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- Author or Editor: R. C. Srivastava x

- Journal of Applied Meteorology and Climatology x

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## Abstract

The effect of discrete sampling on the measurement of the root mean square frequency *f*
_{rms} of the Doppler spectrum by the bipolar video zero-crossing rate method is discussed. It is shown that discrete sampling results in an underestimate of *f*
_{rms}. A method of correcting for the underestimate is suggested. Conditional positive and negative axis-crossing rates, or the rates at which the signal vector crosses a phase angle in the counterclockwise and clockwise directions, are defined and shown to be equal to (*f*
_{rms} + *f̄*)/2 and (*f*
_{rms} − *f̄*)/2, respectively, where *f̄* is the mean of the Doppler spectrum of the signal. This result suggests that the conditional axis-crossing rates may be used for the measurement of spectrum mean and variance. The effect of discrete sampling on the conditional axis-crossing rates is also discussed.

## Abstract

The effect of discrete sampling on the measurement of the root mean square frequency *f*
_{rms} of the Doppler spectrum by the bipolar video zero-crossing rate method is discussed. It is shown that discrete sampling results in an underestimate of *f*
_{rms}. A method of correcting for the underestimate is suggested. Conditional positive and negative axis-crossing rates, or the rates at which the signal vector crosses a phase angle in the counterclockwise and clockwise directions, are defined and shown to be equal to (*f*
_{rms} + *f̄*)/2 and (*f*
_{rms} − *f̄*)/2, respectively, where *f̄* is the mean of the Doppler spectrum of the signal. This result suggests that the conditional axis-crossing rates may be used for the measurement of spectrum mean and variance. The effect of discrete sampling on the conditional axis-crossing rates is also discussed.

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## Abstract

Calculations show that for spherical hail, a negative value of the range derivative of the logarithm of the ratio of the powers received at 10 and 3 cm wavelengths (*P*
_{10} and *P*
_{3}, respectively) is indicative of the presence of large hail (diameter 1 cm). The negative derivative can occur at the far edge of a boundary between a region of large hail and rain, as shown by Eccles and Atlas (1973). Calculations with a simple model of a hail region, having monodisperse spherical hail at any range, show that a negative derivative of log (*P*
_{10}/*P*
_{3}) can occur embedded in a region of hail if the diameter of the hail and/or the thickness of water coat on it is a function of the range. The results of the calculations are supported by vertical incidence dual-wavelength Doppler radar observations of a hailstorm. Negative values of the range derivative of log (*P*
_{10}/*P*
_{3}) are inferred from the Doppler observations, to occur within a region of hail in which the mean hail diameter was a function of the range. Accordingly, a negative range derivative of log (*P*
_{10}/*P*
_{3}), although indicative of the presence of large hail, is not useful for delimiting the boundaries of a hail region.

## Abstract

Calculations show that for spherical hail, a negative value of the range derivative of the logarithm of the ratio of the powers received at 10 and 3 cm wavelengths (*P*
_{10} and *P*
_{3}, respectively) is indicative of the presence of large hail (diameter 1 cm). The negative derivative can occur at the far edge of a boundary between a region of large hail and rain, as shown by Eccles and Atlas (1973). Calculations with a simple model of a hail region, having monodisperse spherical hail at any range, show that a negative derivative of log (*P*
_{10}/*P*
_{3}) can occur embedded in a region of hail if the diameter of the hail and/or the thickness of water coat on it is a function of the range. The results of the calculations are supported by vertical incidence dual-wavelength Doppler radar observations of a hailstorm. Negative values of the range derivative of log (*P*
_{10}/*P*
_{3}) are inferred from the Doppler observations, to occur within a region of hail in which the mean hail diameter was a function of the range. Accordingly, a negative range derivative of log (*P*
_{10}/*P*
_{3}), although indicative of the presence of large hail, is not useful for delimiting the boundaries of a hail region.

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## Abstract

Equations relating the mean of the Doppler spectrum and the distribution of point velocities, and their spectra are derived under the assumptions that: 1) the scatterers follow the air motion faithfully, 2) the reflectivity is constant, and 3) the beam illumination function is separable. It is found that the three-dimensional spectral density function is strongly attenuated at scales small compared to the beam dimensions, and essentially unaffected at scales large compared to the beam dimensions. Relationships between the one-dimensional longitudinal and transverse spectra of the mean velocity and the three-dimensional spectrum of the point velocities are derived. Numerical computations with a model Kolmogorov-Obukhov turbulence spectrum are performed to illustrate the effects of filtering. Energy at scales small compared to the beam dimensions is attenuated. Energy at scales large compared to the beam dimensions is also reduced, in the case of the one-dimensional spectrum, because small scales in the orthogonal directions contributing to the energy are attenuated by the filtering. The energy depleted from the spectrum of the mean velocity appears as an increased variance of the Doppler spectrum. The ratio of the total energy under the measured spectrum to that under the spectrum of the point velocities is computed as a function of beam dimensions. An equivalent rectangular filter approximation is proposed for computing the one-dimensional spectra. Analytical results are obtained for the longitudinal spectrum and are shown to be in excellent agreement with the numerical results for the actual filter. The use of a spherical volume equal to that of the actual radar pulse volume is shown to be invalid.

## Abstract

Equations relating the mean of the Doppler spectrum and the distribution of point velocities, and their spectra are derived under the assumptions that: 1) the scatterers follow the air motion faithfully, 2) the reflectivity is constant, and 3) the beam illumination function is separable. It is found that the three-dimensional spectral density function is strongly attenuated at scales small compared to the beam dimensions, and essentially unaffected at scales large compared to the beam dimensions. Relationships between the one-dimensional longitudinal and transverse spectra of the mean velocity and the three-dimensional spectrum of the point velocities are derived. Numerical computations with a model Kolmogorov-Obukhov turbulence spectrum are performed to illustrate the effects of filtering. Energy at scales small compared to the beam dimensions is attenuated. Energy at scales large compared to the beam dimensions is also reduced, in the case of the one-dimensional spectrum, because small scales in the orthogonal directions contributing to the energy are attenuated by the filtering. The energy depleted from the spectrum of the mean velocity appears as an increased variance of the Doppler spectrum. The ratio of the total energy under the measured spectrum to that under the spectrum of the point velocities is computed as a function of beam dimensions. An equivalent rectangular filter approximation is proposed for computing the one-dimensional spectra. Analytical results are obtained for the longitudinal spectrum and are shown to be in excellent agreement with the numerical results for the actual filter. The use of a spherical volume equal to that of the actual radar pulse volume is shown to be invalid.

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## Abstract

The mean *V*
_{T}, variance σ_{T}
^{2} and signal-to-noise ratio SNRT of Doppler spectra from precipitation in a thunderstorm and from radar signal reflecting chaff have been estimated by an objective thresholding scheme. Using pulse-pair algorithms the mean *V*
_{P} and the variance of the spectra were estimated with and without correction for noise (σ_{2}
^{2} and σ_{1}
^{2}, respectively). A new technique is proposed for estimation of the Doppler variance σ_{3}
^{2} using *R*(τ) and *R*(2τ), and a similar technique for the signal-to-noise ratio, SNR3, using *R*(0), *R*(τ) and *R*(2τ), where *R*(*X*) is the signal autocorrelation at lag *X* and τ the inter-pulse period of the radar. In the derivation of these functions, a Gaussian-Doppler spectrum was assumed. It has been found that *V*
_{P} agrees closely with *V*
_{T}, and σ_{2}
^{2} and σ_{3}
^{2} agree closely with σ_{T}
^{2} for SNRT≳5 dB. The estimators σ_{2}
^{2} and σ_{3}
^{2} are superior to σ_{1}
^{2} for the estimation of Doppler variance. In contrast to σ_{2}
^{2}, σ_{3}
^{2} does not require explicit knowledge of the signal-to-noise ratio. The estimator σ_{1}
^{2} is seen to routinely give a poor estimate of Doppler spectral variance for all SNRT values. The estimators SNRT and SNR3 agree closely with each other for SNRT≳5 dB.

## Abstract

The mean *V*
_{T}, variance σ_{T}
^{2} and signal-to-noise ratio SNRT of Doppler spectra from precipitation in a thunderstorm and from radar signal reflecting chaff have been estimated by an objective thresholding scheme. Using pulse-pair algorithms the mean *V*
_{P} and the variance of the spectra were estimated with and without correction for noise (σ_{2}
^{2} and σ_{1}
^{2}, respectively). A new technique is proposed for estimation of the Doppler variance σ_{3}
^{2} using *R*(τ) and *R*(2τ), and a similar technique for the signal-to-noise ratio, SNR3, using *R*(0), *R*(τ) and *R*(2τ), where *R*(*X*) is the signal autocorrelation at lag *X* and τ the inter-pulse period of the radar. In the derivation of these functions, a Gaussian-Doppler spectrum was assumed. It has been found that *V*
_{P} agrees closely with *V*
_{T}, and σ_{2}
^{2} and σ_{3}
^{2} agree closely with σ_{T}
^{2} for SNRT≳5 dB. The estimators σ_{2}
^{2} and σ_{3}
^{2} are superior to σ_{1}
^{2} for the estimation of Doppler variance. In contrast to σ_{2}
^{2}, σ_{3}
^{2} does not require explicit knowledge of the signal-to-noise ratio. The estimator σ_{1}
^{2} is seen to routinely give a poor estimate of Doppler spectral variance for all SNRT values. The estimators SNRT and SNR3 agree closely with each other for SNRT≳5 dB.

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## Abstract

Other investigators have discussed the effects of wind and reflectivity gradients across the radar beam on Doppler measurements, but have either estimated their magnitude from a simple approximation or set them aside as negligible. This paper deals with the component of the shear vector along the beam. Exact solutions and simple approximations for both the mean and variance of the Doppler spectrum are derived for two types of reflectivity gradients combined with a linear velocity, gradient. In the case of an exponential reflectivity, gradient it is found that the “effective” beam (i.e., the reflectivity-weighed two-way illumination pattern) remains Gaussian with identical beamwidth to the real beam, but its mean is shifted to an angle ϕ*
_{m}
* on the high-reflectivity side of the actual beam. With a linear velocity profile in the ϕ direction, the approximate solution shows that the mean Doppler velocity, is then shifted to the scatterer velocity found at ϕ

*. This approximation is shown to be valid for most physically realizable conditions. Moreover, the spectra variance is found to be essentially independent of the reflectivity gradient and Lhermitte's simple approximation is also generally valid. Analogous results are obtained for a reflectivity profile varying as exp(*

_{m}*cR*

^{2}ϕ

^{2}) where

*R*is range. The effects of reflectivity gradients on the beam-averaged echo power are also discussed.

## Abstract

Other investigators have discussed the effects of wind and reflectivity gradients across the radar beam on Doppler measurements, but have either estimated their magnitude from a simple approximation or set them aside as negligible. This paper deals with the component of the shear vector along the beam. Exact solutions and simple approximations for both the mean and variance of the Doppler spectrum are derived for two types of reflectivity gradients combined with a linear velocity, gradient. In the case of an exponential reflectivity, gradient it is found that the “effective” beam (i.e., the reflectivity-weighed two-way illumination pattern) remains Gaussian with identical beamwidth to the real beam, but its mean is shifted to an angle ϕ*
_{m}
* on the high-reflectivity side of the actual beam. With a linear velocity profile in the ϕ direction, the approximate solution shows that the mean Doppler velocity, is then shifted to the scatterer velocity found at ϕ

*. This approximation is shown to be valid for most physically realizable conditions. Moreover, the spectra variance is found to be essentially independent of the reflectivity gradient and Lhermitte's simple approximation is also generally valid. Analogous results are obtained for a reflectivity profile varying as exp(*

_{m}*cR*

^{2}ϕ

^{2}) where

*R*is range. The effects of reflectivity gradients on the beam-averaged echo power are also discussed.

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## Abstract

Observations by the airborne X-band Doppler radar (known as EDOP) and the NCAR S-band polarimetric (S-Pol) radar from two field experiments are used to evaluate the surface reference technique (SRT) for measuring the path-integrated attenuation (PIA) and to study attenuation in deep convective storms. The EDOP, flying at an altitude of 20 km, uses a nadir beam and a forward-pointing beam. It is found that over land the surface scattering cross section is highly variable at nadir incidence but is relatively stable at forward incidence. It is concluded that measurement by the forward beam provides a viable technique for measuring PIA using the SRT. Vertical profiles of peak attenuation coefficient are derived in two deep convective storms by the dual-wavelength method. Using the measured Doppler velocity, the reflectivities at the two wavelengths, the differential reflectivity, and the estimated attenuation coefficients, it is shown that supercooled drops and (dry) ice particles probably coexisted above the melting level in regions of updraft and that water-coated partially melted ice particles probably contributed to high attenuation below the melting level.

## Abstract

Observations by the airborne X-band Doppler radar (known as EDOP) and the NCAR S-band polarimetric (S-Pol) radar from two field experiments are used to evaluate the surface reference technique (SRT) for measuring the path-integrated attenuation (PIA) and to study attenuation in deep convective storms. The EDOP, flying at an altitude of 20 km, uses a nadir beam and a forward-pointing beam. It is found that over land the surface scattering cross section is highly variable at nadir incidence but is relatively stable at forward incidence. It is concluded that measurement by the forward beam provides a viable technique for measuring PIA using the SRT. Vertical profiles of peak attenuation coefficient are derived in two deep convective storms by the dual-wavelength method. Using the measured Doppler velocity, the reflectivities at the two wavelengths, the differential reflectivity, and the estimated attenuation coefficients, it is shown that supercooled drops and (dry) ice particles probably coexisted above the melting level in regions of updraft and that water-coated partially melted ice particles probably contributed to high attenuation below the melting level.