Pulse-averaging

The wind velocity υ_r(r) (see list of symbol in appendix A) measurement by HDL can be written as the sum of an effective wind velocity υ_m(r) and an error e(r),

υ_rrυ_mrer(1)

According to Frehlich et al. (1998), the effective wind velocity υ_m(r) is a spatial average over the pulse duration and the processing range gate of the true line-of-sight (LOS) wind velocity υ_t(r),

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where the weighting function W(r) is given by

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The parameter G(r) is the power shape of the probing pulse [∫ G(r) dr = 1 for normalization] and Δp is the length of the processing range gate.

In the spectral domain, Eqs. (1) and (2) may be written as

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The quantity κ is the wavenumber, Φ_t(κ), Φ_m(κ), Φ_r(κ), Φ_e(κ), and Φ_W(κ) are the power spectra of υ_t(r), υ_m(r), υ_r(r), e(r), and W(r), respectively (i.e., the squared magnitude of their Fourier transform).

Equation (4) assumes that e(r) and υ_m(r) have no cross correlation. This assumption calls for a justification since it is known that the measurement errors depend statistically on υ_m(r). For instance, the variance of errors increases with strong windshears or strong turbulent fluctuations. The statistical dependence does not imply, however, that there is a significant cross correlation between both quantities. Numerical studies (e.g., Frehlich et al. 1998) support the assumption that the cross-correlation term is negligible in the cloud-free atmosphere.

Equation (2) was proposed by Frehlich (1997) and Frehlich et al. (1998) for 2-μm HDL with Gaussian-shaped pulses with no chirp. The applicability to any HDL (not necessarily Gaussian shaped and linearly chirped) must be considered further.

Equation (2) is tested on simulated signals. The signals are generated by the “feuilleté” (“sliced”) model of Salamitou et al. (1995). The feuilleté model is an end-to-end numerical model that produces realistic HDL signals and may be used to assess the performance of HDL under nonstationary conditions. It combines laser beam propagation, interaction with the atmosphere, optical mixing, quadratic detection, and Doppler frequency estimators. The model slices the scattering medium along the beam axis, with each slice retaining the full transverse geometry of the problem. In such a feuilleté model, the individual particles or targets are first grouped within each slice and then a summation over the wavefields originating from the various slices is performed to determine the HDL signal. Because each slice backscatters an optical wavefield with a random phase delay, the summation displays speckle fluctuations. These fluctuations are independent from one model run to another. The input parameters are the complex amplitude of the laser pulse, the system efficiency accounting for the degrading effect of refractive index turbulence, a “true” velocity profile υ_t(r), and profiles for backscatter coefficient and atmospheric attenuation. Drobinski et al. (1999) have shown that at 10 μm, the system efficiency is degraded for C²_n ≥ 10⁻¹⁴ m^−2/3.

For the present tests, two different types of laser pulses are considered (Fig. 1). One is representative of a 10-μm HDL (Fig. 1, left panels). The laser pulse lasts several microseconds. It starts with a gain-switched-spike followed by a tail. It contains a quadratic frequency chirp (Willets and Harris 1982). The parameters used for the spike, the pulse duration, and frequency chirp are those of the experimental 10-μm HDL of the Laboratoire de Météorologie Dynamique (Delville et al. 1995). This system is described in section 3. The other laser pulse is representative of a 2-μm HDL (right panels of Fig. 1). The pulse shape is Gaussian, and the frequency chirp varies linearly with time. The parameters are taken from Frehlich et al. (1994).

The numerical experiment is based on the simulation of two sets of 500 signals (one set for each HDL). The backscatter, attenuation, and heterodyne system efficiencies are such that the average power of the signals is constant with range. The signals have all been obtained from the same input velocity profile υ_t(r) drawn from a Kolmogorov–Obukhov (Kolmogorov 1941; Obukhov 1941) spectrum

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where α = Cϵ^2/3(2π)^−2/3 (with C ≃ 0.76C_K, where C_K ≃ 2 is the Kolmogorov constant), ϵ is the dissipation rate of TKE, κ_in is the inner scale (κ⁻¹_in is the length of the smallest turbulent eddies, i.e., dissipation scale), and κ_out the κ-spectral peak or “plateau” (κ⁻¹_out is the size of the largest eddies). The quantity κ is in cycles per meter. The procedure used for generating υ_t from Φ_t is explained in appendix B. The range increment between the discrete velocity samples is δr = 1.5 m corresponding to a sampling frequency F_s = 100 MHz. For the present set of simulations, ϵ = 7 × 10⁻³ m² s⁻³, κ_in = 0.33 m⁻¹ (the range increment δr is 1.5 m; it is the largest unaliased wavenumber), and κ_out = 0.002 m⁻¹ (i.e., 500 m). The standard deviation σ_υt of υ_t(r) is then 1 m s⁻¹. In the present experiment, the signal-to-noise ratio (SNR) (ratio of signal power to noise power) is infinite. An infinite SNR has been retained here because it reduces the level of statistical uncertainty.

For each set, υ_r(r) is estimated by the pulse-pair frequency estimator (Miller and Rochwarger 1972). The 500 signals of each set are accumulated to retrieve a single profile. The frequencies are estimated over a sliding range gate of length Δp = 96 m. The number of signal samples per estimate is M = Δp/δr = 64. Considering the large number of accumulations, the precision for υ_r(r) is expected to be ≃0.10 m s⁻¹ for the 10-μm HDL, and ≃0.02 m s⁻¹ for the 2-μm HDL. Here, υ_m(r) is computed from υ_t(r) by using Eq. (2).

The comparison between υ_r(r) and υ_m(r) is presented in Fig. 2. The left and right panels are for the 10-μm and 2-μm HDL, respectively [the υ_t(r) profiles for 10-μm and 2-μm HDL are different because they result from two realizations of the turbulent wind field]. On both panels, υ_r(r) is drawn with a solid line, and υ_m(r) with a dashed line. The agreement between the curves is good, the rms of the differences (0.12 and 0.02 m s⁻¹) are of the same order as the expected precision. Equation (2) is thus validated to the extended case of non-Gaussian and/or chirped laser pulses.

The low-pass filtering effect of pulse averaging is illustrated in Fig. 3 where the transfer function Φ_W(κ) is displayed for the two different HDL pulses (see Fig. 1). A processing range gate Δp = 150 m is assumed. The figure displays a sharp transition between weakly and strongly attenuated wavenumbers at about κ_W = 3.0 × 10⁻³ m⁻¹ and κ_W = 4.5 × 10⁻³ m⁻¹ for the 10-μm and 2-μm HDL, respectively (κ_W is defined as the wavenumber achieving the attenuation of −10 dB). The figure sets the minimum size of turbulent eddies that can be detected by the HDL. In the present case, the sizes are 333 m and 222 m for the 10-μm and 2-μm HDL, respectively. Smaller scale eddies are filtered out. Figure 4 gives κ⁻¹_W as a function of Δ_p for a 10-μm HDL and a 2-μm HDL. For instance, it appears that for a 10-μm HDL a mimimum detectable eddy size of 500 m requires Δp ≃ 300 m, while for a 2-μm HDL, Δp ≃ 370 m.

Errors

The contribution of measurement errors to wind spectra derived from HDL data is a critical issue. In Frehlich et al. (1998) and Mayor et al. (1997), the error contribution is estimated from actual velocity profiles recorded by the HDL, assuming statistical independence of the errors between consecutive velocity profiles.

In this paper, the error contribution is modeled a priori, that is, related to HDL parameters. Presently, the model is only applicable to the pulse-pair frequency estimator. The major advantage is that the error contribution can be predicted prior to any measurements, based on a small number of parameters. Thus it makes possible the determination of a set of processing parameters that will guarantee negligible errors for specified conditions.

Error model

The error model is based on a set of equations that approximate the autocorrelation function (ACF) of errors. The error contribution to wind spectra is then determined by a Fourier transform [see Eq. (23) below]. The ACF of errors is calculated from the ACF of the range-resolved frequency estimates. The velocity υ_r(r_l) and frequency f(l), both measured at a range r_l = lδr are linked by

View Expanded

where λ is the laser wavelength. Under the assumption that the statistical mean of e(r_l) is zero (see next paragraph), we then obtain

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where the overbar denotes the statistical average, and, by substraction

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where f′(l) = f(l) − f(l). Finally,

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where the angle brackets 〈 · · · 〉 indicate statistical average.

The assumption that the errors have a zero mean is not true in reality unless several precautions are taken. For the pulse–pair frequency estimator reasons for the occurence of systematic errors are colored noise, occurence of gross errors at low SNR (Frehlich and Yadlowsky 1994), and a possible asymmetry of the power spectrum of the sounding pulse (Dabas et al. 1998). A colored noise can be avoided in practice by a proper matching of the analog antialiasing filter and the sampling frequency at detection level or by applying a digital whitening filter. Gross errors can also be avoided by restricting the processing of the signals to those with high SNR. As for the errors introduced by the asymetry of the pulse, it can be corrected (Dabas et al. 1998; Dabas et al. 2000). In the following, we assume that no systematic errors occur.

Let us now concentrate on the ACF of frequency estimates. Denoted by s_n(p), the complex value of HDL signal n at range r_p = pδr, the frequency f(l) estimated by the pulse–pair frequency estimator over the range interval [r_l, r_l+M−1] is given by

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with

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Here, Γ(l) is an estimate of the first-lag of the signal ACF averaged over the n_acc accumulated signals. The superscript “*” denotes the complex conjugate.

A first-order expansion of Eq. (11) then provides with an approximate relationship between the random fluctuations of f(l) and Γ(l). Denoting Re and Im the real and imaginary parts of complex numbers, we have

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The correlation between the frequencies is derived in appendix C:

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The first- and second-order moments of f(l) can also be estimated from acquired velocity profiles (Mayor et al. 1997; similarly for water vapor by Kiemle et al. 1997), but they cannot be related to the Γ fluctuations by an analytical equation.

Equation (14) shows that the spatial autocorrelation of the frequency estimates depends on HDL signal statistics that cannot be related to HDL parameters unless some simplifying assumptions are made. In the following, we make the usual assumption of Gaussian HDL signal statistics and stationary processes. The optical wave captured by the HDL results from the addition of a great number of backscattered wavelets, which actually validates the Gaussian property. Stationarity, however, is an ideal property that HDL signals may approach over limited range and time intervals. It is nevertheless a reasonable approximation.

Under these assumptions, all fourth-order moments of the HDL signals can be split into a sum of second-order moments, and the statistics of HDL signals is described by the autocorrelation function of signal n, γ_n(p, q) = 〈s^*_n(p)s_n(q)〉. The first-order moments in Eq. (14) can then be related to the second-order moments of the HDL signal

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The second-order moments are given by

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The ACF of measurement errors is now related to the single function γ_n(p, q). The next step is to determine γ_n(p, q). Two different approaches are possible: 1) estimating γ_n(p, q) from the signals, or 2) relating γ_n(p, q) to HDL and atmospheric parameters. The statistical estimate approach requires a large number of signals acquired under stationary conditions. In practice, this is possible with high repetition rate and stable HDL. We propose here to use the second approach.

A detailed study of the autocorrelation of HDL signals can be found in Frehlich (1994), and Frehlich and Cornman (1999). In both publications, equations are provided to relate the autocorrelation to instrument and atmospheric parameters. Assuming 1) that the atmospheric return (the useful part of the signal) and the contaminating noise are statistically independent, 2) that the contaminating noise is white (this assumption is justified above), and 3) that the optical and dynamical properties of the atmosphere are independent, then

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where δ(p − q) is a Dirac and R(p, q) is the autocorrelation of the atmospheric return given by Frehlich and Cornman (1999):

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The quantity H(r) is the optical responsivity of the atmosphere, and υ′(r) the deviation of the true radial velocity from its linear average inside the probed volume. Equation (19) can be computed numerically assuming that H(r) is known and that wind fluctuations are modeled using Kolmogorov’s theory (e.g., Frehlich et al. 1998; Frehlich and Cornman 1999). However, to simplify the Eq. (19), 〈H(r)〉 can be assumed to be nearly constant with z (which is valid for a measurement inside an optically homogeneous volume), and 〈υ^′2_t(r)〉/ = σ²_υt (Frehlich and Cornman 1999). Then the autocorrelation of the HDL signal becomes

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with

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Equation (20) offers the advantage that it can be conveniently applied since only the shape of the laser pulse and the variance of the radial velocity is required. Equation (20) can be used as a lower bound for the covariance, since σ²_υt is larger than the mean-square velocity fluctuations over the range gate used in Frehlich and Cornman (1999). For a 10-μm HDL, the exponential term accounting for the decorrelation by turbulence can be neglected in most conditions, high turbulence levels are required to make it significant (ϵ ≥ 10⁻² m² s⁻³) (Giez et al. 1999; Dabas et al. 2000). This is at least true for the pulse–pair estimator because only the first lag of the ACF is used (|q − p| = 1). If larger lags are used, the exponential term may impact on the decorrelation and the turbulence contribution may become important. For a 2-μm HDL, turbulence-induced decorrelation cannot be neglected and wind fluctuations must be taken into account using Eqs. (18) and (19).

As previously mentioned, the ACF of frequency estimates can be used to determine the spectral density of measurement errors. A good estimate for Φ_e is

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where n_υ is the number of error profiles e_n over which the estimate is averaged, and L sets the length of the range gate over which the Fourier transform is performed. The mathematical expectation

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is then a good (i.e., nonbiased) prediction for Φ_e.

Derivation of wind spectra

Given a set of n_υ HDL profiles, an estimate for Φ_r(κ) is

View Expanded

According to Eqs. (4) and (5), we should then have

Φ̂_r(κ)Φ̂_t(κ)_WκΦ̃_e(κ)(26)

Below, we test Eq. (26) against the spectra derived from simulated HDL signals.

Figure 9 presents the results. It is based on the simulation of n_υ = 50 sets of 10 signals generated from 50 different input velocity profiles υ_t(r), all drawn from the same Kolmogorov–Obukhov spectrum [see Eq. (6) with ϵ = 7 × 10⁻³ m² s⁻³, κ_in = 0.33 m⁻¹, and κ_out = 0.002 m⁻¹]. The simulations consider a 10-μm HDL (left panels in Fig. 1). White noise is added so that SNR = 0 dB. Each set of 10 signals is accumulated (n_acc = 10) to produce a single-velocity profile υ_r(r) (Δp = 100 m). Then, Φ̂_r(κ) is computed according to Eq. (25) with L = 8192.

Figure 9 shows the contributions of Φ_m(κ) and Φ̃_e(κ) to Φ̂_r(κ). The original spectrum Φ_t(κ) is drawn with a dotted line. The attenuation due to pulse averaging is shown with Φ_m(κ) (dashed line) [see Eq. (5)]. The dash–dotted line is Φ̃_e(κ). It shows that the contribution of the errors dominates above κ ≈ 2 × 10⁻³ m⁻¹. In this region, the good agreement with Φ̂_r(κ) displayed with a light solid line demonstrates the overall level of errors is well predicted. The heavy solid line is Φ_m(κ) + Φ̃_e(κ). It agrees very well with Φ̂_r(κ), which shows that the contributions of the real wind and the errors to the experimental spectrum Φ̂_r(κ) are both well predicted by our model.

Statistical uncertainty

We study here the potential impact of statistical uncertainties on the “inversion” of Φ̂_r(κ). Our aim is the determination of experimental conditions that guarantee a reliable inversion.

Since Φ̂_r(κ) is the average of n_υ spectra, each obtained from the Fourier transform of a single-velocity profile υ_r(r), it follows a chi-square distribution with n_υ degrees of freedom [we assume υ_r(r) is Gaussian]. Its statistical average is 〈Φ̂_r(κ)〉 = Φ_r(κ), and its standard deviation is n^−1/2_υΦ_r(κ).

In comparison, δΦ̃_e(κ) is not random because both Φ̃_e(κ) and Φ_e(κ) are deterministic quantities [Φ̃_e(κ) is determined by a set of analytic equations, and Φ_e(κ) is a statistical average]. Its statistical average is δΦ̃_e(κ), and its variance is zero.

Considering now the second-order moments, we obtain

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where σ_Φ̂m(κ), and σ_Φ̂r(κ) are the standard deviations of Φ̂_m(κ), and Φ̂_r(κ), respectively. In the following, we will consider that the estimate Φ̂_m(κ) is accurate provided σ_Φ̂m(κ) ≪ Φ_m(κ). Thus the condition required for a useful measurement becomes

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Since Φ_e(κ) scales with n⁻¹_acc [this can be seen from Eqs. (16) and (17)], we can write

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where Φ¹_e(κ) is the “true” error spectrum when n_acc = 1. Providing n_υ ≫ 1, Eq. (30) becomes

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In Fig. 9, this condition is fulfilled up to κ = 3.5 × 10⁻³ m⁻¹ (i.e., 286 m). In Fig. 10, we show that an accurate measurement Φ̂_m(κ) of Φ_m(κ) can be made up to that wavenumber. The spectrum Φ_m(κ) is plotted with a dotted line and the solid line shows Φ̂_m(κ) = Φ̂_r(κ) − Φ̃_e(κ) over the frequency range 0 ⩽ κ ⩽ 3.5 × 10⁻³ m⁻¹. The agreement is good. Note that the random fluctuations of Φ̂_m(κ) become larger as the wavenumber increases. The reason is that, in the ratio Φ_e(κ)/Φ_m(κ), Φ_m(κ) tends to zero because of the pulse-averaging while Φ_e(κ) remains strictly positive. This results in the gradual saturation of condition Eq. (30).

Equation (32) sets a minimum value for n_accn^1/2_υ needed to achieve an accurate retrieval of the wind spectrum on a spectral bandwidth [0; κ_W] for a given level of turbulence set by ϵ. Figure 11 displays the minimum value for n_accn^1/2_υ to obtain a reliable spectrum in the frequency range [0, κ_W] for a 10-μm and 2-μm HDL, respectively. Figure 11 has been obtained by computing Φ¹_e from Eq. (23), assuming the error model is accurate, and Φ_m(κ) from Eq. (5). The maximum value of Φ¹_e(κ)/Φ_m(κ) over [0; κ_W] is determined and sets the minimum value for n_accn_υ according to Eq. (32). The quantity n_accn_υ is given as a function of Δp and dissipation rate ϵ [which partially characterizes Φ_m(κ)], considering no noise in HDL signals. Figure 11 can be used to set the minimum number of velocity profiles and signals to be accumulated in order to make an accurate measurement for a specified level of turbulence. Let us consider for instance ϵ = 10⁻⁴ m² s⁻³. For a 10-μm HDL with Δp = 200 m (κ⁻¹_W ≃ 375 m), at least n_υ ≈ 120 for n_acc = 20 is required; that is, an acquisition time of 4 min at a 10-Hz pulse repetition frequency (PRF). For a 2-μm HDL, and Δp = 200 m (κ⁻¹_W ≃ 375 m), Eq. (32) gives n_υ = 1 for n_acc = 20. But, because the equation has been established on the condition that n_υ ≫ 1, n_υ ≈ 10 is preferable, so an acquisition time of 2 s is required for a 10-Hz PRF.

Limitations of the accumulation technique

Since the contribution of HDL shot accumulation may have a large impact on the retrieval of the turbulent wind spectra, two constraints are now required for a relevant measurement. The first constraint sets the condition on the level of measurement errors with respect to the random fluctuations of wind profile υ_m(r). Equation (32) sets the minimum value n_accn^1/2_υ to meet the measurement specifications. The second constraint refers to the number of HDL signals n_acc that must be accumulated, provided that the accumulation period is short enough, that is κ_acc must be large enough to satisfy κ_W ⩽ κ_acc. In summary, the conditions required for a reliable measurement are

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Let us consider two HDL operating at 10 and 2 μm, respectively. From Fig. 4, we have κ⁻¹_W are at best ≃300 m and ≃100 m, respectively. Table 1 gives the maximum number n_acc of HDL shots that must be accumulated in order to satisfy condition 2 [see Eq. (34)] for various PRF and various values for U. Condition 1 sets the minimum number n_υ of spectral densities to average as a function of expected turbulence. The present state of art in HDL technology allows high enough PRF that condition 2 is easily met.

Retrieval of wind spectra

The turbulent wind spectra are estimated with an accumulation factor of n_acc = 20 and by averaging the power spectrum of n_υ = 10 velocity profiles (n_accn1/2υ ≃ 64). According to Eq. (33) and Table 2, Taylor’s hypothesis is satisfied. The average wind speed ranged between 1 and 7 m s⁻¹, with peak values larger than 10 m s⁻¹. For n_acc = 20, condition (2) is always fulfilled. The HDL was looking nearly horizontally in the PBL (elevation angle 0.8°). The SNR is better than 0 dB up to ranges of several kilometers. Thus, we expect that our error model is valid (negligible detection noise). The angle between the streamwise motion of turbulence and the HDL LOS was corrected before the retrieval of wind spectra (Banakh et al. 1995).

Figure 14 shows the spectral density Φ̂_r(κ) (thin line) computed from one set of actual data. The spectral density is computed from the radial velocity profiles over the entire range (2.8 km; 7 km) in Fig. 14a, over (2.8 km; 5 km) range in Fig. 14b and over (5 km; 7 km) range in Fig. 14c to assess the natural variability of the turbulence parameters. The signal frequency is estimated over a sliding window of length Δp = 100 m (M = 17 signal samples per estimate with a sampling frequency F_s = 25 MHz. The mean of the nυ = 20 velocity profiles is removed before each profile is individually Fourier transformed. The Fourier transform is performed over the range interval 2.8–7 km (700 velocity estimates), where SNR ≥ 0 dB (Fig. 13b). The predicted spectrum Φ̃_r(κ) closest to Φ̂_r(κ) is displayed with a thick line. It has been derived by determining the turbulence quantities ϵ and κ_out of the Kolmogorov–Obukhov spectrum Φ_t(κ) that minimize the cost function

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For this example, the minimization over the entire range is reached for ϵ = 4.6 × 10⁻³ m² s⁻³ and κ_out = 0.0022 m⁻¹. The minimization over the subranges (2.8 km; 5 km) and (5 km; 7 km) is reached for ϵ = 3.7 × 10⁻³ m² s⁻³ and κ_out = 0.0021 m⁻¹, and ϵ = 5.2 × 10⁻³ m² s⁻³ and κ_out = 0.0023 m⁻¹, respectively. Figure 14 shows this minimization produces a good agreement between the predicted and the observed spectrum, at least in the spectral domain concerned by the minimization (κ ⩽ κ_W ≃ 0.004 m⁻¹) (in Figs. 14b and c, the smaller number of samples may degrade the precision of the measurement). It can be shown that the contribution of errors is negligible in that particular domain. As a matter of fact, the application of the error-filtering technique from Mayor et al. (1997) to our velocity profiles results in a power spectrum very close to Φ̂_r(κ) in the spectral interval κ ⩽ κ_W, confirming that the contribution of errors is negligible in that particular domain. Sensitivity studies using numerical simulations have shown that the precision on ϵ and κ_out estimates is ≃10% for κ_out ⩽ 0.004 m⁻¹. The procedure used to derive the “instrument” precision was the following: 1) generation of simulated HDL signals (the total length is 8192 samples at 100 MHz sampling frequency, i.e., 12.3 km range, and 2 Hz PRF) accounting for a turbulent wind profile for SNR = +∞; 2) estimation of the radial velocity profile using the pulse–pair frequency estimator (with M = 64) from n_acc = 20 accumulated HDL signals (i.e., 10-s accumulation) simulated with same υ_t(r) radial velocity profiles; 3) estimation of Φ̂_r(κ) from n_υ = 10 radial velocity profile estimates (so one spectrum used to compute ϵ and κ_out is estimated every 100 s); and 4) minimization of the cost function J(ϵ, κ_out), retrieval of the turbulence parameters ϵ and κ_out, and calculation of the departure with respect to the input values. This procedure was applied 10 000 times in order to have good statistics on the accuracy of the model for both ϵ and κ_out retrieval. The statistics are thus derived from 10 000 × n_υ random realizations of the radial velocity profile as well as new speckle signals for each lidar shot. This precision accounts for the uncertainties arising from random fluctuations of the true wind field and of the HDL signals (speckle and noise). In the simulation, the impact of shot-to-shot pulse instabilities was not treated, the transmitted pulses are the same for all simulated signals and the n_acc realizations used to process one velocity profile are uncorrelated. These assumptions have an impact on the calculation of the instrument error, and the value of 10%–20% must be considered as an optimistic bound. The variability of the turbulence parameters shown on Fig. 14 (≃15%–30%) is partly due to instrument statistical uncertainty (≃10%) and partly to natural variability (≃10%–30% assuming decorrelation between instrument precision and natural variability). The atmospheric variability is the dominant source of error in the following.

This value of κ_out corresponds to a length scale that is larger than the height above the boundary (i.e., ≃250 m), and to eddies that transport momentum. Hence, the corresponding eddies are anisotropic. The −5/3 slope of the radial velocity component’s κ spectra is not sufficient evidence for the existence of an “inertial subrange” that depends on the flow being locally isotropic (Kolmogorov 1941). Considering the altitude of the LOS above the surface (see Fig. 13a) or the κ-spectral peak determined from our analysis (see Fig. 15), these structures are likely to be outside the inertial subrange. That the −5/3 regime predicted by Kolmogorov’s theory extends above the inertial subrange has been reported in literature (Busch and Panofsky 1968; Vinnichenko and Dutton 1969; Donelan and Miyake 1973).

The reasons why our model in Fig. 14 underestimates the actual spectrum in the high-wavenumber domain (κ ≥ κ_W) has not been determined precisely. However, it has no impact on the estimation of the turbulence quantities because it is outside the range over which the optimization (36) was performed. A possible explaination is pulse-to-pulse instability of the laser that makes the average frequency, the chirp and the shape of pulse fluctuate randomly from one shot to the other. The magnitude of the measurement errors may also be increased by natural wind variability and probably mainly by wind shear within the range gate. According to Dabas et al. (2000), the range of estimated ϵ values in the ECLAP experiment, using the LMD 10-μm HDL measurements, should not induce significative decorrelation due to turbulence, but Eq. (20) does not account for wind shear which may not be negligible.

Figure 15 is the histogram of κ⁻¹_out for the entire dataset. Two modes appear, one with a Gaussian shape around ≃410 m, and another one between 1 and 1.4 km. This second mode probably reflects the instability of the minimization technique in presence of organized convection [here roll vortices, see Etling and Brown (1993) for details]. The scale of such structures developed in these conditions is typically above 1 km (Drobinski et al. 1998). Table 2 shows that most of the HDL data were collected during slightly unstably stratified PBL when such structures are likely. During M13, periodicities appear in the data recorded by the sonic anemometer causing an approximate 20% variation. Strong peak in the energy appears on M13 with a 1.3-h period suggesting organized wave motion in the PBL flow. On M13 and M14, HDL radial velocity measurements show energy peaks in the spectral analysis corresponding to longitudinal roll wavelengths of 1.4 km (±0.2 km) and 1.6 km (±0.2 km), respectively (Drobinski et al. 1998). On M09 and M10, no organized wave motion was apparent. In the following, the organized large eddies are “removed” by smoothing the spectra, in order to stabilize the minimization.

The mean value for κ-spectral peak is κ_out ≃ 410 m with a standard deviation of ≃100 m, which is partly due to the model accuracy and partly due to natural variability (see Fig. 14). The mean value of κ_outz is ≃0.66 (±0.2). Previous studies have shown that κ_outz depends on stratification (Busch and Panofsky 1968; Van Mieghen 1973; Hojstrup 1982): 0.01 ⩽ κ_outz ⩽ 0.3 for free convection, 0.3 ⩽ κ_outz ⩽ 1 for forced convection and 0.6 ⩽ κ_outz in case of isotropic turbulence. Considering the present values of κ_outz, they are consistent with forced convection and isotropic turbulence regimes.

Measurement of the TKE dissipation rate

The measurements of TKE dissipation rate by HDL are compared with values computed from the turbulent surface fluxes measured by a sonic anemometer at the same time. The surface measurements of energy dissipation rate ϵ are approximately corrected for altitude dependence, using (Hojstrup 1982; Kristensen et al. 1989; Frehlich et al. 1998):

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Hojstrup in 1982 concluded that his model [Eq. (37)] provided a good description of the PBL turbulence up to a height of 0.5z_i under stability conditions ranging from neutral throughout the unstable range, over homegeneous terrain. Our HDL measurements were taken in these conditions with z/z_i ⩽ 0.4 (except on M14 where z/z_i ≃ 0.6–0.7). The model described by Eq. (37) is no more reliable when z/z_i ≥ 0.5, for in the upper half of the PBL, velocity spectra are influenced by entrainment processes and effect of inversion (Hojstrup 1982). These effects were described qualitatively by Hojstrup (1982) and were not included in the model. Studies accounting for shear- and buoyancy-driven PBL flows were conducted since then using large eddy simulations (LES) (Moeng 1984; Moeng and Wyngaard 1988; Mason 1989; Moeng and Sullivan 1994). Moeng and Sullivan (1994) derived from their simulations an analytical model for TKE dissipation rate accounting for shear and buoyancy:

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Figure 17 displays the turbulence parameters from the sonic anemometer measurements on M09, M10, M13, and M14. The thin solid and dashed lines represent u∗ and w∗, respectively. The thick solid and dashed lines represent ϵ at 250-m height computed from Eq. (37) (Hojstrup 1982) and Eq. (38) (Moeng and Sullivan 1994), respectively. According to the relative errors on z_i estimates and on sonic measurements (precision ≃1%), the instrument error bar for ϵ is ≃3%–9% (power of three of the turbulent fluxes). For the four days, the temporal variability of u∗ and w∗ is ≃20% of their mean values. These fluctuations induce a natural variability on the computed TKE dissipation rates of ≃30% of their mean values, showing that natural variability dominates over instrument error (Lenschow and Stankov 1986; Lenschow et al. 1994). The values of ϵ computed from Hojstrup or Moeng and Sullivan models are in good agreement (the difference between the two models is on average ⩽1.5 × 10⁻⁴ m² s⁻³, or ≃6% of the ϵ mean value, that is, within the statistical uncertainty of our measurements).

Figure 18a compares lidar and sonic anemometer measured energy dissipation rates using Eq. (37) (represented with circles) and Eq. (38) (represented with stars). Numerical sensitivity study has shown that the model precision on ϵ estimates is ≃10% for κ_out ⩽ 0.004 m⁻¹ (this values does not account for natural variability). The instrument statistical uncertainty is shown with vertical bars (±1 − σ). The comparison indicates both sensors obtain the same order of magnitude, which is encouraging considering the uncertainties of measurements. The rms error is 1.7 × 10⁻³ m² s⁻³ and 1.4 × 10⁻³ m² s⁻³ for Eqs. (37) and (38), respectively. It is due partly to the intrinsic errors in the measurements by the sonic anemometer and HDL, partly to errors introduced by the height correction, partly to the different geometries of the measurement (point measurements and spatial measurements), partly due to the spatial and temporal variability and, also to the signature of large-scale wind features.

The ratio of Eqs. (37) and (38) ranges between 1 and 2 at most. The difference between the two models does not affect the order of magnitude. In particular, the bias of 2 × 10⁻⁴ m² s⁻³ for ϵ computed with Eq. (37) [8 × 10⁻⁴ m² s⁻³ for ϵ computed with Eq. (38)] is not significant because, according to Fig. 11, n_accn^1/2_υ = 64 allows a reliable measurement for ϵ ≥ 10⁻³ m² s⁻³. This TKE dissipation rate measurement precision is thus not fully adapted to turbulence studies because of the large pulse length. Using an HDL with shorter pulse duration will improve this measurement.

The horizontal velocity standard deviation σ̂_υt, calculated from the turbulent wind spectrum estimates Φ̂_t(κ) [see Eq. (26)] (characterized by ϵ and κ_out) are compared with the best-fit spectral model (Panofsky et al. 1977; Kristensen et al. 1989). This model does not vary with height,

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The spatial integration of Φ̂_t(κ) needed to derive σ̂_υt is equivalent to a 10-min integration time for the sonic anemometer measurements. Figure 18b compares lidar and sonic anemometer measured horizontal velocity standard deviation. The instrument statistical uncertainty is shown with vertical bars (±1 − σ). The comparison gives the right order of magnitude. The bias is 0.11 m s⁻¹, and the rms error is 0.10 m s⁻¹. This comparison looks better than that for ϵ for this is a lower order term than ϵ [see Eqs. (37), (38) and (39), the instrument error bar for σ_υ is ≃1%]. The results are consistent between the lidar and the sonic anemometer measurements and give confidence in the estimated values of ϵ and κ_out. Nevertheless, these are preliminary results and the precision of ϵ and κ_out measurements have to be assessed later using an HDL with shorter pulse length.