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This paper evaluates four wind retrieval methods for micro-scale meteorology applications with volume and time resolution in the order of 30 m^{3} and 5 s. Wind field vectors are estimated using sequential time-lapse volume images of aerosol density fluctuations. Suitably designed mono-static scanning backscatter LIDAR systems, which are sensitive to atmospheric density aerosol fluctuations, are expected to be ideal for this purpose. An important application is wind farm siting and evaluation. In this case, it is necessary to look at the complicated region between the earth’s surface and the boundary layer, where wind can be turbulent and fractal scaling from millimeter to kilometer. The methods are demonstrated using first a simple randomized moving hard target, and then with a physics based stochastic space-time dynamic turbulence model. In the latter case the actual vector wind field is known, allowing complete space-time error analysis. Two of the methods, the semblance method and the spatio-temporal method, are found to be most suitable for wind field estimation.

Space-time volume profiling is expected to be an important tool for designing optimum wind farm parameters for wind power generation, facilitating the next-generation tools for large area 3D wind modeling over complex terrains, improving wind turbine performance, increasing turbine life, and reducing turbine operating and maintenance costs. Presently, anemometers are commonly used to sample wind fields at several selected points or regions. Newer large turbines have hub heights larger than 130 m. Anemometer masts need to be at hub height while masts higher than 100 m are problematic [^{3}. Our work here focuses on software processing from a suitable volume scanning backscatter LIDAR system. Recently, a commercial conical scanning vertical pointing aerosol LIDAR system specifically designed for the wind industry has been introduced [

Measurement of wind fields using elastic backscatter LIDAR with short pulse and angular scanning capability has better time and space resolution than radar or sodar systems [^{3} with ranges of a few kilometers. This opens the door for possible non-Doppler, volumetric three-component LIDAR wind field measurements ideally suited for wind farm siting and evaluation. Most of the required components for such systems are commercially available. Wilkerson [

Features in the LIDAR backscatter patterns are caused by lower atmospheric aerosol particle loading. Particles with diameters _{s}

Motion inferred from time-lapse LIDAR imagery has been developed by many authors. An important early example is by Eloranta [

Here, because of the interdisciplinary nature of this paper, it seems appropriate to include both a summary and comparison of four alternative vector wind field retrieval methods. All methods compare successive time lapse imagery. The methods are (a) cross correlation method (CCM), (b) semblance method (SM), (c) translation phase shift method (TPSM), and (d) a spatio-temporal method (STM). The first three methods use a combination of segmentation and Fourier transform (FFT) processing. STM uses smaller neighborhood processing and therefore applies directly in the space and time domain. Care is taken to make all the methods numerically efficient. CCM is defined in one dimension, but has obvious extension to two and three spatial dimensions using multi-dimensional FFT’s. Vector field examples in two dimensions using synthetic time lapse target imagery are used to illustrate the methods. To compare and evaluate these four methods, we have implemented a version of Stam’s [

Once the dynamic 3D vector field has been estimated, it remains a non-trivial task to visualize the dynamical results. Indeed, computer visualization is intrinsically two-dimensional; the three-dimensional vector field has seven variables, four independent (_{x}, v_{y}, v_{z}_{x}, v_{y}_{h}_{x}_{y}_{n}, y_{m}_{h}_{z}/v_{h}_{z}/v_{h}

Dense rectangular arrays of length-modulated and color-coded arrows are not eye friendly. Streamlines are more intuitive and yield a less cluttered map. Completing the visualization are streamlets, moving along the streamlines with speed proportional to _{h}_{z}/v_{h}

An important processing detail relates to the aerosol backscatter data. Signal and image processing are most directly accomplished in rectangular coordinates. Each volume of spherical coordinate data thus needs to be interpolated onto a rectangular voxel grid. Note 30 m^{3} resolution over volumes in the order of 1 km^{3} could result in a computational bottleneck. Since the scan is doubly periodic, volume interpolation is done every period. The period is in the order of several seconds. An efficient strategy is to pre-compute locations for a given number of nearest neighbor pointers, say _{n}_{m}, m_{m}_{mn(ℓ(m,j))}, _{nm}_{m}_{s}_{c}^{th}^{3} resolution over a 1 km^{3} scan volume, corresponding to 1.0 ^{7} voxels, takes 0.6 s for

The next eight sections summarize the four alternative and complementary methods for estimating vector fields from time lapse imagery. Then we discuss some image processing and filtering of the data that proceeds wind field estimation. This is followed by a derivation of the dynamic turbulent wind field model and a method for advecting aerosol particles in the resulting wind field. The paper concludes with comparisons of the four wind field estimation methods using the modeled dynamic wind field data.

In this and the following three sections, methods are stated for the most simple one-dimensional case. However, the results easily generalize to two and three-dimensions. Details of the implementation of these methods can be found in [_{rs}(_{fg}(

Computation of _{fg}(_{fg}(

Semblance is a generalization of cross-correlation and depends upon relative amplitudes in addition to correlation [_{fg}(_{f}_{g}_{fg}(_{fg}(_{fg}(^{2}) having maximum value of 1/2 for

The underlying idea of the translation phase shift method follows from definition in

From _{1}, _{2}, ⋯, _{n}

For discrete FFT application with (_{x}_{y}

Assume _{n}_{m}_{n}_{m}_{x}_{y}_{x}_{y}

In _{nm}_{xn}_{ym}_{nm}_{x}_{y}

In _{nm}

Minimization of _{x}_{y}

This method directly extends to three dimensions determining _{x}_{y}_{z}_{nmp}

CCM, SM and TPSM as formulated here produce global estimates of translation shifts _{x}_{y}_{x}_{y}

For a segmented implementation, let the digital image _{nm}_{r}_{c}_{f}_{b}_{f}_{rs}_{cs}_{b}_{b}_{f}_{rs}_{cs}

A limitation of CCM, SM and TPSM is that intrinsic resolution can be less than the underlying time-lapse image sequence data. This is a consequence of transform methods requiring minimum sub-interval lengths of 16 or 32 to have reliable central transform values. STM is formulated in the space and time domain. Because of this, STM honors the resolution intrinsic to the data. The method, also called optical flow, is well documented by Lim [

In common with the other three methods, time-lapse image differences are assumed to be caused solely by aerosol feature pattern translation. For three-dimensional motion, with local velocity components (_{x}_{y}_{z}_{n−1} and _{n}_{n−1} + Δ_{n−1} ≤ _{n}_{n}_{n−1} and over source-free regions where aerosol patterns are advected without distortion. As shown in [

_{x}_{y}_{z}_{f}_{m}_{n}_{p}_{ℓ−1}, _{ℓ}. Then define the quadratic cost function _{x}_{y}_{z}

The multi-dimensional sums in _{x}_{y}_{z}

_{n}_{n}

In the three-dimensional case, an equation of motion, namely conservation of mass or mass balance [

This general result often simplifies to the condition of incompressible flow. This approximation of microscale meteorological conditions is valid [_{s}_{s}_{c}^{2} is the typical gravitational constant acceleration at the earth’s surface, _{c}

Implementation of

In order to improve image resolution in the time-lapse data, some image processing and filtering can be applied. For time lapse image analysis for motion retrieval, a good indication of image resolution is the relative area under local semblance peaks in the vicinity of local maxima. An open question is the importance of quantization error in this regard,

In Schols and Eloranta [

The first three methods process in the spatial frequency domain. In these cases, optional FFT based filters are used. Kaiser low pass windows [_{in} has size [_{r}_{c}_{aux}(_{r}n_{c}_{neib}] where _{neib} is the number of neighborhood pixels for the image point _{in} then is simply the dot product
_{neib}) of the filter coefficients for one, two, or three-dimensional filtering and reshape returns the one dimensional output into the original matrix size with dimensions [_{r}_{c}_{aux}(_{aux} as defined by _{in}.

To develop robust wind field estimation methods, it is essential to have a realistic dynamical moving aerosol density model. The model should be stochastic, turbulent, and produce a known underlying wind field allowing space-time error analysis.

Fortunately, such models exist. In particular, we make use of the pioneering and seminal work of Stam [

The turbulent model of Stam incorporated here is an efficient FFT-PC based solution to the nonlinear Navier–Stokes equation for incompressible fluids with periodic boundary conditions. Reference to some of his numerical result graphics confirms their realism. Simulating fluids is one of the more challenging problems in computational physics so this is a non-trivial feature. Even so, the model does not include temperature effects, vertical wind shear caused by surface friction arising from vegetation, terrain,

The wind field is composed of two components, large and small,

The large scale is chosen to be slowly varying and deterministic. The small scale is random and turbulent. Dropping its subscript, this vector field component has the Fourier transform representation

Under the Gaussian assumption, the random component is defined by its first two statistical moments. Without loss of generality, the means are assumed to be zero. The remaining statistic is the space-time cross correlation _{ij}

By the correlation theorem the Fourier transform of _{ij}

The wind field is assumed to be incompressible,

There is a simple way to enforce the incompressibility condition in the Fourier transform domain. By the Helmholtz theorem [

In the Fourier domain this becomes

From this it follows that the projection operator _{ij}_{ij}

Thus for any vector field _{i}_{i}^{−1} is the inverse Fourier transform operator. The mean kinetic energy per unit mass of the vector field is

The kinetic energy spectrum function

In the case of isotropic, locally homogeneous turbulence, Lesieur [

As stated by [_{i}

Using Stam’s method [_{k}

Note that for this choice
_{k}

The second order statistics for the wind field are now determined. It remains to construct an explicit associated wind field _{nℓ}(_{ℓ}(

Thus substitution of _{nm}_{nℓ} are independent. Let H_{12} = H_{13} = H_{23} = 0 resulting in the solution

Using a numerical pseudo-random number generator define the random vectors
_{ℓ} is a normally distributed and _{ℓ} is uniformly distributed on the unit interval [0, 1]. Both are four-dimensional arrays of size (_{x}_{y}_{z}_{t}

It remains to associate with the velocity field

In

In _{p}_{p}_{0}

The spatial wavenumber magnitude is
_{x}_{x}_{x}_{max} = 2^{1/2} = 10.9. From _{max} for particles with _{p}^{−3}_{max} ≈ 2 ^{−6} m^{2}/s so that _{max} _{max} ≈ 2.2 ^{−5}m/s. The maximum Brownian motion transport speed is four orders of magnitude less than usable surface winds. Hence the diffusion term in

Statistically the total mass density of aerosol particles is a measured quantity, ^{3} and standard deviation ^{3}. Our wind field uses ^{3} and ^{3}.

Let _{n}_{n}^{th}

The refresh term in

In _{1}(_{x}_{y}_{z}_{m}_{m}_{m}_{m}^{3}. As seen in _{m}^{3}. At later times, _{t}_{n}

Because of the cited complicated nature of actual aerosol dust sources and their interactions, a simple ad hoc source model is chosen. To maintain aerosol heterogeneity, a small amplitude of random mass density Δ

Recently, the challenge to visualize high-resolution dynamic three-dimensional vector fields in an intuitive manner has been addressed by many authors (see for example [

A streamline or flow line is a path _{0})_{0}. By definition a streamline is everywhere tangent to the vector field

Solutions _{0} are called integral curves. It can be shown that in regions where |^{2} + ^{2} + ^{2})^{1/2}. Units are chosen such that the speed is
_{0} are chosen to be a randomly permuted subset _{0} of the centers of the display grid cells, and then each point is moved to a random location within its cell. Numerical solutions to streamline

Reference [_{x}_{y}_{x}_{y}_{x}_{y}_{f}_{b}_{f}_{b}

_{x}_{y}_{f}_{b}

For this type of motion detection, the wind field estimation uses threshold speeds greater than 2 m/s. To discriminate against noise or and low level eddies, define precomputed local neighborhood sums _{x}_{y}_{z}_{x}_{y}_{z}

For time intervals defined by subscript ℓ, _{x}_{y}_{z}_{x}_{y}_{grd}_{max}

_{max}

A Matlab® [_{x}_{y}^{7}, 320 = 5 × 2^{6}). The inertial wavenumber cutoff in _{i}

The large scale slowly varying wind component _{ℓ}(_{ω}_{n}^{3}. Image processing converts each 640 × 640 spatial grid to a 20 × 20 segmentation. Each of the 400 resulting segments is analyzed by a 64 × 64 FFT for local semblance computation.

_{10} scale so that for example bright red corresponds to approximately 10% relative error with cooler colors less.

The error analysis in

The techniques proposed here fail when the aerosol density is uniform,

Finally some words on integration of this process with hardware and applying it to 4D wind field estimation. Because such systems are monostatic, all signals are returned via a collecting telescope to the detector area in the order of 0.04 mm^{2}. Commercial 12 bit high-speed digitizers can now sample up to 2 GS/s. A cubic km of data at a resolution of 30 m^{3} per voxel is 3.33 ^{7} samples. Assume a 5 s period for the scanning system, and 100 sample averaging. This corresponds to 0.67 GS/s. The signal averaging can also be done on the digitizer card and then stored to a solid state drive with speeds up to 500 Mb/s. All of this is at the front end of the wind field estimation process outside of the CPU. The first step of the CPU process is to interpolate the 3D spherical coordinate data onto rectangular coordinates as described in the introduction. Linearly scaling this interpolation time to 3.33 × 10^{7} samples yields 2 s. Because the data is 12 bit, faster integer arithmetic can be used. Also, as planes of rectangular data become available, the STM processing can begin in the populated planes. As expected, the majority of time is spent with the velocity field estimation step. For example, as presently coded, STM processing of 512 × 512 time lapse images takes 0.5 s. Linear scaling of this time to 3.33 × 10^{7} voxels predicts an STM processing time of 63 s for 1 km^{3} of data at 30 m^{3} voxel size resolution. Efficient coding can be expected to reduce this time by a factor of at least 2. Because the output of processing in parallel planes of data are independent, the STM process could then be implemented using 8 parallel threads to reduce the data cycle time to approximately 5 s.

Another approach is to have a quick-look real-time process that outputs wind field data in several user defined slices through the volume data. The complete space time-data is stored for batch processing.

The trend in wind energy production is to use larger turbines requiring new techniques for wind field measurement. Scanning wind LIDARs are of two types: direct detection and coherent detection. Coherent detection uses relatively expensive transceivers with accurately frequency-modulated laser pulses and measures the Doppler frequency shift in the return signal, thus determining the radial velocity component. In comparison, direct detection aerosol backscatter LIDARs measure all three speed components equally well. This paper compares four methods for the general class of direct detection volume scanning LIDARs. The four methods are CCM, SM, TPSM and STM. The methods are first compared with a simple orbiting cloud model. The methods of SM and STM are found to be most accurate, with averaged mean-speed errors of 0.03 m/s for gradient weighted SM and 0.05 m/s for gradient weighted STM.

A second non-linear turbulent flow model provides for more realistic simulations and incorporates a FFT-PC based solution for the Navier–Stokes equation for isotropic and non-compressible flow. It does not contain terrain profile nor boundary layer effects but is nonetheless adequate for inter-comparison of the four retrieval methods. To simulate LIDAR backscatter data, an associated model for atmospheric aerosol density fluctuations ^{3} and 5 s.

The authors gratefully acknowledge USTAR funding and the support and collaboration of the CASI and EDL staff at Utah State University that made this work possible. The first author thanks the Geophysics Department of UFPa for their support during paper revisions. We also acknowledge the MDPI reviewers and editors for several important comments and suggestions leading to an improved paper.

Diffusion coefficient D as a function of aerosol diameter _{p}

Idealized schematic of atmospheric aerosol cycles including sources, sinks, particle modes and particle creation as a function of particle diameter in

Segmented SM processing result for one frame of stochastic cloud model data. Color bar scale aerosol density units are ^{3}.

Runge–Kutta streamline example from stochastic model using 25 start points shown in yellow. Cubic interpolation is used to smooth the results.

Associated magnitude and direction error analysis for _{10}(_{e}_{e}

Associated summary median percent relative error statistics for 64 time-lapse image frames.

Associated summary of frame mean and median statistics for 64 time-lapse image frames.

Summary statistics for STM processing averaged over all 45 orbiting cloud images.

med(v) | med(v̂_{max}) |
med(v̂_{grd}) |
avg(v) | avg(v̂_{max}) |
avg(v̂_{grd}) |
std(v) | std(v̂_{max}) |
std(v̂_{grd}) | |
---|---|---|---|---|---|---|---|---|---|

_{x} |
−0.5551 | −0.3789 | −0.2495 | −0.0059 | 0.1196 | 0.0716 | 10.1753 | 9.3823 | 8.8252 |

_{y} |
−1.0219 | −0.9864 | −1.0063 | −0.2116 | −0.2216 | −0.2247 | 6.0165 | 6.0044 | 6.0499 |

| |
11.8414 | 11.4639 | 11.0144 | 11.4624 | 10.8603 | 10.4340 | 2.3109 | 1.8587 | 1.7724 |

Summary statistics for SM processing averaged over all 45 orbiting cloud images.

med(v) | med(v̂_{max}) |
med(v̂_{grd}) |
avg(v) | avg(v̂_{max}) |
avg(v̂_{grd}) |
std(v) | std(v̂_{max}) |
std(v̂_{grd}) | |
---|---|---|---|---|---|---|---|---|---|

_{x} |
−0.5551 | 1.5656 | 1.0568 | −0.0059 | 1.6367 | 1.5229 | 10.1753 | 10.0554 | 10.0582 |

_{y} |
−1.0219 | 0.7828 | 0.5480 | −0.2116 | 1.3877 | 1.3432 | 6.0165 | 5.8793 | 5.9780 |

| |
11.8414 | 11.3974 | 11.6369 | 11.4624 | 11.3674 | 11.4055 | 2.3109 | 2.8579 | 2.8310 |