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- Author or Editor: D. H. Lenschow x

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## Abstract

It is determined how long a time series must be to estimate covariances and moments up to fourth order with a specified statistical significance. For a given averaging time *T* there is a systematic difference between the true flux or moment and the ensemble average of the time means of the same quantities. This difference, referred to here as the systematic error, is a decreasing function of *T* tending to zero for *T*→∞. The variance of the time mean of the flux or moment, the so-called error variance, represents the random scatter of individual realizations, which, when *T* is much larger than the integral time scale *T* of the time series, is also a decreasing function of *T*. This makes it possible to assess the minimum value of *T* necessary to obtain systematic and random errors smaller than specified values. Assuming that the time series are either Gaussian processes with exponential correlation functions or a skewed process derived from a Gaussian, we obtain expressions for the systematic and random errors. These expressions show that the systematic error and the error variance in the limit of large *T* are both inversely proportional to *T*, which means that the random error, that is, the square root of the error variance, will in this limit be larger than the systematic error. It is demonstrated theoretically, as well as experimentally with aircraft data from the convective boundary layer over the ocean and over land, that the assumption that the time series are Gaussian leads to underestimation of the random errors, while derived processes with a more realistic skewness and kurtosis give better estimates. For fluxes, the systematic and random errors are estimated when the time series are sampled instantaneously, but the samples separated in time by an amount Δ. It is found that the random error variance and the systematic error increase by less than 8% over continuously sampled data if Δ is no larger than the integral scale obtained from the flux time series and the cospectrum, respectively.

## Abstract

It is determined how long a time series must be to estimate covariances and moments up to fourth order with a specified statistical significance. For a given averaging time *T* there is a systematic difference between the true flux or moment and the ensemble average of the time means of the same quantities. This difference, referred to here as the systematic error, is a decreasing function of *T* tending to zero for *T*→∞. The variance of the time mean of the flux or moment, the so-called error variance, represents the random scatter of individual realizations, which, when *T* is much larger than the integral time scale *T* of the time series, is also a decreasing function of *T*. This makes it possible to assess the minimum value of *T* necessary to obtain systematic and random errors smaller than specified values. Assuming that the time series are either Gaussian processes with exponential correlation functions or a skewed process derived from a Gaussian, we obtain expressions for the systematic and random errors. These expressions show that the systematic error and the error variance in the limit of large *T* are both inversely proportional to *T*, which means that the random error, that is, the square root of the error variance, will in this limit be larger than the systematic error. It is demonstrated theoretically, as well as experimentally with aircraft data from the convective boundary layer over the ocean and over land, that the assumption that the time series are Gaussian leads to underestimation of the random errors, while derived processes with a more realistic skewness and kurtosis give better estimates. For fluxes, the systematic and random errors are estimated when the time series are sampled instantaneously, but the samples separated in time by an amount Δ. It is found that the random error variance and the systematic error increase by less than 8% over continuously sampled data if Δ is no larger than the integral scale obtained from the flux time series and the cospectrum, respectively.

## Abstract

Techniques are presented to obtain vertical velocity in cirrus clouds from in situ aircraft lateral wind measurements and from ground-based remote Doppler lidar measurements. In general, direct measurements of absolute vertical velocity *w* from aircraft are currently not feasible because of offsets in the air velocity sensors. An alternative to direct measurement is to calculate *w* from the integral of the divergence of the horizontal velocity around a closed path. We discuss divergence measurements from both aircraft and Doppler lidar. The principal errors in the calculation of *w* from aircraft lateral wind measurements are bias in the lateral wind, ground speed errors, and error due to vertical shear of the horizontal wind. For Doppler lidar measurements the principal errors are in the estimate of mean terminal velocity and the zeroth order coefficient of the Fourier series that is fitted to the data. The technique is applied to a cirrus cloud investigated during the FIRE (First International Satellite Cloud Climatology Regional Experiment) Cirrus Intensive Field Observation Program. The results indicate that the error in *w* is about ±14 cm s^{−1} from the aircraft technique. We show that this can be reduced to about ±2 to 3 cm s^{−1} with technical improvements in both ground speed and lateral velocity measurements. The error in *w* from Doppler lidar measurements, which is about ±8 cm s^{−1}, can be reduced to about ±5 cm s^{−1} by improvements in the Doppler velocity measurements with technology that is currently available.

## Abstract

Techniques are presented to obtain vertical velocity in cirrus clouds from in situ aircraft lateral wind measurements and from ground-based remote Doppler lidar measurements. In general, direct measurements of absolute vertical velocity *w* from aircraft are currently not feasible because of offsets in the air velocity sensors. An alternative to direct measurement is to calculate *w* from the integral of the divergence of the horizontal velocity around a closed path. We discuss divergence measurements from both aircraft and Doppler lidar. The principal errors in the calculation of *w* from aircraft lateral wind measurements are bias in the lateral wind, ground speed errors, and error due to vertical shear of the horizontal wind. For Doppler lidar measurements the principal errors are in the estimate of mean terminal velocity and the zeroth order coefficient of the Fourier series that is fitted to the data. The technique is applied to a cirrus cloud investigated during the FIRE (First International Satellite Cloud Climatology Regional Experiment) Cirrus Intensive Field Observation Program. The results indicate that the error in *w* is about ±14 cm s^{−1} from the aircraft technique. We show that this can be reduced to about ±2 to 3 cm s^{−1} with technical improvements in both ground speed and lateral velocity measurements. The error in *w* from Doppler lidar measurements, which is about ±8 cm s^{−1}, can be reduced to about ±5 cm s^{−1} by improvements in the Doppler velocity measurements with technology that is currently available.

## Abstract

We present procedures to evaluate air motion measurements on two or more aircraft by flying them in formation at a known lateral displacement. The analysis is applied to two formation flights involving three aircraft—the NCAR Electra, Sabreliner and King Air—in a clear convective boundary layer to compare two types of air motion sensing probes mounted on different aircraft. The lateral separation between the Electra in the center, and the other two aircraft was ≈30 m. One sensing system utilized constrained vanes and the other differential pressure measurements across ports on a nose radome to obtain the two airflow angles that are used to calculate the transverse air velocity components. Both systems used a Pitot-static pressure difference for obtaining the longitudinal velocity component. We compare differences in means and variances, spectra and cospectra, and spatial coherences between the same velocity components measured on the different aircraft. The differences are, in most cases, comparable to what is predicted on the basis of making identical measurements of the same variable laterally displaced by 30 m in a turbulent velocity field. Measurements from a constrained vane gust probe and a differential pressure gust probe mounted less than 0.2 m apart on the Electra noseboom also compared well with each other. Thus, we have some assurance that both systems are measuring the true air velocity components.

## Abstract

We present procedures to evaluate air motion measurements on two or more aircraft by flying them in formation at a known lateral displacement. The analysis is applied to two formation flights involving three aircraft—the NCAR Electra, Sabreliner and King Air—in a clear convective boundary layer to compare two types of air motion sensing probes mounted on different aircraft. The lateral separation between the Electra in the center, and the other two aircraft was ≈30 m. One sensing system utilized constrained vanes and the other differential pressure measurements across ports on a nose radome to obtain the two airflow angles that are used to calculate the transverse air velocity components. Both systems used a Pitot-static pressure difference for obtaining the longitudinal velocity component. We compare differences in means and variances, spectra and cospectra, and spatial coherences between the same velocity components measured on the different aircraft. The differences are, in most cases, comparable to what is predicted on the basis of making identical measurements of the same variable laterally displaced by 30 m in a turbulent velocity field. Measurements from a constrained vane gust probe and a differential pressure gust probe mounted less than 0.2 m apart on the Electra noseboom also compared well with each other. Thus, we have some assurance that both systems are measuring the true air velocity components.

## Abstract

Measurement of air motion relative to an aircraft by a conically scanned optical Doppler technique has advantages over measurements with conventional gust probes for many applications. Advantages of the laser air motion sensing technique described here include calibration based on physical constants rather than experiment for an accurate measurement of mean wind, freedom from flow distortion effects on turbulence measurements, all-weather performance, reduction in error from mechanical vibrations and ability to measure vertical wind shear. An experiment comparing a single-component laser velocimeter and a differential pressure gust probe shows that the optical approach measures the turbulence spectrum accurately at frequencies up to 10 Hz and that the signal-to-noise ratio is not a limiting factor. In addition, we have observed the effect of spectral skewing caused by airflow distortion in cloud.

## Abstract

Measurement of air motion relative to an aircraft by a conically scanned optical Doppler technique has advantages over measurements with conventional gust probes for many applications. Advantages of the laser air motion sensing technique described here include calibration based on physical constants rather than experiment for an accurate measurement of mean wind, freedom from flow distortion effects on turbulence measurements, all-weather performance, reduction in error from mechanical vibrations and ability to measure vertical wind shear. An experiment comparing a single-component laser velocimeter and a differential pressure gust probe shows that the optical approach measures the turbulence spectrum accurately at frequencies up to 10 Hz and that the signal-to-noise ratio is not a limiting factor. In addition, we have observed the effect of spectral skewing caused by airflow distortion in cloud.

## Abstract

Wavelet analysis is applied to airborne infrared lidar data to obtain an objective determination of boundaries in aerosol backscatter that are associated with boundary layer structure. This technique allows high-resolution spatial variability of planetary boundary layer height and other structures to be derived in complex, multilayered atmospheres. The technique is illustrated using data from four different lidar systems deployed on four different field campaigns. One case illustrates high-frequency retrieval of the top of a strongly convective boundary layer. A second case illustrates the retrieval of multiple layers in a complex, stably stratified region of the lower troposphere. The method is easily modified to allow for varying aerosol distributions and data quality. Two more difficult cases, data that contain a great deal of instrumental noise and a cloud-topped convective layer, are described briefly. The method is also adaptable to model analysis, as is shown via application to large eddy simulation data.

## Abstract

Wavelet analysis is applied to airborne infrared lidar data to obtain an objective determination of boundaries in aerosol backscatter that are associated with boundary layer structure. This technique allows high-resolution spatial variability of planetary boundary layer height and other structures to be derived in complex, multilayered atmospheres. The technique is illustrated using data from four different lidar systems deployed on four different field campaigns. One case illustrates high-frequency retrieval of the top of a strongly convective boundary layer. A second case illustrates the retrieval of multiple layers in a complex, stably stratified region of the lower troposphere. The method is easily modified to allow for varying aerosol distributions and data quality. Two more difficult cases, data that contain a great deal of instrumental noise and a cloud-topped convective layer, are described briefly. The method is also adaptable to model analysis, as is shown via application to large eddy simulation data.

## Abstract

The capability of the NCAR 10.6-*μ*m-wavelength CO_{2} Doppler lidar to measure radial air motion is validated by examining hard-target test data, comparing measurements with those from a two-axis propeller anemometer and a 915-MHz profiling radar, and analyzing power spectra and autocovariance functions of the lidar radial velocities in a daytime convective boundary layer. Results demonstrate that the lidar is capable of measuring radial velocity to less than 0.5 m s^{−1} precision from 20 laser pulse averages under high signal-to-noise ratio conditions. Hard-target test data and comparisons with other sensors show that the lidar data can be biased by as much as ±2 m s^{−1} when operating in the coherent oscillator mode and that correlated errors are negligible. Correlation coefficients are as large as 0.96 for 90-min comparisons of horizontal velocities averaged for 1 min from the lidar and anemometer, and 0.87 for 2.5-h comparisons between vertical velocities averaged for 30 s from the lidar and profiler. Comparisons of the lidar and profiler vertical velocities are particularly encouraging for the profiler since these results show that 915-MHz profilers are capable of making good vertical velocity measurements in strong convective boundary layers. The authors conclude that despite the commonplace systematic bias in lidar radial velocity, ground-based operation of the NCAR CO_{2} Doppler lidar can provide valuable velocity data for meso- and microscale meteorological studies. The lidar can also provide filtered velocity statistics that may be useful for boundary layer turbulence research.

## Abstract

The capability of the NCAR 10.6-*μ*m-wavelength CO_{2} Doppler lidar to measure radial air motion is validated by examining hard-target test data, comparing measurements with those from a two-axis propeller anemometer and a 915-MHz profiling radar, and analyzing power spectra and autocovariance functions of the lidar radial velocities in a daytime convective boundary layer. Results demonstrate that the lidar is capable of measuring radial velocity to less than 0.5 m s^{−1} precision from 20 laser pulse averages under high signal-to-noise ratio conditions. Hard-target test data and comparisons with other sensors show that the lidar data can be biased by as much as ±2 m s^{−1} when operating in the coherent oscillator mode and that correlated errors are negligible. Correlation coefficients are as large as 0.96 for 90-min comparisons of horizontal velocities averaged for 1 min from the lidar and anemometer, and 0.87 for 2.5-h comparisons between vertical velocities averaged for 30 s from the lidar and profiler. Comparisons of the lidar and profiler vertical velocities are particularly encouraging for the profiler since these results show that 915-MHz profilers are capable of making good vertical velocity measurements in strong convective boundary layers. The authors conclude that despite the commonplace systematic bias in lidar radial velocity, ground-based operation of the NCAR CO_{2} Doppler lidar can provide valuable velocity data for meso- and microscale meteorological studies. The lidar can also provide filtered velocity statistics that may be useful for boundary layer turbulence research.