1. Introduction
In the last decades coherent Doppler lidars (CDLs) have proven to be important tools for atmospheric wind research (Fujii and Fukuchi 2005). They have been used to measure boundary layer convection (e.g., Frehlich et al. 1998; Ansmann et al. 2010), vertical wind velocities at cloud bases (Lottman et al. 2001; Grund et al. 2001), fluxes of latent heat (Giez et al. 1999; Kiemle et al. 2007), and fluxes of particles (Engelmann et al. 2008). Recently, the application of wind lidars is studied to improve the energy output of wind power plants (Harris et al. 2006; Käsler et al. 2010). They are also regularly employed on research aircraft (Weissmann et al. 2005; Reitebuch et al. 2003).
Here δf is the bandwidth, τ is the duration of the laser pulse, l is the pulse length, and c is the speed of light (Paschotta 2008); γ is a constant that depends on the pulse shape. In our case we will use γ = 0.44 for a Gaussian-shaped pulse.
Thus, for wind lidars a trade-off between range and frequency resolution has to be found. For example, the CDL used in the context of this work has a pulse length of about l = 175 m, which leads to a bandwidth of δf ≈ 750 kHz for one laser shot. However, the uncertainty introduced by the pulse bandwidth can be overcome by averaging over many laser shots (Frehlich 2001; Smalikho et al. 2005), and there are other important influences that determine the measurement error of wind velocity measurements. Examples are the lidar pointing accuracy, turbulent spectral broadening (Banakh and Werner 2005), and the chirp of the laser pulse of solid-state and CO2 lasers. A laser pulse chirp is a gradual change in the frequency of the laser light during pulse emission (see Fig. 1). For solid-state lasers it can result, for example, from nonlinear optical effects within the active medium (e.g., inversion-dependent change of refractive index). For our CDL, the frequency deviation of the laser pulse is shown in Fig. 1.
One laser pulse recorded with the heterodyne reference detector. (left) The intensity (temporal average of the squared raw heterodyne amplitude) is depicted. The intensity FWHM is about 590 ns (arrows). (right) The shift in the main frequency of the raw heterodyne signal determined by a sliding Fourier transform is shown. The pulse chirp is about +0.25 MHz μs−1 before and −0.5 MHz μs−1 after the pulse maximum.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
Around strong signal gradients (e.g., at cloud bases) the chirp effect leads to the detection of artificial velocities (Fig. 2), but it also affects continuous signals of boundary layer aerosol, optically thin clouds, virgae, etc. (Fig. 3). Within such continuous signals the measured velocity is shifted by a constant factor if the light emitted before and after the pulse peak is shifted toward the same side of the spectrum or, in general, when the chirp is nonlinear (Dabas et al. 1998). Hence, light scattered back by an air volume is overlaid by frequency-shifted light from uprange and downrange air volumes leading to a shift of the spectral peak, and hence to the detection of biased wind velocities. This bias and the distortions in the velocity profile have to be taken into account especially when dealing with vertical velocities. Korolev and Isaac (2003) showed that up- or downdrafts of only a few centimeters per second can already decisively influence the meteorological processes within an air parcel.
The chirp effect in vertical wind velocity observation at cloud bases. An optically thick cloud effectively returns a copy of the laser pulse, which is recorded by the data acquisition software. The Fourier analysis now yields a lower Doppler shift at the beginning and at the end of the laser pulse, which could be mistaken as negative vertical velocities. The height bins of the data acquisition are shown (dashed lines). For simplicity it is shown here without any interpolation.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
The chirp effect in vertical wind velocity observation within the boundary layer. An ensemble of particles returns overlapping copies of the laser pulse. Hence, the strong signal of the center frequency overlaps with chirped lower-frequency signals, which effectively shifts the frequency peak of the spectra toward lower frequencies. For our system, a shift in the detected vertical velocity in the order of 0.2 m s−1 is introduced. In the case of a Gaussian pulse with only a chirped tail the effect is the same; for a completely symmetrical chirp, however, the effect cancels out. The height bins of the data acquisition are shown (dashed lines).
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
Up to now much effort has been put into the improvement of laser designs and hardware chirp correction (Wulfmeyer et al. 2000). If hardware changes are neither desired nor applicable, then the correction of the chirp effect can also be done on software basis by deconvolution techniques (Gurdev and Dreischuh 2003, 2008; Zhao and Hardesty 1988). The techniques presented up to now mostly involve the direct deconvolution of the raw heterodyne signal. In this work, however, we show how to apply a two-dimensional deconvolution on the averaged spectra in order to correct the chirp-induced velocity bias while simultaneously improving the range and frequency resolution of our CDL. However, we keep the processing of the recorded heterodyne signal very simple: we only divide the digitally recorded raw heterodyne signal up in overlapping range gates and calculate the Fourier spectrum for each gate.
We treat these averaged height-resolved Doppler spectra like an image blurred by a certain point spread function (PSF). In our case the PSF corresponds to the reference laser pulse spectra recorded at the beginning of the dataset. We show that it is possible to apply two-dimensional deconvolution techniques known from image restoration and enhancement to remove the influence of the chirped laser pulse from the datasets. A major difference of this approach compared to other deconvolution techniques is that it does not need to be employed in real time during data acquisition, and it does not need access to the raw heterodyne data. It can be applied to the averaged Doppler spectra, which significantly reduces both the complexity of the data acquisition software and the amount of data to be stored.
In section 2 we will briefly present our CDL system and discuss the laser pulse characteristics. Next we will introduce the two-dimensional deconvolution (section 3a) and apply it to atmospheric data (section 3b). To further estimate the performance of the two-dimensional deconvolution technique we will apply it on a simulated dataset in section 4.
2. The coherent Doppler wind lidar
The CDL “WiLi” of the Leibniz Institute for Tropospheric Research used for this study is described in detail in Engelmann et al. (2008), so only a brief overview will be given here. The lidar is equipped with a master oscillator–power amplifier (MOPA) design operating at a wavelength of 2022 nm. The bandwidth of the master oscillator (MO) is 150 kHz. The pulse energy of the power oscillator (PO) is 2 mJ at a pulse repetition frequency of 750 Hz. In Fig. 1 the signal intensity of a recorded laser pulse is depicted. The change in frequency was determined by a sliding window Fourier transform and is overlaid in the figure. The trend to negative frequency shifts at the beginning and end of the laser pulse are visible. The pulse chirp is about +0.25 MHz μs−1 before and −0.5 MHz μs−1 after the pulse maximum.
The laser pulse shown in Fig. 1 has a length [intensity full width at half maximum (FWHM), see description of Fig. 1] of 590 ns and a spectral width of 720 ± 20 kHz, which was determined by Gaussian fitting the peak of its Fourier transform. The spectral width is slightly smaller than the theoretical bandwidth of 750 kHz [see Eq. (2)] and might be the result of the studied single pulse not being truly Gaussian (it has rather an asymmetric q-switched pulse shape). However, for the deconvolution method presented here the exact pulse length and spectral shape is not important because the averaged pulse spectrum itself is the reference for the determination of the wind velocities. The measured bandwidth of the laser pulse corresponds to a velocity uncertainty of about ±0.73 m s−1 for one laser shot [Eq. (1)]. However, after averaging over 1000 laser shots the standard deviation of the reference frequency fluctuation has reduced to less than 
While the laser pulse is being emitted by the power amplifier a small fraction is coherently mixed with the light of the master laser at a reference detector. The intermediate frequency of a laser pulse (i.e., the frequency offset between the MO and the PO) is −80 MHz and is introduced by an acousto-optical modulator (AOM). The generated heterodyne signal is recorded with 250 MSamples s−1. The light scattered backward from atmospheric particles is consecutively detected on another heterodyne detector. For signal switching we use a low-noise, high-frequency integrated circuit. The two receivers use photodiodes of the same type and have similar amplifiers, and their signals are recorded consecutively by the same data acquisition system. The equal detection and data processing of the laser pulse and the atmospheric signal is very important for the deconvolution method explained later.
The heterodyne signal is recorded for 100 μs after the release of the laser pulse, which corresponds to a distance of 15 km in the atmosphere. For one laser shot the data acquisition system records 25 000 data points, which are cut into 200 range gates of 250 data points (1 μs), each with an overlap of 125 data points. A Hanning window is applied in order to suppress sidelobes from discontinuities in the signal. In this way one range gate is of 150-m resolution at interpolated height steps of Δr = 75 m. On each range gate a fast Fourier transform (FFT) is performed by extending the dataset to 256 points by zeros. Therefore, the width of one frequency bin is 
The Doppler lidar spectra of the marked profile in Fig. 5 (1740 UTC) is shown (a) before and (b) after deconvolution. It was averaged over 1500 laser shots. Atmospheric features and the laser pulse are labeled.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
The power spectra are calculated for each range gate of every emitted laser pulse and summed for an integration time of 1–5 s before they are stored on a hard disk. This approach greatly reduces the amount of data from 80 Mbyte s−1 to about 100 kbyte s−1 but prohibits any reanalysis of the raw heterodyne signal. The further analysis is done by independent software, which searches for the peak within the averaged spectra and computes their first moments for velocity estimation. Consecutively, the wind velocity in each range gate is calculated by Eq. (1). The top two graphs in Fig. 5 show the signal intensity (in the following we will always use the term “intensity” to describe the estimated height of a peak after background subtraction), the vertical velocity within the planetary boundary layer (0–3-km height), and the terminal velocity of ice particles and water droplets in the virgae of altocumulus clouds (3–6-km height).
(a),(b) The signal intensities and the vertical velocity distribution of an atmospheric measurement of the boundary layer and midlevel cloud layers are depicted (measured at Leipzig on 13 Mar 2011). In the red box the chirp effect is visible above and below the cloud as artificial negative velocities (green color). (c),(d) The same measurement is shown after correction of the chirp effect by deconvolution. At 1740 UTC the position of the example measurement shown Fig. 4 is indicated (dashed red line). In the magnified portion of the velocity plots the chirp effect is visible above and below the cloud layers.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
If the range gates are long enough to include a whole laser pulse (including the chirped tail, not only the FWHM), then the chirp effect does not play a major role. However, when interpolating the heterodyne signal with range gates smaller than the laser pulse itself, chirp-induced artifacts as described in section 1 will appear. Indeed the bias for continuous signals could be avoided by using the average frequency of the whole outgoing laser pulses as reference frequency. This would largely compensate for the bias, but would in turn shift the velocities at cloud bases into the opposite direction toward positive velocities. Therefore, the only possibility to have at the same time correct values at cloud bases and within the boundary layer is to resolve the frequency chirp of the laser pulse in different range gates and use this information to remove the chirp artifacts. In this work we present a deconvolution technique that is capable of removing the chirp artifacts from cloud bases and continuous signals at once.
3. Application of two-dimensional deconvolution to atmospheric Doppler spectra
a. Method
Let D(ρ, ν) (ρ = 0, … , Nρ, ν = 0, … , Nν) contain the pure and undisturbed information about the intensity scattered back from each element of the discrete range–frequency space. The discrete coordinates are connected with the continuous coordinates like r = ρΔr and f = νΔf.
Removing the effect of the chirped laser pulse from the dataset requires the convolution operation to be undone in order to recover the original atmospheric information D(ρ, ν). This is theoretically possible by dividing Eq. (4) by DFT[P(ρ, ν)] and applying an inverse Fourier transformation. However, this approach usually fails because of noise amplification. The direct inversion of the deconvolution by solving the equation system presented by Eq. (3) is also not possible because the equation system is underdetermined. We therefore have to apply iterative deconvolution algorithms to retrieve an approximation of the input information D(ρ, ν).
Iterative two-dimensional deconvolution algorithms are best known from confocal microscopy (Baddeley et al. 2006) and astronomy. For our application we use the Richardson–Lucy (RL) algorithm (Richardson 1972; Lucy 1974), which is famous for its application on blurred images of the Hubble Space Telescope (Stobie et al. 1994). However, this algorithm is not the only one available. There are various other algorithms that could be applied here, too. Most of them are already available in ready-to-use software packages.
The RL algorithm is an iterative algorithm that tries to reproduce the function B(ρ, ν) [Eq. (3)] by making an initial guess of D(ρ, ν) and then convolving this guess with a provided point spread function P(ρ, ν). [The initial guess can, for example, be provided by directly solving Eq. (4) as explained above.] The disparity between the resulting dataset B*(ρ, ν) and B(ρ, ν) is then used to make a better estimate of D(ρ, ν) in the next iteration step. The iteration is stopped after a defined number of steps or when the difference between B*(ρ, ν) and B(ρ, ν) has become smaller than a defined threshold. To increase the convergence speed of the RL algorithm we use a vector extrapolation–based acceleration technique (Biggs and Andrews 1997; Remmele and Hesser 2009).
Many different software solutions exist, which can perform an iterative two-dimensional deconvolution. For this work we used the implementation of the RL algorithm in the image restoration software “BiaQIm” (Tadrous 2011). It was favorable for our purposes because it has command-line access and can read floating point data. For the deconvolution of one dataset with 128 frequency and 196 height bins, the software needs less than 1 s on a desktop computer. Because the acquisition time for one dataset is at minimum 1 s with our CDL, the operation could be performed in real time during measurements, but it can also be applied afterward on the stored spectra.
b. Improvement of the velocity estimation for atmospheric data
Figure 4a presents one profile of atmospheric CDL spectra measured in vertical direction at Leipzig, Germany, on 13 March 2011. The spectra were averaged over 1500 laser shots (2 s). The chirped laser pulse is magnified. Here, the trend to higher-frequency shifts before and after the laser pulse maximum is visible. The dataset was deconvolved with the accelerated RL algorithm (10 iterations) using the recorded laser pulse at the beginning of the dataset as a PSF. The result is shown in Fig. 4b. Figure 6 shows a comparison between the vertical velocities estimated from the untreated (Fig. 6a) and the deconvolved dataset (Fig. 6b) marked in Fig. 5 with a dotted line at 1740 UTC. It is visible that the deconvolution shifts the detected wind velocities about 0.2 m s−1 toward positive values. Artificial velocities at the cloud bases and the cloud tops are no longer present. The method works equally for weak and strong signals. In Fig. 4b it is visible that the recorded reference laser pulse has been reduced to nearly one bin in range and frequency dimensions. That means that now the resolution of the dataset matches the grid and is therefore at its theoretical maximum.
Signal (left) intensities and (right) velocities calculated by means of the peak-finding software from the raw atmospheric dataset (black line) and from the deconvolved dataset (gray line). The input data are depicted in Figs. 4a,b.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
A measurement of the atmospheric vertical velocities over 1 h is shown in Fig. 5b. The corresponding intensities are depicted in Fig. 5a. The lidar was pointing vertically, so the range corresponds directly to the height above ground. The chirp effect is visible above and below clouds (e.g., in the magnified portion) and at positions where the signal-to-noise ratio (SNR) is very high. At the bottom and at the top of those structures negative velocities seem to appear, which could be mistaken for falling particles or downdrafts. In reality these artifacts originate from backscattered light from the beginning (bottom chirp effect) and from the end (top chirp effect) of the laser pulse.
To correct the chirp effect in the entire measurement of Fig. 5 the RL algorithm was applied on each dataset independently with 10 iteration steps. The results are shown in Figs. 5c,d. The velocities within the optically thin cloud layers and the boundary layer are shifted, and there is no longer an overall trend to negative vertical velocity values. The removal of the velocity bias is best visible in the boundary layer from 1730 to 1800 UTC.
c. Improvement of the range resolution
The deconvolution technique presented in this work does not only correct the chirp effect. It also simultaneously improves the range and the frequency resolution of the spectra. The improvement of the range resolution is visible, for example, in Fig. 4 between 3 and 6 km. The single layers are much better separated after the deconvolution method has been applied. In Fig. 5c the features within the cloud layers are also much more distinct than in Fig. 5a. The qualification of the resolution improvement is given in section 4.
d. Improvement of the frequency resolution
In most cases a frequency resolution improvement is not needed because only one spectral peak is present (e.g., Fig. 5). However, there are certain conditions where an improvement of the frequency resolution can be an advantage. Figure 7a shows spectra averaged over 5 s (3600 laser shots) recorded during a measurement of low-level clouds in 1-km height with onsetting rainfall at Ziegendorf, Germany, on 29 March 2006. [The data were obtained during the Lindenberg Campaign for the Assessment of Humidity and Cloud Profiling Systems and its Impact on High-Resolution Modeling (LAUNCH-2005), the first measurement campaign on which the Doppler wind lidar WiLi participated.] At about 1-km height the spectra split and two peaks appear. Here, water drops (drizzle) fall with −3.5 m s−1 out of the cloud, which itself is composed of smaller droplets that move slowly upward at 0.2 m s−1. Figure 8 shows the Doppler frequency shift between the arrows in Figs. 7a,b. After deconvolution the peaks are much better separated.
Comparison between the (a) raw atmospheric and (b) deconvolved Doppler spectra recorded during rainfall on 29 Mar 2006 in Ziegendorf, Germany (LAUNCH-2005). Double peaks resulting from upward-driven aerosol particles and falling raindrops, respectively, are visible in the spectra at about 600–800-m height. The two spectra marked by arrows are shown in Fig. 8. (The structure at 1.5-km height and 15 MHz is an artifact, presumably resulting from a weak parasitic laser line of the master oscillator; however, this does not affect the central part of the spectrum.)
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
Atmospheric Doppler spectra during the rainfall event of Fig. 7. The improvement of frequency resolution between the original dataset (black line) and the deconvolved dataset (gray line) is visible.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
4. Iterative deconvolution of modeled spectra
A major problem in validating atmospheric wind lidar data is the fact that there is no reference accessible. To characterize the performance of the deconvolution technique we therefore created an artificial dataset by convolution of ideal input data with the spectra of a recorded laser pulse as the PSF. Therefore, the modeled spectra and the real datasets are on the same grid. The simulation also indicates the magnitude of the velocity deviations introduced by the chirp effect in our CDL.
The discrete input spectra D(ρ, ν) created for this simulation are depicted in Fig. 9a. They are intentionally similar to the spectra in Fig. 4a. As before, the range corresponds directly to the height above ground. Between 0 and 2 km a planetary boundary layer is simulated with continuously declining intensity at the 0-MHz frequency bin. On the top of the boundary layer two cloud layers separated by one range step of Δr = 75 m were added. In the free troposphere between 3 and 6 km altocumulus clouds with virgae are modeled. All modeled air volumes have a simulated Doppler shift of 0 MHz, which also corresponds to 0 m s−1 vertical velocity. Only the virgae of the altocumulus clouds have a Doppler shift of −0.977 MHz, which corresponds to a terminal velocity of −0.987 m s−1. These modeled ideal spectra D(ρ, ν) were now convolved with the recorded PSF from Fig. 4a according to Eq. (3). Because the PSF and the modeled spectra are on the same grid a discrete convolution like that in Eq. (3) can be performed.
Overview of the different steps of the simulation: (a) ideal input spectra and (b) raw spectra as they would be detected by the wind lidar. Spectra are deconvolved with (c) 5 and (d) 10 iterations.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
Figure 9b shows the modeled artificial spectra after convolution with the PSF from Fig. 4. Figures 10a,b show the intensities and the vertical wind velocities detected by the peak-finding software as a function of height. In this representation the optically thick clouds are delta-function-like point sources (small range extent and high intensity). They effectively return a copy of the laser pulse. Therefore, around the clouds the characteristic chirp effect is best visible. Within the continuous signal of the planetary boundary layer or the virgae the chirped pulse smears the spectra, and the chirp of our laser introduces an offset of about −0.2 m s−1 compared to the input spectra.
(a) Intensities and (b) vertical wind velocities calculated by the evaluation software from the simulated datasets (Figs. 9a,b,d): idealized input data (Fig. 9a; thick gray lines), simulated raw spectra (Fig. 9b; straight black lines), and deconvolution with 10 iterations (Fig. 9d; dotted lines). (c) The difference in the velocity estimation between the idealized input data and the deconvolved spectra. It is visible that the deconvolution removes the chirp effect and the deviation between the ideal input data and the deconvolved data is less than 0.02 m s−1.
Citation: Journal of Atmospheric and Oceanic Technology 29, 8; 10.1175/JTECH-D-11-00144.1
The deconvolution of the spectra in Fig. 9b is shown in Fig. 9c for 5 iterations, and in Fig. 9d for 10 iterations, respectively. The same PSF that was used to generate the modeled spectra was used here to iteratively deconvolve it using the method described in section 3a. Figure 10b shows the detected velocities calculated from the spectra deconvolved with 10 iterations as a dotted line. It is important to consider a reasonable threshold on the signal intensity because the velocity estimation at very small intensities can yield incorrect results. Therefore, in this simulation normalized intensities smaller than 0.05 are considered as noise and are not taken into account for the velocity evaluation process. After 10 iteration steps the velocity readings within the continuous signals are shifted back to their initial values (0 m s−1 in the boundary layer and to −0.987 m s−1 in the virga) and the cloud peaks are reduced to one height step again. After 10 iteration steps the maximum deviation between the input signal and the deconvolved signal is 0.02 m s−1 (see Fig. 10c), which is well below the nominal measurement accuracy of 0.10 m s−1 of our CDL.
5. Conclusions
The application of two-dimensional deconvolution to correct the chirp influence on CDL datasets was demonstrated. The deconvolution is applied to the coherent wind lidar datasets in the range and frequency space. This correction is demanded in the case of profiling of the vertical wind component. It was shown that the chirp effect fundamentally limits the measurement accuracy in the vicinity of clouds and in the boundary layer. The method presented does not only remove chirp effects but also improves the range and frequency resolution of our wind lidar. The influence of the laser chirp effect on the velocity measurement has been explained by convolution of a modeled atmospheric signal with a recorded chirped laser pulse. Two-dimensional deconvolution methods were applied to remove this chirp influence from the modeled dataset. The differences in velocity estimations between the ideal input signal and the deconvolved dataset were found to be smaller than 0.02 m s−1 after 10 iteration steps.
The improvement of frequency resolution by deconvolution depends on the averaging time and the atmospheric turbulence. If small-scale turbulence introduces velocity changes on time scales smaller than the averaging time, then the spectral peak becomes broadened. Up to now it was not possible to evaluate the peak width because the bandwidth of the laser pulse has been too high and has masked the turbulence broadening, but after deconvolution the remaining peak width is only connected to the turbulence and thus could be evaluated.
The method enables CDLs equipped with pulse lasers of low complexity to reach very high velocity precisions in the range of 0.05–0.10 m s−1. In this case the final precision is no longer determined by the chirp effect but by other factors, for example, the pointing stability of the scanning unit. The deconvolution technique presented in this work is applied on the averaged spectra and can be easily applied on stored data or in real time during the measurement. Hence, no additional computational effort is necessary during data acquisition, which is especially interesting for systems with very high pulse repetition rates. Likewise, it could be used to remove the chirp influence even from historical datasets (e.g., from CO2 wind lidars with even larger chirp effects) if only the averaged spectra have been stored. The method works with arbitrary chirp and pulse shapes and can also be employed in chirp-free systems simply to increase the range and frequency resolution.
Acknowledgments
This study was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant AN 258/15.
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