Optimization of the Cross-Correlation Algorithm for Two-Component Wind Field Estimation from Single Aerosol Lidar Data and Comparison with Doppler Lidar
1. Introduction
The cross-correlation algorithm (CCA) is a mainstay in the field of motion estimation. It is used to compute the apparent motion of objects and fluid flows in the fields of robotics, navigation, medical imaging, and geosciences (Murray et al. 2009; Emery et al. 2003; Avants et al. 2008; Schubert et al. 2013; Adrian and Westerweel 2011; Cheng et al. 2005; Antoine et al. 2013). In the atmospheric sciences, the cross-correlation algorithm has been applied to satellite imagery (Leese et al. 1971; García-Pereda and Borde 2014), radar data (Rinehart and Garvey 1978), and lidar data (Eloranta et al. 1975; Shimizu et al. 1981; Kolev et al. 1988; Schols and Eloranta 1992; Piironen and Eloranta 1995). Under the assumption that macroscopic aerosol features are advected by the wind, the wind velocity can be estimated remotely from the apparent motion of the features in lidar image sequences.
The CCA has been the primary numerical method for determining fluid motion in particle image velocimetry (PIV) experiments.1 In most PIV experiments, the fluid is deliberately seeded with very small particles that serve as robust tracers of the local flow. The particles are typically illuminated with a laser light plane and individual particles are discernible in the rapidly collected sequence of images taken by a camera. The particles do not change appreciably in shape nor in brightness as they move. For low and moderate particle density experiments, in which the fluid between the particles appears dark and does not contribute any information, the motion of the small particles is solely relied upon for determining the motion field. As a result, the cross-correlation functions (CCFs) for PIV experiments contain sharp peaks. Furthermore, few particles straddle the edges of the interrogation window. Particles that appear or disappear within the time between two frames, either by moving into or out of the interrogation window, or into or out of the illumination plane, only contribute to incoherent variance confined to the zero lag of the CCF. The application of the CCA to derive air motion from atmospheric aerosol backscatter lidar images raises a set of issues that are not present in most PIV experiments (at least low-density experiments). These issues result from the time required for the lidar to collect a scan, the inability to discern individual particles, and the continuous range-dependent image intensity that is a proxy to particle concentration (Held et al. 2012).
At the heart of the CCA is the computation of a CCF from a pair of images. The location of the peak of the CCF relative to its origin indicates the displacement of dominant image features. A consequence of selecting a single peak is the loss of information that describes the true velocity field that causes the motion of features. As reflected in the title of the paper by Schols and Eloranta (1992), some have regarded the velocity estimate to be the area average of the wind field in the interrogation window. However, Schols and Eloranta (1992) do not elaborate on their use of area average. The wind is a vector and the average of a set of vectors may be computed by vector averaging (summing the components first and then computing the mean vector) or scalar averaging (averaging the speeds and directions that result from each vector in a set). In the former, a mean displacement would result. In the later, the mean speed may be different from the magnitude of the mean displacement, and of more interest to the wind energy community, for example. Moreover, it is possible that, because of the organization of the wind and the aerosol fields within the interrogation window, CCA estimates do not equal an area average at all. To investigate these issues, we generated synthetic aerosol images and velocity fields and applied the CCA to the synthetic aerosol backscatter images. This enabled us to compare the results of the CCA with the mean of the synthetic velocity field that was used to advect the synthetic aerosol structures.
In addition, the synthetic data are used to confirm the necessary sequence of steps within the CCA. Mayor et al. (2012) compared the results of a rudimentary CCA applied to scanning elastic backscatter aerosol lidar data to tower-mounted sonic anemometer wind measurements between 10 and 30 m AGL. The study showed that the use of the CCA for interrogation windows that were 500 m × 500 m and smaller produced plausible but noisy results. However, a large and important part of the spectrum of atmospheric motions exists at these microscales. Thus, a higher spatial resolution of the wind velocity fields is desired. The algorithm used by Mayor et al. (2012) was functional but not optimal. It did not include several steps tested in this paper that may deliver better results. They are 1) zero padding to account for nonperiodic images, 2) use of a window function to reduce the undesirable effects of sharp edges and aerosol features entering and leaving the block area, 3) multipass interrogation for iterative refinement of the motion estimation, 4) multigrid interrogation to improve the spatial resolution of the resulting flow fields, and 5) deformation of images to correct for the mean advection of aerosol features during the time required to complete one scan as described by Sasano et al. (1982).
Finally, because the synthetic images and velocity fields are only an approximation to reality, we also conducted a field experiment to validate the results of the optimized CCA applied to real lidar data. The elastic backscatter aerosol lidar system used was the Raman-shifted Eye-safe Aerosol Lidar (REAL) (Mayor and Spuler 2004; Spuler and Mayor 2005; Mayor et al. 2007). A single compact Doppler lidar was used to validate the wind velocity fields resulting from the optimized CCA applied to the REAL aerosol backscatter images.
This paper is organized as follows: Section 2 introduces the CCA and describes the options that improve the wind velocity estimation. In section 3, the performance of the CCA with these options is evaluated by using synthetic lidar backscatter images and wind velocity fields. In section 4, the performance of the optimized CCA is validated by field experiments using the Doppler lidar (DL) data as a reference.
2. Application of the cross-correlation algorithm to elastic lidar backscatter images
Scanning lidar data are collected in a spherical coordinate system with coordinates of azimuth, elevation, and range. The data are processed and interpolated to a Cartesian grid before the CCA is applied. The “preprocessing” that is applied to each range-dependent lidar backscatter array includes the calculation and subtraction of the raw background signal, multiplication of the waveform by the range squared, conversion to decibels, and the application of a high-pass median filter (Schols and Eloranta 1992). The application of the high-pass median filter is critical. It removes large-scale trends in the backscatter arrays caused by attenuation (or real atmospheric structure) and leaves the smaller-scale backscatter perturbations that are more likely to be advected by the local wind. For the REAL data, we typically use a 333-point filter window (500 m in range) resulting in the removal of features larger than one-half the filter length (250 m). The preprocessed data are then interpolated to a Cartesian grid, which results in a 2D array referred to as the image. In this work, the image is constructed by interpolating to a Cartesian grid with a spacing of 10 m in both the east–west and north–south directions. The CCA can then be applied to any square subset of the image. This subset is referred to as the interrogation window or the block (see Fig. 1). In practice we often start with a 1 km2 block containing 100 × 100 data points. The CCA is applied to as many different blocks as desired to compute a vector flow field.
Diagram defining terminology used and relative locations of each object.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
a. The cross-correlation function
b. Zero padding
The FFT is designed for periodic signals, but real atmospheric lidar data are not periodic. Thus, it is important to circumvent the assumption of periodicity by using “zero padded” arrays (Adrian and Westerweel 2011). Each dimension is padded by zeros over a domain that is twice the size of the original signal, so three-quarters of the arrays for input to the FFT are filled with zeros (Bastiaans 2000). The first input array contains the data from the first interrogation window in the lower-left corner and zeros elsewhere. The second input array contains the data from the second interrogation window in the upper-right corner and zeros elsewhere. Figure 1 shows this arrangement. This is done because the complex conjugate of the FFT of the second input array is used to compute the CCF. According to Hamada (2014), the performance of a CCA using 100 pairs of synthetic backscatter images and synthetic wind velocity fields are slightly increased by using zero-padded images.
c. Histogram equalization
Histogram equalization has been used previously for computing horizontal wind vectors from lidar backscatter images (Schols and Eloranta 1992). Without histogram equalization, the motion of small areas of bright features may dominate the CCF. In that case, the cross correlation may be biased toward the motion of the small and bright features. On the other hand, with histogram equalization, other dimmer features in the image are able to influence the CCF. However, according to Hamada (2014), histogram equalization tends to broaden the peak of the CCF thereby increasing uncertainty of the peak location. Since our experiment to test histogram equilization involved only one relatively strong wind case (mean wind speed of approximately 10 m s−1), the effects of histogram equalization should be investigated for other scenarios in the future to determine whether it is broadly beneficial.
d. Window function
e. Multipass interrogation
The CCF for two consecutive lidar backscatter blocks gathers contributions from aerosol features that appear in both blocks. Because aerosol features are advected by the wind, some aerosol features in the first image block may move out of the interrogation window in the time interval between consecutive scans. During that time, aerosol features initially outside the first block may appear in the second block. In this case, these features do not contribute to the CCF, and the wind velocity may be underestimated. For example, if the wind velocity field is nonuniform within an interrogation window, then the aerosol features with lower velocity tend to remain within two consecutive image blocks, while those moving faster tend to disappear. In this case, the CCA is biased and underestimates the wind velocity field in the interrogation window. A multipass interrogation (Raffel et al. 2007) can minimize such effects. This approach iterates two steps: (i) compute a displacement vector from two image blocks by the cross correlation and then (ii) displace the center of the second block according to this vector. Each vector is an incremental refinement of the solution, and the process repeats until the magnitude of the incremental vector falls below 1 pixel—typically, after two or three iterations. The subpixel location of the peak of the CCF is then estimated by curve fitting following Piironen and Eloranta (1995). Finally, the solution vector is given by the sum of the incremental estimations and the subpixel location. Following this process, the displaced second block contains a similar number of the features that also appeared in the first image block. The CCF is therefore better defined, and the accuracy of the motion estimation increases.
f. Multigrid interrogation
The CCA provides one wind velocity vector per interrogation window. The typical size of a large interrogation window, for elastic lidar backscatter images, is about 1 km × 1 km, which is larger than the size of most turbulent coherent structures. With a large block size, most microscale structures cannot be resolved although they are important meteorological phenomena. In this case, multigrid interrogation can be used to increase the spatial resolution of the wind velocity vector field. Multigrid interrogation is similar to multipass interrogation except that the dimensions of the blocks are reduced after each pass (Adrian and Westerweel 2011). In general, the number of features that appeared in both blocks decreases as the block size is reduced—typically, the size reduction factor is 50%. However, displacing the second block increases the similarity of these blocks and makes it possible to resolve the wind velocity vector for a relatively small region (less than 500 m × 500 m). Mayor and Eloranta (2001) applied multipass interrogation and multigrid interrogation but were not able to validate the resulting flow fields.
g. Image deformation
Lidar scans do not represent an instantaneous distribution of aerosol features as in a “snapshot” because of the time required to complete a scan; that is, during a scan some aerosol features are observed before others, while all features are advected by the wind. The result is a distorted image relative to the ideal snapshot. Because of this, apparent displacements are inaccurate and the motion estimation is biased. This problem was first discussed by Sasano et al. (1982), who proposed an iterative correction of image deformation. In this process, aerosol features that are observed before and after a reference time within a given scan are translated forward and backward, respectively, according to the mean estimated wind velocity in the entire region of the scan. The process is repeated until the mean velocity is changed by less than 1%. After the image correction, the deformed image approximates a snapshot of the true lidar backscatter distribution at the reference time, here corresponding to the time associated with the backscatter arrays collected in the center of the sector. While processing and comparing the time series data presented in section 4, we noticed a clear bias in the velocity estimates when the image deformation was turned off. The magnitude of the bias increased with the mean wind speed and was eliminated with the image deformation turned on.
h. Quality control
In general, the signal-to-noise ratio (SNR) of elastic backscatter data decays as one over the range squared. At long ranges, the noise amplitude may dominate the backscatter from coherent aerosol features. Applying the CCA to such data can result in areas of spurious, random vectors. In other circumstances, the aerosol features can lead to a peak of the CCF at a location that does not represent the actual motion. Such situations are more likely to occur close to the scan edges, or when high wind speeds are involved. They result in isolated spurious vectors, known as outliers. These two distinct sets of erroneous estimates are detected by two different mechanisms inspired from the PIV expertise.
A first step consists of discarding vectors for which the value of the peak of the CCF is below a certain threshold. This test is efficient at detecting erroneous vectors resulting from low SNR backscatter data, typically removing patches of vectors in the far range. The second step is handled once the whole vector field has been estimated. It is the normalized median test, as described in Adrian and Westerweel (2011) and Westerweel and Scarano (2005). It assumes a local, spatial coherence of the vector field and therefore is able to detect isolated outliers that were missed by the previous test.
i. Implementation
A simplified diagram of the operational algorithm is presented in Fig. 2. It combines iterations of the distortion correction (section 2g), the multigrid interrogation (section 2f), and the multipass estimation (section 2e). For each vector, there can be up to 27 iterations total (three for each of the distortion correction, multigrid, and multipass), with as many evaluations of the CCF. To complete the execution of the motion estimation within the time between two scans of the REAL, the core pieces of the method (CCF, histogram equalization, interpolation for the distortion correction) are written in the Compute Unified Device Architecture (CUDA) language (Mauzey et al. 2012). These functions are executed in a massively parallel fashion on specific graphic processing units that are designed for scientific computation, thus enabling real-time execution.
Simplified diagram of the CCA.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
3. Tests using synthetic backscatter images and wind velocity fields
Unlike other wind measurement techniques that sample relatively small volumes of the atmosphere, the CCA relies upon spatial data over a large area to make a velocity estimate that is assigned to a single point (the location of the center of the interrogation window). The spatial data are the aerosol backscatter field in the interrogation window. If the actual wind field is uniform throughout the interrogation window, then the spatial distribution of aerosol features that contribute to the CCF does not matter. However, the wind velocity in the real world is spatially variable. Therefore, it is reasonable to question whether the peak of the CCF represents a central tendency of the actual spatial wind field within the interrogation window. This is because the aerosol features may not be uniformly distributed across the interrogation window, or they may be correlated with velocity structures.
To study this problem, we developed synthetic velocity fields and synthetic aerosol backscatter images. The synthetic velocity fields, although not continuously defined, are high in spectral integrity, of much higher resolution than can be obtained from present observing techniques, and enable us to calculate the spatial mean velocity. The synthetic backscatter field enables us to confirm the previously suggested notion that the CCA results in an area average of the wind field in the interrogation window.
The structure of turbulence depends on the static stability (Moeng and Sullivan 1994; Khanna and Brasseur 1998). Turbulence may be generated by shear or convection or some combination of both. During periods of neutral static stability, turbulence is generated only by shear. Coherent wind and aerosol structures tend to be elongated and aligned with the mean wind direction (streaks). During periods of pure convection in the lower boundary layer, cellular flow structures exist. The cells are characterized by broad areas of subsidence and divergence and share borders of strong convergence and updrafts (Schmidt and Schumann 1989). Often, vortices form near the intersections of the convergence lines (Kanak 2005).
Because these two stability regimes produce different turbulence coherent structures, we created two types of synthetic flow fields for testing. For pure shear-induced turbulent flows, we added small-scale turbulent perturbations to a uniform velocity field. For pure convection, the four fundamental kinematic components (divergence, rotation, stretching deformation, and shearing deformation) (Holton and Hakim 2013, 23–25) of fluid flow fields were generated.
To generate synthetic backscatter and velocity fields, we first defined a Cartesian grid. The grid spacing (
Comparison of (left) a synthetic backscatter image and (right) a REAL backscatter image. The REAL backscatter image was collected at the University Farm at California State University, Chico, on 17 Oct 2013.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
The (a) u and (b) υ components of shear-induced turbulent wind perturbations resulting from the Mann model.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
Streamlined flow fields for pure (a) divergence, (b) rotation, (c) stretching deformation, and (d) shearing deformation. The centers of the functions were deliberately offset because the interrogation window is unlikely to be perfectly centered on such flow structures when applied to real data.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
The CCA was applied to the synthetic pairs of images and velocity vectors were computed. The precise peak location of the CCF is estimated by fitting a 2D surface (polynomial of degree 2) to the peak (5 × 5 pixels). Without this subpixel interpolation, the theoretical error on the estimated motion is 
a. Significance of optional steps
The experiments with synthetic data were used to elucidate two aspects of the CCA. First, they were used to study the significance of the impact of the optional numerical procedures within the CCA (described in section 2 and Table 1). Results are shown in Table 2. The image distortion correction, as described by Sasano et al. (1982), was not included since there is no image distortion for the synthetic lidar backscatter images. The five options of the CCA are the multipass interrogation (MP), the multigrid interrogation (MG), the zero padding (ZP), the Tukey window (TW), and the histogram equalization (HE). None of these options are included for test 1, but all five options are included for test 6. A block size of 25 × 25 pixels, which corresponds to 250 m × 250 m regions of REAL backscatter images, was chosen to evaluate the performance of the CCA in the smallest of the block sizes used in the real-time operational version of our algorithm.
The tests of the CCA using synthetic backscatter images and the velocity fields. The five options of the CCA are MP, MG, ZP, TW, and HE. Symbols ✓ and × represent that these options are turned on and off, respectively.
The results of the tests of the CCA using synthetic backscatter images for light (top two rows), moderate (middle two rows), and strong (bottom two rows) wind cases. The 2D synthetic velocity field and the test results are expressed in meters per second. The first and the second rows show the mean velocity and the standard deviation (SD), respectively, obtained from 100 estimations.
The multipass and multigrid interrogations contribute the most to the performance of the CCA, while the other options bring relatively small improvements. The histogram equalization and the Tukey window tend to slightly underestimate the u component but provide better estimation of the υ component, making a correction of the wind direction estimation. On the other hand, the zero padding tends to reduce the underestimation of the u component and improve the estimation of the wind speed. Overall, the results suggest that the performance of the CCA for the relatively small region is optimized by applying all five options.
b. Accuracy and precision of the wind estimates
The use of a Cartesian grid and a discrete time interval between scans tends to quantize the CCA velocity estimates. Fitting a 2D polynomial function to the 5 × 5 domain around the peak of the CCFs, following Piironen and Eloranta (1995), ameliorates quantization somewhat, but it does not eliminate it. In our use of synthetic data with 
By including turbulence (i.e., including spatial variability of the wind field) and gradually increasing the mean wind speed for neutral conditions, we observed that the oscillations in systematic errors become secondary to random errors resulting from the effects of a nonuniform wind field and the resulting deformation of aerosol features. The result is shown in Fig. 6. The red line was computed from 
Absolute error of the optimized CCA as a function of mean wind speed based on experiments with synthetic flow fields (neutral static stability) and synthetic aerosol distributions. The red line represents accuracy, or the systematic error associated with the mean of 100 trails for a given true mean speed. The light blue shaded region corresponds to one standard deviation and reflects the precision. The oscillations at slow true mean speeds are the result of the less-than-ideal performance of the polynomial fit used to obtain subpixel velocity resolution. The increasing standard deviation of the estimates with speed is the result of increasing turbulence intensity.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
Relative errors of the optimized CCA as a function of mean wind speed based on experiments with synthetic flow fields (neutral static stability) and synthetic aerosol distributions. These are the same data as plotted in Fig. 6, except they have been normalized by the corresponding true mean speed and are therefore unitless.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
The optimized CCA (with all five options) was also applied to the four flow structures that are typically observed in the lower convective boundary layer (divergent, rotational, stretching deformation, and shearing deformation). Table 3 lists the results of the tests for the four flows. By comparing test 6 of Table 2 and Table 3, one can see that the performance of the optimized CCA for convective flows was lower than all three cases of shear-induced turbulent flow. We attribute this to the diversity of wind directions within the interrogation window, sometimes displacing aerosol structures in opposing directions. In the unlikely case where a CCF contains two peaks of equal value, the algorithm will report only one. Therefore, when applied to regions of strong shear (i.e., an interrogation window with half easterly flow and half westerly flow as a worst case), the CCA will fail to provide the vector average of the true velocity field.
The results of the tests of the optimized CCA using synthetic backscatter images for divergent flow, rotational flow, stretching deformation, and shearing deformation wind cases. The 2D synthetic velocity field and the test results are expressed in meters per second. The first and the second rows show the mean velocity and the SD, respectively, obtained from 100 estimations.
While the use of synthetic images and flow fields are powerful tools for testing the algorithm, they also have severe limitations and miss relevant physics that occur in the real world. Foremost, our 2D synthetic images and flow fields lack the realism of the 3D nature of aerosol and wind. In the real world, aerosol features may pass through the scan plane, resulting in false apparent motions. Also, in the real world, air parcels could move circuitously and not at constant velocity during the time between scans.
4. Comparison with Doppler lidar wind measurements
As described by Mayor et al. (2012), a rudimentary CCA was applied to REAL data collected during the Canopy Horizontal Array Turbulence Study (CHATS; Patton et al. 2011), from March to June of 2007 near Dixon, California. The REAL was located 1.61 km north of the National Center for Atmospheric Research (NCAR) Integrated Surface Flux Facility (ISFF) 30-m vertical tower. The tower was surrounded by an orchard (800 m × 800 m) of walnut trees approximately 10 m tall. Five Campbell Scientific CSAT3 3D sonic anemometers were located on the tower at 12.5, 14.0, 18.0, 23.0, and 29.0 m AGL to measure the wind velocity. The REAL scanned the atmosphere nearly horizontally over the orchard, and the rudimentary CCA was applied to estimate the wind velocity in a series of different sized blocks centered on the tower.
The REAL dataset from CHATS was groundbreaking but it had some deficiencies. In particular, the ISFF tower and nearby trees caused hard-target reflections, creating intermittent bright pixels and large shadows in the aerosol backscatter data that prevented optimal retrieval of wind fields by cross correlation. Moreover, the REAL platform settled into the soil during the experiment and precise measurements of the pitch and roll of the instrument were not available. This resulted in uncertainty of the altitude of the lidar scan at the location of the tower. Given the strong speed shear with increasing altitude in the roughness sublayer just above the top of the canopy, Mayor et al. (2012) chose not to calculate and compare temporal mean wind velocity data and instead focused on the instantaneous wind vectors resulting from the CCA.
To move forward, a new field experiment was conducted in Chico, California, from May of 2013 through January of 2014. Chico is 130 km north of Dixon and has less variable relative humidity than Dixon due to its distance from the Sacramento–San Joaquin River delta. However, nearby agriculture activities and boundary layer convection offer good conditions for testing. In the Chico experiment, a Halo Photonics Streamline Doppler lidar (serial number 0811–35, manufactured in November 2011) was employed as the reference system in order to avoid hard target reflections associated with mast-mounted in situ sensors. The DL operated at a wavelength of 1.5 μm, a pulse energy of 20 μJ, a pulse rate of 15 kHz, and a pulse duration of 150 ns [full width at half maximum (FWHM)]. The REAL was on firm ground and the pitch and roll of the platform was recorded to ensure precise knowledge of the altitude of the laser beam as a function of range. Both the REAL and the DL data acquisition systems were synchronized to global positioning system (GPS) time. Since a single DL cannot retrieve a 2D two-component wind velocity field, the following two experiments were conducted.
a. Temporal validation
The DL was located at a range of 1523 m and a heading of 15° azimuth and operated in a mast replacement mode for this part of the field experiment (see Figs. 8–10). Horizontal wind vectors between 30 and 170 m AGL at 10-m intervals were calculated from a sequence of four inclined beams and one vertical beam every 17 s. The inclined beams were directed 43° above horizontal in the cardinal directions, resulting in the sampling of radial velocities at a horizontal distance of 107 m from the DL at 100 m AGL. Typical agreement for equivalent DLs to mast-mounted in situ wind sensors is 1% or better for both speed and direction.
A map showing the experimental setup at the University Farm at California State University, Chico, in 2013. The yellow lines represent the University Farm border. The two red symbols (×) represent the locations of the REAL system and the DL, respectively. The blue region represents the PPI scans collected by the REAL system for the experiment.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
Vertical cross-section diagram for the 2013 Chico field experiment. The REAL scans the atmosphere at 4° elevation. The DL is located 1523 m from the REAL and operated in vertical profile mode. With an elevation angle of 43°, the DL samples at 100 m AGL were 107 m from the center location.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
Diagram of lidar data density in a 250 m × 250 m area at 100 m above the DL location. REAL aerosol backscatter (+) and DL radial velocity measurement (
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
The REAL system scanned the atmosphere between −15° and 45° azimuth, at 4° elevation, in about 15 s. Then, the optimized CCA with all options (MP, MG, HE, ZP, and TW) was applied to estimate the wind velocity in a block of the dimensions 250 m × 250 m centered directly over the DL at the altitude of 100 m AGL. Figure 10 shows the backscatter data density for the REAL system (+) and the DL (
The quality of the aerosol backscatter data depends upon the performance of the instrument and the state of the atmosphere. In the following, for sake of brevity, only three cases featuring different wind conditions are presented as examples: low, moderate, and strong wind speeds. Then, a statistical analysis of 15 days of data that occurred in September and October of 2013 are considered. These 15 days present the best-quality aerosol backscatter data at the DL location during daytime while covering a broad range of wind speeds (0–16 m s−1) and constant to variable wind directions. Therefore, these days constitute the best dataset to analyze the performance of the algorithm while minimizing any weaknesses due to the instrument or nonoptimal atmospheric conditions. They were selected by analyzing the “image SNR” as detailed in Dérian et al. (2015).
1) Light wind case
Figure 11 shows time series of wind speeds and directions, as estimated by the DL (blue) and by the optimized CCA (green) from REAL backscatter images for a 12-h period starting from 1500 UTC 23 October 2013. Skies were clear with a maximum air temperature at the REAL of 29.2°C at 1632 Pacific daylight time (PDT) (2332 UTC). Range–height indicator (RHI) scans made by the REAL reveal a shallow convective boundary layer that did not exceed approximately 300-m depth during the day. Plus symbols represent the individual estimates and the line a 10-min rolling average. This is an example of a light wind case where the wind speed ranges between 0 and 2 m s−1 and the wind direction is variable. The results show that the time series of both methods are very similar except for a 2-h period between 1500 and 1700 UTC, where no coherent aerosol features were present. The correlation coefficients 
Time series of wind speed and direction, as estimated by the DL (blue) and by the optimized CCA (green) from REAL backscatter images for a light wind case starting at 1500 UTC 23 Oct 2013. Each plus (+) represents an individual estimation or measurement, separated by approximately 17 s. Solid lines represent 10-min rolling averages.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
2) Moderate wind case
Figure 12 shows time series of wind speeds and directions, as estimated by the DL (blue) and by the optimized CCA (green) from REAL backscatter images for a 12-h period starting from 1500 UTC 17 September 2013. Skies were clear with a maximum air temperature at the REAL of 27.3°C at 1659 PDT (2359 UTC). RHI scans by the REAL show a mixed layer reaching approximately 1100 m deep in the afternoon. This is an example of a moderate wind case where the wind speed ranges between 0 and 8 m s−1 and the wind direction is approximately constant for the first half of the period but varies after 2000 UTC. The time series of both methods are remarkably similar for both wind speed and direction. Correlation coefficients 
Time series of wind speed and direction, as estimated by the DL (blue) and by the optimized CCA (green) from REAL backscatter images for a moderate wind case starting at 1500 UTC 17 Sep 2013. Each plus (+) represents an individual estimation or measurement, separated by approximately 17 s. Solid lines represent 10-min rolling averages.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
3) Strong wind case
Figure 13 shows the time series of wind speeds and directions, as estimated by the DL (blue) and by the optimized CCA (green) from REAL backscatter images for a 12-h period starting from 1500 UTC 9 October 2013. Skies were clear with a maximum air temperature at the REAL of 24.2°C at 1537 PDT (2237 UTC). RHI scans made by the REAL show a mixed layer approximately 1 km deep. This is an example of a strong wind case where the average wind speed is about 10 m s−1 (with gusts to 14 m s−1) and the wind direction is approximately constant (from the northwest direction). As in the moderate wind case, the time series of both methods are strikingly similar for both wind speed and direction. Correlation coefficients 
Time series of wind speed and direction, as estimated by the DL (blue) and by the optimized CCA (green) from REAL backscatter images for a strong wind case starting at 1500 UTC 9 Oct 2013. Each plus (+) represents an individual estimation or measurement, separated by approximately 17 s. Solid lines represent 10-min rolling averages.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
4) Statistical analysis of temporal validation
Statistical comparisons between optimized CCA and the DL are summarized in Tables 4 and 5 for all three cases (weak, moderate, and strong winds). Figure 14 presents scatterplots for the 10-min means collected over 15 days (891 intervals), and the corresponding statistical results are available in Tables 4 and 5. The root-mean-square (RMS) deviations for components u and υ over 15 days are 0.36 m s−1 and 0.37 m s−1, respectively. The scatterplots show excellent agreement between the cross-correlation motion estimates and the DL measurements. This is confirmed by the slopes of the best-fit lines (0.974 for u and 0.991 for υ) and 
RMS deviation of differences, linear regression variables (slope, offset), R2, number of points, and recovery percentage w.r.t. the DL reference for the 10-min averaged wind component u (west–east), for the three specific cases, and for the 15 days considered for the temporal validation.
RMS deviation of differences, linear regression variables (slope, offset), R2, number of points, and recovery percentage w.r.t. the DL reference for the 10-min averaged wind component υ (south–north), for the three specific cases and the 15 days considered for the temporal validation.
(top) Scatterplots of 10-min averaged u and υ components of the wind velocity estimated by the optimized CCA (vertical axis) vs that estimated by the DL at 100 m AGL (horizontal axis), for 15 days, during daytime (891 intervals). The histogram distributions of differences for the same dataset are shown in the bottom panels.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
The time series of Figs. 11–13 reveal that the wind velocities obtained from the cross correlation have less variability than the DL measurements. Figure 15 shows scatterplots of the turbulent kinetic energy (TKE) measured by the DL and the CCA over the 891 ten-minute intervals. Three sets of results are presented for the CCA, corresponding to three levels of the multigrid estimation: 1000 m × 1000 m, 500 m × 500 m, and 250 m × 250 m. In all three sets, the cross-correlation method underestimated the TKE. However, as the block size is reduced, more TKE is recovered: from ≈25% with the largest blocks to ≈39% with 250 m × 250 m blocks. These results support the notion that smaller block sizes are able to capture smaller-scale velocity perturbations. As stated in Mayor et al. (2012), estimating the wind directly from small block sizes leads to noisier results. However, with the addition of multipass, multigrid, and quality control, the algorithm is now able to increase the resolution of the motion field.
TKE of the cross-correlation wind estimates (vertical axis) vs Doppler wind measurements (horizontal axis) computed from 891 intervals, each lasting 10 min. The gray shading indicates the mean wind speed measured over the interval. The three sets correspond to block sizes of (left) 1000 m × 1000 m , (middle) 500 m × 500 m, and (right) 250 m × 250 m.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
b. Wind velocity fields
Two-component velocity fields can be retrieved from the REAL backscatter images to a range of several kilometers via application of the optimized CCA. Each vector in the field is the result of placing the interrogation window on the region of the preprocessed backscatter intensity image surrounding the desired vector location. Figure 16 shows an example of a strong wind case on 3 October 2013 when the wind direction is relatively uniform. RHI scans near this time indicate mixing to altitudes above 1 km AGL. Mean wind speed at 100 m AGL at the location of the DL was about 13 m s−1. Skies were clear and the high temperature on this day was 24°C. This is an example of a case where there are likely significant contributions from both shear and buoyancy. Figure 17 shows an example of a microscale vortex observed from the light wind case (23 October 2013) discussed in a previous section. The optimized CCA greatly improves the resolution of such velocity structures.
Wind velocity field obtained by the optimized CCA (at 1845:07 UTC 3 Oct 2013), superimposed on the first scan of the pair used for estimation. The turquoise circle represents the conic section sampled by the DL.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
Wind velocity field obtained by the optimized CCA (at 2332:04 UTC 23 Oct 2013), superimposed on the first scan of the pair used for estimation. The turquoise circle represents the conic section sampled by the DL. In (top) a close-up view of a vortex. The radius of the vortex is approximately 200 m.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
c. Spatial validation
For this phase of the experiment, the DL was located on the roof of the REAL system and operated in fixed-beam mode pointing at 45° azimuth and 2° elevation to estimate the radial component of the wind velocity field at the center of the scan sector swept by the REAL. The corresponding radial component of the wind velocity was retrieved from the horizontal components of the wind velocity vector estimated by the optimized cross-correlation algorithm applied to REAL backscatter images.
The radial components of the wind velocity vectors at 45° azimuth and 2° elevation were retrieved by the algorithm with all options (MP, MG, HE, ZP, and TW) applied to image blocks (250 m × 250 m) in the range between 0.5 and 3 km from the REAL system. The REAL system scanned the atmosphere every 17 s (between 15° and 75° azimuth at 2° elevation). The radial component of the wind vectors as a function of time and range were compared. This experiment was conducted in December 2013 and January 2014.
As an example, Fig. 18 shows the radial component of the wind velocity as measured by the DL and estimated by the optimized cross-correlation algorithm, for an 8-h period starting at 1700 UTC 8 January 2014. The intensity of solar radiation on this day was weak due to partly sunny skies and the low sun angle associated with the winter season. The high temperature was about 13°C. Wind speeds at 100 m did not exceed approximately 4 m s−1. The two radial velocity fields shown in the figure bear strong resemblance.
Range vs time images of radial velocity from the (top) DL and (bottom) optimized CCA applied to the REAL backscatter images, for a 8-h period starting at 1700 UTC 8 Jan 2014. Gray shading indicates data discarded by quality control, likely associated with the absence of aerosol structures.
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
Statistics on the 10-min mean radial velocities, computed at different ranges, are presented in Fig. 19. These results were computed over 8-h periods (from 1700 to 0100 UTC) for 8 days selected in December 2013 and January 2014. It shows that 
(a) Scatterplot of 10-min averaged radial component of the wind velocity vector, as estimated by the optimized CCA (vertical axis) vs that estimated by the DL (horizontal axis). Color indicates the range, from blue (0.5 km) to red (3 km). (b) Histogram of difference of radial component of the wind velocity vector. (c) Slope of linear regression (vertical axis) as a function of range (horizontal axis). Dashed red line indicates overall slope. (d) 
Citation: Journal of Atmospheric and Oceanic Technology 33, 1; 10.1175/JTECH-D-15-0009.1
5. Conclusions
This paper describes the results of a research program that utilized two very different approaches to characterize and improve the performance of the CCA as applied to elastic lidar data for remote wind estimation. The first approach, described in detail in Hamada (2014), used synthetic aerosol backscatter fields and synthetic turbulent velocity fields to conduct highly controlled numerical experiments. By using synthetic images and velocity fields, we were able to evaluate the significance of the various procedures included in the algorithm on the resulting motion vectors. An operational real-time version of the optimized CCA is now available for use with the REAL. Moreover, the synthetic test results enabled us to estimate the accuracy and precision of the technique. Results suggest that when the turbulence is shear-generated under neutral conditions, systematic errors dominate at slow mean speeds (
The second approach involved a field experiment. When sufficient small-scale aerosol structures are present, the 10-min mean wind estimates (over a set of 150 h) from the optimized CCA (for a single point within the scan area) match those obtained from a calibrated vertically profiling Doppler lidar. The RMS deviations for the u and υ components are 0.36 m s−1 and 0.37 m s−1, respectively, with 
Overall, the results suggest that while the CCA does exhibit strong skill at extracting motion information that closely approximates the wind, its ability to resolve finescale velocity structures is handicapped by the use of an interrogation window. Its performance is also unreliable when the interrogation window straddles an edge of the lidar sector scan. Implementation of the algorithm proposed by Chen and Katz (2005) could reduce or eliminate the systematic errors we observed at low wind speeds. Alternative motion estimation algorithms, such as wavelet-based optical flow described by Dérian et al. (2015), appear to be better suited to resolve smaller coherent structures at the microscale and are less sensitive to the scan edges. In the future, large-eddy simulations could be used to generate more realistic 3D synthetic wind and aerosol data. Field experiments to test our conclusions should include measurements necessary to calculate the Obukhov length and quantify the contributions of shear and buoyancy.
Acknowledgments
This material is based upon work supported by the National Science Foundation’s Physical and Dynamic Meteorology Program under Grant AGS 1228464. The authors thank Dr. Jakob Mann for kindly sharing his spectral tensor turbulence model with us.
APPENDIX
Nomenclature
L Size of the interrogation window
p x coordinate of the interrogation window center
q υ coordinate of the interrogation window center
Greek
α Constant that determines the width of the cosine lobe of the Tukey window
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Recently, optical flow methods have become an alternative. For example, Corpetti et al. (2006) applies an optical flow scheme to study the plane turbulent mixing layer, the wake of a circular cylinder, and vorticity measurement.
