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## Abstract

The performance of the maximum likelihood (ML) estimates of mean velocity and signal power for Doppler radar and Doppler lidar, assuming known signal spectral width, is presented. The results are compared with the theoretical limit of the Cramer–Rao bound (CRB). The performance of the ML estimator for mean velocity is similar to the performance when the signal power is known ahead of time. For cases of very high signal-to-noise ratio (SNR) and typical values of the spectral width, the performance of the maximum likelihood estimator of signal power, assuming known spectral width, does not approach the CRB for the limit of infinite SNR. The ML estimates of mean power for Doppler radar operated in Doppler lidar mode are more accurate than are traditional estimates.

## Abstract

The performance of the maximum likelihood (ML) estimates of mean velocity and signal power for Doppler radar and Doppler lidar, assuming known signal spectral width, is presented. The results are compared with the theoretical limit of the Cramer–Rao bound (CRB). The performance of the ML estimator for mean velocity is similar to the performance when the signal power is known ahead of time. For cases of very high signal-to-noise ratio (SNR) and typical values of the spectral width, the performance of the maximum likelihood estimator of signal power, assuming known spectral width, does not approach the CRB for the limit of infinite SNR. The ML estimates of mean power for Doppler radar operated in Doppler lidar mode are more accurate than are traditional estimates.

## Abstract

The performance of coherent Doppler lidar in the weak-signal regime is investigated by computer simulations of velocity estimators that accumulate the signal from *N* pulses of zero-mean complex Gaussian stationary lidar data described by a Gaussian covariance function. The probability density function of the resulting estimates is modeled as a fraction *b* of uniformly distributed had estimates or random outliers and a localized distribution of good estimates with standard deviation *g*. Results are presented for various velocity estimators and for typical boundary layer measurements of 2-µm coherent lidars and also for proposed space-based measurements with 2- and 10-µm lidars. For weak signals and insufficient pulse accumulation, the fraction of bad estimates is high and *g* ≈ *W _{V}*, the spectral width of the signal in velocity space. For a large number of accumulated pulses

*N*, there are few bad estimates and

*g*∝

*W*

_{v}N^{−1/2}. The threshold signal energy or average number of coherent photoelectrons per pulse with accumulation is defined by a given fraction of random outliers and is proportional to

*N*

^{−1/2}for large

*N*and decreases faster than

*N*

^{−1/2}for small

*N*. At the threshold level, the standard deviation

*g*of the good estimates is approximately constant for large

*N*. For space-based measurements and with the signal statistics determined by the wind fluctuations over the range gate the, 2- and 10-µm lidars have similar performance when referenced to the average number of photoelectrons detected per velocity estimate. The threshold signal level for large,

*N*can be described by simple empirical functions.

## Abstract

The performance of coherent Doppler lidar in the weak-signal regime is investigated by computer simulations of velocity estimators that accumulate the signal from *N* pulses of zero-mean complex Gaussian stationary lidar data described by a Gaussian covariance function. The probability density function of the resulting estimates is modeled as a fraction *b* of uniformly distributed had estimates or random outliers and a localized distribution of good estimates with standard deviation *g*. Results are presented for various velocity estimators and for typical boundary layer measurements of 2-µm coherent lidars and also for proposed space-based measurements with 2- and 10-µm lidars. For weak signals and insufficient pulse accumulation, the fraction of bad estimates is high and *g* ≈ *W _{V}*, the spectral width of the signal in velocity space. For a large number of accumulated pulses

*N*, there are few bad estimates and

*g*∝

*W*

_{v}N^{−1/2}. The threshold signal energy or average number of coherent photoelectrons per pulse with accumulation is defined by a given fraction of random outliers and is proportional to

*N*

^{−1/2}for large

*N*and decreases faster than

*N*

^{−1/2}for small

*N*. At the threshold level, the standard deviation

*g*of the good estimates is approximately constant for large

*N*. For space-based measurements and with the signal statistics determined by the wind fluctuations over the range gate the, 2- and 10-µm lidars have similar performance when referenced to the average number of photoelectrons detected per velocity estimate. The threshold signal level for large,

*N*can be described by simple empirical functions.

## Abstract

Various methods of estimating the magnitude of the random error of Doppler lidar velocity measurements are compared for typical operating conditions using computer simulations of lidar data. Under certain conditions, the magnitude of the random estimation error can be determined from data without the need for in situ measurements for both ground-based and space-based wind measurement.

## Abstract

Various methods of estimating the magnitude of the random error of Doppler lidar velocity measurements are compared for typical operating conditions using computer simulations of lidar data. Under certain conditions, the magnitude of the random estimation error can be determined from data without the need for in situ measurements for both ground-based and space-based wind measurement.

## Abstract

The locally stationary temperature spectrum in the atmospheric surface layer is estimated using laser scintillation. The fluctuations of the parameters of the turbulence spectrum (the structure constant *C _{T}^{2}* and inner scale

*l*

_{0}) have a lognormal distribution. The average spectrum is calculated by averaging the locally stationary spectrum over these fluctuations. The average spectrum does not have a universal form. The fluctuations in the turbulence parameters produces a bias in measurements of the Obukhov-Corrsin constant and in estimates of energy dissipation rate ε based on average scintillation statistics. The performance of the scintillation technique and the accuracy of scintillation measurements of inner scale and structure constant are estimated using Monte Carlo simulation. One scintillation measurement can provide accurate estimates of the important turbulence parameters and the statistics of the fluctuations of these parameters. The scintillation estimates are true path-averaging estimates and do not require instrumental corrections for the high-frequency region nor the conversion of temporal statistics to spatial statistics.

## Abstract

The locally stationary temperature spectrum in the atmospheric surface layer is estimated using laser scintillation. The fluctuations of the parameters of the turbulence spectrum (the structure constant *C _{T}^{2}* and inner scale

*l*

_{0}) have a lognormal distribution. The average spectrum is calculated by averaging the locally stationary spectrum over these fluctuations. The average spectrum does not have a universal form. The fluctuations in the turbulence parameters produces a bias in measurements of the Obukhov-Corrsin constant and in estimates of energy dissipation rate ε based on average scintillation statistics. The performance of the scintillation technique and the accuracy of scintillation measurements of inner scale and structure constant are estimated using Monte Carlo simulation. One scintillation measurement can provide accurate estimates of the important turbulence parameters and the statistics of the fluctuations of these parameters. The scintillation estimates are true path-averaging estimates and do not require instrumental corrections for the high-frequency region nor the conversion of temporal statistics to spatial statistics.

## Abstract

The effects of wind turbulence on pulsed coherent Doppler lidar performance are investigated theoretically and with computer simulations. The performance of velocity estimators is determined for the case of a single realization of a wind field described by a Kolmogorov spatial spectrum and for an ensemble average over many realizations. The results are compared with previously published data. The important normalized physical parameters are identified to reduce the parameter space. For a given realization of a random wind field, the mean Doppler lidar velocity (an ensemble average over the random aerosol particles) is a function of the lidar parameters and the instantaneous radial velocity over the sensing volume of the lidar pulse. Various approximations for the mean Doppler lidar velocity are compared using computer simulations. The best approximation for the mean Doppler lidar velocity is used to calculate the effects of the spatial averaging of the radial velocity by the lidar pulse on Doppler lidar estimates of the spatial structure function of the velocity and the variance of the velocity. Doppler lidar estimates of point statistics of the wind field (velocity variance, spatial velocity structure function, energy dissipation rate) are possible for certain conditions.

Using numerical simulations, the statistical description of common velocity estimators is determined by performing the ensemble average over many realizations of a stationary wind field. Velocity estimator error extracted from 2-*μ*m coherent Doppler lidar data in the surface layer are shown to be within 5% of simulation results that include the effects of wind turbulence.

For typical surface-layer measurements, a linear array of in situ wind sensors along the lidar propagation direction would be required to produce reliable comparison of in situ measurement with coherent Doppler lidar wind measurements. The effects of wind turbulence on the performance of 2- and 10-*μ*m coherent Doppler lidars for space-based operation are also presented.

## Abstract

The effects of wind turbulence on pulsed coherent Doppler lidar performance are investigated theoretically and with computer simulations. The performance of velocity estimators is determined for the case of a single realization of a wind field described by a Kolmogorov spatial spectrum and for an ensemble average over many realizations. The results are compared with previously published data. The important normalized physical parameters are identified to reduce the parameter space. For a given realization of a random wind field, the mean Doppler lidar velocity (an ensemble average over the random aerosol particles) is a function of the lidar parameters and the instantaneous radial velocity over the sensing volume of the lidar pulse. Various approximations for the mean Doppler lidar velocity are compared using computer simulations. The best approximation for the mean Doppler lidar velocity is used to calculate the effects of the spatial averaging of the radial velocity by the lidar pulse on Doppler lidar estimates of the spatial structure function of the velocity and the variance of the velocity. Doppler lidar estimates of point statistics of the wind field (velocity variance, spatial velocity structure function, energy dissipation rate) are possible for certain conditions.

Using numerical simulations, the statistical description of common velocity estimators is determined by performing the ensemble average over many realizations of a stationary wind field. Velocity estimator error extracted from 2-*μ*m coherent Doppler lidar data in the surface layer are shown to be within 5% of simulation results that include the effects of wind turbulence.

For typical surface-layer measurements, a linear array of in situ wind sensors along the lidar propagation direction would be required to produce reliable comparison of in situ measurement with coherent Doppler lidar wind measurements. The effects of wind turbulence on the performance of 2- and 10-*μ*m coherent Doppler lidars for space-based operation are also presented.

## Abstract

The performance of 2- and 10-µm coherent Doppler lidar is presented in terms of the statistical distribution of the maximum-likelihood velocity estimator from simulations for fixed range resolution and fixed velocity search space as a function of the number of coherent photoelectrons per estimate. The wavelength dependence of the aerosol backscatter coefficient, the detector quantum efficiency, and the atmospheric extinction produce a simple shift of the performance curves. Results are presented for a typical boundary layer measurement and a space-based measurement for two regimes: the pulse-dominated regime where the signal statistics are determined by the transmitted pulse, and the atmospheric-dominated regime where the signal statistics are determined by the velocity fluctuations over the range gate. The optimal choice of wavelength depends on the problem under consideration.

## Abstract

The performance of 2- and 10-µm coherent Doppler lidar is presented in terms of the statistical distribution of the maximum-likelihood velocity estimator from simulations for fixed range resolution and fixed velocity search space as a function of the number of coherent photoelectrons per estimate. The wavelength dependence of the aerosol backscatter coefficient, the detector quantum efficiency, and the atmospheric extinction produce a simple shift of the performance curves. Results are presented for a typical boundary layer measurement and a space-based measurement for two regimes: the pulse-dominated regime where the signal statistics are determined by the transmitted pulse, and the atmospheric-dominated regime where the signal statistics are determined by the velocity fluctuations over the range gate. The optimal choice of wavelength depends on the problem under consideration.

## Abstract

Scanning Doppler lidar is a promising technology for improvements in short-term wind power forecasts since it can scan close to the surface and produce wind profiles at a large distance upstream (15–30 km) if the atmosphere has sufficient aerosol loading and there are no sizable blockages from terrain or large structures. However, successful measurements require a large spatial sampling domain and new estimation algorithms that can perform well in the very weak signal regime. The maximum likelihood (ML) algorithm in the spectral domain and a faster version based on the minimum mean-square-error (MSE) are investigated by numerical simulation and with actual scanning Doppler lidar data from the Lockheed Martin Coherent Technologies WindTracer lidar. In addition, the maximum range can be extended by simultaneous estimation of the wind speed and wind direction from a larger azimuth sector scan if the atmosphere is well behaved. Real-time operation is possible using the spectral data from the WindTracer lidar and a dedicated computer to interface with a data assimilation system. Analysis of the Doppler lidar data in the first few kilometers can be used to extract the turbulence conditions for improvements in real-time wind farm operations.

## Abstract

Scanning Doppler lidar is a promising technology for improvements in short-term wind power forecasts since it can scan close to the surface and produce wind profiles at a large distance upstream (15–30 km) if the atmosphere has sufficient aerosol loading and there are no sizable blockages from terrain or large structures. However, successful measurements require a large spatial sampling domain and new estimation algorithms that can perform well in the very weak signal regime. The maximum likelihood (ML) algorithm in the spectral domain and a faster version based on the minimum mean-square-error (MSE) are investigated by numerical simulation and with actual scanning Doppler lidar data from the Lockheed Martin Coherent Technologies WindTracer lidar. In addition, the maximum range can be extended by simultaneous estimation of the wind speed and wind direction from a larger azimuth sector scan if the atmosphere is well behaved. Real-time operation is possible using the spectral data from the WindTracer lidar and a dedicated computer to interface with a data assimilation system. Analysis of the Doppler lidar data in the first few kilometers can be used to extract the turbulence conditions for improvements in real-time wind farm operations.

## Abstract

The performance of coherent Doppler lidar velocity estimates for a space-based platform are produced using computer simulations of raw data and statistical descriptions of the resulting velocity estimates. The random spatial variability of the wind field and aerosol backscatter is included as well as the improvement using coprocessing of multiple shots for a fixed lidar beam geometry. Performance of velocity estimates is defined as the rms error of the good radial velocity estimates and the fraction of bad estimates in low signal conditions. For a wide range of conditions, performance is effectively described by a few basic parameters that are a function of the atmospheric conditions and the lidar design. The threshold signal level for acceptable velocity measurements scales as *N*
^{−1/2}, where *N* is the number of coprocessed lidar shots. For a 100-km satellite track and *N* = 100 lidar shots, the rms error is typically less than 0.4 m s^{−1} for high signal levels.

## Abstract

The performance of coherent Doppler lidar velocity estimates for a space-based platform are produced using computer simulations of raw data and statistical descriptions of the resulting velocity estimates. The random spatial variability of the wind field and aerosol backscatter is included as well as the improvement using coprocessing of multiple shots for a fixed lidar beam geometry. Performance of velocity estimates is defined as the rms error of the good radial velocity estimates and the fraction of bad estimates in low signal conditions. For a wide range of conditions, performance is effectively described by a few basic parameters that are a function of the atmospheric conditions and the lidar design. The threshold signal level for acceptable velocity measurements scales as *N*
^{−1/2}, where *N* is the number of coprocessed lidar shots. For a 100-km satellite track and *N* = 100 lidar shots, the rms error is typically less than 0.4 m s^{−1} for high signal levels.

## Abstract

Verification of space-based wind measurements will be difficult because of the random variations of the atmospheric velocity field over the measurement volume. The definition of accuracy requires a definition of “truth.” For this work, truth is defined as the spatial average of the random velocity over the measurement volume. The bias of space-based velocity measurements is dominated by the pointing errors of the lidar beam. The bias and random errors of Doppler lidar measurements produced by the estimation algorithm and the random atmospheric parameters are determined with computer simulations for both the sampling errors of the lidar scanning pattern and the estimation error of the radial velocity measurements. The results are compared with the error produced by ideal rawinsonde measurements and aircraft winds. A simple method to verify the random error of Doppler lidar radial velocity measurements using only multiple-shot lidar data (no in situ data are required) is compared with results from computer simulation. The verification of space-based measurements using the correlation between rawinsonde measurements and aircraft winds is also investigated for typical tropospheric conditions. For coherent Doppler lidar, the random errors are dominated by the sampling or representative errors produced by the scanning geometry of the lidar beam. The instrumental random error is typically less than the sampling error.

## Abstract

Verification of space-based wind measurements will be difficult because of the random variations of the atmospheric velocity field over the measurement volume. The definition of accuracy requires a definition of “truth.” For this work, truth is defined as the spatial average of the random velocity over the measurement volume. The bias of space-based velocity measurements is dominated by the pointing errors of the lidar beam. The bias and random errors of Doppler lidar measurements produced by the estimation algorithm and the random atmospheric parameters are determined with computer simulations for both the sampling errors of the lidar scanning pattern and the estimation error of the radial velocity measurements. The results are compared with the error produced by ideal rawinsonde measurements and aircraft winds. A simple method to verify the random error of Doppler lidar radial velocity measurements using only multiple-shot lidar data (no in situ data are required) is compared with results from computer simulation. The verification of space-based measurements using the correlation between rawinsonde measurements and aircraft winds is also investigated for typical tropospheric conditions. For coherent Doppler lidar, the random errors are dominated by the sampling or representative errors produced by the scanning geometry of the lidar beam. The instrumental random error is typically less than the sampling error.

## Abstract

The random estimation or instrument error of coherent Doppler lidar velocity estimates with pulse accumulation (multiple lidar shots per velocity estimate) is determined with computer simulations for general conditions. The sampling errors for overlaid lidar tracks and tropospheric wind field conditions are also calculated for space-based operation. These results permit useful engineering analysis based on the total observation error of the velocity measurements.

## Abstract

The random estimation or instrument error of coherent Doppler lidar velocity estimates with pulse accumulation (multiple lidar shots per velocity estimate) is determined with computer simulations for general conditions. The sampling errors for overlaid lidar tracks and tropospheric wind field conditions are also calculated for space-based operation. These results permit useful engineering analysis based on the total observation error of the velocity measurements.