## 1. Introduction

Weather and climate models still need a good deal of technical refinement in order to meet the accuracy level necessary to resolve the current environmental, economic, and societal issues. The atmosphere is a very complex system and the requirements for improving the physics of the models and the quality of the inputs are numerous. One such requirement is understanding the atmospheric phenomena at increasingly finer temporal and spatial scales, both for research and operational applications. The purpose of this paper is to discuss the potential of multiple field-of-view (MFOV) lidars to provide good quality, high-resolution data on cloud geometry and structure for real time and statistical uses. Such lidars could be easily integrated into ground-based, ship-based, and even airborne remote sensing stations.

Low-level clouds are generally optically dense. In such media, light scattering, which is the driving force of lidars, can occur several times before being collected. These extra scattering events, although considered a hindrance in many applications, are a source of information on particle size, as shown by Benayahu et al. (1995), Roy et al. (1999, 2002), and Eloranta (2002). MFOV lidars were designed to capture, angularly resolve, and make use of the multiple scattering contributions (Bissonnette and Hutt 1990; Hutt et al. 1994). Over the past years, Bissonnette et al. (2002, 2005) have implemented a method to retrieve simultaneously from MFOV lidar returns, the range-resolved extinction coefficient and effective particle diameter. From these primary solutions and the assumption that the form of the droplet size distribution is known or given, for example, a modified gamma function for cloud droplets, other important microphysical parameters such as the liquid water content (LWC) of water clouds can be calculated at the same high temporal and spatial resolutions as those of the original lidar measurements. The subject of this paper is not the retrieval method but its application to measuring important meteorological parameters. In particular, we show through sample results that MFOV lidars can contribute in near–real time significant knowledge on the geometry and structure of water clouds.

We briefly review the retrieval technique in section 2. We describe our current lidar instrumentation in section 3, and in section 4 we give the essentials on the experiment carried out to obtain the data used here. We present sample solution results in section 5, and report examples of cloud structure data that can be derived from the lidar solutions in section 6.

## 2. MFOV lidar retrieval technique

Our retrieval technique is based on the measurement of the multiply scattered lidar returns at multiple receiver fields of view. Figure 1 shows a typical MFOV lidar measurement. On the left panel are plotted the returns as functions of range for different FOVs, and on the right panel the same returns are shown, but as functions of FOV for different ranges. The sensitivity on the extinction coefficient is strongest in the range dependence and the sensitivity on the droplet size is strongest in the FOV dependence. The retrieval method takes advantage of this situation.

Because the dependence on the extinction coefficient and droplet size is nearly separable in the first approximation, the solution is best approached by an iterative algorithm. Also, the iterative method has the advantage of using easier direct-problem calculations of the multiple scattering effects instead of requiring the more difficult mathematical inversion of the implicit equations for the extinction coefficient and droplet diameter that constitute the currently known models of multiply scattered lidar returns (Bissonnette 2005). The principle is as follows: given intermediate solutions for the droplet diameter and extinction coefficients, we use them as inputs to run direct-problem calculations of the multiply scattered returns for comparison with measurements, and thus update the solutions to start a new iteration. We repeat the process until convergence is achieved.

The retrieval technique is fully described in Bissonnette et al. (2005). We only summarize below the principal features.

*d*: where

_{e}*d*is the ratio of the third- to the second-order moments of the size distribution,

_{e}*R*is the range,

*λ*is the lidar wavelength,

*R*

_{0}is the range to cloud base, with

*R*) = ∫

^{R}

_{0}

*α*(

*R*′)

*dR*′ is the one-way optical depth,

*α*is the extinction coefficient, and

*θ*

_{md}(

*R*) is the angle that scales the field-of-view dependence of the measured MFOV returns

*P*(

*R*,

*θ*), where

*θ*is the half-angle receiver field of view. For instance,

*θ*

_{md}(

*R*) is determined from curves such as those depicted on the right-hand-side panel of Fig. 1. Bissonnette et al. (2005) describe in detail how

*θ*

_{md}is calculated. The function

*g*(

*R*) takes into account the extinction effects; it is a weak function of the optical depth and extinction coefficient.

*α*(

*R*). The equation describing the multiply scattered lidar return

*P*(

*R*,

*θ*) is where

*C*is a system constant,

*κ*=

*α*/

*β*is the lidar ratio,

*β*is the backscatter coefficient, and

*M*(

*R*,

*θ*) is the multiple-to-single scattering ratio that is calculated as explained below. Equation (4) can be rearranged as follows: Taking the logarithm of Eq. (5) and differentiating with respect to

*R*, we obtain which has the well-known solution where

*R*is the far range of the lidar return and

_{f}*α*is the specified boundary value at

_{f}*R*. Therefore, we obtain the extinction solution for the current iteration step by simply evaluating the right-hand-side expression of Eq. (7). First, we need to calculate

_{f}*S*(

_{θ}*R*), which depends on the measured quantity

*P*(

*R*,

*θ*), but also on

*M*(

*R*,

*θ*) and

*κ*(

*R*). It is not necessary to express

*M*(

*R*,

*θ*) in explicit mathematical form, we can simply calculate

*M*(

*R*,

*θ*) with any convenient numerical code of the direct multiple scattering problem using as inputs the

*α*solution of the preceding iteration step and the

*d*solution of the current step. Similarly, we run a Mie code to compute the lidar ratio profile

_{e}*κ*(

*R*) with the droplet distribution specified by

*d*and the index of refraction of liquid water. The boundary value

_{e}*α*is determined from the first-iteration solution as explained in Bissonnette et al. (2005). Because we are dealing with dense clouds, the solution of Eq. (7) becomes rapidly independent of

_{f}*α*as

_{f}*R*recedes from

*R*. We use the return

_{f}*P*(

*R*,

*θ*

_{max}) measured at the largest field of view

*θ*

_{max}for which the range

*R*to the end of the usable data is the longest. Typical

_{f}*R*ranges in dense, low-altitude water clouds are 250–300 m, but the range to cloud base can be as far as 3 km for our system. More powerful lidars could probe clouds at greater distances.

_{f}The retrieval method was both successfully tested with Monte Carlo simulations and shown to produce solutions in general agreement with aircraft in situ measurements in stratus clouds (Bissonnette et al. 2002, 2005). The Monte Carlo tests gave an average ratio of the retrieved-to-true parameter value of 1.00, with a standard deviation of 5% for the extinction coefficient and 11% for the effective droplet diameter. However, the agreement with the in situ measurements cannot be quantified in such clear precision intervals because, as will be seen in section 6, the correlation length in clouds is much too short to allow point-by-point comparisons.

*α*and the effective droplet diameter

*d*

_{e}. Other products can be derived from these two, in particular, the LWC given by where

*ρ*is the liquid water density,

*dN*/

*dr*is the droplet radius density distribution, and

*Q*(

_{e}*r*,

*λ*,

*m*) is the Mie extinction efficiency factor at the lidar wavelength

*λ*for a droplet of radius

*r*and complex refractive index

*m*. Here, we calculate LWC as expressed in Eq. (8) using for

*dN*/

*dr*the modified gamma function parameterized by

*d*, but a good approximation for water droplet clouds at the visible and near-infrared wavelengths of practical lidars consists in making the ratio of the two integrals of Eq. (8) equal to unity because

_{e}*Q*≃ 2 for all

_{e}*r*contributing to the integral.

## 3. Instrumentation

The lidar’s main characteristics are an eye-safe wavelength of 1.57 *μ*m, a holographic optical element for the simultaneous separation of the received backscattered radiation into seven fields of view, and a weatherproof hemispherical scanner. The system specifications are summarized in Table 1.

The lidar transmitter is a 10-Hz Nd:YAG laser, pumping an optical parametric oscillation (OPO) cavity that produces a 1-mrad full-angle-divergence (80% of encircled energy) 25-mm-diameter beam of 25-mJ, 10-ns pulses at the eye-safe wavelength of 1.57 *μ*m. The receiver is a 202-mm-diameter, 0.017-m^{2} effective-aperture-area, 977-mm-focal-length Cassegrain-type telescope that images the received radiation on a holographic optical element (HOE) shaped in the form of seven concentric rings. Each ring is a holographically imprinted grating that deflects the incident radiation at 30° from the telescope axis but at different azimuth angles distributed about 40° apart. The ring-shaped holograms were recorded concentrically on a single substrate and the dead gap between the rings is ∼50 *μ*m wide. The deflected beams have a divergence cone of 17° full angle; they are recollimated and reoriented parallel to the telescope axis by an eyepiece lens assembly that produces seven spatially separated parallel beams. The parallel beams are finally focused on different detectors by different optical concentrators. The ring diameters define the FOVs that are, respectively, 0.2, 0.4, 0.8, 2, 4, 8, and 12 mrad, full angle. The detectors are 200-*μ*m-diameter InGaAs avalanche photodiodes (APDs) for the four smallest FOVs and 1-mm-diameter InGaAs positive intrinsic negative (PIN) photodiodes for the largest FOVs. The nonavalanche PIN photodiodes are used because the radiation from the large outer rings cannot be concentrated on the small APDs; there were no InGaAs APDs of a diameter larger than 200 *μ*m at the time of the experiment. Finally, the radiation going through the 0.2-mrad FOV is separated into parallel and perpendicular polarization components.

The main elements of the transceiver are shown in Fig. 2. The laser beam is transmitted coaxially with the receiver telescope. Both the outgoing and returned beams are reflected by a fixed 45° folding mirror and two 45° mirrors forming a periscope spherical scanner. The two scanner mirrors are specially coated to preserve the polarization state of the received radiation with respect to the outgoing radiation at all scanning angles. The scanner is enclosed in a weatherproof housing. Transmission to and from the atmosphere is through an antireflection-coated, 3°-tilted window. The window is equipped with a wiper that can be activated manually or automatically to clean raindrops when necessary. The scanner and folding mirror assembly has a clear aperture slightly greater than 200 mm. The HOE is fixed to the detector block that can be translated axially to adjust the focus distance on the probed cloud. A photograph of the lidar transceiver is shown in Fig. 3.

The detected signals are log amplified and digitized with eight different but slaved 8-bit digitizers at selectable rates between 10 and 125 MHz or range bins between 1.2 and 15 m. The lidar range resolution based on the pulse length is 1.5 m. The usable dynamic range is ∼70 dB.

## 4. Experiment

To demonstrate the potential of the MFOV lidar and retrieval technique for characterizing the geometry and structure of water droplet clouds, we joined in a sea experiment designed to test various radar and infrared sensors. The lidar was mounted on the quarterdeck of the Defense R&D Canada (DRDC) research Canadian Forces Auxiliary Vessel (CFAV) *Quest*, which operated off the coast of Nova Scotia, Canada, 30–50 km outside of Halifax Harbor.

The cloud characterization experiment included both temporal and spatial soundings. For the temporal soundings, the lidar was pointed in a fixed direction, generally the vertical, and bursts of 10 pulses (duration of 0.9 s) were fired at fixed intervals of typically 30 s for total periods of 1–2 h. The 10 pulses were averaged into a single MFOV measurement and the retrievals were performed online. The 30-s intervals provided ample time to complete the calculations that generally took less than 5 s.

The spatial soundings consisted of azimuth-over-elevation scans. The elevation angle was varied between 20° and 70° by steps of 5°, and the azimuth angle was swept between −90° and +80° at each elevation angle. The zero azimuth was chosen in the direction of the ship’s stern. The azimuth sweeps were limited at both ends by ship structures. The duration of a full scan was typically 1 min. During the acceleration and deceleration of the scanner at the start and end of the azimuth sweeps, the data acquisition was stopped to ensure constant azimuth steps of typically 2.5°–3° between the recorded lidar pulses. The selected scan speed was a trade-off between temporal (snapshot) and spatial (tight azimuth–elevation grid) resolutions.

The retrieval processing time varied between 5 and 15 min, depending on the cloud range and the chosen range bin size. The scans were repeated at a rate of about one every 3 min. Hence, the retrievals could not keep up with the acquisitions but lagged the measurements by a few minutes to hours, depending on the number of consecutive scans made in a given measurement set. The adopted procedure was to interlace 5–10 consecutive scans with periods of temporal soundings to keep all retrievals almost on time.

## 5. Sample results

Figure 4 shows examples of time–height plots of the retrieved LWC and effective droplet diameter from a stratus cloud deck that appeared uniform to the ground observer. The solutions are drawn on logarithmic scales to better resolve their wide dynamic range. The results indicate for both LWC and effective droplet diameter a mixture of horizontal stratification and temporal fluctuations that appear as cells of irregular time–height shapes. Figure 4 is a good illustration of the high temporal and altitude resolutions achievable in real time with our lidar. Below the cloud base at ∼540 m, the retrievals show the presence of particles or droplets in the size range of 16–32 *μ*m, but the concentration is very tenuous, with calculated LWC less than 0.004 g m^{−3}. Figure 5 shows similar results but for a more perturbed case.

The azimuth-over-elevation scans provide instantaneous (∼1 min) three-dimensional pictures of the cloud in a spherical coordinate system centered on the lidar position. The solutions are therefore four-dimensional functions that have no straightforward graphical representation. To get around the problem, we cut horizontal planes through the scanned volume on which we interpolate the retrieved solutions. Horizontal planes are physically more meaningful than the series of constant elevation angle surfaces that constitute the scans. However, no compensation is made for the ship motion, so the horizontal planes are actually warped. To minimize the effect, we average over a height of 10–25 m, depending on the prevailing sea state. Examples of interpolated solutions are plotted in Fig. 6 for two altitudes within the same cloud layer, respectively, 325 and 350 m, or approximately 25 and 50 m above the average cloud-base height over the probed area. Both the LWC and effective droplet diameter solutions are shown. The plots illustrate the high degree of spatial resolution achieved, although it should be noted that the resolution is not uniform across the displayed areas because of the scanning geometry. The measurement cells are larger for smaller elevation angles, which occurs at the outer radial edge of the maps.

Figure 6 shows important horizontal variability and a strong vertical dependence. The horizontal variations take the form of irregular cells as small as 100 m in size. The parameter range between the peaks and valleys is high and increases significantly with the altitude from the cloud base. Table 2 lists the low, peak, average, and standard deviation values of *d _{e}* and LWC for the two altitudes of Fig. 6. The average values agree with what is generally measured in such clouds, but the fine spatial resolution of the lidar solutions reveals a high degree of variability with highs and lows alternating on a rather small horizontal scale. This was not anticipated from visual observations that indicated a rather uniform layer.

There is in Fig. 6 a lot more quantitative information on the variability of the parameters and on the size and spatial distribution of the inhomogeneities than the cursory observations discussed above and recorded in Table 2. We will address the analysis of the spatial structure in the next section. As for measuring variability, a suitable tool is a histogram. For example, we plot in Fig. 7 the histograms of the *d _{e}* and LWC solution values of the 350-m-altitude maps of Fig. 6. These histograms not only provide a clear measure of the strength and breadth of the fluctuations, but also show well-defined statistical distributions; in other words, there is statistical order in those fluctuations.

The rapid altitude dependence of the average parameters near the cloud base is consistent with the physics of cloud formation. There are no indications of measurement and solution biases. The intervals of lidar ranges intersecting the 325- and 350-m horizontal planes, respectively, 335–650 and 360–700 m, are very wide compared with the 25-m difference in altitude. Hence, the altitude differences found in Fig. 6 and summarized in Table 2 cannot be related to a spurious lidar range effect on either the measurements or the solution method because, if there were such an effect, it would also show up within the constant-altitude planes.

## 6. Cloud structure

We illustrate in this section how the retrievals from the MFOV lidar measurements can be applied to characterize the structure of dense low-level water clouds. Discussions on the physical implications are left out for a subsequent paper.

One recognized means of quantifying the spatial or temporal structure of a random function are its autocovariance (or structure) function and corresponding power spectrum. We follow here the methodology of Blackman and Tukey (1959).

*C*

_{T,X}(

*τ*) of the true autocovariance function over a given record, denoted by a suitable parameter

*T*, of a one-dimensional statistically homogeneous or stationary random function

*X*(

*t*) of zero mean is defined by where the overbar denotes averaging over a given record of length

*T*, and

*τ*is the lag variable. The function

*C*

_{T,X}(

*τ*) is independent of the value of

*t*where the origin

*τ*= 0 is defined by virtue of the assumption of homogeneity or stationarity with respect to

*t*. Averaging over several records gives a valid estimate of the true autocovariance function. The power spectrum of

*X*(

*t*) is the Fourier transform of Eq. (9). Experimentally,

*C*

_{T,X}(

*τ*) is defined up to a maximum finite lag

*τ*. To eliminate the large sidelobes in the corresponding power spectrum that would result from the unavoidable truncation of

_{m}*C*

_{T,X}(

*τ*) at

*τ*, the common practice consists of multiplying

_{m}*C*

_{T,X}(

*τ*) by a lag window function

*D*(

*τ*) that satisfies the conditions

*D*(0) = 1 and

*D*(

*τ*) = 0 for |

*τ*| >

*τ*. A convenient

_{m}*D*(

*τ*) (Blackman and Tukey 1959) is

The solution maps shown in Figs. 6 are two-dimensional. However, to simplify the analysis, we assume statistical isotropy in these horizontal planes, which means that the autocovariance function is independent of the direction of the lag vector, and is only dependent on its magnitude. We do not claim that the observed clouds are horizontally isotropic at all scales—there is actually organized large-scale motion as will be seen—but the hypothesis is acceptable at small scales (<100–200 m). In any case, our objective here is to illustrate how structure data can be derived from the spatially detailed lidar solutions.

*θ*and height

_{i}*h*constitutes a record. For instance, the autocovariance function of the effective droplet diameter

*d*(elevation, azimuth, range) for the constant elevation angle

_{e}*θ*and height

_{i}*h*is given by where The quantity

*d*(

_{e}*θ*,

*ϕ*,

*R*) is the retrieved lidar solution defined on the spherical coordinate system of the measured scan, (

*M*− 1)Δ

*ϕ*= (

*ϕ*

_{max}−

*ϕ*

_{min}) is the azimuth range of the sweep at the elevation angle

*θ*, Δ

_{i}*ϕ*is the constant azimuth increment, Φ

*= (*

_{k}*k*− 1)Δ

*ϕ*is the azimuth lag,

*k*varies from 1 to

*N*, (

*N*− 1)Δ

*ϕ*= Φ

_{max,}and Φ

_{max}is the maximum lag chosen to be equal to half the interval (

*ϕ*

_{max}−

*ϕ*

_{min}).

*x*–

*y*plane and the relations between

*r*= |

_{k}**r**

*| and Φ*

_{k}*. Next, we interpolate*

_{k}*C*

_{θi, de}(

*r*) on a constant-increment grid of Δ

_{k}*r*=

*h*Δ

*ϕ*/tan

*θ*and Fourier transform the resulting equally spaced function to obtain the power spectrum function

_{i}*P*

_{θi, de}(

*ν*) of frequency

_{l}*ν*given by We repeat the same operations for the LWC solutions.

_{l}We calculate in this fashion the autocovariance and power spectrum functions in a given horizontal plane for all elevation angles *θ _{i}*. We normalize each result by its variance, that is,

*C*

_{θi, de}(

*r*= 0), and average over all

_{k}*θ*to obtain an estimate of the true autocorrelation function over the given probed horizontal plane. Note that the space and frequency resolutions and record lengths derived from Eqs. (15) and (16) are different for each elevation angle

_{i}*θ*. However, there is a wide overlap between the records, and the averaging merges these overlapping functions into a single autocorrelation function and normalized power spectrum for the altitude

_{i}*h.*

For example, the autocorrelation functions obtained for the cases of Fig. 6 are plotted in Fig. 9 for both *d _{e}* and LWC. It turns out that the

*d*and LWC functions are nearly equal up to a separation distance of ∼200 m. The curves reveal a correlation length of about 60 m (defined here as the zero-crossing lag) for both altitudes with only slight differences in the correlation falloff. The 60-m correlation length shows that the fast fluctuations are not random measurements or solution errors because, if they were, we would observe complete decorrelation at Δ

_{e}*r*≃ 15 m, the smallest lag increment, because the measurements at different lag values correspond to different lidar pulses that are solved independently.

Beyond the first correlation falloff, the functions go through a secondary maximum at a separation distance near 300 m. This means that there are near-periodic features at a wavelength of ∼300 m. This length is nearly equal to the cloud-base height, which is certainly a physically meaningful scale characterizing the large inhomogeneities within the cloud. Figure 10 shows the autocorrelation function at an altitude of 600 m for another event with a cloud base at ∼550 m, that is, almost twice as high as in Fig. 9. The correlation length is of the same order of magnitude as in Fig. 9, but the large-scale feature shows up at ∼550 m, the new cloud-base height. We find similar results in almost all stratus cloud events observed during the experiment, although not as clear everywhere as in Figs. 9 and 10.

The power spectra provide an alternate view on the spatial structure of the eddies or inhomogeneities embedded in the cloud, more particularly, on how they are distributed in size or spatial frequency. The spectra calculated at different altitudes for the cloud depicted in Fig. 6 are plotted in Fig. 11. They are stacked on the same graph by shifting the ordinate scale by a multiplicative factor of 10^{0.5} between the adjacent altitudes. These are normalized spectra. To give the dimensional information, we plot in Fig. 12 the vertical profiles of the corresponding average and standard deviation values of LWC and *d _{e}*. The quantities used to normalize the spectra at each altitude are the variances or the square of the standard deviation values. We also show in Fig. 12 the profile of the average optical depth

The spectra span a frequency domain between 0.001 and 0.1 cycles per meter (cpm), or a wavelength domain between 10 and 1000 m. The curves show in all cases, except for the lower altitude of 300 m at or near cloud base, a sizeable range over which the Kolmogorov’s turbulence power law of *ν*^{−5/3} (Lumley and Panofsky 1964) is verified. This is a significant result. It is yet another indication that the solution fluctuations are not random errors, but the manifestation of the well-known physical process of inertial energy transfer between the energy-containing eddies and the energy-dissipating eddies in the cloud. The width of the *ν*^{−5/3} regime varies slightly with altitude, from about 0.01–0.1 cpm near the base of the cloud to 0.007–0.1 cpm at the altitude of 425 or 125 m into the cloud. In terms of eddy sizes, these numbers translate into 10–100 and 10–150 m, respectively.

One important special feature revealed by the spectra of Fig. 11 is that the *ν*^{−5/3} regime is split in two; the transition occurs at frequencies between 0.02 and 0.04 cpm or wavelengths between 25 and 50 m. That split is a recurrent feature of all calculated spectra. We have no ready explanation for this transition behavior except to say that it is not due to ship motion. The prevailing period of the ship’s pitch and roll determined by Fourier transforming the pitch-and-roll time series recorded simultaneously to the lidar measurements is 5–10 s, which, incidentally, is 2–3 times greater than the duration of a complete sweep. This would correspond to wavelengths of ship-induced fluctuations on the order of hundreds of meters. By comparison, the wavelength range of the observed transition region is much smaller at 25–50 m.

The temporal soundings can also be used to calculate temporal spectra that can be further transformed into spatial spectra by use of the Taylor hypothesis. However, this adds little to illustrate the potential of the MFOV lidar, and the physical interpretation of the results is beyond the scope of this paper.

## 7. Conclusions

We have applied the MFOV lidar to measure in near–real time the spatial structure of low-level water clouds. The principal results presented herein are the lidar-derived solutions of the cloud liquid water content and the effective droplet diameter for vertical soundings versus time and for instantaneous (∼1 min) truncated spherical scans. We were able to quantify the spatial distributions of the cloud horizontal inhomogeneities in terms of autocorrelation functions and power spectra. The spectra have a domain of horizontal frequencies ranging from 0.001 to 0.1 cpm or from 10 to 1000 m in wavelengths. We found in all cases the presence of a −5/3 power law over a reasonably wide frequency domain. However, the −5/3 regime is not uniform, but instead is made up of two subregimes separated by a short transition region showing an increase in energy density. There are few reports of such transitions; the only examples known to the authors are those discussed by Penner and Shamanaev (1999) and Penner et al. (2000). Without speculating on the physical interpretation of these observations they, and the various other results presented in this paper, illustrate quite well that significant detailed information on cloud structure can be derived from MFOV lidars. One important practical advantage of lidars is the minimal amount of resources needed to operate over extended periods of time.

## Acknowledgments

We are grateful to the technical staff of DRDC Atlantic, the CFAV *Quest* and the dockyard who made operating our lidar at sea possible. We also thank Sylvain Cantin for his able technical assistance.

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Lidar system technical specifications.

Low, high, average, and standard deviation values of effective droplet diameter (*d _{e}*) and LWC derived from Fig. 6, event recorded at 1342 UTC 19 Sep 2005; cloud base is at ∼300 m.