COHERENT DOPPLER LIDAR MEASUREMENTS OF WIND FIELD STATISTICS ROD FREHLICH Cooperative Institute for Research in the Environmental Sciences (CIRES), University of Colorado, Boulder, CO 80309, U.S.A.
STEPHEN M HANNON and SAMMY W. HENDERSON Coherent Technologies, Inc. 655 Aspen Ridge Drive, Lafayette, CO 80026, U.S.A. (Received in final form 5 September, 1997)
Abstract. Coherent Doppler lidar measurements of wind statistics in the boundary layer are presented. The effects of the spatial averaging by the lidar pulse are removed using theoretical corrections and computer simulations. This permits unbiased estimates of velocity variance, spatial velocity structure functions, energy dissipation rate, and other point statistics of the velocity field. Key words: Lidar, Remote sensing, Turbulence, Velocity statistics
1. Introduction Remote measurements of boundary-layer processes have been produced by radar, lidars, and sodars. Measurements of velocity profiles, fluxes of heat and momentum, and other properties have been produced by remote sensing instruments (Eberhard et al., 1989; Gal-chen et al., 1992; Wilczak et al., 1997). Profiles of the point statistics of the atmospheric parameters provide the basic scaling parameters (Stull, 1988). The three-dimensional velocity spectral tensor (Kristensen et al., 1989) provides a complete description of second-order velocity statistics, however, multiple aircraft flights (Lenschow and Kristensen, 1989) are required to determine the parameters of the spectral tensor. High resolution measurements of the velocity field using Doppler lidar could provide the required information. The quality of statistical estimates for atmospheric turbulent processes depends on the spatial averaging over the finite measurement volume. This is particularly true for the turbulent velocity fields at the dissipation and inertial scales. Comparison with in situ point measurements is difficult above a few hundred metres and limited to airplanes, tethered balloons, and kite platforms. Airplanes are ideal for the velocity statistics in the horizontal plane but impractical for the spatial statistics in non-horizontal directions. Multiple kite-borne instruments (Balsley et al., 1992) could provide the three-dimensional wind statistics but aircraft clearance is required above 300 m. Higher resolution remote sensing measurements and comparison with in situ point measurements are critical needs for understanding boundary-layer processes (Wilczak et al., 1997). The development of solid-state coherent Doppler lidar has produced high resolution measurements of wind fields (Kavaya et al., 1989; Henderson et al., 1991, Boundary-Layer Meteorology 86: 233–256, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands.
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Table I Data runs Run
Start time (GMT)
Beam direction
Beam angle
Plotting symbol
A B C D E F
19:14:40 19:37:49 20:02:21 20:33:39 21:16:50 21:40:21
South East Vertical South East Vertical
45 45 0 45 45 0
# 4 2
N
1993; Targ et al., 1997; Frehlich et al., 1994, 1997). The performance of coherent Doppler lidar velocity estimates have been determined using computer simulations (Frehlich and Yadlowsky, 1994; Frehlich, 1996) and using statistical analysis of actual data (Frehlich et al., 1994, 1997). The effects of wind turbulence over the sensing volume of the lidar pulse were determined (Frehlich, 1997) using theoretical analysis and computer simulations assuming that the turbulent processes have a well-defined statistical description such as a Kolmogorov spatial spectrum with an outer-scale of turbulence. In this paper, we apply these results to velocity statistics from coherent Doppler lidar data.
2. Data Collection All data were collected with a 2-m coherent flash-lamp pumped Doppler lidar (Henderson et al., 1991, 1993) for a fixed lidar beam geometry (non-scanning) to concentrate on the estimation of the spatial velocity statistics for various beam angles. The pulse energy was 28 mJ, the pulse repetition frequency was 4.9 Hz, and the telescope diameter was 10 cm. The data were collected on October 8, 1996 at Lafayette, Colorado, U.S.A. The weather was clear and the surface temperature varied from 22–23 C over the complete observation time. Each data run required 24.5 minutes. The times of the runs and the orientation of the lidar beam are listed in Table I.
3. Coherent Doppler Lidar Velocity Estimates A coherent Doppler lidar estimates the radial velocity of the collection of aerosol particles illuminated by the lidar pulse as it travels through the atmosphere (for details, see Frehlich and Yadlowsky, 1994; Frehlich et al., 1994, 1997; Frehlich, 1997). The lidar beam is typically collimated and has a transverse spatial dimension d 5–10 cm for short ranges where diffraction is negligible. For ranges larger than Rc d2 = the beam dimensions are approximately R=d, where is the
=
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wavelength of the laser and R is the range of the measurement. For the eye-safe wavelength of 2 m the transverse dimension of the lidar beam is less than one metre for ranges less than 50 km. For a fixed instant of time, the spatial extent r of the lidar pulse along the transmit axis is typically 30 metres. For a velocity estimate from one lidar pulse, the illuminated aerosol region typically travels a distance p 50 m and the sensing volume of the lidar pulse is described by a narrow pencil shaped region. Therefore, the estimated velocity is a spatial average of the instantaneous radial velocity v r; t over the sensing volume of the pulse, where r is the distance along the transmitted beam axis and t denotes the time of the measurement. For Doppler radar, the transverse dimensions of the radar beam are large and analysis of wind statistics must include the behaviour of the radial velocity over a three-dimensional region. The spatial extent of the lidar pulse r is determined by the temporal power profile PL of the transmitted pulse, where denotes the time of the transmitted lidar pulse. For a solid-state lidar, the transmitted pulse is well approximated as (Frehlich et al., 1994, 1997)
=
=
( )
()
PL ( ) = P0 exp( 2=2 )
(1)
where P0 is the maximum power and [s] is the pulse 1/e width. The sensing volume of the illuminated aerosol targets is defined by
In(r) = W (r)=
Z
1 W (x)dx; 1
(2)
where
W (r) = PL (2r=c)
(3)
is the range weighting function of the Gaussian lidar pulse as a function of the distance r along the lidar transmit axis and c [m s 1] is the speed of light in the homogeneous atmosphere. For a Gaussian transmitted pulse (see Equations (1)–(3)), 2 In(r) = pc exp[ 4r 2 =( 2 c2 )] : and the Full-Width at Half Maximum (FWHM) of W
r =
p
ln 2c;
(4)
(r) is (5)
which defines the spatial extent of the lidar pulse along the transmit axis. As the pulse propagates through the atmosphere, the Doppler lidar data are digitized and converted to a complex data sequence with a sampling interval TS = 0.02 s. The range gate length p is defined as the distance the laser-illuminated
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aerosol volume moves during the observation time per estimate T M is the number of complex data samples per estimate, i.e.,
= MTS where
p = MTS c=2:
(6)
For all the data presented here, = 0.115 s, r = 28.7 m, M = 16, and p = 48 m. The radial velocity can be estimated over a maximum possible velocity search space
vsearch = 2T ;
(7)
S
which would be 52.25 m s 1 , however, for all the data presented here, vsearch = 26.125 m s 1 was sufficient. The maximum possible velocity search space determines the sampling interval TS and the range-gate length p determines M , the number of samples per velocity estimate. Various estimators for the radial velocity have been investigated (Rye and Hardesty, 1993a, b, 1994, 1997; Frehlich and Yadlowsky, 1994; Frehlich et al., 1994, 1997; Frehlich, 1996; Zarader et al., 1996). The Maximum Likelihood (ML) estimate is the velocity that maximizes the log-likelihood function of the data assuming that the radial velocity and the aerosol distribution over the range-gate are constant (Frehlich and Yadlowsky, 1994). This algorithm is similar to spectral domain estimators (Rye and Hardesty, 1994). The ML estimator has the best performance, i.e., the smallest estimation error. If the signal statistics from multiple pulses are accumulated (Rye and Hardesty, 1993a, b; Frehlich and Yadlowsky, 1994; Frehlich, 1996; Frehlich et al., 1994, 1997), the quality of the estimates improves and the maximum measurement range increases. We use the ML estimator with pulse accumulation for all the data considered here. Velocity estimates v R; t for v z; t are generated for range-gates centered on R k p and for a time interval centered on time t l t, where t is the time interval between velocity estimates. For all the data presented here t 1:02 s. Examples of ML velocity estimates for the vertical pointing lidar (Run C, Table I) are shown in Figure 1 and for a beam angle of 45 to the east (Run E, Table I) are shown in Figure 2. The vertical velocities in Figure 1 are characteristic of a convective boundary layer. If the average vertical velocity is zero, the average radial velocity for the nonvertically pointing beams is given by
=
^( )
( )
=
=
hv^(r)i = sin()hvx(r)i (8) where hi denotes the ensemble average over the random velocity field, the random lidar signal, hvx (r )i, is the component of the average horizontal velocity in the direction of the lidar beam, and is the angle between zenith and the lidar transmit
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Figure 1. Doppler lidar velocity estimates v^(R; t) as a function of range R and time t for a vertically pointing lidar (Run C, Table I). Each trace is offset by 4 m s 1 and zero velocity is indicated by a horizontal line.
axis. Profiles of the east and south component of the horizontal velocity as a function of height z R cos are shown in Figure 3. Also shown are the rawindsonde measurements from Denver taken a few hours after the lidar measurements. There is a well-defined mixed layer with an inversion height zi 1.0 km. The horizontal wind in the mixed layer is approximately 2 m s 1 , which results in few independent
=
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Figure 2. Doppler lidar velocity estimates v^(R; t) as a function of range R and time t for a lidar beam pointed 45 east of zenith (Run E, Table I) (altitude between 0.263 km and 0.636 km). Each trace is offset by 3 m s 1 and a velocity of -2 m s 1 is indicated by a horizontal line.
samples of the convective structures (see Figure 1) and a larger sampling error in the turbulent statistics (Lenschow et al., 1993).
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DOPPLER LIDAR MEASUREMENTS OF WIND FIELD STATISTICS
Figure 3. Profiles of average east component and south component of the horizontal velocity (see Table I). The rawindsonde data from Denver Colorado for Oct. 9, 0:0 GMT are shown as (X) and (+).
4. Statistical Description of Coherent Doppler Lidar Velocity Estimates
^( )
The statistical description of the Doppler lidar velocity estimates v R; t is a function of the lidar parameters and the radial velocity v z; t along the lidar beam axis. The Doppler frequency shift produced by the moving aerosol particles produce random fluctuations of the lidar signal, which is well described as a zeromean Gaussian random process or ‘speckle’ process (Frehlich, 1993) because the total optical field collected by the lidar is the superposition of many randomlyphased scattered fields from the aerosol targets in the measurement volume. The lowest level conditional statistic of v R; t is produced by taking an ensemble average over the random location (random phases of the backscattered laser field) of the illuminated aerosol particles in the range-gate for a given v r; t , lidar design, aerosol distribution, and atmospheric extinction (Frehlich, 1993, 1997). This ensemble average is denoted hia and can be estimated using data from a lidar with a high PRF (10 KHz). The lidar signal from shot to shot is uncorrelated because
( )
^( )
( )
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during the time interval between shots the random aerosols have moved on average more than the wavelength ( 2 m) of the laser light (Churnside and Yura, 1983; Frehlich, 1993). The ensemble average < v R; t >a is a function of the lidar parameters, the random radial velocity v r; t , and the atmospheric backscatter. The random variations of the atmospheric backscatter over the range-gate have been shown to have a negligible effect on the statistics of the velocity estimates (Rye, 1990). The random radial velocity v r; t and the lidar signal power are the most important parameters that govern the statistics of the velocity estimates. When the Doppler lidar signal power is high (see Figures 1 and 2), the velocity estimates can be represented as
=
^( ) ( ) ( )
v^(R; t) = hv^(R; t)ia + e(R; t)
(9)
( )
where e R; t is the estimation error due to the random fluctuations of the lidar signal (Frehlich and Yadlowsky, 1994; Frehlich, 1997). The estimation error is clearly visible in Figures 1 and 2 as the rapid fluctuations in the velocity time series. The statistical properties of some common velocity estimators were investigated using computer simulations (Frehlich, 1997) and assuming a Kolmogorov spectrum for the spatial spectrum of v r; t . The Probability Density Function (PDF) of the velocity estimates is well described by a two-component Gaussian PDF (Frehlich, 1997). For typical atmospheric velocity fields, an excellent approximation for < v R; t >a is the pulse-weighted velocity (Frehlich, 1997)
( )
^( )
< v^(R; t) >a = vwgt(R; t) = 1p
+p=2 v (r; t)dr p=2 pulse
Z R
R
(10)
where
vpulse(r; t) =
Z
1 v(x; t)In (r 1
x)dx :
(11)
( )
( )
The random function vwgt R; t is a convolution of the radial velocity v R; t with respect to the range R for each fixed time t. The kernel of the convolution is the convolution of the normalized pulse profile In r with a normalized rectangle of width equal to the range-gate dimensions p (Frehlich, 1997). The pulse-weighted approximation to Equation (9) is
v^(R; t) = vwgt (R; t) + e(R; t) :
()
(12)
Because of the high spatial and temporal resolution of coherent Doppler lidar velocity measurements, they are ideally suited for estimation of the spatial statistics of the atmospheric wind field. For locally stationary atmospheric turbulence, it is standard practice to decompose the velocity components into a mean velocity and a fluctuating or turbulent component (Monin and Yaglom, 1975; Eberhard et al.,
241
DOPPLER LIDAR MEASUREMENTS OF WIND FIELD STATISTICS
1989; Gal-chen et al., 1992). For the radial velocity along the lidar beam axis, the fluctuating component is
v0 (r; t) = v(r; t) hv(r)i
(13)
()
where the mean velocity hv r i is the total ensemble average over a locally stationary time interval. An important description of the turbulent velocity is the covariance defined by
Bv (s; r) = hv0 (r s=2)v0 (r + s=2)i
(14)
where hi denotes ensemble average over the locally stationary velocity fluctuations,
s is the separation of the two observation points, and r is the centroid of the observation points. The variance of the turbulent velocity v 0 (r ) is then v2 (r) = Bv (0; r) = hv0 (r)2i:
(15)
Another important description of the turbulent velocity is the structure function defined by
Dv (s; r) = h[v0 (r s=2) v0 (r + s=2)]2i:
(16)
For high Reynolds number locally homogeneous and isotropic turbulence (Monin and Yaglom, 1975)
Dv (s; r) = Cv 2=3(r)s2=3
(17)
()
for s in the inertial range, where Cv 2 is the Kolmogorov constant and r [m2 s 3 ] is the local energy dissipation rate centered at r . A useful empirical model for the structure function is
Dv (s; r) = 2v2(r)[s=L0 (r)] where L0
(18)
(r) is a measure of the outer scale of turbulence,
(x) = ( x)2=3 1 + ( x)
2 3
(19)
and h
(r) = 2v2(r)=Cv
i3=2
=L0 (r)
(20)
is required for equivalence of Equations (17) and (18) when s = v2wgt R = Bwgt 0; R and (Frehlich, 1997)
( ) ( ) ^ ( ) ( Bwgt(0; R) = v2 (R)K (; )
)
( )
(43)
where
K (; ) = 2
1
Z 0
F (x; )[1 ( x)]dx
=
p=L0. An unbiased estimate for the variance and fluctuations v 0 R is given by
(44)
v2 (R) of the velocity
( ) ^v2 (R) = ^v2wgt (R)=K (; ) : (45) The scaling factor K (; ) has a weak dependence on the parameter = p=L0 when < 1. The estimate ^v2 (R) is equivalent to an in situ point estimate of v2 (R) whenever p < L0 (R) or when accurate estimates of L0 can be extracted from the data.
4.5. PROFILES OF VELOCITY STATISTICS
( )
Profiles of energy dissipation rate R can be produced from Doppler lidar estimates Dvwgt s; R (Equation (37)) provided that the pulse-weighted approximation vwgt r; t is valid (see Equation (10)), or equivalently, that the theoretical calculation Equation (38) for the measured structure function is valid. For a typical convective boundary layer, R has a weak dependence on the range R. Estimates R for R are produced by minimizing the mean-square-error between the structure function estimates Equation (37) and the pulse-weighted approximation Dwgt s; R Equation (38) over a sliding range window, which was chosen as 340 m for the results presented here. Profiles of velocity variance v2 z and z as a function of the height z are shown in Figures 9 and 10 for all the lidar beam geometries in Table I. The vertical velocity variance in Figure 9 is consistent with a convective boundary layer (Stull, 1988). In the middle of the mixed layer, the velocity variance for the 45 beams are less than the vertical velocity variance. The energy dissipation rate for all three beam geometries are not the same, which implies non-stationarity over the total observation time of 70 min or anisotropy of the wind-field statistics over 100–200 m scales (Kristensen et al., 1989). Near the surface, the velocity variance for the 45 beams increases indicating that the contribution from the horizontal velocity components is becoming larger than that for the vertical velocity. At the beginning of the shear layer at 1.2 km altitude, the velocity variance and energy dissipation rate increase.
^
( )
^( )
(
(
( ) )
)
( )
()
()
DOPPLER LIDAR MEASUREMENTS OF WIND FIELD STATISTICS
251
Figure 9. Profiles of radial velocity variance v2 and energy dissipation rate for data Runs A, B, and C (Table I).
The data in Figure 10 show weaker turbulence conditions, thus indicating a collapse of the convective boundary layer in the late afternoon. As in Figure 9, the velocity variance and energy dissipation rate increase at the beginning of the shear layer at 1.2 km altitude. The energy dissipation rate for all three beam geometries are approximately the same up to an altitude of 0.6 km. At higher altitudes, is different for the different beam geometries. The mixed-layer parameters have been estimated from profiles of vertical velocity variance (Stull, 1988; Seibert and Langer, 1996; Angevine et al., 1994). The profiles of vertical velocity variance have been modeled as (Kristensen et al., 1989)
w2 (z ) = 1:44w2 (z=zi )2=3 (1
0:7z=zi
)2 + 109 wu2
2
(46)
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Figure 10. Profiles of radial velocity variance v2 and energy dissipation rate for data Runs D, E, and F (Table I).
where w is the convective velocity scale (related to the surface heat-flux), zi is the inversion height, and u is the friction velocity. The profiles of energy dissipation rate have been approximated by (Kristensen et al., 1989) 3 w (z) = z 0:75 + 1:84(z=zi ) i
2=3
(1:0
2 3=2 u z=zi ) w2 : 2
(47)
2 for a convective boundary In Figure 11, profiles of vertical velocity variance w layer (data Run C, Table I, vertically pointing lidar beam) are compared with the best-fit model Equation (46). The parameters of this best-fit model are used to predict the profile of energy dissipation rate using Equation (47). The inversion height zi from the best fit agrees with the location of the beginning of the shear layer (see Figure 3). The error bars for the vertical velocity variance are large, which reflects the skewness of the vertical velocity and a low number of independent samples (Lenschow et al., 1993) (see Figure 1). A longer data set is required to
DOPPLER LIDAR MEASUREMENTS OF WIND FIELD STATISTICS
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2 Figure 11. Profiles of vertical velocity variance w and energy dissipation rate with 1- error bars ) Equation (46) and the resulting predictions of for data Run C (Table I) and the best-fit model ( ). Equation (47) (
produce more independent samples of the vertical velocity. The agreement of the data and model for z is poor. This may be due to the high shear layer at the top of the mixed layer or the anisotropy of the wind field statistics, i.e., the magnitude of the structure function in the vertical beam is smaller than the structure function for the horizontal wind components.
()
5. Summary and Discussion Coherent Doppler lidar measurements of the spatial wind-field statistics can be converted to estimates of the point velocity statistics using corrections for the spatial averaging of the velocity field by the finite dimensions of the lidar pulse, if a valid theoretical model of the spatial structure function can be determined. For the convective boundary layer, a Kolmogorov model with an outer-scale of turbulence
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(see Equation (18)) is consistent with the data (see Figures 6–8). Estimates of the energy dissipation rate and the variance v2 of the radial component of the velocity can be extracted when the outer-scale of turbulence L0 is larger than the spatial dimensions r of the lidar pulse. Smaller r is desirable to sample fully the structure function in the inertial range and to permit measurements near the surface where the outer scale L0 is small. Profiles of the variance v2 of the radial velocity and the energy dissipation rate for different beam geometries provide information on the boundary-layer parameters and the spectral velocity tensor (Kristensen et al., 1989). Because the different lidar beam geometries sample different components of the spectral velocity tensor, comparison with in situ measurements will require a suite of suitably spaced point velocity measurements, e.g., from a set of instrumented kite platforms (Balsley et al., 1992). Other scanning configurations such as the velocity azimuth display (Eberhard et al., 1989; Gal-chen et al., 1992) would provide profiles of momentum flux and should produce estimates of velocity statistics with less statistical fluctuations because more independent samples of the velocity fluctuations can be observed (Lenschow et al., 1993). This would require a coherent Doppler lidar with a high pulse repetition frequency. Solid-state coherent Doppler lidar measurements of the troposphere have been demonstrated (Frehlich et al., 1997). Extending the analysis presented here to tropospheric wind statistics would help resolve the statistical description of gravity-wave processes (Gardner, 1996).
Acknowledgements This work was supported by the National Science Foundation, the Army Research Office, and the National Aeronautics and Space Administration, Marshall Space Flight Center under Research Grant NAG8-253 (Michael J. Kavaya, Technical Officer). System improvements and field measurements were funded by Wright Laboratories, WPAFB, OH (Richard D. Richmond, technical monitor). The authors acknowledge the contribution of Jim Magee and Phil Gatt in collecting the data. The interpretation of the data was improved following useful discussion with Don Lenschow and Peggy Lemone.
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