Measurement of Atmospheric Turbulence by 2-μm Doppler Lidar

Igor Smalikho Institute of Atmospheric Physics, DLR, Wessling, Germany

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Friedrich Köpp Institute of Atmospheric Physics, DLR, Wessling, Germany

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Stephan Rahm Institute of Atmospheric Physics, DLR, Wessling, Germany

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Abstract

Two methods for the estimation of the turbulence energy dissipation rate (TEDR) from data measured by a 2-μm coherent Doppler lidar are described in this paper. Based on data measured at the Tarbes-Lourdes-Pyrénées International Airport in summer 2003, height profiles of TEDR have been retrieved. The results of TEDR estimation both from the Doppler spectrum width and from the velocity structure function are compared. Moreover, the experiment has been treated by numerical simulation and the theoretical results have been used for verification of the described methods.

Corresponding author address: Friedrich Köpp, Institut für Physik der Atmosphare, DLR, Oberpfaffenhofen, D-82234 Wessling, Germany. Email: friedrich.koepp@dlr.de

Abstract

Two methods for the estimation of the turbulence energy dissipation rate (TEDR) from data measured by a 2-μm coherent Doppler lidar are described in this paper. Based on data measured at the Tarbes-Lourdes-Pyrénées International Airport in summer 2003, height profiles of TEDR have been retrieved. The results of TEDR estimation both from the Doppler spectrum width and from the velocity structure function are compared. Moreover, the experiment has been treated by numerical simulation and the theoretical results have been used for verification of the described methods.

Corresponding author address: Friedrich Köpp, Institut für Physik der Atmosphare, DLR, Oberpfaffenhofen, D-82234 Wessling, Germany. Email: friedrich.koepp@dlr.de

1. Introduction

The development and employment of remote sensing devices and methods based on the Doppler effect allow new knowledge of dynamic processes in the atmosphere to be gained. Thereby, the coherent Doppler lidars have the best possibilities for such investigations in the clear atmosphere because of the high spatiotemporal resolution of the measured characteristics. These lidars are used for measurements of the wind field (Köpp et al. 1984; Hall et al. 1984; Keeler et al. 1987; Hawley et al. 1993; Frehlich et al. 1994; Banakh et al. 1995; Werner et al. 2001; Reitebuch et al. 2001; Smalikho 2003), the atmospheric turbulence (Eberhard et al. 1989; Cal-Chen et al. 1992; Gordienko et al. 1994; Smalikho 1995, 1997; Banakh and Smalikho 1997; Frehlich et al. 1998; Banakh et al. 1999; Frehlich and Cornman 2002; Davies et al. 2004), and the aircraft wake vortices (Köpp 1994, 1999; Constant et al. 1994; Hannon and Thomson 1994; Brockman et al. 1999; Harris et al. 2000, 2002; Vaughan and Harris 2001; Keane et al. 2002; Köpp et al. 2003, 2004, 2005).

The 2-μm pulsed coherent lidar (Henderson et al. 1993) has proved to be a powerful tool for wake-vortex sounding. Using this device, the lidar group of the German Aerospace Center [Deutsches Zentrum für Luft- und Raumfahrt (DLR)] carried out successful measurements of the characteristics of the wake vortices generated by large transport aircraft (Köpp et al. 2004, 2005). One of the important factors influencing the wake vortices is the atmospheric turbulence. Therefore, the extraction of information about both the wake-vortex parameters and the background wind turbulence can be helpful for the analysis of the atmospheric effect on the wake vortices. In accordance with theory (Holzäpfel et al. 2001, 2003; Gerz et al. 2002; Holzäpfel 2003; Holzäpfel and Robins 2004), the effective dwell time of the wake vortices only depends on one turbulent characteristic of the atmosphere: the turbulence energy dissipation rate (TEDR).

Several methods of TEDR estimation from data measured with a pulsed Doppler lidar are known, for example, the methods based on measurements of the spatial structure function of the wind velocity (Banakh and Smalikho 1997; Frehlich et al. 1998; Frehlich and Cornman 2002) and the width of the Doppler spectrum (Frehlich and Cornman 1999). In this paper, we describe the methods of TEDR estimation from data measured with the 2-μm pulsed Doppler lidar with laser beam scanning in a vertical plane. Such measurement geometry allows the extraction of information about both the aircraft wake vortices and the atmospheric turbulence from the lidar data only. The following two approaches are used: estimation of the TEDR from 1) the velocity structure function (VSF method) and 2) the Doppler spectrum width (DSW method). From the data measured at Tarbes, France, in summer 2003, the TEDR profiles as function of height have been retrieved. In the paper, the results obtained by these methods are compared.

2. Theory

a. Estimation of the signal power spectrum and derived parameters

The principle of pulsed Doppler lidar operation is based on the emission of laser pulses into the atmosphere and the coherent detection of the laser radiation backscattered by aerosol particles moving with the wind. The measured power of the photocurrent arising in the chain of the receiving system in the frequency range around the intermediate frequency contains the information about the speed of the moving aerosol particles in the sensing volume. An example of raw data (one shot) measured with the 2-μm pulsed Doppler lidar from DLR is shown in Köpp et al. (2004, their Fig. 2). The description of the lidar and its parameters is given by Köpp et al. (2004, 2005). The measured data are discrete with the sampling interval Ts = 2 ns and consists of two parts: the monitor signal J(i)M(mTs) (m = 0, 1, 2, . . . , mB − 1) and the backscatter signal (also including the noise component of the photocurrent) J(i)B(mTs) (m = mB, mB + 1, mB + 2, . . .), where the index i indicates the shot number and m is the sample number.

The intermediate frequency δf = | fLfP|, where fL is the frequency of the local oscillator and fP is the frequency of the probing laser pulse, can be estimated from the power spectrum of the monitor signal. Parameters as the duration σp and energy Up of the pulse and the position of the pulse center (sampling number m0) are also estimated from the monitor signal using fast Fourier transform (FFT). For the monitor signal, the spectrum versus frequency f and the power versus time t have an almost Gaussian form. Therefore, the power of the monitor signal PM(t) can be described by the equation
i1520-0426-22-11-1733-e1
and the monitor signal can be represented in the form
i1520-0426-22-11-1733-e2
where t = (mm0)Ts and Ψ0 is a parameter equal to the phase at t = 0. The time dependence of Ψ0 can be neglected. Taking into account the conditions Tsσp and σpδf ≫ 1, and determining the monitor signal power spectrum as SM( f ) = |∫+∞−∞ dt J(i)M(t) exp(−2πjft)|2, where j = −1 and the frequency f ∈ [0, (2Ts)−1], we receive from Eqs. (1) and (2):
i1520-0426-22-11-1733-e3
where
i1520-0426-22-11-1733-e4
is the width of the monitor signal power spectrum.

For estimation of the pulse energy (after calibration), the equation Up = 2−1+∞−∞ dt [J(i)M(t)]2 or Up = ∫(2Ts)−10 df SM( f ) can be used. The parameter σp can be estimated both from the measured power M(t) (using σ̂p = ∫+∞−∞ dt M(t)/[M(0)π]) and from the measured spectrum ŜM( f ) (estimate σ̂p as the second moment of the spectrum). In these two ways, practically the same estimates σ̂p are obtained from the experimental data.

The raw data can be considered as a function of the sampling number (m), or the time after pulse emission into the atmosphere [t = (mm0)Ts], or the range (R = ct/2, where c is the speed of light), that is, the distance between the lidar and the center of the sensing volume. Because Ts = 2 ns, the sampling length cTs/2 = 0.3 m. In our experiments, the minimal range for the measurement of the backscatter signal [J(i)B(mTs)] is (mBm0)Tsc/2 = 370 m.

The velocity of the aerosol particles in the sensing volume is estimated from the power spectrum of the backscatter signal. For the sensing volume centered at the distance R from the lidar we select M samples of the backscatter signal around the sample with number mR = [R/(cTs/2)] + m0 and multiply this array by the window function W(mTs),
i1520-0426-22-11-1733-e5
obtaining the estimate of the signal power spectrum (or Doppler spectrum)
i1520-0426-22-11-1733-e6
where Δf = (MTs)−1 is the width of a single frequency bin; k = 0, 1, 2, . . . , M/2, and the spectral bandwidth ΔfM/2 ≡ (2Ts)−1 = 250 MHz. The frequency resolution can be determined as Δfr = T−1eff, where Teff = ∫+∞−∞ dt W(t)/W(TsM/2) is the effective time of spectrum measurement. In the case of a rectangular window with M elements, Teff = MTs and, therefore, Δfr = Δf. Here the Gaussian window is used,
i1520-0426-22-11-1733-e7
where m′ = 0, 1, 2, . . . , M − 1. For TsσWTsM/2, the effective time of spectrum measurement becomes
i1520-0426-22-11-1733-e8
and therefore the frequency resolution Δfr = (2πσW)−1. Let σW = σp and σp = 0.24 μs (as it was in our experiments), then Δfr = 1.67 MHz. By increasing M, Δf can be unlimitedly decreased at constant Δfr. For the estimation of each Doppler spectrum we used M = 2048. In this case, the frequency bin width Δf = 0.244 MHz, and it is approximately 7 times less than Δfr. If some points of the window are outside the data (backscatter signal), the technique of zero padding can be applied.
To decrease the fluctuations of the Doppler spectrum estimate, the method of spectral accumulation is used:
i1520-0426-22-11-1733-e9
where Na is the number of shots. The obtained spectra are normalized to the mean spectral noise level. This noise level is determined at frequencies outside the region of the peak containing the information about the wind velocity, after averaging a large number of realizations. The noise level can also be determined by the method described by Hildebrand and Sekhon (1974). Let Dfk, R) be the normalized Doppler spectrum. Then, we select Nch = 205 spectral channels around the mean intermediate frequency δf = N−1aΣNai=1δfi. In this way, the spectral bandwidth is reduced from 250 MHz to B = ΔfNch = 50 MHz.
From the measured spectra the normalized signal power S(R), the mean frequency D(R), and the square of the Doppler spectrum width σ̂2f(R) are estimated as zero, first, and second spectral moments, respectively,
i1520-0426-22-11-1733-e10
i1520-0426-22-11-1733-e11
and
i1520-0426-22-11-1733-e12
where k1 (k2) is the number of spectral channels [left (right)] from the channel km at the spectrum estimate maximum, and it is determined at the intersection point of the spectrum estimate and the selected noise threshold nth. In our case, the estimate of the signal-to-noise ratio (SNR; ratio of the signal power to the mean noise power in the spectral bandwidth of 50 MHz) is given by
i1520-0426-22-11-1733-e13
Using the Doppler equation V = (λ/2) f, where V is the radial velocity, λ is the laser wavelength, and f is the frequency shift caused by the motion of the aerosol particles, one can obtain the estimates of the radial velocity
i1520-0426-22-11-1733-e14
and the square of the Doppler spectrum width in the velocity domain
i1520-0426-22-11-1733-e15
For estimation of (R) and D(R) we can chose nth = 1 (mean noise level in the normalized spectrum). However, the noise fluctuations in the measured spectrum can be the source of large errors, when turbulent characteristics are estimated from the measured Doppler spectrum width, especially in the case of very low turbulence. To avoid the effect of these noise fluctuations we set the noise threshold above the mean noise level (nth > 1). Concrete values for nth will be determined by simulation in section 2c.

b. Estimation of turbulence parameters

The estimates D(R) and σ̂2sw(R) are dependent on turbulent variations of the wind. From a set of such estimates one can obtain corresponding statistical characteristics that are related to parameters of the atmospheric turbulence. The effects of spatial averaging of the radial wind velocity over the sensing volume and estimation errors resulting from backscatter signal fluctuations and noise have to be taken into account.

For the spectral accumulation 25 shots (Na = 25) are used. Taking into account that the pulse repetition frequency (PRF) is 500 Hz, the measurement time of the spectrum D( f, R) is Na/PRF = 50 ms. During this time the distribution of the radial velocity inside the sensing volume is almost constant. In this case, it is convenient to represent the spectrum estimate in the form
i1520-0426-22-11-1733-e16
where SD( f, R) = 〈D( f, R)〉P is the spectrum estimate averaged over all random parameters except the radial wind velocity (conditional ensemble averaging; Frehlich 2001) and SD( f, R) = D( f, R) − 〈D( f, R)〉P are the spectrum estimate fluctuations, f = Δfkδf. Because of the accumulation [Eq. (9)], the standard deviation of these fluctuations is approximately Na = 5 times less than the mean spectrum SD( f, R).
By analogy with Eq. (16), the estimates D(R) and σ̂2sw(R) can be represented in the form
i1520-0426-22-11-1733-e17
and
i1520-0426-22-11-1733-e18
where VD(R) and σ2sw(R) are the first and second spectral moments obtained from Eqs. (10)(15) after replacing Dfk, R) − nth by SD( f, R) − 1 and Σk=kmk1km+k2 Δf by ∫−∞ df. The second terms on the right-hand side of Eqs. (17) and (18), e(R) and E(R), are related to the spectrum fluctuations SD( f, R). They will be called random errors of the estimation of VD(R) and σ2sw(R), respectively.
From the equation for the coherently detected backscatter signal J(i)B(t) (Salamitou et al. 1995; Banakh and Smalikho 1997; Frehlich 1997), the equation for the mean spectrum SD( f, R) can be obtained using conditional ensemble averaging. Then we get
i1520-0426-22-11-1733-e19
i1520-0426-22-11-1733-e20
where Vr(z) is the radial wind velocity at point z, QS(z) is the function characterizing the spatial averaging over the sensing volume, σ0 is the spectrum width (in the velocity space) in the case of Vr(z) = constant, and
i1520-0426-22-11-1733-e21
is the term related to the inhomogeneity of the wind field inside the sensing volume. In our case of the Gaussian pulse and window [see Eqs. (1), (7)], the function QS(z) also has a Gaussian shape:
i1520-0426-22-11-1733-e22
where
i1520-0426-22-11-1733-e23
is the longitudinal size of the sensing volume, and σ20 is described by the equation
i1520-0426-22-11-1733-e24
In Eq. (24) we take into account the spectral broadening resulting from the probing pulse (first term in square brackets), the window effect (second term), and the variance of the intermediate frequency (third term).
Let us represent the radial velocity in the form Vr(z) = 〈Vr(z)〉 + V ′r(z), where the angular brackets stand for ensemble averaging and V ′r = Vr − 〈Vr〉 are fluctuations. For the case of wind shear we assume that in the interval [R − Δz/2, R + Δz/2] the mean velocity can be described by the equation
i1520-0426-22-11-1733-e25
where μ = dVr〉/dz. By substitution of Eq. (25) into Eq. (19) and averaging, we find 〈VD(R)〉 = 〈Vr(R)〉. From Eqs. (21) and (25) we obtain
i1520-0426-22-11-1733-e26
where
i1520-0426-22-11-1733-e27
and
i1520-0426-22-11-1733-e28
are terms related to the spectral broadening resulting from wind shear (σ2s) and turbulence (σ2t); V ′D = VD − 〈VD〉.
The spatial structure of the turbulence is described by characteristics like the structure function DV(r), variance σ2V, correlation function BV(r), and spectrum SV(κ) of the wind velocity fluctuations. For the case of radial wind velocity and homogeneous turbulence Dυ(r) and Bυ(r) can be written as
i1520-0426-22-11-1733-e29
i1520-0426-22-11-1733-e30
where σ2V = 〈(V ′r)2〉 = BV(0).
In the inertial subinterval of the turbulence (LηrLV or L−1ηκL−1V, where Lη is the inner scale of turbulence and LV is the integral scale of turbulence), the structure function and the spectrum of wind velocity are only depending on the TEDR ɛ. They are described by the Kolmogorov law (Lumley and Panofsky 1964; Monin and Yaglom 1975)
i1520-0426-22-11-1733-e31
i1520-0426-22-11-1733-e32
where CK ≈ 2 is the Kolmogorov constant.
There are many spectral models describing the full range of turbulent inhomogeneities of the wind, in particular, the von Kármán model (Lumley and Panofsky 1964; Vinnichenko et al. 1973),
i1520-0426-22-11-1733-e33
where the integral scale of turbulence LV is determined as
i1520-0426-22-11-1733-e34
which is proportional to the outer scale of turbulence Lo = 1.34LV (Frehlich 1997).
From Eqs. (32) and (33) we have
i1520-0426-22-11-1733-e35
In the case of homogeneous turbulence at very large separation r→∞, the structure function reaches the saturation: DV(∞) = 2 σ2V.
In Eq. (17), VD(R) and e(R) are statistically independent (Frehlich 2001). Then we obtain the variance and the structure function of the radial velocity of Doppler lidar estimate:
i1520-0426-22-11-1733-e36
and
i1520-0426-22-11-1733-e37
where σ2VD = 〈( V ′D)2〉, σ2e = 〈e2〉 (for unbiased estimate 〈e〉 = 0), DVD(r) = 〈[V ′D(R + r) − V ′D(R)]2〉, and De(r) = 〈[e (R + r) − e (R)]2〉. In accordance with Eqs. (18), (20), and (26), the mean square of the Doppler spectrum width is determined as
i1520-0426-22-11-1733-e38
From Eqs. (19), (29), (30), (33), and (35) we obtain
i1520-0426-22-11-1733-e39
i1520-0426-22-11-1733-e40
where
i1520-0426-22-11-1733-e40a
and
i1520-0426-22-11-1733-e41
where
i1520-0426-22-11-1733-e41a
According to Eqs. (35) and (41), the ratio σ2t/σ2V is a function of the ratio LVz,
i1520-0426-22-11-1733-e42
where FW(LVz) = (1.972)2/3C−1KL−2/3VGWz, LV).

On the basis of the obtained equations it is possible to develop two methods of TEDR estimation from the measured lidar data.

  1. The method of TEDR estimation from the velocity structure function (VSF method) is described.

    This method has been described in detail by Frehlich and Cornman (2002). From the measured data, the estimates of the structure functions D(iΔr) and e(iΔr) can be derived. The measured difference D(iΔr)−e(iΔr) is fitted to the function DVD(iΔr) given by Eq. (40), in order to find ɛ and LV in such a way that the best fitting takes place for both variables. The turbulent parameters (estimates ε̂ and V) are determined by minimizing the functional (Frehlich and Cornman 2002)
    i1520-0426-22-11-1733-e43
    where
    i1520-0426-22-11-1733-e44
    This algorithm is used with values of Δr = 30 m and I = 16.
  2. The method of TEDR estimation from the Doppler spectrum width (DSW method) is described.

    Knowing the parameters σ20, σ2s and 〈E〉, and using Eq. (38) and the measured mean square of the spectrum width 〈σ̂2swE, one can estimate the mean turbulent broadening of the spectrum as
    i1520-0426-22-11-1733-e45
    where 〈xE denotes the estimate of the mean value 〈x〉. The measurement results of the variances σ̂2D and σ̂2e allow the estimate of the radial velocity variance to be obtained [see Eqs. (36) and (39)]:
    i1520-0426-22-11-1733-e46

    Using the ratio σ̂2t/σ̂2V and Eq. (42), the integral scale of turbulence V can be determined. When σ̂2V and V are known, the estimate ε̂ can be calculated by Eq. (35).

Information about σ20 is extracted from the data of the monitor signal [see Eqs. (3) and (24)], and σ2s can be found by using Eq. (27). Only the parameter 〈E〉 in Eq. (45) is not determined. If the noise threshold is set to nth = 1 for the estimation of the second moment in Eq. (12), then 〈E〉 ≠ 0. Although at SNR ≥ 1 this error 〈E〉 ≪ σ20, we cannot neglect it in Eq. (45), especially in the case of weak turbulence. In our work, we select the noise threshold nth for minimization of the error 〈E〉 [when we can neglect 〈E〉 in Eq. (45)] on the basis of numerical simulation results.

c. Numerical simulation

For the computer simulation of random realizations of the backscatter signal J(i)B(mTs) (Banakh and Smalikho 1997; Frehlich 1997) and simulated data processing, the following parameters were used: σp = σW = 0.2488 μs, σδ = 0.1 MHz (i.e., σ0 ≈ 0.647 m s−1 and Δz ≈ 94 m), and Na = 25. Applying the processing procedures given in section 2a, we obtained the estimates D(R) and σ̂2sw(R). In the beginning, lidar signals for the case of homogeneous wind [Vr(z) = constant or ɛ = 0] were simulated, and estimates of σ̂2sw(R) at various noise thresholds nthwere obtained. Then, the error 〈E〉 = 〈σ̂2sw〉 − σ20 as a function of nth was calculated. Figure 1 shows the normalized estimation error of the mean square of the Doppler spectrum width versus the noise threshold at different SNR. The intersection of the curves with the zero level (dashed line) occurs in the interval of 1.12 < nth < 1.14. The optimal noise threshold noptth, determined as 〈E(noptth)〉 = 0, at fixed Na is a function of the SNR. Using this optimal threshold for the estimation of the second spectral moment, 〈E〉 can be neglected in Eq. (45).

On the basis of the numerical simulation results, the accuracy of TEDR estimation by the VSF and DSW methods were investigated. As in the paper of Frehlich and Cornman (2002), the lidar signals were simulated for the case of a laser beam pointing at a fixed elevation angle and frozen homogeneous velocity field, which is moving in the direction perpendicular to the beam with a mean velocity of 3 m s−1. For the simulation of 2D wind field realizations, we assumed isotropic turbulence and used the von Kármán model for a 2D velocity spectrum with LV = 150 m and various ɛ. The parameters D(Rk) and σ̂2sw(Rk) are estimated at ranges Rk = R0 + (k − 1)ΔR, where k = 1, 2, . . . , 40, ΔR = 30 m. For agreement with the conditions of the real experiment, as described in section 3, we assume that the estimates D(Rk) and σ̂2sw(Rk) are obtained every 0.5 s, resulting in 40 × 600 = 24 000 estimates within a period of 5 min. We also take into account the range dependence of the SNR observed on the first day of the experiment (see Fig. 5a), whereby the maximum and minimum SNR equals 3 and 1, respectively. Using 24 000 estimates of D(Rk)[σ̂2sw(Rk)], the TEDR was determined by the VSF (DSW) method described in section 2b. In the case of the DSW method, the estimates σ̂2sw(Rk) were obtained on the basis of the calculated dependence of noptth versus SNR, and at a fixed noise threshold of nth = 1.14 (optimal for SNR = 1) and nth = 1.13 (optimal for SNR = 3).

Assuming that ε̂1 and ε̂2 are TEDR estimates obtained by the VSF (ε̂1) and DSW (ε̂2) methods from the same simulated (measured) lidar data, the data for different true TEDR ɛ was simulated. For each ɛ value, 500 independent estimates for ε̂1 and ε̂2 were used for the statistical analysis of the results. The following statistical characteristics of TEDR estimation were calculated: the relative bias
i1520-0426-22-11-1733-e47
the relative error
i1520-0426-22-11-1733-e48
and the discrepancy parameter of TEDR estimates by VSF and DSW methods
i1520-0426-22-11-1733-e49

Figure 2 shows results for the relative bias of TEDR estimation by the VSF (circles) and DSW (squares) methods. It can be seen that the VSF estimator gives a small bias (less than 3%) for any turbulence strength. There is also an insignificant bias for the case of the DSW method, when the optimal noise threshold noptth is used (squares connected by solid lines). However, a relatively small deviation of the noise threshold from the optimal one can produce a large bias of the TEDR estimation, especially in the case of weak turbulence TEDR < 10−4m2 s−3. There is an overestimation for nth = 1.13 (curve 1) and an underestimation for nth = 1.14 (curve 2).

The relative errors of TEDR estimation by the VSF (full circles) and DSW (squares) methods are shown in Fig. 3. In the case of the DSW estimator, only the optimal noise threshold noptth was used. It can be seen that the relative error of the TEDR estimation from the velocity structure function is almost constant for any TEDR ∈ [10−5−10−2] m2 s−3, and equals approximately 20%. At ɛ > 10−4m2 s−3, the error of the TEDR estimation by DSW (∼10%) is approximately 2 times less than that in the case of the VSF method. It is interesting to note that the standard deviation of σ2u(R) [see Eq. (21)] is also 2 times less than the standard deviation of the velocity difference square [VD(R + r) − VD(R)]2, where r = 30 m. Moreover, Fig. 3 shows the results for the discrepancy parameter d (circles connected by dashed lines). The same parameter can be calculated from the data of the real experiment. The results of the discrepancy parameter d for both the numerical simulation and the field experiment will be compared in section 3.

3. Experiment

a. Measurement and data processing

In summer 2003, the lidar group of DLR carried out measurements with the 2-μm pulsed Doppler lidar at the airfield of the Tarbes-Lourdes-Pyrénées International Airport in France, in the frame of the European Community (EC) Aircraft Wing with Advanced Technology Operation (AWIATOR) project. One of the main tasks of this project was the experimental investigation of wake vortices generated by a large transport aircraft. This experiment offered the possibility to simultaneously measure the atmospheric turbulence and the wake-vortex behavior. The measurements were carried out on 2 days: 27 and 28 August. The lidar worked continuously during the following intervals: 1200–1216, 1300–1316, 1800–2020 (27 August), and 0806–1015 (28 August) local time (LT). The weather conditions favored the success of this experiment because of the rather high concentration of aerosol particles in the atmosphere and fine weather (except for the first 30 min of measurements on 28 August, when the maximum height of probing ∼600 m was determined by the lower bound of clouds).

The measurement geometry is sketched in Fig. 4. The laser beam was scanned in the vertical plane by turning up and down with an angular speed of 20 s−1. The minimum and maximum of the elevation angle ϕ was 00 and 300, respectively (00ϕ ≤ 300). The main parameters of the lidar and the data processing are summarized in Table 1.

From the measured data the Doppler spectra D( f, R) were estimated at ranges of Rk = R0 + kΔR, where R0 = 500 m; ΔR = 30 m; k = 0, 1, 2, . . . , K − 1; and K = 40; and at elevation angles ϕl = lΔϕ, where l = 0, 1, 2, . . . , 300, Δϕ = 0.10 (/dt = 20 s−1, PRF = 500 Hz, Na = 25). Using the estimation algorithms described by Eqs. (10)(15), we obtained a set of data for the signal-to-noise ratio (Rk, ϕl, n)≡ (hkl, n), the radial wind velocity D(Rk, ϕl, n) ≡ D(hkl, n), and the square of the Doppler spectrum width σ̂2sw(Rk, ϕl, n) ≡ σ2sw(hkl, n), where hkl = Rk sinϕl is the height and n = 1, 2, 3, . . . is the scan number.

The mean signal-to-noise ratio was estimated as
i1520-0426-22-11-1733-e50
where Ns is the number of scans of the laser beam during the measurement of the data used for averaging.
Assuming that the mean horizontal wind component in the plane formed by the scanning laser beam (or crosswind velocity) 〈Vc〉 = 〈Vr〉/cosϕ = 〈D〉/cosϕ is only depending on the height h, at first we find an estimate for this characteristic as
i1520-0426-22-11-1733-e51
where, in addition to averaging over the number of scans Ns, we average all estimates D/ cosϕl that are inside the layer of thickness Δh = 2δ = 50 m (shown in Fig. 4 as a gray band), and L is the number of estimates in this layer during one scan. Then, the 2D distribution of the mean radial velocity can be calculated by
i1520-0426-22-11-1733-e52
Using the resulting estimates 〈VD(Rk, ϕl)〉E, we calculate the fluctuations D(Rk, ϕl, n) ≡ D(hkl, n) = D(Rk, ϕl, n) − 〈VD(Rk, ϕl)〉E and the parameter
i1520-0426-22-11-1733-e53
In accordance with Eq. (45), where we take 〈E〉 = 0, the mean turbulent broadening of the Doppler spectrum is estimated, using the equation
i1520-0426-22-11-1733-e54
The statistical characteristics of the velocity estimation error e(R) can be determined by the known methods (Frehlich 2001). From the velocities measured with the Doppler lidar we obtain the differences
i1520-0426-22-11-1733-e55
which are used for estimation of the variance
i1520-0426-22-11-1733-e56
On the basis of Eq. (46), an estimate of the wind velocity variance σ̂2V(h) is calculated by the equation
i1520-0426-22-11-1733-e57
assuming homogeneous and isotropic turbulence within the layer of 50 m thickness.
For estimation of the velocity structure function D(ri, h) and e(ri, h), at first, the squares of the velocity difference D(ri, hkli, n) = [D(Rk + ri, ϕl, n) − D(Rk, ϕl, n)]2 and e(ri, hkli, n) = [ê(Rk + ri, ϕl, n) − ê(Rk, ϕl, n)]2 are obtained, where ri = iΔR, i = 1, 2, . . . , 16, and hkli = (Rk + ri/2) sinϕl. Then, we use the equations
i1520-0426-22-11-1733-e58
i1520-0426-22-11-1733-e59

During the measurement periods, every 8–10 min an overflight of the large transport aircraft at approximately a 300-m height in the plane perpendicular to the scanning plane occurred. After each overflight the effect of the wake vortices on the measured data could be observed for several minutes. For TEDR determination, the data measured in the vicinity of the vortices have been excluded.

The mean square of the Doppler spectrum width and the velocity structure function was estimated at different heights every 50 m, from h = 50 m to h = 500 m for data measured on 27 August, and h = 300 m for data measured on 28 August.

Figure 5 shows two examples of measured SNR versus range R at ϕ = 20 (solid curve) and ϕ = 280. To increase the SNR in the region where wake vortices are expected (R ∼0.5–1 km), the laser beam was focused on an approximately 1-km range. This allowed for a 5-times increase of the SNR in the region of 0.5 km ≤ R ≤ 1 km, but at a range of R ∼3 km the SNR was 2 times less as compared with that of the case of a collimated beam. For processing, we used measured data at ranges up to R = 1700 m. In this case, the condition SNR ≥ 1 is valid for all data measured up to the maximal height of h = 500 m. The second spectral moments were estimated at the optimal noise threshold noptth as a function of SNR in accordance with the result of numerical simulation. In our experiment 1 ≤ SNR ≤ 20 and 1.12 < noptth < 1.14.

b. Measurement results

For retrieving the TEDR height profiles the two approaches described in section 2—the estimation of TEDR from 1) the VSF and 2) the DSW methods—were used. To investigate the influence of turbulence on the aircraft wake vortices, we retrieved the height profiles of TEDR from lidar data measured during 20 scans. In accordance with the results of numerical simulation, this measurement period of 5 min should be optimal, because the level of statistical uncertainty is relatively low (see Fig. 3) and the period between aircraft overflights is ∼8–10 min. Moreover, the dwell time of the wake vortices generated by the large transport aircraft is less than 5 min.

Figures 6 and 7 show examples of height profiles for the TEDR (Figs. 6a and 7a) and integral scale of turbulence (Figs. 6b and 7b) retrieved by the two methods: VSF (circles) and DSW (squares). The results of the measurements at noon (Fig. 6) and in the evening (Fig. 7) of 27 August are presented. In the beginning of the measurement on 28 August, clouds were above a height of ∼500–600 m and, therefore, the TEDR profile retrieval is limited to this height. Then, within approximately half an hour, the clouds lifted and at heights above ∼400 m a jet flow with strong wind shear came up. The attempt to take into account the inhomogeneity of the mean wind [using Eqs. (48)(50)] in the vicinity of the jet flow was not successful, because the jet flow height changed slowly during 5 min of measurement (mesoscale wind variations), resulting in too “smoothed” profiles of the mean radial velocity. As a result, the TEDR obtained by both methods is overestimated. Therefore, from data measured on 28 August, the retrieved height profiles of TEDR were limited to heights below 300 m.

In Fig. 8, the TEDR versus time at a height h = 200 m for measurements on 27 (Fig. 8a) and 28 (Fig. 8b) August is shown. In case (Fig. 8a), the TEDR decreased during the time period of 1800–1950 LT, then it increased very quickly. The reason for this behavior is a wind front coming into the measurement region. Within 10 min, the crosswind velocity Vc increased from ∼1 up to ∼10 m s−1. On the next day (Fig. 8b), after approximately 0920 LT, the TEDR increased at an almost constant wind velocity of ∼2 m s−1. In this time, the sun heated the earth’s surface and the thermal stratification became unstable, which caused the turbulence intensification. From Fig. 8 it follows that the use of data measured during 5 min allows stable TEDR estimates to be derived and the evolution of the small-scale turbulence level to be observed with good resolution.

From data measured on 27 and 28 August 2003 we have retrieved 54 height profiles of TEDR, using both methods considered here (i.e., number of all profiles equals 2 × 54). Each profile is obtained from data measured during 5 min. It is obvious (see Fig. 4) that in Eqs. (51)(59) the total number Ns × L of the corresponding parameters that are used for averaging is different for different heights h. For example, in the case of estimation of σ̂2t(h) or D(ri, h), where ri = 30 m, at heights 0 m ≤ h ≤ 250 m, the number Ns × L ≈ 24 000 (the same number of estimates was used for the numerical simulation in section 2c), and at h = 500 m, Ns × L ≈ 11 000. Because the number Ns × L is largest at low heights, the error of ε̂(h) estimation is increasing with height.

To draw reliable conclusions about the accuracy of the Doppler lidar TEDR measurements, it is necessary to compare the results of simultaneous (at the same heights) measurements where the accuracy is known and is sufficiently high, for example, to compare with measurements from a set of sonic or cup anemometers mounted at different heights of a meteorological mast (Byzova et al. 1989). Because this possibility was not given, we compare the results of the measurement and numerical simulation.

For comparison of the TEDR estimates obtained by the VSF and DSW methods, all data measured on 27 and 28 August were used. Figure 9 shows the results of this comparison. It is seen that the points are located around the line of coincidence of the estimates. From the data of this figure we calculated the discrepancy parameter d given by Eq. (49), resulting in a value of d = 41 %.

In accordance with the simulation results shown in Fig. 3, the parameter d ≈ 25% at ɛ = 10−5 m2 s−3, and d ≈ 15% at ɛ > 10−4 m2 s−3. However, in the case of the real experiment the parameter d = 41% is ∼1.5–2.5 times larger than the level of discrepancy for the estimates (by VSF and DSW methods) obtained from the simulated data. Therefore, the error of the TEDR measurements has to be also significantly larger than the theoretical results shown in Fig. 3.

Possible reasons for the discrepancy of the theoretical and experimental results (in relation to the TEDR measurement accuracy) consist of the following.

  1. Geometrical factor

    In the case of numerical simulation the estimates D and σ̂2sw are uniformly distributed on the area of 1200 m × 900 m (3 m s−1 × 300 s = 900 m). Because of the laser beam scanning, the estimates obtained from the measured data have a 3.4 times larger density at a 500-m range, as compared with the density at a 1700-m range. At a height of 500 m the sampling area within a layer of 50 m is approximately 2 times less than for heights h ≤ 250 m (see Fig. 4). The exclusion of data affected by the wake vortices decreases the sampling area (not more than 10%). Except for a few cases (in particular, the last 20 min of measurements on 27 August), the mean wind velocity was not larger than 3 m s−1. It is obvious, that the effective sampling area in the experiment is less than in the simulation.

  2. Anisotropy and inhomogeneity of the wind field

    In Eqs. (54), (57), and (58) we average the data measured at different elevation angles ϕl. Although in each layer of 50 m (gray belt in Fig. 4) the angle between the maximum and minimum elevation is relatively small (e.g., at h = 200 m it is 60 < ϕl < 270), the anisotropy of large-scale turbulence variations is the reason for the inhomogeneity of the estimates D(Rk, ϕl) inside the layer. Therefore, in Eqs. (54), (57), and (58) we average the measured velocity with different integral scales of turbulence, and the model given by Eq. (33) can inadequately describe the measured data.

  3. Other reasons

    Other reasons that could explain the discrepancy of the theoretical and experimental results are the presence of mesoscale processes in the atmosphere and the fact that the probability density of the velocity difference is not Gaussian (in the inertial subinterval the third moment of the velocity difference ∼ɛ; Monin and Yaglom 1975). These two factors, as well as possible errors in measurement of σ20 and noptth (especially for the case of weak turbulence) are not taken into account in the numerical simulation.

4. Conclusions

Two methods of TEDR estimation from data measured with the 2-μm pulsed Doppler lidar have been considered in this paper: the estimation from 1) the velocity structure function (VSF) and 2) the Doppler spectrum width (DSW). The data measured by scanning the laser beam in a vertical plane were used for estimation of the velocity structure function and the Doppler spectrum width. The algorithms for processing these data are described in detail.

The methods were applied for retrieving TEDR height profiles from lidar data acquired at the Tarbes-Lourdes-Pyrénées International Airport in summer 2003. The measurement geometry and the aircraft flights over the airfield allowed information to be obtained about both the atmospheric turbulence and the aircraft wake vortices. From the data measured during 2 days of the experimental campaign 54 height profiles of TEDR were retrieved. Each set of profiles was derived by two estimation methods. In the mean, the results of TEDR estimation by the VSF and DSW methods are in good agreement. However, the discrepancy parameter d calculated from the experimental data equals 41%, and it is significantly larger than the simulation result (∼25%–15%). Apparently, this is the consequence of different factors (listed in section 3b) that were not taken into account in the numerical simulation.

On the basis of the theoretical results shown in Figs. 2 and 3 we can draw the conclusion about the feasibilities of the VSF and DSW methods for TEDR estimation. For moderate and strong turbulence (ɛ > 5 × 10−4 m2 s−3) the accuracy of TEDR estimation from the Doppler spectrum width is 2 times higher than that in the case of the VSF method. The high sensibility of the DSW estimator to measurement errors of σ20 and noptth in the case of weak turbulence can be the source of large errors for TEDR measurements. Therefore, for measurements at low turbulence level (ɛ ≤ 5 × 10−4 m2 s−3) it is better to use the VSF method for TEDR estimation.

The accuracy of TEDR determination by the methods described in this paper is quite sufficient for the analysis of wake vortex behavior under different turbulent conditions (Köpp et al. 2005). An accuracy investigation of turbulence measurement with the pulsed Doppler lidar cannot be complete without intercomparison with measurement results of independent turbulence sensors.

Acknowledgments

The experimental part of the presented work has been carried out in the framework of the European research program AWIATOR (Contract G4RD-CT-2002-00836). The algorithms for data processing have been developed for the DLR project “Wirbelschleppe.”

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APPENDIX

List of Symbols and Acronyms

  • B  Spectral bandwidth

  • BV(r)  Correlation function of the wind velocity

  • CK ≈ 2  Kolmogorov constant

  • DV(r)  Structure function of the wind velocity

  • DD(r)  Structure function of the Doppler lidar estimate of the radial velocity

  • DVD(r)  Structure function of the radial wind velocity averaged over the sensing volume

  • De(r)  Structure function of random error of the velocity estimation

  • DSW  Method of TEDR estimation from Doppler spectrum width

  • e(R)  Random error of the velocity estimation

  • E(R)  Random error of the Doppler spectrum width estimation

  • J(i)B(mTs)  Backscatter signal

  • J(i)M(mTs)  Monitor signal

  • L  Number of estimates in a layer of 50-m thickness at one scan

  • Lo  Outer scale of turbulence

  • LV  Integral scale of turbulence

  • Na  Number of shots used for spectral accumulation

  • Ns  Number of scans used for averaging

  • nth  Noise threshold used for estimation of the Doppler spectrum parameters

  • s(R)  Estimate of the normalized signal power

  • PM(t)  Monitor signal power

  • PRF  Pulse repetition frequency

  • Qs(z)  Function characterizing the spatial averaging over the sensing volume

  • R  Range (distance between lidar and center of the sensing volume)

  • Ŝ(i)Dfk, R)  Signal power spectrum (or Doppler spectrum) estimate obtained from data of ith shot

  • Dfk, R)  Estimate of Doppler spectrum after accumulation and normalization

  • SD( f, R)  Doppler spectrum estimate averaged over all random parameters except the radial wind velocity

  • SD( f, R)  Doppler spectrum estimate fluctuations

  • SNR  Signal-to-noise ratio: ratio of mean signal power to mean noise power within the spectral bandwidth of 50 MHz

  • SV(κ)  Spatial wind velocity spectrum

  • Ts  Sampling interval (2 ns)

  • Teff  Effective time of Doppler spectrum measurement (one shot)

  • Up  Pulse energy

  • Vc  Crosswind velocity

  • D(R)  Doppler lidar estimate of the radial velocity

  • VD(R)  Radial wind velocity averaged over the sensing volume

  • Vr(z)  Radial wind velocity at point z

  • VSF  Method of TEDR estimation from the velocity structure function

  • W(mTs)  Window function

  • δf  Intermediate frequency

  • Δf  Frequency bin width

  • Δfr  Frequency resolution

  • Δz  Longitudinal size of the sensing volume

  • ɛ  Turbulent energy dissipation rate

  • ε̂1  TEDR estimated from the VSF

  • ε̂2  TEDR estimated from the DSW

  • λ  Laser wavelength

  • μ(R)  Wind shear

  • σ2e  Variance of random error of the velocity estimation

  • σ̂2f  Estimate of square of the Doppler spectrum width

  • σ2δ  Variance of the intermediate frequency

  • σ0  Doppler spectrum width in the case of homogeneous wind

  • σs  Doppler spectrum broadening resulting from wind shear

  • σt  Doppler spectrum broadening resulting from turbulence

  • σu(R)  Doppler spectrum broadening resulting from inhomogeneity of the wind inside the sensing volume

  • σMf  Width of the monitor signal power spectrum

  • σp  Pulse duration

  • σ2V  Wind velocity variance

  • σ2D  Variance of the Doppler lidar estimate of the radial velocity

  • σ2VD  Variance of the radial wind velocity averaged over the sensing volume

  • σW  Width of the Gaussian window

  • ϕ  Elevation angle

  •   Estimate of X

  • X〉  Ensemble averaging of X

  • XE  Estimate of the mean of X (using spatial and temporal averaging)

Fig. 1.
Fig. 1.

Error of the estimation of the mean square of the Doppler spectrum width vs noise threshold at SNR = 1 (curve 1), 2 (curve 2), 3 (curve 3), and 10 (curve 4).

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 2.
Fig. 2.

Bias of TEDR estimates obtained by VSF (circles) and DSW (squares) at optimal noise threshold (squares connected by solid lines), and at nth = 1.13 (curve 1) and nth = 1.14 (curve 2).

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 3.
Fig. 3.

Relative error of TEDR estimation by VSF (full circles) and DSW (squares) methods and the discrepancy parameter d (open circles connected by dashed lines).

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 4.
Fig. 4.

Measurement geometry.

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 5.
Fig. 5.

SNR vs range at elevation angle 20 (solid curves) and 280 (dashed curves). The curves show the results of 30-min measurements on (a) 27 Aug 2003 and (b) 28 Aug 2003.

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 6.
Fig. 6.

Example of height profiles of (a) TEDR and (b) the integral scale of turbulence retrieved by VSF (circles) and DSW (squares) method from data measured at 1204:24–1209:49 LT 27 Aug 2003.

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 7.
Fig. 7.

Example of height profiles of (a) TEDR and (b) the integral scale of turbulence retrieved by VSF (circles) and DSW (squares) method from data measured at 1729:08–1734:42 LT on 27 Aug 2003.

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 8.
Fig. 8.

TEDR vs time at a height of 200 m. The data were measured on (a) 27 and (b) 28 Aug. The TEDR estimates are obtained from VSF (circles) and from DSW methods (squares).

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Fig. 9.
Fig. 9.

Comparison of TEDR estimates obtained from VSF (y axis) and DSW (x axis) methods.

Citation: Journal of Atmospheric and Oceanic Technology 22, 11; 10.1175/JTECH1815.1

Table 1.

Main parameters of the 2-μm coherent Doppler lidar.

Table 1.
Save
  • Banakh, V A. and Smalikho I N. , 1997: Estimation of the turbulent energy dissipation rate from the pulsed Doppler lidar data. Atmos. Oceanic Opt., 10:957965.

    • Search Google Scholar
    • Export Citation
  • Banakh, V A., Smalikho I N. , Köpp F. , and Werner Ch , 1995: Representativeness of wind measurements with a cw Doppler lidar in the atmospheric boundary layer. Appl. Opt., 34:20552067.

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  • Fig. 1.

    Error of the estimation of the mean square of the Doppler spectrum width vs noise threshold at SNR = 1 (curve 1), 2 (curve 2), 3 (curve 3), and 10 (curve 4).

  • Fig. 2.

    Bias of TEDR estimates obtained by VSF (circles) and DSW (squares) at optimal noise threshold (squares connected by solid lines), and at nth = 1.13 (curve 1) and nth = 1.14 (curve 2).

  • Fig. 3.

    Relative error of TEDR estimation by VSF (full circles) and DSW (squares) methods and the discrepancy parameter d (open circles connected by dashed lines).

  • Fig. 4.

    Measurement geometry.

  • Fig. 5.

    SNR vs range at elevation angle 20 (solid curves) and 280 (dashed curves). The curves show the results of 30-min measurements on (a) 27 Aug 2003 and (b) 28 Aug 2003.

  • Fig. 6.

    Example of height profiles of (a) TEDR and (b) the integral scale of turbulence retrieved by VSF (circles) and DSW (squares) method from data measured at 1204:24–1209:49 LT 27 Aug 2003.

  • Fig. 7.

    Example of height profiles of (a) TEDR and (b) the integral scale of turbulence retrieved by VSF (circles) and DSW (squares) method from data measured at 1729:08–1734:42 LT on 27 Aug 2003.

  • Fig. 8.

    TEDR vs time at a height of 200 m. The data were measured on (a) 27 and (b) 28 Aug. The TEDR estimates are obtained from VSF (circles) and from DSW methods (squares).

  • Fig. 9.

    Comparison of TEDR estimates obtained from VSF (y axis) and DSW (x axis) methods.

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