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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

It is suggested that progress in hail suppression research requires simultaneous improvements in methods of evaluating seeding effects and in monitoring the physical structure of the hailstorm and the hail growth processes. On this basis a case is made for the extensive use of multiple Doppler radar and chemical tracer techniques.

## Abstract

It is suggested that progress in hail suppression research requires simultaneous improvements in methods of evaluating seeding effects and in monitoring the physical structure of the hailstorm and the hail growth processes. On this basis a case is made for the extensive use of multiple Doppler radar and chemical tracer techniques.

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## Abstract

It is proposed that the range derivative of the logarithm of the ratio of average echo powers from two (S- and X-band) synchronized and slaved radars would yield a highly reliable indication of the boundaries of hail shafts. In the presence of rain alone, and ignoring fluctuations, this derivative would always be positive and proportional to the incremental difference in attenuation at the smaller wavelength. In general, the derivative has the same sign as the hail concentration gradient and attains negative values on the far side of a hail shaft. Without hail, signal fluctuations are the only possible source of negative derivatives, and so of false alarms. Thus, a small negative threshold level would avoid the identification of the effect of signal fluctuations at the far side of a hail shaft; similarly a large positive threshold would avoid identifying regions of intense rain as the near side of a hail shaft. This approach is capable of detecting smaller concentrations of hail with greater confidence and in larger backgrounds of non-hail precipitation than the use of the dual-wavelength reflectivity ratio alone because 1) it requires a smaller hail reflectivity ratio, at the two wave-lengths; 2) it is not affected significantly by attenuation, end 3) it is independent of absolute radar calibrations. The limitations of the technique are discussed.

## Abstract

It is proposed that the range derivative of the logarithm of the ratio of average echo powers from two (S- and X-band) synchronized and slaved radars would yield a highly reliable indication of the boundaries of hail shafts. In the presence of rain alone, and ignoring fluctuations, this derivative would always be positive and proportional to the incremental difference in attenuation at the smaller wavelength. In general, the derivative has the same sign as the hail concentration gradient and attains negative values on the far side of a hail shaft. Without hail, signal fluctuations are the only possible source of negative derivatives, and so of false alarms. Thus, a small negative threshold level would avoid the identification of the effect of signal fluctuations at the far side of a hail shaft; similarly a large positive threshold would avoid identifying regions of intense rain as the near side of a hail shaft. This approach is capable of detecting smaller concentrations of hail with greater confidence and in larger backgrounds of non-hail precipitation than the use of the dual-wavelength reflectivity ratio alone because 1) it requires a smaller hail reflectivity ratio, at the two wave-lengths; 2) it is not affected significantly by attenuation, end 3) it is independent of absolute radar calibrations. The limitations of the technique are discussed.

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## Abstract

Analytical solutions for the growth and vertical and horizontal motion of a precipitation particle growing by coalescence with cloud drops are derived under simplified steady-state assumptions. An equation is also developed for the concentration density of a continuous distribution of growing particles.

Assuming that the cloud water content varies linearly with height, and that the fall speed of a drop is proportional to the square root of its diameter, it is shown that the combination of a linearly increasing updraft surmounted by a sharply decreasing one sets a sharp upper limit to the particle size, and sorts the particles horizontally. Particles which spend their entire life in regions of horizontal convergence associated with increasing updraft are packed into a narrower shaft than that in which they originated. Initially smaller particles are carried above into the region of horizontal divergence associated with decreasing updraft and are displaced far to the sides of the cloud core. It is found that when the updraft increases sharply there is a very small range of initial sizes which can grow to fall-out size. These facts are used to suggest that a steady “balance level” (equal reflectivity in rising and falling particles) may be maintained at a height near and below an updraft maximum. Particle size spectra computed from the concentration density equation are continuous and well-behaved for rising, floating and falling particles alike, without necessarily even maximizing for the floating size.

## Abstract

Analytical solutions for the growth and vertical and horizontal motion of a precipitation particle growing by coalescence with cloud drops are derived under simplified steady-state assumptions. An equation is also developed for the concentration density of a continuous distribution of growing particles.

Assuming that the cloud water content varies linearly with height, and that the fall speed of a drop is proportional to the square root of its diameter, it is shown that the combination of a linearly increasing updraft surmounted by a sharply decreasing one sets a sharp upper limit to the particle size, and sorts the particles horizontally. Particles which spend their entire life in regions of horizontal convergence associated with increasing updraft are packed into a narrower shaft than that in which they originated. Initially smaller particles are carried above into the region of horizontal divergence associated with decreasing updraft and are displaced far to the sides of the cloud core. It is found that when the updraft increases sharply there is a very small range of initial sizes which can grow to fall-out size. These facts are used to suggest that a steady “balance level” (equal reflectivity in rising and falling particles) may be maintained at a height near and below an updraft maximum. Particle size spectra computed from the concentration density equation are continuous and well-behaved for rising, floating and falling particles alike, without necessarily even maximizing for the floating size.

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A simple method is described for calibrating a weather radar by means of a standard spherical target, thus permitting the radar to be used for quantitative measurements of storm reflectivity. The technique involves determination of that storm reflectivity which provides an echo equivalent to that from the known target. The sphere, suspended from a balloon, is tracked as it leaves the radar site. Its echo is “measured” by reducing the receiver gain control to the threshold of visibility. The threshold gain setting is thereby calibrated and subsequently provides an accurate measure of storm reflectivity. There is no need for any other test equipment such as a microwave-signal generator. Absolute accuracy is greater than that attainable with a signal generator since no reliance need be placed on the generator calibration or upon the specified antenna gain.

A simple method is described for calibrating a weather radar by means of a standard spherical target, thus permitting the radar to be used for quantitative measurements of storm reflectivity. The technique involves determination of that storm reflectivity which provides an echo equivalent to that from the known target. The sphere, suspended from a balloon, is tracked as it leaves the radar site. Its echo is “measured” by reducing the receiver gain control to the threshold of visibility. The threshold gain setting is thereby calibrated and subsequently provides an accurate measure of storm reflectivity. There is no need for any other test equipment such as a microwave-signal generator. Absolute accuracy is greater than that attainable with a signal generator since no reliance need be placed on the generator calibration or upon the specified antenna gain.

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## Abstract

A method of calibrating a fixed vertically pointing radar is presented. The technique involves the firing of B-B shot of known radar cross section through the beam while making successive trajectory corrections until the absolute maximum signal is attained. The results agree closely with an independent calibration of antenna gain. The approach is particularly suited to an FM-CW radar with high range resolution because the pellets reach heights well in excess of the minimum range and errors in range are negligible. Corrections are presented for the reduction in maximum two-way gain resulting from intersecting beams whose full gain is attained only at the point of intersection. It is also shown that Probert-Jones’ *k*
^{2} factor is significantly smaller for this system, and possibly for others, than the generally accepted value of unity. The method can be readily extended to any sufficiently sensitive pulsed radar by using small elevation angles and direct measurements of range rather than those obtained from the echoes.

## Abstract

A method of calibrating a fixed vertically pointing radar is presented. The technique involves the firing of B-B shot of known radar cross section through the beam while making successive trajectory corrections until the absolute maximum signal is attained. The results agree closely with an independent calibration of antenna gain. The approach is particularly suited to an FM-CW radar with high range resolution because the pellets reach heights well in excess of the minimum range and errors in range are negligible. Corrections are presented for the reduction in maximum two-way gain resulting from intersecting beams whose full gain is attained only at the point of intersection. It is also shown that Probert-Jones’ *k*
^{2} factor is significantly smaller for this system, and possibly for others, than the generally accepted value of unity. The method can be readily extended to any sufficiently sensitive pulsed radar by using small elevation angles and direct measurements of range rather than those obtained from the echoes.

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## Abstract

Through the use of observation operators, modern data assimilation systems have the capability to ingest observations of quantities that are not themselves model variables but are mathematically related to those variables. An example of this is the so-called line-of-sight (LOS) winds that a spaceborne Doppler wind lidar (DWL) instrument would provide. The model or data assimilation system ideally would need information about both components of the horizontal wind vectors, whereas the observations in this case would provide only the projection of the wind vector onto a given direction. The estimated or analyzed value is then calculated essentially as a weighted average of the observation itself and the model-simulated value of the observed quantity. To assess the expected impact of a DWL, it is important to examine the extent to which a meteorological analysis can be constrained by the LOS winds. The answer to this question depends on the fundamental character of the atmospheric flow fields that are analyzed, but, just as important, it also depends on the real and assumed error covariance characteristics of these fields. A single-level wind analysis system designed to explore these issues has been built at the NASA Data Assimilation Office. In this system, simulated wind observations can be evaluated in terms of their impact on the analysis quality under various assumptions about their spatial distribution and error characteristics and about the error covariance of the background fields. The basic design of the system and experimental results obtained with it are presented. The experiments were designed to illustrate how such a system may be used in the instrument concept definition phase.

## Abstract

Through the use of observation operators, modern data assimilation systems have the capability to ingest observations of quantities that are not themselves model variables but are mathematically related to those variables. An example of this is the so-called line-of-sight (LOS) winds that a spaceborne Doppler wind lidar (DWL) instrument would provide. The model or data assimilation system ideally would need information about both components of the horizontal wind vectors, whereas the observations in this case would provide only the projection of the wind vector onto a given direction. The estimated or analyzed value is then calculated essentially as a weighted average of the observation itself and the model-simulated value of the observed quantity. To assess the expected impact of a DWL, it is important to examine the extent to which a meteorological analysis can be constrained by the LOS winds. The answer to this question depends on the fundamental character of the atmospheric flow fields that are analyzed, but, just as important, it also depends on the real and assumed error covariance characteristics of these fields. A single-level wind analysis system designed to explore these issues has been built at the NASA Data Assimilation Office. In this system, simulated wind observations can be evaluated in terms of their impact on the analysis quality under various assumptions about their spatial distribution and error characteristics and about the error covariance of the background fields. The basic design of the system and experimental results obtained with it are presented. The experiments were designed to illustrate how such a system may be used in the instrument concept definition phase.

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## Abstract

The probability-matching method (PMM) was used to determine the relation between the distribution of equivalent reflectivity *Z _{e}
* measured by an airborne C-band radar and that for concurrently measured rain rate

*R*by a disdrometer on the same aircraft in the eyewall and outer bands of Hurricane Anita in 1977. When the PMM is applied to the disdrometer population of

*Z*'s and

*R*'s one finds that the

*Z-R*relations differ significantly from those obtained by linear regression of their 1ogarithms. Such regression relations are deceptive. When PMM is applied to the set of

*Z*'s and

_{e}*R*'s we get a family of

*Z*relations as a function of range which differ significantly from the traditional disdrometer-based

_{e}-R*Z-R*relation for hurricanes by Jorgensen and Willis (JW). These new relations are approximate power laws with slope (exponent) which decrease with increasing range. At ranges less than about 35 km the reflectivity in the eyewall exceeds that in the outer bands and is consistent with the expectation from the disdrometer-based relations. At greater ranges the converse is true due to beamwidth averaging over a broader beam and different vertical profiles of reflectivity in the eyewall and outer bands. We also devise a method to obtain an “effective zero-range”

*Z*relation. This differs from the JW relation by −8.2 dBZ and reflects an error in the radar calibration. This approach is a novel way to calibrate an airborne meteorological radar. The methods may be used with any type of rainstorms and provide a means of using airborne radar and disdrometer systems for air-truthing rainfall measurements from space.

_{e}-R## Abstract

The probability-matching method (PMM) was used to determine the relation between the distribution of equivalent reflectivity *Z _{e}
* measured by an airborne C-band radar and that for concurrently measured rain rate

*R*by a disdrometer on the same aircraft in the eyewall and outer bands of Hurricane Anita in 1977. When the PMM is applied to the disdrometer population of

*Z*'s and

*R*'s one finds that the

*Z-R*relations differ significantly from those obtained by linear regression of their 1ogarithms. Such regression relations are deceptive. When PMM is applied to the set of

*Z*'s and

_{e}*R*'s we get a family of

*Z*relations as a function of range which differ significantly from the traditional disdrometer-based

_{e}-R*Z-R*relation for hurricanes by Jorgensen and Willis (JW). These new relations are approximate power laws with slope (exponent) which decrease with increasing range. At ranges less than about 35 km the reflectivity in the eyewall exceeds that in the outer bands and is consistent with the expectation from the disdrometer-based relations. At greater ranges the converse is true due to beamwidth averaging over a broader beam and different vertical profiles of reflectivity in the eyewall and outer bands. We also devise a method to obtain an “effective zero-range”

*Z*relation. This differs from the JW relation by −8.2 dBZ and reflects an error in the radar calibration. This approach is a novel way to calibrate an airborne meteorological radar. The methods may be used with any type of rainstorms and provide a means of using airborne radar and disdrometer systems for air-truthing rainfall measurements from space.

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## Abstract

For a spaceborne meteorological radar, the use of frequencies above 10 GHz may be necessary to attain sufficient spatial resolution. As the frequency increases, however, attenuation by rain becomes significant. To extend the range of rain rates that can be accurately estimated, methods other than the conventional *Z*-*R*, or backscattering method, are needed. In this paper, tests are made of two attenuation-based methods using data from a dual-wavelength airborne radar operating at 3 cm and 0.87 cm. For the conventional dual-wavelength method, the differential attenuation is estimated from the relative decrease in the signal level with range. For the surface reference method, the attenuation is determined from the difference of surface return powers measured in the absence and the presence of rain. For purposes of comparison, and as an indication of the relative accuracies of the techniques, the backscattering, (*Z*-*R*), method, as applied to the 3 cm data, is employed. As the primary sources of error for the *Z*-*R*, dual-wavelength, and surface reference methods are nearly independent, some confidence in the results is warranted when thew methods yield similar rain rates. Cases of good agreement occur most often in stratiform rain for rain rates between a few mm h^{−1} to about 15 mm h^{−1}; that is, where attenuation at the shorter wavelength is significant but not so severe as to result in a loss of signal. When the estimates disagree, it is sometimes possible to identify the likely error source by an examination of the return power profiles and a knowledge of the error sources.

## Abstract

For a spaceborne meteorological radar, the use of frequencies above 10 GHz may be necessary to attain sufficient spatial resolution. As the frequency increases, however, attenuation by rain becomes significant. To extend the range of rain rates that can be accurately estimated, methods other than the conventional *Z*-*R*, or backscattering method, are needed. In this paper, tests are made of two attenuation-based methods using data from a dual-wavelength airborne radar operating at 3 cm and 0.87 cm. For the conventional dual-wavelength method, the differential attenuation is estimated from the relative decrease in the signal level with range. For the surface reference method, the attenuation is determined from the difference of surface return powers measured in the absence and the presence of rain. For purposes of comparison, and as an indication of the relative accuracies of the techniques, the backscattering, (*Z*-*R*), method, as applied to the 3 cm data, is employed. As the primary sources of error for the *Z*-*R*, dual-wavelength, and surface reference methods are nearly independent, some confidence in the results is warranted when thew methods yield similar rain rates. Cases of good agreement occur most often in stratiform rain for rain rates between a few mm h^{−1} to about 15 mm h^{−1}; that is, where attenuation at the shorter wavelength is significant but not so severe as to result in a loss of signal. When the estimates disagree, it is sometimes possible to identify the likely error source by an examination of the return power profiles and a knowledge of the error sources.

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## Abstract

Simultaneous measurements of the radar cross section and fallspeed of 5 cm (and larger) ice spheres falling in free air have been obtained using a high-precision tracking radar operating at a wavelength of 5.47 cm. While they were dry, the spheres fell with supercritical Reynolds numbers and drag coefficients of only 0.24 to 0.30. These coefficients are much smaller than those normally attributed to hailstones under any conditions. The surface of one sphere, 5.1 cm in diameter, became wet during its fall. This was accompanied by a 5 db decrease in its normalized radar cross section and a twofold increase in its drag coefficient. The implications of these observations are discussed.

## Abstract

Simultaneous measurements of the radar cross section and fallspeed of 5 cm (and larger) ice spheres falling in free air have been obtained using a high-precision tracking radar operating at a wavelength of 5.47 cm. While they were dry, the spheres fell with supercritical Reynolds numbers and drag coefficients of only 0.24 to 0.30. These coefficients are much smaller than those normally attributed to hailstones under any conditions. The surface of one sphere, 5.1 cm in diameter, became wet during its fall. This was accompanied by a 5 db decrease in its normalized radar cross section and a twofold increase in its drag coefficient. The implications of these observations are discussed.