rest of the hour,
the six-beam scanning strategy was repeated, wherein each beam was sampled
for 1 s before advancing to the next beam position as described in
Sect. . This 1 h scanning sequence was repeated continuously
for the 17 days at the end of XPIA.
The modest complexity of the terrain and the proximity of the site to the
mountains present complications in calculating turbulence quantities from
remote-sensing data using the techniques described here. Under these
conditions, the flow can exhibit nonturbulent variability along a scan that
contributes to an unknown degree to the calculated variances and covariances
producing larger variance and discrepancies between lidar- and tower-measured
turbulent quantities. This variability is not expected over more homogeneous
topography e.g., or the ocean
e.g.,.
Satellite imagery of the BAO site with the locations of the 300 m
tower and Doppler lidar deployment indicated.
Turbulence statistics comparison
For most measurements, turbulent quantities measured by the Doppler lidar are
not at precisely the same height as the sonic anemometers. This difference in
measurement height is dependent on the type of scan. For instance, the range
gate center for vertical stares is identical to the height the sonic
anemometers at 100–300 m (50 m was below the minimum range). However, the
closest range gate from the six-beam scan to the 300 m sonic was at 282.8 m.
Thus, lidar-measured turbulent quantities are interpolated to the sonic
anemometer heights.
Depending on the application or field of use, different turbulent quantities
may be desired. Within the wind energy industry, turbulence intensity TI
is calculated as
TI=u′2‾U,
where U is the mean horizontal wind speed, which is most often used as it is a
measure of the variability of the inflow into the turbine and affects the
design requirements . In boundary layer meteorology
and air quality, TKE is calculated as
TKE=12u′2‾+v′2‾+w′2‾,
which is often used as a measure of the turbulent mixing in the atmosphere
. Additionally, the covariance terms
u′w′‾ and v′w′‾ are the momentum flux and are
necessary to test and validate models of atmospheric flow. Since these
measures of turbulence are most commonly used, they are the focus of this
section. A complete statistical comparison of each measured variable is
provided in Appendix .
When interpreting the intercomparisons presented here, it is necessary to
consider the statistical uncertainty and representativeness of the
measurements themselves. However, quantifying the sampling error of a
turbulence measurement is not trivial. Numerous studies have been entirely
focused on determining sampling errors of turbulence and flux measurements
from time series analysis alone
e.g.,.
Typically for time series analysis, the magnitude of these sampling errors
largely depends on the record length with respect to the integral timescale
of the measured variable. Sampling errors are reduced for a stationary time
series when a longer record is used for the calculation of the turbulence
quantity. Nevertheless, the record length needs to be short enough to assume
stationarity of turbulence fields. Thus it is difficult to identify a static
record length that is appropriate for all conditions. During strongly
convective conditions with weak winds, using a time series of 1–2 h may
be appropriate e.g.,.
Conversely, turbulence statistics may be more aptly calculated using a record
length of 5–10 min during stable conditions
e.g.,. The scanning strategies evaluated
here use a combination of spatiotemporal sampling to measure turbulent
quantities. To date, no method has been developed to estimate sampling errors
associated with these techniques. As such, and due to the intricate analysis
necessary to determine the appropriate sampling error magnitude, sampling
errors are not quantified for any of the measurements presented. Nevertheless, it is
necessary to interpret the results shown with the understanding that all of
the presented measurements have statistical uncertainty of an unknown
magnitude.
Turbulence kinetic energy
For the six-beam, VAD with multiple ϕ, and RHI–vertical stare
techniques, TKE was directly computed as the sum of the measured velocity
variances using Eq. (). As discussed within
Sect. , TKE can be directly computed from a 35.3∘
ϕ scan without measuring u′2‾, v′2‾, or
w′2‾ directly. From here onward, measured quantities from two
PPI scans at different ϕ are referred to as “VAD” measurements, and
those from one PPI scan at 35.3∘ are “VAD 35.3∘” measurements.
Sample time series of measured TKE on 30 May 2015 at 200 m.
A sample 24 h time series of TKE is provided in Fig. to
demonstrate the ability of the different methods to capture temporal changes.
The diurnal pattern of TKE decreasing overnight between 03:00 and 12:00 UTC is
visible in measurements from the lidar and sonic anemometer. Although all the
lidar techniques capture the decrease in TKE in the evening and early night
hours (00:00–06:00 UTC), the TKE measurement from the RHI and vertical stare
is systematically overestimated later in the night when
TKE <0.1 m2 s-2. The lidar-measured TKE from the VAD, VAD
35.3∘, and six-beam techniques capture the trends in TKE well. When
TKE measured by the sonic anemometer is small (<0.1 m2 s-2) such as
at 12:00 UTC, the measured TKE by the six-beam technique can be negative. Of all
the TKE measurements during the experiment, 0.8 % of the six-beam TKE values
are negative. Since this result is unphysical as discussed in
Sect. , these values have been removed. The other methods
analyzed did not yield negative TKE values.
Scatterplots (a, c, e, g) and histograms (b, d, f, h) showing the relationship between the TKE
measured by the lidar and southeast sonic anemometers at all heights. In (a), (c), (e), and (g), the blue line is the best fit
line given by the equation in the upper left and the black line indicates a one to one relationship. Histograms (b, d, f, h) show
the ratio of the lidar-measured TKE to the sonic-measured TKE with the median ratio, number of points N, and number of
outliers Noutliers that are more than 1 order of magnitude apart. Lidar measurements are from TKE
at ϕ=35.3∘ (a, b), TKE from two ϕ (c, d), the six-beam technique (e, f), and the RHI
and stare combination (g, h).
For a quantitative analysis of the TKE measurements, TKE values from the
Doppler lidar and the southeast sonic anemometers are summarized in
Fig. . These results are from measurements between
100 and 300 m for all the scanning strategies. No TKE measurements are available
below 100 m since the first lidar range gate is 100 m so that no
w′2‾ values are available below this height from the vertical
stares. Comparisons with the northwest anemometers are similar and provide
little additional information; thus, they are not shown. The comparisons shown here
(and throughout the manuscript) are on a logarithmic scale since values can
range several orders of magnitude, generally from 0.01–10 m2 s-2 for
TKE. If the analyses were conducted on a linear scale, large values would
dominate the statistical comparison and the small values would be
overwhelmed. In logarithmic space, the ability to differentiate values of
∼0.01 from ∼0.1 is equally as important as differentiating ∼1
from ∼10, and values across all different orders of magnitude are
weighted equally in determining the trend line. Within each of the scatterplots in Fig. , the best fit lines were determined
fitting
log10(y)=log10(x)+b,
where x and y are data points on their respective axes in linear units
and b is a constant. Transforming the equation back into linear space for
ease of interpretation, the equation shown in the upper left of
Fig. a, c, e, g is y=10bx, where 10b is the
slope of the regression. The slope is related to the bias of the measurement.
For example, a slope less than 1 indicates that measured quantities are
systematically smaller than the reference measurement over all scales.
Each of the techniques evaluated generally shows skill in measuring TKE, as
indicated by r2 values greater than 0.6 in Fig. .
Considering that the sonic anemometer and lidar measurements represent
different spatial areas, which vary according to the scanning technique, and
that each are subject to sampling error, the authors consider the correlation
between the lidar and sonic TKE to be good. In the absence of sampling errors
that would allow a more statistical determination of whether measurements are
in agreement within their respective uncertainties, the relative correlations
between sonic and lidar measurements, and their differences for various scan
strategies, are used to understand the biases and accuracy of each technique.
The six-beam technique demonstrates the best ability to measure TKE overall,
as is evident by the largest r2 and slope of 0.945, which is close to unity.
Additionally, the histogram of the ratio of TKE measurements in
Fig. f shows a distinct peak around 1 with reduced
spread compared to Fig. b, d, h; 83.5 % of six-beam TKE
values are within a factor of 2 of the sonic measurement, which is the largest
proportion of all the techniques analyzed. However, the six-beam technique
also produces the largest number of TKE outliers, defined as being more than
1 order of magnitude different from the sonic-observed TKE. Approximately
half of these outliers are negative TKE values, which were removed as
discussed earlier. The other outliers are when TKE is grossly overestimated,
as is visible in Fig. e. Upon manual inspection of these
high outliers, many are due to contamination of range-folded echoes. When
range-folded echoes appear intermittently, the vr time series within each
beam position changes erratically, resulting in an anomalously large
variance and spuriously increasing the observed TKE. The discontinuity-based
algorithm used to detect range-folded echoes largely relies on contextual
information from proximate beams in time and space not
available from the six-beam technique. These anomalous echoes can typically
be detected and removed in PPI, RHI, and stare scans, but these range-folded
returns persist through the quality-control process of the six-beam
measurements and degrade the accuracy of the calculated variances .
As discussed in Sect. , the VAD technique can be used to
measure TKE from either one scan at 35.3∘ or two scans at different
ϕ, herein 35.3 and 50.8∘. Since the 35.3∘ scan is
used in both approaches, the results from both methods are not independent of
each other. This can be seen by the similar results from both approaches in
Fig. a–d. Although the TKE from both techniques is
highly correlated with the sonic TKE, the VAD and VAD 35.3∘ measured TKE
is systematically biased too low. The low bias is more pronounced for the
two-ϕ technique, although the scatter is reduced slightly as evidenced
by the larger r2 in Fig. c. This reduced scatter is
attributed to smaller sampling errors, since twice the amount of data go into
the measurement. The overall low biases may be due to the inability to
capture all the scales of turbulence; the largest eddies may not be fully
captured and resolved within the scanning circle. This effect would be more
pronounced for higher elevation PPIs, such as at 50.8∘, and explain
the more significant low bias in TKE for the two PPIs used herein. If a lower
ϕ were used (i.e., at 25∘), the low bias may not be as pronounced
for the two-ϕ VAD TKE. Unfortunately, no data are available from this
experiment to validate this hypothesis. Despite these biases, 73.3 %
(35.3∘ VAD) and 71.7 % (two VADs) of the TKE measurements are within a
factor of 2 of the sonic TKE.
Using the RHIs and vertical stares to measure TKE results in the largest
scatter. Nevertheless, 74.7 % of the lidar-measured TKE values are within a factor of 2 of
the sonic measurements, indicating that the technique is still accurate. The
six-beam and VAD techniques show similar scatter for all ranges of TKE, and
the RHI–stare technique typically overestimates TKE when its value is small
(i.e., <0.1 m2 s-2), as apparent in Fig. g.
The cause of the overestimate during weakly turbulent conditions is unclear,
but it may be due to spatial variability of the flow that the sonic cannot
detect. Mean or other nonturbulent variability along the horizontally
oriented vertical bins, such as if the bin is sloped with respect to the
underlying topography, can be a significant contribution to the overestimate
of the calculated variances, which is especially evident in weakly turbulent
conditions. Additionally, random errors are quantified and removed in the VAD
and six-beam techniques as detailed in Sect. , but no
established technique exists to remove these errors from RHI measurements,
which may lead to this high bias in TKE as measured with the RHIs and
vertical stares. Within Fig. h, this high bias under
weakly turbulent conditions manifests itself as right-skewed distribution.
The slope (a) and r2 (b) of the best fit line on a logarithmic scale (similar to those in Fig. )
relating TKE measurements from the sonic anemometer and lidar for each measurement height.
The results shown in Fig. are for all measurement
heights combined. The analysis can be further refined by comparing lidar TKE
measurements at each sonic anemometer height separately.
Figure summarizes this analysis by showing the slope and
r2 of the best fit line at each height. The accuracy of the six-beam
technique is the most consistent at all measurement heights, as the slope and
r2 are nearly constant with height. The value of r2 remains around 0.75
at every height, whereas the slope increases a small amount with height. This
change in slope indicates that TKE is less underestimated above 200 m than it
is closer to the surface.
The bias of TKE measurements using the RHI and vertical stare method is
independent of height and is small (Fig. a), as the slope
is generally around 1. The TKE measurement becomes more accurate with height,
which is
indicated by the increase of r2 with height in
Fig. b. The cause for the increase in accuracy with
height is unclear, but it may be due to the mean flow becoming more homogeneous
aloft. The low bias of VAD and VAD 35.3∘ TKE observations becomes less
significant with height, as shown in Fig. a.
Coincidentally, the VAD and VAD 35.3∘ r2 values increase with
height, representing less scatter and more accurate values at higher
altitudes.
To examine the decrease in the low bias and increase in the accuracy of VAD TKE
measurements with height, the VAD circle diameter is compared with the
typical largest eddy size – the integral length scale of l. First, the
integral timescale tint is calculated from a linearly detrended 20 min
time series of u from the sonic anemometer as
tint=1u′2‾∑τ=0τ(A=0)A(τ)Δτ,
where A(τ) is the autocovariance of u, which is a function of the
time lag τ. Since the time series is discrete, Δτ is the
sampling interval (0.05 s here, since the sonic data rate is 20 Hz). The
median and various percentile values of l, computed as l= Utint, are
shown as a function of height and with reference to PPI scan diameters in
Fig. . Generally, individual 35.3 and 50.8∘
PPI scans do not fully sample the largest turbulent eddies close the ground,
since the scan circle diameter is often less than l. The largest eddies are
better captured by these scans at higher altitudes, especially for the
35.3∘ PPI scan. At 300 m, the integral scale is less than the
35.3∘ scan diameter over 90 % of the time.
The results shown in Fig. explain why VAD and
VAD 35.3∘ TKE measurements become more accurate and less biased with
height, as the largest turbulence scales are more completely captured. These
effects are not important to the RHI method, since the spatial extent of the
average is typically several kilometers, which is much larger than the typical eddy size.
Since the vertical stare and six-beam techniques use time series analysis,
the largest scales of turbulence are observed if the time window length
exceeds the integral timescale, which is often ∼ 10–100 s during daytime
and less than 10 s at night
.
The diameter of the scanning circle of the two PPI scans is shown in comparison to l. The red line
denotes the median l, while the progressively darker contours represent the 40–60, 25–75, and 10–90
percentile intervals of l over the entire 17-day experimental period.
Turbulence intensity
Sample time series of measured TI on 30 May 2015 at 200 m.
Similar to the analysis of TKE presented in Sect. ,
measurements of TI from the six-beam, VAD (using 2ϕ angles), and RHI
techniques are compared and validated here. Since TI→∞ as U→0 following Eq. (), TI is only calculated
when U>1 m s-1. A sample time series of TI is shown in
Fig. . The diurnal trend in TI is clearly visible in the sonic
measurements, as TI is generally low (3–10 %) at night until 12:00 UTC and
TI is larger (20–70 %) during the day. During the morning hours (i.e.,
12:00–18:00 UTC), U was less than 2.5 m s-1, causing TI to become
large. Despite some scatter, TI measurements from the Doppler lidar show a
similar trend with smaller TI values at night and larger ones during the day.
Same as Fig. but for TI instead of TKE. Lidar measurements are from TKE from two-ϕ (a, b),
the six-beam technique (c, d), and the RHI scans (e, f).
Nonphysical negative u′2‾ values due to computational artifacts
as described previously by , caused by the same
effect as the negative TKE measurements discussed in
Sect. , have been removed from the analysis.
Measurements of TI at all heights over the entire experiment are summarized
in Fig. . For each of the three techniques analyzed, the
r2 for TI is ≈0.2 lower than it is for TKE. This indicates the
combined velocity variance components in TKE are more accurately measured
than individual velocity variances separately (see also
Table for u′2‾, v′2‾, and
w′2‾ comparison statistics). Nevertheless, the VAD, six-beam, and RHI
techniques each show skill (i.e., show correlation) in measuring TI. The VAD
and six-beam techniques perform comparably; having a similar r2 and slope
indicated a low bias. also show that the six-beam
technique tends to underestimate u′2‾ by a similar amount. The
RHI TI measurements show more scatter than the other two methods, given the
lower r2, but showed little bias.
The slope and r2 of the best fit line as a function of height is shown in
Fig. . Similarly to TKE, VAD measurements of TI are biased
low near the ground as indicated by the slope of ≈0.7 at 100 m. By
250 m, the bias becomes small as most of the turbulence scales are resolved,
as discussed in Sect. . The accuracy of the VAD TI
measurement does not change significantly with height, as r2 does not
consistently depend on height. The six-beam TI measurement is biased
consistently low regardless of height, as indicated by the slope of
≈0.83 at all heights, and the scatter of the measurements does not
have a consistent trend with height. The slope of RHI TI tends to be larger
near the ground and slowly decrease with height, as evidenced in
Fig. a. Coincidently, the scatter associated with these
measurements decreases significantly with height, as the r2 increases from
0.12 to 0.56.
Stress velocity
Same as Fig. but for TI instead of TKE.
Sample time series of measured u* on 30 May 2015 at 200 m.
The momentum flux terms u′w′‾ and v′w′‾ can be
combined through the calculation of a stress velocity scale u* by
u*=(u′w′‾2+v′w′‾21/4.
Of the techniques analyzed, only the six-beam and VAD methods have a
theoretical basis for measuring the covariances u′w′‾ and
v′w′‾ necessary to compute u*. Each PPI scan at any ϕ can
independently provide a measurement of the covariances, so the u* values
shown here are taken as the average of all PPI scans at both 35.3 and
50.8∘. An example of a 24 h time series of u* is shown in
Fig. . The sonic data are not shown for 00:00–03:00 and
14:00 UTC, since the u* measurements on opposing booms were more than a
factor of 2 different from each other, even though neither sonic was waked.
Thus, neither is taken as a baseline measurement. For this sample period, the
lidar and sonic data show a similar trend, values of u* decreasing for
00:00–12:00 UTC, rapidly increasing for 12:00–15:00 UTC, and remaining
nearly constant after 15:00 UTC. These trends are a result of u* steadily
decreasing overnight, increasing in the morning hours, and remaining steady
over the day.
Same as Fig. but for u* instead of TKE. Lidar measurements
are from TKE from two ϕ (a, b) and the six-beam technique (c, d).
The comparisons between sonic and lidar u* measurements are summarized in
Fig. . Both the VAD and six-beam techniques generally
overestimate u*, as shown in both the histograms and scatterplots. During
time periods when the sonic-estimated u* is small (i.e.,
<0.1 m s-1), the lidar techniques predominately overestimate u* as
indicated by the large number of data points above the one-to-one line in
Fig. a, c. The small r2 for the best fit lines
indicates that there is substantial scatter in the comparison of the lidar
and sonic measurements. Thus, the six-beam and VAD methods show little skill
in being able to accurately measure u* and the covariances, as shown in
Table . These results do not significantly change with
height (not shown), as the r2 remains small for both methods at all
measurement heights between 50 and 300 m. The accuracy of covariance and u*
measurements from the VAD and six-beam methods has not been evaluated in the
past, but here the measurements are found to exhibit large error. Over
simpler topography, present results from a DBS technique that
produced more accurate measurements of u′w′‾ and
v′w′‾.
Discussion
From the results shown in Sect. , it is clear that TKE can
be measured by each of the three techniques analyzed. However, measurements
of each individual term of the Reynolds stress tensor are more difficult to
accurately measure. The velocity covariances are particularly difficult to
quantify, as the six-beam and VAD techniques show little skill in their
measurement. It is thought that the poor comparison for the covariance terms
is due to the fact that the sampling error for the measurement exceeds the
covariance typical dynamic range. Based on sonic anemometer observations,
80 % of |u′v′‾|, |u′w′‾|, and |v′w′‾|
were <0.1 m2 s-2. Additionally, covariance terms having small correlations
take much longer to converge to a stable value .
Since the individual covariance terms do not correlate with sonic anemometer
measurements, it is unsurprising that the u* values computed from either
the six-beam or VAD techniques also show little correlation with u* from
the sonic anemometer (r2= 0.14–0.17).
Strengths and limitations of each strategy
Each of the scanning strategies evaluated herein has its own strengths and
limitations. One of the biggest limitations for all of the techniques except
vertical stares is that turbulence is assumed to be homogeneous over the area
of each scan. Thus, these techniques do not always work well in complex
terrain or differential land use where turbulence can significantly vary
spatially e.g.,. For the VAD and RHI techniques in
particular, spatial variations in the mean wind due to local drainage flows
e.g., can result in large
deviations from the spatially averaged mean wind. Since these methods are
unable to differentiate turbulent deviations from mean deviations, turbulence
is overestimated. In these situations, it may be possible to use arc segments
from the PPI scans to compute TKE over different radials
where the mean flow is homogeneous. With the current technology,
multi-Doppler measurements
e.g., are best able to
quantify turbulence at specific locations in complex terrain.
The spatial height resolution for the PPI, six-beam, vertical stare, and RHI
scans largely depends on scan geometry. Direct measurements of
w′2‾ from vertical stares can only be taken starting at the
height of the lowest range gate, and the spatial resolution is limited to the
range gate size. Since the six-beam technique presented here and in previous
studies e.g., uses a vertical
beam, the spatial resolution is limited by that beam the same as vertical
stares. Future studies may try removing the vertical beam and instead use
six-beams all at ϕ=45∘, or another ϕ, to make measurements at
a lower altitude. The vertical resolution of a PPI scan is dictated by its
ϕ: a larger ϕ results in a higher minimum measurement height,
reduced vertical resolution, greater height coverage, and reduced horizontal
scan footprint compared to typical eddy size. The residuals in the PPI scans
are more sensitive to w′2‾ for a larger ϕ and are more
sensitive to u′2‾ and v′2‾ for a smaller ϕ.
The height resolution of an RHI scan is truly customizable, as
u′2‾ and v′2‾ are computed by user-defined height
bins. Since RHI scans typically start or end at the horizon,
u′2‾ and v′2‾ can be calculated within a few meters
of the surface. On the other hand, this technique is especially susceptible
to nonturbulent horizontal variability along the scan due to complex terrain
and other effects, especially since small ϕ that cover large distances
horizontally are used.
Although a 20 min averaging interval is used here for comparison,
measurements for several of the techniques could be made much quicker.
Turbulence statistics from vertical stares and the six-beam technique are
computed through typical time series analysis; thus, the time series needs to
be long enough to ensure that the largest turbulent eddies pass through the
resolution volume to be captured see yet
short enough that the flow can be considered stationary .
Since the VAD and RHI methods compute turbulence through quantifying spatial
variability, they are not subject to these same sampling error limitations
when all scales of turbulence are captured in the scanning volume. With a
data rate of 2 Hz, each PPI scan with 240 beams takes 2 min to complete;
thus,
TKE can be measured in 2 min with a scan at ϕ=35.3∘, and each
velocity variance and covariance can be measured from two scans, taking
4 min. Since each RHI scan can be conducted in <1 min, u′2‾
and v′2‾ can be measured in ≈2 min.
The methods presented here measure velocity variances, but none currently are
able to distinguish atmospheric turbulence from submeso motions, including
waves. Since the value of TKE is calculated from u′2‾,
v′2‾, and w′2‾, the observed TKE may be a mixture
of turbulent and submeso variances and not always a measure of pure
atmospheric turbulence. Considering these submeso motions have been
predominantly documented within the nocturnal stable PBL when turbulence is
typically weak , the value of TKE defined as a measure
of turbulent motion may be overestimated when waves are present. Numerically
differentiating between nonturbulent and turbulent motions is difficult
and is best done through multiresolution decomposition or
wavelet analysis e.g.,. This
requires a high-resolution (>1 Hz) time series. Thus, out of the methods
analyzed, only the vertical stare has the data necessary to separate
turbulent from nonturbulent motions using established techniques.
Future directions for improving turbulence estimates
One of the main limitations of the six-beam technique using current
commercially available scanning Doppler lidars is the return time between
samples at the same beam position (≈27 s). When the six-beam
technique was performed, the scanner spent 78 % of the time slewing from one
beam to the next. Thus a two-axes hemispheric scanner is not be the best option
for running the six-beam technique. A wedge scanner that can quickly rotate
between beam positions is more appropriate, as the time between beams could
be minimized. This would yield a higher temporal resolution time series for
each beam, enabling a better method of noise removal through a structure-function
fit and possibly differentiating between
turbulent and nonturbulent variances through multiresolution decomposition
or wavelet analysis.
The RHI technique is best suited for measuring u′2‾ and
v′2‾ near the surface (<100 m), since the measurements need to
be made at a low angle. However, there is currently no method to remove
random errors from RHI measurements. Thus, it may be better in the future to
simply perform shallow horizontal stares where ϕ<20∘ to measure the
horizontal variances. Removing noise and correcting for volume averaging
effects would be straightforward , and it also may be able
to distinguish turbulence from nonturbulent motions.
Conclusions
The XPIA field experiment was conducted at the Boulder Atmospheric
Observatory in the spring of 2015. For 17 days at the end of the experiment,
a Leosphere Windcube 200S® continuously
alternated between a PPI, RHI, vertical stare, and six-beam scanning
strategy. Measurements from each scan type were used to calculate components
of the Reynolds stress tensor and other measures of turbulence. These Doppler
lidar turbulence measurements were compared to those from sonic anemometers
on a 300 m tower located 540 m from the lidar to evaluate the accuracy of
each technique.
Overall, TKE and velocity variances (i.e., u′2‾,
v′2‾, w′2‾) were more accurately measured by the
six-beam technique than the other methods. Six-beam measurements showed the
best agreement with the sonic-anemometer data across all ranges of turbulence
magnitude (r2≈0.78). Additionally, the error and bias of the
six-beam turbulence measurements did not significantly change with height. On
the other hand, the VAD measurements of TKE and velocity variances tended to
become more accurate with height. VAD-measured turbulence tended to be biased
low near the surface, and this bias decreased with height. This bias is
attributed to the inability of the PPI scan to resolve all scales of
turbulence near the surface, since the largest eddies extend beyond the
scanning circle. The scanning volume geometrically becomes larger with
height; thus, the PPI is better able to resolve all scales of turbulence and
make more accurate measurements of turbulent quantities farther from the
surface. Although the RHI-measured TKE and TI agreed most poorly with sonic
anemometer observations, it showed little bias (slope of linear regression
for TKE was 1.003) and still showed considerable skill in measuring
turbulence. The inability to quantify and remove random errors from the RHI
measurements led to an overestimate under time periods when turbulence was
weak (TKE <0.1 m2 s-2). The methods evaluated herein showed little
skill in measuring u* and velocity covariances (r2=0.15-0.17).
When selecting a scanning strategy in future experiments, one needs to
consider the desired turbulence measurements. While the RHI technique may be
the least accurate of the three evaluated, it is the only method that can
obtain measurements just above the surface. If a rapid update time is desired
(i.e., <5 min), the VAD technique may best address these needs. Vertical
stares and the six-beam technique use time series analysis to quantify
turbulence. If the temporal resolution is sufficiently high, established
techniques may be used to partition turbulent and nonturbulent variance,
for which no method currently exists for the RHI and VAD data.